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Detection of He${}^{++}$ ion in the star-forming ring of the Cartwheel using MUSE data and ionizing mechanisms
Y. D. Mayya${}^{1\href https://orcid.org/0000-0002-4677-0516{}}$,
A. Plat${}^{2\href https://orcid.org/0000-0003-0390-0656{}}$, V. M. A. Gómez-González${}^{3\href https://orcid.org/0000-0001-8252-6548{}}$, J. Zaragoza-Cardiel${}^{1,4\href https://orcid.org/0000-0001-8216-9800{}}$,
S. Charlot${}^{5\href https://orcid.org/0000-0003-3458-2275{}}$ and G. Bruzual${}^{6\href https://orcid.org/0000-0002-6971-5755{}}$
${}^{1}$Instituto Nacional de Astrofísica, Óptica y Electrónica, Luis Enrique Erro 1, Tonantzintla 72840, Puebla, Mexico
${}^{2}$Steward Observatory, 933 N. Cherry Avenue, University of Arizona, Tucson, AZ 85721, USA
${}^{3}$Institute for Physics and Astronomy, Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany
${}^{4}$Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, 03940, Mexico City, Mexico
${}^{5}$Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
${}^{6}$Instituto de Radioastronomía y Astrofísica, UNAM Campus Morelia, Apartado postal 3-72, 58090 Morelia, Michoacán, Mexico
Email: [email protected]
Abstract
We here report the detection of the nebular He ii $\lambda 4686$ line in 32 H ii regions in the metal-poor collisional ring galaxy Cartwheel using the Multi-Unit Spectroscopic Explorer (MUSE) dataset. The measured I(He ii $\lambda 4686$)/I(H$\beta$) ratio varies from 0.004 to 0.07, with a mean value of 0.010$\pm$0.003. Ten of these 32 H ii regions are coincident with the location of an Ultra Luminous X-ray (ULX) source. We used the flux ratios of important diagnostic lines and results of photoionization by Simple Stellar Populations (SSPs) to investigate the likely physical mechanisms responsible for the ionization of He${}^{+}$. We find that the majority of the regions (27) are consistent with photoionization by star clusters in their Wolf-Rayet (WR) phase with initial ionization parameter $-3.5<$$\log\langle U\rangle$$<-2.0$. Blue Bump (BB), the characteristic feature of the WR stars, however, is not detected in any of the spectra. We demonstrate that this non-detection is due to the relatively low equivalent width (EW) of the BB in metal-poor SSPs, in spite of containing sufficient number of WR stars to reproduce the observed I(He ii $\lambda 4686$)/I(H$\beta$) ratio of $\leq$1.5% at the Cartwheel metallicity of Z=0.004. The H ii regions in the WR phase that are coincident with a ULX source do not show line ratios characteristic of ionization by X-ray sources. However, the ULX sources may have a role to play in the ionization of He${}^{+}$ in two (#99, 144) of the five regions that are not in the WR phase. Ionization by radiative shocks along with the presence of channels for the selective leakage of ionizing photons are the likely scenarios in #17 and #148, the two regions with the highest observed I(He ii $\lambda 4686$)/I(H$\beta$) ratio.
keywords:
galaxies: star clusters – galaxies: individual (ESO 350-G040 or Cartwheel)
1 Introduction
Ionization of He${}^{+}$ requires environments that generate energies in excess of
54.4 eV. Detection of the He ii $\lambda 4686$ nebular line in star-forming galaxies
suggests that processes related to star formation are able to create
such environments. Photoionization by hard ultraviolet photons emanating
from Wolf–Rayet (WR) stars is the front-runner among these processes
(Schaerer, 1996).
Simple stellar population (SSP) models incorporating the state-of-the-art
developments in massive star evolution and modeling of the wind-dominated
atmospheres of O-type and WR stars are able to explain the observed I(He ii $\lambda 4686$)/I(H$\beta$)
intensity ratios (He ii $\lambda 4686$/H$\beta$ for short) in at least some star-forming galaxies (Plat et al., 2019).
These models predict a maximum value of He ii $\lambda 4686$/H$\beta$=0.01 during
the WR phase ($\sim$3–5 Myr) for metallicities $Z\geq$0.004, with a decrease
of this ratio at lower metallicities (see Mayya et al., 2020, and references therein).
The observed values of the He ii $\lambda 4686$/H$\beta$ ratio in metal-poor galaxies often are found to
be above 0.01 (Shirazi & Brinchmann, 2012; Kehrig et al., 2015, 2018; Schaerer, Fragos & Izotov, 2019).
The SSP models of Eldridge et al. (2017) involving binary stars help to alleviate the
problem to some extent, but explaining the presence of He${}^{++}$ ion in high H$\beta$ equivalent width (EW) systems (burst ages $<$3 Myr) remains a challenge.
This has forced exploration of alternative mechanisms of ionization.
Plat et al. (2019) carried out an exhaustive analysis of the physical
mechanisms that can produce the observed ratio of He ii $\lambda 4686$/H$\beta$. They find the
need for one or more of the following processes at work in order to produce
the observed high values:
(1) the presence of stars significantly more massive than 100 M${}_{\odot}$;
(2) extremely high ionization parameter, $\log(U)>-1$;
(3) hard radiation from binary stars (in particular, X-ray binaries);
(4) ionization of He${}^{+}$ by radiative shocks; and/or
(5) ionization of He${}^{+}$ by an active galactic nucleus (AGN), if and when present.
Over the last decade or so, it has been possible to obtain reliable fluxes
of the relatively weak He ii $\lambda 4686$ line for a large sample of objects, thanks to
dedicated spectroscopic surveys such as Sloan Digital Sky Survey (SDSS).
Given that the often-used nebular diagnostic lines
(see e.g. Pérez-Montero, 2017)
are much brighter than the He ii $\lambda 4686$ line, the available dataset has also
allowed accurate determination of nebular metallic abundances.
Dataset obtained from these large surveys are the ones often used to confront
with the predictions of SSP models (e.g. Plat et al., 2019; Schaerer, Fragos & Izotov, 2019).
These data typically sample zones that span several kiloparsecs in size,
centered on the nucleus of the galaxy. Though the diagnostic lines (Baldwin, Phillips, & Terlevich, 1981)
allow the rejection of spectra dominated by an AGN at least at high enough metallicities
($Z>$0.008; see Feltre, Charlot, & Gutkin, 2016),
the presence of a weak AGN in a spectrum dominated by a starburst component
cannot be ruled out. Over the large spatial scales sampled in these studies,
several of the above mentioned five physical mechanisms are likely to be at
work, thus inhibiting discerning the
relative importance of each mechanism. Data on smaller scales, ideally of
selected regions in nearby galaxies, are required so as to explore the role
of each of the above-listed mechanisms in increasing the He ii $\lambda 4686$/H$\beta$ ratio above the
canonical values predicted by the SSP models.
Availability of spectrographs incorporating integral field units (IFUs) on
large telescopes such as Multi Unit Spectroscopic Explorer (MUSE) on the
Very Large Telescope (VLT) (Bacon et al., 2010)
and MEGARA on the Gran Telescopio Canarias (GTC) (Gil de Paz et al., 2018)
is allowing such studies possible in recent years.
Kehrig et al. (2015) and Kehrig et al. (2018) used MUSE data to spatially
map the He ii $\lambda 4686$ line in
two of the most metal-poor galaxies: I Zw 18 and SBS 0335 $-$ 052E, finding that the
observed He${}^{+}$ ionization cannot be explained by the WR stars present in
these galaxies. Recently, Mayya et al. (2020) used MEGARA to map the central
starburst cluster of NGC 1569, a galaxy with the oxygen abundance similar to
that of the Large Magellanic Cloud (LMC), finding that the WR stars in the starburst cluster
are able to completely explain the observed ionization.
Data on regions where hydrogen ionization is dominated by photons
from massive stars of metallicity below that of the LMC are needed to address
the sources of He${}^{+}$ ionization in distant metal-poor galaxies. The collisional-ring galaxy Cartwheel
provides such a laboratory, as explained below.
The Cartwheel is considered as the archetype of the class of collisional ring galaxies
(Appleton & Struck-Marcel, 1996; Struck, 2010). Ring galaxies
are characterized by a ring that harbours a chain of star forming knots.
The star formation in the ring is believed to be triggered by a radially expanding
density wave that was formed as a result of a compact galaxy
plunging through a massive gas-rich disk galaxy close to its center and almost
perpendicular to it (Lynds & Toomre, 1976).
Higdon (1995) found that the H$\alpha$ emission in the Cartwheel, a tracer
of current star formation, is distributed along a ring as predicted by the
collisional scenario of the formation of ring galaxies.
MUSE dataset is available on this galaxy at the
seeing-limited spatial resolution of $\sim$0.6 arcsec. This dataset provides
optical spectra covering a rest wavelength range of $\sim$4600 to 9100 Å over the entire galaxy.
On the H$\alpha$ image constructed using this dataset, we have identified more
than 200 individual H ii regions in and around the ring. A colour-composite image
formed using this H$\alpha$ image is shown in Figure 1.
At the distance of the Cartwheel (128 Mpc using the Hubble constant of
71 km s${}^{-1}$ Mpc${}^{-1}$), MUSE spectra are available at physical
scales of $\sim$370 pc, which is an order of magnitude better as compared
to the typical size scale where He ii $\lambda 4686$ is detected in metal-poor galaxies.
At the spatial resolution of the Wide Field and Planetary Camera 2 (WFPC2) images of
the Hubble Space Telescope (HST), which are the highest resolution images
($\sim$0.2 arcsec=125 pc) available for this galaxy, we can associate
each MUSE-identified H ii region with a population of super star clusters
(SSCs), which provide the ionization of the H ii regions.
Fosbury & Hawarden (1977) have measured an oxygen abundance of 12+log(O/H)$\sim$8.0, corresponding roughly to $Z\sim 0.003$ (see Table 2 of Gutkin et al., 2016),
a value at which the observed He ii $\lambda 4686$/H$\beta$ ratio in galaxy samples is higher than that predicted by most of the SSPs. The wide spectral coverage of MUSE data allowed the flux measurement of nebular lines useful to study the ionization state of the H ii regions using the standard line ratio diagrams (Baldwin, Phillips, & Terlevich, 1981, hereafter BPT).
We used this new dataset to measure an average oxygen abundance for the ring regions of 12+log(O/H)$\sim$8.19$\pm$0.15 (Zaragoza-Cardiel et al., 2022), which is marginally higher than the value reported by Fosbury & Hawarden (1977) for three of the brightest H ii regions.
An additional aspect that makes the Cartwheel a good candidate for
understanding the He ii ionization problem is the presence of more than
15 ultra-luminous X-ray sources (ULXs) in its star-forming ring
(Gao et al., 2003; Wolter & Trinchieri, 2004). In fact the Cartwheel is the record
holder for the maximum number of ULX sources in a single galaxy (Wolter, Fruscione & Mapelli, 2018).
The ULX emission is believed to be originating
in high-mass X-ray binaries (HMXBs), with the compact object most likely an
intermediate-mass black hole (IMBH) (Mapelli et al., 2010; Wolter, Fruscione & Mapelli, 2018).
The presence of these sources allows us to explore the role of HMXBs in the
He${}^{+}$ ionization in regions where massive stars contribute to the ionization of
hydrogen and other ions of ionization potential much lower than 54.4 eV.
In Section 2, we describe the dataset, extraction of individual spectrum, and
details of measurement of line fluxes.
The analysis of nebular line ratios is described in Section 3.
A detailed discussion of results on the He${}^{+}$ ionization in each star-forming complex is carried out in Section 4. Our conclusions are given in Section 5.
2 The sample of He${}^{++}$ nebulae and the control sample of bright H ii regions
2.1 The H ii region parent sample
Given that the ionization potential of hydrogen is four times lower as compared to the second
ionization potential of helium, the He${}^{++}$ nebulae are expected to be a subset of the ionized nebulae.
The outer ring of the Cartwheel is currently experiencing an intense burst of star formation.
Higdon (1995) found that the H$\alpha$ emission, a tracer of current star formation,
is predominantly confined to 29 ionized complexes in the outer ring.
As a first step, we used the MUSE datacube to obtain a narrow-band image at the
red-shifted wavelength of the H$\alpha$ line to locate ionized regions at the
resolution of MUSE (FWHM=0.6 arcsec$\sim$370 pc),
which is $\sim$3 times better as compared to the image used by Higdon (1995).
In Figure 1, we show an RGB image formed by combining
the HST/WFPC2 filters in F814W, pseudo-green and F450W as red, green and blue components, respectively;
the H$\alpha$ image constructed from MUSE data is used as a fourth reddish component.
We identified visually 221 prominent H$\alpha$-emitting knots in the ring or near to it on the MUSE image, which are shown by numbers in this image. At the resolution of the HST/WFPC2 images, most of the H$\alpha$-emitting knots are associated with one or more compact clusters, which are the most likely ionizing sources of the nebulae. Thus, the selected regions are star-forming complexes spread over the entire extracted area. However, at the spatial resolution offered by MUSE, each complex is basically a compact unresolved knot. We hence used a uniform aperture of 0.6 arcsec radius to extract spectrum of each region.
We note that the H$\alpha$ surface brightness in the outer ring is 10 to 100 times brighter than the typical cut-off surface brightness of 2$\times 10^{-17}$ erg cm${}^{-2}$ s${}^{-1}$ arcsec${}^{-2}$ for the Diffuse Ionized Gas (DIG) (see e.g. Belfiore et al., 2022), and hence the DIG contribution to the flux ratios of lines in the extracted spectra can be ignored.
We did not attempt to subtract the disk spectrum from the extracted spectra as it is non-trivial to locate a region for extraction of the local disk spectrum in a galaxy such as the Cartwheel because of its peculiar morphology and star formation history. Thus the extracted spectra contain contributions from any pre-collisional, as well as from all post-collisional, populations in the extracted area. From an analysis of stellar populations in the extracted spectra, Zaragoza-Cardiel et al. (2022) concluded that the spectra lack absorption features characteristics of disk stellar populations. This is understandable because the pre-collsional disk is around a factor of five fainter than the ring (Marcum, Appleton & Higdon, 1992; Higdon, 1995), and hence the extracted spectra are dominated by contribution from all populations formed after the collision around 100 Myr ago (Renaud et al., 2018). The extracted spectra, however, could contain contribution from non-ionizing populations formed over the last 100 Myr. Such a population will not contribute to the measured emission line fluxes, but will make the observed values of emission-line EWs lower that that expected for a single burst young population. We will take this into account while we carry out a detailed comparison of the observed EWs with that predicted from population synthesis models in Sec.4.
2.2 The He${}^{++}$ nebular sample
We measured fluxes of all major nebular lines in all the extracted spectra.
We used the Gaussian profile fitting routine of the splot task in iraf111 IRAF is distributed by the National Optical Astronomy Observatories, which
are operated by the Association of Universities for Research in Astronomy,
Inc., under cooperative agreement with the National Science Foundation.
for this purpose. The measured line fluxes are free from the underlying continuum.
We fitted each extracted spectrum with a Gaussian profile at the redshifted
wavelengths of a list of nebular lines commonly reported in extragalactic H ii regions,
including the He ii $\lambda 4686$ line. The output quantities of the Gaussian fitting task that we used here
are: line flux, FWHM and EW. We measured the root mean square (rms) noise at feature-less parts of the continuum adjacent to the measured line, which we used to determine the signal-to-noise ratio (SNR) of the measured line. An emission line is deemed detected if the fitted FWHM is
comparable to that of the H$\beta$ line (2.5 Å) and that the integrated line flux has
a SNR$\geq$3. Using these line fluxes, we constructed two sub-samples: (1)
a sub-sample of 87 bright H ii regions defined as those with SNR$>$40 in the H$\beta$ line, and
(2) a sub-sample of 32 He ii nebulae defined as those where the He ii $\lambda 4686$ nebular line is detected.
As expected, the latter sample is a subset of the former.
The bright H ii region sample is used as a control sample to understand
the physical conditions that favour the detection of the He ii $\lambda 4686$ line in ionized regions.
The criterion that the FWHM of a line should be comparable to that of the H$\beta$ emission line ensures that the detected line is of nebular origin. Nevertheless, the spectra of the He ii nebulae were visually inspected for the possible presence of an underlying broad He ii $\lambda 4686$ emission feature,
commonly referred to as the ’blue bump’ (BB), from WR stars. None of the spectra showed this feature, and hence any contribution of the WR bump to the measured nebular He ii $\lambda 4686$ fluxes can be ignored.
The He ii nebular sample222Fluxes of all measured lines for the entire sample of H ii regions are presented in
companion paper by Zaragoza-Cardiel et al. (2022), which deals with the nebular elemental abundances. The ratios of fluxes of lines used in this work are given in a Supplementary Electronic Table. is listed in Table 1.
In column 2, we list the cross identification of our sources
with the H ii complexes of Higdon (1995). In other columns,
we give the basic measured quantities of these regions such as the flux, SNR and EW of the H$\beta$ line
(columns 4, 5 and 6, respectively), and the He ii $\lambda 4686$ intensity normalized to I(H$\beta$) and multiplied by 100, along with error (column 9).
The error in measured flux ($\sigma_{\rm l}$) of each line is calculated using
the expression (Tresse et al., 1999):,
$$\sigma_{\rm l}=\sigma_{\rm c}D\sqrt{(2N_{\rm pix}+\frac{EW}{D})},$$
(1)
where $D$ is the spectral dispersion in Å per pixel (1.25 for MUSE), $\sigma_{\rm c}$ is
the mean standard deviation per pixel of the continuum, which is measured in a
line-free part of the continuum adjacent to each line, $N_{\rm pix}$ is the
number of pixels covered by the line, which is equated to the FWHM of the fitted Gaussian profile.
In all these 32 regions, the H$\beta$ flux has SNR$>$50, with a
median SNR of 176. The He ii $\lambda 4686$ line is detected at SNR=3–15.
Detection of a line depends on the SNR of the spectrum. In Figure 2, we show the He ii $\lambda 4686$/H$\beta$ ratio for all the H ii regions with SNR(H$\beta$)$\geq$40.
Detections are shown in blue solid circles, whereas the upper limits, defined as 3-$\sigma$ fluxes where $\sigma$ is calculated using the error equation 1 with $N_{\rm pix}$=FWHM(H$\beta$), are shown by red inverted triangles. As a reference, we show by the dotted line the minimum value of He ii $\lambda 4686$/H$\beta$ ratio that a region should have so that the He ii $\lambda 4686$ line is detected, given the SNR of H$\beta$ in its spectra. The intercept of this line depends slightly on EW(H$\beta$), with the plotted line corresponding to EW(H$\beta$)=125 Å. The line would shift upwards (downwards) by $\sim$0.005 for regions with EWs a factor of two lower (higher).
Our sample of He ii nebulae has a mean value of I(He ii $\lambda 4686$)/I(H$\beta$)=0.010$\pm$0.003,
with the lowest and highest ratios being 0.004 (#112) and 0.07 (#148).
We can expect He ii line detection only in H ii regions with SNR(H$\beta$)$>$100, if their He ii $\lambda 4686$/H$\beta$ ratio is close or higher than the mean value of the sample. Conversely, the detection of He ii $\lambda 4686$ line would require abnormally high ratio of He ii $\lambda 4686$/H$\beta$ in regions with SNR(H$\beta$)$<$100, such as the case in #148 and #17. As expected, the He ii $\lambda 4686$ line is detected in all MUSE spectra with SNR(H$\beta$)$\geq$200 (12 regions). For regions having 100$<$SNR(H$\beta$)$<$200, the He ii $\lambda 4686$ line is detected in $\sim$59 per cent (17 of 29) of the H ii regions.
In Figure 3 we show the blue part of the MUSE spectrum for three representative H ii regions where the He ii $\lambda 4686$ line is detected. The regions selected for illustrations are: #99, the brightest H ii region in the Cartwheel, #112 and #148, the H ii regions with the lowest and the highest He ii $\lambda 4686$/H$\beta$ values, respectively. The He ii $\lambda 4686$ and other nebular lines in the plotted range are indicated.
2.3 Association of He${}^{++}$ nebulae to Star Clusters
We use the astrometrized HST images in the F450W and F814W filters to look for a spatial association between the He${}^{++}$ nebulae and star clusters. In Figure 4, we present a close-up view of the entire ring with the intention of showing the cluster(s) associated to He${}^{++}$ nebulae. The HST images reveal the presence of at least one ionizing cluster inside
the aperture used to extract the MUSE spectrum in all the 32 He${}^{++}$ nebulae.
This association suggests that photoionization from stars in young clusters,
or alternatively any process related to young clusters, is mainly responsible
for the ionization of He${}^{+}$.
The star forming complexes defined by Higdon (1995), after correcting for offsets in their coordinates with respect to our MUSE H$\alpha$ image, are also marked in Figure 4. The radius of the circles used to identify the H ii regions and H ii complexes indicates the FWHM of the point spread function (PSF) of the corresponding observations. With the exception of one source (#148), all are associated with these complexes. In general, these complexes contain multiple H ii regions and stellar clusters. However, only 18 of the complexes are associated with a He${}^{++}$ nebula, with a few of the complexes containing multiple knots of He${}^{++}$ emission. H17 in the southern quadrant of the ring, the brightest complex, contains as much as four knots of He${}^{++}$ emission.
2.4 The He ii $\lambda 4686$ nebular line in the SSP models
Photoionization from massive stars is the dominant source of ionization in H ii regions. Massive stars with zero-age main-sequence mass $M_{\mathrm{ZAMS}}\gtrsim 25$ M${}_{\odot}$ go through the WR phase during their post-main sequence evolution (see Crowther, 2007, and references therein). WR stars are the hottest stars in coeval young clusters whose surface temperature reaches in excess of 10${}^{5}$ K, hot enough to doubly ionize helium. The exact age and duration at which WR stars appear in a coeval population depends on the metallicity, the upper cut-off mass ($M_{\rm u}$) of the Initial Mass Function (IMF), and also the inclusion or not of binary star populations in the SSP models used. In this work, we used the latest version of Bruzual & Charlot (2003) SSP models (Charlot & Bruzual in-preparation, hereafter C&B; see also Plat et al. (2019) for details) for single stars corresponding to Z=0.004, the metallicity in the SSP models closest to that of the Cartwheel.
These updated models have incorporated the theoretical spectra from Potsdam Wolf-Rayet (PoWR) model library (see references in Todt et al., 2015) and the stellar evolutionary tracks from Chen et al. (2015) that were computed with the parsec code of Bressan et al. (2012). Tracks for Very Massive Stars up to initial masses of 600 M${}_{\odot}$ are available in this code. We here use the SSP models adopting a Chabrier (2003) IMF with lower cut-off mass of $M_{\rm l}$=0.1 M${}_{\odot}$. We present the model results for two values of upper cut-off mass: $M_{\rm u}$=100 M${}_{\odot}$ and $M_{\rm u}$=300 M${}_{\odot}$. We investigate the effect of including binary star processes (such as envelope stripping and chemical homogenisation) using the BPASS v2.2.1 stellar population models (Stanway & Eldridge, 2018).
The models are computed for a zero-age volume-average ionization parameter $\log\langle U\rangle$ =$-2$ (see Section 4 for details).
In Figures 5 and 6, we show the evolutionary behaviour of the EWs for the H$\beta$ and He ii $\lambda 4686$ nebular lines, and the intensity ratio, I(He ii $\lambda 4686$)/I(H$\beta$). For C&B models, we show the results for two values of $M_{\rm u}$, 100 and 300 M${}_{\odot}$, whereas the binary models are shown only for $M_{\rm u}$=300 M${}_{\odot}$.
The nebular EW(H$\beta$) steadily decreases with age as massive stars die in a coeval population. On the other hand, the nebular EW(He ii $\lambda 4686$) shows a second peak after the initial steady decrease, with the value of this second peak, which corresponds to the WR phase, higher than the values at the pre-WR phase for the C&B models.
The log(I(He ii $\lambda 4686$)/I(H$\beta$)) has values between $-$3.5 to $-$2.7 when all stars are in the main sequence. However, the highest value is reached during the WR phase of single star evolutionary models, which happens between 2.3 to 4.0 Myr with $M_{\rm u}$=300 M${}_{\odot}$, and between 3.2 to 4.0 Myr with $M_{\rm u}$=100 M${}_{\odot}$. In C&B models, the highest value is close to $-1.8$ for both the values of $M_{\rm u}$, dropping steeply to values below $-5$ at 4 Myr.
The evolutionary trend is notably different in BPASS binary models: the highest values are reached during the main sequence phase, thereafter the value remaining between $-$3.5 and $-$3 to ages even beyond 30 Myr.
Unfortunately, the highest value of I(He ii $\lambda 4686$)/I(H$\beta$) ratio differs in different publicly available SSP models, with the value depending on the details of the stellar evolutionary and atmospheric models used as illustrated in Mayya et al. (2020). The mean value of I(He ii $\lambda 4686$)/I(H$\beta$)=0.0094$\pm$0.0025 for our sample regions is in good agreement with the expected value for ionization from WR stars in C&B models, whereas it is higher than the peak value reached in BPASS models. The systematically low value of this ratio during the WR phase in BPASS single star and binary star models limits the use of BPASS models, as of now, for a robust discussion of WR stars as ionizing source in our He ii nebula. We hence draw conclusions regarding the source of ionization of He${}^{+}$ using the C&B SSP models, and make use of the BPASS models only to discuss qualitative changes in our C&B-based results, if the Cartwheel clusters contain binaries in significant numbers.
In Figure 7, we show the expected number of WR stars of different sub-types during the WR phase (top panel), the ionizing photons rates Q(H${}^{0}$) and Q(He${}^{+}$), as well as the expected EW of the BB (bottom panel) for the Z=0.004 C&B SSP.
Note that these numbers do not take into account the presence of WR-like stripped (or spun-up) binary stars, which are capable of ionizing He${}^{+}$, but do not show strong blue bumps (see e.g. Götberg et al., 2018).
In this figure, all quantities except EW(BB) depend on the cluster mass. We scaled all calculated values to a cluster mass of $10^{6}$ M${}_{\odot}$. In order to calculate the model EW(BB), we obtained the flux in the BB, and the underlying continuum. The bump flux is obtained by integrating the model spectra of pure WR stars between 4570 Å and 4740 Å. The underlying continuum flux is obtained as a mean of continuum fluxes at the blue and red part of the BB. The blue and red continua are obtained over 50 Å width filters centered at 4525 Å and 4905 Å, respectively.
Both the $M_{\rm u}$=100 M${}_{\odot}$ and 300 M${}_{\odot}$ models have a total number of $\sim$100 WR stars/$10^{6}$ M${}_{\odot}$ of stellar mass at any time during the WR phase. The WC and WNL stars are the two main types expected up to around 3.6 Myr, after which WNL type is the dominant type with some contribution from WNE type. However, as illustrated earlier, the WR phase begins earlier for higher $M_{\rm u}$, which leads to earlier availability of the He${}^{+}$ ionizing photons for $M_{\rm u}$=300 M${}_{\odot}$ models.
It can be noticed from the figure that the Q(He${}^{+}$) increases by more than two orders of magnitudes when the WR stars start appearing in the cluster, whereas the Q(H${}^{0}$) continues its steady decrease that started at the end of the pre-WR phase into the WR phase.
Unlike Q(He${}^{+}$) and Q(H${}^{0}$), whose maximum values do not depend on the chosen $M_{\rm u}$ of the IMF, the maximum value of the EW(BB) is very sensitive to the choice of $M_{\rm u}$. The maximum value reached for $M_{\rm u}$=100 M${}_{\odot}$ is 1.2 Å at 3.6 Myr, whereas it is as high as 10 Å for the $M_{\rm u}$=300 M${}_{\odot}$ models. The evolutionary behaviours for $M_{\rm u}$=300 M${}_{\odot}$ and 100 M${}_{\odot}$ are indistinguishable for ages greater than 3.4 Myr.
2.5 Ionizing photon rates and cluster masses
We used the observed luminosity in the H$\beta$ recombination line, L(H$\beta$), to estimate the equivalent number of O7V stars using a typical luminosity of an O7V star of 4.76$\times 10^{36}$ erg s${}^{-1}$ from López-Sánchez & Esteban (2010).
We also calculated the H${}^{0}$ and He${}^{+}$ ionizing photon rates, Q(H${}^{0}$) and Q(He${}^{+}$), respectively, for a Case B photoionized nebula using the basic equations for photoionized nebulae (Osterbrock & Ferland, 2006; Mayya et al., 2020).
We then calculated the stellar mass in the cluster that is ionizing each region, using:
$$\frac{M_{\ast}}{\rm M_{\odot}}=\frac{{\rm Q(H}^{0})}{{{\rm Q(H}^{0})_{\rm SSP}}},$$
(2)
where ${{\rm Q(H}^{0})_{\rm SSP}}=4.94\times 10^{46}~{}{\rm photon~{}s^{-1}}$M${}_{\odot}$${}^{-1}$ is the H${}^{0}$ ionizing photon rate for the Z=0.004 SSP in the C&B models for the characteristic age in the WR phase of 3.4 Myr. The masses would be around a factor of two lower, or higher, for the clusters that are in the pre- or post-WR phases, respectively.
The derived values of N(O7V), M${}_{\ast}$, and Q(He${}^{+}$) are given in columns 10, 7 and 8, respectively, of Table 1.
2.6 Potential ionizing sources of He${}^{++}$ nebulae
2.6.1 Wolf-Rayet stars
From Figure 6, it is apparent that the WR stars are the most likely sources of ionization of He${}^{+}$ in majority of our H ii regions. However, as commented earlier in this section, none of the spectra showed the BB (neither other WR features; see e.g. Gómez-González et al. 2021). In order to understand this apparent contradiction, we estimated an upper limit on the EW of the BB, using the noise measurement in the continuum adjacent to the expected BB and assuming a Gaussian profile of FWHM=20 Å for the BB (see e.g. Gómez-González et al., 2020). The resulting EWs are given in column 11 of Table 1. In the table, we also give the theoretically expected number of WNL-type and all WR stars in each region, assuming a uniform age of 3.4 Myr. The expected number of WR stars is as high as 242 for region#99, and in the 30–40 range for four other regions (#90, 81, 84, 118). In rest of the regions the expected numbers are less than 30.
The observed upper limits on the EW(BB) are compared with those expected in the C&B models in the EW(BB) vs He ii $\lambda 4686$/H$\beta$ plane in Figure 8.
As discussed earlier, the theoretically expected EW(BB) reaches a maximum value of only 1.2 Å for $M_{\rm u}=100$ M${}_{\odot}$, whereas it could be as high as 10 Å for $M_{\rm u}=300$ M${}_{\odot}$.
The observational detection upper limits on the EW(BB) are higher by factors of 2 to 6 as compared to the theoretical values in the absence of Very Massive (mass$>$100 M${}_{\odot}$) Stars. This illustrates that the absence of the blue bump does not necessarily imply the absence of WR stars. At the metallicity of the Cartwheel H ii regions, WR stars in numbers sufficient to doubly ionize helium could be present even when the characteristic broad BB is not detected. This could be the case not only in the Cartwheel regions, but in general in all metal-poor systems that show He ii $\lambda 4686$ nebular line (e.g. Shirazi & Brinchmann, 2012).
It is interesting to note that we would have been able to detect the BB when Very Massive Stars in clusters, if present, go through the WR phase.
It can be inferred from Figure 7, that the $M_{\rm u}$=300 M${}_{\odot}$ models provide He${}^{+}$ ionizing photons for double the duration as compared to the $M_{\rm u}$=100 M${}_{\odot}$ models. This implies that if all the He ii $\lambda 4686$-detected H ii regions had an IMF with $M_{\rm u}$=300 M${}_{\odot}$, the BB would have been at the detectable level in 50% of the cases. The non-detection of the BB in all regions points to the absence of $M_{\rm u}>$100 M${}_{\odot}$ stars in H ii regions of the Cartwheel.
Given that our He ii $\lambda 4686$-line detection criteria are tuned to detect nebular (narrow) lines, there exists a possibility that we inadvertently excluded possible WR sources in spectra where we did not detect the He ii $\lambda 4686$ narrow line. In order to verify this possibility, we analyzed the output results of the Gaussian-fitting for all the regions to look for a broad He ii $\lambda 4686$ component with SNR$\geq$3. None of the spectra showed evidence for it. If stars more massive than 100 M${}_{\odot}$ were common, the BB should have been present at detectable levels in at least a few of the 80 H$\beta$-bright regions. In fact, none of the spectra of our original sample of 221 H ii regions showed the BB. Such a non-detection reinforces the inference drawn from the He ii $\lambda 4686$-detected regions that the upper mass cut-off of the IMF rarely exceeds 100 M${}_{\odot}$ in H ii regions.
In summary, WR stars are viable sources of He${}^{+}$ ionization in He ii $\lambda 4686$-detected regions of the Cartwheel, in spite of the non-detection of the BB. In rest of this paper, we use photoionization models to investigate whether the intensity ratios of bright nebular lines support the scenario of WR stars as the only source of ionization in majority of the regions. Four regions with extreme He ii $\lambda 4686$/H$\beta$ ratio (identified in Figure 8), possibly require alternative sources of ionization, which are also investigated in the paper.
2.6.2 Main sequence stars
The bulk of the ionization of hydrogen in H ii regions is provided by star clusters in their early phase when massive O stars are in the main sequence (MS). Some of these stars are hot enough to doubly ionize helium. The emission EW(H$\beta$) is maximum during this early phase having values larger than 500 Å.
The highest values of He ii $\lambda 4686$/H$\beta$ during this phase are 0.002 and 0.001, respectively in C&B and BPASS models, both with $M_{\rm u}$=300 M${}_{\odot}$. It can be inferred from Figure 2 that we require SNR above 400 to detect He ii $\lambda 4686$ line ionized by the MS stars. There are 5 regions with SNR$>$330, all of which have He ii $\lambda 4686$/H$\beta$$>$0.006, i.e. at least a factor of three higher than the MS values. On the other hand, the region with the lowest value of He ii $\lambda 4686$/H$\beta$ is #112, which is around twice the MS value.
Stripped binary stars, which are not taken into account in the C&B and BPASS models, may have a role in increasing the He ii $\lambda 4686$/H$\beta$ above the calculated values. The MS phase is characterized by high EW(H$\beta$). We analyse all regions in He ii $\lambda 4686$/H$\beta$ vs EW(H$\beta$) plane to address this issue in Sec. 3.
2.6.3 Ultra-luminous X-ray sources
Pointlike non-nuclear X-ray sources with isotropic bolometric luminosity in the the 0.5–10 keV band ($L_{\rm X}$) exceeding 3$\times 10^{39}$ erg s${}^{-1}$are referred to as ULX sources.
Gao et al. (2003) and Wolter & Trinchieri (2004) analyzed the Chandra/Acis
data of Cartwheel finding 31 and 24 ULX sources, respectively, in the
FoV of the Cartwheel, the majority of them coinciding with the star-forming
ring of the Cartwheel. The most luminous of these sources (#11 in Gao et al. 2003
and #10 in Wolter & Trinchieri 2004) has $L_{\rm X}>10^{41}$ erg s${}^{-1}$, thus satisfying
the criterion to be called as a hyperluminous X-ray source (HLX).
However, a one-to-one correspondence with an optical knot in the ring was poor
in both the studies, with offsets between the H ii complexes defined by
Higdon (1995) and the ULX coordinates, generally exceeding the 1 arcsec
beam of the Chandra/Acis observations, even after correcting for zeropoint
offsets in the respective coordinate systems. The reason for
these large offsets is that there is more than one star cluster within the seeing-limited resolution of $\sim$1.7 arcsec (1 kpc) of the H$\alpha$ image of Higdon (1995), with the coordinates referring to that of the brightest H ii region in the complex, which is not necessarily the ULX source.
The astrometrically calibrated MUSE and HST dataset that we use in this study offers $\sim$3 and 8 times better spatial resolutions, respectively, as compared to the H$\alpha$ image of Higdon (1995), which allows us to improve upon the identification of the optical counterpart of the ULX sources.
In Figure 4, we mark the positions of ULX/HLX sources by red circles
overlaid on the HST image. Fourteen of the 17 X-ray sources coincide with
the position of an H ii region to better than an arcsec, the beam of
the X-ray observations. The H ii region closest to a ULX/HLX source
is identified in Table 2, where we also give the
offsets for each source from the coordinates reported by Gao et al. (2003) and
Wolter & Trinchieri (2004). It is worth noting that given the high density of H ii regions in
the ring, more than one H ii region can be associated for sources
with offsets exceeding an arcsec. The offsets are systematically smaller for Gao et al. (2003)
coordinates. Source #111, the identified counterpart of the HLX source (G11) is outside the Chandra/Acis beam, suggesting that the identification is likely to be wrong, and that the source may be associated to a non-H$\alpha$-emitting object. In order to identify such a candidate, we looked for any stellar knot in the HST images. We find a faint red knot at the edge of the Chandra/Acis beam, which could be a likely counterpart of the HLX source (see the region G11 in the figure).
A He${}^{++}$ nebula is present within the beam of the X-ray observations for
ten and seven ULX sources identified by Gao et al. (2003) and Wolter & Trinchieri (2004), respectively (see the last column of the table). We analyse below the possible role of X-rays from the ULX sources in the ionization of He${}^{+}$.
Schaerer, Fragos & Izotov (2019) found that the observed He ii $\lambda 4686$/H$\beta$ ratio in metal-poor galaxies can be explained if the bulk of the He${}^{+}$ ionizing photons is emitted by HMXBs, whose numbers are found to increase with decreasing metallicity. They obtained an empirical relation between Q(He${}^{+}$) and the X-ray luminosity, ${\rm L_{X}}$, suggesting an almost constant ratio $q={\rm Q(He^{+})/L_{X}}=2\times 10^{10}$ photon erg${}^{-1}$, with extreme values of $q$ being 1–3$\times 10^{10}$ photon erg${}^{-1}$. Plat et al. (2019) warned that this process is not efficient at EW(H$\beta$)$>$200 Å, as these systems are too young to form compact objects (neutron stars and stellar mass black holes) necessary for the existence of HMXBs. Only two Cartwheel He ii-emitting regions have EW(H$\beta$)$>$200 Å, and hence ionization of He${}^{+}$ by ULX sources is a possibility in majority of the He ii-emitting regions with an associated ULX source.
The presence of an ULX source in 17 of the Cartwheel H ii regions allows us to calculate the value of $q$ directly for these regions. For this purpose, we use the Q(He${}^{+}$) for each region in column 8 of Table 1 and the ${\rm L_{X}}$ of column 7 of Table 2, which was taken from Wolter & Trinchieri (2004). In Figure 9, we plot the He ii $\lambda 4686$/H$\beta$ ratio against the $q$ values for the 11 sources for which we have well-determined values of ${\rm L_{X}}$. The He ii $\lambda 4686$ line is detected in seven of these sources, with the remaining four only having an upper limit for the detection of the He ii $\lambda 4686$ line. We find a dispersion of more than 2 orders of magnitudes in the value of $q$ for the individual H ii regions in the Cartwheel, with only one of these regions having $q$ in the range found by Schaerer, Fragos & Izotov (2019). This large variation in the $q$ value suggests that the X-rays cannot be the unique source of ionization in these sources. It is likely that not all ULX sources in the Cartwheel are HMXBs, and instead the X-ray luminosity may be originating in supernova (SN) remnants. Wolter, Fruscione & Mapelli (2018) discuss them as HMXBs, whereas in the original detection papers (Gao et al., 2003; Wolter & Trinchieri, 2004), such a possibility was firmly established for the only HLX source (#111) in the Cartwheel. The $q$-value obtained for this source is more than an order of magnitude lower than the value proposed by Schaerer, Fragos & Izotov (2019). We analyse the fluxes of lines from high ionization levels such as [Ar iv] to address the role of X-ray ionization in each of the H ii regions associated with an ULX source.
3 Analysis of nebular line ratios
The wealth of spatial and spectroscopic information contained in the MUSE data of the Cartwheel
offers us a great opportunity to comprehensively address the nature of ionizing
sources based on the ionization state of the nebulae. Specifically, the data
allow us to study whether the H ii regions containing the He ii $\lambda 4686$ line
show any difference with respect to our control sample of 87 H ii regions
in the same galaxy, in any of the diagnostic line ratios commonly used
in ionized nebulae (Baldwin, Phillips, & Terlevich, 1981).
In this section, we discuss the general trends seen in different line ratio diagrams. We also discuss the possible role of ULX sources in the ionization of He${}^{+}$. In Sec.4, we compare the observed trends with that expected from different theoretical scenarios of ionization of He${}^{+}$.
3.1 He ii $\lambda 4686$/H$\beta$, [O iii] $\lambda 5007$/H$\beta$ and EW(H$\beta$)
We start our analysis of the diagnostic line ratios by plotting in Figure 10
the He ii $\lambda 4686$/H$\beta$ ratio versus EW(H$\beta$), which is a standard indicator of age of
stellar populations during their early nebular phase, as illustrated in Figure 5. In this and all the upcoming figures, we distinguish H ii regions with and
without the detection of the He ii $\lambda 4686$ nebular line by circles and inverted triangles, respectively.
Filled symbols show the H ii regions that have an associated ULX.
Error bars are shown only when the errors are significantly
larger than the symbol size. H ii regions with detected He ii $\lambda 4686$ line are
annotated with their number designation from Table 1.
Care is taken to avoid superposition of the annotations, but it was
not always possible due to the crowding of points in some of the plots.
Figure 10 shows that the He ii $\lambda 4686$ line is detected in seven of the 12 H ii regions with EW(H$\beta$)$>$100 Å. Among the high EW(H$\beta$) regions without He ii $\lambda 4686$ detection, region#96 is identified in the plot. This region lies slightly outside the ring in the bright southern arc (see Figure 4). The H ii regions with lower emission EW(H$\beta$) have, in general, fainter nebular lines making the detection of the faint He ii $\lambda 4686$ line dependent of the SNR of each spectra (see Figure 2). Three regions (#144, #17 and #148) standout in the diagram, as they are among the regions with the lowest EW(H$\beta$), but having the highest values of He ii $\lambda 4686$/H$\beta$ ratio. In fact, these three regions exemplify a tendency for the upper boundary of the He ii $\lambda 4686$/H$\beta$ ratio to increase with decreasing EW(H$\beta$).
We show the [O iii] $\lambda 5007$/H$\beta$ ratio against EW(H$\beta$) on the left panel of the
Figure 11. The [O iii] $\lambda 5007$/H$\beta$ ratio is a well-known indicator of the ionization
state of an H ii region, with high ionization regions having a higher value
of the ratio. We clarify that the error bars on the ratio are negligibly small,
including for those regions without the He ii $\lambda 4686$ detection (inverted triangles).
Note that the higher EW(H$\beta$) regions have higher [O iii] $\lambda 5007$/H$\beta$ ratio, independent of whether the He ii $\lambda 4686$ line is detected or not, with region #99 (the brightest H ii region) and #148
(H ii region with the highest He ii $\lambda 4686$/H$\beta$ ratio) lying at the extreme ends of the
relation shown by the rest of the regions. The region #111 stands out from the relation for having a too high ionization for its observed low EW(H$\beta$). We recall that this source is the H ii region nearest to the HLX source. However, its association with the HLX source is doubtful as discussed in Sec. 2.6.3. Dilution of EW(H$\beta$) from a non-ionizing cluster inside the aperture used for extracting the spectrum is the most likely reason for this region to displace from the observed correlation. This is supported by a visual examination of the HST images of this region, which reveals two sources, with the source brighter in the F814W image the likely non-ionizing cluster.
In the right panel, we show the He ii $\lambda 4686$/H$\beta$ ratio against the [O iii] $\lambda 5007$/H$\beta$ ratio.
As expected, He ii $\lambda 4686$-line detection is more frequent in high ionization H ii regions as compared to relatively low ionization regions — it is detected in as much as 75 per cent (15 out of 21) of the high ionization regions (log([O iii] $\lambda 5007$/H$\beta$)$>$0.60). Surprisingly, low ionization H ii regions (log([O iii] $\lambda 5007$/H$\beta$)$<$0.40) have a non-zero detection frequency (10 per cent; two out of 20).
The increasing tendency for the upper boundary of He ii $\lambda 4686$/H$\beta$ value with decreasing EW(H$\beta$) is also manifested in the He ii $\lambda 4686$/H$\beta$ vs [O iii] $\lambda 5007$/H$\beta$ plot, with #99 and #148 marking the endpoints of this tendency.
3.2 Location of He ii-emitting H ii regions in diagnostic diagrams
In order to understand the sources of ionization of He${}^{+}$ in H ii regions of
Cartwheel, we show in Figure 12 all our regions in the most commonly used
BPT diagrams (Baldwin, Phillips, & Terlevich, 1981). In the first three plots (top two and the bottom left), we
use lines of low ionization potential in the x-axis, whereas the y-axis
contains [O iii] $\lambda 5007$ line, which as discussed before originates in the high ionization zone.
The H ii regions lie along a sequence wherein the ratios of [N ii] $\lambda 6583$/H$\alpha$, [S ii] $\lambda 6717+6731$/H$\alpha$ and [O i] $\lambda 6300$/H$\alpha$, systematically increase as the [O iii] $\lambda 5007$/H$\beta$ ratio decreases. The fraction of He ii-emitting regions decreases along the sequence from top-left to bottom-right.
In the bottom right panel, we show line ratios that maximize the values for high ionization regions on the y-axis and low ionization regions on the x-axis. In this plot, the relation is much tighter than in other plots with the He ii-emitting regions having the lowest value of [O i] $\lambda 6300$/[O iii] $\lambda 5007$ ratio for any fixed [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ value.
In order to investigate the possible presence of shock ionization/excitation in H ii regions having He ii $\lambda 4686$ lines, in Figure 13 we plot the He ii $\lambda 4686$/H$\beta$ against [O i] $\lambda 6300$/[O iii] $\lambda 5007$ ratio, which is sensitive to the presence of shocks (see e.g. Figure 15 in Plat et al., 2019). The four regions with the highest values of He ii $\lambda 4686$/H$\beta$ (#148, 17, 144 and 108) are indeed among the H ii regions with the highest values of the shock-sensitive ratios.
The regions ionized by the ULX sources are expected to have [O iii] $\lambda 5007$/H$\beta$$>5$ and [O i] $\lambda 6300$/H$\alpha$$>$0.1, occupying the transition region between the Active Galactic Nuclei (AGNs) and Low-Ionization narrow-emission line regions (LINERs) in the BPT diagrams (Gúrpide et al., 2022). None of the ULX sources in the Cartwheel occupy these zones, suggesting that the ULX sources have very limited role, if any, in the ionization of the nebula with which the ULX source is positionally coincident.
4 Discussion
We now investigate the source of He${}^{+}$ ionization in the Cartwheel using
the trajectory of theoretical models of ionization in various diagnostic diagrams using the theoretical line ratios calculated by Plat et al. (2019).
In particular, we use the diagnostic diagrams involving [O iii] $\lambda 5007$/H$\beta$ ratio plotted against
[N ii] $\lambda 6583$/H$\alpha$ and the EW(H$\beta$), [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ vs. [O i] $\lambda 6300$/[O iii] $\lambda 5007$, He ii $\lambda 4686$/H$\beta$ vs. EW(H$\beta$), He ii $\lambda 4686$/H$\beta$ vs [O i] $\lambda 6300$/[O iii] $\lambda 5007$ and [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ vs [Ar iv] $\lambda 4711+4740$/[Ar iii] $\lambda 7135$. These diagrams are chosen because they are sensitive to the different mechanisms of ionization explored in this work.
4.1 Calculation of theoretical nebular line ratios
We consider two sources of ionization: photoionization, which is the most
dominant source of ionization in H ii regions, and ionization by radiative shocks.
Photoionization by stellar clusters with and without binaries is considered.
Since the objects under study are H ii regions, the ionizing source is better modelled as a single age cluster (SSP), rather than a population of stars formed over a long period of time.
We hence consider only SSP models. However, the dilution of EW(H$\beta$) caused by the presence of any non-ionizing source (e.g. an underlying old stellar population) inside the aperture used for extraction is taken into account.
The emission line fluxes from an H ii region are computed with CLOUDY v17.02 (Ferland et al., 2017) following the approach of Gutkin et al. (2016). We use C&B for single star models and BPASS v2.2.1 for binary population models.
The stellar and interstellar medium (ISM) metallicity is set to $Z=0.003$, which corresponds to the gas phase oxygen abundance of $12+\log{\rm(O/H)}_{\rm gas}\approx 8$ for the dust-to-metal mass ratio, $\xi_{d}$=0.3 (the solar value is 0.36) following Gutkin et al. (2016). The ISM is considered to be of uniform hydrogen density of $n_{\rm H}=10^{2}\,$cm${}^{-3}$, which along with the rate of ionizing photons and filling factor sets the ionization parameter $U$. Models are parametrized in terms of the zero-age volume-averaged ionization parameter $\langle U\rangle$, which is varied by varying the filling factor, see Gutkin et al. (2016) for details.
We note that the values of $12+\log{\rm(O/H)}_{\rm gas}$, $\log{\rm(N/O)}_{\rm gas}$ and $n_{\rm e}$ used here are based on the mean values derived using the same MUSE dataset for the Cartwheel ring H ii regions in a companion paper (Zaragoza-Cardiel et al., 2022).
The H ii regions are assumed to be ionization-bounded, but contain dust grains that compete with gas in the absorption of ionizing photons. We also discuss the effect on the line ratios if the H ii regions are density bounded, i.e. when the size of the H ii region is limited by the gas density, rather than the ionizing photon rate. We account for the possible presence of holes or cavities in the H ii regions by means of an escape fraction of ionization photons. In addition to these two escape geometries, we follow the approach of Ramambason et al. (2020) and compute two-zone models, combining a low and high ionization parameter component. These two zones are either both ionization bounded, or one of them is density bounded.
The emission from fast radiative shocks is added using the models of Alarie & Morisset (2019)333The ISM metallicity in the shock models corresponds to the SMC metallicity, which is slightly lower than that used for our H ii regions. The nitrogen to oxygen abundance ratio is also lower in these models as compared to that used in our H ii region models. for the full set of shock velocities (100 to 1000 km s${}^{-1}$) available in these models. A pre-shock density of $10^{2}\,$cm${}^{-3}$ and transverse magnetic field strength $B=1\mu G$, which are typical values for the diffuse ISM in galaxies, are used. The calculated line ratios depend weakly on these fixed values, as compared to the variation in shock velocities. The effects of shocks are added at representative ages, which allows us to illustrate the effect of shocks on line ratios as the cluster evolves.
4.2 The line ratio sequence from photoionization models
In Figures 14 and 15, we overplot tracks for photoionization models for different initial $\log\langle U\rangle$ values using C&B SSP models without binary stars, and BPASS models with binary stars.
The ratios for leaky H ii regions, as well as for H ii regions
experiencing radiative shocks are also explored. Additionally, the effect of an
underlying old population is illustrated.
We also explored the effect of binaries using BPASS models. We refer the reader to Plat et al. (2019) for a detailed illustration of the sensitivities of the explored parameters on the commonly observed optical and ultraviolet line ratios.
In different plots, we plot sequences of age, or $\log\langle U\rangle$ or shock velocities,
depending on the sensitivity of the plotted quantities on the models.
These are explained in annotations and legends of the corresponding figures.
The cluster evolution is followed up to 50 Myr with three epochs (0, 3, and 4 Myr) marked with differently shaped symbols (in some of the plots, the plotted range covers only the trajectory around 3–4 Myr). The initial $\log\langle U\rangle$ between $-1$ and $-4$ are explored. In shock models, shocks are assumed to provide 25 per cent of the observed H$\beta$ flux. The sequences are formed for shock velocities between 100 to 1000 km s${}^{-1}$. High velocity shocks are expected in H ii regions following the explosion of SN, which start occurring at an SSP age of $\sim$4 Myr. Hence, the shock component is added to the 4 Myr H ii region emission with $\log\langle U\rangle=-2.5$. These are shown in Figures 14, 16 and 17 by purple lines. In models involving escape of ionizing photons, the sequence is formed by varying the fraction of Lyman continuum (LyC) photons escaping the H ii regions. In two-zone models, the sequence is formed by varying the fractional contribution to the H$\beta$ flux from the high $\log\langle U\rangle$ zone. These are shown in Figures 14, 16 (right) and 17 with red lines marked with dots for increase of the escape fraction between 0.1 and 0.9.
We now discuss the results on the ionization mechanisms suggested by the models based on the comparison of the locus of model parameters with observations, in selected line-ratio diagrams. We first discuss the results based on C&B models without binary stars, and at the end, comment on the effect of having binary stars in the SSPs.
Effect of $\log\langle U\rangle$:
The density of the ISM surrounding the cluster, the rate of ionizing photons, and the volume filling factor of the ionized gas fix the initial $\log\langle U\rangle$ of the models. In the top-left panel of Figure 14, it can be seen that the observed sequence of [O iii] $\lambda 5007$/H$\beta$ values can be understood as due to a dispersion in the initial $\log\langle U\rangle$ values, with the ionizing clusters in almost all of the H ii regions having ages between 3 and 4 Myr in C&B models. The observed range of line ratios is covered by the models with $\log\langle U\rangle$=$-1.5$ to $-3.0$, with the small variations in the age being responsible for the observed spread in the direction perpendicular to the sequence. The observed sequence in the [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ vs [O i] $\lambda 6300$/[O iii] $\lambda 5007$ plane (top-right) is also consistent as a sequence of initial $\log\langle U\rangle$. However, the inferred age from this diagram is $\sim$0 Myr, rather than 3–4 Myr inferred from the [O iii] $\lambda 5007$/H$\beta$ vs [N ii] $\lambda 6583$/H$\alpha$ diagram. The latter diagram is sensitive to the selective escape of photons from either the low or the high ionization zones as will be discussed in the two-zone models below.
Effect of dust in H ii regions:
All our models include dust inside the H ii regions. The quantity of dust is parametrized by the dust-to-metal mass ratio $\xi_{\rm{d}}$. The dust competes with gas in the absorption of ionization photons following the scheme proposed by Bottorff et al. (1998). Hence the number of ionizing photons absorbed by hydrogen, and all properties that depend on the ionizing photon flux, predicted by our models is lower than those predicted for dust-free ionization-bounded models. The optical depth of ionizing photons arising from dust is proportional to the total hydrogen column density, which is proportional to the ionization parameter. So as the ionization parameter increases, so does the absorption of ionizing photons by dust rather than hydrogen. This effect can be noticed in Figures 15 and 16, where the EW(H$\beta$) decreases with $\log\langle U\rangle$ at a fixed age (see also Erb et al., 2010; Plat et al., 2019).
Effect of upper cut-off mass of the IMF:
The SSPs we used for the calculation of nebular quantities correspond to $M_{\rm u}$=300 M${}_{\odot}$. The non-detection of BB (see Sec. 2.6.1) in our spectra suggests the absence of stars more massive than 100 M${}_{\odot}$ in Cartwheel H ii regions. However, the results for $M_{\rm u}$=100 M${}_{\odot}$ are identical to that for $M_{\rm u}$=300 M${}_{\odot}$ after 3.2 Myr, as can be inferred from Figures 7 and 8. The expected values of He ii $\lambda 4686$/H$\beta$ before and during the WR phase with $M_{\rm u}$=300 M${}_{\odot}$ is only marginally higher than that for $M_{\rm u}$=100 M${}_{\odot}$.
Thus, the results presented here are not sensitive to the choice of upper cut-off mass as long as the SSP contains hot massive-stars that go through the WR phase ($M_{\rm u}>$25 M${}_{\odot}$).
Effect of cluster evolution:
The rate of ionizing photons emanating from a cluster starts decreasing when the
most massive stars, also the hottest and the most luminous, end their main sequence
life time. For clusters with the highest mass $\sim$100 M${}_{\odot}$, this starts happening
at $\sim$3 Myr at Z=0.003, the metallicity corresponding to the Cartwheel.
This decreases $\log\langle U\rangle$ by $\sim$0.6 dex over the first 10 Myr, which leads to a gradual
decrease of the high ionization line intensity ratios such as [O iii] $\lambda 5007$/H$\beta$.
The EW(H$\beta$) decreases slowly in the first 3 Myr reaching values $\sim$200 Å at 3 Myr,
beyond which it drastically drops by more than an order of magnitude in $\sim$10 Myr.
The decrease of $\log\langle U\rangle$ and the EW(H$\beta$) with age leads to a decrease of [O iii] $\lambda 5007$/H$\beta$ ratio as the EW(H$\beta$) decreases.
The evolutionary track in C&B models for a given initial $\log\langle U\rangle$ in Figure 15 follows the observed relation for clusters younger than $\sim$4 Myr beyond which the model-predicted [O iii] $\lambda 5007$/H$\beta$ ratio is much smaller than that expected from the observed relation. Consequently, H ii regions with EW(H$\beta$)$<$50 Å are not expected to have detectable levels of the [O iii] $\lambda 5007$ line emission. The continuity of the observed sequence in this diagram for the whole range of EW(H$\beta$) suggests that the Cartwheel H ii regions are indeed ionized by clusters younger than 4 Myr, and some physical effect is responsible for lowering the observed values of EW(H$\beta$) compared to that predicted in the SSP models we used. The ionization sequence in Figure 14 also implies that all our H ii regions are younger than 4 Myr. Escape of ionizing photons, presence of density-bounded H ii regions, presence of an older underlying population, are some of the physical processes that we have explored in this work to explain the decrease of EW(H$\beta$) without the corresponding decrease in [O iii] $\lambda 5007$/H$\beta$ ratio.
Effect of an older population:
H ii regions often contain a population other than that is ionizing the surrounding gas inside the aperture used for spectral extraction(e.g. Charlot & Fall, 1993; Mayya & Prabhu, 1996). The non-ionizing population could be a cluster older than the ionizing cluster, or it could be the underlying disk population. Given the recent star formation history of the Cartwheel, it is likely that the apertures used for spectral extraction (740 pc diameter) contain non-ionizing clusters of $\sim$10 Myr or slightly older. Results from the recent numerical simulation of the wave propagation in the Cartwheel by Renaud et al. (2018) support the existence of a spread of this order in the ages of clusters in the outer ring. Multiple clusters within the extracted apertures can be directly seen in Figure 4 at the spatial resolution of the HST images.
The presence of such a cluster would decrease the EW(H$\beta$) without affecting the line ratios, and hence would move the points horizontally in Figure 15.
The EW(H$\beta$) would be affected by a larger amount for larger mass of the non-ionizing cluster. In the left panel, we show the effect of a non-ionizing cluster of 10 Myr age for two values of relative masses: (1) equal masses, and (2) the non-ionizing cluster 10 times more massive than the ionizing cluster. This effect is shown by crosses placed at 3 and 4 Myr of age for the ionizing cluster. A part of the observed horizontal spread could be due to the presence of different amounts of mass in old stellar clusters inside the apertures used for extraction.
Effect of escape of ionizing photons: We have assumed that all the ionizing photons that are emitted by the clusters are either used in the ionization, or absorbed by internal dust. However, there are 2 geometries by which ionizing photons can escape the nebula without
getting absorbed by gas and dust: (1) escape through holes, and (2) escape through density-bounded zones. The former case results in the decrease of the intensity of all lines, hence a decrease in the EW(H$\beta$), without changing the line ratios. Thus, the escape of ionizing photons through holes would move the points horizontally in Figure 15, producing an effect indistinguishable from the presence of a non-ionizing population discussed above. The second case is discussed below.
Density bounded models:
H ii regions have an ionization structure with the lines of high ionization (e.g. [O iii] $\lambda 5007$)
originating in zones closer to the ionizing cluster as compared to the lines of low
ionization (e.g. [N ii]$\lambda$6583). In density-bounded H ii regions, the ionizing
photons are lost due to insufficient amount of gas to trap all the ionizing photons.
Such regions would have a reduced intensity of low-ionization lines and EW(H$\beta$) as compared to
the values for ionization-bounded regions. This would lead to an increase of [O iii] $\lambda 5007$/H$\beta$ as
EW(H$\beta$) decreases, which is just the opposite of what is observed in Figure 15.
Hence, density-bounded models cannot explain the continuation of the sequence to low EW(H$\beta$).
Furthermore, the observed sequence in [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ vs [O i] $\lambda 6300$/[O iii] $\lambda 5007$ ratio is not consistent with density-bounded models.
Two-zone escape models:
In order to reproduce the observed large range of [O i] $\lambda 6300$/[O iii] $\lambda 5007$ ratios in LyC galaxies,
Ramambason et al. (2020) proposed a model wherein the nebulae loose ionizing photons
selectively from the high-$U$ or low-$U$ zones. Such a configuration is naturally expected if the ISM around the clusters is non-uniform, and contains dense clumps and filaments. Results for two-zone models are calculated by combining the spectrum of a high initial $\log\langle U\rangle$ nebula with a second spectrum of a low initial $\log\langle U\rangle$ nebula, with the relative luminosity of the H$\beta$ lines in the two spectra used as a free parameter. This free parameter, $\omega$, varies between 0 and 1, which is defined as the fractional contribution of the high $\log\langle U\rangle$ spectrum to the total H$\beta$ luminosity. Thus, $\omega$=1 corresponds to a nebula containing only high ionization zone and $\omega$=0 corresponds to a nebula containing only low ionization zone. The ionization-bounded or density-bounded status of the two zones are independent of each other. In Figure 14 (right), we show the locus of these models by red lines for three cases, all corresponding to 3 Myr age clusters with initial $\log\langle U\rangle$=$-1$ and $\log\langle U\rangle$=$-3.5$ for the high and low ionization zones, respectively. The 3 cases are: (1) both the zones are ionization-bounded (solid line), (2) combination of an ionization-bounded low $\log\langle U\rangle$ zone, with a density bounded high $\log\langle U\rangle$ zone with an escape fraction, $f_{\rm esc}\sim 30$ percent (dashed line), and (3) combination of an ionization-bounded high $\log\langle U\rangle$ zone, with a density bounded low $\log\langle U\rangle$ zone with $f_{\rm esc}\sim 10$ percent (dotted line). Values for the free parameters are chosen that best illustrate the observed trends in Figure 14 (right).
C&B ionization-bounded models (yellow lines) even for the youngest age lie slightly to the left of the observed points in this figure. The tracks for ages of 3 and 4 Myr are further away from the observed points.
On the other hand, the two zone escape models are able to explain the dataset for cluster ages of 3 Myr — most of the observed points lie between the dashed red lines (3 Myr track with selective escape of ionization photons from high $\log\langle U\rangle$ zone) and the yellow line for the 3 Myr track (ionization bounded regions without any escape).
Thus, with two-zone models, it is possible to obtain consistent ages of $\sim$3–4 Myr for the majority of the H ii regions in the Cartwheel, from both the diagnostic diagrams in Figure 14.
Shocks:
Some of the H ii regions, though principally photoionized by the stars, can also
experience shocks, especially during the post-main sequence evolution and the death of massive
stars. In order to illustrate the effect of shocks in the line ratio diagrams, shock models of Alarie & Morisset (2019) are added to the evolutionary track of SSP models at one particular epoch (4 Myr) with $\log\langle U\rangle=-2.5$. The shocks are parametrized by 3 parameters: the percentage of energy in shocks compared to photoionization energy to ionize hydrogen, the velocity of the shock and the shock age.
We used their grid of truncated shock models. We here plot models where 25 percent of the total H$\beta$ flux is provided through shocks for various values of shock velocities. At each velocity, we plot the ratios as the shock propagates into the ISM.
Increasing the shock velocity increases the line ratios involving high ionization, whereas the low ionization lines become stronger as the shock propagates.
The effect of shocks on the line ratios can be best seen in the [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ vs. [O i] $\lambda 6300$/[O iii] $\lambda 5007$ diagram of Figure 14 (top right panel). High velocity shocks tend to move the points to the right, mostly occupied by H ii regions not showing He ii $\lambda 4686$ line. Among the regions with He ii $\lambda 4686$ emission, shocks could be important in low-ionization H ii regions such as #148. On the other hand, shocks have very little effect in the [O iii] $\lambda 5007$/H$\beta$ vs [N ii] $\lambda 6583$/H$\alpha$ plot (top left panel), and no effect on the EW(H$\beta$).
Cluster evolution with binaries:
We now discuss the effect of binary population on the results obtained from C&B models.
For this purpose, we use the BPASS binary models plotted in the bottom panels of Figures 14, and right panel of Figure 15. The result that the observed sequence of points is a sequence in $\log\langle U\rangle$ holds even after including binary stars in the SSPs.
Star clusters containing binary stars produce the ionizing photons over an extended
period of time ($>$15 Myr) as compared to the evolution without binaries. This
causes both the [O iii] $\lambda 5007$/H$\beta$ and the EW(H$\beta$) to decrease more slowly with evolution as compared to the evolution of clusters without binary stars. The evolutionary locus and $\log\langle U\rangle$ sequences in binary models are almost parallel in Figure 15, both following the ionization sequences seen in these diagrams. Thus, under the binary models, Cartwheel H ii regions could have an age range between 0 and 15 Myr and $\log\langle U\rangle$ range between $-1.5$ to $-3.5$. However, at ages as late as 15 Myr, the EW(He ii $\lambda 4686$) decreases by more than an order of magnitude (see Figure 5). Hence, it is highly unlikely that the He ii $\lambda 4686$ detections correspond to the faint late phase, rather than the luminous early phase.
Thus, the inclusion of binary models does not qualitatively change the conclusions arrived from using C&B models, which do not take into account the possible presence of binary stars.
4.3 On the ionization state of Cartwheel H ii regions
The observed range of line ratios in the H ii regions of the Cartwheel
corresponds to photoionization by young clusters (age$\sim$3–4 Myr in C&B models or 3–15 Myr in BPASS binary models) with initial H ii-averaged ionization parameter $\log\langle U\rangle$ lying between $-1.5$ to $-3.5$. The [O iii] $\lambda 5007$/H$\beta$ ratio, which is a direct measure of the degree of ionization of an H ii region, is correlated with EW(H$\beta$) over the entire observed range of both the quantities. This correlation suggests a systematic decrease of the [O iii] $\lambda 5007$/H$\beta$ ratio, or equivalently $\log\langle U\rangle$, with age. For the ionization-bounded models that we have used, $\log\langle U\rangle$ decreases only by 0.6 dex or alternatively [O iii] $\lambda 5007$/H$\beta$ by 0.2 dex in 4 Myr in C&B SSP models and 0.1 dex in 15 Myr in BPASS binary SSP models. The amount of change of this ratio in SSP models depends on the metallicity, with the values in our models being consistent with the values obtained by Stasinska & Leitherer (1996) for the Cartwheel metallicity. Thus, an age-dependent process of softening of $\log\langle U\rangle$ is required in order to interpret the observed correlation between [O iii] $\lambda 5007$/H$\beta$ and EW(H$\beta$) primarily as an age sequence.
Kim, Kim & Ostriker (2019) found that the fraction of the escape of the ionizing photons systematically increases with age of the cluster due to the increased amount of feedback with age.
The escape of ionizing photons has been long suspected to be one of the reasons for the low EW(H$\beta$) of H ii regions (e.g. Mayya & Prabhu, 1996). Castellanos, Diáz & Tenorio-Tagle (2002) determined escape fraction between 0.1–0.7 for three H ii regions they analysed. The mechanical energy feedback to the ambient ISM can also decrease $\log\langle U\rangle$ due to the decrease in the density following the feedback-driven expansion of the H ii regions. Martín-Manjón et al. (2010) found that this effect can decrease $\log\langle U\rangle$ by as much as 3 dex at the Cartwheel metallicity. All these suggest that the evolution-dependent feedback is driving the observed correlation through the escape of ionizing photons and the decrease in the ISM density. Thus, statistically, the high excitation regions are systematically younger than the low excitation regions.
We also investigated the effect of radiative shocks in H ii regions primarily photoionized by clusters as described in Sec. 4.2 above. High velocity shocks tend to increase the line ratios involving low ionization ions such as the [O i] $\lambda 6300$ line (e.g. Stasinska et al., 2015), which would move the points to the right of the ionization sequence in the right panels of Figure 14. A tendency of broadening of the sequence in this diagram for low-ionization H ii regions is seen, suggesting a possible presence of shocks in some low $\log\langle U\rangle$, which are relatively older, H ii regions.
4.4 The ionizing source of He${}^{+}$ in the Cartwheel H ii regions
Having addressed the ionization mechanism and physical processes prevalent in the Cartwheel H ii regions, we now investigate whether the same physical mechanisms account for the observed He ii $\lambda 4686$/H$\beta$ ratio. We have chosen two plots to verify this: He ii $\lambda 4686$/H$\beta$ vs EW(H$\beta$) and He ii $\lambda 4686$/H$\beta$ vs [O i] $\lambda 6300$/[O iii] $\lambda 5007$. These are shown in Figure 16 for the chosen theoretical tracks from C&B models, along with special scenarios discussed in the paragraphs above. We do not show the plots with BPASS binary models, as the He ii $\lambda 4686$/H$\beta$ ratios produced by these models are systematically lower than the observed values as illustrated in Figure 6.
4.4.1 He${}^{++}$ nebulae photoionized by WR stars
The majority of He ii $\lambda 4686$-emitting regions fall between the tracks corresponding to ionization-bounded H ii regions photoionized by clusters of age between 3 and 4 Myr. This age range corresponds to the WR-phase in C&B models, as can be inferred from Figure 6.
Escape of ionizing photons through holes, and/or the presence of non-ionizing population:
The $\log\langle U\rangle$ values inferred for each region using the He ii $\lambda 4686$/H$\beta$ vs EW(H$\beta$) are systematically higher by around 1 dex as compared to that inferred from the He ii $\lambda 4686$/H$\beta$ vs [O i] $\lambda 6300$/[O iii] $\lambda 5007$ plot. This can be explained as due to the escape of ionizing photons through holes, and/or the presence of a spatially close older cluster in majority of the Cartwheel H ii regions, both of which displace the tracks horizontally without changing the flux ratios of the nebular lines. This inference is consistent with the observed correlation between [O iii] $\lambda 5007$/H$\beta$ ratio and EW(H$\beta$).
Is the ionization by WR stars the only way to explain all the observed line ratios. Is the source of ionization of He${}^{+}$ the same as that of other ions? In order to address these questions, we here summarize the results from other scenarios that we have explored.
Density-bounded models: The loci of density-bounded H ii regions in the line ratio diagrams are shown by green solid lines in the two panels of Figure 16 for C&B models of initial $\log\langle U\rangle$=$-2.5$. The length of the plotted lines correspond to 50 percent of the ionizing photons escaping the nebula from the density bounded zones. The locations of the observed points, with the exception of regions #144, #17 and #148, can be reconciled with this scenario, but for a lower initial $\log\langle U\rangle$ value, as compared to the ionization-bounded case. However, the observed ionization sequences (see Figure 15) are not consistent with a low value of $\log\langle U\rangle$. Thus, we rule out the possibility that the majority of the Cartwheel H ii regions are density-bounded with a low initial $\log\langle U\rangle$.
Role of radiative shocks and two-zone escape models:
In Figure 16, the radiative shocks and two-zone escape models are plotted for a 3 Myr old ionizing cluster in the right panel, with the aim of covering the observed ratios of regions #144, #17 and #148. It can be inferred from the plot that these scenarios, especially with escape from high $\log\langle U\rangle$ zone (dashed red line) would cover the observed range of values for the majority of the regions if the ionizing cluster is $\sim$4 Myr old. In fact, we have considered such a possibility to explain the behaviour of points in the right panel of Figure 16.
However, these scenarios produce higher than the observed values of [Ar iv] $\lambda 4711+4740$/[Ar iii] $\lambda 7135$ line ratio, as will be discussed later in this section. Thus, we do not find it necessary to look beyond the WR stars to explain the observed He${}^{++}$ nebulae. In summary, the majority of the He${}^{++}$ nebulae in the Cartwheel are photoionized by WR stars, with the H ii regions enclosing the He${}^{++}$ nebulae. Different observed quantities can be consistently explained with $\sim$50 percent of the hydrogen ionizing photons escaping through holes, and/or the presence of older non-ionizing populations inside the aperture used for extraction.
4.4.2 He${}^{++}$ nebulae requiring alternative sources of ionization
Exceptions to the above scenario are five H ii regions that standout from the main group. Two (#99 & #112) are at the high-EW(H$\beta$) end, and the other three (#144, #17 & #148) are among the lowest EW(H$\beta$) H ii regions.
Ionization by ULX source and stripped binary stars:
The two high-EW(H$\beta$) regions also have the highest [O iii] $\lambda 5007$/H$\beta$ ratio and the lowest [O i] $\lambda 6300$/[O iii] $\lambda 5007$ ratio, all indicating that these regions have the highest ionization parameter, and the youngest of the sample regions. We infer $\log\langle U\rangle$$\sim-$2.0 and an age corresponding to the pre-WR phase, suggesting that the hot main-sequence stars are the most likely sources of He${}^{+}$ ionization in these two sources.
Region #99 is the brightest, and the most massive H ii region in the Cartwheel with an estimated number of more than 11,000 O stars, assuming O stars are the sole sources of ionization of hydrogen. This region is associated with an ULX source, and hence we discuss here the possible role of this ULX source in the ionization of He${}^{+}$.
The observed [O iii] $\lambda 5007$/H$\beta$ is high enough as expected for the ionization by the hard radiation from a ULX source. However, the observed [O i] $\lambda 6300$/H$\alpha$ ratio for this regions is significantly lower than that expected for the ionization by a ULX source (Gúrpide et al., 2022), and hence ULX cannot be the sole or principal source of ionization of this region. The line ratio diagrams presented suggest that the photoionization by the main-sequence stars is a viable source of ionization. The ULX may provide additional photons for the ionization of He${}^{+}$.
The observed EW(H$\beta$), which in spite of being the highest among the sample regions, is still more than a factor of two lower than that expected for single burst of age$<$3 Myr, suggests the presence of underlying non-ionizing populations. It is likely that the star formation in the region is proceeding for more than 3 Myr or that it had a star formation event in the recent past. These slightly evolved stars had enough time to form HMXBs that are generating the X-rays emitted by the ULX source (Plat et al., 2019).
Region #112 has the lowest value of He ii $\lambda 4686$/H$\beta$ among the regions studied here, with a value intermediate between that of main-sequence and WR phases. Like in the case of #99, all line ratio diagrams presented here suggest ionization of He${}^{+}$ by main sequence stars.
Regions affected by radiative shocks and/or two-zone escape models:
We now discuss the ionization mechanism of He${}^{+}$ in regions #148, #17 and #144, the three low-EW(H$\beta$) regions with the highest ratio of He ii $\lambda 4686$/H$\beta$. These three regions are at the low $\log\langle U\rangle$ end of the ionization sequence. The He ii $\lambda 4686$/H$\beta$ ratios predicted by the traditional ionization-bounded case and photoionized by stellar radiation are much lower at these low $\log\langle U\rangle$ values as compared to the observed values for these three regions. Two of the various processes we have explored produce the observed high He ii $\lambda 4686$/H$\beta$ ratios at low $\log\langle U\rangle$ values. These are (1) radiative shock contribution, and (2) two-zone escape models with $\sim$50 percent escape from the high $\log\langle U\rangle$ zone. Regions #144, #17 and #148, in that order, lie on a sequence of increasing shock velocities with shock velocities in the 100–1000 km s${}^{-1}$ range, with the He ii $\lambda 4686$/H$\beta$ value of #148 only produced by shock models. The two-zone escape models can also explain their location in the plots. Their low EW(H$\beta$) and low $\log\langle U\rangle$ suggest that these three regions are more evolved than the rest of the regions, and are likely to be in the post-WR phase. We hence favour shock ionization associated with SN explosions as the most likely causes of the observed high values of He ii $\lambda 4686$/H$\beta$ ratio. SN explosions can also create escape routes for ionizing photons from the high $\log\langle U\rangle$ zone, and hence shocks and two-zone escape scenarios could be co-existing. Region #144 is associated with a ULX source whose X-ray luminosity is high enough to contribute to He${}^{+}$ ionization. Thus, X-ray ionization could also be prevalent in #144.
Stasinska et al. (2015) has advocated the use of the [Ar iv] $\lambda 4711+4740$/[Ar iii] $\lambda 7135$ ratios to test the hardness of the ionizing spectrum, especially for the He ii $\lambda 4686$-emitting regions. Ar${}^{++}$ has an ionization potential of 40.7 ev and hence the collisionally excited [Ar iv] lines are expected to be present in He ii $\lambda 4686$-emitting regions. The [Ar iv]$\lambda\lambda$4711,4740 doublet is relatively faint. Nevertheless, the doublet is detected at more than 3$\sigma$ levels in 14 of the 32 He ii $\lambda 4686$-emitting regions. In Figure 17, we plot the [O iii] $\lambda 5007$/[O ii] $\lambda 7325$ ratios against the [Ar iv] $\lambda 4711+4740$/[Ar iii] $\lambda 7135$ line ratio. For photoionized nebulae, the locus of points in this diagram is mainly governed by $\log\langle U\rangle$, with the observed points lying between $-3.0<$$\log\langle U\rangle$$<-2.0$. Spectral evolution in the age range 0–4 Myr introduces small spread in the direction perpendicular to the sequence formed by ranges of initial $\log\langle U\rangle$. On the other hand, the presence of radiative shocks and/or selective escape of photons from low or high $\log\langle U\rangle$ zones moves the points to the right (i.e. higher [Ar iv] $\lambda 4711+4740$/[Ar iii] $\lambda 7135$ ratios). Thus, this figure is useful to discriminate the purely photoionized models from the photoionized+shock models, or the models involving selective escape of photons. Unfortunately the last two cases follow similar trajectories in the diagram, and hence cannot be distinguished.
The main-group of the He ii $\lambda 4686$-emitting regions in which [Ar iv] lines have been detected lie along the photoionization sequence by the SSP models, and more importantly, are not consistent with the presence of shocks and/or two-zone escape scenarios, independent of the age of the ionizing clusters. Thus, this diagram helps us to break the degeneracy seen in Figure 16. Region #99 lies clearly to the right of the photoionization sequence, reiterating that the hard radiation from the ULX source plays a role in the ionization of ions that have higher than 40 eV of ionization potential. The second region in which we cannot rule out ionization by the ULX source is #144. The [Ar iv] lines are not detected in this region, but the upper limit on the [Ar iv] $\lambda 4711+4740$/[Ar iii] $\lambda 7135$ ratio is slightly to the right of the photoionization sequence by the SSP models. Unfortunately, we have only upper limits on the detection of the [Ar iv] lines in the remaining two interesting regions #148 and #17. The observed upper limits in #148 and #17 are consistent with the presence of shocks and/or two-zone escape scenarios.
4.5 Regions with non-detection of He ii$\lambda$ 4686 line
We here carry out an analysis on the nature of the H ii regions where the He ii $\lambda 4686$ line could not be detected at the 3-$\sigma$ confidence level.
Given that the WR stars are the principal sources of He${}^{+}$ ionizing photons in our sample regions, and that the WR stars appear only for a short duration in an SSP, this fraction is expected to be a function of age. The EW(H$\beta$) is an excellent proxy for age in young stellar systems (see Figures 5), and hence we analyse the detection fraction as a function of EW(H$\beta$). From the analysis of Figures 15 and 16, we arrived at the conclusion that quantitatively the observed values of EW(H$\beta$) are systematically smaller as compared to the values expected for ionization-bounded H ii regions using C&B models. Continuum contribution from a non-ionizing population and the escape of ionizing photons from the nebula are two principal mechanisms that lead to a decrease in EW(H$\beta$) from those expected for ionization-bounded H ii regions ionized by a single-age population. The factor by which the EW(H$\beta$) is reduced may vary from region to region. For the sake of using EW(H$\beta$) as a proxy for age, we assume a reduction factor anywhere between 0 and 50% for the sample regions.
In Figure 18, we show the fraction of H ii regions detected as a function of the observed EW(H$\beta$). The theoretically expected range of EW(H$\beta$) during the WR phase is shown by the shaded area, which takes into account the reduction of EW(H$\beta$) by 0 to 50% during the WR phase. The distribution of observed EW(H$\beta$) for the whole sample (dotted histogram), and the sample of regions where we have achieved a 3-$\sigma$ sensitivity to detect the He ii $\lambda 4686$ line if they had He ii $\lambda 4686$/H$\beta$$\geq$0.01 (dashed histogram), are shown. For the whole sample (black line), the detection fraction decreases with decreasing EW(H$\beta$). For the subset of H ii regions (red line) that have SNRs sufficient to detect the He ii $\lambda 4686$ line for typical values during WR phase (He ii $\lambda 4686$/H$\beta$$\geq$0.01), the detection fraction peaks at EW(H$\beta$)$\sim$60 Å. The peak value reaches as high as 90%. The EW(H$\beta$) at the peak value corresponds to that during the WR phase. This suggests that our dataset is sensitive enough to detect almost all ($\sim$90%) He${}^{++}$ nebulae ionized by the WR stars. Before the onset of the WR phase (at the high EW(H$\beta$)-end), the detection fraction is nonzero. The He ii $\lambda 4686$ line is detected in two of the three H ii regions, indicating that the He ii $\lambda 4686$/H$\beta$ ratio might be higher than the values predicted in the current SSPs during the main sequence phase of stars. The ULX is the likely source of additional ionization in one of these (#99), whereas stripped binary stars, which are not included in the C&B models, could be the possible source of the weak ionization in the other (#112).
The rest of the non-detections (26) corresponds to low-EW(H$\beta$) regions. In the C&B models, these regions correspond to the post-WR phase. However, the inclusion of the binary channel for the formation of WR stars in the BPASS models extends the duration of WR phase to these low EW(H$\beta$) values, with the expected ratio of He ii $\lambda 4686$/H$\beta$ lower than that during the WR phase in C&B models. Unfortunately, we do not reach the sensitivity to detect the He ii $\lambda 4686$ line with He ii $\lambda 4686$/H$\beta$$<$0.01 for regions with EW(H$\beta$)$<40$ Å regions, and hence the non-detection of the He ii $\lambda 4686$ line could be due to the absence of He${}^{++}$ ions in these regions, or that we do not reach the sensitivity level required to detect weak ionization from the WR stars formed through the binary channel.
In general, the H ii regions with He ii $\lambda 4686$-line detection have higher ionization parameter as compared to the H ii regions without the He ii $\lambda 4686$-line detection, at each EW(H$\beta$) bin.
The location of H ii regions with non-detection of He ii $\lambda 4686$ line in Figure 17 suggests shock and/or two-zone escape scenarios are more prevalent in these regions as compared to the He ii $\lambda 4686$-emitting main-group H ii regions. These processes are strong enough to increase the He ii $\lambda 4686$ lines above the detectable limits in only three cases (#144, #17 and #148).
5 Conclusions
We have carried out a search for He ii $\lambda 4686$ nebular line in the Cartwheel H ii regions using the VLT/MUSE datacube. We detect the He ii $\lambda 4686$ line in 32 H ii regions, with a mean value of He ii $\lambda 4686$/H$\beta$=0.010$\pm$0.003. All the detections are situated in the star-forming ring of the Cartwheel, with ten of these sources coinciding with the location of a ULX source.
We use commonly used diagnostic line ratios to compare the ionization properties of H ii regions with and without the detection of the He ii $\lambda 4686$ line. The He ii $\lambda 4686$ line-emitting regions with and without the ULX sources, in general, show similar ionization properties in the diagnostic diagrams. Hence, the ULX sources are not the principal suppliers of ionizing photons in all the H ii regions containing ULX sources.
Analysis of the diagnostic diagrams using C&B SSPs suggests that the majority (27) of the detections correspond to H ii regions in their WR phase, with two and three detections corresponding to H ii regions in their pre-WR and post-WR phases, respectively.
However, the characteristic BB indicating the presence of WR stars is not detected in our sample regions. We illustrate that this non-detection is due to the relatively low EWs of the BB in SSPs for IMFs with $M_{\rm u}\leq$100 M${}_{\odot}$ at the metallicity of the Cartwheel, even when the SSPs have sufficient number of WR stars to provide the ionization of He${}^{+}$. We suggest that main sequence stars are the major contributors to ionization in the two pre-WR H ii regions, with an additional contribution from other hard sources. In region#99, this additional contribution most likely comes from the ULX source. On the other hand, the three H ii regions in the post-WR phase may be either ionized by radiative shocks or their H ii regions are leaky. We find a correlation between [O iii] $\lambda 5007$/H$\beta$ and EW(H$\beta$), which requires a more rapid softening of the ionization parameter $\log\langle U\rangle$ than that considered in C&B SSP models. This rapid softening can be naturally explained if the H ii regions expand as the cluster evolves due to the feedback from massive stars.
The detection frequency of the He ii $\lambda 4686$ line reaches values as high as 90% for H ii regions that have EW(H$\beta$)=40–70 Å. These values of EW(H$\beta$) correspond to late stages of the WR phase in the C&B models. Our dataset lacks sensitivity to detect the He ii $\lambda 4686$ line from H ii regions with EW(H$\beta$)$<$40 Å, when WR stars formed from the binary channel are expected to dominate the ionization of He${}^{+}$.
Acknowledgements
We thank an anonymous referee for many thoughtful comments that improved the paper. We also thank Gerardo Ramos-Larios who helped us in preparing the images appearing in Figures 1 and 4, and CONACyT for the research grant CB-A1-S-25070 (YDM).
This work is based on data obtained from the ESO Science Archive Facility, program ID: 60.A-9333. Observations made with the NASA/ESA Hubble Space Telescope were obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.
Data availability
The fluxes of principal emission lines used in this work are available
in the article and in its online supplementary material. The reduced fits files
on which these data are based will be shared on reasonable request to the first
author.
The ESO datacubes are in the public domain. The link and observation ID are available in the article.
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MAGiiCAT IV. Kinematics of the Circumgalactic Medium and
Evidence for Quiescent Evolution Around Red Galaxies
Nikole M. Nielsen${}^{1,2}$,
Christopher W. Churchill${}^{2}$,
Glenn G. Kacprzak${}^{1}$,
Michael T. Murphy${}^{1}$,
and
Jessica L. Evans${}^{2}$
${}^{1}$ Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, Hawthorn, Victoria 3122, Australia;
[email protected]
${}^{2}$ Department of Astronomy, New Mexico State University, Las
Cruces, NM 88003, USA
Abstract
The equivalent widths of Mgii absorption in the circumgalactic
medium (CGM) trace the global star formation rate up to $z<6$, are
larger for star-forming galaxies than passively-evolving galaxies, and
decrease with increasing distance from the galaxy. We delve further
into the physics involved by investigating gas kinematics and cloud
column density distributions as a function of galaxy color, redshift,
and projected distance from the galaxy (normalized by galaxy virial
radius, $D/R_{\rm vir}$). For 39 isolated galaxies at $0.3<z_{\rm gal}<1.0$, we have detected Mgii absorption in high-resolution
($\Delta v\simeq 6.6$ km s${}^{-1}$) spectra of background quasars within a
projected distance of $7<D<190$ kpc. We characterize the absorption
velocity spread using pixel-velocity two-point correlation
functions. Velocity dispersions and cloud column densities for blue
galaxies do not differ with redshift nor with $D/R_{\rm vir}$. This
suggests that outflows continually replenish the CGM of blue galaxies
with high velocity dispersion, large column density gas out to large
distances. Conversely, absorption hosted by red galaxies evolves with
redshift where the velocity dispersions (column densities) are smaller
(larger) at $z_{\rm gal}<0.656$. After taking into account larger possible
velocities in more massive galaxies, we find that there is no
difference in the velocity dispersions or column densities for
absorption hosted by red galaxies with $D/R_{\rm vir}$. Thus, a lack
of outflows in red galaxies causes the CGM to become more quiescent
over time, with lower velocity dispersions and larger column densities
towards lower $z_{\rm gal}$. The quenching of star formation appears
to affect the CGM out to $D/R_{\rm vir}=0.75$.
Subject headings:galaxies: halos — quasars: absorption lines
††slugcomment: Submitted to ApJ, October 23, 2015
1. Introduction
Through extensive observations and detailed simulations, it has become
clear that the baryon cycle plays a key role in governing the
evolution of galaxies (e.g., Oppenheimer & Davé, 2008; Lilly et al., 2013). In this scenario, galaxies grow by accreting
pristine gas from the intergalactic medium (IGM), which is the fuel
for star formation. Intense star formation and/or supernovae then
drive outflowing galactic-scale winds, entraining portions of the
interstellar medium (ISM) and coplanar gas along the way. Serving as
the interface between the IGM and the ISM, the circumgalactic medium
(CGM) contains the gas that has yet to accrete onto the galaxy itself
and stores outflowing material until it re-accretes onto the
galaxy. The CGM is also massive, with estimates of $M_{\rm CGM}>10^{9}M_{\odot}$ for $\sim L_{\ast}$ galaxies (Thom et al., 2011; Tumlinson et al., 2011; Werk et al., 2013), a gas mass comparable to the gas in galaxies
themselves (i.e., the ISM).
The CGM has been studied extensively by using quasar absorption lines
in which a background quasar sightline pierces the CGM within a few
hundred kiloparsecs projected on the sky. Much of this work has been
focused on Mgii $\lambda\lambda 2796,2803$ absorption (e.g., Bergeron & Boissé, 1991; Steidel et al., 1994; Guillemin & Bergeron, 1997; Steidel et al., 1997; Barton & Cooke, 2009; Chen et al., 2010; Kacprzak et al., 2011; Lan et al., 2014), which is
observable from the ground at optical wavelengths over a large
redshift range ($0.1<z<2.5$). Mgii has been found to trace two
critical components of the baryon cycle: accretion
(e.g., Steidel et al., 2002; Kacprzak et al., 2010; Ribaudo et al., 2011; Kacprzak et al., 2012b; Martin et al., 2012; Rubin et al., 2012; Bouché et al., 2013; Crighton et al., 2013) and galactic-scale outflowing
winds. (e.g., Weiner et al., 2009; Rubin et al., 2010, 2014; Bouché et al., 2012; Martin et al., 2012; Bordoloi et al., 2014a, b; Kacprzak et al., 2014).
Determining which aspect of the baryon cycle Mgii absorbers trace
has thus far required examining the velocities of the gas with respect
to the galaxy systemic velocity. For example, the signatures of
accreting and/or rotating gas include absorption that is located to
one side (i.e., entirely bluewards or redwards) of the galaxy systemic
velocity (Steidel et al., 2002; Kacprzak et al., 2010; Stewart et al., 2011; Bouché et al., 2013) or observing
redshifted absorption with respect to the galaxy systemic velocity in
a “down-the-barrel” approach (Martin et al., 2012; Rubin et al., 2012). Outflows are commonly observed as blueshifted
absorption (with respect to the galaxy systemic velocity) for the same
down-the-barrel approach (e.g, Weiner et al., 2009; Rubin et al., 2010, 2014; Martin et al., 2012; Bordoloi et al., 2014b). Recently, Fox et al. (2015) found
a quasar whose sightline passes through the “Fermi bubbles” located
near the center of the Milky Way Galaxy. In low ionization absorption,
they found velocity structure in the form of smaller column density,
higher velocity components that were consistent with the front and
back sides of the Fermi bubbles. This result provides hints that the
velocity structure of the absorbers themselves, rather than just the
velocity with respect to the galaxy, is dependent on baryon cycle
processes.
Other works have alluded to the kinematic properties of the absorption
itself in the presence of outflows by examining only the Mgii
equivalent width, $W_{r}(2796)$. This method finds that $W_{r}(2796)$ is
dependent on galaxy color (star formation rate), azimuthal angle,
and/or inclination, where larger values tend to be associated with
blue galaxies and sightlines probing galaxies near their minor axes
(Bordoloi et al., 2011, 2014a; Kacprzak et al., 2011, 2012a; Bouché et al., 2012), both
of which are known to host bipolar outflows. Given that $W_{r}(2796)$
correlates with the number of clouds or Voigt profile components
(Petitjean & Bergeron, 1990; Churchill et al., 2003; Evans, 2011), this indicates that either the
column densities, velocity spreads, or both are larger in the presence
of outflows. The kinematics and column densities may also differ when
associated with accretion, though accreting gas is harder to detect
due to its small covering fraction ($\sim 6\%$; Martin et al., 2012),
and the fact that outflowing gas dominates the absorption signal.
Using $\sim 8500$ strong Mgii absorbers ($0.7<W_{r}(2796)<6.0$ Å) at $0.4<z<1.3$, Ménard et al. (2011) found a $15\sigma$
correlation between $W_{r}(2796)$ and the [Oii] luminosity of the
associated galaxy, which provides an estimate for the star formation
rate. With this correlation, the authors were able to show that the
star formation rate probed by strong Mgii absorption follows the
star formation history up to at least $z\sim 2$. Matejek & Simcoe (2012)
used similar methods for Mgii absorption in infrared wavelengths
with the FIRE spectrograph on Magellan and were able to extend the
redshift range out to $z<6$. They found that the star formation rate
as probed by Mgii traces the global star formation rate out to
$z<6$, including the peak at $z\sim 2-3$. Thus, the Mgii equivalent
width, and possibly the velocity/column density structure, of the
strongest absorbers traces the global star formation rate up to $z=6$.
Additionally, the equivalent width of Mgii absorption has long been
found to anti-correlate with impact parameter at up to a $7.9\sigma$
significance (e.g., Lanzetta & Bowen, 1990; Steidel et al., 1994; Kacprzak et al., 2008; Chen et al., 2010; Nielsen et al., 2013a, b), where the equivalent width decreases with
increasing impact parameter. More recently, Churchill et al. (2013a) found
that $W_{r}(2796)$ anti-correlates with the impact parameter normalized
by the virial radius of the galaxy, $D/R_{\rm vir}$, at the $\sim 9\sigma$ level, a more significant anti-correlation than with $D$
alone. Given that the equivalent width is proportional to the number
of clouds (or velocity components) fit with Voigt profile modeling
(e.g., Petitjean & Bergeron, 1990; Churchill et al., 2003; Evans, 2011), this anti-correlation may
be due to the column densities, velocity spreads, or both diminishing
with projected distance from the galaxy.
These results indicate that examining the kinematic structure and/or
column density distribution of the gas traced by Mgii absorption
over time and space is critical in understanding the detailed physics
of the baryon cycle processes occurring in the CGM. In particular,
studying the detailed velocity structure and column density
distributions of the absorbers constrains the gas physics involved.
Many works examining the detailed Mgii absorber kinematics have
focused on the clustering of VP components in Mgii absorbers by
constructing a two-point correlation function (TPCF) for their samples
using VP component velocities (Sargent et al., 1988; Petitjean & Bergeron, 1990; Churchill, 1997; Churchill & Vogt, 2001; Churchill et al., 2003; Evans, 2011). Churchill et al. (2003) fitted their TPCFs with two
Gaussian components, where the narrower component is associated with
vertical dispersions in face-on galaxy disks. The second and more
broad component may represent the rotational motions in edge-on disks
(Churchill et al., 2003). More recently, Evans (2011) required three
components to fit their TPCF because their sample was more than an
order of magnitude larger than previous works and therefore more
sensitive to an extended tail in the
distribution. Evans (2011) did not try to interpret their
fitted Gaussian components, stating that doing so would be an
oversimplification. This is reasonable since their absorber sample
spans a large redshift range ($0.1<z_{\rm abs}<2.6$) and likely
probes the CGM of a variety of galaxy types.
What these previous absorber kinematics studies lack is the connection
between the detailed kinematics of these absorbers to the properties
of their host galaxies. In a companion paper (Paper V of the
MAGiiCAT series; Nielsen et al., 2015), we examined the kinematics as a
function of galaxy color, inclination, and the azimuthal angle at
which the CGM is probed. We characterized the kinematics by creating
pixel-velocity two-point correlation functions (TPCFs; similar to the
TPCFs used in previous works) for various color and orientation
subsamples. We found that absorbers hosted by blue galaxies in
“face-on” orientations, especially near the projected galaxy minor
axis, have the largest velocity dispersions, while absorbers hosted by
red galaxies for all orientations have small velocity dispersions. We
concluded that for blue galaxies, gas entrained in bipolar outflows
may have large velocity dispersions and may be fragmented into clouds
with smaller column densities, while gas accreting onto or rotating
around the galaxy along the major axis (especially for “edge-on”
orientations) may be more coherent, due to larger cloud column
densities and smaller velocity dispersions. Conversely, we attributed
small velocity dispersions for red galaxies along the minor axis to a
lack of outflows, but larger velocity dispersions along the major axis
may indicate gas that is accreting onto or rotating around the galaxy.
In this paper, we examine the kinematics of Mgii absorption by
using the same pixel-velocity TPCF method as Nielsen et al. (2015), but do
so as a function of galaxy rest-frame $B-K$ color, redshift, $z_{\rm gal}$, and impact parameter normalized by the virial radius,
$D/R_{\rm vir}$. We organize this paper as
follows. Section 2 details our sample, including both
galaxy properties and quasar spectra. Section 3
briefly characterizes quasar absorption line kinematics in terms of
Voigt profile components and then details our methods for calculating
pixel-velocity two-point velocity correlation functions, presenting
only a bivariate analysis of the TPCFs with galaxy rest-frame color,
$B-K$. Section 4 presents a multivariate analysis in
the TPCFs for cuts in galaxy color, redshift, and $D/R_{\rm vir}$. We
discuss our results in Section 5, and summarize and
conclude our findings in Section 6.
2. Sample and Data Analysis
2.1. Galaxy Properties
All 39 galaxies ($0.3<z_{\rm gal}<1.0$) studied here are a subset
of the Mgii Absorber–Galaxy Catalog (MAGiiCAT) and we refer the
reader to Paper I of the series (Nielsen et al., 2013b) for full details of
the data, the selection methods, and how galaxy properties were
determined. To summarize, each galaxy is spectroscopically identified
to be located at the redshift of an associated Mgii absorber
(whether absorption was detected and measured a priori or not)
and within a projected distance $D<200$ kpc from a background
quasar. All galaxies are isolated to the limits of the data available
(for details, see Nielsen et al., 2013b), where isolation is defined as
having no spectroscopically identified galaxy within 100 kpc
(projected) and within a line-of-sight velocity separation of
500 km s${}^{-1}$. For each galaxy, we have spectroscopic redshifts, $z_{\rm gal}$, rest-frame $B$- and $K$-band AB magnitudes and luminosities,
rest-frame $B-K$ colors, and quasar–galaxy impact parameters, $D$. We
also have halo masses, $\log(M_{\rm h}/M_{\odot})$, virial radii,
$R_{\rm vir}$, and maximum circular velocities, $V_{\rm circ}$, from
halo abundance matching, the details for which are presented in Paper
III (Churchill et al., 2013b).
While the data for the absorber–galaxy pairs used here are published
elsewhere (Kacprzak et al., 2011; Evans, 2011; Nielsen et al., 2013b; Churchill et al., 2013b), we
present the galaxy and absorber data for each pair in
Table 1. Columns (1) and (2) are the quasar field names,
columns (3)–(8) are the galaxy properties and columns (9)–(12) are
the absorber properties. Columns (3)–(5) were published in
Nielsen et al. (2013b), while columns (7) and (8) and the $R_{\rm vir}$
values for column (6) were published in Churchill et al. (2013b).
In order to investigate any dependencies of Mgii absorption on the
star formation rate over time, as well as any radial dependencies, we
slice our sample into various subsamples based on median galaxy
rest-frame color, $B-K$, redshift, $z_{\rm gal}$, and impact parameter
normalized by the virial radius, $D/R_{\rm vir}$. The median value is
appropriate here as it allows for roughly equal sample sizes for
comparison. A summary is presented in Table 2, which
details the median value(s) by which the subsamples are defined and
the number of galaxies in each subsample. The table also lists the
median $z_{\rm gal}$ and $D/R_{\rm vir}$ for each subsample after the
full sample cuts are made.
We note that, though the focus of this paper is on galaxy $B-K$
colors, it is difficult to disentangle effects due to color from those
due to galaxy halo masses, $\log(M_{\rm h}/M_{\odot})$. A
Kendall-$\tau$ rank correlation test on color and mass results in a
$2.8\sigma$ correlation such that redder galaxies tend to be more
massive. Figure 1 presents $B-K$ versus $\log(M_{\rm h}/M_{\odot})$ with points colored by $z_{\rm gal}$. Dashed lines
indicate the median color and mass of the sample. Almost all galaxies
in our sample lie within the blue, low mass or red, high mass regions
of Figure 1, with the exception of four blue, high mass
galaxies and four red, low mass galaxies. If we instead conduct our
analysis with $\log(M_{\rm h}/M_{\odot})$, we find no significant
differences in the TPCFs when we compare blue samples to low mass
samples or red to high mass samples. Therefore, any differences we
find in the TPCFs is due to a color–mass dependence rather than just
a color dependence. To mitigate this, we account for the host galaxy
mass by normalizing velocities by the maximum circular velocity,
$V_{\rm circ}$, of the host galaxy.
We also examine possible trends between the properties we use to cut
the sample to rule out the possibility that any differences we may
find between subsamples are mainly due to biases in the data. We ran a
Kendall-$\tau$ rank correlation test between $B-K$ and $z_{\rm gal}$
and find an anti-correlation with a significance of $3.1\sigma$. In
this case, bluer galaxies tend to be located at higher redshift, as
can be seen in Figure 1. Comparing $B-K$ and $D/R_{\rm vir}$ results in an insignificant anti-correlation at $0.8\sigma$,
while we also find an insignificant anti-correlation between $z_{\rm gal}$ and $D/R_{\rm vir}$ at $2.0\sigma$.
2.2. Quasar Spectra
The sample we present here is a Mgii absorption-selected
sample. For each of the 39 isolated galaxies, we have a
high-resolution spectrum of a nearby background quasar in which
absorption is detected at the redshift of the galaxy. Quasar spectra
were observed with HIRES/Keck (Vogt et al., 1994) or UVES/VLT
(Dekker et al., 2000). Most spectra and details of their reduction are
published in Churchill (1997), Churchill & Vogt (2001), Evans (2011),
and/or Kacprzak et al. (2011). We obtained two additional reduced HIRES/Keck
spectra through private communication with C. C Steidel and
J.-R. Gauthier. These latter two spectra were reduced using the Mauna
Kea Echelle Extraction (makee111http://www.astro.caltech.edu/~tb/makee/) package.
Full explanations of how the Mgii absorption systems are identified
in the quasar spectra and Voigt profile fitted are presented in great
detail in Churchill (1997), Churchill & Vogt (2001), Churchill et al. (2003),
Evans (2011), and Kacprzak et al. (2011). We present only a summary
of the process here.
Using Sysanal (Churchill, 1997; Churchill & Vogt, 2001; Evans, 2011), we detect
the Mgii $\lambda\lambda 2796,2803$ absorption doublet in each spectrum with a $5\sigma$
($3\sigma$) significance criterion in the equivalent width spectrum
for the $\lambda 2796$ ($\lambda 2803$) line by following the
formalism of Schneider et al. (1993). Sysanal determines velocity or
wavelength bounds that define regions in which absorption is formally
detected (“kinematic subsystems”) and calculates the rest-frame
equivalent width, $W_{r}(2796)$. The code also defines the absorption
redshift, $z_{\rm abs}$, as the median velocity of the apparent
optical depth distribution of Mgii absorption
(Churchill, 1997). In Figures 2(a) and (b), we present
an example spectrum of quasar Q0235+164 with three Mgii kinematic
subsystems at $z_{\rm abs}=0.852$. The black histogram is the spectrum
for the Mgii $\lambda 2796$ line (panel (a)) and for the
Mgii $\lambda 2803$ line (panel (b)). The shaded regions in panel
(a) designate two of the three kinematic subsystems for this
system. For our TPCF analysis, we use only the pixels inside these
shaded regions.
We then fit all Mgii systems using Voigt profile (VP) decomposition
with Minfit (Churchill, 1997; Churchill & Vogt, 2001; Churchill et al., 2003) and adopt the model
with the fewest statistically significant VP components. Minfit
defines the VP component (cloud) velocities, column densities, and
Doppler $b$ parameters. Full details of Minfit and the fitting
process are described in Evans (2011) and most VP fits are
presented in Kacprzak et al. (2011). An example VP fit is presented in
Figures 2(a) and (b) as the thick red line. Individual
VP components are plotted as thin red lines centered at velocities
indicated by the red, vertical ticks. This system was fitted with
three components in the first shaded region, six in the second shaded
region, and one component at larger velocities.
The absorber data are listed in columns (9)–(12) of
Table 1222We have data for two additional
MAGiiCAT absorber–galaxy pairs but do not include them in the
table nor the analysis. The first is an outlier with
$W_{r}(2796)=4.422$ Å and has no galaxy $B-K$. The second has
$W_{r}(2796)=0.032$ Å, which is lower than our detection
threshold.. The total column densities, $\log N({\hbox{{\rm Mg}\kern 1.0pt{\sc ii}}})$, in
column 11 are calculated by summing the column densities of each
cloud. For a few absorbers, at least one cloud in the absorber does
not have a well constrained column density. In these cases, we report
only the approximate column densities. Column (12) lists the reference
for the absorption data. In several instances, we fit the absorbers
for this work.
To account for differences in the quality of our spectra and to ensure
we can uniformly detect absorption throughout our sample, we use an
equivalent width detection threshold. We calculate the mean
$3\sigma~{}W_{r}(2796)$ detection threshold in each spectrum, defined as
the minimum $W_{r}(2796)$ a kinematic subsystem should have in order to
be detected. Figure 2(c) presents the cumulative
distribution function of the detection threshold in each spectrum in
our sample, which is $\sim 95\%$ complete to roughly 0.04 Å within
$\pm 800$ km s${}^{-1}$ for all absorbers in our sample. We adopt this value
as our equivalent width detection threshold and do not include any
kinematic subsystems with $W_{r}(2796)<0.04$ Å in our analysis. An
example kinematic subsystem that is just below our sensitivity limit
with $W_{r}(2796)=0.03$ Å is presented in Figure 2(a)
at $v\sim 550$ km s${}^{-1}$.
We also have an additional 23 absorber–galaxy pairs with HIRES/Keck
or UVES/VLT quasar spectra in which only an upper limit on the Mgii
equivalent width was measured, though we do not use the data in this
work as we cannot obtain kinematics information for these
“nonabsorbers.” In all but one case, the upper limits on absorption
are lower than our adopted equivalent width detection
threshold. Therefore, if we were able to obtain kinematic information
for these systems, they would not be included in our sample as their
equivalent widths are too low.
3. Characterizing Kinematics
Several methods to examine the kinematics of Mgii absorbers have
been used in the literature; two common methods utilize the velocity
distribution and/or clustering of VP components. These include the
distributions of Voigt profile components and the two-point
correlation function.
3.1. Voigt Profile Component Distributions
In Figure 3, we present the kinematics of our Mgii
absorbers as a function of rest equivalent width, $W_{r}(2796)$, for all
39 absorber–galaxy pairs, including an additional eight pairs for
which we have a high-resolution quasar spectrum but no measured $B-K$
value. We show a simplified spectrum of each Mgii absorber with a
velocity zero point at $z_{\rm abs}$, defined as the optical depth
weighted median of the absorption. Clouds (VP components) are plotted
as points centered at their fitted line-of-sight velocity (see
Figure 2(a)) and total $W_{r}(2796)$ of the associated
absorber. The spread in velocity of the absorbers is plotted as
vertical lines and represents only the largest deviations from $z_{\rm abs}$; gaps between kinematic subsystems (i.e., stretches of
continuum within the extreme velocity bounds of the absorber) are not
presented here. Kinematic subsystems whose equivalent widths are below
our sensitivity cut are plotted as open points with lighter vertical
lines. Point colors indicate the rest-frame $B-K$ color of the host
galaxy (a proxy for star formation rate), with blue points
representing galaxies with $B-K<1.4$, red points as galaxies with
$B-K\geq 1.4$, and gray points are those galaxies for which we do not have a
$B-K$ measurement (8 galaxies). Point types indicate whether the host
galaxy is located at low redshift (circles, $z_{\rm gal}<0.656$), or high redshift
(triangles, $z_{\rm gal}\geq 0.656$).
As shown in Figure 3, large absorber velocity spreads
can be found for absorbers of nearly all equivalent width strengths,
especially when an equivalent width detection threshold is not
enforced (open points). The narrowing of the profiles near
$W_{r}(2796)\sim 1.0-1.2$ Å is due to the absorbers becoming
saturated, which occurs for $\log N({\hbox{{\rm Mg}\kern 1.0pt{\sc ii}}})\sim 13$. Below this
point, large velocity spreads are largely due to several kinematic
subsystems spread out in velocity and may have stretches of continuum
between subsystems. This is most obvious in the absorber presented in
Figure 2(a), which, when including kinematic subsystems
below our detection threshold, has the largest velocity spread in the
sample. Above this point the number of clouds fit to the profile
increases and corresponds to an increasing velocity width. The
degeneracy between velocity spread and equivalent width in this plot
shows that equivalent width is a poor indicator of absorber
kinematics.
We present the kinematics of our Mgii absorbers as a function of
galaxy rest-frame $B-K$ color, redshift, $z_{\rm gal}$, and impact
parameter, $D$, in Figure 4(a). Point colors and
types, along with line colors are similar to those in
Figure 3. We find several qualitative trends in these
results. Clouds are mostly found within $|v_{\rm(cloud)}|=150$ km s${}^{-1}$ of the absorber systemic velocity. As absorption is probed
further from the galaxy (moving outward with increasing $D$), the
velocity spread of absorption may decrease from large velocity spreads
at low $D$ to smaller velocity spreads at higher $D$. The absorbers
may be more extended in velocity for blue galaxies than red. Also, it
appears that most of the highest velocity clouds are located around
galaxies at high redshift.
This method of examining the absorption kinematics has been used
often, though with the velocities shifted to the galaxy systemic
velocity (see e.g., Tumlinson et al., 2013; Werk et al., 2013; Mathes et al., 2014). Though
the method is effective, it is difficult to extract clear kinematic
trends with, for example, galaxy redshift and color, let alone
characterizing the kinematics of the gas itself (rather than with
respect to the galaxy).
3.2. Pixel-velocity Two-point Correlation Functions
We extend beyond the line of work with cloud velocities started by
Petitjean & Bergeron (1990) by examining the pixel-velocity two-point correlation
function (TPCF) for various galaxy subsamples and compare the
resultant line-of-sight velocity dispersions. Previous works
constructed TPCFs using cloud (VP component; the ticks at the top of
Figure 2(a)) velocities, while we use the velocities of
pixels in regions of the spectrum which contribute to the overall
Mgii equivalent width (i.e., kinematic subsystems, see shaded
regions in Figure 2 for examples). Compared to cloud
velocities, pixel velocities better represent the spread in
absorption, provide more velocity pairs for better statistics, and can
be compared more easily to simulations because the absorption profiles
do not need to be Voigt profile modeled. They are also
model-independent, i.e., they do not depend on the fitting method used
and the resulting fit. We study the velocity dispersions of the
absorbers for various galaxy subsamples using the pixel-velocity TPCF.
To first construct the TPCF, which is a measure of the internal
absorber velocity dispersion, we define a subsample of galaxies. From
the spectra of background quasars associated with these galaxies, we
obtain the velocities (where $v=0$ km s${}^{-1}$ represents $z_{\rm abs}$,
the optical depth weighted median of absorption) of pixels located in
regions in which Mgii absorption has been formally detected, also
known as kinematic subsystems (gray shaded regions of
Figure 2(a)). The velocity bounds of these regions are
defined by searching the spectrum redwards and bluewards from the
subsystem velocity centroid until the significance in the per pixel
equivalent width drops below $1\sigma$ (Churchill & Vogt, 2001). We then combine
the pixel velocities from each absorber–galaxy pair in the subsample
and calculate the absolute value of the velocity separations between
each possible pixel velocity pair for the subsample to get $\Delta v_{\rm pixel}$. The TPCF is then created by binning the velocity
separations and normalizing the value in each bin by the total number
of pixel velocity pairs in the subsample to account for differing
numbers of pixels in each subsample when comparing between
subsamples. The TPCF is therefore a probability distribution
function. We use a bin size of 10 km s${}^{-1}$, which corresponds to roughly
one resolution element of both the HIRES/Keck and UVES/VLT
spectrographs (three pixels per resolution element, with a FWHM
resolution of $\sim 6.6$ km s${}^{-1}$).
To determine the uncertainties on the TPCFs, we conduct a bootstrap
analysis. We randomly draw with replacement a sample of kinematic
subsystems from the subsample we are examining which contains the same
number of kinematic subsystems as the original data and construct a
TPCF. We run 1000 bootstrap realizations and then calculate the
$1\sigma$ standard deviations from the mean of the realizations in
each TPCF bin. The bootstrap uncertainties are plotted as shaded
regions around the TPCF.
We also characterize the TPCFs by measuring the velocity separations
within which 50% and 90% of the data reside, $\Delta v(50)$ and
$\Delta v(90)$, respectively. For these TPCFs, $v$ in $\Delta v(50)$ and
$\Delta v(90)$ represents $v_{\rm pixel}$. Uncertainties on these values
are obtained from the bootstrap analysis and represent $1\sigma$
deviations from the mean. These values and their uncertainties are
presented in Table 2 for each subsample.
We present the absorber TPCFs comparing blue and red galaxies in
Figure 4(b). Blue galaxies are presented as the
thick blue line with blue shaded areas indicating the bootstrap
uncertainties, while the thin red line and shading represents red
galaxies. In this panel, we find that the absorption associated with
blue galaxies has a larger velocity dispersion than with red galaxies,
which can also be seen in panel (a). To test whether the two samples
were drawn from the same population, we ran a chi-squared test on the
binned TPCFs (including the uncertainties on the TPCF) and find that
the null hypothesis that the samples were drawn from the same
population can be ruled out at the $10\sigma$ level. The significance,
$\sigma$, the reduced chi-squared, $\chi^{2}_{\nu}$, and the number of
degrees of freedom, $\nu$, is presented in
Figure 4(b). We find that both the $\Delta v(50)$ and
$\Delta v(90)$ measurements for blue galaxies (75 km s${}^{-1}$and 205 km s${}^{-1}$,
respectively) are greater than for red galaxies (60 km s${}^{-1}$and 151 km s${}^{-1}$,
respectively). This indicates that absorbers around blue galaxies have
significantly larger velocity dispersions than those around red
galaxies.
3.3. Cloud Column Densities
Examining the cloud column densities in addition to the absorption
velocity dispersions yields a more complete picture of the physics
involved in placing and maintaining Mgii absorption in the halos of
galaxies. This is especially true considering that column densities
depend on the ionization conditions, temperature, metallicity, and
path length of the gas that is being probed. As stated in
Section 2.2, we obtain cloud column densities for
absorbers using Voigt profile decomposition. We examine the column
density distributions for the same subsamples we use for the TPCFs in
order to obtain a more complete picture of the gas properties as a
function of galaxy properties, and therefore, evolutionary processes.
In Figure 4(c), we present the cloud column density
distributions for blue and red galaxy subsamples. The counts in each
column density bin are normalized to the total number of clouds in
each subsample to create a probability distribution function. We use
the Kolmogorov–Smirnov (KS) test comparing the plotted column density
distributions to determine if the two samples were drawn from the same
population. With a $3.3\sigma$ significance, we find that the null
hypothesis that the cloud column densities for blue and red galaxies
were drawn from the same population can be ruled out. Thus, absorption
associated with blue galaxies tends to have larger column densities
than absorption associated with red galaxies.
3.4. Mass-normalized Pixel-velocity TPCF
To account for the mass of the galaxy hosting absorption, we normalize
the pixel velocities by the maximum circular velocity, $V_{\rm circ}$,
of the host galaxy. We do this because our sample spans a range of
galaxy halo masses and because of the fact that our red galaxies tend
to be more massive than our blue galaxies (see
section 2 and Figure 1). Here we present
methods for constructing normalized absorber TPCFs.
We calculate the mass-normalized absorber TPCF using a similar
procedure as the unnormalized absorber TPCF. Before we calculate the
pixel pair velocity separations for a given subsample, we normalize
each pixel velocity by the $V_{\rm circ}$ of the host galaxy. After
determining the velocity separations, $\Delta(v_{\rm pixel}/V_{\rm circ})$, we bin the values using the same methods used for the
unnormalized TPCFs. The normalized absorber TPCF for blue and red
galaxies is presented in Figure 5. The general
result in this panel that blue galaxies have a larger absorber
velocity dispersion than red galaxies does not differ from the
unnormalized absorber TPCF in Figure 4(b), but the
significance of the chi-squared test is greater here. We also present
measurements of $\Delta v(50)$ and $\Delta v(90)$ for each subsample in the
right-most columns of Table 2, where the $v$ in this case
represents $(v_{\rm pixel}/V_{\rm circ})$. For these subsamples, we
find that the values of $\Delta v(50)$ and $\Delta v(90)$ are very different,
with much larger values for the blue subsample (0.6 and 1.8,
respectively) than the red subsample (0.35 and 0.9, respectively); the
values for the blue subsample are roughly twice as large as for the
red subsample.
4. Multivariate Analysis
In this section, we report on a multivariate analysis of the
kinematics and column density distributions for blue and red galaxies
cut by (1) galaxy redshift, $z_{\rm gal}$, and (2) the projected
radial distance normalized by the virial radius, $D/R_{\rm vir}$.
4.1. Redshift Evolution
While we find significant differences between blue and red galaxies in
the TPCFs in Figure 4(b), the differences may be
washed out by other effects. One such effect is the fact that the star
formation rate has decreased over time to the present day from a peak
at $z\sim 2-3$ (e.g., Hopkins & Beacom, 2006). Therefore, we slice our
blue and red subsamples into low and high $z_{\rm gal}$, using a
median cut of $\langle z_{\rm gal}\rangle=0.656$. The mean redshift
of the low $z_{\rm gal}$ subsample is $z_{\rm gal}=0.469$, while it is
$z_{\rm gal}=0.804$ for the high $z_{\rm gal}$ subsample,
corresponding roughly to a 2 Gyrs time span between mean redshifts. We
present the TPCFs for $B-K$ and $z_{\rm gal}$ subsamples in
Figure 6. In the panels, we list the significance of a
chi-squared test between subsample pairs as well as the reduced
chi-squared value, $\chi^{2}_{\nu}$, and the degrees of freedom, $\nu$
for each panel. Measurements of $\Delta v(50)$ and $\Delta v(90)$ for each
subsample are presented in Table 2.
We find that the TPCF for the red, low $z_{\rm gal}$ subsample is an
outlier such that it has a significantly smaller velocity dispersion
than the rest of the subsamples. In this case, the absorber velocity
dispersion for red galaxies evolves with redshift over a span of
roughly 2 Gyrs ($6.5\sigma$, Figure 6(d)), while there
is no such evolution for blue galaxies ($0.0\sigma$,
Figure 6(c)). At high $z_{\rm gal}$ in
Figure 6(b), absorption in red galaxies has similar
velocity dispersions as in blue galaxies ($1.8\sigma$). However, at
lower $z_{\rm gal}$ in Figure 6(a), the velocity
dispersion for red galaxies decreases, whereas the dispersion for blue
galaxies remains constant ($6.5\sigma$). The values of $\Delta v(50)$ and
$\Delta v(90)$ for all TPCFs are consistent within uncertainties ($\sim 75$ km s${}^{-1}$and $\sim 190$ km s${}^{-1}$, respectively) except the red, low
$z_{\rm gal}$ subsample, which has smaller values than the rest of the
subsamples ($\sim 50$ km s${}^{-1}$and $\sim 120$ km s${}^{-1}$, respectively) and thus
confirms the low velocity dispersions.
In the panels above the TPCFs we plot the cloud column density
distributions for the same subsamples as those in the TPCF plots. The
listed significance is the result of a KS test between plotted
subsamples. The outlying subsample in these panels is the red, high
$z_{\rm gal}$ subsample which has smaller values of $\log N({\hbox{{\rm Mg}\kern 1.0pt{\sc ii}}})$
than both the blue, high $z_{\rm gal}$ subsample ($4.9\sigma$, panel
(b)), and the red, low $z_{\rm gal}$ subsample ($4.1\sigma$, panel
(d)). We find no difference between the column density distributions
associated with blue and red galaxies at low $z_{\rm gal}$
($0.8\sigma$, panel (a)) nor do we find redshift evolution in the
column density distributions for blue galaxies ($1.6\sigma$, panel
(c)).
Figure 7 presents TPCFs in which the pixel
velocities have been normalized by the circular velocity of the host
galaxy. Plotted subsamples are the same as those in
Figure 6. We do not plot the column density
distributions above these TPCF panels because the act of normalizing
the velocities by $V_{\rm circ}$ does not affect the column
densities. In general, we find the same results for the normalized
TPCFs as we did in the unnormalized TPCFs. However, we find that the
redshift evolution present in the red galaxies (panel (d)) is no
longer as strong as it was when the pixel velocities were not
normalized ($3.3\sigma$). For both low and high $z_{\rm gal}$, red
galaxies tend to host absorbers with lower velocity dispersions than
blue galaxies at the $14\sigma$ (panel (a)) and $9.5\sigma$ (panel
(b)) levels, respectively. Lastly, we find that the velocity
dispersion of absorbers hosted by blue galaxies does not evolve with
redshift with a $0\sigma$ significance in panel (c).
To summarize, we find redshift evolution in both the velocity
dispersions and cloud column densities for absorbers associated with
red galaxies. However, the sense of the evolution is reversed in that
the velocity dispersion decreases from higher to lower $z_{\rm gal}$,
while the column densities increase for the same time span. We find no
evolution in either the velocity dispersion or the cloud column
densities for absorbers associated with blue galaxies.
4.2. Radial Dependence
Another effect that may be washing out differences in the TPCFs of
blue and red galaxies is the projected radial distance at which
absorption is found. Many previous works have studied the well-known
anti-correlation between $W_{r}(2796)$ and $D$, which is significant to
the $7.9\sigma$ level (see Nielsen et al., 2013b, and references
therein). Furthermore, since galaxies span a range of
masses, Churchill et al. (2013a) normalized $D$ by the virial radius to
account for the mass of the host galaxy and found an even stronger
anti-correlation between $W_{r}(2796)$ and $D/R_{\rm vir}$ ($8.9\sigma$)
(also see Churchill et al., 2013b). Since $W_{r}(2796)$ depends on column
densities and/or velocity spreads, examining the TPCFs and cloud
column densities as a function of $D/R_{\rm vir}$ may provide insight
into what aspect of the gas physics gives rise to the $W_{r}(2796)$ and
$D/R_{\rm vir}$ anti-correlation. Therefore, we present TPCFs for
subsamples sliced by median values of $\langle B-K\rangle=1.4$ and
$\langle D/R_{\rm vir}\rangle=0.24$ in
Figure 8. The corresponding $\Delta v(50)$ and $\Delta v(90)$
measurements are listed in Table 2.
We find that the internal velocity dispersion of absorbers (TPCF)
around blue galaxies does not depend on where the absorbers are
located in projected distance away from the galaxy ($0\sigma$, panel
(c)), except the dispersion does if the absorbers are located around
red galaxies ($14\sigma$, panel (d)). In red galaxies, the internal
dispersion of absorbers at low $D/R_{\rm vir}$ is comparable to
absorbers in blue galaxies, regardless of where they are being probed
($0.1\sigma$, panel (a)). The outlier of these TPCFs is the high
$D/R_{\rm vir}$, red galaxy subsample, which has a significantly
smaller velocity dispersion than for blue, high $D/R_{\rm vir}$
galaxies ($17\sigma$, panel (b)), or for red, low $D/R_{\rm vir}$
galaxies ($14\sigma$, panel (d)). These results are also represented
in the $\Delta v(50)$ and $\Delta v(90)$ measurements, where all subsamples but
the red, high $D/R_{\rm vir}$ subsample have values of $\Delta v(50)$ and
$\Delta v(90)$ that are consistent within uncertainties ($\sim 70$ km s${}^{-1}$and $\sim 190$ km s${}^{-1}$, respectively). The red, high $D/R_{\rm vir}$
subsample has values that are lower than the rest of the subsamples
($\Delta v(50)$$\sim 50$ km s${}^{-1}$and $\Delta v(90)$$\sim 110$ km s${}^{-1}$). We note that,
although these results are similar to those examining redshift
evolution in which one subsample is a clear outlier from the rest, we
find no significant anti-correlation from a Kendall-$\tau$ rank
correlation test between $z_{\rm gal}$ and $D/R_{\rm vir}$
($2.0\sigma$).
The column density distributions for the TPCF subsamples are plotted
above each panel in Figure 8. Unlike the galaxy color
and redshift subsamples, we find no differences in the column
densities with color or $D/R_{\rm vir}$. The largest significance from
a KS test is between blue and red galaxies at low $D/R_{\rm vir}$ with
$2.5\sigma$, where red galaxies may tend to have smaller cloud column
distributions than blue galaxies. This trend may also be present at
high $D/R_{\rm vir}$ with a $2.4\sigma$ significance. Comparing low
and high $D/R_{\rm vir}$ for blue galaxies and red galaxies, we find
insignificant results from the KS test, with $1.0\sigma$ and
$0.8\sigma$, respectively.
We present the TPCFs normalized by the host galaxy virial radius in
Figure 9 for the same $B-K$ and $D/R_{\rm vir}$
subsamples as in Figure 8. Here we find that red
galaxies have lower velocity dispersions than blue galaxies at all
$D/R_{\rm vir}$, with a significance level of $8.4\sigma$ at low
$D/R_{\rm vir}$ in panel (a) and $17\sigma$ at high $D/R_{\rm vir}$ in
panel (b). The $\Delta v(50)$ and $\Delta v(90)$ values for these subsample
pairs are also not consistent within uncertainties, where the
$\Delta v(50)$ values for blue galaxies ($\sim 0.6$) are roughly twice as
large as those for red galaxies ($\sim 0.3$) for all $D/R_{\rm vir}$. In panel (c) we find no difference in the TPCFs with
$D/R_{\rm vir}$ for blue galaxies ($0\sigma$), with $\Delta v(50)$ and
$\Delta v(90)$ ($\sim 0.6$ and $\sim 1.65$, respectively) echoing this
result. Finally, in panel (d), we find no significant difference in
the TPCFs for red galaxies with $D/R_{\rm vir}$ ($2.1\sigma$);
however, we find that, while $\Delta v(50)$ is consistent within
uncertainties for the two subsamples ($\Delta v(50)$$\sim 0.3$), the
$\Delta v(90)$ is larger for the low $D/R_{\rm vir}$ subsample (0.95) than
the high $D/R_{\rm vir}$ subsample (0.75).
In summary, the absorber velocity dispersion depends on
$D/R_{\rm vir}$ for red galaxies only, where the dispersions are
smaller at larger $D/R_{\rm vir}$, and this difference is present only
in the tails for the normalized TPCFs. At low $D/R_{\rm vir}$, the
velocity dispersions for absorption associated with blue and red
galaxies are comparable in the unnormalized TPCFs. In contrast, the
cloud column densities do not depend on whether the absorption is located
around blue or red galaxies, nor do they depend on $D/R_{\rm vir}$.
4.3. Anti-correlation of $W_{r}(2796)$ and $D/R_{\rm vir}$
The results presented in the previous section (Section 4.2)
are puzzling given the anti-correlation between $W_{r}(2796)$ and
$D/R_{\rm vir}$ (Churchill et al., 2013a, b). Since equivalent width
correlates with the number of clouds (Petitjean & Bergeron, 1990; Churchill et al., 2003; Evans, 2011), the column densities, the velocity spreads, or both
should diminish with increasing $D/R_{\rm vir}$. Therefore, we
expected that the TPCFs and/or the cloud column densities would show a
dependence on $D/R_{\rm vir}$ regardless of color where the cloud
column densities and/or TPCF velocity dispersions would decrease with
increasing $D/R_{\rm vir}$. However, this is not the case. We found no
dependence of the TPCF velocity dispersions on $D/R_{\rm vir}$ for
blue galaxies, but the velocity dispersion for red galaxies is lower
at high $D/R_{\rm vir}$ (as might be expected) in the unnormalized
TPCFs. However, the red galaxy TPCF $D/R_{\rm vir}$ dependence
vanished when we normalized the pixel velocities by $V_{\rm circ}$. Additionally, the cloud column density distributions do not
differ with $D/R_{\rm vir}$ for both blue and red galaxies.
To better understand the sample examined here in the context of the
$W_{r}(2796)$–$D/R_{\rm vir}$ anti-correlation, we present
Figure 10 in which we plot the present sample of
galaxies as solid points, the rest of the MAGiiCAT sample
(Nielsen et al., 2013b) absorbing galaxies as open points, and MAGiiCAT
nonabsorbing galaxies as downward arrows as their absorption is only
known to a $3\sigma$ upper limit. This plot is similar to Figure 1(c)
in Churchill et al. (2013a), though point colors here represent galaxy
colors sliced by the median color, $\langle B-K\rangle=1.4$. The
solid and dashed lines are the fit to the data in
Churchill et al. (2013a). For reference we plot the median $D/R_{\rm vir}$
as a vertical dotted line.
As reported by Churchill et al. (2013a), the anti-correlation is significant
to the $8.9\sigma$ level for the full MAGiiCAT sample using a
BHK-$\tau$ non-parametric rank correlation test to account for the
upper limits on absorption. Since we study only those systems with
detected absorption in high resolution quasar spectra, we ran a
Kendall-$\tau$ rank-correlation test between $W_{r}(2796)$ and $D/R_{\rm vir}$ for the present sample (filled points) and found an
anti-correlation that is significant to the $0.8\sigma$ level. Thus
the data presented here do not exhibit an anti-correlation between
$W_{r}(2796)$ and $D/R_{\rm vir}$ (however, if we examine all
absorbers in the MAGiiCAT sample, we do find an anti-correlation
with $4.6\sigma$ significance). If we examine only blue galaxies, the
significance drops further to $0.2\sigma$ (for red galaxies the
significance remains at $0.8\sigma$). This result is then consistent
with no $D/R_{\rm vir}$ dependence for the cloud column densities
regardless of galaxy color and for the TPCFs of blue
galaxies. However, it does not explain the differences in the TPCFs of
absorbers associated with red galaxies in the unnormalized TPCFs.
We also examined the statistics on this anti-correlation for low and
high $z_{\rm gal}$ subsamples to determine if the anti-correlation was
affecting the redshift evolution results in the TPCFs. Point types in
Figure 10 represent $z_{\rm gal}$ subsamples, with
circles for low $z_{\rm gal}$ and triangles for high $z_{\rm gal}$. The Kendall-$\tau$ rank-correlation test resulted in an
insignificant anti-correlation between $W_{r}(2796)$ and $D/R_{\rm vir}$
for both the low $z_{\rm gal}$ ($1.1\sigma$) and the high $z_{\rm gal}$ ($0.1\sigma$) subsamples.
To ensure that our kinematics sample is not unusual, we randomly drew
39 absorbers from the full MAGiiCAT absorber sample for one million
realizations and ran the rank-correlation test each time. The fraction
of realizations in which the anti-correlation between $W_{r}(2796)$ and
$D/R_{\rm vir}$ is significant (i.e., the significance is greater than
$3\sigma$) is 25%. For blue galaxies (19 absorbers) and similarly
with red galaxies (20 absorbers), this fraction drops to 4%. Finding
no significant anti-correlation is not unusual. Given the history of
absorber-galaxy studies (see e.g., Churchill et al., 2005; Nielsen et al., 2013b, for a list of
references), this is not unexpected. With
larger numbers of absorber–galaxy pairs, the statistics on the
$W_{r}(2796)$–$D$ anti-correlation has steadily become more
significant. The main reason for the lack of an anti-correlation for a
given smaller sample is the large scatter in the relation.
5. Discussion
By examining the kinematics and cloud column densities of Mgii
absorbers, we have observed redshift evolution in the CGM of red
galaxies where the velocity dispersions of absorbers decrease and the
cloud column densities increase with decreasing redshift. When
examining the kinematics as a function of $D/R_{\rm vir}$, we also
find a difference for red galaxies where the velocity dispersions
decrease with increasing $D/R_{\rm vir}$, though the cloud column
densities do not differ at low and high $D/R_{\rm vir}$. The radial
dependence in the velocity dispersions for red galaxies is removed
when we normalize the pixel velocities by $V_{\rm circ}$. Conversely,
we find no redshift or radial dependence of the velocity dispersions
and cloud column densities for blue galaxies. Compared to the red
galaxies, the blue galaxy velocity dispersions and cloud column
densities are larger than for red galaxies.
We found that red galaxies have smaller velocity dispersions than blue
galaxies overall; this is most obvious in
Figure 4. Since blue (less massive) galaxies tend to
have a larger star formation rate than red (more massive) galaxies,
blue galaxies are more likely to experience outflows than red
galaxies. Thus, the large velocity dispersions in blue galaxies may
well be due to outflows induced by star formation which act to “stir
up” the Mgii absorbers, whereas a lack of outflows in red galaxies
likely causes the smaller velocity dispersions in red galaxy
TPCFs. This is consistent with previous works in which outflows were
invoked to explain the presence and properties of Mgii absorption
(e.g., Rubin et al., 2010, 2014; Bouché et al., 2012; Martin et al., 2012; Bordoloi et al., 2014a, b; Kacprzak et al., 2014).
A possible alternative explanation for the large TPCF velocity
dispersions for the blue galaxies is the presence of merging satellite
galaxies. Regardless of the host galaxy type, satellite galaxies
present such a small cross-section that they are unlikely to be a
significant source of Mgii absorption around host galaxies. Several
works have investigated this by comparing the estimated satellite
cross-sections from simulations to the observed incidence of
absorption and found that the satellite cross-sections are much lower
than the absorption incidence rate (e.g., Tumlinson et al., 2013; Gauthier et al., 2010). Thus, satellites are unlikely to explain the properties
of the Mgii absorbers we present here. For more discussion of
possible effects of satellite contributions to Mgii kinematics, see
Paper V of the MAGiiCAT series (Nielsen et al., 2015).
That we find differences in the velocity dispersions and cloud column
densities for absorbers around red galaxies with redshift, but no such
evolution in blue galaxies, may suggest we are observing the
consequences of quenched star formation in red galaxies but ongoing
star formation in blue galaxies. Due to the ongoing star formation in
blue galaxies, the absorbers are likely to be involved in outflows,
accretion, and/or recycling at all redshifts, thus their velocity
dispersions remain large and their cloud column densities remain
unchanged. Outflows may continually replenish the CGM of disturbed,
large column density gas.
At high redshift in Figure 6(b), the absorber velocity
dispersions are similar regardless of galaxy color, indicating that
the red galaxies we observe in the high redshift subsample may have
undergone star formation driven outflows recently. At low redshift in
Figure 6(a), the TPCFs are narrower for red galaxies
than blue, possibly indicating that the outflows at higher redshift
have since shut off. This is also shown in the normalized TPCFs in
Figure 7, where the high $z_{\rm gal}$, red galaxy
TPCF is more narrow than the blue galaxy TPCF, but this difference
increases for low $z_{\rm gal}$ subsamples. Since the cloud column
densities of red galaxies at high $z_{\rm gal}$ are smaller than those
for blue galaxies, some mechanism present only in red galaxies may
break the clouds into smaller column density clouds. At lower
redshift, the cloud column density distributions for blue and red
galaxies are comparable, but similar to the distribution for blue
galaxies at higher redshift.
The quenching of star formation may act to slowly reduce the velocity
dispersion of Mgii absorbers, but the quenching event initially
breaks the clouds into smaller column density clouds, which then
increase over time. This may be explained by a scenario in which star
forming galaxies have outflows driven by active star formation which
agitate and disperse the gas in the CGM to larger velocity
dispersions. Star formation is then shut off via an unknown quenching
mechanism (either AGN activity, intense star formation, a
galaxy–galaxy merger, etc.) which breaks the clouds into smaller
column densities. Over time, the absorbers are allowed to settle to
lower velocity dispersions, which then allows for the individual
clouds to “re-condense” to form larger column density
clouds. Alternatively, the cloud column densities may appear to
increase because as the velocity dispersion decreases, the gas builds
over a narrower velocity range, resulting in larger measured column
densities even if the individual clouds are physically separate with
unchanging column densities. Thus the CGM becomes quiescent.
Initially we found a dependence of the red galaxy TPCFs on $D/R_{\rm vir}$ where the velocity dispersion decreases with increasing
$D/R_{\rm vir}$, which would follow from the $W_{r}(2796)$–$D/R_{\rm vir}$ anti-correlation discussed in Churchill et al. (2013a). This result
and the lack of a dependence of the TPCFs for blue galaxies on
$D/R_{\rm vir}$ would suggest that outflows push disturbed material
out to large distances. Then when star formation is quenched, the
velocity dispersions of gas in the inner CGM remain large, but the
outer region of the CGM is the first to show signs of quenched star
formation in the form of decreasing velocity dispersions. However, by
investigating the $W_{r}(2796)$–$D/R_{\rm vir}$ relation in
Section 4.3 we found that there is no anti-correlation for
the sample presented here with a $0.8\sigma$ significance (same for
red galaxies) nor blue galaxies whose significance drops to
$0.2\sigma$. This is likely due to the small sample size as well as
the fact that we focus only on absorbers, which have only a
$4.6\sigma$ anti-correlation, compared to the full sample that includes
nonabsorbers with $8.9\sigma$. Since the $W_{r}(2796)$–$D/R_{\rm vir}$
anti-correlation is not interfering with, nor contaminating the sample
presented here, the redshift evolution of the TPCFs and cloud column
densities is more strongly explained as being due to the quenching of
star formation rather than an underlying sample bias.
Since the most significant differences in the TPCFs with $D/R_{\rm vir}$ were for red galaxies and we have no $W_{r}(2796)$–$D/R_{\rm vir}$ anti-correlation for this sample, we examined the mass
distributions for each of the $B-K$ and $D/R_{\rm vir}$ subsample
combinations. We found that galaxies in the low $D/R_{\rm vir}$
subsamples tend to be slightly more massive than those in the high
$D/R_{\rm vir}$ subsamples, for both blue and red galaxies. Thus,
normalizing the pixel velocities in the TPCFs by $V_{\rm circ}$
removes the mass bias with velocity, where more massive galaxies can
have higher velocity gas. Doing this resulted in no differences in the
mass-normalized TPCFs with $D/R_{\rm vir}$ for both blue and red
galaxies. However, the TPCFs for red galaxies are still narrower than
for blue galaxies at all $D/R_{\rm vir}$. This may indicate that,
after accounting for the mass of the galaxy in both the size of the
CGM and velocities, the quenching of star formation in red galaxies
affects gas in the CGM at all $D/R_{\rm vir}$, at least out to
$D/R_{\rm vir}=0.75$.
While our findings here provide interesting details into the nature of
the CGM, a large body of work has shown that the characteristics of
the absorbing gas in the CGM depend on both the galaxy inclination and
whether the absorption is located along the galaxy projected major or
minor axes (e.g., Bordoloi et al., 2011, 2014b; Bouché et al., 2012; Kacprzak et al., 2010, 2012a; Lan et al., 2014; Rubin et al., 2014). This is especially true for star
forming galaxies as outflowing gas tends to be found along the minor
axes (e.g., Bouché et al., 2012; Kacprzak et al., 2014) while accretion tends to be
detected along the major axes (e.g., Kacprzak et al., 2010; Bouché et al., 2013). These trends have also been observed in simulations
(e.g., Stewart et al., 2011; Danovich et al., 2012, 2015). In the data
presented here, there may be hints of a bimodality for the blue galaxy
subsample TPCFs and cloud column density distributions, which may be
due to these orientation effects. In fact, we examined these
orientation effects in a companion paper (MAGiiCAT
V; Nielsen et al., 2015) and found that the largest velocity dispersions
were associated with blue, face-on galaxies ($i<57^{\circ}$), and are
likely due to outflowing gas pointed towards the observer. The
smallest velocity dispersions were associated with red, face-on
galaxies, which may be due to a lack of outflows as the star formation
in red galaxies has been quenched. Similar velocity dispersions for
blue and red galaxies that are in edge-on inclinations
($i\geq 57^{\circ}$), probed along the galaxy projected major axis,
indicated that the gas we observed was accreting/rotating around the
galaxies. These orientation results have larger significances in the
chi-squared results than the results we present here. Thus, the
orientation of the host galaxy may be more important than $z_{\rm gal}$ and $D/R_{\rm vir}$ in understanding the processes giving rise
to absorption in the CGM.
6. Summary and Conclusions
Using a subset of MAGiiCAT galaxies (Nielsen et al., 2013b), we examined
the kinematics of gas in the CGM as a function of galaxy color,
redshift, and virial radius-normalized impact parameter. Each galaxy
was spectroscopically identified to be located at the redshift of an
associated Mgii absorber in a high-resolution quasar spectrum
within a projected distance of $D=200$ kpc. Thus the galaxy sample is
an absorption-selected sample, and only those absorption regions with
$W_{r}(2796)\geq 0.04$ Å were included in our analysis. Galaxy
virial radii and circular velocities were obtained using halo
abundance matching (Churchill et al., 2013b) and were used to normalize out
any mass dependence with impact parameter and velocity.
Our main conclusions are as follows:
1.
We find no redshift evolution in the kinematics nor cloud column
densities for absorbers hosted by blue galaxies. This is possibly
due to ongoing star formation, which causes outflows that continue
to agitate and disperse the absorbers to form large velocity
dispersions. Outflows thus continually replenish the CGM with large
column density, high velocity dispersion gas. This result is still
true when we normalize the pixel velocities by $V_{\rm circ}$ to
remove any mass dependence.
2.
Conversely, we find redshift evolution in the kinematics for
absorbers hosted by red galaxies. The quenching of star formation in
red galaxies may shut off outflows, which then may prevent the CGM
from being replenished with the large column density, high velocity
dispersion gas seen in blue galaxies. Because of this, once star
formation has been quenched, absorbers appear to relax into lower
velocity dispersions. The quenching mechanism may act to reduce the
cloud column densities initially, but the column densities increase
towards lower redshifts. Due to the lower velocity dispersions at
lower redshifts, the clouds may be able to “re-condense” into
larger column density clouds, or appear to increase in column
density due to a narrower velocity range over which the clouds are
spread, regardless of the physical distance between clouds. This
result also stands when we normalize the pixel velocities by $V_{\rm circ}$.
3.
Despite an overall anti-correlation between Mgii equivalent
width and $D/R_{\rm vir}$ reported in Churchill et al. (2013b), the sample
we present here (those absorbers for which we have high-resolution
quasar spectra) does not follow a $W_{r}(2796)$–$D/R_{\rm vir}$
anti-correlation. This strengthens the result that the kinematics
and cloud column densities of red galaxies undergo redshift
evolution since an underlying anti-correlation with $D/R_{\rm vir}$
is not interfering with the result.
4.
Neither the TPCFs nor the cloud column density distributions
depend on $D/R_{\rm vir}$ when the absorber pixel velocities are
normalized by $V_{\rm circ}$ (to remove any possible mass bias in
the data). This is consistent with the lack of an anti-correlation
between $W_{r}(2796)$ and $D/R_{\rm vir}$. Since the TPCFs for red
galaxies are more narrow than for blue galaxies at all $D/R_{\rm vir}$, this suggests that quenching affects the CGM out to at
least $D/R_{\rm vir}=0.75$.
This work constitutes our first examination of the kinematics of
Mgii absorbers as a function of galaxy properties. Previous works
had examined the kinematics of the absorbers in a variety of ways but
had not connected their results to the host galaxy properties, at
least in a statistical fashion as is possible with the pixel-velocity
TPCFs. In future work and to further understand the kinematics of gas
in the CGM as a function of galaxy properties, we will shift the pixel
velocities to the galaxy systemic velocity. We will also examine the
gas kinematics as a function of star formation rate (SFR), specific
SFR, SFR density, and galaxy metallicity.
We thank C. Steidel and J.-R. Gauthier for providing reduced
HIRES/Keck quasar spectra. This material is based upon work supported
by the National Science Foundation under Grant No. 1210200 (NSF East
Asia and Pacific Summer Institutes). N.M.N. was also partially
supported through a NMSGC Graduate Fellowship and a Graduate Research
Enhancement Grant (GREG) sponsored by the Office of the Vice President
for Research at New Mexico State University. G.G.K. acknowledges the
support of the Australian Research Council through the award of a
Future Fellowship (FT140100933). M.T.M. thanks the Australian Research
Council for Discovery Project grant DP130100568 which supported this
work. This work is based in part on observations collected at the
European Organisation for Astronomical Research in the Southern
Hemisphere under ESO programs 076.A-0860, 69.A-0371, 68.A-0170,
075.A-0841, 65.O-0158, 072.A-0346, 074.A-0201, 67.A-0146, 278.A-5048,
076.A-0463, 67.A-0022, and 67.C-0157. Some of the data presented
herein were obtained at the W.M. Keck Observatory, which is operated
as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics
and Space Administration. The Observatory was made possible by the
generous financial support of the W.M. Keck Foundation.
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Phase diagram of the ST2 model of water
Frank Smallenburg${}^{a}$${}^{\ast}$,
Peter H. Poole${}^{b}$, and Francesco Sciortino${}^{c}$
${}^{a}$Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine Universität Düsseldorf, Universitätstrasse 1, 40225 Düsseldorf, Germany;
${}^{b}$Department of Physics, St. Francis Xavier University, Antigonish, Nova Scotia B2G 2W5, Canada;
${}^{c}$Dipartimento di Fisica, Università di Roma La Sapienza, Piazzale A. Moro 5, 00185 Roma, Italy
${}^{\ast}$Corresponding author. Email: [email protected]
Abstract
We evaluate the free energy of the fluid and crystal phases for the ST2 potential [F.H. Stillinger and A. Rahman, J. Chem. Phys. 60, 1545 (1974)] with reaction field corrections for the long-range interactions. We estimate the phase coexistence boundaries in the
temperature-pressure plane, as well as the gas-liquid critical point and gas-liquid coexistence conditions.
Our study frames the location of the previously identified liquid-liquid critical point relative to the crystalline phase boundaries, and opens the way for exploring crystal nucleation in a model where the metastable liquid-liquid critical point is computationally accessible.
I Introduction
The thermodynamic behavior of water at low temperatures is unconventional. Several quantities, e.g. the isobaric density $\rho$, the isothermal compressibility $K_{T}$, and the constant-pressure specific heat $C_{P}$, are characterized by
non-monotonic temperature or pressure dependence Debenedetti and Stanley (2003).
Over the past decades, the anomalous behavior of these quantities has attracted the attention of numerous researchers. In 1992, a numerical investigation of the equation of state (EOS) suggested the presence of a liquid-liquid (LL) critical point Poole et al. (1992) in the
ST2 model Stillinger and Rahman (1974), an interaction potential that describes water as a classical, rigid, non-polarizable molecule.
The presence of a LL critical point, located in the supercooled region, provides an elegant explanation of the thermodynamic anomalies that characterize liquid water and which become more pronounced close to such a critical point Xu et al. (2005).
The conceptual novelty of a one-component system with more than one liquid phase
has stimulated the scientific community to deeply probe the physical
origin of this phenomenon Mishima and Stanley (1998); Soper and Ricci (2000); Katayama et al. (2000); Kurita and Tanaka (2004); Taschin et al. (2013); Pallares et al. (2014); Amann-Winkel et al. (2013); Azouzi et al. (2013); Sellberg et al. (2014).
It is now clear that a LLCP, while common in tetrahedral network-forming liquids Saika-Voivod et al. (2001); Vasisht et al. (2011); Hsu et al. (2008); Abascal and Vega (2010); Smallenburg et al. (2014); Starr and Sciortino (2014), can also be observed in complex one-component fluids when the (spherically symmetric) interaction potential generates two competing length scales Jagla (1999); Franzese
et al. (2001a); Xu et al. (2006); Gallo and Sciortino (2012).
In the last few years the interest has shifted towards the interplay between
the liquid-liquid critical point and crystal nucleation Limmer and Chandler (2011); Palmer et al. (2014); Smallenburg et al. (2014); Singh and Bagchi (2014); Buhariwalla et al. (2015). Indeed, in experiments,
crystallization has so far prevented direct observation of this phenomenon in a one-component bulk system.
Only recently have computer simulations demonstrated the possibility of generating a
thermodynamically stable liquid-liquid critical point (as opposed to a metastable one) in models of network-forming liquids Smallenburg et al. (2014); Starr and Sciortino (2014).
Accurate information on the phase coexistence boundaries between disordered and
ordered phases is relevant not only to establish the thermodynamic fields of stability
of the different phases, but also as a reference for estimating when the liquid
becomes metastable. In turn, this has relevance for estimating when the barrier
to crystallization becomes finite and how rapidly the barrier decreases on
supercooling Romano et al. (2011).
Except for one early report focussing on the liquid-ice I${}_{h}$ boundary reh ,
none of the coexistence lines between the gas, liquid, and the many phases of crystalline ice have been accurately determined for the ST2 model.
In this article we fill this gap and evaluate these coexistence boundaries by calculating the
fluid chemical potential (via thermodynamic integration) and the crystal
chemical potential (via the Frenkel-Ladd method Frenkel and Ladd (1984), extended to molecules Vega et al. (2008)). We test
several crystals (ice I${}_{h}$, I${}_{c}$, VI, VII, and VIII)
and find that in the region of pressure where thermodynamic
anomalies appear (e.g. near the lines of maxima of $C_{P}$ and $K_{T}$) ice I${}_{h}$ and I${}_{c}$ have the same free energy within our numerical precision. Unexpectedly, we discover that for the ST2 model, on increasing pressure, the stable phase is a dense tetragonal crystal
with partial proton order. This structure has a free energy about 0.4 $k_{B}T$
lower than
ice VII, the structure obtained by interspersing two I${}_{c}$ lattices. (Here $T$ is the temperature and $k_{B}$ is the Boltzmann constant.)
We also evaluate the (metastable) line of coexistence for the recently reported
ice $0$ lattice Russo et al. (2014); Quigley et al. (2014), a structure which could act (according to the
Ostwald rule) as the intermediate phase in the process of nucleating the stable
ice I${}_{h/c}$ crystal from the fluid. For completeness, we determine the
location of the gas-liquid critical point, which is found to be at $T_{c}=558.0\pm 0.3$K and $\rho_{c}=0.265\pm 0.005$ g/cm${}^{3}$.
II Model and simulation methods
We study, via Monte Carlo (MC) simulations, the original ST2 potential as defined by Rahman and Stillinger Stillinger and Rahman (1974), with reaction field corrections to approximate the long-range contributions to the electrostatic interactions. ST2 models water as a rigid body with an oxygen atom at the center and four charges $q=\pm 0.4e$ (where $e$ is the electron charge), two positive and two negative, in a tetrahedral geometry. The distances from the oxygen to the positive and negative charges are 0.1 and 0.08 nm respectively. The oxygen-oxygen interaction is modeled via a standard Lennard-Jones potential truncated at $2.5\sigma_{LJ}$, with $\sigma_{LJ}=0.31$ nm and $\epsilon_{LJ}=0.31694$ kJ/mol. The Lennard-Jones
residual interactions are handled through standard long-range corrections, i.e. by assuming that
the radial distribution function is unity beyond the cutoff. The charge-charge interactions are smoothly switched off both at small and large distances via a tapering function, as in the original model Stillinger and Rahman (1974). Complete details of the simulation procedure are as described in Ref. Poole et al. (1992). In the following, we use $\sigma=1$ nm as unit of length.
II.1 Thermodynamic integration: Fluid free energy
To evaluate the fluid free energy we perform thermodynamic integration
along a path of constant reference density $\rho_{\mathrm{ref}}$ for a modified pair potential,
$$V=\min(V_{ST2},200\mathrm{\;kJ/mol}).$$
(1)
This potential coincides with the ST2 potential for all intermolecular distances and orientations where
$V_{ST2}<200$ kJ/mol, and is constant and equal to 200 kJ/mol otherwise. Note that in the temperature range where we investigate the phase behavior,
molecules never approach close enough to reach this limit.
In this way, the divergence of the potential energy for configurations in which some intermolecular separations vanish
(which would otherwise be probed at very high
temperatures) is eliminated and the infinite temperature limit is properly
approximated by an ideal gas of molecules at the same density.
The fluid free energy (per particle) is calculated as
$$\beta f_{ST2}^{\mathrm{fluid}}(\beta,\rho_{\mathrm{ref}})=\beta f_{\mathrm{ig}%
}(\beta,\rho_{\mathrm{ref}})+\int_{0}^{\beta}\left\langle V(\beta,\rho_{%
\mathrm{ref}})\right\rangle d\beta,$$
(2)
where $\beta=1/k_{B}T$ and $\beta f_{\mathrm{ig}}(\beta,\rho)=\log(\rho_{n}\sigma^{3})-1$ is the ideal gas free energy and
$\rho_{n}$ is the number density. Fig. 1 shows the average modified pair potential energy $\left\langle V(\beta,\rho)\right\rangle$ and the
interpolating (spline) continuous curve used to numerically evaluate the integral.
The free energy at different densities along a constant-$T$ path is evaluated via thermodynamic integration of the equation of state
$$\beta f_{ST2}^{\mathrm{fluid}}(T,\rho_{n})=\beta f_{ST2}^{\mathrm{fluid}}(T,%
\rho_{n,\mathrm{ref}})+\int_{\rho_{n,\mathrm{ref}}}^{\rho_{n}}\frac{\beta P(%
\rho_{n}^{\prime})}{\rho_{n}^{\prime}}d\ln(\rho_{n}^{\prime}),$$
(3)
where $P(\rho_{n})$ is the equation of state for the pressure $P$ at fixed $T$.
II.2 Crystal free energy
To evaluate the free energy of a selected crystalline structure
we follow the methodology reviewed in Ref. Vega et al. (2008).
We define an Einstein crystal in which each molecule interacts, in addition to the ST2 potential, with a
Hamiltonian, composed of a
translational ($H_{\mathrm{trans}}$) and a rotational ($H_{\mathrm{rot}}$) part, that attaches each molecule to a reference position and orientation.
For each particle we define two unit vectors: the (normalized) HH vector and dipole vector, named respectively $\vec{a}$ and $\vec{b}$. The reference configuration is defined by the reference position of the oxygen atom ${\bf r_{0}}$ and
the reference position of $\vec{a}$ and $\vec{b}$ Vega et al. (2008); Noya et al. (2008).
In the following we indicate with ${\bf r}-{\bf r}_{0}$ the
displacement of a particle located at ${\bf r}$ from
its reference position, and with $\phi_{a}$ and
$\phi_{b}$ the angles between $\vec{a}$ and $\vec{b}$
and their reference values. More precisely,
$$H_{\mathrm{Einstein}}=H_{\mathrm{trans}}+H_{\mathrm{rot}}$$
(4)
with
$$H_{\mathrm{trans}}=\lambda_{t}({\bf r-r_{0}})^{2}/\sigma^{2}$$
(5)
and
$$H_{\mathrm{rot}}=\lambda_{r}\left[\sin^{2}\phi_{a}+\left(\frac{\phi_{b}}{\pi}%
\right)^{2}\right].$$
(6)
Here $\lambda_{t}$ and $\lambda_{r}$ indicate the strength of the coupling to the reference configuration.
Again following Ref. Vega et al. (2008),
the free energy (per particle) of a crystal structure $f^{\mathrm{xt}}$, in the limit of large $\lambda_{r}$ and $\lambda_{t}$ is calculated as,
$$\beta f^{\mathrm{xt}}=\beta f_{1}+\beta f_{2}+\beta f_{3}+\beta f_{4}+\beta f_%
{5}+\beta f_{6}$$
(7)
where, indicating with $N$ the number of molecules in the system,
$$\displaystyle\beta f_{1}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{N}\ln\left[\left(\frac{\pi}{\beta\lambda_{t}}\right)^{%
\frac{3(N-1)}{2}}N^{\frac{3}{2}}\frac{1}{\rho_{n}\sigma^{3}}\right]$$
(8)
$$\displaystyle\beta f_{2}$$
$$\displaystyle=$$
$$\displaystyle-\ln{\frac{\sqrt{\pi}}{4}}+1.5\ln(\beta\lambda_{r})$$
$$\displaystyle\beta f_{3}$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\lambda_{t}}\left\langle\beta H_{trans}\right\rangle_{%
\lambda}d\ln\lambda$$
$$\displaystyle\beta f_{4}$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\lambda_{r}}\left\langle\beta H_{rot}\right\rangle_{%
\lambda}d\ln\lambda$$
$$\displaystyle\beta f_{5}$$
$$\displaystyle=$$
$$\displaystyle-\frac{\ln\left\langle e^{-\beta V_{ST2}}\right\rangle_{\lambda_{%
r},\lambda_{t}}}{N}$$
$$\displaystyle\beta f_{6}$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{ll}\ln[1.5]&\text{(full proton-disordered %
crystal)}\\
0&\text{(proton-ordered crystal).}\end{array}\right.$$
The symbols $\left\langle H_{\mathrm{rot}}\right\rangle_{\lambda}$ and $\left\langle H_{\mathrm{trans}}\right\rangle_{\lambda}$ indicate
the average values of $H_{\mathrm{rot}}$ and $H_{\mathrm{trans}}$ calculated from a
MC simulation of particles interacting via the ST2 potential complemented by
$H_{\mathrm{Einstein}}$. The symbol $\left\langle e^{-\beta V_{ST2}}\right\rangle_{\lambda_{r},\lambda_{t}}$
indicates the average value of $e^{-\beta V_{ST2}}$ (where $V_{ST2}$ is the
system ST2 potential energy) in a simulation in which the particles interact with each other via the ST2 potential and with
the Einstein Hamiltonian with values $\lambda_{r}$ and $\lambda_{t}$.
In all simulations carried out to perform the integration, the center of mass of the system
is kept fixed Smith and Frenkel (1996).
Finally, $\beta f_{6}$ indicates the contribution of proton disorder, evaluated according to Pauling’s
estimate Pauling (1945). More recent calculations have essentially confirmed
Pauling’s value Berg et al. (2007).
Table 1 reports the values of $\beta f_{j}$ for a few representative cases.
II.3 Grand canonical simulation: Gas-liquid phase coexistence
To evaluate the gas-liquid coexistence and the location of the gas-liquid critical point,
we perform grand-canonical MC simulations to evaluate
at fixed $T$, volume $v$, and chemical potential $\mu$, the
probability $p$ of observing $N$ particles in the simulated volume. To overcome the
large free energy barriers separating the gas and liquid phases we
implement the successive umbrella sampling (SUS) technique Virnau and Müller (2004).
Since this method has been applied previously to ST2 Sciortino et al. (2011) to estimate the
liquid-liquid coexistence conditions, and has been documented in detail in these
works, we refer the interested reader to the original literature.
II.4 Proton position in the crystal structures
To generate proton-disordered crystals, such as ice I${}_{h/c}$ and ice VII,
one needs to assign protons to the oxygens, located at the lattice positions,
so as to satisfy the ice rules. To this end, we first calculate a list of all bonded oxygen neighbours (where four bonds connect to each oxygen atom) and then decorate the oxygen lattice by assigning the proton for each bond to one of the two bonded atoms, iterating the following procedure:
(i) Randomly select one oxygen with less than two hydrogens and one of the remaining
undecorated bonds emanating from the selected oxygen.
(ii) Randomly follow the path of undecorated bonds until the path loops back to the original oxygen.
(iii) Decorate all bonds of the selected path with one proton each, associating the protons
to the oxygens encountered in the path. The procedure is iterated until all oxygens have two protons associated with them. Paths in which the initial and final oxygen atoms coincide only via periodic images produce a non-zero dipole moment and should be rejected if the net dipole moment of the cell is to vanish.
To account for all possible proton realizations one needs to investigate large systems
or average over several configurations. Indeed, we find that
there is a significant correlation between the proton realization and the average potential
energy $E$ and average pressure $P$ (at constant volume). Fig. 2
correlates $P$ and $E$ for each realization, while the inset
shows
$P$ in different realizations for system sizes from
$N=512$ to $N=21952$ molecules.
Only for 8000 or more particles is the variance between different realizations
within a few MPa and a tenth of a kJ/mol, the tolerance required to allow for a precise determination
of the thermodynamic variables entering into the free-energy calculation.
Unless otherwise stated, we have analyzed configurations with 8000 or more particles
for all proton-disordered crystals.
III Results
III.1 Gas-liquid coexistence
Fig. 3 shows the results of the SUS calculations. Panel (a) shows the
probability $p$ of finding $N$ particles at fixed $T$ and $v$ at
the coexistence chemical potential $\mu_{c}$ for different $T$. $\mu_{c}$ is
evaluated by reweighting the histogram $p(N)$ with respect to $N$, such that the area below the gas and the
liquid peak is identical (0.5). At low $T$, the probability minimum separating the two phases is more than 50 orders of magnitude lower than the peak heights,
highlighting the need for a numerical technique (like SUS) that allows the
observation of rare states. Close to the critical point [panel (b)], the probability of exploring
intermediate densities between the gas and the liquid becomes significant and $p(N)$ [or $p(\rho)$]
assumes the characteristic shape typical of all systems belonging to the same universality class.
Panel (c) compares $p(N+sE)$, where $E$ is the potential energy of the configuration and $s$ is the so-called
mixing field parameter Wilding (1995), with the theoretical expression for the magnetization in the Ising model.
To reinforce the identification of the critical point with the Ising universality class, the inset shows
the finite size scaling of the critical $T$ (defined as the $T$, for each size, at which the fluctuations in
$N+sE$ are best fitted with the Ising form) as a function of $L^{(1+\theta)/\nu}=L^{-2.448}$, with $\theta=0.54$ and $\nu=0.630$ Ferrenberg and Landau (1991); Pelissetto and Vicari (2002).
The extrapolation to $L\rightarrow\infty$ suggests that the gas-liquid critical point for the
reaction field ST2 model is $T_{c}=558.0\pm 0.3$K and $\rho_{c}=0.265\pm 0.005$ g/cm${}^{3}$.
Finally, panel (d) shows the gas-liquid coexistence in the $\rho-T$ plane. A clear nose appears around $T=300$ K,
signaling the onset of the network of hydrogen bonds (HB).
Indeed, strong directional interactions (such as the HB), impose a strong coupling between density and energy.
The formation of a fully bonded tetrahedral network (the expected thermodynamically stable state at low $T$)
requires a well-defined minimum local density, which for the present model is approximately $\rho=0.8$ g/cm${}^{3}$.
Hence, at low $T$, the density of the network coexisting with the gas must approach this value.
For completeness, the inset in panel (d) reports the value of $\beta\mu_{c}$ along the coexistence line.
III.2 Fluid-crystal coexistence
We have investigated the stability of crystal phases that may coexist with the fluid at low $T$. In particular, we have determined the free energies of ices I${}_{c}$, I${}_{h}$, VI, VII, and VIII, as well as the recently proposed metastable ice 0 structure Russo et al. (2014). Note that with the exception of ice VIII, all these phases have disordered hydrogen bonding. Examples of our thermodynamic integration results are reported in
Fig. 4, where we plot the reduced chemical potential $\beta\mu\equiv\beta f+\beta P/\rho_{n}$
of different phases at two selected $T$. For each pressure interval, the lowest chemical potential phase is the
thermodynamically stable one. Intersections of different curves locate coexistence
points, either stable or metastable. We then interpolate the fluid and crystal free energies based on the equation of state
to draw the coexistence lines in the phase diagram.
The complete phase diagram is reported in Fig. 5. At low $T$ and low $P$, the most stable crystal structure is the ice I lattice. From our simulations, the cubic (I${}_{c}$) and hexagonal (I${}_{h}$) ice structures have the same free energy within our numerical accuracy. At positive pressures, the liquid phase coexisting with ice I is always denser than ice, and as a result, the melting temperature of ice I decreases with increasing $P$. At negative $P$ (near $P=-80$ MPa), the ice I and liquid phases coexist at the same density, and the melting temperature reaches a maximum. We note that we have confirmed the ice I${}_{h/c}$ melting temperature calculated via thermodynamic integration
at two separate pressures using direct coexistence simulations, and find good agreement. We note that for the ST2-Ewald model, the melting temperature of I${}_{c}$ at the single pressure of 260 MPa was estimated to be around 274 K, consistent with the present estimate Palmer et al. (2014).
At high pressure, the main candidate structures are the proton-ordered ice VIII structure, and the proton-disordered ice VII structure. Both structures consist of two interpenetrating I${}_{c}$ lattices (somewhat distorted in the case of proton-ordering), where the oxygen positions form a BCC lattice. According to our free-energy calculations, the disordered ice VII is the more stable one in the region where coexistence with the fluid might occur. However, when trying to confirm the accuracy of our predicted liquid-ice VII coexistences using direct coexistence simulations, we observed crystal growth at temperatures significantly above the melting temperature predicted from free energy calculations. The newly grown parts of the crystal still display the BCC topology of the oxygen atoms, but the crystal shrinks by a few percent in the direction perpendicular to the growth direction, leading to a slight distortion of the lattice, that we refer to in the following as ice VII${}^{*}$.
As this distortion does not occur in fully disordered ice VII, we attribute the unexpectedly high stability of the ice VII${}^{*}$ lattice to the emergence of partial proton ordering, which decreases the crystal free energy.
To confirm this, we created a fully regrown ice VII${}^{*}$ configuration by alternately melting and regrowing the two halves of an ice VII configuration in an elongated simulation box. When measuring the proton-proton and dipole-dipole correlation functions for both the original ice VII structure and the regrown ice VII${}^{*}$, we see only minor changes in the proton-proton correlation function in the region 3 Å$<r<$ 4 Å [see
Fig. 6(a)]. In contrast, the dipole-dipole correlation function [see
Fig. 6(b)]
shows significant additional signal which although weak, extends up to long spatial scales. Using the Frenkel-Ladd method, we calculate the free energy of this configuration (assuming full proton disorder), and find that it is indeed lower than that of the original crystal by $\approx 0.4~{}k_{B}T$ per particle, confirming that the lower melting temperature observed in our direct coexistence simulations can be attributed to the (slight) change in crystal structure. The difference in free energy mainly results from the lower potential energy of the regrown crystal. We note here that partial proton ordering would reduce the contribution of the residual entropy to the free energy of the crystal, causing us to underestimate the ice VII${}^{*}$ free energy. On the other hand, the presence of defects in the system is expected to cause an overestimate in the crystal free energy. It is thus not a priori obvious that this free energy can be used to predict coexistences. Nonetheless, comparing the melting temperature predicted from the free energy and equations of state of the regrown crystal with the melting temperature taken from the direct coexistence simulations, we find good agreement ($T\approx 320K\pm 5K$ at $P=250$ MPa). Calculating the rest of the coexistence lines for this crystal using thermodynamic integration, we observe that ice VII${}^{*}$ has a significantly larger stability region than the original ice VII (see Fig. 5).
We note that neither ice VI nor ice 0 are ever the most thermodynamically stable phase in the investigated region.
As it may be relevant in future nucleation studies, we include the metastable coexistence line of the liquid with ice 0 in the phase diagram (Fig. 5).
IV Conclusions
Recently, the ST2 potential has been at the centre of renewed interest in connection to the debate on the origin of the
liquid-liquid critical point Debenedetti and Stanley (2003); Franzese
et al. (2001b); Sciortino et al. (2003); Fuentevilla and
Anisimov (2006); Holten et al. (2012, 2014).
This model exhibits known deficiencies in accurately modelling water properties, e.g. it
overemphasizes the tetrahedrality of the liquid structure, thus shifting all water anomalies to higher temperatures.
Despite these deficiencies, the ST2 model plays a key role as a prototype system in many studies related to the presence of a liquid-liquid critical point.
We report here fundamental properties of the ST2 model, by evaluating the location of the
gas-liquid critical point and the gas-liquid coexistence curve, as well as the coexistence lines between the liquid
and several crystal structures, allowing us to map out the phase diagram of the ST2 model in the low-temperature regime.
We find a stable ice I phase at low pressure and temperature, with both the hexagonal and cubic stackings approximately equal in free energy.
Differently from real water, the high-pressure phase behavior of the model is dominated by a new crystal whose growth is templated by
the ice VII interface. This ice VII${}^{*}$ tetragonal crystal is composed of a lattice in which the oxygens have the same topology as ice VII
but in which the protons are not completely randomly distributed. We have not been able to identify a small unit cell for this new crystal, but inspection of the HH
radial distribution function indicates minute but observable differences in the region around $3.3$ Å, accompanied
by weak but long ranged correlations in the dipole-dipole correlation function.
This structure, despite the small partial proton order, has a significant lower potential energy than VII (approximately 1.2 kJ/mol). As a result, ice VII${}^{*}$
is significantly more stable than the fully proton-disordered ice VII phase at all pressures and it
dominates the high-pressure phase behavior of the model. The liquid-liquid critical point for this model lies, according to the most recent estimates, inside the region of stability of the ice VII${}^{*}$ crystal phase and is metastable with respect to ice I${}_{h}$ or I${}_{c}$ as well as to ice VII.
Our results provide a starting point for the study of nucleation in the ST2 model, as well as for the exploration of modifications to the model Smallenburg and Sciortino (2015) that could make the liquid-liquid critical point more accessible Smallenburg et al. (2014).
V Acknowledgments
We thank L. Filion, A. Geiger, A. Rehtanz, and I. Saika-Voivod
for useful discussions. We are honored to dedicate this study to
Prof. Jean-Pierre Hansen, from whom we have learned liquid state theory.
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On maximum-likelihood estimation in the all-or-nothing regime
Luca Corinzia, and
Paolo Penna
Department of Computer Science
ETH Zürich, Switzerland
Email: {luca.corinzia,paolo.penna}@inf.ethz.ch
Wojciech Szpankowski
Department of Computer Science
Purdue University, USA
Email: [email protected]
Joachim M. Buhmann
Department of Computer Science
ETH Zürich, Switzerland
Email: [email protected]
Abstract
We study the problem of estimating a rank-1 additive deformation of a Gaussian tensor according to the maximum-likelihood estimator (MLE).
The analysis is carried out in the sparse setting, where the underlying signal has a support that scales sublinearly with the total number of dimensions.
We show that for Bernoulli distributed signals, the MLE undergoes an all-or-nothing (AoN) phase transition, already established for the minimum mean-square-error estimator (MMSE) in the same problem.
The result follows from two main technical points: (i) the connection established between the MLE and the MMSE, using the first and second-moment methods in the constrained signal space, (ii) a recovery regime for the MMSE stricter than the simple error vanishing characterization given in the standard AoN, that is here proved as a general result.
I Introduction
A fundamental question in information theory, statistics, and machine learning is to establish the computational limits of estimation problems and determine the statistical limit as the inviolable benchmark for the same problem. The common picture that arises in many problems is given by the presence of phase transitions where the behaviour of the optimal estimators changes abruptly with the variation of the parameter of the problem. Typically, at least two phase transitions are present: the statistical phase transition that establishes the limit of any estimator, and the computational phase transition at higher signal strength that establishes the limit of tractable estimators, despite the two can coincide.
In the sparse setting of many estimation problems, a rather different picture emerges, as the statistical and computational phase transitions are situated at different scales of the parameters space and such gap diverges in the limit of vanishing sparsity. Moreover, the statistical phase transition is characterized by the so-called all-or-nothing phenomenon (AoN): below a critical signal strength, the recovery of the planted signal is impossible, above the threshold is possible and with vanishing error. Although the AoN is conjectured to extend to the behaviour of any optimal estimator, the analysis has been so far focused on the minimum mean-square-error (MMSE) estimation, that is typically a bulk estimator and hence can be too coarse for specific applications in the sparse setting. Hence, it would be desirable to extend the analysis of this phenomenon to other estimators, like the maximum-likelihood estimator (MLE), which recently received attention for showing optimal performance in retrieving the planted signal [1] in a tensor-PCA model.
I-A Contribution
In this work, we provide new results on the AoN phenomenon in the sparse estimation setting, with the following main contributions:
•
In Theorem 1, we generalize the AoN phenomenon proved in [2] for the additive Gaussian noise model and the MMSE to arbitrarily asymptotics in the recovery regime. The proof follows a conditional second-moment method argument [3] and extends the proof given in [2] with a careful control of the asymptotics of the bounds. This result is of independent interest as more stringent conditions than the simple error vanishing characterization are needed in specific applications.
•
As an application of the first result, we study the maximum-likelihood estimator in the sparse tensor-PCA problem and show in Theorem 2 that also this estimator undergoes a weak AoN transition.
•
The proof of the latter results is of independent interest, as it exploits the relations between different estimators with first and second-moment methods, crucially introducing the analysis of estimators constrained in the signal hypothesis space. As a side result, the weak AoN phenomenon is proved for the constrained MMSE in Theorem 3.
I-B Related work
The problem of high dimensional statistical estimation that we study here has received much attention recently, with considerable progress obtained in the last years in understanding planted matrix and tensor models. Early works on statistical and computational limits of estimation focused on dense problems where the signal effective dimensionality scales linearly with the problem’s dimensionality. Examples include: (i) compressed sensing [4] and matrix-PCA [5, 6] where the approximated message passing (AMP) algorithms are introduced and demonstrated to match the statistical phase transition; (ii) the tensor-PCA extension [7] where the statistical and computational transitions are currently separated by a gap, considering a wide range of algorithms, i.e. spectral [8, 9], AMP [10], Sum-of-Square [11] and gradient descent [12, 13]. Many of these works focused on the mean-square-error and (high dimensional, i.e., matrix or tensorial) posterior average estimator. Nonetheless, recently more attention has been given to other estimators, e.g., in [1] where the vectorial maximum-likelihood estimator has been shown to reach optimal correlation with the planted signal in the tensor-PCA model. See [14] for a thorough review in the field.
In the sparse regime in which the hidden signal’s dimensionality is sublinear to the problem’s dimensionality, the AoN phenomenon emerges. This phenomenon have been shown recently to hold in a wide range of problems, i.e., for sparse linear regression [15], sparse matrix-PCA [16] and sparse tensor-PCA [2] according to the mean-square-error loss. However, to the best of our knowledge, only a few other works studied how the same AoN phenomenon extends to other estimators. Examples include [17, 18], where non-matching upper and lower bounds are provided for the transition of the vectorial-MLE in the sparse planted hypergraph problem (equivalent to sparse tensor-PCA up to a reparameterization of the dimensionality of the problem), and [19] where the AoN is proved in the sparse linear regression model for the vectorial-MLE estimator.
II Setting
We study the estimation problem with observations given by the Gaussian additive model
$$\mathbf{Y}=\sqrt{\lambda}\mathbf{X}+\mathbf{Z}$$
(1)
where the signal to be estimated $\mathbf{X}\in\mathbb{R}^{n}$ is corrupted by Gaussian noise $\mathbf{Z}$, with the collection $\{Z_{i}\}_{i=1}^{n}\overset{iid}{\sim}\mathbb{P}_{z}=\mathcal{N}(0,1)$.
We further assume that the prior distribution of $\mathbf{X}$, denoted by $\mathbb{P}_{n}$, is uniform and discrete with support $supp(\mathbb{P}_{n})\subset\mathcal{S}_{n-1}$, where $\mathcal{S}_{n-1}$ is the unit sphere in $\mathbb{R}^{n}$.
We denote by $M_{n}=|supp(\mathbb{P}_{n})|$ the cardinality of the support of $\mathbf{X}$ and by ${\mathbb{Q}}_{y|x}(\mathbf{Y}|\mathbf{X})=\mathcal{N}(\mathbf{Y}|\sqrt{\lambda}\mathbf{X},\bm{1}_{n\times n})$ the conditional distribution of $\mathbf{Y}$ given $\mathbf{X}$, where $\bm{1}_{n\times n}$ is the identity matrix.
We hence define
$${\mathbb{Q}}_{\lambda,n}(\mathbf{Y})={\mathbb{E}}_{\mathbb{P}_{n}}[{\mathbb{Q}}_{y|x}(\mathbf{Y}|\mathbf{X})]$$
as the distribution over the observations $\mathbf{Y}$, highlighting the respective signal-to-noise-ratio (snr) $\lambda$ and the problem dimension $n$ for convenience.
Definition 1.
Let us denote as $\bar{\mathcal{S}}_{n-1}=\{\mathbf{X}\in\mathbb{R}^{n}\colon\|\mathbf{X}\|\leq 1\}$ the unit ball. For any set $A$ we use the short notation $\min_{\hat{\mathbf{X}}(\mathbf{Y})\in A}$ for $\min_{\hat{\mathbf{X}}:\mathbf{Y}\to\hat{\mathbf{X}}(\mathbf{Y})\in A}$ and analogously for other operators. For a generic bounded loss function $\operatorname{L}$ we define the respective optimal estimator as
$$\mathbf{X}_{\operatorname{L}}(\mathbf{Y})=\operatorname*{argmin}\limits_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}{\mathbb{E}}\left[\operatorname{L}(\mathbf{X},\hat{\mathbf{X}}(\mathbf{Y}))\right].$$
The minimum loss achieved by such estimator is the quantity
$$\operatorname{L}_{n}(\lambda)={\mathbb{E}}\left[\operatorname{L}(\mathbf{X},\mathbf{X}_{\operatorname{L}}(\mathbf{Y}))\right].$$
We further define the best estimator that solves the optimization problem constrained to set $A$ with $supp(\mathbb{P}_{n})\subset A\subset\bar{\mathcal{S}}_{n-1}$ as
$$\mathbf{X}_{\operatorname{C-L}}(\mathbf{Y})=\operatorname*{argmin}\limits_{\hat{\mathbf{X}}(\mathbf{Y})\in A}{\mathbb{E}}\left[\operatorname{L}(\mathbf{X},\hat{\mathbf{X}}(\mathbf{Y}))\right],$$
and the respective minimum loss achieved as
$$\operatorname{C-L}_{n}(\lambda)={\mathbb{E}}\left[\operatorname{L}(\mathbf{X},\mathbf{X}_{\operatorname{C-L}}(\mathbf{Y}))\right].$$
We can easily observe that since $A\subset\bar{\mathcal{S}}_{n-1}$ then $\operatorname{C-L}_{n}(\lambda)\geq\operatorname{L}_{n}(\lambda)$ for any loss function $\operatorname{L}$.
Definition 2.
Denote by $\operatorname{L}$ a bounded loss function $\operatorname{L}\colon\bar{\mathcal{S}}_{n-1}^{\otimes 2}\to\mathbb{R}_{+}$ with $\operatorname{L}(\mathbf{X},\mathbf{X})=0$.
Denote by $c>0$ the optimal error obtained by the estimator independent on the observations $\mathbf{Y}$ as $c=\lim_{n\to\infty}\operatorname{L}_{n}(0)$.
The estimation problem with observations given by Equation 1 with prior $\mathbb{P}_{n}$ satisfies the all-or-nothing phenomenon (AoN) with respect to the loss $\operatorname{L}$, with recovery asymptotics at least $\tau_{n}\in o(1)$ and critical snr $\lambda_{n}$ if
$$\lim_{n\to\infty}\operatorname{L}_{n}(\beta\lambda_{n})=\begin{cases}\begin{aligned} &\ c\ \ &\beta<1\\
&\ 0\ \ &\beta>1\end{aligned}\end{cases}$$
(2)
and moreover
$$\operatorname{L}_{n}(\beta\lambda_{n})\in o(\tau_{n})$$
for $\beta>1$, where $\beta$ is a constant independent on $n$.
Intuitively, in the AoN phenomenon the estimation is impossible for a normalized snr smaller then a critical value $\beta_{c}=1$, as the loss converges to the error achieved by an uninformative estimator, equivalent to the loss given by $\beta=0$, while for higher snr the estimation is almost perfect, with error smaller then a given $\tau_{n}\to 0$. Note the difference of this definition from the one given in [2] in the recovery regime $\beta>1$. In latter case the simpler condition $\operatorname{L}_{n}(\beta\lambda_{n})\to 0$ is given such that the asymptotics of the loss in the recovery regime in no further characterize. In the following we denote by $\tau_{n}$ a vanishing sequence such that $\tau_{n}\in o(1)$.
III Generalized all-or-nothing phenomenon
In this section, we consider the mean-square-error (MSE), ${\mathbb{E}}[\|\mathbf{X}-\hat{\mathbf{X}}\|^{2}]$ where the expectation is taken with respect to $\mathbb{P}_{n}$ and $\mathbb{P}_{z}$, and $\hat{\mathbf{X}}\coloneqq\hat{\mathbf{X}}(\mathbf{Y})$ is an estimator of the signal given the observation $\mathbf{Y}$.
The MSE is minimized by the posterior average $\mathbf{X}_{l_{2}}={\mathbb{E}}[\mathbf{X}|\mathbf{Y}]$, with the average taken respect to the posterior $\mathbb{P}_{n}(\mathbf{X}|\mathbf{Y})$ (see [20] and Lemma 12).
The minimum mean-square-error (MMSE) is then $\operatorname{MMSE}_{n}(\lambda)\coloneqq{\mathbb{E}}[\|\mathbf{X}-{\mathbb{E}}[\mathbf{X}|\mathbf{Y}]\|^{2}]$. This error is the minimum square-error achievable by any estimator that has access to the observations $\mathbf{Y}$. In the following, we denote by $D(p||q)$ the KL divergence between the distribution $p$ and $q$, and by $\pm o(\tau_{n})$ a sequence $f_{n}\in o(\tau_{n})$ that is respectively non-negative and non-positive. We further use the scaling $\lambda_{n}=2\log M_{n}$. A sufficient condition for having the AoN phenomenon is given by the property of the overlap rate function defined here.
Definition 3.
For any $t\in[-1,1]$ define the overlap rate function between two independent instances of the signal $\mathbf{X}$ and $\mathbf{X}^{\prime}$ as
$$r_{n}(t)=-\frac{1}{\log M_{n}}\log\mathbb{P}_{n}^{\otimes 2}[\langle\mathbf{X},\mathbf{X}^{\prime}\rangle\geq t].$$
where $\langle\mathbf{X},\mathbf{X}^{\prime}\rangle$ is the scalar product of two vectors.
Intuitively, the rate function describes the rate of the exponential decay of the overlap $\langle\mathbf{X},\mathbf{X}^{\prime}\rangle$.
The following theorem shows that a simple lower bound on the overlap rate function is sufficient to establish the AoN with recovery asymptotics $\tau_{n}$ if the latter is not too small.
Theorem 1.
For any $\epsilon>0$ constant, if $\lambda_{n}^{-1/2+\epsilon}\in o(\tau_{n})$ and the overlap rate function $r_{n}(t)$ satisfies
$$r_{n}(t)\geq\frac{2t}{1+t}-o(\tau_{n}),$$
then the probability $\mathbb{P}_{n}$ of the problem defined in Equation 1 satisfies the AoN in Definition 2 with recovery asymptotics at least $\tau_{n}$ according the mean-square-error.
Proof.
The first part of Definition 2 related to Equation 2 follows easily noting that the assumption given here is stricter than the one given in [2]. We hence have to prove only that the stronger asymptotics holds in the recovery regime. The proof follows the steps of the proof in [2] and mainly uses the widely known I-MMSE relation that relates the MMSE to the mutual information $I(\mathbf{X},\mathbf{Y})$ and hence to the $D({\mathbb{Q}}_{\lambda,n}||{\mathbb{Q}}_{0,n})$. It then uses the conditional second-moment method to bound such divergence. We first have the following bound that connects the KL divergence and the MMSE, that is proved in the appendix using the I-MMSE relation.
Lemma 1.
If $\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\lambda_{n},n}||{\mathbb{Q}}_{0,n})\in o(\tau_{n})$ then for any $\beta>1$ constant,
$$\operatorname{MMSE}_{n}(\beta\lambda_{n})\in o(\tau_{n}).$$
We can now bound the KL divergence $\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\lambda_{n},n}||{\mathbb{Q}}_{0,n})$ conditioning on a high probability event defined as follows.
Definition 4.
Let ${\mathbb{Q}}_{xy}=\mathbb{P}_{n}\otimes{\mathbb{Q}}_{y|x}$ the joint probability distribution of the vectors $(\mathbf{X},\mathbf{Y})$ of problem defined in Equation 1 with snr $\lambda_{n}$. A series of events $\Omega_{n}\subset supp(\mathbb{P}_{n})\otimes supp({\mathbb{Q}}_{\lambda_{n},n})$ occurs with high probability $1-o(\tau_{n})$ uniformly over $\mathbf{X}$ if
$${\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]=1-o(\tau_{n})$$
(3)
for any $\mathbf{X}\in supp(\mathbb{P}_{n})$.
Let us define as $\tilde{{\mathbb{Q}}}_{\lambda,n}$ the probability distribution of $\mathbf{Y}$ condition on a high probability event $\Omega_{n}$. Then the following bound holds.
Lemma 2.
If $\Omega_{n}$ is an event that occurs with uniform high probability $1-o(\tau_{n})$ then
$$\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\leq\frac{1}{\lambda_{n}}D(\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})+o(\tau_{n})$$
To complete the proof, we need a claim that relates the KL given by the conditional distribution $\tilde{{\mathbb{Q}}}_{\lambda,n}$ of $\mathbf{Y}$ given a high probability event, to the overlap rate function of the problem. We first introduce the high probability events as follows.
Lemma 3.
Given a sequence $C_{n}$ with $\frac{1}{C_{n}}e^{-C_{n}^{2}/2}\in o(\tau_{n})$, the event
$$\Omega_{n}=\{(\mathbf{X},\mathbf{Y})\colon|\langle\mathbf{X},\mathbf{Y}\rangle-\sqrt{\lambda_{n}}|\leq C_{n}\}$$
satisfies Definition 4.
We can hence prove the following.
Lemma 4.
For any $\epsilon>0$ constant, if $\lambda_{n}^{-1/2+\epsilon}\in o(\tau_{n})$, conditioning on the event $\Omega_{n}=\{(\mathbf{X},\mathbf{Y})\colon|\langle\mathbf{X},\mathbf{Y}\rangle-\sqrt{\lambda_{n}}|\leq\sqrt{\log\lambda_{n}}\}$, the following bound holds:
$$\frac{1}{\lambda_{n}}D(\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\leq\sup_{t\in[0,1]}\left[\frac{t}{t+1}-\frac{r_{n}(t)}{2}\right]+o(\tau_{n}).$$
We can then finally bound the KL divergence conditioning on the events $\Omega_{n}$ defined in Lemma 4 as
$$\displaystyle\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\lambda_{n},n}||{\mathbb{Q}}_{0,n})$$
$$\displaystyle\leq\frac{1}{\lambda_{n}}D(\tilde{{\mathbb{Q}}}_{\lambda_{n},n}||{\mathbb{Q}}_{0,n})+o(\tau_{n})$$
$$\displaystyle\leq\sup_{t\in[0,1]}\left[\frac{t}{t+1}-\frac{r_{n}(t)}{2}\right]+o(\tau_{n})$$
$$\displaystyle\in o(\tau_{n})$$
where the first inequality comes from Lemma 2, Lemma 3 and the assumption that $\lambda_{n}^{-1/2+\epsilon}\in o(\tau_{n})$, the second inequality comes from Lemma 4 and the final inclusion is due to the assumption of the theorem on the rate function $r_{n}(t)$. The missing proofs of the lemmas are postponed to the appendix. ∎
IV Maximum-likelihood estimation
We here study the MLE, showing that a weaker AoN phenomenon extends to the behaviour of this estimator in the case of the sparse tensor-PCA model.
Definition 5.
The MLE for the generic model in Equation 1 is the estimator that maximizes the likelihood as
$$\mathbf{X}_{\operatorname{MLE}}=\operatorname*{argmax}\limits_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}{\mathbb{Q}}_{y|x}(\mathbf{Y}|\mathbf{X}(\mathbf{Y}))$$
The following characterization of the MLE follows easily from the definition.
Lemma 5.
The MLE minimizes the probability of error
$$\operatorname{EP}_{n}(\hat{\mathbf{X}})\coloneqq\mathbb{P}_{n}{\mathbb{Q}}_{y|x}[\hat{\mathbf{X}}(\mathbf{Y})\neq\mathbf{X}]={\mathbb{E}}\left[\mathbbm{1}_{\{\|\hat{\mathbf{X}}(\mathbf{Y})-\mathbf{X}\|^{2}>0\}}\right]$$
According to the latter lemma and Definition 1, we hence characterize the MLE as the optimal estimator according to the 0-1 loss, hence we can denote
$\mathbf{X}_{\operatorname{0-1}}=\mathbf{X}_{\operatorname{MLE}}$ and by
$$\displaystyle\operatorname{MEP}_{n}(\lambda)$$
$$\displaystyle=\mathbb{P}_{n}{\mathbb{Q}}_{y|x}[\mathbf{X}_{\operatorname{MLE}}(\mathbf{Y})\neq\mathbf{X}]$$
$$\displaystyle={\mathbb{E}}\left[\mathbbm{1}_{\{\|\mathbf{X}_{MLE}-\mathbf{X}\|^{2}>0\}}\right]$$
the minimum error probability obtained by such estimator. Based on the same definition, the constrained version is further defined.
IV-A Application to the sparse tensor-PCA problem
In the following we assume for $d\geq 2$ the following sparse tensor-PCA model with observations
$$\mathbf{Y}=\sqrt{\lambda}\mathbf{x}^{\otimes d}+\mathbf{Z},$$
(4)
that corresponds to the additive Gaussian model defined in Equation 1 using $\mathbf{X}=\mathbf{x}^{\otimes d}$, with $\mathbf{x}\in\mathbb{R}^{p}$ and $n=p^{d}$, and considering the Frobenius norm for tensors in $\mathbb{R}^{n}$ 111Note that this problem can also encompass the planted problem in hypergraph, with observations in the upper-triangular part of the tensor as
$\mathbf{Y}=(\sqrt{\lambda}\mathbf{x}^{\otimes d}+\mathbf{Z})\mathbbm{1}_{\{i_{1}<\dots<i_{d}\}}$
and with $n=\binom{p}{d}$. Results easily extend to the hypergraph variation seamlessly.. A discrete uniform prior $\tilde{\mathbb{P}}_{p}$ over $\mathcal{S}_{p-1}$ induces a discrete uniform prior $\mathbb{P}_{n}$ over $\mathcal{S}_{n-1}$, hence the assumption of the model defined in Equation 1 are satisfied. Here and in the following we assume $\tilde{\mathbb{P}}_{p}$ to be a Bernoulli prior over the subset of the unit sphere with $k$ binary entries, hence
$$\mathbf{x}\in\left\{0,\frac{1}{\sqrt{k}}\right\}^{p}\cap\mathcal{S}_{p-1}=\mathcal{C}_{p,k}=supp(\tilde{\mathbb{P}}_{p})$$
The cardinality of the hypothesis space is hence $M_{p}=\binom{p}{k}$, and $\lambda_{n}=\log M_{p}=k\log\left(\frac{p}{k}\right)(1+o(1))$. The prior $\tilde{\mathbb{P}}_{p}$ maps to the uniform prior $\mathbb{P}_{n}$ over the space $supp(\mathbb{P}_{n})\subsetneq\mathcal{C}_{n,s}=\{0,s^{-1/2}\}^{n}\cap\mathcal{S}_{n-1}$, where $s=k^{d}$. Note here the difference between the $supp(\mathbb{P}_{n})$, that is the set of tensors formed as $\mathbf{x}^{\otimes d}$ with $\mathbf{x}\in\mathcal{C}_{p,k}$, and the set $\mathcal{C}_{n,s}$, that is the set of tensors with any $s$ entries equal to $s^{-1/2}$. We here study the constrained estimators $\operatorname{C-MMSE}$ and $\operatorname{C-MEP}$ on the set $\mathcal{C}_{n,s}$, as it allows an easy characterization in terms of the unconstrained one. The main theorem of this section gives a sufficient condition for an AoN phenomenon to hold for the MEP.
Theorem 2.
For the sparse Bernoulli tensor-PCA model defined in Equation 4, with $k\in o\left(\log^{\frac{1}{4d-1}}p\right)$, the $\operatorname{MEP}$ satisfies the weak AoN transition as:
$$\displaystyle\liminf\limits_{n\to\infty}\operatorname{MEP}_{n}(\beta\lambda_{n})\geq\frac{1}{4}\ \qquad\beta<1$$
$$\displaystyle\lim_{n\to\infty}\operatorname{MEP}_{n}(\beta\lambda_{n})\ \ =0\ \ \qquad\beta>1$$
The same transition holds for the $\operatorname{C-MEP}_{n}(\beta\lambda_{n})$.
We conjecture that the MLE undergoes a strict AoN transition, but further work is necessary to establish the full characterization of the MLE in the impossibility regime.
Proof.
The main idea of the proof is to relate the MMSE to the MEP studying the constrained counterpart of both. For these latter two quantities, a simple first-moment method can be applied, as there exists a minimum non-vanishing distance between any two points in the constrained set $\mathcal{C}_{n,s}$. Hence, for any estimator $\hat{\mathbf{X}}(\mathbf{Y})$, using the Markov inequality we can derive the following bound on its error probability:
$$\displaystyle\operatorname{EP}_{n}(\hat{\mathbf{X}})={\mathbb{E}}\left[\mathbbm{1}_{\{\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}>0\}}\right]$$
$$\displaystyle=\mathbb{P}\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}\geq\frac{2}{s}\right]$$
$$\displaystyle\leq\frac{s}{2}{\mathbb{E}}\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}\right]$$
Note that this is only possible if the estimator is constrained in $\mathcal{C}_{n,s}$. The following bounds can hence be derived with the full proof given in the appendix.
Lemma 6.
Given the problem in Equation 4, the following bounds hold:
$$\frac{s}{2}\operatorname{C-MMSE}_{n}(\lambda)\geq\operatorname{C-MEP}_{n}(\lambda)$$
(5)
$$\displaystyle\operatorname{C-MEP}_{n}(\lambda)$$
$$\displaystyle\geq\frac{1}{4}\operatorname{C-MMSE}_{n}(\lambda)^{2}$$
(6)
$$\displaystyle\operatorname{MEP}_{n}(\lambda)$$
$$\displaystyle\geq\frac{1}{4}\operatorname{MMSE}_{n}(\lambda)^{2}$$
(7)
The inequality given in Equation 5 relates now the $\operatorname{C-MMSE}$ to $\operatorname{C-MEP}$, such that if the $\operatorname{C-MMSE}$ is small enough, then the $\operatorname{MEP}$ is small too. We can further have a bound that relates the $\operatorname{MMSE}$ to the $\operatorname{C-MMSE}$ in the same direction, so as to derive a chain of inequalities between the $\operatorname{MMSE}$ and the $\operatorname{MEP}$. This is given by the following lemma.
Lemma 7.
For any $\epsilon>0$, $\operatorname{MMSE}_{n}(\lambda)<\epsilon$ if and only if $\operatorname{C-MMSE}_{n}(\lambda)<4\epsilon s$.
Proof.
Let us define the (constrained-) MSE distance as, respectively,
$$\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})=\|\mathbf{X}_{\operatorname{C-\textit{l}}_{2}}(\mathbf{Y})-\mathbf{X}\|^{2}$$
and
$$\operatorname{MSE}_{p}(\mathbf{X},\mathbf{Y})=\|\mathbf{X}_{l_{2}}(\mathbf{Y})-\mathbf{X}\|^{2}.$$
We can write the expectation conditioning on the event $A$ that the first distance is smaller then a given $\delta<\frac{1}{2s}$,
$$A=\{(\mathbf{X},\mathbf{Y})\colon\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})\leq\delta\}$$
as:
$$\displaystyle\operatorname{C-MMSE}_{n}(\lambda)$$
$$\displaystyle={\mathbb{E}}\left[\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})\right]$$
$$\displaystyle={\mathbb{E}}\left[\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})|A\right]\mathbb{P}[A]+$$
$$\displaystyle\hskip 28.45274pt+{\mathbb{E}}\left[\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})|A^{c}\right]\mathbb{P}[A^{c}].$$
(8)
We now characterize the optimal constrained $l_{2}$ estimator as the simple rounding of the top entries of the standard posterior average estimator.
Lemma 8.
The optimal estimator constrained in the hypothesis space $\mathcal{C}_{n,s}$ for the problem in Equation 4 for the MSE reads
$$\mathbf{X}_{\operatorname{C-\textit{l}}_{2}}=\operatorname*{argmin}\limits_{\hat{\mathbf{X}}(\mathbf{Y})\in\mathcal{C}_{n,s}}{\mathbb{E}}[\|\hat{\mathbf{X}}(\mathbf{Y})-\mathbf{X}\|^{2}]=\operatorname{Top}_{s}\left({\mathbb{E}}[\mathbf{X}|\mathbf{Y}]\right),$$
where the $\operatorname{Top}_{s}(\cdot)$ operator rounds the top $s$ entries of $\mathbf{X}$ to $s^{-1/2}$, and zeros out all other entries.
Now we can easily note that the following geometrical lemma:
Lemma 9.
For any integer $n$ and $s$ and $\mathbf{U}\in\mathcal{C}_{n,s}$, $\mathbf{V}\in\left[0,s^{-1/2}\right]^{n}$ and $\delta<\frac{1}{2s}$, such that $\|\mathbf{U}-\mathbf{V}\|^{2}\leq\delta$,
$$\operatorname{Top}_{s}(\mathbf{V})=\mathbf{U}.$$
Combining Lemma 8 and Lemma 9 it follows that
$${\mathbb{E}}\left[\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})|A\right]=0$$
(9)
as $\operatorname{Top}_{s}(\mathbf{X}_{\operatorname{C-\textit{l}}_{2}})=\mathbf{X}_{\operatorname{C-\textit{l}}_{2}}$.
Using the same decomposition for the MMSE with respect to the event
$$B=\{(\mathbf{X},\mathbf{Y})\colon\operatorname{MSE}_{n}(\mathbf{X},\mathbf{Y})\leq\delta\}$$
we get
$$\displaystyle\operatorname{MMSE}_{n}(\beta)$$
$$\displaystyle={\mathbb{E}}[\operatorname{MSE}_{n}(\mathbf{X},\mathbf{Y})|B]\mathbb{P}[B]+$$
$$\displaystyle\hskip 28.45274pt+{\mathbb{E}}[\operatorname{MSE}_{n}(\mathbf{X},\mathbf{Y})|B^{c}]\mathbb{P}[B^{c}]<\epsilon$$
(10)
Bounding as ${\mathbb{E}}[\operatorname{MSE}_{n}(\mathbf{X},\mathbf{Y})|B]\geq 0$, $\mathbb{P}[B]\geq 0$ and ${\mathbb{E}}[\operatorname{MSE}_{n}(\mathbf{X},\mathbf{Y})|B^{c}]\geq\delta$ we get from Section IV-A
$$\mathbb{P}[B^{c}]<\frac{\epsilon}{\delta}.$$
(11)
From Lemma 8, we have further that for $\delta<\frac{1}{2s}$,
$$\displaystyle B$$
$$\displaystyle\subset\{\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})=0\}\subset A$$
hence that
$$\mathbb{P}[A^{c}]\leq\mathbb{P}[B^{c}].$$
(12)
Plugging Equations 9, 11 and 12 into the decomposition in Section IV-A and using the fact that $\operatorname{C-MSE}_{n}(\mathbf{X},\mathbf{Y})\leq 2$ we finally get
$$\operatorname{C-MMSE}_{n}(\lambda)\leq\frac{2\epsilon}{\delta}.$$
The theorem follows from the arbitrariness of $\delta<\frac{1}{2s}$.
∎
Plugging in the result of Lemma 7 and Equation 5 we get the further lemma that relates the MMSE in the recovery regime to the C-MMSE and the MEP.
Lemma 10.
$\operatorname{MMSE}_{n}(\lambda_{n})\in o(1/s)$ if and only if $\operatorname{C-MMSE}_{n}(\lambda_{n})\in o(1)$. If $\operatorname{MMSE}_{n}(\lambda_{n})\in o(1/s^{2})$ then $\operatorname{MEP}_{n}(\lambda_{n})\in o(1)$
In the same regime $\beta>1$, we can hence use the results on the generalized AoN, Theorem 1, to have the MMSE to be $o(1/s^{2})$. For such theorem to hold, we use the following lemma on the overlap rate function of the sparse tensor-PCA problem.
Lemma 11 (Proof given in Proposition 3, [2]).
For the Bernoulli sparse tensor-PCA problem with signal $\mathbf{X}=\mathbf{x}^{\otimes d}$, $d\geq 2$, and $\mathbf{x}\in\{0,1/\sqrt{k}\}^{p}\cap\mathcal{S}_{p-1}$ the following bound on the overlap rate function of the tensors $\mathbf{X},\mathbf{X}^{\prime}$ holds for any $t\in[0,1]$:
$$r_{n}(t)\geq\sqrt{t}-\frac{\mathcal{O}(1)}{\lambda_{n}}$$
Combining Lemma 11, Theorem 1, Lemma 7 and Lemma 6 we finally get the claim of the main theorem as
$$\sqrt{t}\geq\frac{2t}{1+t}$$
for $t\in[0,1]$ and as $k\in o\left(\log^{\frac{1}{4d-1}}p\right)$ implies
$$\lambda_{n}^{-1/2+\epsilon}=(k\log(p/k)+o(1))^{-1/2+\epsilon}\in o(1/s^{2})=o(1/k^{2d})$$
∎
Analogously, we can prove the following transition for the constrained MMSE.
Theorem 3.
For the sparse Bernoulli tensor PCA model with $k\in o\left(\log^{\frac{1}{2d-1}}p\right)$, the $\operatorname{C-MMSE}$ satisfies the AoN transition
$$\displaystyle\liminf\limits_{n\to\infty}\operatorname{C-MMSE}_{n}(\beta\lambda_{n})\geq 1\ \qquad\beta<1$$
$$\displaystyle\lim_{n\to\infty}\operatorname{C-MMSE}_{n}(\beta\lambda_{n})\ \ =0\ \ \qquad\beta>1$$
Proof.
The proof follows the same steps of the proof of Theorem 2, with only two main differences. First, in the impossibility regime, we can use the stronger bound $\operatorname{C-MMSE}_{n}(\lambda)\geq\operatorname{MMSE}_{n}(\lambda)$ in place of Equation 6 to get the first part of the theorem. In the recovery regime, we can use the weaker requirement $\operatorname{MMSE}_{n}(\beta\lambda_{n})\in o(1/s)$ that is satisfied by the assumption of the theorem $k\in o\left(\log^{\frac{1}{2d-1}}p\right)$.
∎
V Conclusion
In this paper, we analysed the maximum-likelihood estimator for the sparse tensor-PCA problem with Bernoulli prior. We established that this estimator undergoes a weak AoN transition and conjectured that this transition is equivalent to the MMSE transition.
The proof follows from the connection of the MLE to the MMSE using the first and second-moment method in the constrained signal space, and hence it is of independent interest as it can lead to further results from the community.
While this paper sets a first step in understanding a wider range of optimal estimators in sparse high-dimensional inference problems, a general theory of the all-or-nothing statistical transition is still lacking. This theory could provide a wider understating of the phenomenon, including the analysis of vectorial estimators for planted matrix and tensor PCA problems, that is not here considered and is carried out in the dense setting using rigorous tools of statistical physics and replica methods (see [10]).
The same methods established recently that tractable estimators, like the approximate-message-passing algorithms, undergo the same all-or-nothing transition in the sparse matrix PCA problem [16]. The extension of these results to the sparse tensor-PCA problem and currently optimal algorithms for this problem, like averaged gradient descent [13] and sum-of-squares algorithms [11], is of crucial importance.
Acknowledgement
This work was supported in part by
NSF Center on Science of Information
Grants CCF-0939370 and NSF Grants CCF-1524312, CCF-2006440, CCF-2007238.
Appendix A Postponed proofs of Section III
Proof of Lemma 1.
Using, respectively, the maximum entropy bound and 1/2 Lipschitz-continuity of the function (see Lemma 14 and [2]) we can see first that
$$\displaystyle 0\leq\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})-\frac{1}{2}(\beta-1)\leq$$
$$\displaystyle\leq\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\lambda_{n},n}||{\mathbb{Q}}_{0,n})\in o(\tau_{n})$$
where the first inequality comes from the maximum entropy bound, the second from Lipschitz-continuity, and the inclusion follows from the assumption of the theorem. For convenience let us denote
$$\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})=\frac{1}{2}(\beta-1)+f_{n}(\beta)$$
where $\lim_{n\to\infty}\frac{f_{n}(\beta)}{\tau_{n}}=0$ for any $\beta>1$.
We can now use the I-MMSE relation (see [21, 2]), such that $\operatorname{MMSE}_{n}(\beta\lambda_{n})=1-2\frac{d}{d\beta}\frac{1}{\lambda_{n}}D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})$.
$$\displaystyle\operatorname{MMSE}(\beta\lambda_{n})$$
$$\displaystyle=1-2\frac{d}{d\beta}\left(\frac{1}{2}(\beta-1)+f_{n}(\beta)\right)$$
$$\displaystyle=-2\frac{d}{d\beta}f_{n}(\beta)$$
hence
$$\displaystyle\lim_{n\to\infty}\frac{\operatorname{MMSE}(\beta\lambda_{n})}{\tau_{n}}$$
$$\displaystyle=\lim_{n\to\infty}-2\frac{1}{\tau_{n}}\frac{d}{d\beta}f_{n}(\beta)$$
$$\displaystyle=\lim_{n\to\infty}-2\frac{d}{d\beta}\frac{1}{\tau_{n}}f_{n}(\beta)=0$$
where the second equality follows from the linearity of differentiation and the third from the interchanging of limit and differentiation under uniform convergence.
∎
Proof of Lemma 2.
Following the proof of Theorem 5 in [22], and defining the function $Z(\mathbf{Y})=\frac{{\mathbb{Q}}_{\beta\lambda_{n},n}(\mathbf{Y})}{{\mathbb{Q}}_{0,n}(\mathbf{Y})}$ we have that
$$\displaystyle D(\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})-D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\geq$$
$$\displaystyle\geq{\mathbb{E}}_{\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}}\log Z(\mathbf{Y})-{\mathbb{E}}_{{\mathbb{Q}}_{\beta\lambda_{n},n}}\log Z(\mathbf{Y})$$
(13)
Using the definition of the conditional pdf, we have
$$\displaystyle\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}(\mathbf{Y})$$
$$\displaystyle={\mathbb{Q}}_{\beta\lambda_{n},n}(\mathbf{Y}|\Omega_{n})={\mathbb{E}}_{\mathbb{P}_{n}}{\mathbb{Q}}_{y|x}(\mathbf{Y}|\Omega_{n},\mathbf{X})$$
$$\displaystyle={\mathbb{E}}_{\mathbb{P}_{n}}\frac{{\mathbb{Q}}_{y|x}(\mathbf{Y}|\mathbf{X}){\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X},\mathbf{Y}]}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}$$
$$\displaystyle={\mathbb{E}}_{\mathbb{P}_{n}}\frac{{\mathbb{Q}}_{y|x}(\mathbf{Y}|\mathbf{X})\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]},$$
(14)
where $\mathbbm{1}_{A}(\cdot)$ is the indicator function of set $A$. Plugging Appendix A into Appendix A we get
$$\displaystyle D(\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})-D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\geq$$
$$\displaystyle\geq{\mathbb{E}}_{\mathbb{P}_{n}}\frac{{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[(\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}])\log Z(\mathbf{Y})\right]}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}$$
(15)
Using the Cauchy-Schwartz inequality we can bound the expectation over $\mathbf{Y}$ as
$$\displaystyle\left|{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[(\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}])\log Z(\mathbf{Y})\right]\right|\leq$$
$$\displaystyle\sqrt{{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}(\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}])^{2}\cdot{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[\log^{2}Z(\mathbf{Y})\right]}$$
(16)
It is easy to see that
$$\displaystyle{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}$$
$$\displaystyle\left[(\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}])^{2}\right]=$$
$$\displaystyle={\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\big{[}(\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})+{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]^{2}+$$
$$\displaystyle\hskip 28.45274pt-2\cdot\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y}){\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]^{2}\big{]}$$
$$\displaystyle={\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})\right]+{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]^{2}+$$
$$\displaystyle\hskip 28.45274pt-2\cdot{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})\right]{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]=$$
$$\displaystyle=Q_{xy}[\Omega_{n}|\mathbf{X}]-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]^{2}$$
hence recombining the latter and Appendix A and Appendix A we get
$$\displaystyle D($$
$$\displaystyle\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})-D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\geq$$
$$\displaystyle\geq-{\mathbb{E}}_{\mathbb{P}_{n}}\sqrt{\frac{1-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[\log^{2}Z(\mathbf{Y})\right]}.$$
Using again the Cauchy-Schwartz inequality over the expectations on $\mathbb{P}_{n}$ we finally get
$$\displaystyle D($$
$$\displaystyle\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})-D({\mathbb{Q}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\geq$$
$$\displaystyle-\sqrt{{\mathbb{E}}_{\mathbb{P}_{n}}\left(\frac{1-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}\right)^{2}}\cdot\sqrt{{\mathbb{E}}_{\mathbb{P}_{n}}{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\left[\log^{2}Z(\mathbf{Y})\right]}$$
$$\displaystyle=-o(\tau_{n})\cdot\sqrt{{\mathbb{E}}_{{\mathbb{Q}}_{\beta\lambda_{n},n}}\left[\log^{2}Z(\mathbf{Y})\right]}$$
where the equality comes from the assumption on the event $\Omega_{n}$. Using now the result from Proposition 3 in [2],
$$\sqrt{{\mathbb{E}}_{{\mathbb{Q}}_{\beta\lambda_{n},n}}\left[\log^{2}Z(\mathbf{Y})\right]}=\mathcal{O}(\log M_{n})$$
we get the claim.
∎
Proof of Lemma 3.
We can see easily that
$$\displaystyle{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]$$
$$\displaystyle={\mathbb{Q}}_{y|z}[\Omega_{n}|\mathbf{X}]$$
$$\displaystyle=\mathbb{P}_{z}[|\langle\mathbf{X},\mathbf{Z}\rangle|\leq C_{n}]$$
$$\displaystyle=2\phi(C_{n})-1$$
$$\displaystyle=1-\sqrt{\frac{2}{\pi}}\frac{1}{C_{n}}e^{-C_{n}^{2}/2}\left(1+\mathcal{O}\left(\frac{1}{C_{n}}\right)\right)$$
where in the second equality we used the fact that $\mathbf{Y}=\sqrt{\lambda_{n}}\mathbf{X}+\mathbf{Z}$ and that $\|\mathbf{X}\|^{2}=1$, the third follows from the fact that $\langle\mathbf{X},\mathbf{Z}\rangle$ is a univariate Gaussian random variable distributed as $\mathcal{N}(0,1)$, and $\phi(\cdot)$ is the cdf of the standard Gaussian, with asymptotics given, for large $x$, as $\phi(x)=1-\frac{1}{\sqrt{2\pi}x}e^{-x^{2}/2}\left(1+\mathcal{O}\left(\frac{1}{x^{2}}\right)\right)$. We hence have the claim as
$$1-{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]=\mathcal{O}\left(\frac{1}{C_{n}}e^{-C_{n}^{2}/2}\right)\in o(\tau_{n}).$$
∎
Proof of Lemma 4.
Using the Jensen inequality, it can be easily seen that for any two distributions
$$\displaystyle D(p||q)={\mathbb{E}}_{p}\log\frac{p(x)}{q(x)}\leq\log\left({\mathbb{E}}_{p}\frac{p(x)}{q(x)}\right)$$
$$\displaystyle=\log\left({\mathbb{E}}_{q}\left(\frac{p(x)}{q(x)}\right)^{2}\right)$$
To bound the KL, we can hence study the ratio $p/q$. Using appendix A we can hence write
$$\frac{\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}(\mathbf{Y})}{{\mathbb{Q}}_{0,n}(\mathbf{Y})}={\mathbb{E}}_{\mathbb{P}_{n}}\frac{1}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]}\frac{{\mathbb{Q}}_{\beta\lambda_{n},n}(\mathbf{Y}|\mathbf{X})}{{\mathbb{Q}}_{\beta 0,n}(\mathbf{Y})}\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y}).$$
Plugging in the definition of the model for $\mathbf{Y}$ we can easily see that
$$\displaystyle\frac{{\mathbb{Q}}_{\beta\lambda_{n},n}(\mathbf{Y}|\mathbf{X})}{{\mathbb{Q}}_{0,n}(\mathbf{Y})}$$
$$\displaystyle=\frac{\exp(-\frac{1}{2}\|\mathbf{Y}-\sqrt{\lambda_{n}}\mathbf{X}\|^{2})}{\exp(-\frac{1}{2}\|\mathbf{Y}\|^{2})}$$
$$\displaystyle=\exp\left(\sqrt{\lambda_{n}}\langle\mathbf{X},\mathbf{Y}\rangle-\frac{\lambda_{n}}{2}\right)$$
where in the second inequality we used the fact that $\|\mathbf{X}\|^{2}=1$. Using the latter, we can obtain
$$\displaystyle\left(\frac{\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}(\mathbf{Y}|\mathbf{X})}{{\mathbb{Q}}_{0,n}(\mathbf{Y})}\right)^{2}=$$
$$\displaystyle={\mathbb{E}}_{\mathbb{P}_{n}^{\otimes 2}}\frac{\exp\left(\sqrt{\lambda_{n}}\langle\mathbf{X}+\mathbf{X}^{\prime},\mathbf{Y}\rangle-\lambda_{n}\right)}{{\mathbb{Q}}_{xy}[\Omega_{n}|\mathbf{X}]Q_{xy}[\Omega_{n}|\mathbf{X}^{\prime}]}\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})\mathbbm{1}_{\Omega_{n}}(\mathbf{X}^{\prime},\mathbf{Y})$$
Using the fact that for $C_{n}=\sqrt{\log\lambda_{n}}$ and $\lambda_{n}^{-1/2+\epsilon}\in o(\tau_{n})$ we can satisfy the assumption of Lemma 3, we can exchange the small-o notation and the integrals as
$$\displaystyle\left(\frac{\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}(\mathbf{Y}|\mathbf{X})}{{\mathbb{Q}}_{0,n}(\mathbf{Y})}\right)^{2}=$$
$$\displaystyle=(1+o(\tau_{n})){\mathbb{E}}_{\mathbb{P}_{n}^{\otimes 2}}\big{[}\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})\mathbbm{1}_{\Omega_{n}}(\mathbf{X}^{\prime},\mathbf{Y})$$
$$\displaystyle\times\exp\left(\sqrt{\lambda_{n}}\langle\mathbf{X}+\mathbf{X}^{\prime},\mathbf{Y}\rangle-\lambda_{n}\right)\big{]}$$
We hence have a bound for the KL that reads
$$\frac{1}{\lambda_{n}}D(\tilde{{\mathbb{Q}}}_{\beta\lambda_{n},n}||{\mathbb{Q}}_{0,n})\leq\frac{1}{\lambda_{n}}\log\left[{\mathbb{E}}_{\mathbb{P}_{n}^{\otimes 2}}m_{n}(\mathbf{X},\mathbf{X}^{\prime})\right]+o(\tau_{n})$$
where $m_{n}$ is defined as
$$\displaystyle m_{n}(\mathbf{X},\mathbf{X}^{\prime})\coloneqq{\mathbb{E}}_{{\mathbb{Q}}_{0,n}}\big{[}\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})\mathbbm{1}_{\Omega_{n}}(\mathbf{X}^{\prime},\mathbf{Y})$$
$$\displaystyle\times\exp\left(\sqrt{\lambda_{n}}\langle\mathbf{X}+\mathbf{X}^{\prime},\mathbf{Y}\rangle-\lambda_{n}\right)\big{]}$$
and we used the fact that $\frac{o(\tau_{n})}{\lambda_{n}}\in o(\tau_{n})$ and where we used Fubini’s theorem to exchange the order of the integrals.
Note that ${\mathbb{Q}}_{0,n}=\mathbb{P}_{z}=\mathcal{N}(0,1)$. Now it is sufficient to prove that
$$\frac{1}{\lambda_{n}}\log\left[{\mathbb{E}}_{\mathbb{P}_{n}^{\otimes 2}}m_{n}(\mathbf{X},\mathbf{X}^{\prime})\right]\leq\sup_{t\in[0,1]}\left(\frac{t}{t+1}-\frac{r_{n}(t)}{2}\right)+o(\tau_{n})$$
to get the claim. We can readily see that the function $m_{n}$ depends only on the overlap $\rho=\langle\mathbf{X},\mathbf{X}^{\prime}\rangle$ due to the rotational invariance of the Gaussian pdf. Using Lemma 15, the definition of $r_{n}$, the monotonicity of the exponential function and the simple inequality
$$sup(f+g)\leq\sup f+\sup g$$
we get
$$\displaystyle\frac{1}{\lambda_{n}}\log{\mathbb{E}}_{\mathbb{P}_{n}^{\otimes 2}}m_{n}(\rho)$$
$$\displaystyle\leq\frac{\log(2L_{n})}{\lambda_{n}}+$$
$$\displaystyle+\sup_{t\in[-1,1]}\left(\left(\frac{t}{1+t}\right)_{+}-\frac{r_{n}(t)}{2}\right)+\frac{C_{n}}{\lambda_{n}^{1/2}}+\mathcal{O}\left(\frac{1}{L_{n}}\right)$$
We can easily observe that the supremum can be limited to the interval $t\in[0,1]$ noting that $r_{n}(-1)=0$ and $r_{n}(t)$ is a non-negative function.
The claim then follows easily from the assumption of $\lambda_{n}^{-1/2+\epsilon}\in o(\tau_{n})$ and choosing $L_{n}=\lfloor\lambda_{n}^{1/2}\rfloor$ and $C_{n}=\sqrt{\log\lambda_{n}}$.
∎
Appendix B Postponed proofs of Section IV
Proof of Lemma 5.
$$\displaystyle\mathbb{P}_{n}{\mathbb{Q}}_{y|x}[\hat{\mathbf{X}}(\mathbf{Y})\neq\mathbf{X}]$$
$$\displaystyle={\mathbb{E}}_{P_{n}}{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\mathbbm{1}_{\{\hat{\mathbf{X}}(\mathbf{Y})\neq\mathbf{X}\}}$$
$$\displaystyle=1-{\mathbb{E}}_{P_{n}}{\mathbb{E}}_{{\mathbb{Q}}_{y|x}}\mathbbm{1}_{\{\hat{\mathbf{X}}(\mathbf{Y})=\mathbf{X}\}}$$
$$\displaystyle=1-\int d\mathbf{Y}{\mathbb{Q}}_{y|x}(\mathbf{Y}|\hat{\mathbf{X}}(\mathbf{Y}))\mathbb{P}_{n}(\hat{\mathbf{X}}(\mathbf{Y}))$$
hence $\operatorname*{argmin}\limits_{\hat{\mathbf{X}}(\cdot)}\mathbb{P}_{n}{\mathbb{Q}}_{y|x}[\hat{\mathbf{X}}(\mathbf{Y})\neq\mathbf{X}]$ satisfies for every $\mathbf{Y}$
$$\displaystyle\left(\operatorname*{argmin}_{\hat{\mathbf{X}}(\cdot)}\mathbb{P}_{n}{\mathbb{Q}}_{y|x}[\hat{\mathbf{X}}(\mathbf{Y})\neq\mathbf{X}]\right)(\mathbf{Y})=$$
$$\displaystyle\hskip 56.9055pt=\operatorname*{argmax}_{\hat{\mathbf{X}}(\mathbf{Y})}{\mathbb{Q}}_{y|x}(\mathbf{Y}|\hat{\mathbf{X}}(\mathbf{Y}))\mathbb{P}_{n}(\hat{\mathbf{X}}(\mathbf{Y})),$$
that, for uniform prior, corresponds to the MLE estimator.
∎
Proof of Lemma 6.
The two bounds are, respectively, given by the first and second-moment methods. Equation 5 follows easily from from the Markov inequality as for any estimator $\hat{\mathbf{X}}(\mathbf{Y})$,
$$\displaystyle\operatorname{EP}_{n}(\hat{\mathbf{X}})={\mathbb{E}}\left[\mathbbm{1}_{\{\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}>0\}}\right]$$
$$\displaystyle=\Pr\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}\geq\frac{2}{s}\right]$$
$$\displaystyle\leq\frac{s}{2}{\mathbb{E}}\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}\right]$$
hence
$$\displaystyle\operatorname{C-MEP}_{n}(\lambda)$$
$$\displaystyle=\min_{\hat{\mathbf{X}}(\mathbf{Y})\in\mathcal{C}_{n,s}}{\mathbb{E}}\left[\|\hat{\mathbf{X}}-\mathbf{X}\|_{0}\right]$$
$$\displaystyle\leq\frac{s}{2}\min_{\hat{\mathbf{X}}(\mathbf{Y})\in\mathcal{C}_{n,s}}{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}]=\frac{s}{2}\operatorname{C-MMSE}_{n}(\lambda).$$
To prove the second bound, we use the Paley–Zygmund inequality that reads for a general positive random variable $Z$ and $0\leq\theta\leq{\mathbb{E}}[Z]$
$$\Pr[Z>\theta]\geq\frac{({\mathbb{E}}[Z]-\theta)^{2}}{{\mathbb{E}}[Z^{2}]}.$$
(17)
Using Equation 17 for the random variable $\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}$ we obtain
$$\operatorname{EP}_{n}(\hat{\mathbf{X}})=\Pr\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}>0\right]\geq\frac{{\mathbb{E}}\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}\right]^{2}}{{\mathbb{E}}\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{4}\right]}$$
from which we get
$$\displaystyle\operatorname{MEP}_{n}(\lambda)$$
$$\displaystyle=\min_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}\Pr\left[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}>0\right]$$
$$\displaystyle\geq\min_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}\frac{{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}]^{2}}{{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{4}]}$$
$$\displaystyle\geq\frac{\left(\min_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}]\right)^{2}}{\max_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{4}]}$$
$$\displaystyle\geq\frac{1}{4}\operatorname{MMSE}_{p}(\lambda)^{2}$$
where in the last inequality we used the definition of the $\operatorname{MMSE}$ and the fact that for any two vectors
$$\bm{a},\bm{b}\in\bar{\mathcal{S}}_{n-1},\quad\|\bm{a}-\bm{b}\|^{2}\leq 2.$$
The third inequality follows analogously.
∎
Proof of Lemma 8.
$$\displaystyle{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}]={\mathbb{E}}\left[\|\hat{\mathbf{X}}\|^{2}\right]+{\mathbb{E}}\left[\|\mathbf{X}\|^{2}\right]-2{\mathbb{E}}\left[\sum_{i}^{n}\hat{X}_{i}X_{i}\right].$$
(18)
Given the constraint on the estimator and on the fact that $\|\mathbf{X}\|^{2}=\|\hat{\mathbf{X}}\|^{2}=1$, the optimization problem becomes:
$$\operatorname*{argmax}_{\hat{\mathbf{X}}(\mathbf{Y})\in\mathcal{C}_{n,s}}{\mathbb{E}}_{{\mathbb{Q}}_{\lambda,n}}\sum_{i=1}^{n}\hat{\mathbf{X}}_{i}(\mathbf{Y}){\mathbb{E}}[X_{i}|\mathbf{Y}]$$
hence for every fixed $\mathbf{Y}$ the optimal estimator reads
$$\operatorname*{argmax}_{\hat{\mathbf{X}}(\mathbf{Y})\in\mathcal{C}_{n,s}}\sum_{i}^{n}\hat{\mathbf{X}}_{i}(\mathbf{Y}){\mathbb{E}}[\mathbf{X}|\mathbf{Y}]_{i}.$$
The theorem follows easily from linearity and from $\mathcal{C}_{n,s}$ being a binary set for which the greedy algorithm is optimal.
∎
Proof of Lemma 9.
$$\displaystyle\delta\geq\|\mathbf{U}-\mathbf{V}\|^{2}$$
$$\displaystyle=\sum_{i\colon U_{i}=s^{-1/2}}\left(s^{-1/2}-V_{i}\right)^{2}+\sum_{i\colon U_{i}=0}V_{i}^{2}$$
$$\displaystyle\geq\left(\max_{i\colon U_{i}=s^{-1/2}}\left(s^{-1/2}-V_{i}\right)\right)^{2}+\left(\max_{i\colon U_{i}=0}V_{i}\right)^{2}$$
$$\displaystyle=\left(s^{-1/2}-\min_{i\colon U_{i}=s^{-1/2}}V_{i}\right)^{2}+\left(\max_{i\colon U_{i}=0}V_{i}\right)^{2}$$
Multiplying both sides of the latter inequality by $s$ we get a inequality of the form $\left(1-a\right)^{2}+b^{2}\leq s\delta<\frac{1}{2}$ for $a,b\in\left[0,1\right]$. It is easy to observe using simple calculus that this implies $a>b$ and hence
$$\min_{i\colon U_{i}=s^{-1/2}}V_{i}>\max_{i\colon U_{i}=0}V_{i}.$$
Given this condition, and the definition of the $\operatorname{Top}_{s}(\cdot)$ operator, it follows that $\operatorname{Top}_{s}(\mathbf{V})=\mathbf{U}$. Note that the strict inequality $\delta<\frac{1}{2s}$ is essential to guarantee the strict inequality above and hence that there are no ties in the selection of the top $s$ entries.
∎
Appendix C Useful lemmas
Lemma 12.
The posterior average ${\mathbb{E}}[\mathbf{X}|\mathbf{Y}]$ is the optimal estimator (MMSE) for the $l_{2}$ loss $l_{2}(\hat{\mathbf{x}},\mathbf{x})=\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}$ and the generic model in eq. 1, formally
$$\operatorname*{argmin}_{\hat{\mathbf{X}}(\mathbf{Y})\in\bar{\mathcal{S}}_{n-1}}{\mathbb{E}}[\|\hat{\mathbf{X}}(\mathbf{Y})-\mathbf{X}\|^{2}]={\mathbb{E}}[\mathbf{X}|\mathbf{Y}]$$
Proof.
$$\displaystyle\frac{\partial}{\partial\hat{\mathbf{X}}}{\mathbb{E}}[\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}]$$
$$\displaystyle={\mathbb{E}}_{{\mathbb{Q}}_{\lambda,n}}\sum_{\mathbf{X}}\ \mathbb{P}_{n}(\mathbf{X}|\mathbf{Y})\frac{\partial}{\partial\hat{\mathbf{X}}}\|\hat{\mathbf{X}}-\mathbf{X}\|^{2}=$$
(19)
$$\displaystyle={\mathbb{E}}_{{\mathbb{Q}}_{\lambda,n}}\sum_{\mathbf{X}}\mathbb{P}_{n}(\mathbf{X}|\mathbf{Y})2(\hat{\mathbf{X}}-\mathbf{X})=0$$
(20)
from which it follows easily that for every $\mathbf{Y}$, the gradient is equal to zero if $\hat{\mathbf{X}}(\mathbf{Y})={\mathbb{E}}[\mathbf{x}|\mathbf{Y}]$. It can be easily shown that the Hessian of the loss is positive semidefinite and hence satisfies the property of having a global minimum.
∎
We here further characterize the MLE estimator as following:
Lemma 13 (equivalent to Theorem 1 in [17]).
For the model defined in Equation 1, the MLE estimator reads:
$$\mathbf{X}_{\operatorname{0-1}}(\mathbf{Y})=\operatorname*{argmax}\limits_{\mathbf{X}\in\bar{\mathcal{S}}_{n}}\sum_{i_{1},\dots,i_{d}}Y_{i_{1},\dots,i_{d}}X_{i_{1},\dots,i_{d}}$$
(21)
Proof.
$$\displaystyle\log{\mathbb{Q}}_{y|x}$$
$$\displaystyle(\mathbf{Y}|\mathbf{X})=\sum_{i_{1},\dots,i_{d}}\log{\mathbb{Q}}_{y|x}(Y_{i_{1},\dots,i_{d}}|x_{i_{1}}\cdot\dots\cdot x_{i_{d}})$$
$$\displaystyle=\sum_{i_{1},\dots,i_{d}}-\frac{1}{2}\log 2\pi-\frac{1}{2}(Y_{i_{1},\dots,i_{d}}-\beta x_{i_{1}}\cdot\dots\cdot x_{i_{d}})^{2}$$
$$\displaystyle=\sum_{i_{1},\dots,i_{d}}-\frac{1}{2}\log 2\pi-\frac{1}{2}\beta^{2}x_{i_{1}}^{2}\cdot\dots\cdot x_{i_{d}}^{2}+$$
$$\displaystyle\hskip 56.9055pt-\frac{1}{2}Y_{i_{1},\dots,i_{d}}^{2}+\beta Y_{i_{1},\dots,i_{d}}x_{i_{1}}\cdot\dots\cdot x_{i_{d}}$$
$$\displaystyle=-\frac{n}{2}\log 2\pi-\frac{1}{2}\lambda d!\binom{k}{d}-\frac{1}{2}\sum_{i_{1},\dots,i_{d}}Y_{i_{1},\dots,i_{d}}^{2}+$$
$$\displaystyle\hskip 56.9055pt+\sqrt{\lambda}\sum_{i_{1},\dots,i_{d}}Y_{i_{1},\dots,i_{d}}x_{i_{1}}\cdot\dots\cdot x_{i_{d}}.$$
The theorem follows easily noting that only the last term depends on $\mathbf{X}$.
∎
Lemma 14.
Given the setting in Equation 1, for all $n$ and $\lambda>0$, the function $\beta\to\frac{1}{\lambda}D({\mathbb{Q}}_{\beta\lambda,n}||{\mathbb{Q}}_{0,n})$ is nonnegative, nondecreasing, 1/2-Lipschitz and satisfies the bound
$$\frac{1}{\lambda}D({\mathbb{Q}}_{\beta\lambda,n}||{\mathbb{Q}}_{0,n})\geq\frac{1}{2}-\frac{\log M_{n}}{\lambda}.$$
Proof.
The proof is given in Lemma 2 and Lemma 3 in [2].
∎
Lemma 15.
Given the setting of the problem defined in Equation 4 and the function
$$\displaystyle m_{n}(\mathbf{X},\mathbf{X}^{\prime})\coloneqq{\mathbb{E}}_{{\mathbb{Q}}_{0,n}}\big{[}\mathbbm{1}_{\Omega_{n}}(\mathbf{X},\mathbf{Y})\mathbbm{1}_{\Omega_{n}}(\mathbf{X}^{\prime},\mathbf{Y})$$
$$\displaystyle\times\exp\left(\sqrt{\lambda_{n}}\langle\mathbf{X}+\mathbf{X}^{\prime},\mathbf{Y}\rangle-\lambda_{n}\right)\big{]},$$
there exist a constant $C>0$ such that for any integer sequence $L_{n}$ the following bound holds:
$$\displaystyle{\mathbb{E}}_{\mathbb{P}_{n}^{\otimes 2}}m_{n}(\rho)$$
$$\displaystyle\leq 2L_{n}\sup_{t\in[-1,1]}\exp\bigg{(}\lambda_{n}\left(\frac{t}{1+t}\right)_{+}+$$
$$\displaystyle\hskip 28.45274pt+\log\mathbb{P}[\rho\geq t]+C_{n}\lambda_{n}^{1/2}+\mathcal{O}\left(\frac{\lambda_{n}}{L_{n}}\right)\bigg{)}$$
Proof.
The proof follows, mutatis mutandis, the proof of Theorem 4 in [2], with the minor change of the definition of the events $\Omega_{n}$ that are here defined as $\{|\langle\mathbf{X},\mathbf{Z}\rangle-\sqrt{\lambda_{n}}|\leq C_{n}\}$.
∎
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Entanglement Hamiltonian of the 1+1-dimensional free, compactified boson conformal field theory
Ananda Roy
[email protected]
Department of Physics, T42, Technische Universität Mun̈chen, 85748 Garching, Germany
Frank Pollmann
Department of Physics, T42, Technische Universität Mun̈chen, 85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
Hubert Saleur
Institut de Physique Théorique, Paris Saclay University, CEA, CNRS, F-91191 Gif-sur-Yvette.
Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA
Abstract
Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1+1 space-time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs.
As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson CFT over a finite spatial interval.
We compare the analytical results obtained for the continuum theory with numerical simulations of a lattice-regularized model for the CFT using density matrix renormalization group technique. To that end, we use a lattice regularization provided by superconducting quantum electronic circuits, built out of Josephson junctions and capacitors.
Up to non-universal effects arising due to the lattice regularization, the numerical results are compatible with the predictions of the exact computations.
1 Introduction
Entanglement plays an indispensable role in the analysis of correlations present in quantum field theories. The von-Neumann entanglement entropy, $S(\rho_{A})=-{\rm Tr}\rho_{A}\ln\rho_{A}$, is one of the most popular measures of bipartite entanglement [1]. Here, $\rho_{A}$ is the reduced density matrix of the subsystem A: $\rho_{A}={\rm Tr}_{B}\rho$, where $\rho$ is the total density matrix of the system composed of parts A and B.
The entanglement entropy is crucial in the characterization of quantum field theories in 1+1 space-time dimensions. The scaling of the entanglement entropy in critical systems in 1+1 dimensions has been predicted using conformal field theory (CFT) techniques [2, 3, 4]. These have been used to describe the quantum critical phenomena in 1D spin-chains [5, 6] as well as observe boundary-RG flow between different conformal invariant boundary conditions [7, 8, 9].
In 1+1 dimensional CFTs, the scaling of the entanglement entropy with the size of the subsystem A enables the determination of the central charge or the conformal anomaly parameter without the need to determine the velocity of sound in the theory [3]. However, the full operator content of the CFT remains elusive. The latter can be probed through the spectrum of the entanglement or modular Hamiltonian of the subsystem A (see Fig. 1) defined as [10, 11]
$${\cal H}_{A}=-\frac{1}{2\pi}\ln\rho_{A},$$
(1)
where we follow the convention of Ref. [12].
It turns out that the spectrum of the entanglement Hamiltonian of a CFT is given by the physical spectrum of a corresponding boundary CFT [12, 13]. The relationship of the entanglement Hamiltonian to the physical Hamiltonian of a boundary CFT opens the possibility to determine exactly the spectrum of the entanglement Hamiltonian. This can be done by computing boundary/Ishibashi states of the theory and subsequently, the partition function of the boundary CFT (see Chap. 11 of Ref. [14] or Ref. [15] for details of the formalism).
The main goal of this work is perform this computation for the free, compactified boson CFT and provide numerical data obtained with density matrix renormalization group (DMRG) technique for a lattice-regularized model. We note that DMRG results were obtained earlier for the critical transverse-field Ising chain and the Bose-Hubbard model in Ref. [16], which had suggested the boundary CFT structure of the entanglement Hamiltonian. In this work, we go a step further and perform analytical computations for the different boundary conditions and conduct a careful comparison of the DMRG results for finite system sizes.
We focus on the case when the system under investigation is finite (with length $L_{0}$) with a certain prescribed boundary condition, $\alpha$ at its ends. We treat only the case of identical boundary conditions at both ends (the case with different boundary conditions at the two ends has additional technical complications [12], which we leave for a later work) for a system at zero temperature and consider a subsystem A of length $r$. For this case, the spectrum of ${\cal H}_{A}$ is determined by that of the Hamiltonian $H_{\alpha\beta}$ of the boundary CFT with boundary conditions $\alpha,\beta$, where $\alpha\neq\beta$ in general [12]. The first boundary condition $\alpha$ is inherited from the original system, while the second $\beta$ originates from the entanglement cut and is usually the free boundary condition. Thus, for the case when $\alpha$ corresponds to free boundary conditions, the boundary CFT also has free boundary conditions at both ends. On the other hand, if $\alpha$ corresponds to fixed boundary conditions, then the corresponding boundary CFT has fixed and free boundary conditions at its ends. The final result for the entanglement Hamiltonian is given by [12]
$${\cal H}_{A}=-\frac{1}{2\pi}\ln\frac{e^{-2\pi H_{\alpha\beta}}}{{\rm Tr}\ e^{-%
2\pi H_{\alpha\beta}}},$$
(2)
where the denominator inside the logarithm originates from the fact that the reduced density matrix $\rho_{A}$ should be normalized. The above equation is to be understood as an equality of the eigenvalues of the two sides the equation up to overall shifts and rescalings, which can be absorbed by rescaling the velocity of sound in the corresponding boundary CFT.
To illustrate the basic principle of the analysis, we start by deriving the entanglement Hamiltonian of the Ising CFT using the exact results for the Ishibashi states obtained by Cardy [15] and compare with DMRG results of the corresponding lattice model of the critical transverse-field Ising chain. Then, we present the exact results for the free compactified boson CFT. Generalizing the computation done in Ref. [17], we provide an explicit closed form expression for the spectrum of the entanglement Hamiltonian in terms of the compactification radius and the system size. In order to compare our analytical predictions of the continuum theory for a lattice model, we analyze a lattice regularized model using quantum electronic circuits [18, 19, 20, 21, 22] using DMRG. This should be contrasted with usual lattice-regularizations of the free, compactified boson CFT using the paramagnetic phase of the XXZ spin chain [23, 24]. 111We focus on the compact boson case. The non-compact case can be treated by considering massless harmonic oscillator chains [13, 25]. In contrast to the latter model, we start from compact, bosonic lattice degrees of freedom. Thus, we avoid nonlinear and nonlocal transformations like Jordan-Wigner transformations and bosonization, which do not faithfully capture the entanglement spectrum. The quantum circuit is a 1D array of superconducting islands, separated by tunnel junctions, realizing a generalized Bose-Hubbard model in the limit of high-occupancy of each site [26]. Each superconducting island has a finite charging energy $E_{C_{0}}=2e^{2}/C_{0}$, where $C_{0}$ is the capacitance to the ground plane. The Josephson junction separating two such islands has a junction energy $E_{J}$ and charging energy $E_{C_{J}}=2e^{2}/C_{J}$. The lattice model has a rich phase-diagram comprising Mott-insulating, charge-density-wave and the free, compactified boson phases. We focus on the latter phase, which occurs in the regime $E_{C_{J}},E_{C_{0}}\ll E_{J}$. The compactification radius $R=1/\beta=1/\sqrt{\pi K}$, where $K$ is the Luttinger parameter of the system. The compactified bosonic field $\phi(x,t)$ is the Josephson phase on the island at position $x$ at time $t$. We analyze the entanglement Hamiltonian of the CFT under consideration using DMRG.
The article is organized as follows. In Sec. 2, we derive the spectrum of the entanglement Hamiltonian of the Ising CFT and compare with that obtained using DMRG. Subsequently in Sec. 3, we present our results for the free, compactified boson CFT. The analytical, exact results are presented in Sec. 3.1. The details of the computation of the boundary states are provided in A, where we compute the partition function of the compactified boson CFT in the presence of different boundary conditions. In Sec. 3.2, we provide DMRG analysis of free, compactified boson phase of the lattice quantum circuit model. The detailed phase-diagram of the lattice model, while interesting on its own, is not relevant for the main goal of the work, and is given in B. In Sec. 4, we summarize our findings and provide a concluding perspective.
2 A simple test case: the Ising CFT
To illustrate the basic principle of the analysis, in this section, we derive the spectrum of the entanglement Hamiltonian for the Ising CFT. The latter is the unitary, minimal model ${\cal M}(4,3)$ with central charge $c=1/2$ (see, for example, Chapters 7 and 8 of Ref. [14]). It contains three primary fields, $I,\sigma,\epsilon$, with conformal dimensions: $h_{0}=0,h_{\sigma}=1/16$ and $h_{\epsilon}=1/2$. 222We have used $\epsilon$ with a subscript to denote the eigenvalues of the Hamiltonian of the corresponding boundary CFTs in A, but there should not be any confusion.
2.1 Exact results
We will consider two cases: (i) free/Neumann (N) boundary conditions at both ends ($\alpha={\rm N}$) and (ii) fixed/Dirichlet (D) boundary conditions at both ends ($\alpha={\rm D}$). Thus, the entanglement Hamiltonian for the subsystem $A$ (see Fig. 1) will be given by the spectrum of the boundary CFT over a length $L$ with boundary conditions $\alpha=\beta={\rm N}$ in the first case and $\alpha={\rm D}$, $\beta={\rm N}$ in the latter. We emphasize that $L$ is not the length of the subsystem A, the latter being denoted by $r$. The relation between $L_{0},L$ and $r$, to leading order, is given by
$$L=\ln\Big{(}\frac{2L_{0}}{\pi a}\sin\frac{\pi r}{L_{0}}\Big{)},$$
(3)
with correction arising at ${\cal O}(a)$. Here, $a$ is the lattice spacing. The partition function for the boundary CFTs can be expressed in terms of the parameters $q,\tilde{q}$:
$$q=e^{-2\pi^{2}/L},\ \tilde{q}=e^{-2L}.$$
(4)
Note that as $L\rightarrow\infty$, $q\rightarrow 1$ and $\tilde{q}\rightarrow 0$. So, it is more convenient to express final results as series in $\tilde{q}$ rather than $q$ for better convergence. The boundary states for the different boundary conditions are given by [15]
$$\displaystyle|\tilde{0}\rangle$$
$$\displaystyle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|\epsilon\rangle+%
\frac{1}{2^{1/4}}|\sigma\rangle,$$
(5)
$$\displaystyle\Big{|}\tilde{\frac{1}{2}}\Big{\rangle}$$
$$\displaystyle=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|\epsilon\rangle-%
\frac{1}{2^{1/4}}|\sigma\rangle,$$
(6)
$$\displaystyle\Big{|}\tilde{\frac{1}{16}}\Big{\rangle}$$
$$\displaystyle=|0\rangle-|\epsilon\rangle,$$
(7)
where the first two correspond to Dirichlet boundary conditions and the last corresponds to Neumann boundary condition.
For case (i), the corresponding partition function can be written as a sum over characters of the Ising CFT:
$$\displaystyle Z_{\rm{NN}}(q)$$
$$\displaystyle={\rm Tr}e^{-2\pi H_{\rm NN}}=\sum_{j=0,\sigma,\epsilon}\Big{|}%
\Big{\langle}\tilde{\frac{1}{16}}\Big{|}j\Big{\rangle}\Big{|}^{2}\chi_{j}(%
\tilde{q})$$
(8)
$$\displaystyle=\chi_{0}(\tilde{q})+\chi_{\epsilon}(\tilde{q})$$
$$\displaystyle=\chi_{0}(q)+\chi_{\epsilon}(q),$$
(9)
where in the last line, we have used the explicit form of the modular S-matrix of the Ising CFT [15]. Thus, we find that the partition function gets contribution from two primary fields: $I,\epsilon$. We use the explicit formulas for the characters (see Chapter 8 of Ref. [14]):
$$\displaystyle\chi_{0}(q)$$
$$\displaystyle=\frac{1}{\eta(q)}\sum_{n\in\mathbb{Z}}\Big{[}q^{(24n+1)^{2}/48}-%
q^{(24n+7)^{2}/48}\Big{]},$$
(10)
$$\displaystyle\chi_{\epsilon}(q)$$
$$\displaystyle=\frac{1}{\eta(q)}\sum_{n\in\mathbb{Z}}\Big{[}q^{(24n+5)^{2}/48}-%
q^{(24n+11)^{2}/48}\Big{]},$$
(11)
where $\eta(q)$ is the Dedekind function defined as
$$\eta(q)=q^{1/24}\varphi(q)=q^{1/24}\prod_{n>0}(1-q^{n}).$$
(12)
Expanding in $q$, we get
$$\displaystyle\chi_{j}(q)$$
$$\displaystyle=q^{-1/48+h_{j}}\sum_{n\geq 0}p_{j}(n)q^{n},\ j=0,\epsilon,$$
(13)
where $p_{0,\epsilon}(i)$ are obtained to be
$$\displaystyle p_{0}(n)$$
$$\displaystyle=1,0,1,1,2,2,3,\ldots,$$
(14)
$$\displaystyle p_{\epsilon}(n)$$
$$\displaystyle=1,1,1,1,2,2,3,\ldots.$$
(15)
Thus, the entanglement energies, labeled by two indices: $(j,n)$, are given by
$$\displaystyle\varepsilon_{\rm{N}}(j,n)$$
$$\displaystyle=-\frac{1}{2\pi}\ln\frac{q^{-1/48+h_{j}+n}}{\tilde{q}^{-1/48}\sum%
\limits_{k=0,\epsilon}\sum\limits_{m\geq 0}p_{k}(m){\tilde{q}}^{h_{k}+m}}$$
(16)
$$\displaystyle=\frac{L}{48\pi}+\frac{\pi}{L}\Big{(}-\frac{1}{48}+h_{j}+n\Big{)}$$
$$\displaystyle\quad+\frac{1}{2\pi}\ln\sum\limits_{k=0,\epsilon}\sum\limits_{m%
\geq 0}p_{k}(m)e^{-2L(h_{k}+m)}$$
with degeneracy at the level $(j,n)$ being given by $p_{j}(n)$. The lowest entanglement energy level is given by
$$\displaystyle\varepsilon_{\rm{N}}(0,0)$$
$$\displaystyle=\frac{L}{48\pi}-\frac{\pi}{48L}+\frac{1}{2\pi}\ln\sum\limits_{k=%
0,\epsilon}\sum\limits_{m\geq 0}p_{k}(m)e^{-2L(h_{k}+m)}.$$
(17)
With respect to this lowest level, the entanglement energies are given by
$$\Delta\varepsilon_{\rm{N}}(j,n)\equiv\varepsilon_{\rm{N}}(j,n)-\varepsilon_{%
\rm{N}}(0,0)=\frac{\pi}{L}\big{(}h_{j}+n\big{)},$$
(18)
and thus, occur at integer (half-integer) values in units of $\pi/L$ for $j=0(\epsilon)$.
For case (ii), the analysis proceeds analogously. The partition function for the corresponding boundary CFT with Dirichlet and Neumann boundary conditions at the ends is
$$\displaystyle Z_{\rm{DN}}(q)$$
$$\displaystyle=\chi_{\sigma}(q)=\frac{1}{\sqrt{2}}[\chi_{0}(\tilde{q})-\chi_{%
\epsilon}(\tilde{q})].$$
(19)
Here,
$$\displaystyle\chi_{\sigma}(q)$$
$$\displaystyle=\frac{1}{\eta(q)}\sum_{n\in\mathbb{Z}}\Big{[}q^{(24n-2)^{2}/48}-%
q^{(24n+10)^{2}/48}\Big{]}$$
(20)
$$\displaystyle=q^{-1/48+h_{\sigma}}\sum_{n\geq 0}p_{\sigma}(n)q^{n},$$
where we obtain
$$p_{\sigma}(n)=1,1,1,2,2,3,\ldots.$$
(21)
In this case, the entanglement energies, indexed by $n$, are given by
$$\displaystyle\varepsilon_{\rm{D}}(n)$$
$$\displaystyle=-\frac{1}{2\pi}\ln\frac{\sqrt{2}q^{-1/48+h_{\sigma}+n}}{\tilde{q%
}^{-1/48}\sum\limits_{k=0,\epsilon}e^{2\pi ih_{k}}\sum\limits_{m\geq 0}p_{k}(m%
){\tilde{q}}^{h_{k}+m}}$$
(22)
$$\displaystyle=\frac{L}{48\pi}-\frac{1}{4\pi}\ln 2+\frac{\pi}{L}\Big{(}\frac{1}%
{24}+n\Big{)}$$
$$\displaystyle\quad+\frac{1}{2\pi}\ln\sum\limits_{k=0,\epsilon}e^{2\pi ih_{k}}%
\sum\limits_{m\geq 0}p_{k}(m)e^{-2L(h_{k}+m)},$$
where the degeneracy at level $n$ is given by $p_{\sigma}(n)$. The lowest energy level $\varepsilon(0)$ is given by
$$\displaystyle\varepsilon_{\rm D}(0)$$
$$\displaystyle=\frac{L}{48\pi}-\frac{1}{4\pi}\ln 2+\frac{\pi}{24L}$$
(23)
$$\displaystyle\quad+\frac{1}{2\pi}\ln\sum\limits_{k=0,\epsilon}e^{2\pi ih_{k}}%
\sum\limits_{m\geq 0}p_{k}(m)e^{-2L(h_{k}+m)},$$
with respect to which the entanglement energies are given by
$$\Delta\varepsilon_{\rm{D}}(n)\equiv\varepsilon_{\rm{D}}(n)-\varepsilon_{\rm{D}%
}(0)=\frac{\pi}{L}n$$
(24)
Recall that the entanglement entropy for the subsystem A is given by [3]
$${\cal S}(L)=\frac{c}{6}L+{\cal O}(1),$$
(25)
where $c=1/2$. The last term contains the boundary terms predicted by Affleck and Ludwig as well as a non-universal correction. Comparing the leading order terms in either of Eqs. (17,23) with Eq. (25), we get the expected relation between the entanglement entropy and the single-copy entanglement [27]
$$\varepsilon_{\rm{D/N}}(0,0)=\frac{1}{4\pi}{\cal S}+{\cal O}(1),$$
(26)
where we have an extra factor of $2\pi$ due to our definition of Eq. (1). Furthermore, it is useful to compare the lowest entanglement energies for the two cases obtained in Eqs. (17, 23). For $L\rightarrow\infty$, the difference between the two is given by the term of ${\cal O}(1)$ in Eq. (23):
$$\varepsilon_{\rm{N}}(0,0)-\varepsilon_{\rm{D}}(0)=\frac{1}{4\pi}\ln 2.$$
(27)
This difference is the change in the Affleck-Ludwig boundary entropy as we go from Neumann-Neumann to Dirichlet-Neumann boundary conditions in the boundary CFT. Note that the relationship between the single-copy entanglement and the entanglement entropy [Eq. (26)] does not hold for the ${\cal O}(1)$ term. Furthermore, there is a difference by a factor of $2\pi$ with the original work [7] due to conventions chosen in Eq. (1).
2.2 DMRG results
In this section, we compute using DMRG, the entanglement spectrum of the critical transverse-field Ising chain and compare with the analytical CFT predictions derived above. We show the DMRG results for the cases when either Neumann or Dirichlet boundary conditions are imposed on both ends of the chain [referred to as cases (i) or (ii) above]. We chose the system size to be $L_{0}=1600$ and a bond-dimension of $600$ to keep truncation errors below $10^{-12}$. We verify the central charge ($c$) to be $\simeq 1/2$. This is done by evaluating the entanglement entropy ${\cal S}$ for a finite block (of length $r$) within the system (of length $L_{0}$), using Eqs. (3, 25). Explicitly,
$${\cal S}(r,L_{0})=\frac{c}{6}\ln\Big{(}\frac{2L_{0}}{\pi a}\sin\frac{\pi r}{L_%
{0}}\Big{)}+{\cal S}_{0},$$
(28)
where ${\cal S}_{0}$ contains the contribution from the boundary as well as non-universal terms. By changing the boundary conditions, we obtain a change in entropy that is very close to the expected value of $(\ln 2)/2$ [see Fig. 2 (a)] (note that the change in entropy is $2\pi$ times the change in the lowest entanglement energy in our convention).
In Fig. 2(b), we show the rescaled and shifted entanglenent spectrum obtained for Neumann boundary conditions for a partitioning at the center of the center of the chain ($r=L_{0}/2$). As predicted by Eq. (16), the spectrum is indeed split into two Virasoro blocks, corresponding to the primary fields $0$ and $\epsilon$. The rescaled spectrum is close to the CFT predictions indicated by the dashed lines.
In Fig. 2(c), we show the same for Dirichlet boundary conditions partitioning the chain in two halves. From Eq. (22), there is only one Virasoro block corresponding to the primary field $\sigma$. The obtained entanglement spectrum is close to the CFT predictions (shown in dashed lines). The finite size effects are larger in this case, compared to the Neumann case, but we verified that the discrepancy between the DMRG results and the CFT predictions diminish upon increasing the system size.
Next, we compare the DMRG results for the lowest entanglement spectrum eigenvalue with that obtained from CFT (Fig. 3). We do this for the Neumann case, similar results were obtained for the Dirichlet case. Up to an overall shift, which for large system sizes is a constant $\simeq 0.24$, the DMRG results exhibit the same asymptotic behavior as system sizes are increased [note that the variation is plotted with respect to $L=\ln(2L_{0}/\pi)$].
3 The free compactified boson CFT
In this section, we compute the spectrum of the entanglement Hamiltonian of the free compactified boson CFT. The compactification radius is given by $R=1/\sqrt{\pi K}$, where $K$ is the Luttinger parameter of the theory. The analytical results were obtained by extending the calculations of Ref. [17], for more details see A.
3.1 Exact results
We consider a finite system of size $L_{0}$ and analyze the cases when the system has (i) free/Neumann boundary conditions and (ii) fixed/Dirichlet boundary conditions at both ends. Thus, the spectrum of the boundary CFT that needs to be evaluated has Neumann-Neumann and Dirichlet-Neumann boundary conditions at two ends of the interval of length $L$ for the two cases respectively. These can be obtained from the corresponding expressions in A by setting $T=2\pi$.
Consider case (i). In terms of these two parameters $q,\tilde{q}$, the corresponding partition function for the boundary CFT is given by
$$\displaystyle Z_{\rm{NN}}$$
$$\displaystyle={\rm Tr}\ e^{-2\pi H_{\rm{NN}}}$$
(29)
$$\displaystyle=\sum_{k\in\mathbb{Z}}\sum_{l\geq 0}p(l)q^{-\frac{1}{24}+\frac{K}%
{2}k^{2}+l}$$
(30)
$$\displaystyle=\frac{1}{\sqrt{K}\eta(\tilde{q})}\sum_{m}\tilde{q}^{\frac{m^{2}}%
{2K}}$$
(31)
$$\displaystyle=\sum_{h}n^{h}_{\rm{NN}}\chi_{h}(q),$$
(32)
where $\eta(q)$ is the Dedekind function and and $p(l)$ is the number of integer partitions of the integer $l$. In the last line, we have expressed the partition function as a sum over the Virasoro characters for different primary fields with dimension $h$ [15](see A for more details). Here, we have also used the fact that the central charge for the CFT is 1. The expression in Eq. (30) shows that the spectrum is composed of Virasoro towers built on primary fields with dimension $Kk^{2}/2$, $k\in\mathbb{Z}$. The towers on top of each primary are themselves built out of the descendants indexed by $l$ and each level has degeneracy $p(l)$. Denote the eigenvalues of the entanglement Hamiltonian ${\cal H}_{A}$ by $\varepsilon_{\rm{N}}(k,l)$ where it is implied that the degeneracy for each $l$ is $p(l)$. Then,
$$\displaystyle\varepsilon_{\rm{N}}(k,l)$$
$$\displaystyle=-\frac{1}{2\pi}\ln\frac{q^{-\frac{1}{24}+\frac{K}{2}k^{2}+l}}{%
\frac{1}{\sqrt{K}\eta(\tilde{q})}\sum_{m\in\mathbb{Z}}\tilde{q}^{\frac{m^{2}}{%
2K}}}$$
(33)
$$\displaystyle=\frac{L}{24\pi}-\frac{1}{4\pi}\ln K+\frac{\pi}{L}\Big{(}-\frac{1%
}{24}+\frac{K}{2}k^{2}+l\Big{)}$$
$$\displaystyle\quad-\frac{1}{2\pi}\Bigg{[}\sum_{n>0}\ln(1-e^{-2Ln})-\ln\sum_{m%
\in\mathbb{Z}}e^{-\frac{Lm^{2}}{K}}\Bigg{]}.$$
The smallest eigenvalue $\varepsilon_{\rm{N}}(0,0)$ is given by
$$\displaystyle\varepsilon_{\rm{N}}(0,0)$$
$$\displaystyle=\frac{L}{24\pi}-\frac{1}{4\pi}\ln K-\frac{\pi}{24L}$$
(34)
$$\displaystyle-\frac{1}{2\pi}\Bigg{[}\sum_{n>0}\ln(1-e^{-2Ln})-\ln\sum_{m\in%
\mathbb{Z}}e^{-\frac{Lm^{2}}{K}}\Bigg{]}.$$
The last two terms in the expression for $\varepsilon_{\rm N}(0,0)$ contribute to the ‘unusual’ corrections to the entanglement entropy obtained in Ref. [28]. The scaling of the higher entanglement energies is given by
$$\Delta\varepsilon_{\rm{N}}(k,l)\equiv\varepsilon_{\rm{N}}(k,l)-\varepsilon_{%
\rm{N}}(0,0)=\frac{\pi}{L}\Big{(}\frac{K}{2}k^{2}+l\Big{)}.$$
(35)
Now consider case (ii). The relevant partition function [see Eq. (80)] is given by
$$\displaystyle Z_{\rm DN}(q)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}\eta(\tilde{q})}\sum_{n}(-1)^{n}\tilde{q}^{n^{2}%
}=q^{1/48}\sum_{n\geq 0}q^{n/2}p_{\sigma}(n)$$
(36)
where the $p_{\sigma}(n)$ is defined in Eq. (21).
Using Eq. (2), the spectrum of the entanglement Hamiltonian is given by
$$\displaystyle\varepsilon_{\rm{D}}(n)$$
$$\displaystyle=-\frac{1}{2\pi}\ln\frac{\sqrt{2}\eta(\tilde{q})q^{\frac{1}{48}+n%
/2}}{\sum\limits_{n\in\mathbb{Z}}(-1)^{n}\tilde{q}^{n^{2}}}$$
(37)
$$\displaystyle=\frac{L}{24\pi}-\frac{1}{4\pi}\ln 2+\frac{\pi}{L}\Big{(}\frac{1}%
{48}+\frac{n}{2}\Big{)}$$
$$\displaystyle\quad-\frac{1}{2\pi}\Bigg{[}\sum_{n>0}\ln(1-e^{-2Ln})-\ln\sum_{m%
\in\mathbb{Z}}(-1)^{m}e^{-2Lm^{2}}\Bigg{]},$$
where we used the definitions in Eq. (4) and $n\geq 0$. The degeneracy at level $n$ is given by $p_{\sigma}(n)$. The smallest eigenvalue is given by
$$\displaystyle\varepsilon_{\rm{D}}(0)$$
$$\displaystyle=\frac{L}{24\pi}-\frac{1}{4\pi}\ln 2+\frac{\pi}{48L}-\frac{1}{2%
\pi}\Bigg{[}\sum_{n>0}\ln(1-e^{-2Ln})$$
(38)
$$\displaystyle\quad-\ln\sum_{m\in\mathbb{Z}}(-1)^{m}e^{-2Lm^{2}}\Bigg{]},$$
and the higher entanglement energies are given by
$$\displaystyle\Delta\varepsilon_{\rm{D}}(n)$$
$$\displaystyle\equiv\varepsilon_{\rm{D}}(n)-\varepsilon_{\rm{D}}(0)=\frac{\pi}{%
2L}n.$$
(39)
We note again the corresponding relation between the single-copy entanglement and the entanglement-entropy as obtained for the Ising CFT in Eq. (26). Furthermore, by comparing the lowest entanglement energies for the two cases, in the limit of $L\rightarrow\infty$, we find
$$\varepsilon_{\rm{N}}(0,0)-\varepsilon_{\rm{D}}(0)=-\frac{1}{4\pi}\ln\frac{K}{2},$$
(40)
which is the change in the boundary entropy in this case [29]. Fermionizing the compactified boson action in the presence of boundary fields at $K=1$ results in the same change in the boundary entropy as in the Ising case [see Eq. (27)], a reflection of the well-known correspondence of this model (with boundary fields) with two uncoupled Ising chains with boundary magnetic field on one of them [29].
So far, we have computed exactly the spectrum of the entanglement Hamiltonian. Next, we provide a numerical test for our computations by performing DMRG calculations on a suitably regularized lattice model.
3.2 DMRG analysis of the quantum circuit model
3.2.1 Description of the model:
The lattice model for the free-compactified boson CFT comprises a 1D array of mesoscopic superconducting islands separated by tunnel junctions (see Fig. 4). Each unit cell contains a capacitor (with capacitance $C_{0}$) on the vertical link and a Josephson junction (with junction energy $E_{J}$ and junction capacitance $C_{J}$). Throughout this work, we assume the absence of disorder in the model. Here, we choose the parameters such that $E_{C_{J}}\ll E_{C_{0}}\ll E_{J}$, where $E_{C_{0,J}}=2e^{2}/C_{0,J}$. In this limit, the phase-slips across the array are exponentially suppressed by a WKB factor $\sim e^{-\sqrt{E_{J}/E_{C_{J}}}}$. This leads to the low-energy properties of the theory being described by a Luttinger liquid or equivalently a free, compactified boson CFT [20, 21, 22]. Here, the superconducting phase, $\phi(x,t)$, at node $(x,t)$ is bosonic field under consideration.
The effective euclidean action for the theory describing the low-energy physics is given by [23]
$$\displaystyle S_{\rm{array}}=\frac{1}{2\pi K}\int dt\int_{0}^{L}dx\Big{[}\frac%
{1}{u}(\partial_{t}\phi)^{2}+u(\partial_{x}\phi)^{2}\Big{]},$$
(41)
Here, the plasmon velocity, $u$, and the Luttinger parameter, $K$, are given by [21]
$$u\simeq a\sqrt{2E_{C_{0}}E_{J}},\ K\simeq\frac{1}{2\pi}\sqrt{\frac{2E_{C_{0}}}%
{E_{J}}},$$
(42)
where $a$ is the lattice spacing. We note that these analytical expressions are only asymptotically true since the lattice model, to the best of our knowledge, is not exactly solvable. In our work, we extract the relevant properties of the model using DMRG. This is done by computing the ground state properties of the following lattice Hamiltonian:
$$\displaystyle H_{\rm{array}}$$
$$\displaystyle=E_{C_{0}}\sum_{i=1}^{L}n_{i}^{2}+\delta E_{C_{0}}\sum_{i=1}^{L-1%
}n_{i}n_{i+1}-E_{J}\sum_{i=1}^{L-1}\cos(\phi_{i}-\phi_{i+1}).$$
(43)
Here, the first term arises due to the finite charging energy of the mesoscopic islands and $n_{i}$ is the excess number of Cooper pairs on the $i^{\rm{th}}$ island 333Note that $n_{i}$ can be both positive or negative, the latter corresponding to removal of a Cooper-pair from the superconducting condensate on the $i^{\rm th}$ island.. The finite junction capacitance $C_{J}$ leads to, in principle, infinite-range interaction between any two islands with a magnitude that decays exponentially with distance [21]. However, for realistic system parameters [30], it suffices to include only the nearest neighbor interaction [20], indicated by the second term in Eq. (43) with $\delta$ being a small parameter $<1$. The last term in Eq. (43) describes the coherent tunneling of Cooper-pairs between neighboring islands. Here, the operators $n_{i},\phi_{j}$ are canonically conjugate satisfying $[n_{i},e^{\pm i\phi_{j}}]=\pm e^{\pm i\phi_{j}}\delta_{ij}$. The Hamiltonian can be viewed as a Bose-Hubbard model with nearest neighbor interaction, in the limit of very high-occupancy at each site and zero gate voltage [31] (see B for more details) 444In general, there is a term $E_{g}\sum_{i}n_{i}$ in the lattice Hamiltonian, which corresponds to ‘chemical potential’ in the Bose-Hubbard language and $E_{g}$ is the gate voltage, see B. In this section, this term is set to zero by choosing the ‘chemical potential’ appropriately..
The boundary conditions have simple physical interpretations for the circuit model. The Dirichlet boundary condition at an end corresponds to a fixed superconducting phase at that end. As a result, there is no voltage drop at the: $V\sim\partial_{t}\phi=0$ and there is short-circuit at the boundaries [32, 33, 34]. This can be achieved by adding a Josephson junction with a very large junction energy compared to its charging energy at the boundary. The Neumann boundary condition at and end corresponds to leaving the end open. As a result, no current can flow $I\sim\partial_{x}\phi=0$ [33, 34]. In the DMRG simulations, we implement the boundary conditions by appropriately choosing the boundary interaction terms.
3.2.2 DMRG results:
The DMRG simulations were performed using the TeNPy package [35]. The local Hilbert space on each island was truncated to 9: $n_{i}=-4,-3,\ldots,3,4$. For definiteness, we chose $\delta=0.2$ and $\langle n_{i}\rangle=0$ by choosing an appropriate ground state sector. Furthermore, we chose a maximum bond-dimension of $500$ to keep the errors in truncation below $10^{-9}$. Here, we provide only the results relevant for the free, compactified boson CFT (the details of the phase-diagram can be found in B). We perform DMRG simulations for both open Neumann and Dirichlet boundary conditions at the ends of the lattice.
First, we obtain the two main characteristics of the free, compactified boson CFT: the central charge and the compactification radius or equivalently the Luttinger parameter. The first is obtained by computing the entanglement entropy for the subsystem A as a function of the subsystem size $r$ [see Eq. (28)]. The results are shown in Fig. 5(a) for both Neumann (dark green) and Dirichlet (maroon) boundary conditions for a system size $L_{0}=400$. The central charge is extracted from the data for the Neumann boundary conditions and is obtained to be $\simeq 1$ as expected. As the boundary condition is changed, the entanglement entropy changes by the expected amount of $\ln(2/K)/2$, where $K$ is the Luttinger parameter. This the contribution from the boundary entropy [7, 29] as computed in Eq. (40) [note the extra factor of $1/2\pi$ in the latter equation due to conventions chosen in Eq. (1)].
To obtain the Luttinger parameter, we compute the particle number fluctuations within the subsystem as a function of $r$. This yields the Luttinger parameter through the following relation [36]
$$\displaystyle(\Delta N_{A})^{2}$$
$$\displaystyle=\langle N_{A}^{2}\rangle-\langle N_{A}\rangle^{2}=\frac{1}{2\pi^%
{2}K}\ln\Big{(}\frac{2L_{0}}{\pi a}\sin\frac{\pi r}{L_{0}}\Big{)}+\Delta N_{0},$$
(44)
where $\Delta N_{0}$ is some non-universal contribution. The result for Neumann boundary condition is shown in Fig. 5(b), where the Luttinger parameter is obtained to be $K\simeq 0.192$, with error bars occurring in the third decimal place.
Next, we compute the entanglement spectrum for the Neumann and Dirichlet boundary conditions using DMRG. The results are shown in Fig. 5(c) and (d) respectively. The entanglement spectrum is computed by partitioning the system in half: $r=L_{0}/2$. To relate to Eqs. (35, 39), we plot the rescaled entanglement energies. For the Neumann case, we plot $[\varepsilon_{N}(k,l)-\varepsilon_{N}(k,0)]/\Delta\varepsilon_{N}(0,0)$ [Fig. 5(c)] as a function of $k$. The latter determine the dimension of the primary fields, which are $Kk^{2}/2$. From Eq. (35), the y-axis is the level of the descendant field, indexed by $l$. The CFT predictions for each level is indicated by dashed dark green lines and the corresponding degeneracies are given by the integer partitioning of $l$, denoted by $p(l)$. Fig. 5(d) shows the rescaled entanglement energies $\Delta\varepsilon_{D}(n)/\Delta\varepsilon_{D}(1)$ for the Dirichlet boundary conditions. The corresponding CFT predictions for the entanglement energies and the corresponding degeneracies are indicated by dashed maroon lines and $p_{\sigma}(n)$ respectively.
Now, we analyze the finite size dependence of the lowest few entanglement energies for the Neumann case (similar analysis was done for the Dirichlet case and are not shown for brevity). First, consider the lowest entanglement energy $\varepsilon_{N}(0,0)$. We again choose the subsystem size $r=L_{0}/2$ and plot the variation with respect to $L=\ln(2L_{0}/\pi)$ [see Eq. (3)]. The variation of the lowest entanglement energy with $L$ is shown in maroon in Fig. 6. As the subsystem size is increased, $\varepsilon_{N}(0,0)$ shows the expected linear dependence [see Eq. (34)] with slope $0.015\sim 1/24\pi$. The corresponding values of $\varepsilon_{N}(0,0)$ obtained from the CFT predictions of Eq. (34) is plotted in blue. In the limit of large $L$, we see that there is a non-universal constant shift between the CFT prediction and the DMRG result. This non-universal shift is generically present in the equality given in Eq. (2). This is because this shift can be absorbed by rescaling the speed of sound of the boundary CFT (the latter is set to unity in the computations of A).
Finally, we analyze the variation of the higher entanglement energy levels, $\varepsilon_{N}(k,l)$, in each Virasoro tower [see Fig. 5(c)] as a function of system size. From Eq. (35), for the Virasoro tower built on top each primary field indexed by $k$, $\varepsilon_{N}(k,l)-\varepsilon_{N}(k,0)=\pi l/L$, where $L$ is defined in Eq. (3). Thus, from this dependence we can get the dimension of the descendants indexed by $l$, which should exhibit the degeneracy given by $p(l)$. We do this for $k=0$ (similar results were obtained for other $k$-s and are not shown for brevity). On the other hand, the variation of $\Delta\varepsilon_{N}(k,0)$ vs $1/L$ yields the Luttinger parameter $K$ [see Eq. (35)]. We do this analysis for $k=1$. In order to remove the non-universal effects due to the lattice, we needed to normalize the obtained values of $l$ and $K$ by a non-universal parameter given by the slope of the variation of $\Delta\varepsilon_{N}(0,1)$ vs $l/L$. The latter is not the expected value of $\pi$ as predicted by the CFT computations. This is because the values of $L=\ln(2L_{0}/\pi)$ are quite small despite the overall system size being $L_{0}$ up to 800. Fig. 6(b) shows the expected degeneracies for $l=1,2,3$. Fig. 6(c) shows the variation of $\Delta\varepsilon_{N}(0,2)$ vs $1/L$, which after the normalization process described above yields $l\simeq 1.935$ which is close to the expected value of 2. Finally, Fig. 6(d) shows the variation of $\Delta\varepsilon_{N}(1,0)$ vs $1/L$, which yields a Luttinger parameter of $K\simeq 0.188$ which is close to the value obtained earlier using particle number fluctuations (see Fig. 5).
4 Summary and Perspectives
In this work, we have analytically computed the spectra of the entanglement/modular Hamiltonian of the Ising and the free, compactified boson CFTs in terms of the spectra of corresponding boundary CFTs. The boundary CFTs were analyzed by computing the corresponding partition functions using the relevant Ishibashi states. We compared the analytical predictions for the continuum theory by numerically analyzing corresponding lattice regularized models. In contrast to traditional approaches of using the XXZ chain as a lattice-regularization for the compactified boson CFT, in this work, we analyzed a quantum circuit model using an array of Josephson junctions. While the quantum circuit lattice model is non-integrable to the best of our knowledge, in the long-distance limit, for appropriate choice of parameters, it gives rise to the relevant CFT. The advantage of this approach is that here, we start with lattice degrees of freedom that are directly the discretized, compact bosonic fields being simulated. This avoids nonlocal and nonlinear transformations like Jordan-Wigner transformation and bosonization, necessary for the XXZ or related spin chains, which inevitably do not correctly capture the spectrum of the entanglement Hamiltonian of the CFT. We investigate the lattice model with DMRG. We showed that the CFT and the DMRG predictions are compatible with each other, up to non-universal renormalization of the entanglement spectrum including overall shifts and scale factors. These non-universal effects can be absorbed by rescaling the velocity of sound in the boundary CFT computations.
Exact computation of entanglement Hamiltonians can also be performed in the case when there is a bulk perturbation to the CFT. In particular, consider the case when the perturbation preserves a subset of the infinite set of integrals of motion of a CFT, i.e., the resultant is an integrable quantum field theory [37]. In this case, the problem reduces to the computation of properties of integrable, perturbed, boundary-interacting CFTs [38]. In many cases, this computation is also analytically tractable. As a concrete example, consider the case of the quantum sine-Gordon model with Neumann boundary conditions at the ends. Then, the entanglement spectrum of the quantum sine-Gordon model is given by the spectrum of the boundary sine-Gordon model [39]. The boundary sine-Gordon model has Neumann boundary condition at one end (this is inherited from the original model with bulk perturbation, see Fig. 1) and a cosine potential at the other end (this boundary condition arises from the entanglement cut). The spectrum and the boundary S-matrices for this model are well-known. We aim to analyze the problem in the context of the entanglement Hamiltonian of the quantum sine-Gordon model in the near future.
Finally, we note that the entanglement Hamiltonians provide a fruitful method to investigate the physical spectrum of perturbed CFTs using DMRG. In general, with the latter numerical tool, the computational complexity to evaluate the physical spectrum beyond the first few levels grows rapidly. However, the entanglement spectrum, which is contained in the Schmidt decomposition of the system at various bipartitionings, is the key ingredient of the DMRG analysis and can be evaluated with much more accuracy more easily.
5 Acknowledgments
Discussions with Johannes Hauschild are gratefully acknowledged. FP and AR are funded by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No. 771537). FP acknowledges the support of the DFG Research Unit FOR 1807 through grants no. PO 1370/2-1, TRR80, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germanyâs Excellence Strategy EXC-2111-390814868. HS was supported in part by the Advanced ERC NuQFT.
Appendix A Boundary CFT of the free compactified boson
In this section, we provide a derivation of the boundary states and the partition function of the free compactified boson theory using the general formalism of boundary CFTs [15, 14], extending the calculation done in Ref. [17]. We start with the Euclidean action for the free compactified boson CFT on a finite length interval $L$ given by
$$S=\frac{g}{2}\int_{0}^{T}dt\int_{0}^{L}dx\Big{[}(\partial_{t}\phi)^{2}+(%
\partial_{t}\phi)^{2}\Big{]}.$$
(45)
Periodic boundary condition is imposed in the imaginary time direction, the length of which is given by $T$. The boundary condition at $x=0(L)$ is given by $\alpha(\beta)$ (see Fig. 7).
The compactification radius is $R$, so that $\phi(x,t)$ may be identified by $\phi(x,t)+2\pi R$. The partition function with boundary condition $(\alpha,\beta)$ is given by
$$Z_{\alpha\beta}(q)\equiv{\rm Tr}\ e^{-TH_{\alpha\beta}}={\rm Tr}\ q^{LH_{%
\alpha\beta}/\pi},$$
(46)
where $q\equiv e^{2\pi i\tau}=e^{-\pi T/L}$, where the modular parameter $\tau=iT/2L$. Furthermore, $H_{\alpha\beta}$ is the Hamiltonian of the boundary CFT with boundary conditions $\alpha,\beta$. The spectrum of $H_{\alpha,\beta}$ falls into irreducible representations of the Virasoro algebra [14]
$$Z_{\alpha\beta}(q)=\sum_{i}n^{i}_{\alpha\beta}\chi_{i}(q),$$
(47)
where the Virasoro character $\chi_{i}(q)$ is given by
$$\chi_{i}(q)=q^{-1/24}{\rm Tr}_{i}\ q^{L_{0}},$$
(48)
where we have used the fact that the central charge is 1 and $L_{0}$ is the relevant Virasoro generator. The partition function can equally well be written in the modular transformed picture $\tau\rightarrow-1/\tau$. In this picture,
$$Z_{\alpha\beta}(q)=\langle\alpha|e^{-L\bar{H}}|\beta\rangle,$$
(49)
where $\bar{H}$ is the Hamiltonian of the interval $T$ with periodic boundary conditions and $|\alpha,\beta\rangle$ are the corresponding boundary states. Below, we compute these boundary states and evaluate this partition function. It is convenient to map the cylinder to the plane using $z=e^{T(t-ix)/2\pi}$. Then,
$$\bar{H}=\frac{2\pi}{T}\Big{(}L_{0}^{z}+\bar{L}_{0}^{z}-\frac{1}{12}\Big{)},$$
(50)
where we have labeled the Virasoro generators in the $z$-plane by a superscript to emphasize the fact that on the $z$-plane, the holomorphic and the anti-holomorphic components propagate separately,
which is different from those defined in Eq. (47). This leads to
$$Z_{\alpha,\beta}(q)=\langle\alpha|(\tilde{q}^{1/2})^{L_{0}^{z}+\bar{L}^{z}_{0}%
-1/12}|\beta\rangle,\ \tilde{q}\equiv e^{-4\pi L/T}.$$
(51)
Expanding in normal modes (see Chap. 6.3.5 of Ref. [14]), we can write
$$\displaystyle\phi(z,\bar{z})$$
$$\displaystyle=\phi_{0}-i\Big{(}\frac{n}{4\pi gR}+\frac{mR}{2}\Big{)}\ln z$$
(52)
$$\displaystyle\quad-i\Big{(}\frac{n}{4\pi gR}-\frac{mR}{2}\Big{)}\ln\bar{z}$$
$$\displaystyle\quad+\frac{i}{\sqrt{4\pi g}}\sum_{k\neq 0}\frac{a_{k}}{k}z^{-k}+%
\frac{i}{\sqrt{4\pi g}}\sum_{k\neq 0}\frac{\bar{a}_{k}}{k}\bar{z}^{-k},$$
where $m$ is the winding number and $n$ is the quantization of the zero-mode momenta. Here, we have defined
$$\displaystyle a_{k}$$
$$\displaystyle=-i\sqrt{k}\tilde{a}_{k},\ k>0,$$
(53)
$$\displaystyle=i\sqrt{-k}\tilde{a}_{-k}^{\dagger},\ k<0,$$
$$\displaystyle\bar{a}_{k}$$
$$\displaystyle=-i\sqrt{k}\tilde{a}_{-k},\ k>0,$$
(54)
$$\displaystyle=i\sqrt{-k}\tilde{a}_{k}^{\dagger},\ k<0,$$
(55)
where $\tilde{a}_{k}$ are the original bosonic operators satisfying $[\tilde{a}_{k},\tilde{a}_{l}]=0$, $[\tilde{a}_{k},\tilde{a}_{l}^{\dagger}]=\delta_{kl}$.
Straightforward computations show that
$$\displaystyle L_{k}$$
$$\displaystyle=\frac{1}{2}\sum_{l}:a_{l}a_{k-l}:,\ \bar{L}_{k}=\frac{1}{2}\sum_%
{l}:\bar{a}_{l}\bar{a}_{k-l}:,$$
(56)
where we have defined
$$\displaystyle a_{0},\bar{a}_{0}=\sqrt{4\pi g}\Big{(}\frac{n}{4\pi gR}\pm\frac{%
mR}{2}\Big{)}.$$
(57)
The Hamiltonian, $\bar{H}$, is given by
$$\displaystyle\bar{H}$$
$$\displaystyle=\frac{2\pi}{T}\Big{[}\Big{(}\frac{n^{2}}{4\pi gR^{2}}+m^{2}R^{2}%
\pi g\Big{)}-\frac{1}{12}$$
(58)
$$\displaystyle\qquad+\sum_{k>0}k\big{(}a_{-k}a_{k}+\bar{a}_{-k}\bar{a}_{k}\big{%
)}\Big{]}$$
In the boundary CFT, a boundary state, $|B\rangle$, satisfies the Ishibashi condition: [40]
$$(L_{k}-\bar{L}_{-k})|B\rangle=0,\ \forall k.$$
(59)
For the compactified boson, Ishibashi conditions for the the Dirichlet (D) and Neumann (N) boundary conditions are satisfied if: [17]
$$\displaystyle\Big{[}a_{k}-(+)\bar{a}_{-k}\Big{]}|D(N)\rangle=0,$$
(60)
where we have denoted the corresponding boundary states by $|D(N)\rangle$. The boundary states can be constructed by applying appropriate operators on the vacua labeled by the zero-mode indices $(n,m)$. Since the Dirichlet (Neumann) states have to satisfy the Ishibashi condition [Eq. (59)] for $k=0$, this implies the Dirichlet (Neumann) states are built on vacua labeled by $|n,0\rangle$ ($|0,m\rangle$). By using the bosonic commutation relations, it is easy to see that
$$\displaystyle|D\rangle$$
$$\displaystyle=\sum_{n}c_{n}{\rm{exp}}\Big{[}-\sum_{k>0}\tilde{a}_{-k}^{\dagger%
}\tilde{a}_{k}^{\dagger}\Big{]}|n,0\rangle$$
(61)
$$\displaystyle|N\rangle$$
$$\displaystyle=\sum_{m}d_{m}{\rm{exp}}\Big{[}+\sum_{k>0}\tilde{a}_{-k}^{\dagger%
}\tilde{a}_{k}^{\dagger}\Big{]}|0,m\rangle,$$
(62)
where $c_{n},d_{m}$ are coefficients that need to be determined. The overall normalization of the two states are fixed by imposing the Cardy consistency condition that exactly one dimension zero character. [15] We directly provide the boundary states following Ref. [17] and check for the consistency afterwards 555We correct some errors in the cited reference.. The Dirichlet and Neumann boundary states are given by
$$\displaystyle|D(\phi_{0})\rangle$$
$$\displaystyle=\frac{1}{\sqrt{2R\sqrt{\pi g}}}\sum_{n}e^{-\frac{in\phi_{0}}{R%
\sqrt{\pi g}}}{\rm{exp}}\Big{[}-\sum_{k>0}\tilde{a}_{-k}^{\dagger}\tilde{a}_{k%
}^{\dagger}\Big{]}|n,0\rangle,$$
(63)
$$\displaystyle|N(\tilde{\phi}_{0})\rangle$$
$$\displaystyle=\sqrt{R\sqrt{\pi g}}\sum_{m}e^{-\frac{im\tilde{\phi}_{0}R}{2%
\sqrt{\pi g}}}{\rm{exp}}\Big{[}+\sum_{k>0}\tilde{a}_{-k}^{\dagger}\tilde{a}_{k%
}^{\dagger}\Big{]}|0,m\rangle,$$
(64)
where we have also used the duality between Neumann and Dirichlet boundary conditions: $R\leftrightarrow 2/R$ and $\tilde{\phi}_{0}$ is the field dual to $\phi_{0}$. It is easy to check that under Dirichlet (Neumann) boundary conditions, the field $\phi(\tilde{\phi})$ is pinned to the value $\phi_{0}(\tilde{\phi}_{0})$ at the boundary. Next, we compute the partition functions for different combinations of boundary conditions.
A.1 Dirichlet-Dirichlet boundary condition
Consider the case when Dirichlet boundary conditions are imposed both at $x=0$ and $x=L$. Denote the corresponding boundary states by $|D(\phi_{0})\rangle,|D(\phi_{0}^{\prime})\rangle$. Then,
$$\displaystyle Z_{\rm{DD}}(q)$$
$$\displaystyle=\langle D(\phi_{0})|e^{-L\bar{H}}|D(\phi_{0}^{\prime})\rangle$$
(65)
$$\displaystyle=\frac{1}{2R\sqrt{\pi g}}\sum_{n}e^{\frac{in\Delta\phi_{0}}{R%
\sqrt{\pi g}}}e^{-\frac{2\pi L}{T}\big{(}\frac{n^{2}}{4\pi gR^{2}}-\frac{1}{12%
}\big{)}}\prod_{k>0}\frac{1}{1-e^{-\frac{4\pi Lk}{T}}}$$
$$\displaystyle=\frac{1}{2R\sqrt{\pi g}}\frac{1}{\eta(\tilde{q})}\sum_{n}e^{%
\frac{in\Delta\phi_{0}}{R\sqrt{\pi g}}}\tilde{q}^{\frac{n^{2}}{8\pi gR^{2}}},$$
where $\Delta\phi_{0}=\phi_{0}-\phi_{0}^{\prime}$ and $\eta(q)$ is the Dedekind function defined in Eq. (12). In order to express the result in terms of $q$, we use the following relation [41]:
$$\eta(\tilde{q})=\sqrt{\frac{T}{2L}}\eta(q)$$
(66)
and the Poisson summation formula:
$$\sum_{n}e^{-\pi an^{2}+bn}=\frac{1}{\sqrt{a}}\sum_{k}e^{-\frac{\pi}{a}\big{(}k%
+\frac{b}{2i\pi}\big{)}^{2}}.$$
(67)
After simple manipulations, we arrive at
$$Z_{\rm{DD}}(q)=\frac{1}{\eta(q)}\sum_{k}\big{(}q^{2\pi gR^{2}}\big{)}^{\big{(}%
k+\frac{\Delta\phi_{0}}{2\pi R\sqrt{\pi g}}\big{)}^{2}}.$$
(68)
Now, consider the case when $\Delta\phi_{0}=0$. For this case,
$$\displaystyle Z_{\rm{DD}}(q)$$
$$\displaystyle=\frac{1}{\eta(q)}\sum_{k}q^{2\pi gR^{2}k^{2}}$$
(69)
$$\displaystyle=\sum_{h}n^{h}_{\rm{DD}}\chi_{h}(q)=\sum_{h}n^{h}_{\rm{DD}}\frac{%
q^{h}}{\eta(q)},$$
where we have used Eq. (47) and the definition of $\eta(q)$. From the above equation, we see that $n^{h=0}_{\rm{DD}}=1$, as expected from the Cardy consistency relations [15]. The other primary fields have dimensions $2\pi gR^{2}k$ where $k\neq 0$.
Using the explicit expression of $\eta(q)$:
$$\displaystyle\frac{1}{\eta(q)}\equiv q^{-1/24}\frac{1}{\varphi(q)}\equiv\sum_{%
l\geq 0}p(l)q^{-1/24+l},$$
(70)
where $p(l)$ is the number of ways to partition the integer $l$, we arrive at
$$\displaystyle Z_{\rm{DD}}(q)$$
$$\displaystyle={\rm Tr}\ q^{LH_{\alpha,\beta}/\pi}$$
(71)
$$\displaystyle=\sum_{k}\sum_{l\geq 0}p(l)q^{-1/24+2\pi gR^{2}k^{2}+l}.$$
Thus, the boundary CFT has the spectrum given by
$$\epsilon_{\rm{DD}}(k,l)=\frac{\pi}{L}\Big{(}-\frac{1}{24}+2\pi gR^{2}k^{2}+l%
\Big{)},\ k\in\mathbb{Z},l\geq 0,$$
(72)
with degeneracy $p(l)$. The spectrum can be thought of as composed of towers on each primary field with dimension $2\pi gR^{2}k$ and the descendants being indexed by $l$. For each tower, the descendants at level $l$ have degeneracy $p(l)$. The compactification radius is related to the Luttinger parameter $K$ by
$$R=\frac{1}{\sqrt{\pi K}}.$$
(73)
Furthermore, it is convenient to choose $g=1$ as the normalization of the free-boson action. Then, the spectrum is given by
$$\epsilon_{\rm{DD}}(k,l)=\frac{\pi}{L}\Big{(}-\frac{1}{24}+\frac{2}{K}k^{2}+l%
\Big{)},\ k\in\mathbb{Z},l\geq 0.$$
(74)
A.2 Neumann-Neumann boundary conditions
Now, consider the case when Neumann boundary conditions is imposed on both ends $x=0,L$. Denote the boundary states by $|N(\tilde{\phi}_{0})\rangle,|N(\tilde{\phi}_{0}^{\prime})\rangle$. Similar calculation as in the Dirichlet case leads to
$$Z_{\rm{NN}}(q)=\frac{R\sqrt{\pi g}}{\eta(\tilde{q})}\sum_{m}e^{\frac{imR\Delta%
\tilde{\phi}_{0}}{2\sqrt{\pi g}}}\tilde{q}^{\frac{m^{2}R^{2}\pi g}{2}},$$
(75)
which can be rewritten in terms of $q$ as
$$Z_{\rm{NN}}(q)=\frac{1}{\eta(q)}\sum_{m}\big{(}q^{\frac{1}{2R^{2}\pi g}}\big{)%
}^{\big{(}m+\frac{R\Delta\tilde{\phi}_{0}}{4\pi\sqrt{\pi g}}\big{)}^{2}}.$$
(76)
For $\Delta\tilde{\phi}_{0}=0$, using Eq. (73), we get
$$\displaystyle Z_{\rm{NN}}(q)$$
$$\displaystyle=\frac{1}{\eta(q)}\sum_{k}q^{\frac{K}{2g}k^{2}}=\sum_{h}n^{h}_{%
\rm{NN}}\frac{q^{h}}{\eta(q)}$$
(77)
$$\displaystyle=\sum_{k}\sum_{l\geq 0}p(l)q^{-\frac{1}{24}+\frac{Kk^{2}}{2g}+l}$$
(78)
We see that the Cardy consistency condition is again satisfied: $n^{h=0}_{\rm{NN}}=1$ and the spectrum of the boundary CFT is given by
$$\epsilon_{\rm{NN}}(k,l)=\frac{\pi}{L}\Big{(}-\frac{1}{24}+\frac{K}{2}k^{2}+l%
\Big{)},\ k\in\mathbb{Z},l\geq 0$$
(79)
with degeneracy $p(l)$.
A.3 Dirichlet-Neumann boundary conditions
Finally, we consider the case when Neumann boundary condition is imposed on the end $x=0$ and Dirichlet on the end $x=L$. In this case, the partition function is computed by evaluating $\langle D(\phi_{0})|e^{-L\bar{H}}|N(\tilde{\phi}_{0})\rangle$. The rest of the steps are identical as in the previous cases and lead to
$$\displaystyle Z_{\rm DN}(q)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\sqrt{2}\eta(\tilde{q})}\sum_{n}(-1)^{n}\tilde{q}^{n^{2}}$$
(80)
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\eta(q)}\sum_{k}q^{\frac{1}{4}\big{(}k+\frac{1}{2}\big{%
)}^{2}}$$
(81)
$$\displaystyle=$$
$$\displaystyle\frac{q^{1/48}}{2\varphi(q)}\sum_{k}q^{\frac{k^{2}+k}{4}}=q^{1/48%
}\sum_{n\geq 0}q^{n/2}p_{\sigma}(n)$$
(82)
where $p_{\sigma}(n)$ are the degeneracy factors that occur in the Ising CFT with Dirichlet boundary conditions, given in Eq. (21).
Thus, the spectrum in this case is given by
$$\epsilon_{\rm{DN}}(n)=\frac{\pi}{L}\Big{(}\frac{1}{48}+\frac{n}{2}\Big{)},$$
(83)
where $n\geq 0$ and the degeneracies are given by $p_{\sigma}(n)$ at level $n$.
Appendix B Phase-diagram of the quantum circuit model
In this section, we analyze the phase-diagram of the quantum circuit model which, for appropriate choice of parameters, provides as a lattice-regularization of the free, compactified boson CFT. To that end, we start with the Hamiltonian of the model, given by (see Sec. 3.2.1)
$$\displaystyle H_{\rm{array}}$$
$$\displaystyle=$$
$$\displaystyle E_{C_{0}}\sum_{i=1}^{L}n_{i}^{2}+\delta E_{C_{0}}\sum_{i=1}^{L-1%
}n_{i}n_{i+1}-E_{g}\sum_{i=1}^{L}n_{i}$$
(84)
$$\displaystyle\quad-E_{J}\sum_{i=1}^{L-1}\cos(\phi_{i}-\phi_{i+1}),$$
where compared to Eq. (43), we have included an additional term corresponding to the gate-voltage at each superconducting island. This model can be viewed as a generalized Bose-Hubbard model in the limit of high-occupancy of the sites, where the role of bosons in played by Cooper-pairs. We work with the case when there is no disorder in the system. The charging energy terms (proportional to $E_{C_{0}}$) corresponds to onsite and nearest-neighbor repulsion terms, while the gate-voltage plays the role of the chemical potential. Finally, the Josephson tunneling term (proportional to $E_{J}$) gives rise to nearest-neighbor hopping. Note a crucial difference with the conventional Bose-Hubbard model. Here, $n_{i}$-s can be both positive and negative. Physically, $n_{i}$ corresponds to the excess number of Cooper-pairs on the $i^{\rm th}$ island. Thus, a negative $n_{i}$ would correspond to the removal of $|n_{i}|$ Cooper-pairs from the condensate on the $i^{\rm th}$ island. The phase-diagram of this model has been analyzed using perturbative analytical methods [20, 42]. In what follows, we analyze the phase-diagram using DMRG generalizing the methods described in Ref. [43]. The local Hilbert space at each site was taken to be 9: $n_{i}=-4,-3,\ldots,3,4$. Furthermore, we chose $\delta=0.2$.
The phase-diagram obtained using DMRG is shown in Fig. 8(a). Within the maroon lobes, the system is in a Mott-insulating (MI) phase. In this phase, the occupation of Cooper-pairs at each site is pinned to an integer, as shown in Fig. 8(b). We only show the lobes for $\rho=0,1$ for brevity. Note that, in contrast to the conventional Bose-Hubbard model, the lobes extend in the negative $E_{g}/E_{C_{0}}$ regime because $n_{i}$-s can be negative. In addition to the MI lobes, the system can also be in a charge-density-wave (CDW) phase, that occurs in between two successive MI phases. This phase occurs due to the presence of nearest-neighbor repulsion in the model [$\delta\neq 0$ in Eq. (84)]. In this phase, the average densities of Cooper-pairs are half-integers, the case of $\rho=1/2$ is shown in dark-orange in Fig. (8)(a,b), where the system shows an alternating Neel order for the Cooper-pair occupation, given by $010101\ldots$. Changing $E_{J}/E_{C_{0}}$ or $E_{g}/E_{C_{0}}$ causes the system to undergo a phase-transition into a charge-$2e$ Luttinger liquid (LL) phase, where the system is described by the free, compactified boson CFT.
The compactification radius is determined by the Luttinger parameter $K$ [see Eq. (73)]. The latter determines the exponent of algebraic decay of correlations of the Cooper-pair creation and annihilation operators:
$$\langle e^{i\phi(0)}e^{-i\phi(r)}\rangle\sim\frac{1}{|r|^{K/2}}$$
(85)
The transition through the tip of the lobes occurs at constant density of Cooper-pairs and is of the type Kosterlitz-Thouless. The location of the tip of the lobe was computed by locating the location where the $K$ crossed $1/2(2)$ for the MI(CDW) lobes. The phase-transition across the sides of the lobes occur with a change in the Cooper-pair density. The location of the sides of the lobe was computed by computing the values of $E_{J}$ for which the cost of adding/removing a particle is zero [43].
This characteristic decay in the LL phase and the obtained values of $K$ are shown in Fig. 8(c) for two points, indicated by maroon and dark-orange circles in panel (a). The computation of the entanglement spectrum in the main-text was done for $E_{g}=0,E_{J}/E_{C_{0}}=8$.
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Automatic Argumentative-Zoning Using Word2vec
Haixia Liu
School Of Computer Science, University of Nottingham Malaysia Campus, Jalan Broga, 43500 Semenyih, Selangor Darul Ehsan.
11email: [email protected]
Abstract
In comparison with document summarization on the articles from social media and newswire, argumentative zoning (AZ) is an important task in scientific paper analysis. Traditional methodology to carry on this task relies on feature engineering from different levels. In this paper, three models of generating sentence vectors for the task of sentence classification were explored and compared. The proposed approach builds sentence representations using learned embeddings based on neural network. The learned word embeddings formed a feature space, to which the examined sentence is mapped to. Those features are input into the classifiers for supervised classification. Using 10-cross-validation scheme, evaluation was conducted on the Argumentative-Zoning (AZ) annotated articles. The results showed that simply averaging the word vectors in a sentence works better than the paragraph to vector algorithm and by integrating specific cuewords into the loss function of the neural network can improve the classification performance. In comparison with the hand-crafted features, the word2vec method won for most of the categories. However, the hand-crafted features showed their strength on classifying some of the categories.
1 Introduction
One of the crucial tasks for researchers to carry out scientific investigations is to detect existing ideas that are related to their research topics. Research ideas are usually documented in scientific publications. Normally, there is one main idea stated in the abstract, explicitly presenting the aim of the paper. There are also other sub-ideas distributed across the entire paper. As the growth rate of scientific publication has been rising dramatically, researchers are overwhelmed by the explosive information. It is almost impossible to digest the ideas contained in the documents emerged everyday. Therefore, computer assisted technologies such as document summarization are expected to play a role in condensing information and providing readers with more relevant short texts. Unlike document summarization from news circles, where the task is to identify centroid sentences [1] or to extract the first few sentences of the paragraphs [2], summarization of scientific articles involves extra text processing stage [3]. After highest ranked texts are extracted, rhetorical status analysis will be conducted on the selected sentences. Rhetorical sentence classification, also known as argumentative zoning (AZ) [4], is a process of assigning rhetorical status to the extracted sentences. The results of AZ provide readers with general discourse context from which the scientific ideas could be better linked, compared and analyzed. For example, given a specific task, which sentences should be shown to the reader is related to the features of the sentences. For the task of identifying a paper’s unique contribution, sentences expressing research purpose should be retrieved with higher priority. For comparing ideas, statements of comparison with other works would be more useful. Teufel et. al. [3] introduced their rhetorical annotation scheme which takes into account of the aspects of argumentation, metadiscourse and relatedness to other works. Their scheme resulted seven categories of rhetorical status and the categories are assigned to full sentences. Examples 111These texts were randomly selected from Argumentative Zoning Corpus, which is described in dataset section. of human annotated sentences with their rhetorical status are shown in Table. 1. The seven categories are aim, contrast, own, background, other, basis and textual.
Analyzing the rhetorical status of sentences manually requires huge amount of efforts, especially for structuring information from multiple documents. Fortunately, computer algorithms have been introduced to solve this problem. With the development of artificial intelligence, machine learning and computational linguistics, Natural Language Processing (NLP) has become a popular research area [5, 6]. NLP covers the applications from document retrieval, text categorization [7], document summarization [8] to sentiment analysis [9, 10]. Those applications are targeting different types of text resources, such as articles from social media [11] and scientific publications [3]. There are several approaches to tackle these tasks. From machine learning prospective, text can be analysed via supervised [3], semi-supervised [12] and unsupervised [13] algorithms.
Document summarization from social media and news circles has received much attention for the past decades. Those problems have been addressed from many angles, one of which is feature extraction and representation. At the early stage of document summarization, features are usually engineered manually. Although the hand-crafted features have shown the ability for document summarization and sentiment analysis [14, 10], there are not enough efficient features to capture the semantic relations between words, phrases and sentences. Moreover, building a sufficient pool of features manually is difficult, because it requires expert knowledge and it is time-consuming. Teufel et. al. [3] have built feature pool of sixteen types of features to classify sentences, such as the position of sentence, sentence length and tense. Widyantoro et. al. used content features, qualifying adjectives and meta-discourse features [15] to explore AZ task. It took efforts to engineer these features and it is also time consuming to optimize the combination of the entire features.
With the advent of neural networks [16], it is possible for computers to learn feature representations automatically. Recently, word embedding technique [17] has been widely used in the NLP community. There are plenty of cases where word embedding and sentence representations have been applied to short text classification [18] and paraphrase detection [19]. However, the effectiveness of this technique on AZ needs further study. The research question is, is it possible to extract word embeddings as features to classify sentences into the seven categories mentioned above using supervised machine learning approach?
2 Related Work
The tool of word2vec proposed by Mikolov et al. [17] has gained a lot attention recently. With word2vec tool, word embeddings can be learnt from big amount of text corpus and the semantic relationships between words can be measured by the cosine distances between the vectors. The idea behind word embeddings is to use distributed representation [20] to map each word into k-dimension vector. How these vectors are generated using word2vec tool? The common method to derive the vectors is using neural probabilistic language model [21]. The underlying word representations for each word are obtained while training the language model. Similar to the mechanism in language model, Mikolov et al. [17] introduced two architectures: Skip-gram model and continuous bag of words (CBOW) model. Each of the model has two different training strategies, such as hierarchical softmax and negative sampling. Both these two models have three layers: input, projection and output layer. The word vectors are obtained once the models are optimized. Usually, this optimizing process is done using stochastic gradient descent method. It doesn’t need labels when training the models, which makes word2vec algorithm more valuable compared with traditional supervised machine learning methods that require a big amount of annotated data. Given enough text corpus, the word2vec can generate meaningful representations.
Word2vec has been applied to sentiment analysis [22, 23, 24] and text classification [25]. Sadeghian and Sharafat [26] explored averaging of the word vectors in a sentiment review statement. Their results indicated that word2vec models significantly outperform the vanilla bag-of-words model. Amongst the word2vec based models, softmax provides the best form of classification. Tang et al. [22] used the concatenation of vectors derived from different convolutional layers to analyze the sentiment statements. They also trained sentiment-specific word embeddings to improve the twitter sentiment classification results. This work is aiming at learning word embeddings for the task of AZ. The results were compared from three aspects: the impact of the training corpus, the effectiveness of specific word embeddings and different ways of constructing sentence representations based on the learned word vectors.
Le and Mikolov [27] introduced the concept of word vector representation in a formal way:
Given a sequence of training words $w=<w_{1},x_{2},...,w_{n}>$, the objective of the word2vec model is to maximize the average log probability:
$\frac{1}{T}$ $\sum^{T-k}_{t=k}$ $log$ p$(w_{t}|w_{t-k},...,w_{t+k})$ (1)
Using softmax technique, the prediction can be formalized as:
p$(w_{t}|w_{t-k},...,w_{t+k})$ = $\frac{e^{y_{w_{t}}}}{\sum e^{y_{w_{y}}}}$ (2)
Each of $y_{i}$ is un-normalized log probability for each output word $i$:
$y=b+Uh(w_{t-k},...,w_{t+k};W)$ (3)
3 Methodology
3.1 Models
In this study, sentence embeddings were learned from large text corpus as features to classify sentences into seven categories in the task of AZ. Three models were explored to obtain the sentence vectors: averaging the vectors of the words in one sentence, paragraph vectors and specific word vectors.
The first model, averaging word vectors ($AVGWVEC$), is to average the vectors in word sequence $w=<w_{1},x_{2},...w_{n}>$. The main process in this model is to learn the word embedding matrix $W_{w}$:
$V_{avgwvec}(w)=$ $\frac{1}{n}$ $\sum$ $W^{x_{i}}_{w}$ (4)
where $W_{w}$ is the word embedding for word $x_{i}$, which is learned by the classical word2vec algorithm [17].
The second model, $PARAVEC$, is aiming at training paragraph vectors. It is also called distributed memory model of paragraph vectors (PV-DM) [27], which is an extension of word2vec. In comparison with the word2vec framework, the only change in PV-DM is in the equation (3), where $h$ is constructed from $W$ and $D$, where matrix $W$ is the word vector and $D$ holds the paragraph vectors in such a way that every paragraph is mapped to a unique vector represented by a column in matrix $D$.
The third model is constructed for the purpose of improving classification results for a certain category. In this study specifically, the optimization task was focused on identifying the category $BAS$ 222This is a general case to show how to improve the classification result by integrating cuewords to the embeddings.. In this study, $BAS$ specific word embeddings were trained ($BSWE$) inspired by Tang et al. [22]’s model: Sentiment-Specific Word Embedding (unified model: $SSWE_{u}$). After obtaining the word vectors via $BSWE$, the same scheme was used to average the vectors in one sentence as in the model $AVGWVEC$.
3.2 Classification and evaluation
The learned word embeddings are input into a classifier as features under a supervised machine learning framework. Similar to sentiment classification using word embeddings [22], where they try to predict each tweet to be either positive or negative, in the task of AZ, the embeddings are used to classify each sentence into one of the seven categories.
To evaluate the classification performance, precision, recall and F-measure were computed.
4 Experimental Evaluation
4.1 Training Dataset
$ACL$ collection. ACL Anthology Reference Corpus 333$http://acl$-$arc.comp.nus.edu.sg/$ contains the canonical 10,921 computational linguistics papers, from which 622,144 sentences were generated after filtering out sentences with lower quality.
$MixedAbs$ collection contains 6,778 sentences, extracted from the titles and abstracts of publications provided by WEB OF SCIENCE 444$webofknowledge.com$.
4.2 Test Dataset
Argumentative Zoning Corpus ($AZ$ corpus) consists of 80 AZ$-$annotated conference articles in computational linguistics, originally drawn from the Cmplg arXiv. 555$http://www.cl.cam.ac.uk/$~$sht25/AZ\_corpus.html$. After Concatenating sub-sentences, 7,347 labeled sentences were obtained.
4.3 Training strategy
To compare the three models effectiveness on the AZ task, the three models on a same ACL dataset (introduced int he dataset section) were trained. The word2vec were also trained using different parameters, such as different dimension of features. To evaluate the impact from different domains, the first model was trained on different corpus.
The characteristics of word embeddings based on different model and dataset are listed in Table. 2.
4.4 Parameters
Inspired by the work from Sadeghian and Sharafat [26] 666$https://www.kaggle.com/c/word2vec-nlp-tutorial/details/part-2-word-vectors$, the word to vector features were set up as follows: the Minimum word count is 40; The number of threads to run in parallel is 4 and the context window is 10.
4.5 Strategy of dealing with unbalanced data
In imbalanced data sets, some classes are significantly outnumbered by other classes [28], which affects the classification results. In this experiment, the test dataset is an imbalanced data set. Table. 3 shows the distribution of rhetorical categories from the $AZ$ test dataset. The categories OWN and OTH are significantly outnumbering other categories.
To deal with the problem of classification on unbalanced data, synthetic Minority Over-sampling TEchnique (SMOTE) [29] were performed on the original dataset. 10-cross validation scheme was adopted and the results were averaged from 10 iterations.
4.6 Results of classification for per category
Table. 4 and 5 show the classification performance of different methods.
777Note that it is not completely compatible with Teufel 2002 results, since the dataset is different due to the sentence concatenation in this paper. But Teufel’s reports could be a reference.
The results were examined from the following aspects:
When the feature dimension is set to 100 and the training corpus is ACL, the results generated by different models were compared (AVGWVEC,
PARAVEC and AVGWVEC+BSWE for BAS category only). Looking at the F-measure, AVGWVEC performs better than PARAVEC, but PARAVEC gave a better precision results on several categories, such as AIM, CTR, TXT and OWN. The results showed that PARAVEC model is not robust, for example, it performs badly for the category of BAS. For specific category classification, take the BAS category for example, the BSWE model outperforms others in terms of F-measure.
When the model is fixed to AVGWVEC and the training corpus is ACL, the feature size impact (300 and 100 dimensions) was investigated. From the F-measure, it can be seen that for some categories, 300-dimension features perform better than the 100-dimension ones, for example, CTR and BKG, but they are not as good as 100-dimension features for some categories, such as BAS.
When the model is set to AVGWVEC and the feature dimension is 100, the results computed from different training corpus were compared (ACL+AZ, MixedAbs and Brown corpus). ACL+AZ outperforms others and brown corpus is better than MixedAbs for most of the categories, but brown corpus is not as good as MixedAbs for the category of OWN.
Finally, the results were compared between word embeddings and the methods of cuewords, Teufel 2002 and baseline. To evaluate word embeddings on AZ, the model AVGWVEC trained on ACL+AZ was used for the comparison. It can be seen from the table. 4, the model of word embeddings is better than the method using cuewords matching. It also outperforms Teufel 2002 for most of the cases, except AIM, BAS and OWN. It won baseline for most of the categories, except OWN.
5 Discussion
The classification results showed that the type of word embeddings and the training corpus affect the AZ performance. As the simple model, $AVGWVEC$ performs better than others, which indicate averaging the word vectors in a sentence can capture the semantic property of statements. By training specific argumentation word embeddings, the performance can be improved, which can be seen from the case of detecting BAS status using $BSWE$ model.
Feature dimension doesn’t dominate the results. There is no significant difference between the resutls generated by 300-dimension of features and 100 dimensions.
Training corpus affects the results. ACL+AZ outperforming others indicates that the topics of the training corpus are important factors in argumentative zoning. Although Brown corpus has more vocabularies, it doesn’t win ACL+AZ.
In general, the classification performance of word embeddings is competitive in terms of F-measure for most of the categories. But for classifying the categories AIM, BAS and OWN, the manually crafted features proposed by Teufel et al. [3] gave better results.
6 Conclusion
In this paper, different word embedding models on the task of argumentative zoning were compared . The results showed that word embeddings are effective on sentence classification from scientific papers. Word embeddings trained on a relevant corpus can capture the semantic features of statements and they are easier to be obtained than hand engineered features.
To improve the sentence classification for a specific category, integrating word specific embedding strategy helps. The size of the feature pool doesn’t matter too much on the results, nor does the vocabulary size. In comparison, the domain of the training corpus affects the classification performance.
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Spin splitting with persistent spin textures induced by the line defect in 1T-phase of monolayer transition metal dichalcogenides
Moh. Adhib Ulil Absor
[email protected]
Department of Physics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Sekip Utara BLS 21 Yogyakarta 55186, Indonesia.
Iman Santoso
Department of Physics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Sekip Utara BLS 21 Yogyakarta 55186, Indonesia.
Naoya Yamaguchi
Nanomaterial Reserach Institute (NANOMARI), Kanazawa University, 920-1192 Kanazawa, Japan.
Fumiyuki Ishii
Nanomaterial Reserach Institute (NANOMARI), Kanazawa University, 920-1192 Kanazawa, Japan.
(November 22, 2020)
Abstract
The spin splitting driven by spin-orbit coupling in monolayer transition metal dichalcogenides (TMDCs) family has been widely studied only for the 1H-phase structure, while it is not profound for the 1T-phase structure due to the centrosymmetric of the crystal. On the basis of first-principles calculations, we show that significant spin splitting can be induced in the 1T-phase of monolayer TMDCs by introducing the line defect. Taking the monolayer PtSe${}_{2}$ as a representative example, we considere the most stable formation of the line defects, namely Se-vacancy line defect (Se-VLD). We find that large spin splitting is observed in the defect states of the Se-VLD, exhibiting a highly unidirectional spin configuration in the momentum space. This peculiar spin configuration may yield the so-called ”persistent spin textures (PST)”, a specific spin mode resulting in a protection against spin-decoherence and supporting an extraordinarily long spin lifetime. Moreover, by using $\vec{k}\cdot\vec{p}$ perturbation theory supplemented with symmetry analysis, we clarified that the emerging of the spin splitting maintaining the PST in the defect states is originated from the inversion symmetry breaking together with one dimensional nature of the Se-VLD engineered monolayer PtSe${}_{2}$. Our finding pave a possible way to induce the significant spin splitting in the 1T-phase of the monolayer TMDCs, which could be of highly important for designing spintronic devices.
††preprint: APS/123-QED
I INTRODUCTION
Since the experimental isolation of graphene in 2004Novoselov et al. (2004), significant research efforts have been devoted to the investigation of two-dimensional (2D) materials with atomically-thin crystalsTan et al. . Here, growing research attention has been focused on monolayer transition metal dichalcogenides (TMDCs) family due to a high possibility to be used in the future nanoelectronic devicesManzeli et al. (2017). Most of the monolayer TMDCs families have a graphene-like hexagonal crystal structures where transition metal atoms ($M$) are sandwiched between layers of chalcogen atoms ($X$) with $MX_{2}$ stoichiometry. However, due to the local coordination of the transition metal atoms, they admit two different stable formations in the ground state, namely a 1H-phase structure having trigonal prismatic symmetry, and a 1T-phase structure that consists of distorted octahedral symmetry Cudazzo et al. (2014). The different coordination environments in the monolayer TMDCs lead to distinct crystal field splitting of the $d$-like bands, thus induces very different electronic properties such as a direct semiconductor for 1H-phase as recently predicted on WS${}_{2}$ Absor et al. (2016) and an indirect semiconductor for 1T-phase as recently reported on PtSe${}_{2}$ Yao et al. (2017); Wang et al. (2015). Furthermore, various physical properties such as tunability of band gap by electric field and strain effects Absor et al. (2016); Ramasubramaniam et al. (2011), high carrier mobilityWang et al. (2015); Zhang et al. (2014), and superior surface reactivity Chia et al. (2016) are established, evidenting that the monolayer TMDCs is an ideal platform for next-generation technologies.
One of the important feature found in the monolayer TMDCs is the strong spin-orbit coupling (SOC), which is mainly noticeable in the 1H-phase such as molybdenum and tungsten dichalcogenides (Mo/W$X_{2}$, $X$=S,Se)Zhu et al. (2011); Liu et al. (2013); Absor et al. (2016). Here, the absence of inversion symmetry in the crystal structure together with the strong SOC in the 5$d$ orbitals of transition metal atoms leads to the large spin splitting in the electronic band structures, which is believed to be responsible for inducing some of interesting phenomena such as spin Hall effect Cazalilla et al. (2014), spin-dependent selection rule for optical transitions Chu et al. (2014), and magneto-electric effect in the monolayer TMDCs Z.Gong et al. (2013). Moreover, due to the presence of the in-plane mirror symmetry in the crystral structures of the monolayer TMDCs, a unidirectional out-of-plane spin polarization is preserved in the spin-split bands around the $K$ point in the first Brillouin zone (FBZ), resulting in a Zeeman-type spin splittingZhu et al. (2011); Bragança et al. (2019), which is predicted to exhibit a highly spin coherence and a long-lived spin relaxation of electrons Bragança et al. (2019). Furthermore, an electrically controllable spin splitting and spin polarization in the monolayer TMDCs has been reported, making the monolayer TMDCs is suitable for spintronics devices such as spin-field effect transistor (SFET)Radisavljevic et al. (2011).
Compared with the 1H-phase of the monolayer TMDCs, the effect of SOC in the 1T-phase is equally interesting. Especially monolayer PtSe${}_{2}$ attracted much scientific attentions since it has been successfully synthesized by a direct selenization at the Pt(111) substrateWang et al. (2015); Yao et al. (2017). Recently, Yao $et$. $al$., by using spin- and angle-resolved photoemission spectroscopy (spin-ARPES), reported spin-layer locking phenomena in the monolayer 1T-PtSe${}_{2}$, which is manifestation of the SOC effect and local dipole field Yao et al. (2017). The similar phenomena has also been theoretically predicted on the 1T-phase of the monolayer zirconium and hafnium dichalcogenides (Zr(Hf)$X_{2}$, $X$=S,Se)Cheng et al. (2018). However, in the 1T-phase of the monolayer TMDCs, spin degeneracy remains in the electronic band structures Yao et al. (2017); Cheng et al. (2018), which is protected by the centrosymmetric of the crystal, thus limiting its spintronics functionality. Considering the fact that the 1T-phase possess superior transport properties due to the high electron mobility Wang et al. (2015); Zhang et al. (2014), lifting the spin degeneracy could be the important key for their realization in the spintronics devices. Therefore, finding a feasible method to induce the significant spin splitting in the 1T-phase of the monolayer TMDCs is highly desirable.
In this paper, by using density-functional theory (DFT) calculations, we show that the significant spin splitting can be induced in the 1T-phase of monolayer TMDCs by introducing the line defect. By using the monolayer PtSe${}_{2}$ as a representative example, we investigate the most stable formation of the line defects, namely Se-vacancy line defect (Se-VLD). We find that a sizable spin splitting is observed in the defect states of the Se-VLD, exhibiting a highly unidirectional spin configuration in the momentum space. This peculiar spin configuration gives rise to the so-called ”persistent spin textures (PST)”Schliemann (2017); Bernevig et al. (2006), a specific spin mode which protects the spin from decoherence and induces an extremely long spin lifetimeDyakonov and Perel (1972); Altmann et al. (2014). Moreover, by using $\vec{k}\cdot\vec{p}$ perturbation theory supplemented with symmetry analysis, we clarified that the emerging of the spin splitting maintaining the PST in the defect states is originated from the inversion symmetry breaking and one dimensional nature of the Se-VLD engineered monolayer PtSe${}_{2}$. Finally, a possible application of the present system for spintronics will be discussed.
II Computational Details
To model the vacancy line defect (VLD) in the 1T-phase of the monolayer TMDCs, we considered the monolayer PtSe${}_{2}$ as a representative example since it has been successfully synthesized by a direct selenization at the Pt(111) substrate Wang et al. (2015); Yao et al. (2017). We performed first-principles electronic structure calculations based on the DFT within the generalized gradient approximation (GGA) Perdew et al. (1996) implemented in the OpenMX code Ozaki et al. (2009). Here, we adopted norm-conserving pseudopotentials Troullier and Martins (1991) with an energy cutoff of 350 Ry for charge density. The wave functions are expanded by the linear combination of multiple pseudoatomic orbitals (LCPAOs) generated using a confinement scheme Ozaki (2003); Ozaki and Kino (2004). The orbitals are specified by Pt7.0-$s^{2}p^{2}d^{2}$ and Se9.0-$s^{2}p^{2}d^{1}$, which means that the cutoff radii are 7.0 and 9.0 bohr for the Pt and Se atoms, respectively, in the confinement scheme Ozaki (2003); Ozaki and Kino (2004). For the Pt atom, two primitive orbitals expand the $s$, $p$, and $d$ orbitals, while, for the Se atom, two primitive orbitals expand the $s$ and $p$ orbitals, and one primitive orbital expands $d$ orbital. The SOC was included in the DFT calculations by using $j$-dependent pseudopotentials (Theurich and Hill, 2001). The spin textures in the momentum space were calculated using the spin density matrix of the spinor wave functions obtained from the DFT calculations as we applied recently on various 2D materials Absor and Ishii (2019a, b).
In this work, the VLD were constructed by using supercell model of the pristine monolayer PtSe${}_{2}$ [Fig. 1(a)]. We build the supercell model from the minimum rectangular cell where the optimized lattice parameters obtained from the primitive cell (hexagon) are used [Fig. 1(a)]. As a consequence, the folding cell from the hexagonal to rectangular cells in the FBZ is expected [Fig. 1(b)-(c)]. Here, we used the axes system where layers are chosen to sit on the x-y plane, where the $x$ ($y$) axis is taken to be parallel to the zigzag (armchair) direction. We considered two different configurations of the VLD, namely the Se-VLD and Pt-VLD, where their relaxed structures are displayed in Figs. 1(d) and 1(e), respectively. To model these VLDs, we extend the supercell size of the monolayer by 10 times in the $x$-direction, which is perpendicular to the direction of the extended defect line along the $y$-direction in order to eliminate interaction between periodic image of the line defect [see Fig. 1(d)-(e)].
In our DFT calculations, we used a periodic slab where a sufficiently large vacuum layer (20 Å) is used to avoid interaction between adjacent layers. The $3\times 12\times 1$ k-point mesh was used and the geometries were fully relaxed until the force acting on each atom was less than 1 meV/Å. We calculate formation energy to confirm energetic stability of th VLD through the following formula Freysoldt et al. (2014):
$$E_{f}=E_{\texttt{VLD}}-E_{\texttt{Pristine}}+\sum_{i}n_{i}\mu_{i}.$$
(1)
In Eq. (1), $E_{\texttt{VLD}}$ is the total energy of the defective system, $E_{\texttt{Pristine}}$ is the total energy of the pristine system, $n_{i}$ is the number of atom being removed from the pristine system, and $\mu_{i}$ is the chemical potential of the removed atoms corresponding to the chemical environment surrounding the system. Here, $\mu_{i}$ obtains the following requirements:
$$E_{PtSe_{2}}-2E_{Se}\leq\mu_{Pt}\leq E_{Pt},$$
(2)
$$\frac{1}{2}(E_{PtSe_{2}}-E_{Pt})\leq\mu_{Se}\leq E_{Se}.$$
(3)
Under Se-rich condition, $\mu_{Se}$ is the energy of the Se atom in the bulk phase (hexagonal Se, $\mu_{Se}=\frac{1}{3}E_{Se-hex}$) which corresponds to the lower limit on Pt, $\mu_{Pt}=E_{PtSe_{2}}-2E_{Se}$, where $E_{PtSe_{2}}$ is the total energy of the monolayer PtSe${}_{2}$ in the primitive unit cell. On the other hand, in the case of the Pt-rich condition, $\mu_{Pt}$ is associated with the energy of the Pt atom in the bulk phase (fcc Pt, $\mu_{Pt}=\frac{1}{4}E_{Pt-fcc}$) corresponding to the lower limit on Se, $\mu_{Se}=\frac{1}{2}(E_{PtSe_{2}}-E_{Pt})$.
III RESULT AND DISCUSSION
First, we briefly discuss the structural symmetry and electronic properties of the pristine monolayer PtSe${}_{2}$. As we mentioned previously that the pristine monolayer PtSe${}_{2}$ belongs to the 1T-phase of the monolayer TMDCs having centrosymmetric crystal. Here, one Pt atom (or Se atom) is located on top of another Pt atom (or Se atom) forming an octahedral coordination, while it shows trigonal structure when projected to the (001) plane [Fig. 1(a)]. As a result, a polar group $C_{3v}$ and a symmorphic group $D_{3d}$ are identified for the Se and Pt sites, respectively. We find that the calculated lattice constant of the pristine monolayer PtSe${}_{2}$ in the primitive unit cell is 3.75 Å, which is in good agreement with the experiment (3.73 ÅWang et al. (2015)) and previous theoretical calculations (3.75 ÅZulfiqar et al. (2016); Zhang et al. (2016)).
Figure 2(a) shows the calculated result of the electronic band structures of the ($1\times 1$) primitive unit cell of the pristine monolayer PtSe${}_{2}$. Consistent with previous theoreticalZhang et al. (2016) and experimentalWang et al. (2015) results, we find that the the pristine monolayer PtSe${}_{2}$ is an indirect semiconductor with the band gap of 1.38 eV, where the valence band maximum (VBM) is located at $\Gamma$ point, while the conduction band minimum (CBM) is located at the $\vec{k}$ along the $\Gamma-M$ line. Our calculated results of the density of states (DOS) projected to the atomic orbitals confirmed that the VBM is predominantly contributed from the Se-$p$ orbitals, while the CBM is mainly originated from the Pt-$d$ orbitals [Fig. 2(b)]. Turning the SOC, lifting spin degeneracy of the electronic band structure is expected, which is dictated by the lack of the inversion symmetry Rashba (1960); Dresselhaus (1955). However, in the monolayer PtSe${}_{2}$, we find that all the bands are doubly degenerated, which is protected by the centrosymmetric of the crystal [Fig. 2(c)]. Evidently, the spin degeneracy observed in the electronic band structures of the pristine monolayer PtSe${}_{2}$ is consistent with the recent experimental results reported by Yao $et$. $al$., by using spin-ARPESYao et al. (2017).
Next, we consider the effect of the VLD on the structural symmetry of the monolayer PtSe${}_{2}$. To examine the optimized structures of the VLDs, we show in Table 1 the calculated results of the Pt-Se bond length around the VLD site corresponding to their symmetry group. In the case of the Se-VLD, removing one Se atom from the supercell breaks the inversion symmetry of the monolayer PtSe${}_{2}$. Consquently, three Pt atoms surrounding the vacancy line are found to be relaxed, implaying that the Pt-Se bond lengths around the VLD for the nearest neighbor (NN) Se atoms (2.553 Å) is lower than that for the next-nearest neighbor (NNN) Se atoms (2.580 Å). As a result, only a mirror symmetry plane ($M_{yz}$) along the the extended vacancy line is preserved [Fig. 1(d)], leading to the fact that the symmetry of the Se-VLD belongs to $C_{s}$ point group. In contrast to the Se-VLD case, the inversion symmetry remains in the case of the Pt-VLD. Here, we find that the Pt-Se bond lengths around the vacancy line for the NN and NNN Se atoms are 2.576 Å and 2.692 Å, respectively, which are larger than that of the Se-VLD as well as the pristine system (2.548 Å). Moreover, removing the Pt atom from the supercell leads to the fact that both mirror symmetry plane $M_{yz}$ and $C_{2}$ rotation around the extended vacancy line retain [Fig. 1(e)], resulting in that the Pt-VLD exhibits $C_{2h}$ point group symmetry.
The significant structural changes induced by the VLDs strongly affect the energetic stability of the monolayer PtSe${}_{2}$, which can be confirmed by the calculated formation energy $E_{f}$. As shown in Table 1, we find that the $E_{f}$ of the Se-VLD is much lower than that of the Pt-VLD, indicating that the Se-VLD is more favorable to be formed in the monolayer PtSe${}_{2}$. This is consistent with the general fact that the chalcogen vacancies and their alignment into extended line defect can be easily formed in the monolayer TMDCs as recently predicted on MoS${}_{2}$Komsa and Krasheninnikov (2015); Noh et al. (2014) and WS${}_{2}$Li et al. (2016), and PtSe${}_{2}$Absor et al. (2017); Zhang et al. (2016). On the other hand, due to the covalent bonding between the Pt atom and the six neighboring Se atoms, removing the Pt atom is stabilized by increasing the $E_{f}$. Moreover, we also compare the calculated $E_{f}$ of the Se-VLD with those of the Se single vacancy defect (Se-SVD) and Pt single vacancy defect (Pt-SVD) [see Table 1]. We found that the $E_{f}$ of the Se-VLD system is comparable to that of the Se-SVD, but is much lower than that of the Pt-SVD, suggesting that the formation of the Se-VLD is energetically accessible. In fact, agregration of the chalcogen vacancy and their alignment into extended line defect has been experimentally reported on the monolayer MoS${}_{2}$ by using electron irradiation techniqueKomsa et al. (2013); Chen et al. (2018); Wang et al. (2016), indicating that the Se-VLD engineered monolayer PtSe${}_{2}$ is experimentally feasible. Since the Se-VLD has the lowest formation energy in the VLD systems, in the following discussion, we will concentrate only for the electronic properties of the Se-VLD engineered monolayer PtSe${}_{2}$.
Figure 3(a) shows the calculated electronic band structures of the Se-VLD corresponding to the DOS projected to the atoms near the VLD site without including the SOC. Compared with the pristine supercell system, we identify three defect states induced by the Se-VLD, which are located inside the band gap. Here, a single occupied defect state (DS-1) is located above the VBM while two unoccupied defect states (DS-2 and DS-3) are located below the CBM. Our spin polarized calculations on the Se-VLD system confirmed that there is no lift of the spin degeneracy in any eigenstate, indicating that the Se-VLD engineered monolayer PtSe${}_{2}$ remains non-magnetic as the defect free one. Moreover, according to the DOS projected to the atoms near the VLD site, we found that the domination of the NNN Se-$p$ orbitals with small contribution of the NN Se-$p$ and the Pt-$d$ orbitals characterizes the single occupied defect state, while the two unoccupoied states are mostly contributed from the Pt-$d$ orbital mixed with the NNN Se-$p$ orbital [Fig. 3(b)]. Remarkably, the observed defect states together with non-magnetic character of the Se-VLD engineered monolayer PtSe${}_{2}$ are consistent with previous results of a single chalcogen vacancy in various monolayer TMCDsLi et al. (2016); Absor et al. (2017); Noh et al. (2014).
It is pointed out here that a single chalcogen vacancy in the monolayer TMCDs is known to have dispersionless midgap defect states due to the localized wavefunction around the vacancy site Li et al. (2016); Zhang et al. (2016). However, in our proposed system, the interactions between the neighboring Se-VLD along the extended defect line [see Fig. 1(d)] result in dispersive defect states, in particularly, in the bands along the $\Gamma-Y$ ($k_{y}$) direction [Fig. 3(a)]. On the other hand, the defect states are dispersionless in the bands along the $\Gamma-X$ ($k_{x}$) direction since the Se-VLD is isolated far from each other along the $x$-direction. The strong dispersive character of the defect states along the $\Gamma-Y$ line is further confirmed by the calculated results of the wavefunctions shown in Figs. 3(d)-(f). Here, we find that all the defect states are delocalized, forming a quasi-1D states oriented along the defect line in the $y$-direction, and thus suppresses the bands to exhibit strong dispersion along the $\Gamma-Y$ direction. The strong dispersive defect states found in the Se-VLD engineered monolayer PtSe${}_{2}$ are expected to enhance carrier mobilityFishchuk et al. (2016), which plays an important role in the transport-based electronic devices.
When the SOC is taken into account, we find a sizable spin splitting in the defect states since the inversion symmetry of the monolayer PtSe${}_{2}$ is already broken by the formation of the Se-VLD [Fig. 3(c)]. In particularly, the spin splitting is observed in the bands along the $\Gamma-Y$ line, while the spin degeneracy remains in the bands along the $\Gamma-X$ line, indicating a strongly anisotropic spin splitting. We emphasized here that the significantly large spin splitting observed in the defect states can be atributed by the strong coupling between atomic orbitals due to the non-zero SOC matrix element, $\zeta_{l}\left\langle\vec{L}\cdot\vec{S}\right\rangle_{u,v}$, where $\zeta_{l}$ is angular momentum resolved atomic SOC strength with $l=(s,p,d)$, $\vec{L}$ and $\vec{S}$ are the orbital angular momentum and Pauli spin operators, respectively, and $(u,v)$ is the atomic orbitals. By calculating orbital-resolved of the electronic band structures projected to the atoms around the VLD site, we find that the large spin splitting in the defect states (DS-1, DS-2, DS-3)) is mainly originated from the strong hybridization between the NNN Se-$p_{x}+p_{y}$, NN Se-$p_{x}+p_{y}$, and Pt-$d_{x^{2}-y^{2}}+d_{xy}$ orbitals [Fig. 4]. This is consistent with the fact that the $p-d$ orbitals coupling plays an important role for inducing the large spin spliting in the defect states of a single chalcogen vacancy in the monolayer TMDCsLi et al. (2016); Absor et al. (2017).
To further demonstrate the nature of the spin-split defect states, we investigate the spin textures in the momentum $k$-space. Here, we calculate the spin polarization of each eigenstate $\psi(\vec{k})$ for a given $\vec{k}$ defined as $S_{i}(\vec{k})=\left\langle\psi(\vec{k})|\sigma_{i}|\psi(\vec{k})\right\rangle$, where $\sigma_{i}$ is the i-direction component ($i=x,y,z$) of the Pauli matrix. The resulting spin textures calculated around the $\Gamma$ point in the spin-split occupied defect states is shown in Fig. 4(a) for the DS-1, while in the spin-split unoccupied defect states are given in Fig. 4(b) and 4(c) for the DS-2 and DS-3, respectively. It is found that all the defect states (DS-1, DS-2, DS-3) exhibit a highly uniform spin configuration, which is oriented along the $x$-direction. This peculiar pattern of the spin textures gives rise to the so-called persistent spin textures (PST)Schliemann (2017); Bernevig et al. (2006), which is similar to those observed on the bulk BiInO${}_{3}$ Tao and Tsymbal (2018) and CsBiNb${}_{2}$O${}_{7}$Autieri et al. (2019). Specifically, the PST observed in the present system leads to the spatially periodic mode of the spin polarization to form the persistent spin helix (PSH) states Schliemann (2017); Bernevig et al. (2006), protecting the spin from the decoherence based on the Dyakonov-Perel (DP) mechanism of spin-relaxationDyakonov and Perel (1972); Altmann et al. (2014). As a result, an extremely long spin lifetime is expectable, thus offering a promising platform to realize an efficient spintronics devices.
The origin of the spin splitting and the peculiar spin configuration observed in the defect states of the Se-VLD engineered monolayer PtSe${}_{2}$ can be clarified in term of $\vec{k}\cdot\vec{p}$ perturbation theory combined with group theory analysis to deduce the effective SOC Hamiltonian. Although the inversion symmetry of the monolayer PtSe${}_{2}$ is already broken by formation of the Se-VLD, it’s time reversal symmetry is conserved, leading to the fact that the spin degeneracy remains, in particularly, at the high symmetry point in the FBZ such as the $\Gamma$ (0,0,0) and Y (0,$\pm$ 0.5, 0) points. By introducing the SOC, the doublet splits at the $\vec{k}$ away from the time-reversal-invariat points, in which the effective SOC Hamiltonian $\hat{H}_{\textbf{SOC}}$ can be derived by $\vec{k}\cdot\vec{p}$ theory. Here, according to Vajna. $et$. $al$., it is possible to construct $\hat{H}_{\textbf{SOC}}$ from the following invariance formulation (Vajna et al., 2012):
$$H_{\textbf{SOC}}(\vec{k})=\alpha\left(g\vec{k}\right)\cdot\left(\det(g)g\vec{%
\sigma}\right),$$
(4)
where $\vec{k}$ and $\vec{\sigma}$ are the electron’s wavevector and spin vector, respectively, and $\alpha\left(g\vec{k}\right)=\det(g)g\alpha\left(\vec{k}\right)$, where $g$ is the element of the point group characterizing the small group wave vector $G_{\vec{Q}}$ of the high symmetry point $\vec{Q}$ in the first Brillouin zone. By sorting out the components of $\vec{k}$ and $\vec{\sigma}$ according to irreducible representation (IR) of $G_{\vec{Q}}$, we can decompose again their direct product into IR. On the basis of the Eq. (4), only the total symmetric IR from this decomposition contributes to the $\hat{H}_{\textbf{SOC}}$. Therefore, by using the corresponding tables of the point group, one can easily construct the possible term of $\hat{H}_{\textbf{SOC}}$.
Now, let us apply the above mentioned procedure to predict the possible term of the $\hat{H}_{\textbf{SOC}}$ in our defective system. As previously mentioned that the symmetry of the Se-VLD belongs to $C_{s}$ point group. Therefore, the group of the wave vector at the $\Gamma$ point also belongs to $C_{s}$ point group. This group has two elements: identity operation $E:(x,y,z)\rightarrow(x,y,z)$ and mirror symmetry operation $M_{yz}:(x,y,z)\rightarrow(-x,y,z))$. Accordingly, there are two one-dimensional IRs, namely $A^{\prime}$ and $A"$, where character and direct product tables are given in Table 2 and 3, respectively. Since $\vec{k}$ and $\vec{\sigma}$ can be transformed as polar and axial vectors, respectively, a comparison with the character table allow us to sort out the components of these vectors according to the IR as $A^{\prime}$: $k_{y}$, $k_{z}$, $\sigma_{x}$ and $A"$: $k_{x}$, $\sigma_{y}$, $\sigma_{z}$. Moreover, from the corresponding table of direct products, we obtain the third order terms of $\vec{k}$ as $A^{\prime}$: $k^{3}_{y}$, $k^{3}_{z}$, $k^{2}_{y}k_{z}$, $k^{2}_{x}k_{z}$, $k^{2}_{z}k_{y}$ and $A"$: $k^{3}_{x}$, $k^{2}_{y}k_{x}$, $k_{y}k_{x}k_{z}$, $k^{2}_{z}k_{x}$. However, due to the 1D nature of the defect states, all the terms containing $k_{x}$ and $k_{z}$ should vanish. Therefore, according to the table of direct products, the combination of the first order of $\vec{k}$ as well as the third order of $k^{3}$ with the components of the spin vector $\vec{\sigma}$ that belongs to $A^{\prime}$ IR are $k_{y}\sigma_{x}$ and $k^{3}_{y}\sigma_{x}$. This combinations can be generalized for the higher odd order $n$ in $\vec{k}$, where the only nonzero term is $k^{n}_{y}\sigma_{x}$. By collecting all these terms, the effective SOC Hamiltonian $\hat{H}_{\textbf{SOC}}$ of the Se-VLD engineered monolayer PtSe${}_{2}$ up to $n$th-order in $\vec{k}$ near the $\Gamma$ point can be written as
$$H_{\textbf{SOC}}(\vec{k})=\alpha_{1}k_{y}\sigma_{x}+\alpha_{3}k^{3}_{y}\sigma_%
{x}+\alpha_{5}k^{5}_{y}\sigma_{x}+...+\alpha_{n}k^{n}_{y}\sigma_{x},$$
(5)
where $\alpha_{n}$ is the $n$th-order in $\vec{k}$ SOC parameter.
It is revealed from Eq. (5) that $\hat{H}_{\textbf{SOC}}$ is characterized only by $\sigma_{x}$ term, indicating that the spin textures around the $\Gamma$ point are uniform and oriented along the $x$-direction, thus maintaining the PST. This is in fact consistent well with our spin textures shown in Fig. 4 obtained from the DFT calculations. Moreover, at the $\vec{k}$ point that is far away from the $\Gamma$ point, the higher-order terms of the $\vec{k}$ should be taken into account. However, it has been demonstrated from Eq. (5) that the only $\sigma_{x}$ component of the spin vector is also conserved at the higher-order terms of the $\vec{k}$, preserving the PST at the larger wave vector $\vec{k}$ in the FBZ. This is in contrast to the widely studied PST in various IIIâV semiconductor quantum well (QW) systems such as GaAs/AlGaAs Walser et al. (2012); Schönhuber et al. (2014) and InAlAs/InGaAs Sasaki et al. (2014); Ishihara et al. (2014) where the higher order term of the $\vec{k}$ in the $\hat{H}_{\textbf{SOC}}$ usually breaks the formation of the PST. Similarly, the broken formation of the PST by the higher-order terms of the $\vec{k}$ is also observed in the bulk systems such as BiInO${}_{3}$Tao and Tsymbal (2018) and CsBiNb${}_{2}$O${}_{7}$ Autieri et al. (2019). Remarkably, the preserving formation of the PST in the Se-VLD engineered monolayer PtSe${}_{2}$ is expected to significantly reduce the spin decoherence in the diffusive transport regime, which is important in the operation of the spintronics devices.
For a quantitative analysis of the predicted spin splitting in the Se-VLD engineered monolayer PtSe${}_{2}$, we focus on the linear term of $\hat{H}_{\textbf{SOC}}$ given in Eq. (5) to calculate the SOC parameter $\alpha_{1}$. By fitting the DFT bands of the defect states along the $\Gamma-Y$ line, it is found that the calculated values of $\alpha_{1}$ are 1.14 eVÅ for the DS-1, which is larger than that for the DS-2 (0.20 eVÅ) and DS-3 (0.28 eVÅ) [see Table III]. However, these values are much larger than that observed on the PST in various QWs systems such as GaAs/AlGaAs Walser et al. (2012); Schönhuber et al. (2014) and InAlAs/InGaAs Sasaki et al. (2014); Ishihara et al. (2014), ZnO (10-10) surfaceAbsor et al. (2015), strained LaAlO3/SrTiO3 (001) interface Yamaguchi and Ishii (2017), and bulk CsBiNb${}_{2}$O${}_{7}$Autieri et al. (2019). In addition, for the case of the DS-1, the calculated $\alpha_{1}$ is comparable with that observed on the bulk BiInO${}_{3}$ Tao and Tsymbal (2018), monolayer WO${}_{2}$Cl${}_{2}$Ai et al. (2019), and monolayer group IV monochalcogenide [see Table III]. The associated SOC parameters found in the defect states of the Se-VLD engineered monolayer PtSe${}_{2}$ are sufficient to support the room temperature spintronics functionality.
Importantly, the formation of the PST observed in our defective system leads to the fact that carriers can move in the spatially periodic mode of the spin polarization to form the PSH statesSchliemann (2017); Bernevig et al. (2006) with the wavelength of $\lambda=(\pi\hbar^{2})/(m_{\Gamma-Y}^{*}\alpha_{1})$. Here, $m_{\Gamma-Y}^{*}$ is the carrier effective mass along the $\Gamma-Y$ direction, which can be estimated by fitting the band dispersion in the defect states along the $\Gamma-Y$ line. We find that the calculated effective mass $m_{\Gamma-Y}^{*}$ is -0.21$m_{0}$ for the DS-1, while it is 0.25$m_{0}$ and 0.19$m_{0}$ for the DS-2 and DS-3, respectively, where $m_{0}$ is the free electron mass. The negative (positive) value of $m_{\Gamma-Y}^{*}$ characterizes the hole (electron) carriers in the occupied (unoccupied) defect states. Therefore, the resulting wavelength $\lambda$ is 6.33 nm for the DS-1, which is smaller than that for the DS-2 (29.47 nm) and DS-3 (28.12 nm) [see Table III]. Specifically, the small value of $\lambda$ observed in the DS-1 is comparable with that reported on the bulk BiInO${}_{3}$ Tao and Tsymbal (2018), monolayer WO${}_{2}$Cl${}_{2}$ Ai et al. (2019), and monolayer group IV monochalcogenide, rendering that the present system is promising for nanoscale spintronics devices.
Thus far, we have predicted that the large spin splitting maintaining the PST is achieved in the defect states of the Se-VLD engineered monolayer PtSe${}_{2}$. In particularly, we find that the largest strength of the spin splitting ($\alpha=1.14$ eVÅ) is observed in the occupied defect state (DS-1), indicating that $p$-type defective system for spintronics is expected to be realized. In fact, $p$-type defective system with the higher carriers mobility and concentration in the monolayer TMDCs has been reported Yuan et al. (2014). Considering the fact that the calculated wavelength ($\lambda=6.33$ nm) of the PSH states in the DS-1 is substantially small, it is possible to resolve the features down to tens-nm scale with sub-ns time resolution by using near-filled scanning Kerr microscopyRudge et al. (2015), thus calling for experimentall confirmations of our theoretical predictions. As such, our findings of the large spin splitting with small wavelength of the PSH is useful for realization of an effcient and highly scalable spintronics devices.
IV CONCLUSION
The effect of the line defect on the electronic properties of the 1T-phase of the monolayer TMDCs is systematically investigated by employing the first-principles density functional theory (DFT) calculations. As a representative example, we have considered the Se-VLD in the the monolayer PtSe${}_{2}$ since it has the lowest formation energy of the line defect. By taking into account the SOC, we have found that introducing the Se-VLD in the the monolayer PtSe${}_{2}$ leads to a sizable spin splitting in the defect states. Our analysis using the orbital-resolved of the electronic band structures have confirmed that the strong $p-d$ orbitals coupling in the defect states plays a significant role for inducing such spin splitting. Specifically, we have identified a highly unidirectional spin configuration in the momentum space, giving rise to the so-called persistent spin textures (PST)Schliemann (2017); Bernevig et al. (2006), a specific spin mode resulting in a protection against spin-decoherence, which induces an extraordinarily long spin lifetime. Moreover, by using $\vec{k}\cdot\vec{p}$ perturbation theory supplemented with symmetry analysis, we have demonstrated that the emerging of the spin splitting with the highly PST in the defect states is subjected to the inversion symmetry breaking together with the 1D-nature induced by the Se-VLD in the monolayer PtSe${}_{2}$. Recently, the defective of the 1T-phase of the monolayer TMDCs has been extensively studied Absor et al. (2017); Zhang et al. (2016); Zulfiqar et al. (2016); Kuklin and Ågren (2019). Our study clarifies that the line defect plays an important role in the spin-splitting properties of the 1T-phase of the monolayer TMDCs, which could be of highly important for designing spintronic devices.
We emphasized here that our proposed approach for inducing the large spin splitting with the highly PST by using the line defects is not only limited on the monolayer PtSe${}_{2}$ ML, but also can be extendable to other 1T-phase of the monolayer TMDCs systems such as paladium dichalcogenidesKuklin and Ågren (2019), tin dichalcogenideGonzalez and Oleynik (2016), hafnium and rhenium dichalcogenides Horzum et al. (2014), where the structural and electronic structure properties are similar. In fact, manipulation of the electronic properties of these particular materials by introducing the defect has been recently reportedKuklin and Ågren (2019). Therefore, it is expected that our predictions will stimulate further theoretical and experimental efforts in the exploration of the spin-splitting properties of the the monolayer TMDCs, broadening the range of the 2D materials for future spintronic applications.
Acknowledgements.
M.A.U. Absor and I. Santoso would like to thanks the Indonesian Ministry of Research and Technology, and Indonesian Ministry of Education and Culture for partially support through World Class University (WCU) Program managed by Institute Technologi Bandung. This work was partly supported by Grants-in-Aid on Scientific Research (Grant No. 16K04875) from the Japan Society for the Promotion of Science (JSPS) and a JSPS Grant-in-Aid for Scientific Research on Innovative Areas âDiscrete Geometric Analysis for Materials Designâ (Grant No. 18H04481). The computation in this research were performed using the supercomputer facilities at RIIT, Kyushu University, Japan.
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Resolving the Disc-Halo Degeneracy II: NGC 6946
S. Aniyan${}^{1}$, A. A. Ponomareva${}^{2,1,3}$, K. C. Freeman${}^{1}$, M. Arnaboldi${}^{4}$, O. E. Gerhard${}^{5}$, L. Coccato${}^{4}$, K. Kuijken${}^{6}$ & M. Merrifield${}^{7}$
${}^{1}$Research School of Astronomy & Astrophysics, Australian National University, Canberra, ACT 2611, Australia
${}^{2}$Oxford Astrophysics, Denys Wilkinson Building, University of Oxford, Keble Rd, Oxford, OX1 3RH, UK
${}^{3}$Kapteyn Astronomical Institute, University of Groningen, Postbus 800, NL-9700 AV Groningen, The Netherlands
${}^{4}$European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching, Germany
${}^{5}$Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, 85741 Garching, Germany
${}^{6}$Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands
${}^{7}$School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Email: [email protected]: [email protected]
(Accepted XXX. Received YYY; in original form ZZZ)
Abstract
The mass-to-light ratio ($M/L$) is a key parameter in decomposing galactic rotation curves into contributions from the baryonic components and the dark halo of a galaxy. One direct observational method to determine the disc $M/L$ is by calculating the surface mass density of the disc from the stellar vertical velocity dispersion and the scale height of the disc. Usually, the scale height is obtained from near-IR studies of edge-on galaxies and pertains to the older, kinematically hotter stars in the disc, while the vertical velocity dispersion of stars is measured in the optical band and refers to stars of all ages (up to $\sim$ 10 Gyr) and velocity dispersions. This mismatch between the scale height and the velocity dispersion can lead to underestimates of the disc surface density and a misleading conclusion of the sub-maximality of galaxy discs. In this paper we present the study of the stellar velocity dispersion of the
disc galaxy NGC 6946 using integrated star light and individual planetary nebulae as dynamical tracers. We demonstrate the presence of two kinematically distinct populations of tracers which contribute to the total stellar velocity dispersion. Thus, we are able to use the dispersion and the scale height of the same dynamical population to derive the surface mass density of the disc over a radial extent. We find the disc of NGC 6946 to be closer to maximal with the baryonic component contributing
most of the radial gravitational field in the inner parts of the galaxy ($\rm V_{max}(bar)=0.76(\pm 0.14)V_{max}$).
keywords:
Galaxies: kinematics and dynamics – Galaxies: evolution – Galaxies: spiral – dark matter
††pubyear: 2020††pagerange: Resolving the Disc-Halo Degeneracy II: NGC 6946–LABEL:lastpage
1 Introduction
The "disc-halo" degeneracy is an important issue when decomposing H I rotation curves of galaxies, where the contribution from the baryonic matter is mostly determined by the stellar mass-to-light ratio ($M/L$). Unfortunately, the $M/L$ is uncertain due to the challenges involved in traditional methods used to determine it (van Albada et al., 1985; Maraston, 2005; Conroy
et al., 2009). Accurate rotation curve decomposition is crucial in determining the potential of a galaxy and the parameters of its dark matter (DM) halo. The densities and scale radii of dark haloes are known to follow well-defined scaling laws and can therefore be used to measure the redshift of assembly of haloes of different masses (Macciò et al., 2013; Kormendy &
Freeman, 2016; Somerville
et al., 2018).
A rather direct observational technique of measuring the disc $M/L$ is from its surface mass density, which can be estimated from the vertical velocity dispersion and the vertical scale height of the disc (van der Kruit
& Freeman, 1984; Bottema et al., 1987; Herrmann et al., 2008; Bershady et al., 2010a). The 1D Jeans equation in the vertical direction can be used to estimated the surface mass density ($\Sigma$) of the disc via the relation:
$$\rm\Sigma=f\sigma_{z}^{2}/Gh_{z}$$
(1)
where $h_{z}$ is the vertical scale height and $\sigma_{z}$ is the integrated vertical velocity dispersion of the exponential disc, $\rm G$ is the gravitational constant and $f$ is a geometric factor, known as the vertical structure constant, that depends weakly on the adopted vertical structure of the disc.
Having estimated $\Sigma$, we can infer stellar mass surface density ($\Sigma_{\star}$) and the $M/L$ as $\Upsilon_{\star}=\Sigma_{\star}/\mu$, where $\mu$ is the surface brightness of a galaxy in physical units ($\rm L_{\odot}~{}pc^{2}$), thus breaking the disc-halo degeneracy.
In Aniyan
et al. 2018 (hereafter An18), we argued that for the Jeans equation (Eqn. 1) to work effectively, the scale height and dispersions must refer to the same population of stars, and in this case the $\sigma_{z}$ is expected to fall exponentially with twice the galaxy scale length. However, usually the vertical dispersions are obtained from measurements of near-face-on galaxies in the optical bands, whereas the scale heights are obtained from studies of edge-on galaxies in the red and NIR bands (Kregel et al., 2002)111 See also Kregel et al. (2005) for a detailed study of the correlation between the intrinsic properties of edge-on galaxies, i.e. central surface brightness, scale length and scale height in the I-band..Thus, the spectra of star-forming face-on disc galaxies include light from the younger, kinematically colder population of stars. On the other hand, the scale heights obtained from observations of edge-on discs are primarily for the kinematically hotter stars which are above the dust lane in the galaxies. This mismatch can lead to the surface mass density being underestimated which can in turn lead to galaxies’ discs being incorrectly classified as sub-maximal. This issue has plagued most of the previous measurements of the surface mass density of face-on discs (Herrmann &
Ciardullo, 2009b; Bershady et al., 2011).
In An18 we used integrated light data as well as planetary nebulae (PNe) as tracers of the kinematics of the disc in the face-on spiral NGC 628 to demonstrate the presence of two kinematically distinct populations of disc stars in this galaxy. We also showed the effect of the cold layer on the total surface mass density of the disc (see Appendix in An18). This is based on an exact solution for an isothermal sheet of older stars with an embedded very thin layer of younger stars and gas. Its density distribution is a modified version of the familiar ${\rm sech}^{2}(z/2h_{z})$ distribution. Following this solution equation 1 becomes:
$$\Sigma_{T}=\Sigma_{D}+\Sigma_{C,*}+\Sigma_{C,gas}=\sigma_{z}^{2}/(2\pi Gh_{z}),$$
(2)
where $\Sigma_{T}$ is the total surface density of the disc and $\Sigma_{D}$ is the surface density of the older stellar component which is used as the dynamical tracer (its scale height is $h_{z}$ and its isothermal vertical velocity dispersion is $\sigma_{z}$). $\Sigma_{C,*}$ and $\Sigma_{C,gas}$ are the surface densities of the cold thin layers of young stars and gas respectively. An independent measurement of $\Sigma_{C,gas}$ is available from 21-cm and mm radio observations. As a result, in An18 we showed that the disc which was primarily considered sub-maximal (Herrmann &
Ciardullo, 2009b) appears maximal with $V_{\rm baryonic}=(0.78\pm 0.11)V_{\rm max}$.
In this paper, we extend our previous work to describe the kinematics of the nearby disc galaxy NGC 6946 with the purpose to establish the method’s general viability. The choice of NGC 6946 is motived by its inclination ($\sim$ 37${}^{\circ}$), as it is not as face-on as NGC 628, which allows us to derive a more reliable rotation curve, as well as makes more targets accessible for the analysis. We observe NGC 6946 with the VIRUS-W integral field unit spectrograph in the inner regions (30-125”) to study stellar kinematics through the absorption lines, and with the Planetary Nebula Spectrograph (PN.S), which allows us to study the kinematics of the stellar component in galaxies at low surface brightness values in ellipticals (Pulsoni
et al., 2018), lenticulars (Cortesi
et al., 2013) and discs (An18) using PNe as discrete tracers out to large radii. The use of PNe to map kinematics to large radii in discs showed the presence of a cold younger disc and a older hotter thicker disc in the Andromeda galaxy in the radial range 14-28 kpc (Bhattacharya
et al., 2019). NGC 6946 is a nearby (D = 6.1 Mpc, $1^{\prime\prime}=29.6$ pc) disc galaxy with a high star formation rate (SFR $=$ 4.8 M${}_{\odot}$ yr ${}^{-1}$ Lee, 2006), which is $\sim$ 4 times the SFR of NGC 628. This contributes to the importance of the NGC 6946 analysis to access differences between the systems. Unfortunately, the inclination of this galaxy brings in some challenge, as it gets harder to separate the in-plane dispersion components ($\sigma_{R}$ and $\sigma_{\phi}$) from the vertical dispersion ($\sigma_{z}$). However, NGC 6946 is closer than NGC 628, and thus we can achieve higher S/N data for the PNe which helps in our analysis. With the PN.S., PNe in NGC 6946 can be detected and their velocities measured out to $388^{\prime\prime}=11.5$ kpc (equivalent to 4.1 disc scale lengths) reaching a surface brightness value of 25 mag in B-band.
This paper is organised as follows: Section 2 describes the observations and data reduction for VIRUS-W. Section 3 summarises the same for the PN.S. Section 4 discusses the photometric properties and derives scale height of the galaxies. Section 5 discusses our analysis to derive the surface mass density of the cold gas in this galaxy. Section 6 presents the analysis involved in the extraction of a double Gaussian model from our data. Section 7 discusses the vertical dispersion profile of the hot and cold stellar components. Section 8 describes the calculation of the stellar surface mass density. Section 9 explains the rotation curve decomposition using the calculated surface mass densities. Section 10 presents our conclusions and scope for future work.
2 VIRUS-W Spectrograph
2.1 Observations
VIRUS-W is an IFU spectrograph on the 2.7-m telescope at McDonald Observatory designed for relatively high resolution spectroscopy of low surface brightness regions of galaxies (Fabricius
et al., 2012). VIRUS-W observations for NGC 6946 were carried out in October 2014. We were able to get good quality data for 4 fields which were positioned on the galaxy at a luminosity weighted radius of about 1 radial scale length along the major and minor axis, and cover the radial extend of up to 175” which corresponds to $\approx 20.6$ $\rm mag/arcsec^{2}$ in I-band, and the surface brightness in the V-band
(in which the Virus W observations were made) is about a magnitude fainter than the dark V-band sky (see Figure 4). The positions of the IFU fields are shown in Figure 1.
The coordinates and exposure time at each position are given in Table 1. The observations were carried out in a sky – galaxy – sky observing sequence. This sequence was repeated at least three times at each field, as indicated in Table 1. This enabled very good sky subtraction using the automated pipeline developed for VIRUS-W (see An18 for further details).
We then used the CURE data reduction pipeline (Goessl
et al., 2006) to get the sky subtracted 1D spectra at each of the four fields. Since the IFU fields cover a large radial extent of NGC 6946 (Figure 1), we split the data into two radial bins at luminosity-weighted mean radii of $54^{\prime\prime}$ and $98^{\prime\prime}$ respectively. After further processing (see Section 6.1), we obtain a 1D summed spectrum for each radial bin that we use to extract the velocity dispersions from the absorption lines.
3 Planetary Nebula Spectrograph
3.1 Observations and Velocity Extraction
The PN.S is a double counter-dispersed wide-field spectrograph designed to discover extragalactic planetary nebulae and measure their radial velocities in a single observation, used on the William Herschel Telescope on La Palma (Douglas
et al., 2002).
The data for NGC 6946 were acquired in September 2014. The weather during the run was excellent, with typical seeing of $\sim 1^{\prime\prime}$. We obtained 11 images centred on the centre of the galaxy, each with an exposure time of 1800s following the strategy adopted in An18. We use the PN.S data reduction pipeline (Douglas
et al., 2007) to get the final stacked ’left’ and ’right’ image for the
$\rm[OIII]$ data. The unresolved objects are candidate planetary nebulae, and their relative positions on the left and right stacked images give us their radial velocities. We also imaged NGC 6946 in H$\alpha$, using the H$\alpha$ narrow band filter on the undispersed H$\alpha$ arm of the PN.S. We then used the [OIII] stacked images along with the H$\alpha$ stacked image to identify the PNe candidates in this galaxy.
3.2 Identification of Sources
To identify our PNe and separate them from the HII regions which also emit in O[III], we use the luminosity function for all spatially unresolved [OIII] emitters identified in the combined left and right images of the PN.S. Then we introduce the expected bright luminosity cut-off for PNe. We include only objects fainter than this value in our analysis. Objects brighter than the cut-off are mostly obvious bright HII regions. For more details please see An18. Consequently we identified 444 unresolved objects with [OIII] emission in this galaxy. From the measured positions of these sources on the left and right images, astrometric positions and line-of-sight (LOS) velocities were derived simultaneously. The typical measurement error associated with the PNe radial velocities is $<$ 9 km s${}^{-1}$ (see Section 3.3 in An18).
We then converted our instrumental magnitudes on to the $m_{5007}$ magnitude scale $m_{5007}=m_{0}+25.16$, using our spectrophotometric standards. These magnitudes were corrected for foreground extinction using the Schlafly &
Finkbeiner (2011) dust maps. At the distance of NGC 6946, the bright luminosity cut-off for PNe in this galaxy is expected to be at $m_{5007}=23.22$. Figure 2 shows the luminosity function for NGC 6946, including all identified unresolved sources. The dashed line in the plot shows the position of the bright luminosity cut-off for the PNe.
After the identification of 444 sources, the resulting sample represents a mix of HII regions and PNe since both can have strong [OIII] emission. We again use our empirical relationship between the ([OIII] – H$\alpha$) colour and the m${}_{5007}$ magnitude to discriminate between PNe and HII regions (Arnaboldi et al., 2020). Figure 3 shows the colour-magnitude diagram used to identify PNe. Thus, we obtain 375 unresolved [OIII] sources classified as PNe: 125 per each radial bin. Earlier observations by Herrmann &
Ciardullo (2009a) found $\approx$ 70 PNe candidates in this galaxy.
In addition to HII regions, the historical supernovae are a potential source of contamination which can bias the planetary nebulae luminosity function (Kreckel
et al., 2017). According to the IAU Central Bureau for Astronomical Telegrams (CBAT) List of Supernovae, there are nine known historical supernovae in NGC 6946, which is an unusually high number. However, none of these objects made it into our PNe sample. We had one unresolved [OIII] source at $\sim$ $3^{\prime\prime}$ from the historical supernova SN 2002hh. Yet, this object was classified as an HII region after applying our colour-magnitude cut. Thus, we don’t have any of these contaminants in our final PNe sample.
4 Photometry and Scale Height
NGC 6946 is a late-type spiral galaxy which shows the presence of a small bulge. Figure 4 shows the surface brightness profiles of NGC 6946 in four photometrical bands: BVI from Makarova (1999) and 3.6 $\rm\mu$m from Muñoz-Mateos
et al. (2009). We use spatial information from these profiles later in the paper to calculate the mass-to-light ratio as a function of radius. We also use the 3.6 $\rm\mu$m profile to perform bulge-disc decomposition (shown with red and blue lines in Figure 4) in order to account for the bulge contribution during the mass modelling.
The disc scale height ($\rm h_{z}$) cannot be directly measured for face-on galaxies. Thus, the scaling relation between the scale height and scale length ($\rm h_{R}$) of spiral galaxies is often used to infer $\rm h_{z}$ of a face-on system. Kregel et al. (2002) studied the scale heights of a sample of edge-on galaxies in the I-band and found strong correlations with the scale lengths. The use of the I-band is justified as it is not sensitive to the contribution of dust and PAHs, and it traces the older thicker stellar disc population. Bershady
et al. (2010b) found that $\log(h_{R}/h_{z})=0.367\log(h_{R}/{\rm kpc})+0.708\pm 0.095$ by fitting the data from Kregel et al. (2002).222Bershady
et al. (2010b) scaling is used to convert the scale length into scale height for an old population of stars. Thus, using the disc fit to the I-band surface brightness profile (Figure 4) we measure the disc scale length to be equal to 95" or 2.8 kpc. Throughout the paper we adopt the distance to the galaxy D = $6.1\pm 0.6$ Mpc from Herrmann et al. (2008) to be consistent with our previous study of NGC 628 (An18).
We note that this distance is a bit smaller than the distance determined by Anand
et al. (2018) ($7.72\pm 0.32$ Mpc) who used the tip of the giant branch method. Finally, we obtain the scale height of the disc $h_{z}=376\pm 75$ pc. We use this value throughout the paper when we refer to the calculation of the surface mass density and mass modelling. Following de Grijs et al. (1997) we assume that the scale height $h_{z}$ for the disc of this late-type spiral is independent of the galactic radius.
Table 2 summarises the values of the various parameters of this galaxy that we use in our analysis of the surface mass density and mass modelling.
Inclination and position angles (PA) are measured with the tilted-ring modelling of the THINGS HI data (Walter et al. 2008, see Section 9.1).
These values agree well with the values obtained from the PNe. This indicates that there is no evident disconnect between the gaseous HI disc and the stellar disc, at least within our covered radii.
5 Surface Mass Densities of Cold Gas
To account for the gas contribution to the measured total surface mass density, and to able to separate stellar from gaseous components, we derive total cold gas surface
density using HI data from the THINGS survey (Walter et al., 2008) and CO data from the HERACLES survey (Leroy
et al., 2009).
The HI radial surface density profile was derived from the integrated column-density HI map, constructed by summing the primary beam corrected channels of the clean data cube.
We use the same radial sampling, position and inclination as for the tilted ring modelling (Section 9.1). The resulting flux (Jy/beam) was converted to mass densities ($\rm M_{\odot}/\rm pc^{2}$) using Eqn. 5 and 6 in Ponomareva et al. (2016). The error on the HI surface mass density was determined as the difference in the profile between approaching and receding sides of the galaxy, and does not exceed $\sim 0.4~{}$$\rm M_{\odot}/\rm pc^{2}$.
The H${}_{2}$ surface mass density profile was obtained from the CO total intensity map with the same radial sampling, position and inclination angles as for the HI profile. The resulting CO intensities were converted into the H${}_{2}$ surface mass density using the prescription by Leroy
et al. (2009). The error on the H${}_{2}$ surface mass density was obtained from the HERACLES error maps and does not exceed $\sim 0.7~{}$M${}_{\odot}/\rm pc^{2}$.
The resulting HI and H${}_{2}$ surface mass density profiles together with the total cold gas profile are shown in Figure 5 with dot dashed, long-dashed and solid curves respectively. All profiles are corrected for the presence of metals and helium and de-projected so as to be face-on. It is worth mentioning the large amount of molecular gas in NGC 6946,
which significantly dominates the amount of atomic gas in the inner parts. We note that Crosthwaite &
Turner (2007) find similar results and report the total H${}_{2}$ mass to be equal to $3\times 10^{9}$ $\rm M_{\odot}$, approximately one-third of the total interstellar hydrogen gas mass.
6 Extracting Velocity Dispersions of the Hot and Cold Components
In this section we describe in detail the procedure of extracting accurate velocity dispersions of two stellar components. We present the different techniques used to derive dispersions from the stellar absorption line spectra in the inner parts, and from the PNe data in the outer regions.
6.1 Stellar Absorption Spectra
6.1.1 Removing Galactic rotation
To remove the galactic rotation and to get rid of any large scale streaming motions across the IFU, we measure the local HI velocity at the position of each fibre from the THINGS data (Walter et al., 2008) and shift each fibre spectrum by this local HI velocity. Figure 6 shows the DSS image of NGC 6946 with the overlapped velocity contours from the HI velocity field. We note that our IFU observations lie within the stellar disc of the galaxy, as shown in Figure 1, while the HI disc is more extended.
This method proved to be preferable to removing the galactic rotation by modelling the rotation field over the entire IFU using the observed rotation curve, as the latter gives larger line-of-sight velocity dispersions ($\sigma_{LOS}$).
However, our technique introduces an additional small velocity dispersion component from the HI itself, for which a correction is needed (see Section 6.1.2).
The VIRUS-W IFU covers a large radial extent of the galaxy. Since the vertical velocity dispersion ($\sigma_{z}$) is expected to fall exponentially with twice the galaxy’s scale length, we don’t want each radial bin to be too large. We divide our IFU data into two radial bins corresponding to luminosity weighted mean radii of $54^{\prime\prime}$ and $98^{\prime\prime}$. The stellar component tends to rotate more slowly than the gaseous component and neglecting this effect can lead to overestimated $\sigma_{LOS}$. NGC 6946 has an inclination of $37^{\circ}$, so this asymmetric drift may affect the measurement significantly.
To remove the effects of differential asymmetric drift, we split the 267 fibres in each IFU field into a grid of six cells, each with about 44 fibres.
The gradient in asymmetric drift is negligible across the small area of a cell. We sum the spectra from all fibres in a cell and cross-correlate five of the summed spectra against the sixth. This gives the shift in velocity to apply to the five spectra to match the spectrum from the sixth grid. This procedure gets rid of any differential asymmetric drift across the IFU fields. We repeat this exercise for all four IFU fields. The cross-correlated, asymmetric-drift-corrected spectra were then shifted to redshift $z=0$, before summing up to the final spectra.
This approach has its caveats, as the hot and cold disc components will have slightly different asymmetric drifts. The difference in asymmetric drift between the hot and cold components is 5 kms${}^{-1}$ evaluated at the mean radius for the inner VIRUS-W region and 6 kms${}^{-1}$ for the outer region. At 37${}^{\circ}$ inclination, these differences become 3 kms${}^{-1}$ and 4 kms${}^{-1}$ respectively. These values are uncertain because of the observation errors in the dispersion, but seem unlikely to make a significant contribution to the measured velocity dispersions.
Although this effect is expected to be minor for a galaxy with an inclination of 37${}^{\circ}$, it could be an issue for more inclined galaxies.
The final summed spectra from the inner and outer radial bins respectively have an SNR of 105 and 77 per wavelength pixel (each wavelength pixel is $\sim$ 0.19 Å; the resolving power $R=8700$, so the Gaussian $\sigma$ of the PSF is $14.7$ km s${}^{-1}$). We only use the region between wavelengths of about 5050 – 5300 Å in our analysis, since it has the highest resolution and avoids the emission lines at
lower wavelengths. The [NI] doublet emission lines together with the third peak can be seen at $\sim$ 5200 Å (see Figure 7).
Although we exclude the emission line region from our velocity dispersion fits, the presence of these three lines is unexpected.
It is hard to definitely conclude that they come from the galaxy interstellar medium (ISM), because the heliocentric radial velocity of
NGC 6946 is only 40 km s${}^{-1}$.
In An18, NGC 628 (radial velocity 657 km s${}^{-1}$) showed the two [NI]
emission lines at $\lambda=5197.9$ and $\lambda=5200.3$ Å, and they were clearly at the
velocity of the galaxy. The strengths of the two lines are approximately equal.
The sky [NI] lines, which typically have line ratios 5198/5200 $\sim$ 1.7, have been
successfully subtracted in the reduction pipeline and do not appear.
In NGC 6946, our strategy for removing galactic rotation means that all lines
from the galactic plane have velocities near zero, relative to the systemic velocity
of the galaxy. We see three emission lines at observed $\lambda$ 5196.2, 5198.4,
5200.4 Å . Their relative line strengths can be seen in Figure 7.
NGC 6946 is known to have a significant amount of halo HI at velocities up to $\pm 100$ km s${}^{-1}$,
in addition to its planar HI (Boomsma et al., 2008)333We thank the anonymous referee for pointing out the presence of the "beard" in this galaxy, which is usually associated with the extraplanar gas.. Thus, we speculate that the three
observed lines come from a superposition of two pairs of [NI] lines: a weaker
pair at near-systemic velocity from the planar gas, and a stronger pair blue shifted
by about 2 Å associated with the negative velocity gas. The superposition of
the two pairs gives three lines as observed, with their apparent line ratios.
6.1.2 LOSVD and the Vertical Velocity Dispersion
We use pPXF (Cappellari, 2017) to obtain the first two moments (the mean line-of-sight velocity and the $\sigma_{LOS}$) for the two Gaussians for the two VIRUS-W spectra. We fit two and single Gaussian components to the data for the comparison. The resulting fits are shown in Figure 7. Further, we use the Bayesian Information Criterion (BIC; Schwarz 1978) to judge the preferred model (one-component or two-component) for the spectra (see Eqn. 3 and Eqn. 4 in An18 on how to calculate BIC). The BIC applies a penalty for models with a larger number of fitted parameters. Thus, between two models, the model with the lower BIC value is preferred. However, we note that in case of the similar BIC values, the BIC can only be considered as a suggested preference. For our data the two component fit is preferred over the single component fit in both of the radial bins, this result is also consistent with the reduced $\chi^{2}$. The results for both fits are tabulated in Table 3.
pPXF uses a best-fit linear combination of stellar templates to directly fit the spectrum in pixel space and to recover the line of sight velocity distribution (LOSVD). We observed template stars of different spectral types with VIRUS-W as our list of stellar templates, to avoid resolution mismatch between the stellar templates and the galaxy spectrum. We assume the two components of the LOSVD to be Gaussian for this close-to-face-on galaxy, and therefore retrieved only the first and second moment parameters from pPXF.
pPXF finds an excellent fit to our spectrum, as shown in Figure 7. It also returns the adopted spectra of the individual components, which are consistent with the spectra of red giants. The mean contributions of the cold and hot disc components to the total light are 52% and 48% in the inner radial bin and 46% and 54% in the outer bin. To correct $\sigma_{LOS}$ values for the contribution of the HI velocity dispersion we adopt a correction value to be $\sim$ 6 km s${}^{-1}$ (see Section 6.1.1).
We then calculate the vertical component of the stellar velocity dispersion $\sigma_{z}$ from the line of sight component $\sigma_{LOS}$ using the following equation:
$$\sigma_{LOS}^{2}=\sigma_{\theta}^{2}\cos^{2}\theta.\sin^{2}i+\sigma_{R}^{2}%
\sin^{2}\theta.\sin^{2}i+\sigma_{z}^{2}\cos^{2}i+\sigma_{meas}^{2}$$
(3)
where $\sigma_{R}$, $\sigma_{\theta}$ and $\sigma_{z}$ are the three components of the dispersion in the radial, azimuthal and vertical direction, $\sigma_{meas}$ is the measurement error on the velocity and $i$ is the inclination of the galaxy ($i=0$ is face-on). Using the epicyclic approximation in the part of the rotation curve that is close to solid body, we adopt $\sigma_{R}=\sigma_{\theta}$,
and for the part of the rotation curve that is flat, we adopt $\sigma_{R}$ = $\sqrt{2}\sigma_{\theta}$. Based on an examination of the THINGS HI velocities along the galaxy’s kinematic major axis (last panel in Figure 12) we determine this galaxy’s rotation curve at radius $\leq 200^{\prime\prime}$ to be solid body and beyond $200^{\prime\prime}$ to have a flat rotation curve. We also adopt the stellar velocity ellipsoid parameter $\sigma_{z}/\sigma_{R}$ ratio to be $0.6\pm 0.15$ following the result from Shapiro et al. (2003). This value is consistent with the value used by Bershady
et al. (2010b) and the value found in the solar neighbourhood by Aniyan et al. (2016). For the discussion regarding the uncertainties of $\sigma_{z}/\sigma_{R}$ measurements for external galaxies and its dependence on the morphological type please see An18, Section 5. The uncertainty in $\sigma_{z}/\sigma_{R}$ is included in the error of $\sigma_{z}$ as described in Section 7.1.2 of An18. We correct $\sigma_{z}$ values for the small broadening introduced by subtracting the local HI velocity to remove galactic rotation. Our results for the stellar $\sigma_{z}$ values for both stellar components are presented in Table 3. The errors are the 1$\sigma$ errors obtained using Monte Carlo simulations. This was done by running 1000 iterations where, in each iteration random Gaussian noise appropriate to the observed SN of the IFU data was added to the best fit spectrum originally returned by pPXF. Then, pPXF was run again on the new spectrum produced in each iteration. The errors are the standard deviations of the distribution of values obtained over 1000 iterations.
6.2 Planetary Nebulae
6.2.1 Removing Galactic Rotation
From the PNe velocity field we also remove the effects of galactic rotation, similarly to the analysis of the IFU integrated light absorption spectra.
We use HI velocity at the position of each of our PNe from the THINGS first moment map and then subtract local HI velocities from the PNe velocities.
These velocities, corrected for the galactic rotation, are henceforth denoted $\rm v_{LOS}$. We use $\rm v_{LOS}$ to calculate the velocity dispersions. As for the VIRUS-W data, the radius and azimuthal angle ($\theta$) of the PNe in the plane of the galaxy were calculated using our estimated PA and angle of inclination, and the $\rm v_{LOS}$ data were then radially binned into 3 bins, each with about 125 PNe. Figure 8 shows the $v_{LOS}$ vs $\theta$ plots in each radial bin before and after the HI velocities were subtracted off. The distribution of the $v_{LOS}$ after the correction for the HI velocities in shown Figure 9 for 3 radial bins respectively. The distributions of the data points already suggest the presence of the two kinematically distinct components as it can not be well fit with the single Gaussian distribution, as shown in Figure 9, and show a clear indication of a cold kinematic component in $v_{LOS}$ in each radial bin.
6.2.2 LOSVD and the Vertical Velocity Dispersion
In each radial bin we remove a few obvious outliers, based on a visual inspection of the velocity histogram of the objects.
A maximum likelihood estimator (MLE) routine was then used to calculate the LOS velocity dispersions and the subsequent $\sigma_{z}$ in each radial bin.
The first iteration in this routine estimates $\sigma_{LOS}$ for the kinematically cold and hot component by maximizing the likelihood for the two-component probability distribution function given by Eqn. 6 in An18.
In order to calculate the surface mass density using Eqn. 2, we need the vertical velocity dispersion of the hot component ($\sigma_{z}$). To determine it, we use a second MLE routine. Two parameters are passed to the function in this stage: $\sigma_{z1}$ and $\sigma_{z2}$ which are the vertical velocity dispersions of the cold and hot components respectively. The $\sigma_{LOS}$ values obtained using Eqn. 6 in An18 are passed to the routine as initial guesses, since the $\sigma_{z}$ will be very close to the value of $\sigma_{LOS}$ for this galaxy. We assume $f=\sigma_{R}/\sigma_{z}=1.7\pm 0.42$ (see Section 6.1.2) and use inclination $i=37^{\circ}$ (see Section 9.1). The PN.S data for the first radial bin are all at radii $<200^{\prime\prime}$, where the rotation curve is close to solid body. In this bin, we assume the radial and azimuthal components of the velocity dispersion are equal i.e. $\sigma_{R}=\sigma_{\theta}$. The PN.S data for the second and third radial bin are all at radii $>200^{\prime\prime}$ where the rotation curve is flat and we use: $\sigma_{R}=\sqrt{2}\sigma_{\theta}$, where $\sigma_{R}$ and $\sigma_{\theta}$ are the in-plane dispersions in the radial and azimuthal directions.
Finally, once the initial guesses are passed to the routine, it calculates the expected $\sigma_{LOS}$ for the hot and cold component at each azimuthal angle ($\theta$) using the relation:
$$\begin{split}&\displaystyle\sigma_{LOS1}^{2}=\frac{\sigma_{\text{z1}}^{2}f^{2}%
}{2}\cos^{2}\theta.\sin^{2}i+\sigma_{\text{z1}}^{2}f^{2}\sin^{2}\theta.\sin^{2%
}i+\sigma_{\text{z1}}^{2}\cos^{2}i\\
&\displaystyle\sigma_{LOS2}^{2}=\frac{\sigma_{\text{z2}}^{2}f^{2}}{2}\cos^{2}%
\theta.\sin^{2}i+\sigma_{\text{z2}}^{2}f^{2}\sin^{2}\theta.\sin^{2}i+\sigma_{%
\text{z2}}^{2}\text{cos}^{2}i\end{split}$$
The subscript 1 and 2 refers to the components of the cold and hot populations respectively. This step depends on the theta-distribution of the PN in each bin. Thus, the routine then proceeds to calculate the probability of a particular v${}_{LOS}$ to be present at that azimuthal angle via the equation:
$$\begin{split}\displaystyle P=&\displaystyle\frac{N}{\sigma_{LOS1}\sqrt{2\pi}}%
\exp\left(\frac{-(\rm v_{LOS}-\mu_{1}\cos\theta.\sin i)^{2}}{2\sigma_{LOS1}^{2%
}}\right)+\\
&\displaystyle\frac{1-N}{\sigma_{LOS2}\sqrt{2\pi}}\exp\left(\frac{-(\rm v_{LOS%
}-\mu_{2}\cos\theta.\sin i)^{2}}{2\sigma_{LOS2}^{2}}\right)\end{split}$$
(4)
In Eqn. 4 $N$ is the value of the fraction of the cold population returned by the routine that calculated $\sigma_{LOS}$ described earlier; $\mu_{1}$ and $\mu_{2}$ are associated with the means of the two components; the $\cos\theta.\sin i$ term factors for any asymmetric drift present in the data. Whether we let the code estimate the terms $\mu_{1}$ and $\mu_{2}$ or if we fix these terms at 0, it made negligible difference to the dispersions. This shows that there is negligible contribution from the asymmetric drift in the data in each of our radial annuli. Eqn. 4 is maximised to return the best fit values for $\sigma_{z}$ for the hot and cold population of PNe. The 1$\sigma$ errors are calculated similar to the method used in the analysis of the VIRUS-W data. We carry out our Monte Carlo error estimation by using the double Gaussian distribution found by our MLE code, to extract about 125 random velocities (i.e same number of objects as in each of our bins). We then use this new sample to calculate the $\sigma_{z}$ of the hot and cold component using our MLE routines. This whole process was repeated 1000 times, recording the dispersions returned in each iteration. The 1$\sigma$ error is then the standard deviation of the distribution of the dispersions returned from these 1000 iterations. The vertical velocity dispersions for the two components returned from the MLE routine is given in Table 4. After extracting $\sigma_{z}$ from $\sigma_{LOS}$, the presence of the cold component seems to remain prominent in bins 1 and 2, but not in bin 3, where the one component model is preferred. However, the distribution of the PNe line-of-sight velocities for bin 3 (Figure 9) still suggests the presence of the two kinematically distinct components.
The dispersions for the PNe were also corrected for the HI dispersion, similarly to the spectra analysis by quadratically subtracting HI velocity dispersion ($\sim$ 6 kms${}^{-1}$) from the values of the vertical velocity dispersions returned by the MLE routine. This dispersion is evaluated by fitting a plane function to the HI velocities over the IFU and calculating the rms scatter of the HI velocities about this plane. We note that this scatter is not the same as the local HI velocity dispersion as measured from the HI profile width. We also correct the measured dispersions for the measurement errors of the individual PNe, as shown in Figure 3 in An18. The rms measuring error in each radial bin is about 6 km s${}^{-1}$. The formal variance of the corrected cold dispersion value at all PNe radial bins is negative. Hence we show its 90% confidence upper limit in Figure 10 and Table 4.
We carried out a comprehensive analysis using the colour-magnitude cut to discriminate between the HII regions and PNe (Figure 3). Unfortunately, removing HII contaminants in disc galaxies is very challenging. There is a small probability that we may still have some HII contaminants in our cold component. However, we do not use the dispersion of our cold component in any further analysis. Our aim is to separate out the cold component (whether cold PNe or HII regions) from the kinematically hot component and then use the $\sigma_{z,\,hot}$ with the scale height for the same population to calculate the surface mass density. Hence, the nature of the objects that contribute to our cold dispersion is irrelevant as long as they are separated out effectively.
7 Vertical velocity dispersion profile
Figure 10 shows the velocity dispersion results obtained from the integrated light VIRUS-W data (points at R = $54^{\prime\prime}$ and $98^{\prime\prime}$) and the planetary nebulae from the PN.S data (3 outer points) in each radial bin. At each radius we show the one component dispersion (cyan markers) and then the hot and cold thin disc dispersion from the double Gaussian fit (black and grey markers).
The solid curve in Figure 10 is an exponential fit to the hot component data in the form $\sigma_{z}\,(R)=\sigma_{z}\,(0)\exp\,(-R/2h_{dyn})$, as expected from Eq. 2 if the total surface density $\Sigma_{T}$ is exponential in radius and has a radially constant scale height $h_{z}$. From this exponential fit we obtain the best-fit dynamical scale length to be $h_{dyn}$ = 92" which is in excellent agreement with the scale length measured for the 3.6 $\rm\mu$m band as $\rm I\,(R)=I\,(0)\exp\,(-R/h_{r,[3.6]})$ (Table 2).
The fitted central velocity dispersion for the hot component is $\sigma_{z}\,(0)=87.8\pm 4.9$ km s${}^{-1}$. We recall that it is this hot component dispersion that should be used with the derived scale height from Section 4 to estimate the total surface mass density of the disc. If we assume a single homogeneous population of tracers and fit the same exponential function to the one-component dispersions (cyan markers in Figure 10), we find the best-fit dynamical scale length to be $h_{dyn}$ = 120" and the central vertical dispersion $\sigma_{z}\,(0)=49.2\pm 5.1$ km s${}^{-1}$, which is $\sim 50\%$ smaller than the central velocity dispersion from the fit to the hot component. The use of this one-component dispersion for the calculation of the surface mass density would underestimate the surface density by a factor of $\sim 2$, which would be enough to make the maximal disc look sub-maximal, but with a gradient in the mass-to-light ratio (see the results from Herrmann &
Ciardullo (2009b) about the increase in the disk mass-to-light ratio in the outer disc).
We note that the difference between the one component dispersions and the hot component dispersions decreases with radius and is almost zero in the outermost radial bin (Figure 10). This is consistent with the BIC values shown for the outer radial bin, which favours the one component model (Table 4).
8 Stellar Surface Mass Density
Using Eqn. 1 with $f=1/2\pi$ (isothermal model, see Appendix 1 in An18) and measured $\rm h_{z}$ (Table 2) and $\sigma_{z}$ from the hot disc (Table 5, column 2), we calculate the total surface mass density ($\Sigma_{T}$) in each radial bin (Table 5, column 3).
The stellar mass surface density of the hot disc ($\Sigma_{D}$) is $\Sigma_{D}=\Sigma_{T}-(\Sigma_{C,*}+\Sigma_{C,gas})$, where $\Sigma_{C,*}+\Sigma_{C,gas}$ is the surface density of a cold thin layer made up of the cold thin disc of stars and the gaseous disc of the galaxy. While we can directly measure $\Sigma_{C,gas}$ (see Figure 5 and Table 5, column 4), the cold stellar population contribution $\Sigma_{C,\,*}$ is not known directly from our data.
We can write the equation for the surface densities of the stellar disc components as:
$$\Sigma_{D}=\frac{\Sigma_{T}-\Sigma_{C,\,\rm gas}}{1+F_{C}},~{}~{}{\rm where}~{%
}~{}~{}F_{C}\,=\frac{\Sigma_{C,\,*}}{\Sigma_{D}}.$$
(5)
We can estimate the ratio of luminosities $L_{C}$/$L_{D}$ of the cold and hot stellar layers from the integrated spectra. These luminosity ratios are given in Table 5 (column 5) for the first two radial positions where the dispersions come from integrated light. From these ratios, we can then estimate the ratios of the surface densities of the cold and hot layers, using stellar population models Bruzual &
Charlot (2003).
We showed in An18 (Figure 12) that the thin disc stars in the solar neighbourhood older than about 3 Gyr have vertical velocity dispersions almost independent of age, at about $20$ km s${}^{-1}$. Stars with ages younger than about 3 Gyr show a strong age-velocity dispersion relation, rising from about $10$ km s${}^{-1}$ for the youngest stars to about $20$ km s${}^{-1}$ at 3 Gyr. We identify stars with ages $>3$ Gyr as the old hot population denoted D in Table 5, and stars with ages $<3$ Gyr as the young cold populations denoted C${}_{*}$ in Table 5.
The age difference of the cold and hot stellar populations is also indicated visually by the spectra of the best-fit stellar templates used in the spectral fitting analysis (Figure 7): the colder population looks younger, with its stronger $H_{\beta}$ and (slightly) weaker metallic lines. Figure 11 shows an extended section of the pPXF spectra including the H$\beta$ region. This region was excluded from the kinematical analysis, thus is not shown in Figure 7. Three trends are evident in the ratio of the cold/hot spectra: a) H$\beta$ is stronger in the spectrum for the kinematically colder component, and shows broader wings; b) the metallic lines such as the Mgb triplet and the 5270 Å and 5320 Å Fe-lines are stronger in the kinematically hotter component and c) the continuum for the colder component is somewhat bluer than for the hotter component, as seen by reference to the horizontal dashed line. These three trends are consistent with the colder population being younger, and follow naturally if the mean temperatures of the stars which dominate the spectrum are lower in the hotter component. We note that only the region of spectrum from 5040 to 5310 $\rm\mathring{A}$ was used in the pPXF fit.
In An18, we then showed that, if the star formation rate in the disc has decayed with time like $\exp\,(-t/\tau)$, and if $\tau>3$ Gyr, then the ratio of ($M/L$ for the cold component) to ($M/L$ for the hot component) is about $0.167$, almost independent of $\tau$.
Assuming NGC 6946 has a similar age-velocity relation as in the solar neighbourhood, i.e the old thin disc is composed of stars with ages between 3 – 10 Gyr and assuming a star formation rate that decays with time like $\exp(-t/\tau$), then we can derive the ratio of surface densities of young and old populations given as $F_{C}$ in Table 5 (column 6) for the first two radial positions. For the three radial positions using planetary nebulae as tracers ($R=4.3$ to $9.9$ kpc), we cannot use these arguments, because of possible contamination of the sample by HII regions, and because the lifetimes of planetary nebulae vary strongly with progenitor mass (Miller
Bertolami, 2016). For columns 5 and 6 of Table 5, we therefore adopt the means of the values found from our integrated light spectra in the first two radial positions. The values of $\Sigma_{D}$ and $\Sigma_{C,*}$ in columns 7 and 8 follow from Eqn. 5. The errors for $\Sigma_{T}$, shown in Table 5 are relatively large (30 to 40 % for most radial bins) - these errors include errors in all spectroscopic and photometric parameters (scale length and scale height) used to evaluate $\Sigma_{T}$, and therefore following simple error propagation overestimate the relative errors between radial bins.
Table 5 (columns 9 to 12), gives the total stellar mass-to-light ratios for ($\Sigma_{D}+\Sigma_{C,*})$ at each radius in BVI and 3.6 $\mu$m photometric bands (Figure 4). Prior to derivation of the mass-to-light ratio all photometric magnitudes were corrected for the Galactic extinction and inclination effects.
From Table 5 it is visible that values of the mass-to-light ratio (M/L, $\Upsilon_{\star}$) vary with radius in every band differently from some previous studies (Martinsson
et al., 2013; Swaters et al., 2014). In our approach, even if the light declines in the same way as mass ($\rm h_{phot}=h_{dyn}$), the ratio of the cold-to-hot component and the contribution from gas mass also changes with radius, contributing to radial variation of the M/L values. However, we also do not exclude the contribution of the observational errors to this radial change in M/L. Interestingly, the mean value of the $\Upsilon_{\star}$[3.6] is equal to 0.4, which is lower in comparison with the current stellar population models which assume constant $\Upsilon_{\star}$[3.6]=0.6 (Meidt
et al., 2012; Röck et al., 2015). This lower value of $\Upsilon_{\star}$[3.6] can be explained with the very high star formation rate for this galaxy ($4.76\,M_{\odot}yr^{-1}$, Walter et al., 2008). As was shown by Querejeta
et al. (2015) the flux of the Spitzer 3.6 $\mu$m band represents not only the light from the old stellar population, but also that emitted by the warm dust heated by young stars and re-emitted at longer wavelengths, and its contribution can be as high as 30% at the regions of high star formation activity. Moreover AGB stars also peak at 3.6 $\mu$m, significantly contributing into the the total flux, but not into the stellar mass of a galaxy. Ponomareva et al. (2017) also shown that the use of the 3.6 $\mu$m luminosities corrected for the non-stellar contamination can even decrease the scatter in the Tully-Fisher relation. Thus, if we assume the non-stellar contamination of $\sim$ 30 % for NGC 6946, the values of the $\Upsilon_{\star}$[3.6] will increase towards the mean value of $\sim$ 0.7. Moreover, the mean value of $\Upsilon_{\star}$[3.6]=0.4 is in agreement with the $\Upsilon_{\star}$[3.6] derived as a function of the [3.6]–[4.5] colour for a sample of spiral galaxies (Ponomareva et al., 2018, see Eqn.13.)
In comparison with NGC 628 (An18) we find the mean $\Upsilon_{\star}$ to be lower for the NGC 6946 independently of a photometrical band. This is consistent with the difference in total star formation rate between the two galaxies: Walter et al. (2008) give the SFR for NGC 628 as $1.21M_{\odot}\,yr^{-1}$ and for NGC 6946 as $4.76M_{\odot}\,yr^{-1}$, indicating that the stellar population in NGC 628 is likely to be older in the mean. This is also consistent with the above-mentioned non-stellar contamination in the 3.6 $\mu$m band.
9 Rotation Curve Decomposition
9.1 Observed HI rotation curve
We derive the observed rotation curve of NGC 6946 using THINGS data (Walter et al., 2008). As the galaxy is large and well-resolved we use the 2D tilted-ring modelling approach (Begeman, 1989) to derive the rotation curve from the observed velocity field. First the velocity field was constructed using the Gauss-Hermit polynomial fitting function and then corrected for the skewed velocity profiles which reflect random motions (Ponomareva et al., 2016). The results of the tilted-ring modelling are shown in Figure 12. We find the position and inclination angles to be 242 and 37 degrees respectively. We fix these values to derive the final rotation curve for the receding and approaching sides of the galaxy, shown in the bottom panel of Figure 12. It is clear that the rotation curve of NGC 6946 is well-behaved, reaching the flat part at $\sim$ 170". The difference between the approaching and receding sides of the rotation curve was adopted as the error on the rotational velocity.
9.2 Stellar distribution
The rotation curve and the 3.6$\mu$m surface brightness profile of NGC 6946 indicate the presence of the small bulge component. As our spectral observations do not cover the central region of the galaxy, the bulge does not contribute to the derived total surface mass density. Therefore, to include the contribution from the bulge to the total observed rotation curve, we fit the bulge component to the 3.6 $\mu$m profile (Figure 4) and then convert its luminosity into mass using $\Upsilon_{\star}[3.6]=0.6$ (Meidt
et al., 2012; Querejeta
et al., 2015; Röck et al., 2015).
Thus, we have all of the baryonic components contributions and we model their rotation curves using a spherical potential for the bulge and the exponential total disc component, using the derived central surface mass density ($\Sigma_{0}=758.84\pm 162$ M${}_{\odot}$ pc${}^{-2}$), dynamical scale length ($\rm h_{dyn}=2.72$ kpc) and I-band scale height ($\rm h_{z}=376$ pc), inferred from the measured I-band scale length (Section 4). We model the rotation curves of all the baryonic components with the same radial sampling as was used for the derivation of the observed rotation curve.
9.3 Mass modelling
The total rotational velocity of a spiral galaxy can be presented as the quadratic sum of the rotational velocity for its baryonic components and a dark halo:
$$V^{2}_{tot}=V^{2}_{bar}+V^{2}_{halo},$$
(6)
where $V_{bar}$ is the rotational velocity of the baryonic components of a galaxy (stars and gas) and $V_{halo}$ is the velocity of a dark matter halo.
From our analysis we have direct measurements on all baryonic components of the disc, using dynamically obtained surface mass density, and of the bulge from the 3.6$\mu$m profile. Thus we can model the total baryonic rotational velocity curve and then fit the rotation curve of the dark matter halo so that $V_{tot}$ matches $V_{obs}$ as closely as possible.
Since we have directly measured total surface mass density of the disc we do not have the usual uncertainties related to the disc $M/L$ that are a major concern in decomposing rotation curves.
We estimate the maximum rotational velocity of the baryonic rotation curve (blue line in Figure 13) at $\rm 2.2h_{dyn}$ to be equal to $\rm V_{max}(bar)=130$ km s${}^{-1}$ with the typical error of 13% due to the error on the central surface brightness and the negative covariance between the $\rm h_{R,dyn}$ and the $\sigma_{z}(0)$ (see An18 for more details). In comparison with the maximum velocity of the total rotation curve at $\rm 2.2h_{dyn}$ ($\rm V_{max}=170\pm 10$ km s${}^{-1}$) we find our disc to be closer to maximal with $\rm V_{max}(bar)=0.76(\pm 0.14)V_{max}$.
For our further analysis we use two different dark matter halo models: the pseudo isothermal (pISO) and the NFW halo.
The pISO halo rotation curve is parametrised by its central core density $\rho_{0}$ and its core radius $R_{c}$:
$$V_{DM}^{pISO}(R)=\sqrt{4\pi G\rho_{0}R^{2}_{c}\Big{[}1-\frac{R_{c}}{R}{\rm tan%
}^{-1}\Big{(}\frac{R}{R_{c}}\Big{)}\Big{]}},$$
(7)
while the NFW halo (Navarro
et al., 1997) is parameterised by its circular velocity $V_{200}$ at the virial radius $R_{200}$ and its central concentration $c$:
$$V_{DM}^{NFW}(R)=V_{200}\bigg{[}\frac{ln(1+cx)-cx/(1+cx)}{x[ln(1+c)-c/(1+c)]}%
\bigg{]}^{1/2}$$
(8)
where $x=R/R_{200}$.
Thus, we keep the rotation curve of the baryons fixed, and use the Gipsy (van der Hulst et al., 1992) task ROTMAS to fit the rotation curves for the dark haloes. The results from this modelling is presented in Figure 13 (the top panel is for the pISO halo and the bottom panel for the NFW halo). The derived parameters of the fitted dark matter rotation curves are presented in Table 6. The error on the total rotational velocity (blue line in Figure 13) follows from the error on the central surface density.
Although the rotation curve shapes for the pISO and NFW halos are similar, the pISO rotation curve is a slightly better fit, with $\chi_{red}^{2}=0.93$ versus $\chi_{red}^{2}=1.1$ for the NFW halo.
Figure 14 shows the $\chi^{2}$ maps of the 2D parameter space between the main parameters of the dark matter haloes. It is clear that derived dark halo parameters for the pseudo isothermal and NFW models have significant covariance: the coloured ellipses in each panel represent the 1$\sigma$-5$\sigma$ values from inside to outside. The errors quoted in Table 6 are the 1$\sigma$ uncertainties.
10 Conclusions
We use absorption line spectra in the inner regions and planetary nebulae in the outer regions of the galaxy NGC 6946 to trace the kinematics of the disc. We show that there exists a younger, kinematically colder population of tracers
within an older and hotter component.
When attempting to break the disc-halo degeneracy by measuring the surface mass density of the disc, using the velocity dispersion and the estimated scale height of the disc, it is crucial that the dispersion and the scale height pertain to the same population of stars. The scale height, obtained from NIR studies of edge-on galaxies is for the older population of thin disc stars. We use this scale height with the dispersion of the hotter component to calculate the surface mass density. This density is a factor of 2.3 times greater than the surface density we would get if we assume a single homogeneous population of tracers. This factor is large enough to make the difference between concluding that a disc is maximal or sub-maximal.
We find that the observed vertical velocity dispersion of the hotter component follows an exponential radial decrease. In comparison, the central velocity dispersions for the hot components in NGC 628 and NGC 6946 are not differ by much: 73.6 $\pm$ 9.8 km s${}^{-1}$ for NGC 628 and 87.8 $\pm$ 4.9 for NGC 6946. The B-band magnitudes from Walter et al. (2008) are -19.97 and -20.61 respectively, suggest that the brighter galaxy has a higher central $\sigma_{z}$.
The dynamical scale length of the galaxy (derived by fitting $\sigma_{z}\,(R)=\sigma_{z}\,(0)\exp\,(-R/2h_{dyn})$ to the hot component velocity
dispersions in Figure 10) agrees well with the photometric scale length of the galaxy (derived by fitting $\rm I\,(R)=I\,(0)\exp\,(-R/h_{r,[3.6]})$ to the 3.6 $\mu$m surface brightness distribution in Figure 4). In this case the expected $M/L$ in this galaxy should be close to constant over the radial region probed by our study (see Section 8). We find that in all four photometric bands (BVI and 3.6 $\mu$m) the $M/L$ varies with radius, but within the errors being consistent with a radially constant value. Moreover, we find that $M/L$ in the 3.6 $\mu$m band has a lower value than assumed by single stellar population models (Meidt
et al., 2012; Querejeta
et al., 2015; Röck et al., 2015). Interestingly, its mean value over all radii agrees well with the $\Upsilon_{\star}$[3.6] derived as a function of the [3.6]–[4.5] colour for a sample of spiral galaxies (Ponomareva et al., 2018). We suggest that lower values of the $\Upsilon_{\star}$[3.6] are due to the contamination from dust and AGB stars of the 3.6 $\mu$m flux. The maximum correction of 30 % (Querejeta
et al., 2015) would increase the mean $\Upsilon_{\star}$[3.6] by a factor of two.
Decomposing the rotation curve of this galaxy, after taking into account the hot and cold stellar components, leads to a maximal disc ($\rm V_{max}(bar)=0.76(\pm 0.14)V_{max}$). The disc contributes about 76% of the rotation curve at its peak, which is consistent with our previous study of NGC 628 (78%). The molecular gas makes an unusually large contribution in the inner parts of this galaxy, and the baryons together dominate the radial component of the gravitational field out to a radius of about 8 kpc.
acknowledgements
We thank Maximilian Fabricius for his help with the VIRUS-W
data reduction and analysis. We thank the
anonymous referee for comments and suggestions which significantly
improved this paper. We acknowledge comments and suggestions from Matthew Bershady.
SA would like to thank ESO for the ESO studentship that helped
support part of this work. AAP, KCF, MA, OEG acknowledge the support
of the Australian Research Council Discovery Project grant
DP150104129. AAP acknowledges the support of the STFC consolidated grant
ST/S000488/1, the VICI grant 016.130.338
of the Netherlands Foundation for Scientific Research (NWO),
and the Leids KerkhovenBosscha Fonds (LKBF) for travel support.
The authors are very grateful to the staff at McDonald
Observatory for granting us the observing time and support on the 107
inch telescope. The authors would also like to thank the Isaac Newton
Group staff on La Palma for supporting the PN.S over the years, and
the Swiss National Science Foundation, the Kapteyn Institute, the
University of Nottingham, and INAF for the construction and deployment
of the H$\alpha$ arm.
Data availability
All data used in this paper is available on request to the corresponding authors.
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Determining pressure-temperature phase diagrams of materials
Robert J. N. Baldock${}^{*}$
${}^{*}$Cavendish, ${}^{\dagger}$University Chemical and ${}^{\ddagger}$Engineering Laboratories, University of Cambridge, Cambridge, United Kingdom
Lívia B. Pártay${}^{\dagger}$
${}^{*}$Cavendish, ${}^{\dagger}$University Chemical and ${}^{\ddagger}$Engineering
Laboratories, University of Cambridge, Cambridge, United Kingdom
Albert P. Bartók${}^{\ddagger}$
${}^{*}$Cavendish, ${}^{\dagger}$University Chemical and ${}^{\ddagger}$Engineering
Laboratories, University of Cambridge, Cambridge, United Kingdom
Michael C. Payne${}^{*}$
${}^{*}$Cavendish, ${}^{\dagger}$University Chemical and ${}^{\ddagger}$Engineering
Laboratories, University of Cambridge, Cambridge, United Kingdom
Gábor Csányi${}^{\ddagger}$
${}^{*}$Cavendish, ${}^{\dagger}$University Chemical and ${}^{\ddagger}$Engineering
Laboratories, University of Cambridge, Cambridge, United Kingdom
(December 3, 2020)
Abstract
We extend the Nested Sampling algorithm to simulate materials under periodic boundary and constant pressure conditions, and show how it can be efficiently used to determine the phase diagram directly from the potential energy in a highly automated fashion. The only inputs required are the composition and the desired pressure and temperature ranges, in particular solid-solid phase transitions are recovered without any a priori knowledge about the structure of solid phases. We apply the algorithm to the Lennard-Jones system, aluminium, and the NiTi shape memory alloy.
Phase diagrams of materials describe the regions of stability and equilibria of structurally distinct phases and are crucial in both fundamental and industrial materials science. In order to augment experiments, computer simulations and theoretical calculations are often used to provide reference data and describe phase transitions. Although there exist a plethora of methods to determine phase boundaries, such as Gibbs ensemble Monte CarloPanagiotopoulos (1987), Gibbs–Duhem integrationKofke (1993a), thermodynamic integration or even direct coexistence simulations, they all require specific expertise and separate setup for each type of phase transition. Moreover, in case of solid phases, where most of the interest lies, advance knowledge of the crystal structure of each phase is required.
Methods that systematically explore the potential energy landscape, such as parallel tempering (also known as replica exchange)Swendsen and Wang (1986); Frantz et al. (1990) and Wang-LandauWang and Landau (2001), are potential alternatives, but are invariably hampered by convergence problems due to the entropy jump at a first order transition: the probability distributions (parametrised in terms of temperature in case of
parallel tempering or energy in case of Wang-Landau) on the two sides of the phase transition have very little overlap resulting in a combination of low acceptance rates and poor exploration.
The Nested Sampling (NS) algorithmSkilling (2004, 2006) was designed to solve this problem. It constructs a sequence of uniform distributions bounded from above by a sequence of decreasing potential energy levels, $\{E_{i}\}$, with the property that each level encloses a volume $\chi_{i}$ of configuration space that is approximately a constant factor smaller than the volume, $\chi_{i-1}$, corresponding to the previous level. Hence, each distribution will have an approximately constant fractional overlap with the one immediately before and after, ensuring fast convergence of the sampling and allowing an accurate evaluation of phase space integrals. In particular, the energy level spacings near the phase transition will be very narrow. The sequence of energy levels comprise a discretisation of the cumulative density of states $\chi(E)$, which allows the evaluation of the partition function at arbitrary temperatures,
$$\displaystyle Z(N,V,\beta)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{N!}\left(\frac{2\pi m}{\beta h^{2}}\right)^{3N/2}\int dEe%
^{-\beta E}\chi^{\prime}(E)$$
(1)
$$\displaystyle\approx$$
$$\displaystyle Z_{m}(N,\beta)\sum_{i}(\chi_{i-1}-\chi_{i})e^{-\beta E_{i}}$$
(2)
where $N$ is the number of particles of mass $m$, $V$ is the volume, $\beta$ is the inverse temperature, $h$ is Planck’s constant, the density of
states $\chi^{\prime}$ is the derivative of $\chi$, and we labelled the factor resulting from the momentum integral as $Z_{m}$. The total phase space volume is $\chi_{0}=V^{N}$ corresponding to the ideal gas limit. Note that the sequence of energies and volumes are independent of temperature, so the partition function can
be evaluated at any temperature by changing $\beta$ in (2).
Since its inception Nested Sampling has been used successfully in astrophysics Feroz and Hobson (2008), and also to investigate the potential energy landscapes of atomistic systems ranging from clusters to proteinsPártay et al. (2010, 2014); Burkoff et al. (2012); Do et al. (2011, 2012, 2013); Martiniani et al. (2014).
Here we modify the algorithm to allow for a variable unit cell with periodic boundary conditions, thus enabling the determination of constant-pressure heat capacities and hence pressure-temperature phase diagrams of materials directly from the potential energy function without recourse to any other a priori knowledge.
The basic NS algorithm is as follows. We initialise by generating a pool of $K$ uniformly random configurations and iterate the following loop starting at $i=1$,
1.
Record the energy of the sample with the highest energy as $E_{i}$, and use it as the new energy limit, $E_{\mathrm{limit}}\leftarrow E_{i}$. The corresponding phase space volume is $\chi_{i}\approx\chi_{0}[K/(K+1)]^{i}$.
2.
Remove the sample with energy $E_{i}$ from the pool and generate a new configuration uniformly random in the configuration space, subject to the constraint that its energy is less than $E_{\mathrm{limit}}$. One way to do this is to clone a randomly chosen existing configuration and make it undergo a random walk of $L$ steps, subject only to the energy limit constraint.
3.
Let $i\leftarrow i+1$, and return to step 1.
At each iteration, the pool of $K$ samples are uniformly distributed in configuration space with energy $E<E_{\mathrm{limit}}$. The finite sample size leads to a statistical error in $\log\chi_{i}$, and also in the computed observables, that is asymptotically proportional to $1/\sqrt{K}$, so any desired accuracy can be achieved by increasing $K$. Note that for any given $K$, the sequence of energies and phase volumes converge exponentially fast (the number of
iterations required to obtain results shown below never exceeded $2000\cdot K$), and increasing $K$ necessitates a new simulation from scratch. In this, NS is similar to the Wang-Landau method, and in contrast with the case of parallel tempering in which an existing Markov chain can be extended to an arbitrary number of steps to improve convergence.
We now modify the algorithm for the constant pressure case. The integration in (1) needs to be extended over all volumes and all shapes of a periodic unit cell.
The partition function describing the system at isotropic pressure $p$ isMartyna et al. (1994); Yu et al. (2010),
$$\displaystyle\Delta(N,p,\beta)$$
$$\displaystyle=Z_{m}\beta p\int_{0}^{\infty}\!\!dVV^{N}\int d\mathbf{h}_{0}%
\delta\left(|\mathbf{h}_{0}|-1\right)\times$$
(3)
$$\displaystyle\int\!d\mathbf{s}\,e^{-\beta(E(\mathbf{s},V,\mathbf{h}_{0})+pV)}.$$
where $\mathbf{h}$ is the $3\times 3$ matrix of lattice vectors relating the Cartesian positions of the atoms $\mathbf{r}$ to the fractional coordinates $\mathbf{s}$ via $\mathbf{r}=\mathbf{hs}$, $V=|\mathbf{h}|$ is the volume, and $\mathbf{h_{0}}=\mathbf{h}V^{-1/3}$ is the image
of the unit cell normalised to unit volume. NS is performed at fixed pressure to generate a sequence of enthalpies, $H_{i}$, where $H=E\left(\mathbf{s},V,\mathbf{h}_{0}\right)+pV$. We split the volume integral into two by imposing an upper limit of $V_{0}$ (approximating the dilute limit of the ideal gas) on the numerical integration and incorporate the factor $V^{N}$ into the measure by drawing samples
with volumes proportional to $V^{N}$. Together with the contribution of the tail, corresponding to the ideal gas is, we have
$$\displaystyle\Delta(N,p,\beta)=Z_{m}\beta p\Biggl{(}\Delta_{\mathrm{NS}}\left(%
N,p,\beta,V_{0}\right)+$$
(4)
$$\displaystyle{\cal O}(1)\times\frac{\Gamma(N+1,\beta pV_{0})}{(\beta p)^{N}}%
\Biggr{)}$$
where $\Gamma$ is the upper incomplete Gamma function, and the ${\cal O}(1)$ factor arises from the integration over lattice shapes as explained in the Supplementary Information, and has no material bearing on the numerical results we report below. The first term is computed using the samples generated by NS, as
$$\Delta_{\mathrm{NS}}\left(N,p,\beta,V_{0}\right)\approx\frac{V_{0}^{N+1}}{N+1}%
\sum_{i}(\chi_{i-1}-\chi_{i})e^{-\beta H_{i}}.$$
(5)
The precise setting of $V_{0}$ is not
important as long as the $pV$ term dominates over
the potential term in the enthalpy for volumes beyond $V_{0}$, which in principle does depend on the pressure and temperature range of interest, but in practice it is easy to find values suitable for physically relevant conditions. We typically use $V_{0}=10^{7}N$.
We use single atom Monte Carlo (MC) moves in fractional coordinates with the amplitude updated every $\frac{K}{2}$ iterations to maintain a good acceptance rate. Uniform sampling of lattice shapes was achieved by independent
shearing and stretching moves which do not change the volume, while $\mathbf{h}_{0}$
was also constrained not to be too oblique by rejecting moves that would result in the height of the unit cell (normalised to unit volume) less than 0.7. The ratios of the atom, volume, shear and stretch moves were $N:10:1:1$. Derivations of the above formulae and further details on the MC moves are given in the Supplementary Information. Given the partition function, phase transitions can be easily located by finding the peaks of response functions such as the heat capacity, given by
$$c_{p}=\left(\frac{\partial H}{\partial T}\right)_{p}=\left(-\frac{\partial}{%
\partial T}\right)_{p}\frac{\partial\ln\Delta(N,p,\beta)}{\partial\beta}.$$
(6)
By performing separate NS simulations for a range of pressures and combining the pressure and temperature values corresponding to the heat capacity peaks one can straightforwardly construct the entire phase diagram including all thermodynamically stable phases.
To demonstrate the efficacy of NS, we show the phase diagram of the periodic Lennard-Jones model in Figure 1. Most of the phase diagram is accurately recovered using just 64 particles, with finite size errors only apparent for sublimation at low pressures. NS provides a reasonable estimate of the melting and boiling points using only $\sim 10^{8}$ energy evaluations, while parallel tempering needs many orders of magnitude more computational effort than NS to find the evaporation transition and almost two orders of magnitude more computational effort to find the melting transition. (A similar increase in computational efficiency compared with parallel tempering was found for LJ clustersPártay et al. (2010) and hard spheresPártay et al. (2014); Odriozola (2009).)
For our next example we consider aluminium. As one of the most commonly used metals, the thermodynamic properties of
aluminium have been extensively studied. The melting line of aluminium has
been measured up to 125 GPaErrandonea (2010); Boehler and Ross (1997); Hänström and Lazor (2000); Shaner et al. (1984),
with good agreement between the different experimental techniques, and theoretical
calculations were performed using embedded-atom type potentials Foiles and Daw (1986); Voter and Chen (1987); Oh and Johnson (1988); Mei and Davenport (1992); Morris et al. (1994); Ercolessi and Adams (1994); Liu et al. (2004); Mishin et al. (1999)
and ab initio methodsde Wijs et al. (1998); Vočadlo and Alfè (2002); Alfè et al. (2004),
the latter providing melting temperatures up to 350 GPaBouchet et al. (2009).
At ambient conditions aluminium crystallises in the face-centred-cubic (fcc) structure, but a phase transition to the hexagonal-close-packed (hcp) structure at 217 GPa has been revealed by X-ray diffraction experiments Akahama et al. (2006) and the body-centred-cubic (bcc) phase has been also produced in laser-induced microexplosionsVailionis et al. (2011).
The critical points of most metals are not amenable to conventional experimental study and thus estimation of their properties is usually based upon empirical relationships between the critical temperature and other measured thermodynamic properties.
In case of aluminium these result in predictions in a wide temperature and pressure range Renaudin et al. (2003); Likalter (2002); Fortov and Iakubov (2000); Young and J (1971).
We chose four widely used models all based on
the embedded-atom method (EAM): (1) the model developed by Liu et al.Liu et al. (2004) (LEA-EAM), which is an improved version of the original potential of Ercolessi and AdamsErcolessi and Adams (1994), (2) the model developed by Mishin et al.Mishin et al. (1999) using experimental and ab initio data as well (Mishin-EAM), (3) the EAM of Mei and DavenportMei and Davenport (1992) (MD-EAM) and (4) the recently modified version of the MD-EAM, reparametrised by Jasper et al. to accurately reproduce the DFT energies for Al clusters and nanoparticles of various sizes (NPB-EAM)Jasper et al. (2005).
The phase diagrams for all four models based on NS simulations with 64 particles are shown in Figure 2.
The resulting critical parameters vary over a wide range for the different models.
Above the critical point, the heat capacity peak corresponding to evaporation does not diminish immediately but broadens gradually resulting in the Widom-line, shown by the points with large error bars that correspond to the width of the peak.
The melting lines are in a good agreement with the available experimental data up to the pressure value $p\approx 25$ GPa. Above that the melting curves of the different potentials diverge from the experimental results, except for the MD-EAM potential, which reproduces melting curve remarkably well.
At higher pressures a small peaks appear on the heat capacity curves below the melting temperature for
all models, indicating solid-solid phase transitions. We post processed the samples from the NS simulations which revealed that while at low pressures the fcc structure is the most stable for all four models as expected, the models differ markedly in their predictions for high pressure phases, with the only commonality being that their prediction for the upper critical pressure for the stability of the fcc phase is far too low in comparison with experiment and density functional theoryAkahama et al. (2006); Boettger and Tricke (1996); Sinâko and Smirnov (2002).
Finally, we show preliminary results for a problem of current scientific interest, the NiTi shape memory alloyBuehler et al. (1963). The shape memory effect relies on the structural phase transition from the high temperature austenitic phase (cubic B2 structure) to the low temperature martensitic phaseBhattacharya (2003). Figure 3 shows the pressure-temperature-composition phase diagram corresponding to a recent EAM modelZhong et al. (2011). The NS results for the phase transition temperature are within 50 K of the experimental value, reproduce the trend with compositional change, and predict a decreasing critical temperature with increasing pressure. It is notable that this EAM model seems successful here despite not reproducing the experimentally observed B19’ structure at low temperature. By inspecting the configurations near the end of the NS simulation, we found that the potential has a number
of different low symmetry minima with energies all within a few meV of each other. A more detailed study of NiTi will be presented elsewhere.
In summary, we have extended the Nested Sampling algorithm to allow simulations of periodic systems under constant pressure conditions and demonstrated how it can be used to determine pressure-temperature-composition phase diagrams. In contrast to existing methods for comparing specific phases, NS explores the entire configuration space without requiring any prior knowledge about the structures of different solid phases with the only necessary input
being the composition and the desired pressure and temperature ranges. We
suggest that this makes it eminently suitable for validating materials models,
and in the future could even play a role
in the automatic optimisation of empirical models.
Acknowledgements.
RJNB acknowledges support from the EPSRC. LBP acknowledges support from St. Catharine’s College, Cambridge and to the Royal Society. APB acknowledges support from Magdalene College, Cambridge, the Leverhulme Trust and the Isaac Newton Trust. GC acknowledges EPSRC grant EP/J010847/1. Computer time was provided via the UKCP consortium funded by EPSRC grant ref. EP/K013564/1 and via the Darwin supercomputer in the University of Cambridge High Performance Computing Service funded under EPSRC grant EP/J017639/1..
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Strong approximation of Bessel processes††thanks: This work has been supported by the project PERISTOCH ANR–19–CE40–0023, 2020–2024 of the French National Research Agency (ANR)
Madalina Deaconu${}^{1}$ and Samuel Herrmann${}^{2}$
${}^{1}$Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France,
[email protected]
${}^{2}$Institut de Mathématiques de Bourgogne (IMB) - UMR 5584, CNRS,
Université de Bourgogne Franche-Comté, F-21000 Dijon, France
[email protected]
Abstract
We consider the path approximation of Bessel processes and develop a new and efficient algorithm. This study is based on a recent work by the authors, on the path approximation of the Brownian motion, and on the construction of specific own techniques. It is part of the family of the so-called $\varepsilon$-strong approximations. More precisely, our approach constructs jointly the sequences of exit times and corresponding exit positions of some well-chosen domains, the construction of these domains being an important step. Based on this procedure, we emphasize an algorithm which is easy to implement. Moreover, we can develop the method for any dimension. We treat separately the integer dimension case and the non integer framework, each situation requiring appropriate techniques. In particular, for both situations, we show the convergence of the scheme and provide the control of the efficiency with respect to the small parameter $\varepsilon$. We expand the theoretical part by a series of numerical developments.
Key words: Strong approximation, path simulation, Bessel process, Brownian exit time.
2010 AMS subject classifications: primary
65C05; secondary
60J60, 60J25, 60G17, 60G50.
Introduction
Diffusion processes play a central role in the modelling and study of the behaviour of physical phenomena, of biological problems or of financial products pricing, it is thus of prime interest to develop numerical approaches to characterize and analyze their stochastic trajectories. However, a trajectory is an infinite mathematical object which cannot be generated directly, an approximation procedure and its corresponding error control need therefore to be emphasized.
The Euler scheme is one of the classical standard schemes to get numerical approximated solutions of stochastic differential equations. Indeed, a first method to approximate stochastic processes is the common time-discretization procedure: only the values of the diffusion process on some finite deterministic time grid $t_{1}<t_{2}<\ldots<t_{n}$ are described (or approximated), as in the usual Euler or modified Euler scheme and the literature contains many convergence results in the small time step limit.
In the standard case, that is under the conditions that ensure the existence and uniqueness of the solution of the SDEs, the numerical analysis is well-developed and a large variety of different numerical approximation schemes is available. We refer, in this framework of time splitting procedure to the important work [13].
A huge literature is studying this approach and we can find results on the weak convergence [1] or the strong convergence. It is thus well-known, that under suitable conditions on the coefficients of the SDEs, the Euler scheme has strong rate of convergence $1/2$, [13]. A very good review on the results on the Euler method and its higher-order extensions can be found in [11], and for particular diffusions interesting techniques are developed for instance in [8], [9], [14], and many others.
Such schemes often suffer in terms of efficiency as the computational time is high and it is difficult to solve the trade-off between reducing the error and obtaining a satisfactory computational time. Thus, in non standard cases, other methods need to be developed.
An alternative approach for one-dimensional diffusions is to squeeze the stochastic trajectory $(X_{t})_{t\geq 0}$ under observation inbetween two simple to exhibit paths, depending on a small parameter $\varepsilon>0$: an upper and a lower trajectory $(X_{t}^{\uparrow,\varepsilon})_{t\geq 0}$ respectively $(X_{t}^{\downarrow,\varepsilon})_{t\geq 0}$. Obviously, these two curves need first to be easy to generate numerically (we should avoid infinite mathematical objects) and secondly be such that their difference can be controlled with respect to the parameter $\varepsilon$: on any finite time interval $[0,T]$,
$$X_{t}^{\uparrow,\varepsilon}\geq X_{t}\geq X_{t}^{\downarrow,\varepsilon},\quad\forall t\in[0,T]\quad\mbox{and}\quad\lim_{\varepsilon\to 0}\sup_{t\in[0,T]}(X_{t}^{\uparrow,\varepsilon}-X_{t}^{\downarrow,\varepsilon})=0\quad\mbox{a.s.}$$
There is a challenging dual objective: to point out some upper and lower convergent bounding processes $X^{\uparrow,\varepsilon}$ and $X^{\downarrow,\varepsilon}$ on one hand, and to get a precise convergence result on the other hand. One interesting approach is to link the construction of the bounds with the simulation of the diffusion exiting from thin horizontal layers. Such an approach can be seen through to its successful completion using both the precise description of the Brownian paths, in particular Brownian meanders, and the exact simulation method (rejection sampling), see the seminal paper of Chen and Huang [2] and subsequent developments concerning killed diffusions [4], jump diffusions [10] or further techniques linked to the $\varepsilon$-strong approximation [15].
These approaches concern mainly classical diffusions or jump diffusions with regular coefficients.
The aim of this paper is to develop a different construction for the $\varepsilon$-strong approximation
of Bessel trajectories. The stochastic differential equation satisfied by the Bessel process
presents a singular drift coefficient. We shall thus emphasize an alternative approach which
is not based on exit times of thin horizontal layers but rather on exit times of specific
spheroids (see Figure 1). There is a substantial numerical gain to adopt this new point of
view since the spheroid exit times are easy to generate and the rejection sampling linked to
the exact simulation of a paths skeleton can be avoided. The new algorithm we propose concerns Bessel
processes of integer or non integer dimensions and is based on observations of the Brownian
trajectories.
Let us first introduce the definition of the $\varepsilon$-strong approximation that we shall consider
throughout our study.
Definition 0.1. —
The random process $(y_{t}^{\varepsilon})$ is an $\varepsilon$-strong approximation of the diffusion process $(X_{t})$ if there exists $(x^{\varepsilon}_{t})$ satisfying
$$\sup_{t\in[0,T]}|X_{t}-x^{\varepsilon}_{t}|\leq\varepsilon\quad\mbox{a.s.}$$
(0.1)
such that $(y^{\varepsilon}_{t})$ and $(x^{\varepsilon}_{t})$ are identically distributed.
In the Brownian framework, such an approximation is available using the exit times
of specific spheroids [7]. Let us consider the curve directly linked to the shape of the
$\delta$-dimensional heat ball:
$$\phi_{\delta,\varepsilon}(t):=\sqrt{\delta t\ln\Big{(}\frac{e\varepsilon^{2}}{\delta t}\Big{)}},\quad\mbox{for}\ t\in I_{\delta,\varepsilon}:=[0,r_{\delta,\varepsilon}]\ \mbox{with}\ r_{\delta,\varepsilon}=\frac{e\varepsilon^{2}}{\delta}.$$
(0.2)
We can notice that the maximum of the curve is equal to $\varepsilon$ and is reached for $t_{\rm max}=\varepsilon^{2}/\delta$. The approximation is based on the so-called Brownian skeleton $({\rm BS})_{\eta}$. We recall now the particular situation $\eta\equiv 1$.
Brownian Skeleton $({\rm BS})_{1}$
1.
Let $(A_{n})_{n\geq 1}$ be i.i.d. random variables with gamma distribution ${\rm Gamma}(3/2,2)$.
2.
Let $(Z_{n})_{n\geq 1}$ be i.i.d. Rademacher random variables (taking values +1 or -1 with probability 1/2). The sequences $(A_{n})_{n\geq 1}$ and $(Z_{n})_{n\geq 1}$ are independent.
Definition: Let $\varepsilon>0$. The Brownian skeleton $({\rm BS})_{1}$ is defined by
$\Big{(}(u_{n}^{\varepsilon},s_{n}^{\varepsilon})_{n\geq 1},(x_{n}^{\varepsilon})_{n\geq 0}\Big{)}\quad\mbox{with}\quad\left\{\begin{array}[]{l}u_{n}^{\varepsilon}=\varepsilon^{2}\,e^{1-A_{n}},\quad s^{\varepsilon}_{n}=\displaystyle\sum_{k=1}^{n}u_{k}^{\varepsilon},\\[18.0pt]
x_{n}^{\varepsilon}=x_{n-1}^{\varepsilon}+Z_{n}\,\phi_{1,\varepsilon}(u_{n}^{\varepsilon}),\ \forall n\geq 1\\
\end{array}\right.$
and $x_{0}^{\varepsilon}=x$.
The authors proved in [7] that the piecewise constant function
$$x^{\varepsilon}_{t}=\sum_{n\geq 0}x_{n}^{\varepsilon}1_{\{s_{n}^{\varepsilon}\leq t<s^{\varepsilon}_{n+1}\}}$$
is an $\varepsilon$-strong approximation of the Brownian paths starting in $x$. Moreover the number of random points used to approximate the trajectory on a fixed time interval $[0,T]$, denoted by $N_{T}^{\varepsilon}$
$$N_{T}^{\varepsilon}:=\inf\{n\geq 1:\ s_{n}^{\varepsilon}\geq T\},$$
(0.3)
satisfies
$$\lim_{\varepsilon\to 0}\varepsilon^{2}\mathbb{E}[N^{\varepsilon}_{T}]=\kappa\cdot T\quad\mbox{with}\quad\kappa=3^{3/2}e^{-1}.$$
This means that, the cost of the numerical approximation is of the order $T/\varepsilon^{2}$.
The approximation procedure presented in [7] is based on a piecewise constant function whose intersection with the Brownian paths corresponds to the set of points $\{(s_{n}^{\varepsilon},x_{n}^{\varepsilon}):\ n\geq 0\}$. The sequence $(s_{n}^{\varepsilon})_{n\geq 0}$ is built using the random variables $(u_{n}^{\varepsilon})_{n\geq 1}$ which represent Brownian exit times from some typical spheroid defined by $\phi_{1,\varepsilon}$. Thus, the sequence of points $\{(s_{n}^{\varepsilon},x_{n}^{\varepsilon}):\ n\geq 0\}$ is obtained as the successive Brownian exit points of time-space spheroids of size $\varepsilon$: the Brownian path is therefore completely controlled inbetween two successive points of the skeleton.
In this paper we aim to adapt and develop this technique to the approximation of Bessel processes. Whenever the dimension of the Bessel process is an integer, the paths are distributed like the norm of a multidimensional Brownian motion. Consequently, the successive exit points of spheroids for that Brownian trajectory allows to build a Bessel skeleton, the main ingredient of the $\varepsilon$-strong approximation. In the framework of Bessel processes with non integer dimension, the construction of the algorithm is more difficult: we still use successive exit points of spheroids but we first have to decompose the Bessel paths into two independent parts following the flagship identity of Shiga and Watanabe [16].
In both cases, for integer or non integer dimensions, we develop the corresponding approximation scheme and prove the results that characterize and control its behaviour.
1 Bessel processes with integer dimension
The Bessel process of dimension $\delta$ is the unique solution of the following stochastic differential equation:
$$Z_{t}^{\delta,y}=y+\frac{\delta-1}{2}\int_{0}^{t}(Z_{s}^{\delta,y})^{-1}\,ds+B_{t},\quad t\geq 0,$$
(1.1)
where $y\geq 0$ is the deterministic initial value of the process and $(B_{t})_{t\geq 0}$ stands for a one dimensional standard Brownian motion. In particular, for integer values of $\delta$, the Bessel process and the norm of a $\delta$-dimensional Brownian motion are identically distributed. This classical property shall play a crucial role in the sequel. Let us denote by $\nu$ the so-called Bessel index related to the dimension by the following relation $\nu=\frac{\delta}{2}-1$.
Bessel Skeleton $({\rm BeS})_{\delta}$ – integer dimension
1.
Let $(A_{n})_{n\geq 1}$ be a sequence of i.i.d. random variables with gamma distribution ${\rm Gamma}(\nu+2,1/(\nu+1))$ that is the shape equals $\nu+2$ and the scale $1/(\nu+1)$.
2.
Let $(V_{n})_{n\geq 1}$ be a sequence of i.i.d. uniformly distributed random vectors on the boundary of the unitary sphere $\mathcal{S}^{\delta}$. We denote by $\pi_{1}(V_{n})$ the projection on
the first coordinate. The sequences
$(A_{n})_{n\geq 1}$ and $(V_{n})_{n\geq 1}$ are assumed to be independent.
Definition: For $\varepsilon>0$, the Bessel skeleton $({\rm BeS})_{\delta}$ is given by $\Big{(}(u_{n}^{\varepsilon},s_{n}^{\varepsilon})_{n\geq 1},(y_{n}^{\varepsilon})_{n\geq 0}\Big{)}$
$$\quad\mbox{with}\ \left\{\begin{array}[]{l}u_{n}^{\varepsilon}=\frac{\varepsilon^{2}}{\delta}\,e^{1-A_{n}},\quad s^{\varepsilon}_{n}=\displaystyle\sum_{k=1}^{n}u_{k}^{\varepsilon},\\[18.0pt]
y_{n}^{\varepsilon}=\Big{(}(y_{n-1}^{\varepsilon})^{2}+2\pi_{1}(V_{n})\,y_{n-1}^{\varepsilon}\,\phi_{\delta,\varepsilon}(u_{n}^{\varepsilon})+\phi_{\delta,\varepsilon}^{2}(u_{n}^{\varepsilon})\Big{)}^{1/2},\forall n\geq 1\\
\end{array}\right.$$
and $y_{0}^{\varepsilon}=y$.
The Bessel skeleton permits to construct an approximation of the Bessel trajectory. The main idea leading to this construction is first to relate the Bessel process to the norm of a $\delta$-dimensional Brownian motion. Secondly we replace the Brownian trajectory by a Brownian skeleton: a random walk corresponding to the successive exits of a sequence of small time-space spheroids.
Theorem 1.1. —
Let $\varepsilon>0$ and let us consider a Bessel skeleton $({\rm BeS})_{\delta}$ with $\delta\in\mathbb{N}^{*}$. Then $y^{\varepsilon}_{t}=\sum_{n\geq 0}y_{n}^{\varepsilon}1_{\{s_{n}^{\varepsilon}\leq t<s^{\varepsilon}_{n+1}\}}$ is an $\varepsilon$-strong approximation of the Bessel paths starting in $y$, solution of (1.1).
The number of approximation points $N_{T}^{\varepsilon}$ on the fixed interval $[0,T]$ satisfies:
$$\lim_{\varepsilon\to 0}\varepsilon^{2}\,\mathbb{E}[N^{\varepsilon}_{T}]=\frac{\delta T}{e}\Big{(}\frac{\nu+2}{\nu+1}\Big{)}^{\nu+2}.$$
(1.2)
Moreover the following CLT is observed:
$$\displaystyle\lim_{\varepsilon\to 0}\ \frac{\sqrt{e}}{\varepsilon\sigma\sqrt{\delta T}}\ \Big{(}\frac{\nu+1}{\nu+2}\Big{)}^{3\nu/2+3}\Big{[}\varepsilon^{2}N^{\varepsilon}_{T}-\frac{\delta T}{e}\Big{(}\frac{\nu+2}{\nu+1}\Big{)}^{\nu+2}\Big{]}=G\quad\mbox{in distribution,}$$
with $G$ a $\mathcal{N}(0,1)$ standard Gaussian random variable and
$$\sigma^{2}=\Big{(}\frac{\nu+1}{\nu+3}\Big{)}^{\nu+2}-\Big{(}\frac{\nu+1}{\nu+2}\Big{)}^{2\nu+4}.$$
(1.3)
It is important to notice that Theorem 1.1 leads to confidence intervals for the number of approximated points which represents the efficiency of the approximation algorithm.
Proof.
We construct the proof in several steps.
Step 1: $\varepsilon$-strong approximation.
Let us consider the $\delta$-dimensional Bessel process $(Z^{\delta,y}_{t},\,t\geq 0)$ starting in $y$. We introduce the vector $\hat{y}=(y,0,\ldots,0)\in\mathbb{R}^{\delta}$. It is well-known that $(Z^{\delta,y}_{t},\,t\geq 0)$ has the same distribution as $(\|y+W_{t}\|,\,t\geq 0)$ where $W$ stands for a standard $\delta$-dimensional Brownian motion. It suffices therefore to strongly approximate the Brownian norm since the strong approximation is based on an identity in law.
Let us now build a sequence of points $(t_{n},z_{n})_{n\geq 0}$ belonging to the trajectory of the $\delta$-dimensional Brownian motion $(t,\hat{y}+W_{t})_{t\geq 0}$ and satisfying $t_{n+1}\geq t_{n}$ for any $n\geq 0$. The sequence of times is defined by
$$t_{n+1}:=\inf\{t>t_{n}:\,\|W_{t}-W_{t_{n}}\|=\phi_{\delta,\varepsilon}(t-t_{n})\}\quad\mbox{and}\quad t_{0}=0.$$
(1.4)
These times represent the successive exit times of spheroids sequence (also called heat balls) whose boundary shape corresponds to the function $\phi_{\delta,\varepsilon}$ defined by (0.2). In order to observe points belonging to the path, we set $z_{n}:=\hat{y}+W_{t_{n}}$. Due to the definition of the stopping times and since the maximum of the function $\phi_{\delta,\varepsilon}$ equals $\varepsilon$, we get
$$\|\hat{y}+W_{t}-z_{n}\|\leq\varepsilon,\quad\forall t\in[t_{n},t_{n+1}],\quad\forall n\geq 0.$$
This means that, if we denote by $(z_{t})_{t\geq 0}$ the random function satisfying
$$z_{t}:=\sum_{n\geq 0}z_{n}1_{\{t_{n}\leq t<t_{n+1}\}},\quad t\geq 0,$$
then we have that $\|\hat{y}+W_{t}-z_{t}\|\leq\varepsilon$, for all $t\geq 0$, almost surely. Moreover $\|\hat{y}+W_{t}-z_{t}\|=0$ as soon as $t=t_{n}$ and $n\geq 0$. This approximation of the $\delta$-dimensional Brownian trajectory obviously allows to approximate its Euclidean norm. The second triangle inequality leads to
$$|\|\hat{y}+W_{t}\|-\|z_{t}\||\leq\|\hat{y}+W_{t}-z_{t}\|\leq\varepsilon,\quad\forall t\geq 0.$$
Since the strong approximation is based on an identity in distribution, $(\|z_{t}\|)_{t\geq 0}$ is an $\varepsilon$-strong approximation of the Bessel path.
Step 2: Relation to the Bessel skeleton ${\rm(BeS)}_{\delta}$.
To construct a typical approximated trajectory, it suffices to generate the sequence of successive times $(t_{n})_{n\geq 0}$ and the associated sequence $(z_{n})_{n\geq 0}=(W_{t_{n}})_{n\geq 0}$. It corresponds in fact to the sequences of exit times and exit locations of spheroids. In [5], the authors described the distribution of these two sequences. We note that
•
the random variables $(t_{n+1}-t_{n})_{n\geq 0}$ are independent and identically distributed. Moreover $\delta(t_{n+1}-t_{n})/(e\varepsilon^{2})$ has the same distribution as $e^{-A}$ where $A\sim{\rm Gamma}(\nu+2,1/(\nu+1))$.
•
the $\delta$-dimensional Brownian motion satisfies the rotational invariance property. Therefore $z_{0}=\hat{y}$ and $z_{n+1}$ is uniformly distributed on the sphere of center $z_{n}$ and radius $\phi_{\delta,\varepsilon}(t_{n+1}-t_{n})$. Consequently
$$\|z_{n+1}\|^{2}=\|z_{n}\|^{2}+2\pi_{1}(V)z_{n}\phi_{\delta,\varepsilon}(t_{n+1}-t_{n})+\phi_{\delta,\varepsilon}^{2}(t_{n+1}-t_{n}),$$
where $V$ is uniformly distributed on the unitary sphere of dimension $\delta$ and $\pi_{1}$ stands for the projection on the first coordinate.
We deduce that $(t_{n},\|z_{n}\|)_{n\geq 0}$ and the Bessel skeleton (BeS) $(s_{n}^{\varepsilon},y_{n}^{\varepsilon})_{n\geq 0}$ are identically distributed and consequently, the process $(y^{\varepsilon}_{t})_{t\geq 0}$ defined in the statement of Theorem 1.1 defines an $\varepsilon$-strong approximation of the Bessel process.
Step 3: Number of points needed to cover $[0,T]$.
Let us now focus our attention on the number of spheroids used until a fixed time $T$, defined by $N^{\varepsilon}_{T}:=\inf\{n\geq 0:\ t_{n}\geq T\}$. The arguments used here are similar to those developed in [7] (Proposition 2.2). We denote by $(\hat{N}_{t})_{t\geq 0}$ the Poisson process with independent and identically distributed arrivals $(e^{1-A_{n}})_{n\geq 1}$, defined by the Bessel skeleton. Then the classical asymptotic result holds
$$\lim_{t\to\infty}\frac{\mathbb{E}[\hat{N}_{t}]}{t}=\frac{1}{\mathbb{E}[e^{1-A_{1}}]}=\frac{1}{e\mathcal{L}_{A_{1}}(1)}=e^{-1}\Big{(}\frac{\nu+2}{\nu+1}\Big{)}^{\nu+2}.$$
(1.5)
Here $\mathcal{L}_{A_{1}}(s)$ stands for the Laplace transform of the variate $A_{1}\sim{\rm Gamma}(\nu+2,1/(\nu+1))$, that is
$$\mathcal{L}_{A_{1}}(s)=\Big{(}\frac{s}{\nu+1}+1\Big{)}^{-\nu-2},\quad\forall s\geq 0.$$
Furthermore, the central limit theorem holds: if we denote by $\mu=\mathbb{E}[e^{1-A_{1}}]=e\Big{(}\frac{\nu+1}{\nu+2}\Big{)}^{\nu+2}$ and use the parameter $\sigma$ defined in (1.3), then ${\rm Var}(e^{1-A_{1}})=e^{2}\sigma^{2}$ and
$$\lim_{t\to\infty}\sqrt{\frac{t\mu^{3}}{e^{2}\sigma^{2}}}\Big{(}\frac{\hat{N}_{t}}{t}-\frac{1}{\mu}\Big{)}=G,\quad\mbox{in distribution,}$$
(1.6)
where $G$ is a $\mathcal{N}(0,1)$ standard Gaussian random variable. These two asymptotic results described in (1.5) and (1.6) are related to the behaviour of $N_{T}^{\varepsilon}$ in the $\varepsilon$ small limit since
$$N_{T}^{\varepsilon}=\hat{N}_{\frac{T\delta}{\varepsilon^{2}}},\quad\forall\varepsilon>0.$$
The announced statement is therefore an easy consequence of the previous identity.
∎
2 Bessel processes with non integer dimension
In the previous section, it was crucial that the dimension $\delta$ of the Bessel process was an integer: this allows in particular to associate the Bessel paths with the norm of the $\delta$-dimensional Brownian motion. In the general case the dimension of the Bessel process defined in (1.1) is just a real valued parameter so we also need to develop an $\varepsilon$-strong approximation procedure for noninteger dimensions. In the particular case: $\delta\in[1,\infty[\setminus\mathbb{N}$, the crucial tool is the argument developed by Shiga and Watanabe [16] and already used for simulation purposes in [6]. The Bessel process of dimension $\delta$ starting in $y\geq 0$ has the same distribution as the sum of two independent processes:
•
a Bessel process of dimension $\delta_{i}:=\lfloor\delta\rfloor$ (integer dimension) starting in $y$ (the corresponding index is denoted $\nu_{i}$),
•
a Bessel process of dimension $\delta_{f}:=\delta-\lfloor\delta\rfloor$ (fractional dimension) starting in $0$ (the corresponding index is denoted $\nu_{f}$).
A wise combination of the construction developed in Section 1 on one hand, and the identity of Shiga-Watanabe on the other hand, allows to develop an adapted procedure in the general framework.
A rejection sampling algorithm
Before defining the general Bessel skeleton, we need to introduce the generation of a particular family of random variables already mentioned in [6]. The probability distribution under consideration is deeply related to the Bessel process of dimension $\delta$ exiting from a spheroid of size $\varepsilon$. Nevertheless, we prefer to use in this paragraph generic constants $\alpha>0$ and $\beta>0$ for notational simplicity. In the sequel we are going to fix $\alpha=\delta/2$ and $\beta=2e\varepsilon^{2}/\delta$. Let us introduce the function $u$ defined by
$$u_{\alpha,\beta}(t,x)=\frac{1}{t^{\alpha}}\,\exp\Big{\{}-\frac{x^{2}}{t}\Big{\}}-\frac{1}{\beta^{\alpha}},\quad\forall(t,x)\in\mathbb{R}_{+}^{*}\times\mathbb{R}_{+}^{*},$$
(2.1)
and the associated normalization constant $\kappa_{\alpha,\beta}$:
$$\kappa_{\alpha,\beta,t}:=\int_{0}^{\rho_{\alpha,\beta,t}}u_{\alpha,\beta}(t,y)\,y^{2\alpha-1}\,\mathrm{d}y\quad\mbox{with}\ \rho_{\alpha,\beta,t}:=\sqrt{\alpha t\ln\Big{(}\frac{\beta}{t}\Big{)}}.$$
(2.2)
The constant $\rho_{\alpha,\beta,t}$ corresponds to the positive zero of the function $x\mapsto u_{\alpha,\beta}(t,x)$. We deduce therefore that
$$x\mapsto\kappa^{-1}_{\alpha,\beta,t}\,u_{\alpha,\beta}(t,x)x^{2\alpha-1}\,1_{[0,\rho_{\alpha,\beta,t}]}(x)$$
(2.3)
is a probability distribution function. A random variable whose density is given by (2.3) can be generated using the following rejection sampling.
Conditional distribution $({\rm CD})_{\alpha,\beta}^{t}$
1.
Let $(R_{n})_{n\geq 1}$ be a sequence of uniformly distributed i.i.d. random variables on the interval $[0,1]$.
2.
Let $(V_{n})_{n\geq 1}$ be another sequence of i.i.d. uniformly distributed random variables on $[0,1]$. The sequences
$(R_{n})_{n\geq 1}$ and $(V_{n})_{n\geq 1}$ are assumed to be independent.
Initialization: $n=1$.
While $u_{\alpha,\beta}(t,0)\,R_{n}>u_{\alpha,\beta}(t,\rho_{\alpha,\beta,t}\,V_{n}^{1/(2\alpha)})$ set $n\leftarrow n+1$;
Outcome: $\mathcal{Z}=\rho_{\alpha,\beta,t}\,V_{n}^{1/(2\alpha)}$.
This algorithm is of prime importance in the study of Bessel processes. Indeed let us consider a Bessel process $(Z^{\delta,0}_{t})_{t\geq 0}$ starting in $0$ and with dimension $\delta>0$ and let us denote $\tau_{\phi}$ the first passage time through the curved boundary given by (0.2). We omit the dependence with respect to the parameters $\delta$ and $\varepsilon$ for notational simplicity. The following identity in distribution holds.
Lemma 2.1. —
Let $0<t<r_{\delta,\varepsilon}=e\varepsilon^{2}/\delta$. The outcome of Algorithm $({\rm CD})_{\alpha,\beta}^{2t}$, with $\alpha=\delta/2$ and $\beta=2e\varepsilon^{2}/\delta$, has the same distribution as the conditional distribution of $Z_{t}^{\delta,0}$ given $\tau_{\phi}>t$.
Proof.
Since Algorithm $({\rm CD})_{\alpha,\beta}^{t}$ is an acceptance-rejection sampling, we can easily describe the distribution of its outcome. Let $\psi$ be any non negative measurable function. We consider $R$ and $V$ two independent uniformly distributed r.v., then
$$\displaystyle\mathbb{E}[\psi(\mathcal{Z})]=\frac{\nu(\psi)}{\nu(1)}\quad\mbox{where}\ \nu(\psi):=\mathbb{E}\Big{[}\psi(\rho_{\alpha,\beta,t}\,V^{1/(2\alpha)})1_{\{u_{\alpha,\beta}(t,0)\,R\leq u_{\alpha,\beta}(t,\rho_{\alpha,\beta,t}\,V^{1/(2\alpha)})\}}\Big{]}.$$
(2.4)
Using the change of variables $y=\rho_{\alpha,\beta,t}\,x^{1/(2\alpha)}$ permits to obtain
$$\displaystyle\nu(\psi)$$
$$\displaystyle=\int_{0}^{1}\psi(\rho_{\alpha,\beta,t}\,x^{1/(2\alpha)})\,\frac{u_{\alpha,\beta}(t,\rho_{\alpha,\beta,t}\,x^{1/(2\alpha)})}{u_{\alpha,\beta}(t,0)}\,\mathrm{d}x$$
$$\displaystyle=\frac{2\alpha}{\rho_{\alpha,\beta,t}^{2\alpha}}\int_{0}^{\rho_{\alpha,\beta,t}}\psi(y)\,\frac{u_{\alpha,\beta}(t,y)}{u_{\alpha,\beta}(t,0)}\,y^{2\alpha-1}\mathrm{d}y.$$
(2.5)
Combining (2.4) and (2.5) proves that the p.d.f. of the random variable $\mathcal{Z}\sim({\rm CD})_{\alpha,\beta}^{t}$ corresponds to the function introduced in (2.3). After setting $\alpha=\delta/2$ and $\beta=2e\varepsilon^{2}/\delta$, we deduce that the density of $\mathcal{Z}\sim({\rm CD})_{\alpha,\beta}^{2t}$ corresponds to the function
$$x\mapsto\left(\frac{2}{(2t)^{\delta/2}\Gamma(\delta/2)}\exp\Big{\{}-\frac{x^{2}}{2t}\Big{\}}-\frac{2}{\Gamma(\delta/2)}\Big{(}\frac{\delta}{2e\varepsilon^{2}}\Big{)}^{\delta/2}\right)\,x^{\delta-1}1_{\{0<x<\phi_{\delta,\varepsilon}(t)\}}$$
which is exactly the conditional density of $Z_{t}^{\delta,0}$, given $\tau_{\phi}>t$ (see, for instance, [6]).
∎
Remark 2.2. —
The algorithm $({\rm CD})^{t}_{\alpha,\beta}$ is based on a rejection sampling method, it is therefore straightforward to describe the efficiency of the procedure. It is well known that the number of trials corresponds to a geometrically distributed random variable denoted by $N$, with parameter $\nu(1)$, $\nu$ being defined in (2.4). We deduce from (2.5) that
$$\displaystyle\mathbb{E}[N]=\frac{1}{\nu(1)}=\frac{\rho_{\alpha,\beta,t}^{2\alpha}}{2\alpha}\frac{u_{\alpha,\beta}(t,0)}{\kappa_{\alpha,\beta,t}}.$$
An integration by parts allows to compute the value of the constant $\kappa_{\alpha,\beta,t}$, by introducing the incomplete Gamma function:
$$\kappa_{\alpha,\beta,t}=\frac{1}{2\alpha}\,\gamma\Big{(}\alpha+1,\alpha\ln\frac{\beta}{t}\Big{)},\quad\mbox{where}\quad\gamma(a,x)=\int_{0}^{x}y^{a-1}e^{-y}\,\mathrm{d}y.$$
Finally the average number of steps equals
$$\mathbb{E}[N]=\frac{\alpha^{\alpha}}{\gamma\Big{(}\alpha+1,\alpha\ln\frac{\beta}{t}\Big{)}}\ \Big{(}\ln\frac{\beta}{t}\Big{)}^{\alpha}\ \Big{(}1-\frac{t^{\alpha}}{\beta^{\alpha}}\Big{)},\quad\mbox{for}\ t<\beta.$$
The Bessel skeleton (non integer dimension $\delta>1$)
As already mentioned, our approach for the general case is based on Shiga-Watanabe’s identity in order to split the simulation challenge into two parts: a Bessel process of integer dimension on one hand and a Bessel process of dimension less than $1$ on the other hand. In the sequel, for an easy identification of these two parts, we shall use for most of the parameters either the index $i$ corresponding to the integer part or the index $f$ for the fractional one.
Let us fix two parameters $w_{i}\in]0,1[$ and $w_{f}\in]0,1[$ satisfying the following relation
$$w_{f}+2\sqrt{w_{i}}=1.$$
(2.6)
Let us also define the general Bessel skeleton for a non integer dimension $\delta>1$. We need to introduce the following constants:
$$\alpha_{i}:=\delta_{i}/2,\quad\alpha_{f}:=\delta_{f}/2,\quad\beta_{i}:=2ew_{i}\varepsilon^{2}/\delta_{i}\quad\mbox{and}\quad\beta_{f}:=2ew_{f}\varepsilon^{2}/\delta_{f}.$$
We approximate a Bessel path, with starting value $y$, by constucting the following algorithm.
Bessel Skeleton $({\rm BeS})_{\delta}^{w}$ – non integer dimension $\delta>1$
1.
Let $(A^{(i)}_{n})_{n\geq 1}$ be a sequence of i.i.d. random variables with gamma distribution ${\rm Gamma}(\nu_{i}+2,1/(\nu_{i}+1))$.
2.
Let $(A^{(f)}_{n})_{n\geq 1}$ be a sequence of i.i.d. random variables with gamma distribution ${\rm Gamma}(\nu_{f}+2,1/(\nu_{f}+1))$.
3.
Let $(V_{n})_{n\geq 1}$ be a sequence of i.i.d. uniformly distributed random vectors on the boundary of the unitary sphere $\mathcal{S}^{\delta_{i}}$. We denote by $\pi_{1}(V_{n})$ the projection on
the first coordinate.
The sequences
$(A^{(i)}_{n})_{n\geq 1}$, $(A^{(f)}_{n})_{n\geq 1}$ and $(V_{n})_{n\geq 1}$ are assumed to be independent.
Initialization: $n=0$, $y^{\varepsilon}_{n}=y$, $u_{n}^{\varepsilon}=0$, $s_{n}^{\varepsilon}=0$.
Step 1. Set $n\leftarrow n+1$.
Step 2. If $A^{(f)}_{n}-A^{(i)}_{n}<\ln\Big{(}\frac{w_{f}}{w_{i}}\Big{)}+\ln\frac{\delta_{i}}{\delta_{f}}$ then
•
Set $u_{n}^{\varepsilon}=\frac{\varepsilon^{2}w_{i}}{\delta_{i}}\,e^{1-A^{(i)}_{n}}$ and $\mathcal{Y}=\phi_{\delta_{i},\varepsilon\sqrt{w_{i}}}(u_{n}^{\varepsilon})$
•
Generate $\mathcal{Z}\sim({\rm CD})^{2u_{n}^{\varepsilon}}_{\alpha_{f},\beta_{f}}$
else
•
Set $u_{n}^{\varepsilon}=\frac{\varepsilon^{2}w_{f}}{\delta_{f}}\,e^{1-A^{(f)}_{n}}$ and $\mathcal{Z}=\phi_{\delta_{f},\varepsilon\sqrt{w_{f}}}(u_{n}^{\varepsilon})$
•
Generate $\mathcal{Y}\sim({\rm CD})^{2u_{n}^{\varepsilon}}_{\alpha_{i},\beta_{i}}$
Step 3. Set $y_{n}^{\varepsilon}=\Big{(}(y_{n-1}^{\varepsilon})^{2}+2y_{n-1}^{\varepsilon}\pi_{1}(V_{n})\,\mathcal{Y}+\mathcal{Y}^{2}+\mathcal{Z}^{2}\Big{)}^{1/2}$ and $s_{n}^{\varepsilon}=s_{n-1}^{\varepsilon}+u_{n}^{\varepsilon}$.
Return to Step 1.
Definition: The Bessel skeleton $({\rm BeS})_{\delta}^{w}$ corresponds to $\Big{(}(u_{n}^{\varepsilon},s_{n}^{\varepsilon})_{n\geq 1},(y_{n}^{\varepsilon})_{n\geq 0}\Big{)}$.
The algorithm is based on the construction of a sequence of points $(s_{n}^{\varepsilon},y_{n}^{\varepsilon})_{n\geq 0}$ which essentially permit to emphasize an approximated Bessel path. This sequence is obtained in a Markovian step by step procedure. With a starting time and location $(s_{n}^{\varepsilon},y_{n}^{\varepsilon})$, corresponding to the value of the $\delta$-dimensional Bessel process, we associate two sets composed of a starting point and a spheroid: one intended for a Bessel process of integer dimension and the other one for a Bessel process of fractional dimension. These two paths have been carefully observed until one of them exits from its spheroid. At that random time $s_{n+1}$, both paths are stopped and a combination of their position at that stage permits to compute $y_{n+1}$. To sum up, each step of the algorithm starts with a splitting of the Bessel paths and ends up with a regluing procedure. The sequence $(s_{n}^{\varepsilon},y_{n}^{\varepsilon})_{n\geq 0}$ is crucial for the path approximation as pointed out in the following statement.
Theorem 2.3. —
Let $\varepsilon>0$ and let $w=(w_{i},w_{f})\in]0,1[^{2}$ be a couple of weights satisfying the condition (2.6). Consider a Bessel skeleton $({\rm BeS})_{\delta}^{w}$ with a non integer dimension $\delta>1$. Then $y^{\varepsilon}_{t}=\sum_{n\geq 0}y_{n}^{\varepsilon}1_{\{s_{n}^{\varepsilon}\leq t<s^{\varepsilon}_{n+1}\}}$ is an $\varepsilon$-strong approximation of the Bessel paths starting in $y$, solution of (1.1).
The number of approximation points $N_{T}^{\varepsilon}$ on the fixed interval $[0,T]$ satisfies:
$$\lim_{\varepsilon\to 0}\varepsilon^{2}\,\mathbb{E}[N^{\varepsilon}_{T}]=\frac{T}{\mu}:=\frac{T\delta_{i}}{ew_{i}}\ \mathcal{F}\Big{(}\frac{w_{f}}{w_{i}}\frac{\delta_{i}}{\delta_{f}},\nu_{f}+2,\frac{1}{\nu_{f}+1},\nu_{i}+2,\frac{1}{\nu_{i}+1}\Big{)}^{-1},$$
(2.7)
where $\mathcal{F}(x,a,\lambda,b,\mu)=\mathbb{E}[\min(xe^{-A},e^{-B})]$. Here $A$ and $B$ stand for two independent Gamma distributed random variables with parameters (shape $a$ and scale $\lambda$) and, respectively $(b,\mu)$.
Moreover the following CLT is observed:
$$\displaystyle\lim_{\varepsilon\to 0}\frac{1}{\varepsilon\sqrt{T}}\frac{\mu^{3/2}\delta_{i}}{ew_{i}\sigma}\Big{(}\varepsilon^{2}\,N^{\varepsilon}_{T}-\frac{T}{\mu}\Big{)}=G\quad\mbox{in distribution},$$
with $G$ a $\mathcal{N}(0,1)$ standard Gaussian random variable and
$$\sigma^{2}=\mathcal{F}\Big{(}\frac{w_{f}^{2}}{w_{i}^{2}}\frac{\delta_{i}^{2}}{\delta_{f}^{2}},\nu_{f}+2,\frac{2}{\nu_{f}+1},\nu_{i}+2,\frac{2}{\nu_{i}+1}\Big{)}-\mathcal{F}\Big{(}\frac{w_{f}}{w_{i}}\frac{\delta_{i}}{\delta_{f}},\nu_{f}+2,\frac{1}{\nu_{f}+1},\nu_{i}+2,\frac{1}{\nu_{i}+1}\Big{)}^{2}.$$
(2.8)
Corollary 2.4. —
There exists $\varepsilon_{0}>0$, such that the average number of approximation points $N_{T}^{\varepsilon}$, on the fixed interval $[0,T]$, satisfies:
$$\varepsilon^{2}\mathbb{E}[N_{T}^{\varepsilon}]\leq\frac{T}{e}\max\Big{(}\frac{\delta_{i}}{w_{i}},\frac{\delta_{f}}{w_{f}}\Big{)}\ \Big{(}\frac{\nu_{i}+2}{\nu_{i}+1}\Big{)}^{\nu_{i}+2}\Big{(}\frac{\nu_{f}+2}{\nu_{f}+1}\Big{)}^{\nu_{f}+2},\quad\mbox{for all}\quad\varepsilon\leq\varepsilon_{0}.$$
(2.9)
The right hand side of (2.9) can be minimized with the optimal choice: $w_{i}=\Big{(}\frac{\sqrt{\delta_{i}\delta}-\delta_{i}}{\delta_{f}}\Big{)}^{2}$.
Proof of Corollary 2.4.
The statement is a direct consequence of the convergence result (2.7), combined with the properties of the function $\mathcal{F}$. More precisely, for $x\in(0,1]$, the independence of the variates $A$ and $B$ leads to
$$\displaystyle\mathcal{F}(x,a,\lambda,b,\mu)$$
$$\displaystyle=\mathbb{E}[\min(xe^{-A},e^{-B})]=\mathbb{E}[\exp\{-\max(A-\log(x),B)\}]$$
$$\displaystyle>\mathbb{E}[\exp\{-(A+B-\log(x))\}]=x\mathbb{E}[e^{-A}]\mathbb{E}[e^{-B}]$$
$$\displaystyle=x\mathcal{L}_{A}(1)\mathcal{L}_{B}(1)=x(1+\lambda)^{-a}(1+\mu)^{-b},$$
since $\max(A-\log(x),B)<A+B-\log(x)$ as soon as $A>0$ and $B>0$, that is almost surely.
Here $\mathcal{L}_{A}(s)$ stands for the Laplace transform of the variate $A$, that is
$\mathcal{L}_{A}(s)=(1+\lambda s)^{-a}$. Moreover, if $x\geq 1$, then similar computations lead to
$$\displaystyle\mathcal{F}(x,a,\lambda,b,\mu)$$
$$\displaystyle=x\mathbb{E}[\min(e^{-A},x^{-1}e^{-B})]>x\cdot x^{-1}(1+\lambda)^{-a}(1+\mu)^{-b}=(1+\lambda)^{-a}(1+\mu)^{-b}.$$
Consequently, for any $x>0$, we get
$$\displaystyle\mathcal{F}(x,a,\lambda,b,\mu)^{-1}$$
$$\displaystyle<\max(1,x^{-1})\,(1+\lambda)^{a}(1+\mu)^{b}.$$
(2.10)
Combining (2.10) with the limiting value (2.7) leads therefore to the announced upper-bound (2.9).
∎
Proof of Theorem 2.3.
The structure of the proof is similar to Theorem 1.2. First we replace the paths of the Bessel process by some other paths with the same distribution. Then, on the new paths, we introduce a skeleton. Finally we count the number of points needed to cover a deterministic time interval $[0,T]$.
Step 1: $\varepsilon$-strong approximation.
Let us consider a Bessel process of non integer dimension $\delta>1$, that is the solution of equation (1.1) with initial value $y\geq 0$. Let us denote the distribution of the squared process by $\mathbb{Q}^{\delta,y^{2}}$. We recall that the dimension can be decomposed as follows: $\delta=\delta_{i}+\delta_{f}$ where $\delta_{i}=\lfloor\delta\rfloor$. Using the identity in law pointed out by Shiga and Watanabe [16], we obtain:
$$\mathbb{Q}^{\delta,y^{2}}=\mathbb{Q}^{\delta_{i},y^{2}}\star\mathbb{Q}^{\delta_{f},0},$$
(2.11)
where $\star$ stands for the convolution of the probability distributions. Thus, by introducing two independent Bessel processes $\overline{Z}$ and $\widehat{Z}$, one of integer dimension $\delta_{i}$ starting in $y$: $(\overline{Z}_{t}(y))_{t\geq 0}$, and the other of non integer dimension $\delta_{f}$, starting in $0$: $(\widehat{Z}_{t}(0))_{t\geq 0}$ (when the starting value is equal to $0$, we shall drop the dependence for notational simplicity), then (2.11) leads to the identity
$$(Z^{\delta,y}_{t})_{t\geq 0}\overset{(d)}{=}\Big{(}\overline{Z}_{t}(y)^{2}+\widehat{Z}_{t}(0)^{2}\Big{)}^{1/2}_{t\geq 0}=:\Big{(}\overline{Z}_{t}(y)^{2}+\widehat{Z}_{t}^{2}\Big{)}^{1/2}_{t\geq 0}.$$
(2.12)
Bessel processes with integer dimension $\delta_{i}$ play an important role since they can be represented as the norm of the $\delta_{i}$-dimensional Brownian motion. We just note that the standard Brownian motion $(W_{t})$ is rotational invariant and moreover:
$$(W_{t})_{t\geq 0}=\Big{(}\Theta_{t}\cdot\|W_{t}\|\Big{)}_{t\geq 0},$$
where $(\Theta_{t})_{t>0}$ is a continuous stochastic process valued in the unitary sphere $\mathcal{S}^{\delta_{i}}$, independent of $(\|W_{t}\|)_{t\geq 0}$. We fix $\Theta_{0}=0$ (not continuous for $t=0$) and observe that $\Theta_{t}$ is uniformly distributed at any fixed time $t>0$.
Let us denote $\underline{y}=(y,0,\ldots,0)\in\mathbb{R}^{\delta_{i}}$. We deduce that
$$\displaystyle(\overline{Z}_{t}(y))_{t\geq 0}$$
$$\displaystyle\overset{(d)}{=}\Big{\|}\underline{y}+\Theta_{t}\|W_{t}\|\,\Big{\|}_{t\geq 0}=\Big{(}y^{2}+2y\,\pi_{1}(\Theta_{t})\|W_{t}\|+\|W_{t}\|^{2}\Big{)}^{1/2}_{t\geq 0},$$
(2.13)
where $\pi_{1}$ corresponds to the projection on the first coordinate. Combining (2.12) and (2.13) leads to
$$(Z^{\delta,y}_{t})_{t\geq 0}\overset{(d)}{=}\mathcal{X}_{t}:=\Big{(}y^{2}+2y\,\pi_{1}(\Theta_{t})\overline{Z}_{t}+\overline{Z}_{t}^{2}+\widehat{Z}_{t}^{2}\Big{)}^{1/2}_{t\geq 0},$$
(2.14)
where the processes $(\overline{Z}_{t})_{t\geq 0}$, $(\widehat{Z}_{t})_{t\geq 0}$ and $(\Theta_{t})_{t\geq 0}$ are independent. Using the strong Markov property of the Bessel process, we can propose a more complex identity. If $s_{1}$ is a stopping time with respect to the filtration $\mathcal{F}^{(1)}:=(\mathcal{F}_{t}^{(1)})_{t\geq 0}$ induced by $(W,\widehat{Z})$ (also denoted in the sequel $(W^{(1)},\widehat{Z}^{(1)})$) then the conditional distribution of $(\mathcal{X}_{s_{1}+t})_{t\geq 0}$ given $\mathcal{F}_{s_{1}}$ is identical to the distribution
$$(\mathcal{X}_{t}^{(2)})_{t\geq 0}:=\Big{(}\mathcal{X}_{s_{1}}^{2}+2\mathcal{X}_{s_{1}}\,\pi_{1}(\Theta_{t}^{(2)})\overline{Z}_{t}^{(2)}+(\overline{Z}_{t}^{(2)})^{2}+(\widehat{Z}_{t}^{(2)})^{2}\Big{)}^{1/2}_{t\geq 0},$$
where $((\overline{Z}^{(k)},\widehat{Z}^{(k)},\Theta^{(k)})_{t\geq 0})_{k\geq 2}$ is a family of independent copies of $(\overline{Z}^{(1)},\widehat{Z}^{(1)},\Theta^{(1)})_{t\geq 0}$. So we can build a particular stochastic process $(\mathcal{X}_{t})_{t\geq 0}$ combining $\mathcal{X}$ (also denoted $\mathcal{X}^{(1)}$) and $\mathcal{X}^{(2)}$ by the following identity
$$\overline{\mathcal{X}}_{t}^{(2)}:=\mathcal{X}^{(1)}_{t}1_{\{t<s_{1}\}}+\mathcal{X}^{(2)}_{t-s_{1}}1_{\{t\geq s_{1}\}},\quad t\geq 0.$$
Let us note that both $(\overline{\mathcal{X}}_{t})_{t\geq 0}$ and $(Z^{\delta,y}_{t})_{t\geq 0}$ are identically distributed. Let us go on with the modification of the process. To that end, we denote by $(\mathcal{F}_{t}^{(2)})_{t\geq 0}$ the filtration generated by the following stochastic processes: $(W^{(1)},\widehat{Z}^{(1)})_{t\wedge s_{1}}$ and $(W^{(2)},\widehat{Z}^{(2)})_{(t-s_{1})\vee 0}$. For any $\mathcal{F}^{(2)}$-stopping time $s_{2}>s_{1}$, we can define
$$\overline{\mathcal{X}}_{t}^{(3)}:=\mathcal{X}^{(1)}_{t}1_{\{t<s_{1}\}}+\mathcal{X}^{(2)}_{t-s_{1}}1_{\{s_{1}\leq t<s_{2}\}}+\mathcal{X}^{(3)}_{t-s_{2}}1_{\{t\geq s_{2}\}},\quad t\geq 0,$$
where $\mathcal{X}^{(3)}$ is defined by
$$(\mathcal{X}_{t}^{(3)})_{t\geq 0}:=\Big{(}(\mathcal{X}^{(2)}_{s_{2}-s_{1}})^{2}+2\mathcal{X}^{(2)}_{s_{2}-s_{1}}\,\pi_{1}(\Theta_{t}^{(3)})\overline{Z}_{t}^{(3)}+(\overline{Z}_{t}^{(3)})^{2}+(\widehat{Z}_{t}^{(3)})^{2}\Big{)}^{1/2}_{t\geq 0}.$$
The procedure continues step by step in this way. For any increasing sequence of stopping time $(s_{n})_{n\geq 1}$, satisfying $\lim_{n\to\infty}s_{n}=+\infty$, we construct the stochastic process:
$$\overline{\mathcal{X}}_{t}^{(\infty)}=\mathcal{X}_{t-s_{n}}^{(n+1)},\quad\mbox{if}\ s_{n}\leq t<s_{n+1},$$
(2.15)
with the definition
$$(\mathcal{X}_{t}^{(n+1)})_{t\geq 0}:=\Big{(}(\mathcal{X}^{(n)}_{s_{n}-s_{n-1}})^{2}+2\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}\,\pi_{1}(\Theta_{t}^{(n+1)})\overline{Z}_{t}^{(n+1)}+(\overline{Z}_{t}^{(n+1)})^{2}+(\widehat{Z}_{t}^{(n+1)})^{2}\Big{)}^{1/2}_{t\geq 0}.$$
(2.16)
By construction, we observe that $(\overline{\mathcal{X}}_{t}^{\infty})_{t\geq 0}$ and $(Z^{\delta,y}_{t})_{t\geq 0}$ are identically distributed. Since the definition of the $\varepsilon$-strong approximation only depends on the distribution of the stochastic process, it suffices therefore to point out an approximation of $(\overline{\mathcal{X}}_{t}^{\infty})_{t\geq 0}$ in order to prove the statement.
Step 2: Bessel skeleton
Let us now consider a particular increasing family of stopping times. Let $w\in]0,1[$ be a fixed parameter. We define $\overline{\tau}_{n}$ (respectively $\widehat{\tau}_{n}$), the first passage time of the Bessel process $(\overline{Z}_{t}^{(n)})_{t\geq 0}$ (resp. $(\widehat{Z}_{t}^{(n)})_{t\geq 0}$), through the curved boundary $\phi_{\delta_{i},\varepsilon\sqrt{w_{i}}}$ (resp. $\phi_{\delta_{f},\varepsilon\sqrt{w_{f}}}$), defined in (0.2). We construct a new stopping time $u_{n}$ and the associated cumulative time $s_{n}$, as follows:
$$u_{n}:=\overline{\tau}_{n}\wedge\widehat{\tau}_{n}\quad\mbox{and}\quad s_{n}=s_{n-1}+u_{n},\quad n\geq 1,$$
(2.17)
with the initial value $s_{0}=0$. The sequence of stopping times $(s_{n})_{n\geq 0}$ satisfies the conditions developed in the previous paragraph Step 1. We can therefore construct the continuous process $(\mathcal{X}^{\infty}_{t})_{t\geq 0}$ using (2.15)–(2.16) and the particular sequence $(s_{n})_{n\geq 0}$, just described. Since the maximal value of the curved boundary $\phi_{\delta,\varepsilon}$ equals $\varepsilon$, we can emphasize a crucial upper-bound of the difference $\mathcal{D}^{(n)}_{t}:=|\overline{\mathcal{X}}^{\infty}_{t}-\overline{\mathcal{X}}^{\infty}_{s_{n}}|$. For any $s_{n}\leq t<s_{n+1}$,
$$\displaystyle\mathcal{D}_{t}^{(n)}=\Big{|}\Big{(}(\mathcal{X}^{(n)}_{s_{n}-s_{n-1}})^{2}+2\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}\,\pi_{1}(\Theta_{t}^{(n+1)})\overline{Z}_{t}^{(n+1)}+(\overline{Z}_{t}^{(n+1)})^{2}+(\widehat{Z}_{t}^{(n+1)})^{2}\Big{)}^{1/2}-\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}\Big{|}$$
$$\displaystyle=\Big{|}\Big{(}(\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}+\pi_{1}(\Theta_{t}^{(n+1)})\overline{Z}_{t}^{(n+1)})^{2}+(\overline{Z}_{t}^{(n+1)})^{2}(1-\pi_{1}^{2}(\Theta_{t}^{(n+1)}))+(\widehat{Z}_{t}^{(n+1)})^{2}\Big{)}^{1/2}-\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}\Big{|}.$$
Let us consider $a$ and $b$ two non negative numbers, then for any $x\in\mathbb{R}$, we have $|\sqrt{a+b}-x|\leq|\sqrt{a}-x|+\sqrt{b}$, $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$ and finally $||a+b|-|a||\leq|b|$. Applying to the previous expression of $\mathcal{D}_{t}^{(n)}$, we obtain
$$\displaystyle\mathcal{D}^{(n)}_{t}$$
$$\displaystyle\leq\Big{|}|\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}+\pi_{1}(\Theta_{t}^{(n+1)})\overline{Z}_{t}^{(n+1)}|-\mathcal{X}^{(n)}_{s_{n}-s_{n-1}}\Big{|}+\sqrt{w_{i}\varepsilon^{2}+w_{f}\varepsilon^{2}}$$
$$\displaystyle\leq|\pi_{1}(\Theta_{t}^{(n+1)})\overline{Z}_{t}^{(n+1)}|+\varepsilon\sqrt{w_{i}+w_{f}}\leq\varepsilon(\sqrt{w_{i}}+\sqrt{w_{i}+w_{f}})=\varepsilon.$$
The last equality is a consequence of the particular relation between $w_{i}$ and $w_{f}$ introduced in (2.6). We deduce therefore that the stochastic process defined by $\widehat{y}_{t}=\sum_{n\geq 0}\overline{\mathcal{X}}^{\infty}_{s_{n}}1_{\{s_{n}\leq t<s_{n+1}\}}$, is an $\varepsilon$-strong approximation of the Bessel paths (see Definition 0.1). In order to prove the statement of Theorem 2.3, it suffices to check that $(y_{t}^{\varepsilon})_{t\geq 0}$ defined in the statement and $(\widehat{y}_{t})_{t\geq 0}$, are identically distributed. Let us therefore describe the joint distribution of $(u_{n})_{n\geq 1}$, $(s_{n})_{n\geq 0}$ and $(\overline{\mathcal{X}}^{\infty}_{s_{n}})_{n\geq 1}=(\mathcal{X}^{(n)}_{s_{n}-s_{n-1}})_{n\geq 1}$ and compare it to the Bessel skeleton.
•
Using the definition of the stopping times $u_{n}$ in (2.17), we observe that $(u_{n})_{n\geq 0}$ is a sequence of independent and identically distributed random variables.
Moreover, on one hand, the distribution of the first passage time $\overline{\tau}_{n}$ is identical to that of $\frac{\varepsilon^{2}w_{i}}{\delta_{i}}\,e^{1-A^{(i)}}$, where $A^{(i)}$ stands for a Gamma distributed r.v of parameters $\nu_{i}+2$ and $1/(\nu_{i}+1)$ (see for instance [6]).
On the other hand, $\widehat{\tau}_{n}$ and $\frac{\varepsilon^{2}w_{f}}{\delta_{f}}\,e^{1-A^{(f)}}$ are identically distributed. Here $A^{(f)}$ corresponds to Gamma distributed r.v. with parameters $\nu_{f}+2$ and $1/(\nu_{f}+1)$. The stopping time $u_{n}$ is the minimum of these two first passage times and matches the stopping time $u_{n}^{\varepsilon}$ introduced in Algorithm $({\rm BeS})^{w}_{\delta}$.
Consequently $(s_{n})_{n\geq 0}$ and $(s_{n}^{\varepsilon})_{n\geq 0}$ are identically distributed.
•
Let us now describe the sequence $(\mathcal{X}^{(n)}_{s_{n}-s_{n-1}})_{n\geq 1}$. It is defined recursively by (2.16). In this equation, we need to know the value of three random variables: $\Theta_{u_{n}}$, $\overline{Z}^{(n)}_{u_{n}}$, $\widehat{Z}^{(n)}_{u_{n}}$. Since $u_{n}$ is only linked to stopping times defined on the processes $\overline{Z}^{(n)}$ and $\widehat{Z}^{(n)}$, which are independent from $\Theta$, and since $\Theta_{t}$ is uniformly distributed for any $t>0$, we obtain that $\Theta_{u_{n}}$ is uniformly distributed on the sphere and independent of both $\overline{Z}^{(n)}_{u_{n}}$ and $\widehat{Z}^{(n)}_{u_{n}}$. Moreover the definition (2.1) implies to take into account two different cases: either $u_{n}=\overline{\tau}_{n}<\widehat{\tau}_{n}$ or $u_{n}=\widehat{\tau}_{n}<\overline{\tau}_{n}$. In the first case, we have, on the event $u_{n}=t$, $\overline{Z}^{(n)}_{u_{n}}=\phi_{\delta_{i},\varepsilon\sqrt{w_{i}}}(t)$ and the distribution of $\widehat{Z}^{(n)}_{u_{n}}$ corresponds to $({\rm CD})_{\alpha_{f},\beta_{f}}^{2t}$ as announced in Lemma 2.1: a Bessel process conditioned not to have reach a curved boundary. In the second case, we observe the reverse situation: on the event $u_{n}=t$, $\widehat{Z}^{(n)}_{u_{n}}=\phi_{\delta_{f},\varepsilon\sqrt{w_{f}}}(t)$ and the distribution of $\overline{Z}^{(n)}_{u_{n}}$ corresponds to $({\rm CD})_{\alpha_{i},\beta_{i}}^{2t}$ as announced in Lemma 2.1. To sum up,
$$(\Theta_{u_{n}},\overline{Z}^{(n)}_{u_{n}},\widehat{Z}^{(n)}_{u_{n}})\overset{(d)}{=}(V_{n},\mathcal{Y},\mathcal{Z}),$$
where $V_{n}$,$\mathcal{Y}$ and $\mathcal{Z}$ correspond to the variables introduced in Algorithm $({\rm BeS})_{\delta}^{w}$. Due to (2.16), we deduce quite easily that $(y_{t}^{\varepsilon})_{t\geq 0}$ and $(\widehat{y}_{t})_{t\geq 0}$ are identically distributed. We conclude that $(y_{t}^{\varepsilon})_{t\geq 0}$ is an $\varepsilon$-strong approximation of the Bessel paths.
Step 3: Number of points necessary to cover the time interval $[0,T]$.
The arguments for the description of the number of points have already been introduced in the proof of Theorem 1.1. We introduce $(\widehat{N}_{t})_{t\geq 0}$ a Poisson process with independent and identically distributed arrivals $(M_{n})_{n\geq 1}$ where
$$M_{n}=\Big{(}\frac{w_{i}}{\delta_{i}}\,e^{1-A^{(i)}_{n}}\Big{)}\wedge\Big{(}\frac{w_{f}}{\delta_{f}}\,e^{1-A^{(f)}_{n}}\Big{)},$$
with $A^{(i)}$ and $A^{(f)}$ defined in Algorithm $({\rm BeS})_{\delta}^{w}$. We denote $\mu=\mathbb{E}[M_{1}]$. The classical asymptotic result holds:
$$\lim_{t\to\infty}\frac{\mathbb{E}[\widehat{N}_{t}]}{t}=\frac{1}{\mu}=\frac{\delta_{i}}{ew_{i}}\mathcal{F}\Big{(}\frac{w_{f}}{w_{i}}\frac{\delta_{i}}{\delta_{f}},\nu_{f}+2,\frac{1}{\nu_{f}+1},\nu_{i}+2,\frac{1}{\nu_{i}+1}\Big{)}^{-1},$$
(2.18)
where $\mathcal{F}$ is defined in the statement of Theorem 2.3. The mean of $M_{1}$ plays an important role in the limit so do the variance for the confidence interval. Due to the scaling property of the Gamma distribution, we notice that ${\rm Var}(M_{1})=\frac{e^{2}w_{i}^{2}}{\delta_{i}^{2}}\sigma^{2}$, where $\sigma^{2}$ is defined by (2.8). The central limit theorem, applied in the counting process context, leads to
$$\lim_{t\to\infty}\sqrt{\frac{t\mu^{3}\delta_{i}^{2}}{e^{2}w_{i}^{2}\sigma^{2}}}\Big{(}\frac{\widehat{N}_{t}}{t}-\frac{1}{\mu}\Big{)}=G\quad\mbox{in distribution,}$$
where $G$ is a $\mathcal{N}(0;1)$ standard Gaussian variate. Let us observe that the number of approximation points $N^{\varepsilon}_{T}$ is directly linked in distribution to the Poisson process just defined. More exactly, we have $N^{\varepsilon}_{T}\overset{(d)}{=}\widehat{N}_{\frac{T}{\varepsilon^{2}}}$, which gives directly the statement: the limit with respect to the time variable is replaced by the limit with respect to the parameter $\varepsilon$.
∎
3 Related processes and numerical illustration
3.1 Numerics: Bessel processes
Let us first illustrate the strong approximation of Bessel processes. We choose to observe the paths on some given time interval $[0,T]$. In particular, we are able to present a Bessel skeleton and the corresponding upper and lower bounds for some small precision value $\varepsilon$.
In Figure 2, the skeletons correspond for instance to a Bessel process either of dimension $10$ or of dimension $2$. As we can observe, the variations of the skeleton when time elapses are obviously smaller than the limit $\varepsilon$ chosen for the approximation. We can interpret this as: even if the maximal size of the spheroids corresponds to this specific value $\varepsilon$, the difference between the values of two successive points of the Bessel skeleton is not often near to the maximum. Indeed the spheroid is applied to the $\delta$-dimensional Brownian motion in a first step, and then, in a second step, a random projection is applied, see the algorithm $({\rm BeS})_{\delta}$ for integer dimensions. So denoting by $\tau$ the first Brownian exit time of the spheroid (0.2) and by $p_{\tau}$ its probability density function, we can compute the following average size
$$\displaystyle\mathbb{E}[\phi_{\delta,\varepsilon}(\tau)]$$
$$\displaystyle=\int_{0}^{\frac{e\varepsilon^{2}}{\delta}}\phi_{\delta,\varepsilon}(t)\,p_{\tau}(t)\,\mathrm{d}t=\int_{0}^{\frac{e\varepsilon^{2}}{\delta}}\sqrt{\delta t\ln\Big{(}\frac{e\varepsilon^{2}}{\delta t}\Big{)}}\frac{1}{t\Gamma(\delta/2)}\left(\frac{\delta^{2}t}{2e\varepsilon^{2}}\,\ln\Big{(}\frac{e\varepsilon^{2}}{\delta t}\Big{)}\right)^{\delta/2}\,\mathrm{d}t$$
$$\displaystyle=\varepsilon\frac{\sqrt{e}}{\Gamma(\delta/2)}\Big{(}\frac{\delta}{2}\Big{)}^{\delta/2}\int_{0}^{1}\frac{1}{u}\,\Big{(}u\ln\frac{1}{u}\Big{)}^{(\delta+1)/2}\,\mathrm{d}u=:\varepsilon\eta(\delta).$$
We can evaluate this last integral
$$\displaystyle\int_{0}^{1}\frac{1}{u}\,\Big{(}u\ln\frac{1}{u}\Big{)}^{(\delta+1)/2}\,\mathrm{d}u$$
$$\displaystyle=\Big{(}\frac{2}{\delta+1}\Big{)}^{\frac{\delta+3}{2}}\Gamma\left(\frac{\delta+3}{2}\right),$$
and by using the properties of the Gamma function obtain the explicit form:
$$\eta(\delta)=\sqrt{2\pi e}\frac{\Gamma(\delta)}{\left[\Gamma\left(\frac{\delta}{2}\right)\right]^{2}}\frac{\delta^{\frac{\delta}{2}}}{(\delta+1)^{\frac{\delta+1}{2}}}2^{1-\delta}.$$
The average is obviously proportional to $\varepsilon$ and the constant $\eta(\delta)$ can be evaluated easily. We can observe that $\eta$ is a non decreasing function of the dimension $\delta$ on the interval $[2,+\infty)$ starting with an estimated value $\eta(2)=0.7953$. This function is represented on the opposite figure. Let us note that for high dimensions the average size $\mathbb{E}[\phi_{\delta,\varepsilon}(\tau)]$ is close to $\varepsilon$, which is the optimal size for the strong approximation procedure.
On the one hand, the increments of the Bessel skeleton depend on the Brownian exit time of the spheroid. On the other hand, they are also strongly related to the projection on the first coordinate of a random variable $V$ uniformly distributed on the sphere of dimension $\delta$: $\pi_{1}(V)$. For $\delta>2$, using the spherical coordinates, we obtain
$$\displaystyle\mathbb{E}[|\pi_{1}(V)|]$$
$$\displaystyle=\frac{2^{\delta}}{(\delta-1)(2\pi)^{\delta/2}}\left(\int_{0}^{\infty}r^{\delta-1}e^{-\frac{r^{2}}{2}}\,\mathrm{d}r\right)\times\prod_{k=0}^{\delta-3}W_{k},$$
(3.1)
where $W_{k}$ stands for Wallis’ integrals $W_{n}:=\int_{0}^{\pi/2}\sin^{n}(x)\,\mathrm{d}x$.
Let us note that the integral appearing in (3.1) can be related to the moments of a standard Gaussian variate.
We deduce that
$\int_{0}^{\infty}r^{2k}e^{-\frac{r^{2}}{2}}\,\mathrm{d}r=\sqrt{\frac{\pi}{2}}\,\frac{(2k)!}{2^{k}k!}$
and
$\int_{0}^{\infty}r^{2k+1}e^{-\frac{r^{2}}{2}}\,\mathrm{d}r=2^{k}k!.$
We can therefore compute the average size of the projection which of course depends on the dimension. Let us just note that the particular dimension $\delta=2$ leads to $\mathbb{E}[|\pi_{1}(V)|]=\frac{2}{\pi}\approx 0.6366$. The opposite figure gives this dependence: for large dimensions the projection procedure reduces the difference between two successive points of the skeleton.
We notice that this reduction is not too strong, for $\delta=20$ for instance the reduction corresponds to a division by $5$.
The efficiency of the approximation is deeply related to the number of spheroids used to cover the time interval $[0,T]$. Theorem 1.1 (Central Limit Theorem) points out the asymptotic result as $\varepsilon$ tends to $0$ for Bessel processes with integer dimensions. Numerical experiments permit to obtain an histogram of the number of points for the generation of $10\,000$ skeletons, see Figure 3.
A characteristic of the asymptotic behaviour is the mean number of spheroids necessary to cover some time interval $[0,T]$. We propose here to estimate it by using an empirical mean issued from a sample of $1\ 000$ trajectories. As already mentioned, we observe a dependence with respect to the Bessel dimension, the number of spheroids used by the algorithm increases as $\delta$ increases. Figure 4 emphasizes that this dependence looks linear. Moreover the estimation of the average permits to illustrate the asymptotic linear dependence with respect to the parameter $1/\varepsilon^{2}$, here $\varepsilon$ stands for the accuracy of the strong approximation.
In order to completely illustrate the strong approximation of the Bessel processes, let us consider numerical experiments for non integer dimensions. In this case, Algorithm $({\rm BeS})_{\delta}^{w}$ permits to generate the Bessel skeletons. Of course, due to the decomposition related to Shiga-Watanabe’s property, we need to observe both a sequence of spheroids for the Bessel process corresponding to the integer part of the dimension and a sequence of spheroids for the fractional part. That’s why it is reasonable to see a large number of skeleton points in order to approximate the paths. For instance, for a Bessel process of dimension $\delta=2.2$, the average of this random number represented by the histogram of Figure 5 (left) is about $68\,130$
while the average in the particular $d=2$ dimension (Figure 3 – left) is approximately equal to $1767$. This sharp increase strongly depends on the value of the parameter $w$ which determines the size of the spheroids of both the integer and fractional part of the algorithm. The challenge is therefore to obtain a balanced repartition. The optimal choice of the parameters $(w_{i},w_{f})$, satisfying the identity $w_{f}+2\sqrt{w_{i}}=1$, is illustrated by different numerical experiments in Figure 5 (right). We observe that this optimal choice depends on the Bessel dimension and can be compared to the heuristic choice suggested in Corollary 2.4 which is represented by a vertical line in the figure.
3.2 Related processes
Several stochastic processes related to the Bessel one play an important role in the finance literature. Here the aim of the discussion is not to present a complete overview of financial models which could be concerned by our approximation procedure but rather to present few examples. Let us first recall the statement of Definition 0.1: $(y_{t}^{\varepsilon})$ is an $\varepsilon$-strong approximation of the diffusion process $(X_{t})$ on the fixed time interval $[0,T]$ if there exists $(x^{\varepsilon}_{t})$ satisfying
$$\sup_{t\in[0,T]}|X_{t}-x^{\varepsilon}_{t}|\leq\varepsilon\quad\mbox{a.s.}$$
such that $(y^{\varepsilon}_{t})$ and $(x^{\varepsilon}_{t})$ are identically distributed. Consequently, as a by-product, any approximation of the Bessel path $(Z^{\delta,y}_{t},\,t\leq T)$ leads to an approximation of the path $(Y_{t},\,t\leq T_{0})$ defined by
$$Y_{t}:=f(t,Z^{\delta,y}_{\rho(t)}),$$
(3.2)
with $f:\mathbb{R}_{+}^{2}\to\mathbb{R}$ a continuous function and $\rho:\mathbb{R}_{+}\to\mathbb{R}_{+}$ a strictly monotonous time change function. Of course the identity (3.2) implies a change of accuracy for the approximation and of course a change in the time interval under consideration. This adaptation is rather immediate and permits to handle with a large class of processes. In the family of financial term structure models, we can for instance focus our attention on the square-root process or CIR model (Cox-Ingersoll-Ross). This process appearing in the seminal paper of Cox et al. [3] is the object of many studies and is simply defined as the positive solution of
$$dY_{t}=k(\theta-Y_{t})\,dt+\sigma\sqrt{Y_{t}}\,dB_{t},\quad Y_{0}=x,$$
(3.3)
under the conditions $k\theta>0$ and $\sigma>0$. Using stochastic calculus permits to point out that the process $Y$ satisfies (not especially with respect to the same Brownian motion) (3.2) with
$$f(t,x)=e^{-kt}x^{2},\quad\rho(t)=\frac{\sigma^{2}}{4k}\,(e^{kt}-1),\quad\delta=\frac{4k\theta}{\sigma^{2}}\ \ \mbox{and}\ \ y=\sqrt{x}.$$
Let us note that the coefficients of the diffusion (3.3) are time-homogeneous. It is possible to extend this family of term structure models to inhomogeneous processes (see, for instance [12]) solution to
$$dY_{t}=(a-\lambda(t)Y_{t})\,dt+\sigma\sqrt{Y_{t}}\,dB_{t},\quad Y_{0}=x,$$
where $\lambda$ is a continuous function. We are still able to emphasize a relation like (3.2) with the following functions and parameters (see, for instance Theorem 6.3.5.1 in [12]):
$$f(t,x)=\frac{\sigma^{2}}{4\rho^{\prime}(t)}\,x^{2},\quad\rho(t)=\frac{\sigma^{2}}{4}\,\int_{0}^{t}\exp\Big{\{}\int_{0}^{s}\lambda(u)\,du\Big{\}}\,ds,\quad\delta=\frac{4a}{\sigma^{2}}\ \ \mbox{and}\ \ y=\sqrt{x}.$$
Both the homogeneous and the inhomogeneous CIR models are related to the squared Bessel process through a time dependent linear transformation. Modelling the volatility in finance actually requires to handle with other process: the CEV model (Constant Elasticity of Variance) which satisfies:
$$dY_{t}=Y_{t}(\mu\,dt+\sigma Y_{t}^{\beta}\,dB_{t}),\quad t\geq 0,\quad Y_{0}=x.$$
Under particular conditions, the process $(Y_{t})_{t\geq 0}$ satisfies (3.2) with $f(t,x)=e^{\mu t}x^{\alpha}$, $\alpha$ depending on $\beta$ and being different from the square (see for instance [12]). For option pricing in finance, it is therefore of prime interest to simulate precisely trajectories of underlying assets which follow CIR or CEV models. It permits to estimate the prices of derivatives like European options but also paths dependent options like Asian or barrier options.
As already seen, families of stochastic models are directly related to the Bessel process through the identity (3.2). If the function $f$ is globally Lipschitz continuous with respect to the space variable then the Bessel $\varepsilon$-strong approximation $(y_{t}^{\varepsilon})_{t\geq 0}$ allows to generate a $\varepsilon^{\prime}$-approximation of $(Y_{t})_{t\geq 0}$ which is given by $(f(t,y_{\rho(t)}^{\varepsilon}))_{t\geq 0}$, the parameters $\varepsilon$ and $\varepsilon^{\prime}$ being related through the Lipschitz constant and the time interval under consideration.
If the transformation $f$ is not uniformly Lipschitz with respect to the space variable (CIR and CEV models, for instance), then the Bessel $\varepsilon$-strong approximation permits to obtain a lower-bound and an upper-bound of any path $(Y_{t})_{t\geq 0}$ depending on $\varepsilon$. These bounds imply a precise estimation of path-dependent characteristics and play therefore a crucial role for applications. Let us consider the following example: a CIR model observed on the time interval $[0,2]$ with the parameters: $k=2$, $\theta=1/3$, $\sigma=1$ and the starting value $x=1$. It is therefore expressed by $Y_{t}=f(t,Z_{\rho(t)}^{\delta,y})$ for all $t\in[0,2]$. Introducing the $\varepsilon$-strong approximation of the Bessel process $(y^{\varepsilon}_{t})_{t\geq 0}$, based on the Bessel skeleton $({\rm BeS})_{\delta}$ or $({\rm BeS})_{\delta}^{w}$ that is $((u_{n}^{\varepsilon},s_{n}^{\varepsilon})_{n\geq 1},(y_{n}^{\varepsilon})_{n\geq 0})$, we obtain the almost surely bounds:
$$f(t,y_{\rho(t)}^{\varepsilon}-\varepsilon)\leq Y_{t}\leq f(t,y_{\rho(t)}^{\varepsilon}+\varepsilon),\quad\forall t\in[0,2],$$
since the function $x\mapsto f(t,x)$ is increasing. In Figure 6 (right), one generation of the upper and lower bounds is represented for any $t\in\{\rho^{-1}(s_{n})\}_{n\geq 1}\cap[0,2]$. The accuracy of the approximation is not uniform since it depends on the value of the process and on the time variable. More precisely, we propose to define the precision variable $P_{\varepsilon}$ by
$$P_{\varepsilon}:=\sup_{t\in[0,2]}\Big{|}f(t,y_{\rho(t)}^{\varepsilon}+\varepsilon)-f(t,y_{\rho(t)}^{\varepsilon}-\varepsilon)\Big{|}.$$
(3.4)
Using the explicit expression of the function $f$ associated with the CIR model, we obtain an explicit expression of the accuracy depending on the Bessel skeleton:
$$P_{\varepsilon}=4\,\varepsilon\sup\Big{\{}y_{n}\,e^{-2\rho^{-1}(s_{n})}\ \mbox{s.t.}\ s_{n}\leq\rho^{-1}(2)\Big{\}}.$$
The probability distribution of the ration $P_{\varepsilon}/\varepsilon$ is represented in Figure 6 (left): we observe that the accuracy is close to four times the initial condition of the Bessel process.
Of course the difference between the lower and upper paths is not uniformly bounded. This accuracy is nevertheless sufficient in many applications but if the challenge is to reach a uniform bound, then we suggest another approach. The key is to let the size of the spheroids used in the Bessel approximation depend on the space variable: the size is no more fixed once for ever and equal to $\varepsilon$. Such an approach was presented in detail in [7] for processes defined by $Y_{t}:=f(t,B_{\rho(t)})$, transformations of the time-changed Brownian motion and can be adapted to the Bessel case.
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Correlating $\epsilon^{\prime}/\epsilon$ to hadronic $B$ decays via $U(2)^{3}$ flavour symmetry
Andreas Crivellin
Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Christian Gross
Dipartimento di Fisica dell’Università di Pisa and INFN, Sezione di Pisa, Pisa, Italy
Theoretical Physics Department, CERN, 1211 Geneve 23, Switzerland
Stefan Pokorski
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5,
PL-02-093 Warsaw, Poland
Leonardo Vernazza
Nikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands
Abstract
There are strong similarities between charge-parity (CP) violating observables in hadronic $B$ decays (in particular $\Delta A^{-}_{\rm CP}$ in $B\to K\pi$) and direct CP violation in Kaon decays ($\epsilon^{\prime}$): All these observables are very sensitive to new physics (NP) which is at the same time CP and isospin violating (i.e. NP with complex couplings which are different for up quarks and down quarks). Intriguingly, both the measurements of $\epsilon^{\prime}$ and $\Delta A^{-}_{\rm CP}$ show deviations from their Standard Model predictions, calling for a common explanation (the latter is known as the $B\to K\pi$ puzzle). For addressing this point, we parametrize NP using a gauge invariant effective field theory approach combined with a global $U(2)^{3}$ flavor symmetry in the quark sector (also known as less-minimal flavour violation). We first determine the operators which can provide a common explanation of $\epsilon^{\prime}$ and $\Delta A^{-}_{\rm CP}$ and then perform a global fit of their Wilson coefficients to the data from hadronic $B$ decays. Here we also include e.g. the recently measured CP asymmetry in $B_{s}\to KK$ as well as the purely isospin violating decay $B_{s}\to\phi\rho^{0}$, finding a consistent NP pattern providing a very good fit to data. Furthermore, we can at the same time explain $\epsilon^{\prime}/\epsilon$ for natural values of the free parameters within our $U(2)^{3}$ flavour approach, and this symmetry gives interesting predictions for hadronic decays involving $b\to d$ transitions.
††preprint: PSI-PR-19-18, UZ-TH 41/19, CERN-TH-2019-142, Nikhef/2019-041, INT-PUB-19-041
I Introduction
Even though the Standard Model (SM) of particle physics has been tested to an astonishing precision within the last decades, it cannot be the ultimate theory describing the fundamental constituents and interactions of matter. For example, in order to generate the matter anti-matter asymmetry of the universe, the Sakharov criteria Sakharov (1967) must be satisfied. One of these requirements is the presence of CP violation, which is found to be far too small within the SM Cohen et al. (1993); Gavela et al. (1994a); Huet and Sather (1995); Gavela et al. (1994b, c); Riotto and Trodden (1999) whose only source of CP violation is the phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Therefore, physics beyond the SM with additional sources of CP violation is needed.
Thus, CP violating observables are promising probes of new physics (NP) as they could test the origin of the matter anti-matter asymmetry of the universe. In this respect, direct CP violation in Kaon decays ($\epsilon^{\prime}/\epsilon$) is especially relevant, as it is very suppressed in the SM, extremely sensitive to NP and can therefore test the multi TeV scale Buras et al. (2014). Furthermore, recent theory calculations from lattice and dual QCD Buras and Gérard (2015); Buras et al. (2015a); Bai et al. (2015); Kitahara et al. (2016a) show intriguing tensions between the SM prediction and the experimental measurement. In order to explain this tension,111Calculations using chiral perturbation theory Cirigliano et al. (2012); Pich (2004); Pallante et al. (2001); Gisbert and Pich (2018a, b) are consistent with the experimental value but have large errors. NP must not only violate CP but in general also isospin Branco et al. (1983) (i.e. couple differently to up quarks as to down quarks) in order to give a sizeable effect in $\epsilon^{\prime}/\epsilon$ Aebischer et al. (2019a).
Interestingly, there are also tensions between theory and data concerning CP violation in hadronic $B$ meson decays, including the long-standing $B\to K\pi$ puzzle Gronau and Rosner (1999); Buras et al. (2003, 2004a, 2004b); Baek and London (2007); Fleischer et al. (2007). Recently, LHCb data Aaij et al. (2018) increased this tension Fleischer et al. (2018a, b), and also the newly measured CP asymmetry in $B_{s}\to K^{+}K^{-}$ Aaij et al. (2018) points towards additional sources of CP violation, renewing the theoretical interest in these decays Datta et al. (2019); Faisel and Tandean (2019). Like for $\epsilon^{\prime}/\epsilon$, both CP and isospin violation are in general required for solving this tension.
This can be achieved with NP in electroweak penguin operators Fleischer (1996); Fleischer et al. (2008); Baek et al. (2009) that may for instance be generated in $Z^{\prime}$ models Barger et al. (2009a, b). Furthermore, the same NP effects can be tested in the theoretically clean purely isospin violating decays $B_{s}\to\phi\rho^{0}$ and $B_{s}\to\phi\pi^{0}$ Fleischer (1994); Buras et al. (2003); Hofer et al. (2011); Hofer and Vernazza (2012) where the former one has been measured recently Aaij et al. (2017), putting additional constraints on the parameter space.
These intrinsic similarities between $\epsilon^{\prime}/\epsilon$ and hadronic $B$ decays suggest a common origin of the deviations from the SM predictions resulting in correlations among them. This can be studied in a model independent way within an effective field theory (EFT) approach. In order to connect $\epsilon^{\prime}/\epsilon$ ($s\to d$ transitions) to hadronic $B$ decays ($b\to s,d$ transitions) a flavour link
is obviously necessary. Here, we assume a global $U(2)^{3}$ flavor symmetry in the quark sector Barbieri et al. (1996, 1997, 2011a, 2011b); Crivellin et al. (2011); Barbieri et al. (2012a, b).222The $U(2)^{3}$ flavour symmetry is analogous to Minimal Flavour Violation Chivukula et al. (1987); Hall and Randall (1990); Buras et al. (2001) (MFV) which uses a global $U(3)^{3}$ flavour symmetry instead D’Ambrosio et al. (2002). However, $U(3)^{3}$ flavour is anyway strongly broken by the third generation Yukawa couplings to $U(2)^{3}$.
As we will see, this flavour symmetry yields the desired flavour structure for the Wilson coefficients: it predicts a large phase (equal to the CKM phase) in Kaon decays, and the effect in $B$ physics only differs by a relative order one factor (if the corresponding CKM elements are factored out) but contains an additional free phase.
II Setup and Observables
Here we discuss our setup and the predictions for the observables. The strategy for this is the following: We will start with $\epsilon^{\prime}/\epsilon$ where we want to explaining the difference between experiment and the SM prediction. This will allow us to restrict ourselves to the limited set of operators which are capable of achieving this. We will then move to hadronic $B$ decays, pointing out the striking similarities with $\epsilon^{\prime}/\epsilon$, and then establish our $U(2)^{3}$ flavour setup.
The experimental value for direct CP violation in Kaon decays Batley et al. (2002); Alavi-Harati et al. (2003); Abouzaid et al. (2011),
$$\left({\epsilon^{\prime}}/{\epsilon}\right)_{\rm exp}=(16.6\pm 2.3)\times 10^{%
-4}\,,$$
(1)
lies significantly above the SM prediction from lattice QCD Bai et al. (2015); Buras et al. (2015a); Kitahara et al. (2016a) which is in the range $\left({\epsilon^{\prime}}/{\epsilon}\right)_{\rm SM}\simeq(1-2)\times 10^{-4}$, with an error of the order of $5\times 10^{-4}$. Note that the lattice estimate is consistent with the estimated upper limit from dual QCD Buras and Gérard (2015).
In the past years, many NP explanations of the ${\epsilon^{\prime}}/{\epsilon}$ discrepantly have been put forward (see e.g. Buras and De Fazio (2016a, b); Bobeth et al. (2017a); Endo et al. (2017); Bobeth et al. (2017b); Blanke et al. (2016); Buras et al. (2015b); Buras (2016); Tanimoto and Yamamoto (2016); Kitahara et al. (2016b); Endo et al. (2016); Crivellin et al. (2017); Endo et al. (2018); Chen and Nomura (2018a, b); Haba et al. (2018a, b); Matsuzaki et al. (2018); Aebischer et al. (2019b); Chen and Nomura (2019); Iguro and Omura (2019)).
Since here we want to perform an EFT analysis we consider the impact of the operators listed in Ref. Aebischer et al. (2019a). First of all, one sees that there are eight operators (plus their chirality flipped counter parts) which give numerically large effects in $\epsilon^{\prime}/\epsilon$.
We will focus on these operators in the following since, requiring an explanation of $\epsilon^{\prime}/\epsilon$, the NP scale for the other operators must be so low that it would be in conflict with direct LHC searches. Furthermore – since we will consider a $U(2)^{3}$ setup – the Wilson coefficients of scalar and tensor operators contributing to Kaon physics are suppressed by the corresponding tiny Yukawa couplings of the first and second generation. Therefore, we are left with the Lagrangian
$$\displaystyle{\cal L}_{\epsilon^{\prime}/\epsilon}=C_{q}^{VLR}O_{q}^{VLR}+%
\tilde{C}_{q}^{VLR}\tilde{O}_{q}^{VLR}+L\leftrightarrow R$$
(2)
with $q=u,d$ and the operators
$$\displaystyle O_{q}^{VLR}$$
$$\displaystyle=(\bar{s}_{\alpha}\gamma^{\mu}P_{L}d_{\alpha})(\bar{q}_{\beta}%
\gamma_{\mu}P_{R}q_{\beta})\,,$$
(3)
$$\displaystyle\tilde{O}_{q}^{VLR}$$
$$\displaystyle=(\bar{s}_{\alpha}\gamma^{\mu}P_{L}d_{\beta})(\bar{q}_{\beta}%
\gamma_{\mu}P_{R}q_{\alpha})\,,$$
plus their chirality flipped counterparts. Here, $\alpha$ and $\beta$ are color indices and therefore $O_{q}^{VLR}$ ($\tilde{O}_{q}^{VLR}$) is a color singlet (triplet) operator. However, noting that one needs a violation of isospin (which is conserved in the left-handed quark current due to $SU(2)_{L}$ gauge invariance) we can omit the operators with flipped chiralities and the NP contribution to $\epsilon^{\prime}/\epsilon$ is approximately given by Aebischer et al. (2019a, b)
$$\displaystyle\left(\dfrac{\epsilon^{\prime}}{\epsilon}\right)_{\rm NP}\approx$$
$$\displaystyle\ 1\,{\rm TeV^{2}}\big{(}124\ \textrm{Im}(C_{d}^{VLR}-C_{u}^{VLR})$$
(4)
$$\displaystyle\qquad\left.+432\ \textrm{Im}(\tilde{C}_{d}^{VLR}-\tilde{C}_{u}^{%
VLR})\right)\,.$$
for a NP scale of 1 TeV.333Here we took again into account that for an enhanced effect NP should be isospin violating and neglected small isospin conserving contributions in the numerical factors.
As outlined in the introduction, we want to study correlations between hadronic $B$ decays and $\epsilon^{\prime}/\epsilon$ using a $U(2)^{3}$ flavour symmetry. In particular we want to address the $B\to K\pi$ puzzle. Here the experimental value for
$$\Delta A^{-}_{\rm CP}\equiv A_{\rm CP}(B^{-}\to\pi^{0}K^{-})-A_{\rm CP}(\bar{B%
}^{0}\to\pi^{+}K^{-})\,,$$
(5)
is Amhis et al. (2017)
$$\Delta A^{-}_{\rm CP}|_{\rm exp}=(12.4\pm 2.1)\%\,,$$
(6)
which deviates from the SM prediction Hofer et al. (2011)
$$\Delta A^{-}_{\rm CP}|_{\rm SM}=(1.8^{+4.1}_{-3.2})\%\,,$$
(7)
at the 2$\sigma$ level.444Ref. Beaudry et al. (2018) performed a fit to all $B\to\pi K$ data and finds that the p-value crucially depends on the ratio of the color-suppressed to the color-allowed tree amplitudes. Since an acceptably good fit can be achieved if this ratio is somewhat larger than what is predicted from QCD factorization it is not absolutely clear that $B\to\pi K$ data points to NP, but it certainly leaves room for it. In the following we will investigate how NP can account for the measurement. In addition, one has to take into account also other CP asymmetries and total branching rations of hadronic $B$ decays involving $b\to s$ transitions. Here, the experimental measurements of Aaij et al. (2018, 2017)
$$\displaystyle A_{\rm CP}[B_{s}\to K^{+}K^{-}]_{\rm exp}$$
$$\displaystyle=(-20.0\pm 6.0\pm 2.0)\%\,,$$
(8)
$$\displaystyle{\rm Br}[B_{s}\to\phi\rho^{0}]_{\rm exp}$$
$$\displaystyle=(2.7\pm 0.7\pm 0.2\pm 0.2)\times 10^{-7}\,,$$
which agree with the SM predictions
$$\displaystyle A^{B_{s}}_{\rm CP}|_{\rm SM}$$
$$\displaystyle=(-5.9^{+26.6}_{-5.1})\%\,,$$
(9)
$$\displaystyle{\rm Br}[B_{s}\to\phi\rho^{0}]_{\rm SM}$$
$$\displaystyle=(5.3^{+1.8}_{-1.3})\times 10^{-7}\,,$$
at the 1–2 $\sigma$ level, are two of the most important examples in with respect to SM accuracy and experimental precision.
For hadronic $B$ decays it is standard to use the effective Hamiltonian
$${\cal H}_{\rm eff}^{\rm NP}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V^{*}_{ts}\!\!\!\!%
\sum_{q=u,d,s,c}\left(C^{q}_{5}O^{q}_{5}+C^{q}_{6}O^{q}_{6}\right)+\mbox{h.c.}\,,$$
(10)
for $b\to s$ transitions where the four-quark operators are defined as
$$\displaystyle O_{5}^{q}$$
$$\displaystyle=(\bar{s}_{\alpha}\gamma^{\mu}P_{L}b_{\alpha})\,(\bar{q}_{\beta}%
\gamma_{\mu}P_{R}q_{\beta}),$$
(11)
$$\displaystyle O_{6}^{q}$$
$$\displaystyle=(\bar{s}_{\alpha}\gamma^{\mu}P_{L}b_{\beta})\,(\bar{q}_{\beta}%
\gamma_{\mu}P_{R}q_{\alpha})\,.$$
The corresponding expressions for $b\to d$ transitions follow by replacing $\bar{s}$ with $\bar{d}$ and $V_{tb}V^{*}_{ts}$ by $V_{tb}V^{*}_{td}$. Here, we consider only the operators motivated by $\epsilon^{\prime}/\epsilon$, as discussed in the last subsection, and neglect the numerically very small contributions of $q=c,s$ in Eq. (10).
Under the assumption of a global $U(2)^{3}$ flavour symmetry (to be discussed later on) the NP Wilson coefficients carry a common new weak phase $\phi$ and we parameterise them as
$$C^{d,u}_{5}=c^{d,u}_{5}\,e^{i\phi}\,,\quad\qquad C^{d,u}_{6}=c^{d,u}_{6}\,e^{i%
\phi}\,.$$
(12)
Like for $\epsilon^{\prime}/\epsilon$, the leading effect which is necessary to account for the $K\pi$ puzzle is isospin violating. This can be easily seen by using an intuitive notation, similar to the one used in Ref. Hofer et al. (2011). We parameterize the NP contribution to $K\pi$ decays in terms of $r_{\rm NP}^{q}$ ($r_{\rm NP}^{{\rm A},q}$), representing the ratio of NP penguin (annihilation) amplitudes with respect to the dominant QCD penguin amplitude of the SM. Therefore, one has for instance
$$\displaystyle\Delta A_{\textrm{CP}}^{-}$$
$$\displaystyle\simeq-2{\rm Im}(r_{\rm C})\sin\gamma$$
(13)
$$\displaystyle+2\left[{\rm Im}(r_{\rm NP}^{d})-{\rm Im}(r_{\rm NP}^{u})+{\rm Im%
}(r_{\rm NP}^{{\rm A},d})-{\rm Im}(r_{\rm NP}^{{\rm A},u})\right]\sin\phi,$$
where $r_{\rm C}$ originating from the color suppressed tree topology amplitude of the SM. Here $\gamma$ is the CKM phase defined as $V_{ub}=|V_{ub}|e^{-i\gamma}$ and $\phi$ a generic weak phase of the NP contribution. We see that isospin violation is needed to get an effect in $\Delta A_{\textrm{CP}}^{-}$. Thus, interesting effects are expected in other hadronic $B$ decays sensitive to isospin violations, such as the analogues of $\Delta A^{-}_{\rm CP}$ with $PV$ (pseudo-scalar and vector) and $VV$ (two vector) mesons in the final state (e.g. decays in which one replaces $\pi$ and $K$ in eq. (5) with $\rho$ or $K^{*}$). Furthermore, an equivalent difference of direct CP asymmetries constructed for $B_{s}\to KK$ decays, i.e. $\Delta A_{\textrm{CP}}^{\rm KK}\equiv A_{\rm CP}(\bar{B}_{s}\to\bar{K}^{0}K^{0%
})-A_{\rm CP}(\bar{B}_{s}\to K^{-}K^{+})$, and the purely isospin violating decays $B_{s}\to\phi\pi^{0}$ and $B_{s}\to\phi\rho^{0}$ are sensitive to isospin violating NP as well.
The amplitudes of hadronic $B$ decays, like the ones involved in ratios $r_{\rm NP}^{q}$ and $r_{\rm NP}^{{\rm A},q}$ in Eq. (13) contain strong phases originating from QCD effects. These phases can be calculated at next-to-leading order using QCD factorisation Beneke et al. (1999, 2001); Beneke and Neubert (2003). This calculation is rather technical and involves many input parameters (see e.g. Refs. Beneke et al. (2009); Hofer et al. (2011) for a detailed discussion on the calculation of NP operators matrix elements in the context of QCD factorisation). Thus we provide here semi-numerical formulas which describe the NP effect in the observables in appendix A based on Eq. (12) as input. However, these formula only serve as an illustration of impact of NP while in the phenomenological analysis we will perform a global fit (including also theory errors of the NP contributions), as done in Ref. Hofer et al. (2011), to take all measurements consistently into account.
Let us now turn to the connection between $\epsilon^{\prime}/\epsilon$ and hadronic $B$ decays. For this we consider the $SU(2)_{L}$ invariant operators Buchmuller and Wyler (1986); Grzadkowski et al. (2010)
$$\displaystyle{\cal L}_{\rm SMEFT}$$
$$\displaystyle=\dfrac{1}{\Lambda^{2}}\left(C^{(1)ijkl}_{Qq}O^{(1)ijkl}_{Qq}+C^{%
(3)ijkl}_{Qq}O^{(3)ijkl}_{Qq}\right)$$
(14)
with
$$\displaystyle\begin{aligned} \displaystyle O^{(1)ijkl}_{Qq}=\bar{Q}_{i}^{%
\alpha}\gamma^{\mu}P_{L}Q_{j}^{\alpha}\bar{q}_{k}^{\beta}\gamma_{\mu}P_{R}q_{l%
}^{\beta}\,,\\
\displaystyle O^{(3)ijkl}_{Qq}=\bar{Q}_{i}^{\alpha}\gamma^{\mu}P_{L}Q_{j}^{%
\beta}\bar{q}_{k}^{\beta}\gamma_{\mu}P_{R}q_{l}^{\alpha}\,,\end{aligned}$$
(15)
where $i,j,k,l$ are flavour indices, $q=u,d$ and $Q$ stands for the quark $SU(2)_{L}$ doublet. Depending on the flavour structure, these operators enter $\epsilon^{\prime}/\epsilon$ or hadronic $B$ decays.
Now, we employ the $U(2)^{3}$ flavour symmetry in the quark sector in order to link Wilson coefficients with different flavours to each other. First of all, note that with respect to the right-handed current we are only interested in the flavour diagonal couplings to $u,d$ and do not need to consider the couplings to heavier generations due to their suppressed effects in the observables. Concerning the left-handed current, $U(2)^{3}$ flavour with a minimal spurion sector predicts that $s\to d$ transitions are proportional to $V_{ts}^{*}V_{td}$ while $b\to s(d)$ are proportional to $V_{ts(d)}^{*}V_{tb}$ and the relative effect is governed by an order one factor $x_{B}$ and a free phase $\phi$ Barbieri et al. (2012a). Thus, Eq. (15) can be written as
$$\displaystyle\begin{aligned} \displaystyle C_{Qq}^{(a)2111}&\displaystyle=V_{%
td}V_{ts}^{*}c_{q}^{\left(a\right)}\\
\displaystyle C_{Qq}^{(a)2311}&\displaystyle=V_{tb}V_{ts}^{*}{x_{B}}{e^{i\phi}%
}c_{q}^{\left(a\right)}\\
\displaystyle C_{Qq}^{(a)1311}&\displaystyle=V_{tb}V_{td}^{*}{x_{B}}{e^{i\phi}%
}c_{q}^{\left(a\right)}\end{aligned}$$
(16)
with $a=1,3$ (denoting the color singlet and triplet structure) and $q=u,d$. Note that due to the hermiticity of the operators in Eq. (15) $c_{q}^{\left(1,3\right)}$ must be real and that conventional MFV (based on $U(3)$ flavour) is obtained in the limit $\phi\to 0$ and $x_{B}\to 1$. Therefore, using MFV instead of $U(2)^{3}$ would provide an effect in $\epsilon^{\prime}/\epsilon$ but no source of CP violation in hadronic $B$ decays.
With these conventions we obtain for the Wilson coefficients entering $\epsilon^{\prime}/\epsilon$ and hadronic $B$ decays
$$\displaystyle C_{q}^{VLR}$$
$$\displaystyle=\frac{{V_{ts}^{*}V_{td}c_{q}^{\left(1\right)}}}{{{\Lambda^{2}}}}\,,$$
$$\displaystyle\!\!\!\!\!\!\tilde{C}_{q}^{VLR}$$
$$\displaystyle=\frac{{V_{ts}V_{td}^{*}c_{q}^{\left(3\right)}}}{{{\Lambda^{2}}}}\,,$$
(17)
$$\displaystyle C_{5}^{q}$$
$$\displaystyle=\frac{{\sqrt{2}}}{{4{G_{F}}{\Lambda^{2}}}}{x_{B}}{e^{i\phi}}c_{q%
}^{\left(1\right)}\,,$$
$$\displaystyle\!\!\!\!\!\!C_{6}^{q}$$
$$\displaystyle=\frac{{\sqrt{2}}}{{4{G_{F}}{\Lambda^{2}}}}{x_{B}}{e^{i\phi}}c_{q%
}^{\left(3\right)}\,.$$
III Phenomenological Analysis
Here we present the results of the global fit to the data from hadronic $B$ decays. Taking into account that NP must have a common weak phase $\phi$ originating from $U(2)^{3}$ symmetry breaking we define
$${x^{\left(a\right)}}\equiv c_{d}^{\left(a\right)}-c_{u}^{\left(a\right)}\,,%
\qquad{z^{\left(a\right)}}\equiv c_{d}^{\left(a\right)}+c_{u}^{\left(a\right)}\,,$$
(18)
for future convenience where ${x^{\left(a\right)}}$ (${z^{\left(a\right)}}$) parametrizes the isospin violating (conserving) effects. Marginalizing over ${z^{\left(a\right)}}$ in the ranges from $-0.12<z^{\left(1\right)}<0.12$, and $-0.04<z^{\left(3\right)}<0.04$ we have three degrees of freedom for both the singlet scenario (1) and the triplet scenario (3). While the $\chi^{2}$ of the SM is 18.8, the best fit points for our two scenarios are
$$\displaystyle\begin{aligned} \displaystyle x_{B}x^{(1)}=0.306\,,\quad{x_{B}z^{%
\left(1\right)}}=-0.12\,,\\
\displaystyle x_{B}x^{(3)}=0.144\,,\quad{x_{B}z^{\left(3\right)}}=-0.04\,,\\
\end{aligned}$$
(19)
with a phase of
$$\phi^{(1)}=157.6^{\circ}\,,\qquad\phi^{(3)}=169.0^{\circ}\,,$$
(20)
and
$$\Delta\chi^{2}(1)=16.5\,,\quad\Delta\chi^{2}(3)=13.7\,.$$
(21)
This corresponds to pulls of $3.3\,\sigma$ for (1) and $2.9\,\sigma$ (3) with respect to the SM. Let us also consider the case in which $z^{\left(a\right)}=0$, which corresponds to the scenario of maximal isospin violation. In this case the best fit points are $x_{B}x^{(1)}=0.312$, $\phi^{(1)}=163.3^{\circ}$, and $x_{B}x^{(3)}=0.142$, $\phi^{(3)}=-146.1^{\circ}$. The $\chi^{2}$ difference with respect to the SM are now $\Delta\chi^{2}(1)=15.3$ and $\Delta\chi^{2}(3)=12.9$, which corresponds to pulls of $3.5\,\sigma$ for (1) and $3.0\,\sigma$ for (3) with respect to the SM for two degrees of freedom.
Now, we can correlate hadronic $B$ decays to $\epsilon^{\prime}/\epsilon$. For this we observe that the NP contribution to $\epsilon^{\prime}/\epsilon$ can be directly expressed in terms of $x^{(a)}$ as
$$\displaystyle\left({\frac{{\epsilon^{\prime}}}{\epsilon}}\right)_{{\rm{NP}}}\!%
\!\!\!\!\!\approx\frac{0.018\,x^{(1)}}{(\Lambda/\mathrm{TeV})^{2}}\,,\quad%
\left({\frac{{\epsilon^{\prime}}}{\epsilon}}\right)_{{\rm{NP}}}\!\!\!\!\!\!%
\approx\frac{0.062\,x^{(3)}}{(\Lambda/\mathrm{TeV})^{2}}\,,$$
(22)
for the color singlet and triplet case, respectively. Note that the phase of the contribution to $\epsilon^{\prime}/\epsilon$ is fixed by the $U(2)^{3}$ flavour symmetry such that $\phi$ only enters in hadronic $B$ decays. Furthermore, $z^{(a)}$ is not correlated to $\epsilon^{\prime}/\epsilon$ where only the difference $x^{(a)}$ enters and just a free parameter over which we will marginalize as described above. Therefore, we can express $x^{(a)}$ in terms of the NP contribution to $\epsilon^{\prime}/\epsilon$ and show the effects in hadronic $B$ decays as a function of $x_{B}\times(\epsilon^{\prime}/\epsilon)_{\rm NP}/10^{-3}$ and $\phi$.
The corresponding result is depicted in Fig. 1 where the preferred regions from hadronic $B$ decays are displayed. Note that all regions are consistent with each other (i.e. all overlap at the $1\,\sigma$ level), such that one can account for the deviations (mainly in $A_{\rm CP}[B_{s}\to K^{+}K^{-}]_{\rm exp}$ and $\Delta A^{-}_{\rm CP}$) without violating bounds from other observables.
From Fig. 1 one can also see that a natural order one value of $x_{B}$ can not only account the tensions in hadronic $B$ decays but also give a NP contribution to $\epsilon^{\prime}/\epsilon$ of the order of $10^{-3}$ as needed to explain the tension.
In Fig. 2 we show the predictions for various (differences of) CP asymmetries within the SM compared to the one of the best fit points for the two scenarios as well as and the corresponding experimental results. We use our $U(2)^{3}$ flavour symmetry to give predictions for hadronic $B$ decays involving $b\to d$ transitions as well. Although the fit clearly indicates isospin violating NP as the preferred solution to the $\Delta A_{\rm CP}$ problem, we notice that the errors of the theory predictions are still quite large, calling for future improvements in the calculational methods. Similarly, a clearer picture could be obtained with more precise experimental measurements, in particular for the $PV$
and $VV$ decay modes.
IV Conclusions and Outlook
In this article we pointed out intrinsic analogies between $\epsilon^{\prime}/\epsilon$ and CP violation in hadronic $B$ decays, in particular $\Delta A^{-}_{\rm CP}$: These observables are all sensitive to 4-quark operators with flavour changing neutral currents in the down sector and test the combined effects of CP and isospin violation. Therefore, the $B\to K\pi$ puzzle increases the interest in $\epsilon^{\prime}/\epsilon$ and vice versa, calling for a combined explanation.
After identifying the two operators which are capable of explaining the $\epsilon^{\prime}/\epsilon$ anomaly within an $U(2)^{3}$ flavour setup we performed a global fit to the data from hadronic $B$ decays. We find that both operators provide a consistent pattern in hadronic $B$ decays resulting in a very good fit which is more than 3$\,\sigma$ better than the one of the SM. Furthermore, the $U(2)^{3}$ flavour symmetry is consistent with a common explanation of the anomalies in $\epsilon^{\prime}/\epsilon$ and hadronic $B$ decays, providing at the same time interesting predictions for hadronic decays involving $b\to d$ transitions (such as $B\to K^{+}K^{-}$ and $B\to\pi\pi$) which can be tested experimentally in the near future by LHCb. However, further progress of the theory side is crucial in order to improve the precision of the theoretical results.
Acknowledgments — We thank Andrzej Buras, Robert Fleischer and David Straub for useful discussions. We are grateful to Greg Landsberg for bringing the LHCb measurement of $B_{s}\to\phi\rho$ to our attention. The work of A.C. is supported by a Professorship Grant (PP00P2_176884) of the Swiss National Science Foundation. C.G. is supported by the European Research Council grant NEO-NAT. S.P. research was supported by the Alexander von Humboldt Foundation. He thanks Slava Mukhanov for his hospitality at the LMU, Munich. L.V. is supported by the D-ITP consortium, a program of NWO funded by the Dutch Ministry of Education, Culture and Science (OCW). C.G. and L.V. thank the Mainz Institute for Theoretical Physics (MITP) of the DFG Cluster of Excellence PRISMA${}^{+}$ (Project ID 39083149) for hospitality. A. C. thanks the INT at the University of Washington for its hospitality and the DOE for partial support during the completion of this work.
Appendix A Additional non-leptonic decay observables
In this appendix we collect semi-numerical formulae for other non-leptonic decay observables which are sensitive to isospin violating NP for the case of an $U(2)^{3}$ flavour symmetry. First of all, we list results for $\Delta A_{\textrm{CP}}^{-}$, and the corresponding observable obtained for $PV$ and $VV$ decays. One has
$$\displaystyle\Delta A_{\textrm{CP}}^{-,\pi K}$$
$$\displaystyle\simeq 0.02^{+0.04}_{-0.03}+\big{[}13(c_{5}^{d}-c_{5}^{u})+34(c_{%
6}^{d}-c_{6}^{u})\big{]}\sin\phi-\big{[}2(c_{5}^{d}-c_{5}^{u})+5(c_{6}^{d}-c_{%
6}^{u})\big{]}\cos\phi,$$
(23)
$$\displaystyle\Delta A_{\textrm{CP}}^{-,\rho K}$$
$$\displaystyle\simeq 0.11^{+0.11}_{-0.45}+\big{[}21(c_{5}^{d}-c_{5}^{u})+39(c_{%
6}^{d}-c_{6}^{u})\big{]}\sin\phi-\big{[}12(c_{5}^{d}-c_{5}^{u})+10c_{6}^{d}-1.%
1c_{6}^{u}\big{]}\cos\phi,$$
$$\displaystyle\Delta A_{\textrm{CP}}^{-,\pi K^{*}}$$
$$\displaystyle\simeq 0.09^{+0.23}_{-0.29}+\big{[}23(c_{5}^{d}-c_{5}^{u})+45(c_{%
6}^{d}-c_{6}^{u})\big{]}\sin\phi+\big{[}-6c_{5}^{d}+8c_{5}^{u}-2c_{6}^{d}+7c_{%
6}^{u}\big{]}\cos\phi,$$
$$\displaystyle\Delta A_{\textrm{CP}}^{-,\rho K^{*}}$$
$$\displaystyle\simeq 0.01^{+0.15}_{-0.10}+\big{[}(c_{5}^{d}-c_{5}^{u})-20c_{6}^%
{d}+25c_{6}^{u}\big{]}\sin\phi-\big{[}10(c_{5}^{d}-c_{5}^{u})+2.5c_{6}^{d}+2.5%
c_{6}^{u}\big{]}\cos\phi.$$
These formulae already include the evolution of the Wilson coefficients $C_{5,6}^{u}$ and $C^{d}_{5,6}$ in Eq. (10) from the electroweak scale to the scale $m_{B}$ and the numerical evaluation of the matrix elements using QCD factorization. Note also that the term $\propto\cos\phi$ in the direct CP asymmetries Eq. (23) originate from the interference between amplitudes proportional to $\gamma$ and $\phi$.
Next, we consider $B_{s}\to KK$ and related $VV$ decays. The CP observable in eq. (8) $A_{\rm CP}[B_{s}\to K^{+}K^{-}]$ is given by
$$A_{\rm CP}[B_{s}\to K^{+}K^{-}]\simeq-0.06^{+0.27}_{-0.05}+\big{[}-0.3c_{5}^{d%
}+2.6c_{5}^{u}-1.6c_{6}^{d}+7.1c_{6}^{u}\big{]}\sin\phi+\big{[}-0.75c_{5}^{u}+%
0.2c_{6}^{d}-2.3c_{6}^{u}\big{]}\cos\phi\,.$$
(24)
More sensitive to isospin violation is the difference of direct CP asymmetries
$$\Delta A_{\textrm{CP}}^{\rm KK}\equiv A_{\rm CP}(\bar{B}_{s}\to\bar{K}^{0}K^{0%
})-A_{\rm CP}(\bar{B}_{s}\to K^{-}K^{+})\,,$$
(25)
and the equivalent difference defined for $VV$ modes. One has
$$\displaystyle\Delta A_{\textrm{CP}}^{\rm KK}$$
$$\displaystyle\simeq 0.06^{+0.05}_{-0.26}+\big{[}3\big{(}c_{5}^{d}-c_{5}^{u}%
\big{)}+9\big{(}c_{6}^{d}-c_{6}^{u}\big{)}\big{]}\sin\phi+\big{[}c_{5}^{u}+2c_%
{6}^{u}\big{]}\cos\phi,$$
(26)
$$\displaystyle\Delta A_{\textrm{CP}}^{\rm K^{*}K^{*}}$$
$$\displaystyle\simeq-0.32^{+0.39}_{-0.05}+\big{[}(c_{5}^{d}-c_{5}^{u})-4c_{6}^{%
d}+3c_{6}^{u}\big{]}\sin\phi+\big{[}0.3c_{5}^{u}-0.5c_{6}^{d}-2.0c_{6}^{u}\big%
{]}\cos\phi.$$
Last, we have the $B_{s}$ decays to $\pi,\phi$ and $\rho,\phi$, for which we have
$$\displaystyle{\rm Br}[B_{s}\to\phi\pi^{0}]$$
$$\displaystyle\simeq\Big{\{}0.18^{+0.06}_{-0.05}-\big{[}25\big{(}c_{5}^{d}-c_{5%
}^{u}\big{)}+8\big{(}c_{6}^{d}-c_{6}^{u}\big{)}\big{]}\cos\phi-\big{[}10\big{(%
}c_{5}^{d}-c_{5}^{u}\big{)}+2\big{(}c_{6}^{d}-c_{6}^{u}\big{)}\big{]}\sin\phi%
\Big{\}}\times 10^{-6},$$
(27)
$$\displaystyle{\rm Br}[B_{s}\to\phi\rho^{0}]$$
$$\displaystyle\simeq\Big{\{}0.53^{+0.18}_{-0.13}\big{[}56\big{(}c_{5}^{d}-c_{5}%
^{u}\big{)}+18\big{(}c_{6}^{d}-c_{6}^{u}\big{)}\big{]}\cos\phi+\big{[}22\big{(%
}c_{5}^{d}-c_{5}^{u}\big{)}+6\big{(}c_{6}^{d}-c_{6}^{u}\big{)}\big{]}\sin\phi%
\Big{\}}\times 10^{-6}\,.$$
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An Experimental Design Perspective on
Model-Based Reinforcement Learning
Viraj Mehta, Biswajit Paria, & Jeff Schneider
Robotics Insitute & Machine Learning Department
Carnegie Mellon University
Pittsburgh, PA, USA
\asciifamily{virajm, bparia, schneide}@cs.cmu.edu
&Stefano Ermon & Willie Neiswanger
Computer Science Department
Stanford University
Stanford, CA, USA
\asciifamily{ermon, neiswanger}@cs.stanford.edu
Abstract
In many practical applications of RL, it is expensive to observe state transitions from the environment. For example, in the problem of plasma control for nuclear fusion, computing the next state for a given state-action pair requires querying an expensive transition function which can lead to
many hours of computer simulation or dollars of scientific research.
Such expensive data collection prohibits application of standard RL algorithms which usually require a large number of observations to learn. In this work, we address the problem of efficiently learning a policy while making a minimal number of state-action queries to the transition function.
In particular, we leverage ideas from Bayesian optimal experimental design to guide the selection of state-action queries for efficient learning.
We propose an acquisition function that quantifies how much information a state-action pair would provide about the optimal solution to a Markov decision process. At each iteration, our algorithm maximizes this acquisition function, to choose the most informative state-action pair to be queried, thus yielding a data-efficient RL approach. We experiment with a variety of simulated continuous control problems and show that our approach learns an optimal policy with up to $5$ – $1,000\times$ less data than model-based RL baselines and $10^{3}$ – $10^{5}\times$ less data than model-free RL baselines. We also provide several ablated comparisons which point to substantial improvements arising from the principled method of obtaining data.
1 Introduction
Reinforcement learning (RL) has suffered for years from a curse of poor sample complexity.
State-of-the-art model-free reinforcement learning algorithms routinely take tens of thousands of sampled transitions to solve very simple tasks and millions to solve moderately complex ones (Haarnoja et al., 2018; Lillicrap et al., 2015).
The current best model-based reinforcement learning (MBRL) algorithms are better, requiring thousands of samples for simple problems and hundreds of thousands of samples for harder ones (Chua et al., 2018). In settings where each sample is expensive, even this smaller cost can be prohibitive for the practical application of RL.
For example, in the physical sciences, many simulators require the solution of computationally demanding spatial PDEs in plasma control (Breslau et al., 2018; Char et al., 2019) or aerodynamics applications (Jameson & Fatica, 2006). In robotics, due to the cost of simulating more complicated objects (Heiden et al., 2021), RL methods are typically constrained to fast but limited rigid-body simulators (Todorov et al., 2012).
These costly transition functions prompt the question:
“If we were to collect one additional datapoint from anywhere in the state-action space to best improve our solution to the task, which one would it be?” An answer to this question can be used to guide data collection in RL.
Across the fields of black-box optimization and experimental design, techniques have been developed which choose data to collect that are particularly useful in improving the value of the objective function of the problem.
For example, Bayesian optimization (BO) focuses on maximizing an unknown (black-box) function where queries are expensive (Frazier, 2018; Shahriari et al., 2015).
More generally, Bayesian optimal experimental design (BOED) aims to choose data to collect which are maximally informative about the value of some derived quantity (Chaloner & Verdinelli, 1995). We aim to leverage these ideas for data-efficiency in reinforcement learning. Along these lines,
several works in the realm of Bayesian RL address this problem in the sequential setting. A Bayes-adaptive MDP (Ross et al., 2007) constructs a modified MDP by augmenting the state space with the posterior belief of the MDP, leading to a policy that can optimally trade off between acquiring more information and exploiting the knowledge it already has. However such an MDP is intractable to exactly solve in large spaces so approximations and heuristics have been developed for the solution (Smith, 2007; Guez et al., 2012).
A particularly relevant heuristic is the value of perfect information (VPI) from Dearden et al. (1998), which attempts to capture the potential change in value of a state if the value of a particular action at that state was perfectly known, specifically in a tabular setting. However, VPI doesn’t attempt to distinguish between states that are visited during the execution of the optimal policy and those that aren’t.
This is a critical distinction when collecting data in continuous spaces, as queries may be wasted learning an optimal policy in irrelevant parts of the state space.
Motivated by our opening question, in this paper we study the setting where the agent collects data by sequentially making queries to the transition function with free choice of both the initial state and the action. We refer to this setting as transition-query reinforcement learning (TQRL) and formally define it in Section 3.1.
Although this setting has been studied in the tabular case, to the best of our knowledge it has not been studied in the continuous MDP literature.
In this work, we draw a connection between MBRL and the world of BOED by deriving an acquisition function that quantifies how much information a state-action pair would provide about the optimal solution to a MDP.
Like the techniques in Bayesian RL, our acquisition function is able to determine which state-action pairs are worth acquiring in a way which takes into account the reward function and the uncertainty in the dynamics. However, like the Bayes-Adaptive MDP and unlike the VPI heuristic, this function takes into account the current estimates of which states the optimal policy will visit and values potential queries accordingly. Furthermore, our acquisition function is scalable enough to apply to multidimensional continuous control problems.
In particular, our acquisition function is the expected information gain (EIG) about the trajectory taken by an optimal policy in the MDP that would be achieved
if we were to query the transition function at a given state-action pair.
Finally, we assess the performance of our acquisition function as a data selection strategy in the TQRL setting.
Using this method we are able to solve several continuous reinforcement learning tasks (including a nuclear fusion example) using orders of magnitude less data than a variety of competitor methods.
In summary, the contributions of our paper are:
•
We construct a novel acquisition function that quantifies how much information a state-action pair would provide about the optimal solution to a continuous MDP if the next state were observed from the ground truth transition function. Our acquisition function is able to select relevant datapoints for control purposes leading to improved data efficiency.
•
We propose a practical algorithm for computing this acquisition function and use it to solve continuous MDPs in the TQRL setting.
•
We evaluate the algorithm on five diverse control tasks, where it is often orders of magnitude more sample-efficient than competitor methods and reaches similar asymptotic performance.
2 Related Work
Transition Query Reinforcement Learning
In many RL algorithms, data is collected by initializing a policy at a start state and executing actions in the environment in an episodic manner.
Kearns et al. (2002) introduced the setting where the agent collects data by sequentially sampling transitions from the ground truth transition model by querying at a state and action of its choice, which they refer to as RL with access to a generative model. We refer to this setting for brevity as TQRL.
This setting is relevant in a variety of real-world applications where there is a simulator of the transition model available. In particular, we see the setting in nuclear fusion research, where plasma dynamics are modeled by solving large partial differential equations where 200ms of plasma time can take up to an hour in simulation (Breslau et al., 2018).
There is substantial theoretical work on TQRL for finite MDPs. In particular, Azar et al. (2013) give matching log-linear upper and lower PAC sample complexity bounds,
a substantial speedup to the upper bound for the standard problem which is quadratic in state size (Kakade, 2003).
This is achieved simply by the naive algorithm of learning a transition model by uniformly sampling the space and then performing value iteration on the estimate of the MDP for an optimal policy. More recently, the bound for this setting was tightened to hold for smaller numbers of samples by Li et al. (2020), meaning that for any dataset size in a continuous problem, the PAC performance can be quantified. Finally, Agarwal et al. (2020) show that the naive ‘plug-in’ estimator used in the previous works is minimax optimal for this setting. In summary, this setting is thoroughly understood for finite MDPs and it gives a sample complexity reduction from quadratic to linear in the state space size.
To our knowledge there do not exist works specifically solving the TQRL setting for continuous
MDPs. In this work, we give an algorithm specifically designed for this setting,
which shows sample complexity benefits reminiscent of those theoretically
shown in the tabular setting.
Exploration in Reinforcement Learning
To encourage exploration in RL, agents often use an $\epsilon$-greedy approach (Mnih et al., 2013), upper confidence bounds (UCB) (Chen et al., 2017), Thompson sampling (TS) (Osband et al., 2016), added Ornstein-Uhlenbeck action noise (Lillicrap et al., 2015), or entropy bonuses (Haarnoja et al., 2018) to add noise to a policy which is otherwise optimizing the RL objective.
Although UCB, TS, and entropy bonuses all try to adapt the exploration strategy to the problem, they all tackle which action to take from a predetermined state and don’t explicitly consider which states would be best to acquire data from.
An ideal method of exploration would be to solve the intractable Bayes-adaptive MDP (Ross et al., 2007), giving an optimal tradeoff between exploration and exploitation.
Kolter & Ng (2009); Guez et al. (2012) show that even approximating these techniques in the sequential setting can result in substantial theoretical reductions in sample complexity compared to frequentist PAC-MDP bounds as in Kakade (2003). Other methods stemming from Dearden et al. (1998; 1999) address this by using the myopic value of perfect information as a heuristic for similar Bayesian exploration. However, these methods don’t scale to continuous problems and don’t provide a way to choose states to query.
These methods were further extended with the development of knowledge gradient policies (Ryzhov et al., 2019; Ryzhov & Powell, 2011), which approximate the value function of the Bayes-adaptive MDP, and information-directed sampling (IDS) (Russo & Van Roy, 2014), which takes actions based on minimizing the ratio between squared regret and information gain over dynamics. This was extended to continuous-state finite-action settings in Nikolov et al. (2019). However, this work doesn’t solve fully continuous problems, operates in the rollout setting rather than TQRL, and computes the information gain with respect to the dynamics rather than some notion of the optimal policy.
In a similar spirit, Arumugam & Van Roy (2021) provide a further generalization of IDS which can also be applied to RL.
One recent work very close to ours is Lindner et al. (2021), which actively queries an expensive reward function (instead of dynamics as in this work) to learn a Bayesian model of reward. Another very relevant recent paper (Ball et al., 2020) gives an acquisition strategy in policy space that iteratively trains a data-collection policy in the model that trades off exploration against exploitation using methods from active learning. Achterhold & Stueckler (2021) use techniques from BOED to efficiently calibrate a Neural Process representation of a distribution of dynamics to a particular instance, but this calibration doesn’t include information about the task. A tutorial on Bayesian RL methods can be found in Ghavamzadeh et al. (2016) for further reference.
Separate from the techniques used in RL for a particular task, several methods tackle the problem of unsupervised exploration (Schmidhuber, 1991), where the goal is to learn as much as possible about the transition model without a task or reward function. One approach synthesizes a reward from modeling errors (Pathak et al., 2017). Another estimates learning progress by estimating model accuracy (Lopes et al., 2012). Others use an information gain-motivated formulation of model disagreement (Pathak et al., 2019; Shyam et al., 2019) as a reward. Other methods incentivize the policy to explore regions it hasn’t been before using hash-based counts (Tang et al., 2017), predictions mimicking a randomly initialized network (Burda et al., 2019), a density estimate (Bellemare et al., 2016), or predictive entropy (Buisson-Fenet et al., 2020).
However, these methods all assume that there is no reward function and are inefficient for the setting of this paper, as they spend time exploring areas of state space which can be quickly determined to be bad for maximizing reward on a task.
Bayesian Algorithm Execution and BOED
Recently, a flexible framework known as Bayesian algorithm execution (BAX) (Neiswanger et al., 2021) has been proposed for efficiently estimating properties of expensive black-box functions,
which builds off of a large literature from Bayesian Optimal Experiment Design (Chaloner & Verdinelli, 1995).
The BAX framework gives a general procedure for sampling points which are informative about the future execution of an algorithm.
In this paper, we extend this framework to the setting of model-predictive control, when we have expensive dynamics (i.e. transition function) which we treat as a black-box function in the BAX framework.
Via this strategy, we are able to use similar techniques to develop acquisition functions for data collection in reinforcement learning.
Gaussian Processes (GPs) in Reinforcement Learning
There has been substantial prior work using GPs in reinforcement learning. Most well-known is PILCO (Deisenroth & Rasmussen, 2011), which computes approximate analytic gradients of policy parameters through the GP dynamics model while accounting for uncertainty. Most related to our eventual MPC method is (Kamthe & Deisenroth, 2018), which gives a principled probabilistic model-predictive control algorithm for GPs.
3 Preliminaries
In this work we deal with finite-horizon discrete-time Markov decision processes (MDPs) which consist of a tuple $\langle\mathcal{S},\mathcal{A},T,r,p_{0},H\rangle$
where $\mathcal{S}$ is the state space, $\mathcal{A}$ is
the action space, $T$ is the transition function
$T:\mathcal{S}\times\mathcal{A}\to P(\mathcal{S})$ (using the convention that $P(\mathcal{X})$ is the set of probability measures over $\mathcal{X}$), $r:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\to\mathbb{R}$ is a reward function, $p_{0}(s)$ is a distribution over $\mathcal{S}$ of start states, and $H\in\mathbb{N}$ is a horizon. We always assume $\mathcal{S},\mathcal{A},p_{0},H$ are known. We also assume the reward $r$ is known, though our development of the method can easily be generalized to the case where $r$ is unknown. Our primary function of interest is the transition function $T$, which we learn from data. Our aim is to find a policy $\pi:\mathcal{S}\to P(\mathcal{A})$ that maximizes
the objective given below.
We can describe the execution of $\pi$ in the MDP as a finite collection of random variables generated by $s_{0}\sim p_{0}$ and $a_{i}\sim\pi(s_{i}),s_{i}\sim T(s_{i-i},a_{i-1})$ for $i\in 1,\dots,H$. Then this objective can be written
$$J_{T}(\pi)=\mathbb{E}_{p(s_{0:H},a_{0:H-1})}\left[\sum_{i=0}^{H-1}r(s_{i},a_{i},s_{i+1})\right].$$
(1)
We aim to maximize this objective while minimizing the number of samples from the ground truth transition function $T$ that are required to reach good performance.
We denote the optimal policy as $\pi^{*}=\operatorname{argmax}_{\pi}J_{T}(\pi)$, which we can assume to be
deterministic (Sutton & Barto, 1998) but not necessarily unique.
Finally, we assume that there is some prior $P(T)$ for which the posterior $P(T\mid D)$ is available for sampling given a dataset $D$.
3.1 Transition Query Reinforcement Learning (TQRL)
In the standard online RL setting, one assumes data $D=\{(s_{i},a_{i},s^{\prime}_{i})\}_{i\in[n]}$ must be collected in length-$H$ trajectories (rollouts) where the initial state $s_{0}\sim p_{0}$, and after an action $a_{i}$ is chosen, the next state $s^{\prime}_{i}=s_{i+1}$ is sampled from $T(s_{i},a_{i})$ up to $i=H$, at which point the process repeats.
In this work, we consider the TQRL setting, where the agent sequentially acquires data
$(s_{i},a_{i},s^{\prime}_{i})$ in arbitrary order by querying a state action pair $(s_{i},a_{i})$ from $\mathcal{S}\times\mathcal{A}$ and recieving a sample $s^{\prime}_{i}\sim T(s,a)$ from the black-box transition function $T$ (Kearns et al., 2002; Kakade, 2003; Azar et al., 2013). The goal in both settings is to find a policy which optimizes the objective in Equation (1).
It has been shown for finite MDPs in (Azar et al., 2013) that the PAC sample complexity, which is the number of samples required to identify with high probability a policy that achieves almost optimal value,
of this setting is $\tilde{O}(|\mathcal{S}||\mathcal{A}|)$, ignoring the PAC factors.
This is notably better than the bound of $\tilde{O}(|\mathcal{S}|^{2}|\mathcal{A}|)$ in the online RL setting given in Section 8.3 of Kakade (2003). The improvement shown in finite cases suggests that there could be similar reductions available in a continuous setting.
3.2 Model-Predictive Control
Though our acquisition function is derived for any policy search procedure including planning or model-free reinforcement learning algorithms, we focus on model-predictive control (MPC) as it is simple and requires minimal components to function.
MPC is a standard technique in model-based reinforcement learning (Chua et al., 2018; Wang & Ba, 2020). Using an estimated dynamics model $\hat{T}:\mathcal{S}\times\mathcal{A}\to\mathcal{S}$
and an optimization algorithm, an MPC strategy with a planning horizon $h\in\mathbb{N}$ choosing an action at a state $s_{0}$ solves the planning problem
$$\max_{a_{0},\dots,a_{h}\in\mathcal{A}}\mathbb{E}_{s_{i+1}\sim\hat{T}(s_{i},a_{i})}\left[\sum_{i=0}^{h}r(s_{i},a_{i},s_{i+1})\right].$$
(2)
Planning is typically redone periodically, often every timestep, as actions are executed in the real environment.
The cross-entropy method (CEM) is often used to solve this optimization problem. In this work, we use an improved variant of CEM described in Pinneri et al. (2020). Here we will refer to $\pi_{T}$ as the stochastic policy obtained by running MPC over a dynamics function $T$.
The randomness in the policy is due to any randomness in $T$ and the randomness used by the optimizer.
We give details on the hyperparameters involved in this optimization problem and their values in Section B.
4 An Acquisition Function for Model-Based RL
We draw inspiration from BO and BOED in constructing an acquisition function suitable for control applications. For our purposes, an acquisition function is a computationally tractable function $\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ that describes the marginal improvement in the performance of the policy on the MDP (conditioned on all previously observed data) when observing one additional state-action pair $(s,a)\in\mathcal{S}\times\mathcal{A}$.
As acquisition functions are greedy, they aren’t necessarily optimal data-selection strategies given a fixed budget, compared to non-myopic strategies such as solving the Bayes-adaptive MDP. However, greedy strategies using mutual information are tractable and are often
effective due to the submodularity of the expected information gain.
More specifically, our acquisition function is an expected information gain (EIG) or equivalently the mutual information (MI) between a query of our transition model and a representation of the optimal policy, as we elaborate in this section.
A typical approach in a Bayesian setting might be to gather data such that the entropy $\mathbb{H}[\pi^{*}]$ of the belief of the optimal policy $\pi^{*}$ is minimized. However, a full specification of $\pi^{*}$ includes the behavior of the policy in all parts of the state space including states that are not visited at all, or visited less often in rollouts of $\pi^{*}$ when the start state is sampled from $p_{0}$. As a result, not all points in the state space are equally important when learning an optimal policy aimed at maximizing the expected reward. Optimizing the entropy of $\pi^{*}$ would lead to a uniform treatment of all the points in the state space, and hence would be far from optimal for the standard goal in RL.
We instead propose to minimize the entropy of the optimal trajectory $\tau^{*}=\{s_{i}\}_{i=1}^{H}$ defined as a random vector of states generated by first sampling $s_{0}\sim p_{0}$ then sampling $a_{i}=\pi^{*}(s_{i}),s_{i+1}\sim T(s_{i},a_{i})$ for $H$ timesteps. The optimal trajectory is completely specified by $\pi^{*}$ and the randomness arising from the MDP. Furthermore, $\tau^{*}$ contains the necessary information needed about the transition function $T$ to solve the MDP, since any state that could ever be visited by $\pi^{*}$ is in the support of $\tau^{*}$.
We empirically observe that this leads to an efficient strategy for active RL.
The randomness in $\tau^{*}$ arises from three sources: the start state distribution $p_{0}$, the dynamics $T$ constituting the aleatoric uncertainty, and the uncertainty in our estimate of the model $T$ due to our limited experience which constitutes the epistemic uncertainty. The first two sources of uncertainty being aleatoric in nature cannot be reduced by experience. Our proposed acquisition function based on information gain naturally leads to reduction in the epistemic uncertainty about $\tau^{*}$ as desired.
Finally, our acquisition function for a given state-action pair $(s,a)$ is given as
$$\displaystyle\begin{aligned} \operatorname{EIG}_{\tau^{*}}(s,a)&=\mathbb{E}_{s^{\prime}\sim T(s,a\mid D)}\Big{[}\mathbb{H}[\tau^{*}\mid D]-\mathbb{H}[\tau^{*}\mid D\cup\{(s,a,s^{\prime})\}]\Big{]}\\
&=\mathbb{E}_{s_{0}\sim p_{0}}\Big{[}\mathbb{E}_{s^{\prime}\sim T(s,a\mid D)}\left[\mathbb{H}[\tau^{*}\mid D,s_{0}]-\mathbb{H}\big{[}\tau^{*}\mid D\cup\{(s,a,s^{\prime})\},s_{0}\big{]}\Big{]}\right].\end{aligned}$$
(3)
Here we assume a posterior model of the dynamics $T(s,a\mid D)$ for a dataset $D$ we have observed. The second equality is true because $s_{0}\perp s^{\prime}\mid s,a$. In this paper, we assume the MPC policy using the ground truth transition function is approximately optimal, i.e. $\pi_{T}\approx\pi^{*}$, though in principle $\pi^{*}$ could be approximated using any method. Of course, our method never actually has access to $\pi_{T}$ or $\pi^{*}$.
4.1 Estimating $\operatorname{EIG}_{\tau^{*}}$ via Posterior Function Sampling
For $\operatorname{EIG}_{\tau^{*}}$ to be of practical benefit, we must be able to tractably approximate it. Here we show how to obtain such an approximation.
By the symmetry of MI, we can rewrite Equation (3) as
$$\operatorname{EIG}_{\tau^{*}}(s,a)=\mathbb{E}_{s_{0}\sim p_{0}}\left[\mathbb{E}_{\tau^{*}\sim P(\tau^{*}\mid D)}\left[\mathbb{H}[s^{\prime}\mid s,a,D,s_{0}]-\mathbb{H}[s^{\prime}\mid s,a,\tau^{*},D,s_{0}]\right]\right].$$
(4)
Since $\mathbb{H}[s^{\prime}\mid s,a,D,s_{0}]=\mathbb{H}[s^{\prime}\mid s,a,D]$ doesn’t depend on $\tau^{*}$ or $s_{0}$, we can simply compute it as the entropy of the posterior predictive distribution $P(s^{\prime}\mid s,a,D)$ given by our posterior over the transition function $P(T\mid D)$.
In order to compute the other term, we must take samples $\tau^{*}_{ij}\sim P(\tau^{*}\mid D)$ . To do this, we first sample $m$ start states $s_{0}^{i}$ from $p_{0}$ (we always set $m=1$ in experiments but derive the procedure in general) and for each start state independently sample $n$ posterior functions $T^{\prime}_{ij}\sim P(T^{\prime}\mid D)$ from our posterior over dynamics models. We then run the MPC procedure on each of the posterior functions from $s_{0}^{i}$ using $T^{\prime}_{ij}$ for $T$ and $\pi_{T^{\prime}_{ij}}$ for $\pi^{*}$ (using our assumption that $\pi^{*}\approx\pi_{T}$), giving our sampled $\tau^{*}_{ij}$. This is an expression of the generative process for $\tau^{*}$ as described in the previous section that accounts for the uncertainty in $T$.
Formally, we can approximate $\operatorname{EIG}_{\tau^{*}}$ via Monte-Carlo as
$$\operatorname{EIG}_{\tau^{*}}(s,a)\approx\mathbb{H}[s^{\prime}\mid s,a,D]-\frac{1}{mn}\sum_{i\in[m]}\sum_{j\in[n]}\mathbb{H}[s^{\prime}|s,a,\tau^{*}_{ij},D].$$
(5)
Finally, we must calculate the entropy $\mathbb{H}[s^{\prime}|s,a,\tau^{*}_{i},D]$. For this, we follow a similar strategy as Neiswanger et al. (2021). In particular, since $\tau^{*}_{i}$ is a set of states output from the transition model, we can treat them as additional noiseless datapoints for our dynamics model and condition on them. In the following section we describe our instantiation of this EIG estimate, and how we can use it in reinforcement learning procedures.
Though inspired by the work cited here, we modify the computation of the acquisition function to factor $p_{0}$ as an irreducible source of uncertainty. We also extend the function being queried to be vector-valued.
5 Bayesian Active Reinforcement Learning
In this work, we take
a simple
approach for nonlinear control in continuous spaces and assume a Gaussian process (GP) prior $P(T)$ to model the dynamics. Though computationally expensive, this choice ensures that we
can easily approximate
all necessary quantities.
However, we note that the development of the acquisition function is general and any Bayesian model could be used in principle.
The transition function $T:\mathcal{S}\times\mathcal{A}\to p(\mathcal{S})$ (dynamics) can be modeled with a GP due to its non-parametric nature and ability to capture uncertainties in $T$. The transition function takes a state action pair $(s,a)\in\mathbb{R}^{d+n}$ as input, and produces a $d$-dimensional output denoting the next state. We model each of the $d$ dimensions of the output as independent GPs. More specifically, we model the change in state $\Delta(s,a)=T(s,a)-s$ rather than the final state $T(s,a)$ directly. This is helpful for continuous control problems since the state often changes by only a small magnitude. Given observations $D=\{(s_{i},a_{i},s^{\prime}_{i})\}$, our approach requires a posterior sample of the transition function conditioned on $D$.
We follow the approach of Wilson et al. (2020), based on sparse-GPs and random fourier approximations of kernels (Rahimi et al., 2007), allowing us to approximately but efficiently sample from the GP posterior conditioned on the observations.
Assuming access to a generative model and an initial dataset $D$ (for which, in practice, we use one randomly sampled datapoint $(s,a,s^{\prime})$), we compute $\operatorname{EIG}_{\tau^{*}}$ for $D$ by running MPC on posterior function samples and approximate $\operatorname{argmax}_{s\in\mathcal{S},a\in\mathcal{A}}\operatorname{EIG}_{\tau^{*}}(s,a)$ by zeroth order approximation.
Then we query $s^{\prime}\sim T(s,a)$ and add the subsequent triple to the dataset $D$ and repeat the process. To evaluate, we simply perform the MPC procedure in Equation (2) and execute $\pi_{\mathbb{E}[T\mid D]}$ on the real environment. We refer to this procedure as Bayesian active reinforcement learning (BARL). Details are given in Algorithm 1 (here, $U$ denotes the uniform distribution) and a schematic diagram in Figure 0(a). We discuss details of training hyperparameters and the GP model in Appendix A.
6 Experiments
The aim of our study of acquisition functions for RL is to reduce the sample complexity of learning good policies in continuous spaces, under expensive dynamics.
Here, we demonstrate the effectiveness of using $\operatorname{EIG}_{\tau^{*}}$ to leverage transition queries by comparing
against a variety of state-of-the-art RL algorithms.
In particular, we compare the average return across five evaluation episodes across five runs with differing random seeds of each algorithm on five continuous control problems as data is collected. We also assess the amount of data taken by each algorithm to ‘solve’ the problem, which is taken to mean performing as well as our MPC procedure using the ground truth dynamics. Our proposed method, BARL, greatly outperforms other methods across the board. In particular, BARL uses $5$ – $1,000\times$ less data to solve problems than state-of-the-art model-based RL algorithms and $10^{3}$ – $10^{5}\times$ less data than model-free RL algorithms.
In this section we primarily focus on the performance of the controller, and in section A.1 we also discuss the runtime of the algorithm.
Comparison Methods.
We use as our model-based comparison methods in this work PETS (Chua et al., 2018) as implemented by Pineda et al. (2021), which does MPC using a probabilistic ensemble of neural networks and particle sampling for stochastic dynamics and a similar MPC method using the mean of the same GP model we use for BARL to execute $\pi_{\hat{T}}$ to collect data as in the standard RL setting.
We also compare against PILCO (Deisenroth & Rasmussen, 2011), which also leverages a GP to directly optimize a policy that maximizes an uncertainty-aware long term reward.
For model-free methods, we use Soft Actor-Critic (SAC) (Haarnoja et al., 2018), which is an actor-critic method that uses an entropy bonus for the policy to encourage evaluation, TD3 (Fujimoto et al., 2018) which addresses the stability questions of actor-critic methods by including twin networks for value and several other modifications, and Proximal Policy Optimization (PPO) (Schulman et al., 2017), which addresses stability by forcing the policy to change slowly in KL so that the critic remains accurate. As a baseline TQRL method and to better understand the GP performance, we use a method we denote $\operatorname{EIG}_{T}$, which chooses points which maximize the predictive entropy of the transition model to collect data. We believe that when given access to transition queries many unsupervised exploration methods like Pathak et al. (2019); Shyam et al. (2019) or methods which value information gain over the transtion function (Nikolov et al., 2019) would default to this behavior.
Control Problems.
We tackle five control problems: the standard underactuated pendulum swing-up problem (Pendulum-v0 from Brockman et al. (2016)), a cartpole swing-up problem, a 2D lava path navigation problem, a 2-DOF robot arm reacher problem with 8-dimensional state (Reacher-v2 from Brockman et al. (2016)), and a simplified beta tracking problem from plasma control (Char et al., 2019; Mehta et al., 2020) where the controller must maintain a fixed normalized plasma pressure using as GT dynamics a model learned similarly to Abbate et al. (2021).
The lava path is intended to test stability and exploration of algorithms. The goal is to reach a fixed goal state from a narrow uniform distribution over start states. As shown in Figure 0(b), the state space contains a ‘lava’ region which gives large negative rewards for every timestep. When not in lava, the reward is simply the negative squared distance to the goal, forcing the agent to navigate to the goal as quickly as possible. Since there is a narrow path through the lava, we want to explore a policy which crosses efficiently and safely.
Agents who fail to find this solution will be forced to go around, incurring penalties.
We see in both the sample complexity figures in Table 1, the learning curves in Figure 3, and visually in Figure 2 that BARL leverages $\operatorname{EIG}_{\tau^{*}}$ to
significantly reduce the data requirements of learning controllers on the problems presented. We’d like to additionally point out several failure cases of related algorithms that BARL avoids. Though it performs well on the simplest environments (pendulum and cartpole), $\operatorname{EIG}_{T}$ suffers from an inability to focus on acquiring data relevant to the control problem and not just learning dynamics as the state space becomes higher-dimensional in the reacher problem, or less smooth as in the beta tracking problem. The MPC method performs reasonably well across the board and is competitive with BARL on the plasma problem but requires relatively more samples in smaller environments where the model uncertainty can point to meaningfully underexplored areas. PETS is strong across the board but suffers from more required samples due to both its neural network dynamics model and its inability to make transition queries.
All algorithms besides BARL suffer substantial instability on the lava path problem, which is designed to be challenging to explore in a sequential fashion and require a precise understanding of which areas are safe to enter. BARL manages to learn where it is safe to operate in a handful of queries, which is an exciting result and will bear further investigation.
Figure 0(b) gives some intuition as to why: points are initally queried close to the start and as those dynamics are understood they are subsequently queried farther and farther along the execution paths. This allows BARL to use transition queries to avoid traversing well-understood areas of state space to reach the areas which are worth learning. We see a speedup in sample complexity reminiscent of a move from quadratic to linear, which mirrors some of the theoretical improvements given in the prior work discussed on tabular methods.
We further support our assertion that BARL is picking ‘meaningful’ points to the control problem by the evidence in Figure 4. Here, BARL is able to solve the reacher problem while $\operatorname{EIG}_{T}$ is not.
However, BARL has much worse model predictions on random data than $\operatorname{EIG}_{T}$ while doing a much better job modeling data used by the MPC procedure. Clearly, the $\operatorname{EIG}_{\tau^{*}}$ acquisition function captures in some way which data would be valuable to acquire to not just learn about the transition function but actually solve the control problem. We see this pattern across other tasks as well. In section B.1, we study whether the acquisition function we see here is able to work with a suboptimal controller on posterior samples of the dynamics. Our experiments show that $\operatorname{EIG}_{\tau^{*}}$ seems to work well even when the policy used to generate $\tau^{*}$ is suboptimal.
7 Discussion and Future Work
In this work, we proposed an acquisition function for reinforcement learning, and applied it to the setting of TQRL, leading to a novel algorithm for addressing the problem of efficient data collection in RL. We experimented with several control problems and demonstrated that this approach leads to substantial improvements with respect to the sample efficiency.
However, there are some drawbacks to this method as well. Computing the proposed acquisition function relies on executing the entire control algorithm over a set of posterior function samples, leading to high computational requirements. In our current implementation we use Gaussian processes to model the transition function, which are computationally expensive and do not scale well to higher dimensions and large datasets. In the future, we plan to extend this idea to use other types of Bayesian models such as Bayesian neural networks and take advantage of GPU compute for better scalability.
Acknowledgments
We would like to acknowledge anonymous reviewers for valuable feedback.
VM acknowledges Swapnil Pande for providing the idea for and implementation of the Lava Path environment and Ian Char for providing the trained model for the Beta Tracking environment. This work was funded in part by DOE grant number DE-SC0021414.
WN acknowledges the helpful feedback from members of the Ermon Group studying RL. WN was supported in part by NSF (#1651565), ONR (N000141912145), AFOSR (FA95501910024), ARO (W911NF-21-1-0125), DOE (DE-AC02-76SF00515) and Sloan Fellowship.
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Appendix A Training Details
We vary our budget based on our understanding of how much data would be required to solve the problem. The other hyperparameters of the BARL algorithm are constant but listed for completeness in Table 2.
For all of our experiments, we use a squared exponential kernel with automatic relevance determination (MacKay et al., 1994; Neal, 1995). The parameters of the kernel were estimated by maximizing the likelihood of the parameters after marginalizing over the posterior GP (Williams & Rasmussen, 1996).
To optimize the transition function, we simply sampled a set of points from the domain, evaluated the acquisition function, and chose the maximum of the set. This set was chosed uniformly for every problem but Reacher, for which we chose a random subset of $\cup_{i}\cup_{j}\tau^{*}_{ij}$ (the posterior samples of the optimal trajectory) since the space of samples is 10-dimensional and uniform random sampling will not get good coverage of interesting regions of the state space.
A.1 Runtime Details
Based on these choices and the MPC hyperparameters below in Section B, each of these problems results in a varying runtime for the BARL algorithm. In Table 3, we report the time taken for an iteration of BARL and how it breaks down by step. We give these as ranges, as the computational time requires increases as the learning process continues since GP computational costs scale with the size of the dataset. We also include for completeness the time taken to execute the MPC policy on the ground truth problem, which is not strictly part of the BARL algorithm but still relevant to practitioners.
Clearly, BARL is a relatively slow algorithm computationally. But in settings where samples are scarce, BARL is much cheaper than alternative methods which might use less compute for the RL algorthms but require many more samples. When compared to the costs of running an hour-long simulation or running a costly experiments, spending a few minutes computing the acquisition function seems like a good use of resources.
Appendix B MPC Details
As we’ve discussed, we use model-predictive control in this work to choose actions which maximize future reward given a model of the dynamics. In particular we use the improved Cross-Entropy Method from Pinneri et al. (2020) to solve the optimization problem in Equation 2, which uses several tricks including colored noise samples and caching to reduce the number of queries to the planning model. There is a natural trade-off in any search method between computational cost and quality of actions found in terms of predicted reward. In this work, we chose hyperparameters for each task that were as computationally light as possible which attained a similar reward to larger hyperparameters when executing MPC using the ground-truth model ($\pi_{T}$, in our terms). As recommended by the original paper, we use $\beta=3$ for the scaling exponent of the power spectrum density of sampled noise for action sequences, $\gamma=1.25$ for the exponential decay of population size, and $\xi=0.3$ for the amount of caching.
We manually tuned the base number of samples, planning horizon, number of elites to take from the sampled action sequences, number of iterations of planning, and the replanning period of the model. Here we give the ultimate values for those parameters, which were used for all ablations using our GP model and MPC. The values we used across all experiments for each problem are given in Table 4.
B.1 Robustness of $\operatorname{EIG}_{\tau^{*}}$ to a suboptimal controller
In order to compute $\operatorname{EIG}_{\tau^{*}}$ in this work, we perform model-predictive control on posterior transition function samples (execute $\pi_{T^{\prime}_{\ell}}$ on $T^{\prime}_{\ell}$, in our notation). We assume that $\pi_{T^{\prime}_{\ell}}$ is close to the optimal policy for the MDP with transition function $T^{\prime}_{\ell}$. However, this assumption could lead to pathologies in the method if it doesn’t hold in practice. In this section, we emprically investigate the consequences of using a suboptimal controller when finding samples of $\tau^{*}$ on posterior samples of the transition function.
In order to understand the sensitivity of $\operatorname{EIG}_{\tau^{*}}$ to the MPC policy executed on posterior samples, we ran experiments where we reduced the planning budget or horizon for the posterior function policy in order to see whether the acquisition function fails. In particular, we ran the reacher and cartpole experiments from the main paper with varying MPC budgets for the posterior function policy $\pi_{T^{\prime}_{\ell}}$ and a fixed MPC budget at test time. This allows us to isolate the effect of a suboptimal policy generating samples of $\tau^{*}$.
On the reacher experiment, we vary the number of CEM iterations ranging from 1 to 5. This is straightforwardly linked to the amount of search the policy conducts before executing an action. On the cartpole experiment, we varied the length of the planning horizon, which affects how far in the future the policy will consider actions as it is deciding what is the good immediate next action. In both cases, we see in Figure 5 that performance hardly changes as the budget for MPC is reduced. Only on the reacher problem when the number of CEM iterations is reduced to 1 (effectively reducing CEM to simple random search) do we see a significant drop in performance. This supports the notion that the quality of the approximation of the optimal policy in $\operatorname{EIG}_{O}$ is not critical to the performance of the acquisition function as a data selection strategy. We also plot in the figure the performance of an MPC controller on the ground truth dynamics with these reduced MPC budgets. It is clear that if we were to execute these degraded policies at test time, they would be much worse. We find it interesting and intend to further study the robustness of using cheaper policies to decide where to acquire data.
Appendix C Description of Continuous Control Problems
Lava Path.
The lava path has 4-dimensional state (position and velocity) and 2-dimensional action (an applied force in the plane). The goal is to reach a fixed goal state from a relatively narrow uniform distribution over start states. As shown in Figure 0(b), the state space contains a ‘lava’ region which gives very large negative rewards for every timestep. Other than when in lava, the reward is simply the negative squared distance to the goal, forcing the agent to navigate to the goal as quickly as possible. Since the lava has a narrow path through, the actor is forced to explore a policy which will realize that it is safe and efficient to cross. Agents who fail to find this solution will be forced to go around, incurring penalties.
Pendulum.
The pendulum swing-up problem is the standard one found in the OpenAI gym (Brockman et al., 2016). The state space contains the angle of the pendulum and its first derivative and action space simply the scalar torque applied by the motor on the pendulum. The challenge in this problem is that the motor doesn’t have enough torque to simply rotate the pendulum up from all positions and often requires a back-and-forth swing to achieve a vertically balanced position. The reward function here penalizes deviation from an upright pole and squared torque.
Cartpole.
The cartpole swing-up problem has 4-dimensional state (position of the cart and its velocity, angle of the pole and its angular velocity) and a 1-dimensional action (horizontal force applied to the cart). Here, the difficulty lies in translating the horizontal motion of the cart into effective torque on the pole. The reward function here is a negative sigmoid function penalizing the distance betweent the tip of the pole and a centered upright goal position.
Reacher.
The reacher problem simulates a 2-DOF robot arm aiming to move the end effector to a randomly resampled target provided. The problem requires joint angles and velocities as well as an indication of the direction of the goal, giving an 8-dimensional state space with the mentioned 2-D control. Our results on this problem are particularly encouraging as they show that BARL can scale to some problems with higher dimensionalities.
Beta Tracking (Nuclear Fusion).
Finally, the beta tracking problem has 4-dimensional state consisting of the current normalized plasma performance $\beta_{N}$ in the DIII-D tokamak. $\beta_{N}$ is given by an appropriately normalized ratio between the plasma pressure and the magnetic pressure and is a common figure of merit in fusion energy research.
In addition to $\beta_{n}$ the state space contains its most recent change as well as the current power injection level and its most recent change. The action is the next change in the power injection level. The “ground-truth” dynamics for this problem are given by a neural network model learned from data processed as in Abbate et al. (2021). Control is done at a timestep of 200ms and the reward function is the negative absolute deviation from $\beta_{n}=2$.
Reliably controlling plasmas to sustain high performance is a major goal of research efforts for fusion energy, and though this is very much a simplification of the problem, we intend to extend and apply BARL to more realistic settings in the immediate future. |
A Modified Nonlinear Conjugate Gradient Algorithm for Functions with Non-Lipschitz continuous Gradient
Bingjie Li
Department of Statistics & Data Science
National University of Singapore, Singapore
[email protected]
& Tianhao Ni
School of Mathematical Science
Zhejiang University, China
[email protected]
& Zhenyue Zhang
Nanjing Center for Applied Mathematics, China
School of Mathematical Science, Zhejiang University, China
[email protected]
Abstract
In this paper, we propose a modified nonlinear conjugate gradient (NCG) method for functions with a non-Lipschitz continuous gradient. First, we present a new formula for the conjugate coefficient $\beta_{k}$ in NCG, conducting a search direction that provides an adequate function decrease. We can derive that our NCG algorithm guarantees strongly convergent for continuous differential functions without Lipschitz continuous gradient. Second, we present a simple interpolation approach that could automatically achieve shrinkage, generating a step length satisfying the standard Wolfe conditions in each step. Our framework considerably broadens the applicability of NCG and preserves the superior numerical performance of the PRP-type methods.
Keywords nonlinear conjugate gradient $\cdot$
line search $\cdot$
strongly convergent $\cdot$
non-Lipschitz gradient
1 Introduction
Consider the unconstrained optimization problem:
$$\displaystyle\min f(x)\quad x\in\mathbb{R}^{n},$$
(1)
where $f(x)$ is continuously differentiable. The nonlinear conjugate gradient (NCG) method provides an iterative scheme for minimizing $f(x)$ via the two steps: starting at a point $x_{0}$ and setting $k=0$ and $d_{0}=-\nabla f(x_{0})$, the NCG updates the current point $x_{k}$ via a line search along the direction $d_{k}$ as that
$$\displaystyle x_{k+1}=x_{k}+\alpha_{k}d_{k},$$
(2)
and then updates the direction $d_{k}$ in a linear combination of the previous direction $d_{k}$ and the current gradient $g_{k+1}=\nabla f(x_{k+1})$ for next search,
$$\displaystyle d_{k+1}=-g_{k+1}+\beta_{k}d_{k}.$$
(3)
In each iteration step, the step length $\alpha_{k}$ and conjugate coefficient $\beta_{k}$ determine the convergence behavior of the NCG. The sequence is called globally convergent if $\liminf_{k\to\infty}\|g_{k}\|=0$, and called strongly convergent if $\lim_{k\to\infty}\|g_{k}\|=0$.
1.1 The step length
Basically, given a search direction $d_{k}$, a step length $\alpha_{k}$ is chosen to yield a smaller value of the objective function $f$ at the updated point $x_{k+1}$. The ideal line search set the $\alpha_{k}$ that minimizes $f(x_{k}+\alpha d_{k})$,
$$\alpha_{k}=\arg\min_{\alpha}f(x_{k}+\alpha d_{k}).$$
Since $f$ is nonlinear and not convex in many application, the ideal search may cost much. As a substitute, an inexact search approach is commonly used. In an inexact line search, the step length $\alpha_{k}$ is chosen to satisfy Armijo-Goldstein Condition [1]
$$\displaystyle f(x_{k}+\alpha_{k}d_{k})\leq f(x_{k})+\rho\alpha_{k}\langle g_{k},d_{k}\rangle$$
(4)
$$\displaystyle f(x_{k}+\alpha_{k}d_{k})\geq f(x_{k})+(1-\rho)\alpha_{k}\langle g_{k},d_{k}\rangle,$$
(5)
or the standard Wolfe-Powell conditions [2]
$$\displaystyle f(x_{k}+\alpha_{k}d_{k})\leq f(x_{k})+\rho\alpha_{k}\langle g_{k},d_{k}\rangle$$
(6)
$$\displaystyle\langle\nabla f(x_{k}+\alpha_{k}d_{k}),d_{k}\rangle\geq\sigma\langle g_{k},d_{k}\rangle,$$
(7)
or the strong Wolfe-Powell conditions[3]
$$\displaystyle f(x_{k}+\alpha_{k}d_{k})\leq f(x_{k})+\rho\alpha_{k}\langle g_{k},d_{k}\rangle$$
(8)
$$\displaystyle|\langle\nabla f(x_{k}+\alpha_{k}d_{k}),d_{k}\rangle|\leq-\sigma\langle g_{k},d_{k}\rangle.$$
(9)
Here, the parameters $\rho$ and $\sigma$ are positive and $\rho<\sigma<1$. Generally, $\alpha_{k}$ can be obtained via bisection [4] or interpolation [5], or combination of the two approaches [6].
1.2 The search direction
In nonlinear conjugate gradient methods, the behavior of $d_{k+1}$ is determined by $\beta_{k}$. There are many approaches in the literature to setting $\beta_{k}$. Classical formulas for $\beta_{k}$ are called Fletcher-Reeves (FR) [7], Hestenes-Stiefel (HS) [8], Polak-Ribiere-Polyak (PRP) [9]. They are given by
$$\displaystyle\beta_{k}^{\rm FR}=\frac{\|g_{k+1}\|_{2}^{2}}{\|g_{k}\|_{2}^{2}},\quad\beta_{k}^{\rm HS}=\frac{\langle g_{k+1},y_{k}\rangle}{\langle d_{k},y_{k}\rangle},\quad\beta_{k}^{\rm PRP}=\frac{\langle g_{k+1},y_{k}\rangle}{\|g_{k}\|_{2}^{2}},$$
where $y_{k}=g_{k+1}-g_{k}$. In practice, the PRP method outperforms others in many optimization problems because it can immediately recover after generating a tiny step. However, the PRP method only guarantees global convergence for strictly convex functions, limiting its applicability. To improve it, Gilbert and Nocedal [10] modified the PRP method by setting
$$\displaystyle\beta_{k}^{\rm PRP+}=\max\{\beta_{k}^{\rm PRP},0\},$$
(10)
and showed that this modification of the PRP method, called PRP+, is globally convergent if the search direction is sufficient descending and the step length satisfies the standard Wolfe conditions.
In recent years, a variety of new nonlinear conjugate gradient methods have been proposed to find a search direction satisfying the descending condition
$\langle d_{k+1},g_{k+1}\rangle<0$
or the sufficient descending condition
$\langle d_{k+1},g_{k+1}\rangle<-c\|g_{k+1}\|^{2},$
where $c$ is a positive number. In [11], Dai and Yuan proposed a formula with
$$\displaystyle\beta_{k}^{\rm DY}$$
$$\displaystyle=\frac{\langle g_{k+1},d_{k+1}\rangle}{\langle g_{k},d_{k}\rangle},$$
(11)
and it provides a descending direction. In [12], Hager and Zhang modified HS method to
$$\displaystyle\beta_{k}^{\rm HZ}=\max\{\beta_{k}^{\rm HS}-\frac{2\|y_{k}\|^{2}\langle g_{k+1},d_{k}\rangle}{\langle d_{k},y_{k}\rangle^{2}},-\frac{1}{\|d_{k}\|\min\{\eta,\|g_{k}\|\}}\},$$
(12)
where $\eta>0$ is a constant. Similar modification on PRP method was proposed by Yuan [13], that is,
$$\displaystyle\beta_{k}^{\rm PRP-Y}=\max\{\beta_{k}^{\rm PRP}-\frac{\nu\|y_{k}\|^{2}}{\|g_{k}\|^{4}}\langle g_{k+1},d_{k}\rangle,0\}\quad\nu>\frac{1}{4}.$$
(13)
Both $\beta_{k}^{\rm HZ}$ and $\beta_{k}^{\rm MPRP}$ provide a sufficient descent direction.
The convergence of the above nonlinear conjugate gradient methods requires the gradient $g(x)$ of the objective function $f(x)$ to be Lipschitz continuous. That is, there exists a constant $L>0$ such that
$$\displaystyle\|g(x)-g(y)\|\leq L\|x-y\|,\quad\mbox{for all}\quad x,y\in\mathbb{R}^{n}.$$
(14)
This requirement of the Lipschitz continuous gradient limits the application of nonlinear conjugate gradient methods when faced with complicated practical problems.
1.3 Our contribution
In this paper, we propose a modified nonlinear conjugate gradient method, which does not require the gradient $g(x)$ to be Lipschitz continuous. The novelty of our approach comes from two aspects:
•
We propose a new formula for $\beta_{k}$, called MPRP, obtaining an adequate descending direction. The strong convergence of our approach is guaranteed even though $f(x)$ is just a continuous differential function with a non-Lipschitz gradient.
•
We suggest a more straightforward line search method for a step length that satisfies the standard Wolfe conditions in finite iterations. In practice, it works very well. The line search iteration terminates within one or two iterations generally in our experiments.
This paper is organized as follows. In Section 2, we propose our new formula for $\beta_{k}$, and the line search approach is given in Section 3. We discuss the convergence of our method in Section 4. The numerical experiment is also given in Section 5, to show the performance of our approach. At last, we end the article with a conclusion in Section 6.
2 The new formula for $\beta_{k}$
Besides the Lipschitz continuity of the gradient, the convergence of the PRP-Y method [13] requires that the step length $\{\alpha_{k}\}$ have a positive lower bound. This requirement is difficult to guarantee in practice. To weaken the conditions required for the convergence of the PRP-Y method, we provide a formula for $\beta_{k}$ as
$$\displaystyle\beta_{k}=\min\Big{\{}\frac{\langle g_{k+1},g_{k+1}-g_{k}-\frac{\nu\|g_{k+1}-g_{k}\|^{2}}{\|g_{k}\|^{2}}d_{k}\rangle_{+}}{\|g_{k}\|_{2}^{2}},\frac{\kappa\|g_{k+1}\|_{2}}{\|d_{k}\|_{2}}\Big{\}}.$$
(15)
where $\nu>\frac{1}{4}$ as in (13) and $\kappa>0$. The modification can guarantee a stronger sufficient descent condition than that of the PRP-Y method. That is,
Lemma 2.1.
Let $\beta_{k}$ be defined by (15) and $\mu=\frac{4\nu-1}{4\nu(1+\kappa)}$. Then
$$\langle d_{k+1},g_{k+1}\rangle\leq-\mu\|d_{k+1}\|_{2}\|g_{k+1}\|_{2}.$$
(16)
Proof.
Let $\tilde{\beta}_{k}=\beta_{k}^{\rm PRP}$ for short, and let $\tilde{d}_{k+1}=-g_{k+1}+\tilde{\beta}_{k}d_{k}$. We rewrite
$$\beta_{k}=\rho_{k}\tilde{\beta}_{k},\quad d_{k+1}=\rho_{k}\tilde{d}_{k+1}+(\rho_{k}-1)g_{k+1}$$
with a scale $\rho_{k}\in[0,1]$ since $\beta_{k}\leq\tilde{\beta}_{k}$. At first, we require the
inequality
$$\displaystyle\tilde{\beta}_{k}\langle d_{k},g_{k+1}\rangle\leq\frac{1}{4\nu}\|g_{k+1}\|^{2},$$
(17)
concluded by the definition
$\tilde{\beta}_{k}=\frac{\langle g_{k+1},g_{k+1}-g_{k}\rangle}{\|g_{k}\|_{2}^{2}}-\frac{\nu\|g_{k+1}-g_{k}\|^{2}}{\|g_{k}\|^{4}}\langle g_{k+1},d_{k}\rangle$.
It gives
$$\displaystyle\tilde{\beta}_{k}\langle d_{k},g_{k+1}\rangle$$
$$\displaystyle=\frac{\langle g_{k+1},g_{k+1}-g_{k}\rangle}{\|g_{k}\|_{2}^{2}}\langle g_{k+1},d_{k}\rangle-\frac{\nu\|g_{k+1}-g_{k}\|^{2}}{\|g_{k}\|^{4}}\langle g_{k+1},d_{k}\rangle^{2}$$
$$\displaystyle=\Big{\langle}g_{k+1},\frac{\langle g_{k+1},d_{k}\rangle}{\|g_{k}\|_{2}^{2}}(g_{k+1}-g_{k})\Big{\rangle}-\frac{\nu\langle g_{k+1},d_{k}\rangle^{2}}{\|g_{k}\|^{4}}\|g_{k+1}-g_{k}\|^{2}.$$
Let $q_{k}=\frac{\langle g_{k+1},d_{k}\rangle}{\|g_{k}\|_{2}^{2}}(g_{k+1}-g_{k})$ for simplicity. Then
$$\tilde{\beta}_{k}\langle d_{k},g_{k+1}\rangle=\langle g_{k+1},q_{k}\rangle-\nu\|q_{k}\|^{2}=\frac{\|g_{k+1}\|^{2}}{4\nu}-\|(2\sqrt{\nu})^{-1}g_{k+1}-\sqrt{\nu}q_{k}\|^{2}.$$
Therefore, (17) is true. Following it, we get that
$\langle\tilde{d}_{k+1},g_{k+1}\rangle\leq(\frac{1}{4\nu}-1)\|g_{k+1}\|^{2}$
and
$$\displaystyle\langle d_{k+1},g_{k+1}\rangle=$$
$$\displaystyle\ \rho_{k}\langle\tilde{d}_{k+1},g_{k+1}\rangle+(\rho_{k}-1)\|g_{k+1}\|^{2}$$
$$\displaystyle\leq$$
$$\displaystyle\ \big{(}\rho_{k}(\frac{1}{4\nu}-1)+(\rho_{k}-1)\big{)}\|g_{k+1}\|^{2}\leq\frac{1-4\nu}{4\nu}\|g_{k+1}\|^{2}.$$
Here we have used $\rho_{k}\leq 1$.
On the other hand, since $|\beta_{k}|\leq\frac{\kappa\|g_{k+1}\|}{\|d_{k}\|}$, we also have that
$$\|d_{k+1}\|=\|-g_{k+1}+\beta_{k}d_{k}\|\leq\|g_{k+1}\|+|\beta_{k}|\|d_{k}\|\leq(1+\kappa)\|g_{k+1}\|.$$
Therefore, (16) holds
since $\|g_{k+1}\|_{2}^{2}\geq\frac{1}{1+\kappa}\|d_{k+1}\|_{2}\|g_{k+1}\|_{2}$ and $\frac{1-4\nu}{4\nu}<0$.
∎
We call a search direction $d_{k+1}$ an adequate descending direction if it satisfies (16). Obvious, the gradient itself satisfies (16) with $\mu=1$. To the best of our knowledge, (15) is the first conjugate gradient method that provides an adequate descending direction.
3 A simple interpolation line search approach
For a general continuously differentiable $f(x)$, the interpolation method does not guarantee capturing required $\alpha_{k}$ satisfying the standard Wolfe conditions since it asks for a three times continuously differentiable [5]. One can get $\alpha_{k}$ by the combination method [6] that is more efficient than the bisection approach [4]. In [6], the bisection is combined with the interpolation in a bit complicated way for interval shrinking. Here we give a simpler approach for determining $\alpha_{k}$ satisfying the weak Wolfe-Powell conditions.
Theoretically, at a current point $x=x_{k}$ with the conjugate direction $d=d_{k}$, the required inexact line search $\alpha=\alpha_{k}$ satisfying the weak Wolfe-Powell conditions (6-7) can be chosen as
$$\displaystyle\alpha^{*}=\sup\big{\{}\hat{\alpha}:\mbox{the Wolfe-Powell condition (\ref{wolfe1}) holds over $(0,\hat{\alpha})$ }\big{\}}.$$
(18)
It exists, is positive, and satisfies (6-7). To verify this claim, let’s consider the function
$$g(\alpha)=f(x)+\rho\alpha\langle\nabla f(x),d\rangle-f(x+\alpha d).$$
Clearly, (6) is equivalent to $g(\alpha)\geq 0$, and meanwhile, (7) holds if $g^{\prime}(\alpha)\leq 0$. By the definition and the continuity of $f$, (6) is true for $0<\alpha\leq\alpha^{*}$. The supremum in (18) implies that $g(\alpha^{*})=0$ and $g^{\prime}(\alpha^{*})\leq 0$. Hence, (7) is also satisfied for $\alpha=\alpha^{*}$. Practically, there is a relative large sub-interval of $(0,\alpha^{*}]$ in which both (6) and (7) are true. For instance, if $\hat{\alpha}\in(0,\alpha^{*}]$ is the largest point such that $g(\alpha)$ is a local maximum, then $g^{\prime}(\alpha)\leq 0$ in $[\hat{\alpha},\alpha^{*}]$. Therefore, (6-7) hold for $\alpha\in[\hat{\alpha},\alpha^{*}]$.
An ideal choice of $\alpha$ is the minimizer $\alpha_{\min}$ of $f(x+\alpha d)$ over $(0,\alpha^{*}]$ since it decreases $f$ as small as possible, while both (6) and (7) are still satisfied.
In this subsection, we give a simple rule for pursuing $\alpha_{\min}$ via a quadratic interpolation to $f(x+\alpha d)$, assuming $f$ is continuously differentiable. It generates a nested and shrunk interval sequence containing the required $\alpha$. The pursuing terminates as soon as a point satisfying (6-7) is found.
Initially, we set $\alpha_{0}^{\prime}=0$ that satisfies (6) but (7), and choose a relatively large $\alpha_{0}^{\prime\prime}>0$ that does not satisfy (6). A simple choice of $\alpha_{0}^{\prime\prime}$ will be given later. Starting with $[\alpha_{0}^{\prime},\alpha_{0}^{\prime\prime}]$, we generate a sequence of intervals $[\alpha_{0}^{\prime},\alpha_{0}^{\prime\prime}]$ iteratively such that each $\alpha_{\ell}^{\prime}$ satisfies (6) but $\alpha_{\ell}^{\prime\prime}$ does not, and meanwhile, $\alpha_{\ell}^{\prime}$ doesn’t satisfy (7). That is, for $x_{\ell}^{\prime}=x+\alpha_{\ell}^{\prime}d$ and $x_{\ell}^{\prime\prime}=x+\alpha_{\ell}^{\prime\prime}d$
$$\displaystyle f(x_{\ell}^{\prime})\leq f(x)+\rho\alpha_{\ell}^{\prime}\langle g,d\rangle,\ f(x_{\ell}^{\prime\prime})>f(x)+\rho\alpha_{\ell}^{\prime\prime}\langle g,d\rangle,\ \langle\nabla f(x_{\ell}^{\prime}),d\rangle<\sigma\langle g,d\rangle,$$
(19)
where $g=\nabla f(x)$. The third inequality above implies that $\langle\nabla f(x_{\ell}^{\prime}),d\rangle<0$.
Furthermore, by the first two inequalities in (19), we have that
$$\displaystyle f(x_{\ell}^{\prime\prime})>f(x_{\ell}^{\prime})+\rho(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle g,d\rangle>f(x_{\ell}^{\prime})+\frac{\rho}{\sigma}(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),d\rangle.$$
(20)
In the current interval, we consider a quadratic function $q(\alpha)$ with interpolation conditions
$$q(\alpha_{\ell}^{\prime})=f(x_{\ell}^{\prime}),\quad q^{\prime}(\alpha_{\ell}^{\prime})=\langle\nabla f(x_{\ell}^{\prime}),d\rangle,\quad q(\alpha_{\ell}^{\prime\prime})=f(x_{\ell}^{\prime\prime}),$$
It can be represented as
$$\displaystyle q(\alpha)=$$
$$\displaystyle\ f(x_{\ell}^{\prime})+(\alpha-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),d\rangle$$
$$\displaystyle+\big{(}f(x_{\ell}^{\prime\prime})-f(x_{\ell}^{\prime})-(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),d\rangle\big{)}\frac{(\alpha-\alpha_{\ell}^{\prime})^{2}}{(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})^{2}}$$
with the minimizer $c_{\ell}=\arg\min_{\alpha}q(\alpha)$ given by
$$\displaystyle c_{\ell}=\alpha_{\ell}^{\prime}+\frac{\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime}}{2}\frac{-(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),d\rangle}{f(x_{\ell}^{\prime\prime})-f(x_{\ell}^{\prime})-(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),d\rangle}>\alpha_{\ell}^{\prime}.$$
(21)
By the Mean-Value Theorem for derivatives and the second inequality in (20),
$$\displaystyle 0<(1-M_{\ell})^{-1}\leq$$
$$\displaystyle\ \Big{(}1-\frac{\langle\nabla f(\bar{x}_{\ell}),d\rangle}{\langle\nabla f(x_{\ell}^{\prime}),D_{k}\rangle}\Big{)}^{-1}$$
$$\displaystyle=$$
$$\displaystyle\ \frac{-(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),D_{k}\rangle}{f(x_{\ell}^{\prime\prime})-f(x_{\ell}^{\prime})-(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})\langle\nabla f(x_{\ell}^{\prime}),D_{k}\rangle}<\frac{\sigma}{\sigma-\rho},$$
(22)
where $\bar{x}_{\ell}=x+\bar{\alpha}_{\ell}d$ with
$\bar{\alpha}_{\ell}\in[\alpha_{\ell}^{\prime},\alpha_{\ell}^{\prime\prime}]$ and
$$M_{\ell}=\min_{\alpha\in[\alpha_{\ell}^{\prime},\alpha_{\ell}^{\prime\prime}]}\frac{\langle\nabla f(x+\alpha d),d\rangle}{\langle\nabla f(x_{\ell}^{\prime}),d\rangle}\leq\frac{\langle\nabla f(\bar{x}_{\ell}),d\rangle}{\langle\nabla f(x_{\ell}^{\prime}),d\rangle}<\frac{\rho}{\sigma}.$$
Hence, if $0<2\rho<\sigma$, we have that
$$\displaystyle\alpha_{\ell}^{\prime}<\alpha_{\ell}^{\prime}+\frac{1}{2(1-M_{\ell})}(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})<c_{\ell}<\alpha_{\ell}^{\prime}+\frac{\sigma}{2(\sigma-\rho)}(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})<\alpha_{\ell}^{\prime\prime}.$$
(23)
We may shrink $[\alpha_{\ell}^{\prime},\alpha_{\ell}^{\prime\prime}]$ to $[c_{\ell},\alpha_{\ell}^{\prime\prime}]$ or $[\alpha_{\ell}^{\prime},c_{\ell}]$, if $\alpha=c_{\ell}$ satisfies (6) or does not. However, if (6) is satisfied, the interval length is
$\alpha_{\ell}^{\prime\prime}-c_{\ell}\leq\frac{1-2M_{\ell}}{2-2M_{\ell}}(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime}).$
When $M_{\ell}<0$ and $|M_{\ell}|$ is large, $\frac{1-2M_{\ell}}{2-2M_{\ell}}\approx 1$. The interval shrinking is inefficient in this case.
To avoid this phenomenon, we slightly modify $c_{\ell}$ as that with $\eta=\frac{\sigma}{2(\sigma-\rho)}$
$$\displaystyle\tilde{c}_{\ell}=\max\big{\{}c_{\ell},\ \eta\alpha_{\ell}^{\prime}+(1-\eta)\alpha_{\ell}^{\prime\prime}\big{\}}\in(\alpha_{\ell}^{\prime},\alpha_{\ell}^{\prime\prime}).$$
(24)
Since $\tilde{c}_{\ell}\geq\eta\alpha_{\ell}^{\prime}+(1-\eta)\alpha_{\ell}^{\prime\prime}$ and $c_{\ell}<\alpha_{\ell}^{\prime}+\eta(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})$ by (23), we get
$$\alpha_{\ell}^{\prime\prime}-\tilde{c}_{\ell}\leq\eta(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime}),\quad\tilde{c}_{\ell}-\alpha_{\ell}^{\prime}\leq\max\big{\{}\eta,1-\eta\big{\}}(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})=\eta(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime}).$$
The last equality holds since $\eta>1/2$. Hence, if the Wolfe-Powell conditions (6-7) are satisfied for $\alpha=\tilde{c}_{\ell}$, we get the required $\alpha_{k}=\tilde{c}_{\ell}$. Otherwise, shrink $[\alpha_{\ell}^{\prime},\alpha_{\ell}^{\prime\prime}]$ as
$$\displaystyle[\alpha_{\ell+1}^{\prime},\alpha_{\ell+1}^{\prime\prime}]=\left\{\begin{array}[]{ll}[\alpha_{\ell}^{\prime},\tilde{c}_{\ell}],&\ \mbox{if (\ref{wolfe1}) does not hold for $\alpha=\tilde{c}_{\ell}$};\\
\mbox{$[\tilde{c}_{\ell},\alpha_{\ell}^{\prime\prime}]$},&\ \mbox{otherwise}.\end{array}\right.$$
(27)
The interval length is significantly decreased as $0<\alpha_{\ell+1}^{\prime\prime}-\alpha_{\ell+1}^{\prime}\leq\eta(\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime})$,
where $\eta<1$ since $2\rho<\sigma$. Hence, $\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime}\to 0$ as $\ell\to\infty$.
A good choice of $\alpha_{0}^{\prime\prime}$ helps to pursue the minimizer $\alpha_{\min}$.
Motivated by the above analysis on the estimation of the shrinking rate $\eta_{\ell}$, we suggest the experiential setting
$$\displaystyle\alpha_{0}^{\prime\prime}=\min\big{\{}\alpha=2^{p}\eta:\ \mbox{(\ref{wolfe1}) is not satisfied for $\alpha=2^{p}\eta$ with integer $p\geq 0$}\big{\}}.$$
(28)
Algorithm 1 gives the details of the procedure for determining an inexact line search $\alpha_{k}$, given $x_{k}$, $f_{k}$, $g_{k}$, the conjugate direction $d_{k}$.
4 Convergence of the Algorithm
Combining the formula (15) and line search Algorithm 1, we are able to provide our modified PRP-type (MPRP) nonlinear conjugate gradient method, as shown in Algorithm 2. To show the convergence of the MPRP method, we first prove that the line search Algorithm 1 will converge to a step length that satisfies the standard Wolfe condition.
Lemma 4.1.
If $f$ is lower bounded and continuously differentiable, an $\alpha=\tilde{c}_{\ell^{*}}$ satisfying (6-7) can be obtained within a finite iterations of (27) if $0<2\rho<\sigma<1$.
Proof.
If (6-7) do not hold for all $\tilde{c}_{\ell}$, the updating rule (27) yields a sequence of nested intervals $\{[\alpha_{\ell}^{\prime},\ \alpha_{\ell}^{\prime\prime}]\}$. Since $0<2\rho<\sigma<1$, the intervals tend to a single point $\alpha_{*}$ and both $\{x_{\ell}^{\prime}\}$ and $\{x_{\ell}^{\prime\prime}\}$ tend to $x_{*}=x+\alpha_{*}d$. Hence, by (20) and the Taylor extension of $f(x+\alpha d)$ at $\alpha=\alpha_{*}$, we get
$$\displaystyle\langle\nabla f(x_{*}),d\rangle=\lim_{\ell\to\infty}\frac{f(x_{\ell}^{\prime\prime})-f(x_{\ell}^{\prime})}{\alpha_{\ell}^{\prime\prime}-\alpha_{\ell}^{\prime}}\geq\rho\langle\nabla f(x),d\rangle>\sigma\langle\nabla f(x),d\rangle$$
(29)
since $\langle\nabla f(x),d\rangle<0$ and $\rho<\sigma$. However, by (19),
$\langle\nabla f(x_{*}),d\rangle\leq\sigma\langle\nabla f(x),d\rangle$, a contradiction with (29).
∎
Because the search direction of MPRP is adequate descending, the proof of convergence of the algorithm is simple, similar to the proof of the steepest descent method. We have
Theorem 4.2.
Assume that $f$ is lower bounded and continuously derivative. If the inexact line search $\{\alpha_{k}\}$ satisfies the weak Wolfe-Powell condition (6-7) and
$$\langle d_{k},g_{k}\rangle\leq-\mu\|d_{k}\|_{2}\|g_{k}\|_{2}$$
(30)
for a constant $\mu>0$, then the NCG converges: $\{f(x_{k})\}$ is monotone decreasing and converges, and $\nabla f(x_{k})\!\to\!0$.
Proof.
We assume $g_{k}=\nabla f(x_{k})\neq 0$ for each $k$ without loss of generalities, and let $s_{k}=\alpha_{k}d_{k}$. The condition (30) becomes
$\langle g_{k},s_{k}\rangle\leq-\mu\|g_{k}\|\|s_{k}\|\leq 0$. Hence, the Wolfe-Powell condition (6) gives the monotone decreasing of $\{f(x_{k})\}$,
$$f(x_{k+1})-f(x_{k})\leq\rho\langle g_{k},s_{k}\rangle\leq-\rho\mu\|g_{k}\|\|s_{k}\|\leq 0,$$
and $\{f(x_{k})\}$ is convergent since $f$ itself is lower bounded. We also conclude from the convergence and the above inequality that $\|g_{k}\|\|s_{\!k}\|\to 0$.
We further show that $\|g_{k}\|\to 0$. Otherwise, there is a subsequence $\{\|g_{k_{i}}\|\}$ that has a positive lower bound. The lower bound implies that $\|s_{\!k_{i}}\|\to 0$ since we also have $\|g_{k_{i}}\|\|s_{\!k_{i}}\|\to 0$. Note that $s_{k_{i}}=x_{k_{i}+1}-x_{k_{i}}$ is also the gap vector between $x_{k_{i}}$ and $x_{k_{i}+1}$, $f(x_{k_{i}})$ and $f(x_{k_{i}+1})$ can be represented each other in terms of $s_{k_{i}}$ via the Taylor extensions
$$\displaystyle f(x_{k_{i}+1})=f(x_{k_{i}})+\langle g_{k_{i}},s_{k_{i}}\rangle+o\big{(}\|s_{k_{i}}\|\big{)},$$
$$\displaystyle f(x_{k_{i}})=f(x_{k_{i}+1})-\langle g_{k_{i}+1},s_{k_{i}}\rangle+o\big{(}\|s_{k_{i}}\|\big{)}.$$
Clearly, the each other implies that $\langle g_{k_{i}},s_{k_{i}}\rangle-\langle g_{k_{i}+1},s_{k_{i}}\rangle=o(\|s_{k_{i}}\|)$.
Turn back to the Wolfe-Powell condition (7). Since it gives $\langle g_{k_{i}+1},s_{k_{i}}\rangle\geq\sigma\langle g_{k_{i}},s_{k_{i}}\rangle$,
$$o(\|s_{k_{i}}\|)=\langle g_{k_{i}},s_{k_{i}}\rangle-\langle g_{k_{i}+1},s_{k_{i}}\rangle\leq(1-\sigma)\langle g_{k_{i}},s_{k_{i}}\rangle\leq-(1-\sigma)\mu\|g_{k_{i}}\|\|s_{k_{i}}\|.$$
Here we have used the inequality $\langle g_{k_{i}},s_{k_{i}}\rangle\leq-\mu\|g_{k_{i}}\|\|s_{k_{i}}\|$ from the condition (30) and $\sigma<1$. Hence,
$(1-\sigma)\mu\|g_{k_{i}}\|\leq-\frac{o(\|s_{k_{i}}\|)}{\|s_{k_{i}}\|}$, and
$$0\leq(1-\sigma)\mu\liminf\|g_{k_{i}}\|\leq 0.$$
It implies $(1-\sigma)\mu\leq 0$, a contradiction, since $\liminf\|g_{k_{i}}\|>0$ by assumption.
∎
5 Numerical Experiments
In this section, we show the performance of our MPRP in Algorithm 2, compared with three other PRP-type NCG methods: the classical PRP method [9], the PRP+ method [10] and the PRP-Y method [13].
The numerical experiments are divided into 3 parts. In the first part, we test these PRP-type methods on $84$ unconstrained optimization problems with Lipschitz continuous gradient from [14]. The second part aims to show the enhancements brought by our line search method, compared with the bisection line search method. In the third part, we adopt our algorithm on a regression problem with an objective function whose gradient is non-Lipschitz continuous. The following parameters were adopted in our implementation
$$\displaystyle\nu=0.8,\ \kappa=10,\ \rho=0.1,\ \sigma=0.4.$$
We set the termination criterion as $\|\nabla f(x)\|_{\infty}\leq 10^{-5}$ and the maximum number of iterations as $20000$. All compared algorithms are executed on the Windows system in a PC with Intel Core i5-8250U [email protected] and 8GB RAM.
We adopt the commonly used performance profile of Dolan and Moré proposed in [15], to display the performance of compared NCG methods in terms of CPU time and the number of iterations. Take the CPU time as an example, let $S$ and $P$ be the set of solvers and problems, respectively, and denote $n_{s}=|S|$ and $n_{p}=|P|$. For each solver $s$ and problem $p$, let $t_{p,s}$ be the computing time requiring by solver $s$ to solve problem $p$. For each problem $p$, define the performance ratio as $r_{p,s}=\frac{t_{p,s}}{\min\{t_{p,s},s\in S\}}$ and the ratio $r_{p,s}\geq 1$ obviously for all $p$ and $s$. If a solver fails to solve a problem, the ratio $r_{p,s}$ is set to a large enough positive number $M$ that larger than $r_{p,s}$ of problem $p$ that can be solved by solver $s$. Finally, the performance profile is defined by
$$\rho(\tau)=\frac{1}{n_{p}}|\{p\in P:r_{p,s}\leq\tau\}|$$
. The performance profile of the number of iterations is similar.
5.1 The performance of compared NCG methods on tested functions
This subsection shows the performance of 4 tested NCG algorithms on 84 unconstrained optimization problems drawn from [14]. The performance profile of CPU time and number of iterations is illustrated in Figure 1.
Among the four algorithms, MPRP and PRP-Y have significant advantages over the PRP and PRP+ method in the CPU Time and the number of iterations. However, PRP-Y performs slightly better than our MPRP. This is due to two reasons. For one thing, MPRP controls the angle between the search direction and the negative gradient, weakening the effect of the conjugate direction. Hence, the zig-zag phenomenon may occur in a small part of optimization problems. For another thing,
MPRP requires an additional computation of $\|d_{k}\|$ at each step, which also increases the CPU time slightly.
5.2 Line searches via bisection and interpolation
In this subsection, we test the performance of our interpolation line search in Algorithm 1, compared with the classical bisection line search [4]. The two line search approaches are used for the 4 tested NCG methods, on the 84 unconstrained optimization problems in [14]. The Comparison is shown on CPU time and the number of iterations in Figure 2.
As shown in Figure 2, the interpolation line search performs much better than the bisection line search in most problems regardless of the NCG methods we adopt. It has been shown that our interpolation line search method saves a significant amount of computation both in terms of the number of iterations and the CPU time.
5.3 Performance of NCG methods on a function with non-Lipschitz continuous gradient
In this subsection, we consider a linear regression model with a regular term as follows:
$$\displaystyle\min_{x}\frac{1}{2}\|Ax-b\|_{2}^{2}+\frac{\lambda}{2}\|x\|_{p}^{p},$$
(31)
where $\|x\|_{p}^{p}=|x_{1}|^{p}+...+|x_{n}|^{p}$. The model (31) becomes the lasso regression model or the ridge regression model, if one select $p=1$, or $p=2$, respectively. Here we set $p=1.5$ so that $f(x)=\frac{1}{2}\|Ax-b\|_{2}^{2}+\frac{\lambda}{2}\|x\|_{p}^{p}$ is continuous differential with a non-Lipschitz gradient, and test the compared NCG methods on it.
In our experiment, we set $A$ as a random matrix in $\mathbb{R}^{10\times 50}$ with entries drawn from the uniform distribution in $[0,1]$, and $b=Au$ with $u$ is sparse with $10\%$ non-zero entries drawn from the standard normal distribution. We also set $\lambda=0.01$. We test the four algorithms 10 times, each with different random seeds.
Table 1 shows the average CPU time and the average number of iterations for the 10 runs. It can be seen that the performance of PRP-Y and MPRP is slightly better than that of the other two algorithms. Surprisingly, PRP, PRP+ and PRP-Y all converge on this problem, even though they do not theoretically have a guarantee of convergence. One conjecture is that these algorithms skip the non-Lipschitz region of the gradient and converge to a stationary point with a neighbor where the gradient is Lipschitz continuous.
6 Conclusion
This paper proposed a modified nonlinear conjugate gradient method for continuous differential function. The strong convergence is guaranteed without the condition of the Lipschitz continuous gradient. Furthermore, a simpler but more efficient interpolation Wolfe line search method is also introduced. The numerical results demonstrate the feasibility of the new NCG method and the new line search method. However, although in theory our algorithm gains a greater range of applicability, this advantage does not manifest itself numerically. The reasons behind this are worth further investigation.
Acknowledgments
The work was supported by NSFC project 11971430.
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Unifying Brane World Inflation with Quintessence
M.Sami
[
[email protected]
N. Dadhich
[email protected]
IUCAA, Post Bag 4, Ganeshkhind,
Pune 411 007, India.
Abstract
We review the recent attempts of unifying inflation with quintessence. It appears natural to join the two ends in the framework
of brane world cosmology. The models of quintessential inflation belong to the class of non-oscillatory models for which
the mechanism of conventional reheating does not work. Reheating through gravitational particle production is inefficient and
leads to the excessive production of relic gravity waves which results in the violation of nucleosynthesis constraint.
The mechanism of instant preheating is quite efficient
and is suitable for brane world quintessential inflation. The model
is shown to be free from the problem of excessive production of
gravity waves. The prospects of Gauss-Bonnet brane world inflation are also briefly indicated.
pacs: 98.80.Cq, 98.80.Hw, 04.50.+h
On leave from:] Department of Physics, Jamia Millia, New Delhi-110025
I INTRODUCTION
Universe seems to exhibit an interesting symmetry with regard to accelerated expansion. It has gone
under inflation at early epochs and is believed to be accelerating at present.
The inflationary paradigm was originally introduced to address the initial value problems of the standard hot big bang model. Only later it became clear that the scenario
could provide important clues for the origin of structure in the universe. The recent measurement of the
Wilkinson Microwave Anisotropy Probe (WMAP) WMAP1 ; WMAP2 in
the Cosmic Microwave Background (CMB) made it clear that (i) the
current state of the universe is very close to a critical density
and that (ii) primordial density perturbations that seeded
large-scale structure in the universe are nearly scale-invariant
and Gaussian, which are consistent with the inflationary paradigm.
Inflation is often implemented with a single or multiple scalar-field modelsLR (also see the excellent review
on inflation by Shinji Tsujikawashinji ). In most of these models,
the scalar field undergoes a slow-roll period allowing an accelerated expansion of the universe. After drawing the required amount of inflation, the inflaton enters the regime of quasi-periodic
oscillation where it
quickly oscillates and decays into particles leading to (p)reheating.
As for the current accelerating of universe, it is supported by observations of high
redshift type Ia supernovae treated as standardized candles and, more indirectly,
by observations of the cosmic microwave background and galaxy clustering.
Within the framework of general relativity, cosmic acceleration should be
sourced by an energy-momentum tensor which has a large negative
pressure (dark energy)phiindustry .
Therefore, the standard model should, in order to comply with the logical consistency and observation, be sandwiched between inflation at early epochs and quintessence at late times. It is natural to
ask whether one can build a model with scalar fields to join the two ends without disturbing the thermal
history of universe. Attempts have been made to unify both these concepts using models with a single scalar
field unifiedmodels .In these models, the scalar field exhibits the properties of tracker field. As a result it goes into hiding after the commencement of radiation domination; it emerges from the shadow only at late times to account
for the observed accelerated expansion of universe. These models belong to the category of
non oscillating models in which the standard reheating mechanism does not work. In this case, one can employ an alternative mechanism of
reheating via quantum-mechanical particle production in time varying gravitational field at the end of inflation ford . However,
then the
inflaton energy density should
red-shift faster than that of the produced particles so that radiation domination could commence. And this requires a steep
field potential, which of course, cannot support inflation in the standard FRW cosmology. This is precisely where the branerandall ; h assisted
inflation comes to the rescue.
The presence of the quadratic density term (high energy corrections) in the Friedman equation on the brane changes the
expansion dynamics at early epochs cline (see Refroy rev for details on the dynamics of brane worlds)
Consequently, the field experiences greater damping and rolls down its potential slower than it would during the conventional inflation.
Thus, inflation in the brane world scenario can successfully occur for very steep potentialsbasset ; liddle . The model of
quintessential inflation based upon reheating via gravitational particle production is faced with difficulties associated with excessive production of gravity waves. Indeed the reheating mechanism based upon this process is extremely inefficient. The energy density of so produced radiation
is typically one part in $10^{16}$liddle to the scalar-field
energy density at the end of inflation. As a result, these models
have prolonged kinetic regime during which the amplitude of primordial
gravity waves enhances and violates the nucleosynthesis constraintvst (see also star79 ). Hence, it
is necessary to look for alternative mechanisms more efficient than
the gravitational particle production to address the problem.
A proposal of reheating with Born-Infeld matter was made
in Refbisn (see also Refbi1 ; bi2 on the related theme). It was shown that reheating is quite efficient and
the model does not require any additional fine tuning of parametersbisn .
However, the model works under several assumptions which are not easy to justify.
The problems associated with reheating mechanisms discussed above can be circumvented if
one invokes an alternative method of reheating,
namely ‘instant preheating’ proposed by Felder, Kofman and Linde
FKL (see also Refshtanov on the related theme. For other approaches to reheating in quintessential inflation see
curvaton ).
This mechanism is quite efficient and robust, and is well suited to non-oscillating models. It describes a new method of realizing quintessential inflation on the brane
in which inflation is followed by ‘instant preheating’.
The larger reheating temperature in this model results in a
smaller amplitude of relic gravity waves which is consistent with the nucleosynthesis boundssamiv . However,
the recent measurement of CMB anisotropies
by WMAP places fairly strong constraints on inflationary models
spergel03 ; tegmark03 .
It seems that the steep brane world inflation is on the the verge of being ruled out by the observationssuji04 .
Steep inflation in a Gauss-Bonnet braneworld may appear to be
in better agreement with observations than inflation in a RS scenario lidsey .
II QUINTESSENTIAL INFLATION
Quintessential inflation aims to describe a scenario in which both inflation and dark energy (quintessence) are described by the same scalar field. The unification of these concepts in a single scalar field model imposes certain constraints which were spelled out in the introduction. These concepts can be put together consistently in context with brane world inflation. Let us below list the building blocks of such a model.
$\bullet$ Alternative Mechanisms of Reheating
(i) Reheating via gravitational particle production.
(2) Curvaton reheating.
(3) Born-Infeld induced reheating.
(4) Instant preheating.
$\bullet$ Steep Inflaton Potential.
$\bullet$ Brane World Assisted Inflation.
$\bullet$ Tracher Field.
$\bullet$ Late Time Features in the Potential
(1) Potentials which become shallow at late time (such as inverse power law potentials)
(2) Potentials reducing to particular power law type at late times.
II.1 Steep Brane World Inflation
In what follows we shall work with the steep exponential potential which exhibits the aforementioned
features necessary for the description of inflationary as well as post inflationary regimes. The
brane world inflation with steep potentials becomes possible due to high energy corrections in the Friedmann equation. The exit from inflation also takes place naturally when the high energy corrections become
unimportant.
In the 4+1 dimensional brane scenario
inspired by the Randall-Sundrum randall model, the standard Friedman
equation is modified to cline
$$H^{2}=\frac{1}{3M_{p}^{2}}\rho\left(1+\frac{\rho}{2\lambda_{b}}\right)+\frac{%
\Lambda_{4}}{3}+\frac{\cal E}{a^{4}}$$
(1)
where ${\cal E}$ is an integration constant which transmits bulk graviton
influence onto the brane and
$\lambda_{b}$ is the three dimensional brane tension which provides a
relationship between the four and five-dimensional Planck masses
and also relates the four-dimensional cosmological constant $\Lambda_{4}$
to its five-dimensional counterpart.
The four dimensional cosmological constant $\Lambda_{4}$ can be made to vanish by appropriately tuning the brane tension.
The “dark radiation”
${\cal E}/a^{4}$ is expected to rapidly disappear once inflation has
commenced so that we effectively get cline ; basset
$$H^{2}=\frac{1}{3M_{p}^{2}}\rho\left(1+\frac{\rho}{2\lambda_{b}}\right),$$
(2)
where $\rho\equiv\rho_{\phi}={1\over 2}{\dot{\phi}}^{2}+V(\phi)$, if one is dealing with a
universe dominated by a single minimally coupled scalar field.
The equation of motion of a scalar field propagating on the brane is
$${\ddot{\phi}}+3H{\dot{\phi}}+V^{\prime}(\phi)=0.$$
(3)
From (2) and (3) we find that the presence of the additional term
$\rho^{2}/\lambda_{b}$
increases the damping experienced by the scalar field as it rolls down its
potential. This effect is reflected in the slow-roll parameters which
have the form basset ; liddle
$$\displaystyle\epsilon$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{\rm FRW}\frac{1+V/\lambda_{b}}{\left(1+V/2\lambda_{b}%
\right)^{2}},$$
$$\displaystyle\eta$$
$$\displaystyle=$$
$$\displaystyle\eta_{\rm FRW}\left(1+V/2\lambda_{b}\right)^{-1},$$
(4)
where
$$\epsilon_{\rm FRW}=\frac{M_{p}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2},\,%
\,\eta_{\rm FRW}={M_{p}^{2}}\left(\frac{V^{\prime\prime}}{V}\right)$$
(5)
are slow roll parameters in the absence of brane corrections.
The influence of the brane term becomes important when $V/\lambda_{b}\gg 1$
and in this case we get
$$\epsilon\simeq 4\epsilon_{\rm FRW}(V/\lambda_{b})^{-1},\,\eta\simeq 2\eta_{\rm
FRW%
}(V/\lambda_{b})^{-1}.$$
(6)
Clearly slow-roll ($\epsilon,\eta\ll 1$) is easier to achieve when
$V/\lambda_{b}\gg 1$ and on this basis one can expect inflation
to occur even for relatively steep
potentials, such the exponential and the inverse power-law which we discuss
below.
II.2 Exponential Potentials
The exponential potential
$$V(\phi)=V_{0}e^{\alpha\phi/M_{P}}$$
(7)
with ${\dot{\phi}}<0$
(equivalently $V(\phi)=V_{0}e^{-\alpha\phi/M_{P}}$
with ${\dot{\phi}}>0$)
has traditionally played an important role
within the inflationary framework since, in the absence of matter,
it gives rise to power law inflation $a\propto t^{c}$,
$c=2/\alpha^{2}$
provided $\alpha\leq\sqrt{2}$.
For $\alpha>\sqrt{2}$
the potential becomes too steep to sustain inflation and for
larger values $\alpha\geq\sqrt{6}$ the field enters a
kinetic regime during which
field energy density $\rho_{\phi}\propto a^{-6}$.
Thus within the standard general relativistic framework,
steep potentials
are not capable of sustaining inflation.
However
extra-dimensional effects lead to interesting new possibilities for the
inflationary scenario. The increased damping of the scalar field
when $V/\lambda_{b}\gg 1$ leads to
a decrease in the value of the slow-roll parameters
$\epsilon=\eta\simeq 2\alpha^{2}\lambda_{b}/V$,
so that slow-roll ($\epsilon,\eta\ll 1$) leading to inflation
now becomes possible even for large values of $\alpha$.
The steep exponential potentials satisfies the post inflationary requirements mentioned earlier. Infact, the cosmological dynamics with
steep exponential potential in presence of background (radiation/matter) admits scaling solution as the attractor of the system. The
attractor is characterized by the tracking behavior of the field energy density $\rho_{\phi}$. During the ’tracking regime’, the ratio
of $\rho_{\phi}$ to the background energy density $\rho_{B}$ is held fixed
$${\rho_{\phi}\over{\rho_{\phi}+\rho_{B}}}={{3\left(1+w_{B}\right)}\over\alpha^{%
2}}\lower 3.87pt\hbox{ $\buildrel<\over{\sim}$}~{}0.2$$
(8)
where $w_{B}$ is the equation of state parameter for background ($w_{B}=0,~{}1/3$ for matter and radiation respectively) and the inequality (8) reflects the nucleosynthesis constraint which requires
$\alpha\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}5$. It is therefore clear that the field energy density in the post inflationary regime would keep tracking the
background being subdominant such that it does not interfere with the thermal history of the universe.
Within the framework of the braneworld scenario, the field equations
(2) and (3) can be solved exactly in the
slow-roll limit when
$\rho/\lambda_{b}\gg 1$. In this case
$$\frac{\dot{a}(t)}{a(t)}\simeq\frac{1}{\sqrt{6M_{P}^{2}\lambda_{b}}}V(\phi),$$
(9)
which, when substituted in
$$3H\dot{\phi}\simeq-V^{\prime}(\phi)$$
(10)
leads to
$${\dot{\phi}}(t)=-\alpha\sqrt{2\lambda_{b}/3}$$
(11)
The expression for number of inflationary e-foldings is easy to establish
$$\displaystyle{\cal N}$$
$$\displaystyle=$$
$$\displaystyle\log{\frac{a(t)}{a_{i}}}=\int_{t_{i}}^{t}H(t^{\prime})dt^{\prime}$$
$$\displaystyle=$$
$$\displaystyle\frac{V_{0}}{2\lambda_{b}\alpha^{2}}(e^{\alpha\phi_{i}}-e^{\alpha%
\phi(t)})$$
$$\displaystyle=$$
$$\displaystyle\frac{V_{i}}{2\lambda_{b}\alpha^{2}}\left[1-\exp{\{-\sqrt{\frac{2%
\lambda_{b}}{3M_{P}^{2}}\alpha^{2}}(t-t_{i})\}}\right],$$
(13)
where $V_{i}=V_{0}e^{\alpha\phi_{i}}$.
From Eq. (13) we find that the expansion factor passes
through an inflection point marking the end of inflation and leading to
$$\phi_{\rm end}=-\frac{M_{P}}{\alpha}\log{\bigg{(}\frac{V_{0}}{2\lambda_{b}%
\alpha^{2}}\bigg{)}},$$
(14)
$$V_{\rm end}\equiv V_{0}e^{\alpha\phi_{\rm end}/M_{P}}=2\lambda_{b}\ \alpha^{2}%
\\
.$$
(15)
The COBE normalized value for the amplitude of scalar density perturbations allows to estimate $V_{end}$ and the brane tension
$\lambda_{b}$
$$\displaystyle V_{end}$$
$$\displaystyle\simeq$$
$$\displaystyle{{3\times 10^{-7}}\over\alpha^{4}}\left({M_{p}\over{\cal N}+1}%
\right)^{4}$$
$$\displaystyle\lambda_{b}$$
$$\displaystyle\simeq$$
$$\displaystyle{{1.3\times 10^{-7}}\over\alpha^{6}}\left({M_{p}\over{\cal N}+1}%
\right)^{4}~{},$$
(16)
We work here under the assumption that scalar density
perturbations are responsible for most of the COBE signal. We shall, however, come
back to the important question about the tensor perturbations later in our discussion.
The scenario of quintessential inflation we are discussing here belongs to the class of
non-oscillatory models where the conventional reheating mechanism does not work. We can use
the gravitational particle production to do the required. This is a democratic process which leads
to the production of a variety of species quantum mechanically at the end of inflation when
the space time geometry suffers a crucial change. Unlike the conventional reheating mechanism,
this process does not requite the introduction of extra fields. The radiation density created
via this mechanism at the end of inflation is given by
$$\rho_{r}\sim 0.01\times g_{p}H_{end}^{4}$$
(17)
where $g_{p}\sim 100$ is the number of different particle species created from vacuum. Using the relation (17) and the expressions of $\lambda_{b}$ and $V_{end}$ obtained above, it can easily be shown that
$$\left(\frac{\rho_{\phi}}{\rho_{r}}\right)_{\rm end}\sim 2\times 10^{16}\left(%
\frac{{\cal N}+1}{51}\right)^{4}g_{p}^{-1}$$
(18)
This leads to a prolonged ‘kinetic regime’ during which scalar matter has
the ‘stiff’ equation of state .
Using Eqs. (II.2) & (13)
one can demonstrate
that inflation proceeds at an exponential rate during early epochs which plays an important role
for the generation of relic gravity waves during inflation.
III Late Time Evolution
As discussed above, the scalar field with exponential potential (20) leads to a viable
evolution at early times. We should, however, ensure that the scalar field becomes quintessence at late times
which demands a particular behavior of the scalar field potential as discussed above. Indeed,
any scalar field potential which interpolates between an exponential at early epochs and the power law type potential at late times could lead to a viable cosmological evolution.
The cosine
hyperbolic potential provides one such example
sahni
$$V(\phi)=V_{0}\left[\cosh(\tilde{\alpha}\phi/M_{p})-1\right]^{p},~{}~{}~{}~{}p~%
{}>0$$
(19)
which has asymptotic forms
$$V(\phi)={V_{0}\over 2^{p}}e^{{\alpha}\phi/M_{p}},~{}~{}~{}\tilde{\alpha}\phi/M%
_{p}~{}>>1,~{}~{}\phi~{}>0$$
(20)
$$V(\phi)={V_{0}\over 2^{p}}\left(\tilde{\alpha}\phi\over M_{p}\right)^{2p}~{}~{%
}~{}~{}~{}~{}|\tilde{\alpha}\phi/M_{p}|~{}<<1$$
(21)
where ${\alpha}=p\tilde{\alpha}$.
As the cosine
hyperbolic potential (19) exhibits power law type of behavior near the origin, field oscillations build up in the system
at late times. For a particular choice of power law, the average equation of state parameter may turn negativesahni ; turner
$$\left<w_{\phi}\right>=\left<{{{\dot{\phi}^{2}\over 2}-V(\phi)}\over{{\dot{\phi%
}^{2}\over 2}+V(\phi)}}\right>={{p-1}\over{p+1}}$$
(22)
As a result the scalar field energy density and the scale factor have the following behavior
$$\rho_{\phi}\propto a^{-3(1+\left<w\right>)},~{}~{}~{}~{}~{}~{}~{}~{}a\propto t%
^{{2\over 3}(1+\left<w\right>)^{-1}}.$$
The average equation of state
$\left<w(\phi)\right><-1/3$ for $p<1/2$ allowing the scalar field to play the role of dark energy.We have numerically solved for the behaviour of this model after
including a radiative term (arising from inflationary particle
production discussed in the previous section) and standard cold dark matter.
Our results for a particular realization of the model
are shown in figures 1 & 2.
We find that, due to the very large value of the scalar field kinetic energy
at the commencement of the radiative regime,
the scalar field density
overshoots the radiation energy density.
After this, the value
of $\rho_{\phi}$ stabilizes and
remains relatively unchanged
for a considerable length of time during which the scalar field
equation of state is $w_{\phi}\simeq-1$.
Tracking commences late into
the matter dominated epoch and the universe accelerates today during rapid
oscillations of the scalar field.
This model provides an interesting example of ‘quintessential
inflation’. However as we shall discuss next,
the long duration of the kinetic regime in this model results in a
large gravity wave background which comes into conflict with
nucleosynthesis constraints.
III.1 Relic Gravity Waves and Nucleosynthesis Constraint
The tensor perturbations or gravity waves get quantum mechanically generated during inflation and leave imprints
on the micro-wave background.
Gravity waves in a spatially homogeneous and isotropic background geometry
satisfy the minimally coupled Klein-Gordon equation $\Box h_{ik}=0$ ,
which, after a separation of variables
$h_{ij}=\phi_{k}(\tau)e^{-i{\bf k}{\bf x}}e_{ij}$
($e_{ij}$ is the polarization tensor) reduces to
$${\ddot{\phi}_{k}}+2\frac{\dot{a}}{a}{\dot{\phi}}_{k}+k^{2}\phi_{k}=0$$
(23)
where $\tau=\int dt/a(t)$ is the conformal time coordinate
and $k=2\pi a/\lambda$ is the comoving wavenumber.
Since brane driven inflation is near-exponential we can write
$a=\tau_{0}/\tau$ $(|\tau|<|\tau_{0}|)$, in this case
normalized positive frequency solutions of (23)
corresponding to the adiabatic
vacuum in the ‘in state’ are given by vst
$$\phi_{\rm in}^{+}(k,\tau)=\left(\frac{\pi\tau_{0}}{4}\right)^{1/2}\left(\frac{%
\tau}{\tau_{0}}\right)^{3/2}H_{3/2}^{(2)}(k\tau)F(H_{\rm in}/\tilde{\mu})$$
(24)
where $\tilde{\mu}=M_{5}^{3}/M_{p}^{2}$ and $H_{\rm in}\equiv-1/\tau_{0}$ is the
inflationary Hubble parameter.
The term
$$F(x)=\left(\sqrt{1+x^{2}}-x^{2}\log{\{\frac{1}{x}+\sqrt{1+\frac{1}{x^{2}}}\}}%
\right)^{-1/2}$$
(25)
is responsible for the increased gravity wave amplitude
in braneworld inflation lmaartans .The ‘out state’ is described by a linear superposition of positive and negative
frequency solutions to (23). For power law expansion
$a=(t/t_{0})^{p}\equiv(\tau/\tau_{0})^{1/2-\mu}$, we have
$$\phi_{\rm out}(k,\tau)=\alpha\phi^{(+)}_{\rm out}(k\tau)+\beta\phi^{(-)}_{\rm
out%
}(k\tau)$$
(26)
The energy density of relic gravity waves is given by vst
$$\rho_{\rm g}=\langle T_{0}^{0}\rangle=\frac{1}{\pi^{2}a^{4}}\int dkk^{3}|\beta%
|^{2},$$
(27)
Computing the Bogolyubov coefficient $\beta$, one can show that the spectral energy
density of gravity waves produced during slow-roll inflation isvst ; samiv
$$\rho_{g}(k)\propto k^{2\left(\frac{w-1/3}{w+1/3}\right)}~{}.$$
(28)
where $w$ is the equation of state parameter which characterizes the post-inflationary epoch.
In braneworld model under consideration,
$w\simeq 1$ during the kinetic regime, consequently the gravity wave background
generated during this epoch will have a
blue spectrum $\rho_{g}(k)\propto k$.
We imagine that radiation through some mechanism was generated at the end of inflation with radiation density $\rho_{r}$. Then the ratio of energy in gravity waves to $\rho_{r}$
at the commencement of
radiative regime is given byvst
$$\left({\rho_{g}\over\rho_{r}}\right)_{eq}={64\over{3\pi}}h_{GW}^{2}\left({T_{%
kin}\over T_{eq}}\right)^{2}$$
(29)
where $h_{GW}$ is the dimensionless amplitude of gravity waves (from COBE normalization,
$h_{GW}^{2}\simeq 1.7\times 10^{-10}$, for ${\cal N}\simeq 70$).
We should mention that the commencement of the kinetic regime is not instaneous and the brane effects petrsist for some time after inflation has ended. The temperature at the commencement of the kinetic regime $T_{kin}$ is related to the temperature at the end of inflation as
$$T_{kin}=T_{end}\left(a_{end}\over a_{kin}\right)=T_{end}F_{1}(\alpha)$$
(30)
where $F_{1}(\alpha)=\left(c+{d\over\alpha^{2}}\right)$, $c\simeq 0.142$, $d\simeq-1.057$ and
$T_{end}=\left(\rho_{r}^{end}\right)^{1/4}$. The equality between scalar field matter and radiation
takes place at the temperature
$$T_{eq}=T_{end}{F_{2}(\alpha)\over{\left(\rho_{\phi}/\rho_{r}\right)_{end}^{1/2%
}}}$$
(31)
with $F_{2}(\alpha)=\left(e+{f\over\alpha^{2}}\right)$, $e\simeq 0.0265$, $f\simeq-0.176$. The
fitting formulas (30) and (31) are obtained by numerical integration of equations of motion.
Using equations (30), (31) and (29) we obtain the ratio of scalar field energy density to radiation ener density at the end of inflation
$${\left(\rho_{\phi}\over\rho_{r}\right)_{end}}={3\pi\over 64}\left({1\over{h_{%
GW}^{2}\left(F_{1}(\alpha)/F_{2}(\alpha)\right)^{2}}}\right)\left({\rho_{g}%
\over\rho_{r}}\right)_{eq}$$
(32)
Equation (32 ) is an important result which sets a limit on the ratio of scalar field energy density to radiation energy
density at the end of inflation. Indeed, For the nucleo-synthesis constraint to be respected, the ratio of energy density in gravity waves to radiation energy density at equality $(\rho_{g}/\rho_{r})_{eq}\lower 3.87pt\hbox{ $\buildrel<\over{\sim}$}~{}0.2$.
For a generic steep exponential potential (($\alpha\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}5$), we have
$$\left(\rho_{\phi}/\rho_{r}\right)_{end}\lower 3.87pt\hbox{ $\buildrel<\over{%
\sim}$}~{}10^{7}$$
(33)
As emphasized earlier, this ratio is of the order of $10^{16}$ in case gravitational particle production and
exceeds the nucleo-synthesis constraint by nine orders of magnitudes.
An interesting proposal which can circumvent this difficulty has recently been suggested by Liddle and Lopezcurvaton .
The authors have employed
a new method of reheating via curvaton to address the problems associated with gravitational particle
production mechanism. The curvaton model as shown in Refcurvaton can in principal resolve the difficulties related to excessive amplitude of
short-scale gravitational waves. Although this model is interesting, it operates through a very complex network of constraints dictated by the fine tuning
of parameters of the model.
In the following section, we shall examine an alternative mechanism based upon Born-Infeld reheating.
IV Born-Infeld Brane Worlds
The D-branes are fundamental objects in string theory. The end points of the open string to which the gauge fields are attached are
constrained to lie on the branes. As the string theory contains gravity, the D-branes are the dynamical objects. The effective
D-brane action is given by the Born-Infeld action
$$S_{BI}=-\lambda_{b}\int{d^{4}x\sqrt{-{\rm det}\left(g_{\mu\nu}+F_{\mu\nu}%
\right)}}$$
(34)
where $F_{\mu\nu}$ is the elecromagnetic field tensor (Non-Abelian gauge fields could also be included in the action) and
$\lambda_{b}$ is the brane tension. The Born-Infeld action, in general, also includes Fermi fields and scalars which have been dropped here for simplicity. In the brane
world scenario a la Randall-Sundrum one adopts the Nambu-Goto action instead of the Born-Infeld action. Shiromizu
et al have suggested that in the true spirit of the string theory, the total action in the brane world cosmology be composed of the bulk and D-brane
actionsbi1
$$S=S_{bulk}+S_{BI},$$
(35)
where $S_{bulk}$ is the five dimensional Einstein-Hilbert action with the negative
cosmological constant.
The stress tensor appearing on the right hand side (RHS) of the Einstein equations on the brane will now
be sourced by the Born-Infeld
action. The modified Friedman equation on a spatially flat FRW brane acquires the form
$$H^{2}={1\over 3M_{p}^{2}}\rho_{\rm BI}\left(1+{\rho_{\rm BI}\over 2\lambda_{b}%
}\right)$$
(36)
with $\rho_{\rm BI}$ given by
$$\rho_{\rm BI}=\epsilon+{\epsilon^{2}\over{6\lambda_{b}}}$$
(37)
where $E^{2}=B^{2}=\epsilon$
The tension $\lambda_{b}$ is tuned so that the
net cosmological constant on the brane vanishes.
We have dropped the ‘dark radiation’ term in the equation
(36) as it rapidly disappear once inflation
sets in. Spatial averaging is assumed while computing $\rho_{\rm BI}$
and $P_{BI}$ from the stress-tensor
corresponding to action (34). The scaling of energy density of the
Born-Infeld matter, as usual, can be established from the conservation equation Born-Infeld matter, as usual, can be established from the conservation equation
$$\dot{\rho}_{\rm BI}+3H(\rho_{\rm BI}+P_{\rm BI})=0$$
(38)
where
$$P_{BI}={\epsilon\over 3}-{\epsilon^{2}\over{6\lambda_{b}}}$$
(39)
Interestingly, the pressure due to the Born-Infeld matter becomes negative in the high energy regime allowing the accelerated expansion
at early times without the introduction of a scalar field. As shown in bi1 , the energy density $\rho_{\rm BI}$ scales as radiation when
$\epsilon<<6\lambda_{b}$. For $\epsilon>6\lambda_{b}$, the Born-Infeld matter energy density starts scaling slowly (logarithmically)
with the scale factor to mimic the cosmological constant like behavior. The point is that the Born-Infeld
matter is subdominant during the inflationary stage. It comes to play the important role after the end of inflation
when it behaves like radiation and hence serves as an alternative to reheating mechanism.
The brane world cosmology based upon the Born-Infeld action looks promising as it is perfectly tuned with the D-brane ideology. But
since the Born-Infeld action is composed of the non-linear elecromagnetic field, the D-brane cosmology proposed in
Refbi1 can not accommodate
density perturbations at least in its present formulation. One could include a scalar field in the Born-Infeld action, say, a tachyon condensate to correct the situation. However, such a scenario faces the difficulties associated with reheatingreh ; linde and formation of
acoustics/kinkskink . We shall therefore not follow this track. We shall assume that the scalar field driving the inflation (quintessence) on the
brane is described by the usual four dimensional action for the scalar fields. We should remark here that the problems faced by rolling tachyon models are beautifully circumvented in the scenario
based upon massive Born-Infeld scalar
field on the $\bar{D}_{3}$ brane of KKLT vacuagarousi .
The total action that we are trying to motivate here is given by
$$S=S_{bulk}+S_{BI}+S_{\rm 4d-scalar}$$
(40)
where
$$S_{\rm 4d-scalar}=-\int{\left({1\over 2}g^{\mu\nu}\partial_{\mu}\phi\partial_{%
\nu}\phi+V(\phi)\right)\sqrt{-g}d^{4}x}$$
(41)
The energy momentum tensor for the field $\phi$ which arises from the action (41) is given by
$$T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-g_{\mu\nu}\left[{1\over 2}g^{%
\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+V(\phi)\right]$$
(42)
The scalar field propagating on the brane modifies the Friedman equation to
$$H^{2}={1\over 3M_{p}^{2}}\rho_{\rm tot}\left(1+{\rho_{tot}\over 2\lambda_{b}}\right)$$
(43)
where $\rho_{tot}$ is given by
$$\rho_{tot}=\rho_{\phi}+\rho_{\rm BI}$$
(44)
As mentioned earlier, in the scenario based upon reheating via quantum mechanical
particle production during inflation, the radiation density is very small, typically one part in $10^{16}$ and the ratio of the field
energy density to that of radiation has no free parameter to tune. This leads to long kinetic regime which results in an unacceptably
large gravity background. The Born-Infeld matter which behaves like radiation (at the end
of inflation) has no such problem and can be used for reheating
without conflicting with the nucleosynthesis constraint. Indeed, at the end of inflation $\rho_{\rm BI}$ can be chosen such that $\rho_{\rm BI}^{end}<<6\lambda_{b}$. Such an initial condition
for $\rho_{\rm BI}$ is consistent with the nucleo-synthesis constraintbisn . In that
case the Born-Infeld matter energy density would scale like radiation at the end of inflation. At this epoch the scale factor will be initialized at
$a_{end}=1$. The energy density $\rho_{\rm BI}$ would continue scaling as $1/a^{4}$ below $a=a_{end}$. The scaling would slow down as $\rho_{\rm BI}$
reaches $6\lambda_{b}$ which is much smaller than $V_{end}$ for generic steep potentials, say for $\alpha\geq 5$. Hence
$\rho_{\rm BI}$ remains
subdominant to scalar field energy density $\rho_{\phi}$ for the entire inflationary evolution. The Born-Infeld matter comes to play the
important role only at the end of inflation which is in a sense similar to curvaton. But unlike curvaton, it does not
contain any new parameter. The numerical results for a specific choice of parameters is shown in
Fig. 3. In contrast to the ‘quintessential
inflation’ based upon the gravitational particle production mechanism where the scalar field spends long time in the kinetic regime and makes deep undershoot followed by long locking period with very brief tracking, the scalar field in the present scenario tracks
the background for a very long time (see figure 3). This pattern of evolution is consistent with the
thermal history of the universe. We note that
‘quintessential inflation’ can also be implemented by inverse power law potentials. Unfortunately,
one has to make several assumptions to make the scenario working:
i) The tension
of the D3 brane appearing in the Born-Infeld action is treated as constant and is identified with the brane tension in the Randall-Sundrum
scenario. (ii) The fluctuations in the Born-Infeld matter are neglected.
(iii)The series expansion of the Born-Infeld action is truncated beyond
a certain order.
In what follows, we shall examine the instant reheating mechanism
discovered by Felder, Kofman and Linde and show that their mechanism is superior to
other reheating mechanism mentioned above.
V Braneworld Inflation Followed by Instant Preheating
Braneworld Inflation induced by the steep exponential potential (7) ends when
$\phi=\phi_{end}$, see (14).
Without loss of generality, we can make the inflation end at the
origin by translating the field
$$V(\phi^{\prime})\equiv V(\phi)=\tilde{V_{0}}e^{\alpha\phi^{\prime}/M_{p}}~{},$$
(45)
where $\tilde{V_{0}}=V_{0}e^{+\alpha\phi_{end}}/M_{p}$ and $\phi^{\prime}=\phi-\phi_{end}$.
In order to achieve reheating after inflation has ended we assume that the inflaton
$\phi$ interacts with
another scalar field $\chi$ which has a Yukawa-type interaction with
a Fermi field $\psi$. The interaction Lagrangian is
$$L_{int}=-{1\over 2}g^{2}\phi^{\prime 2}\chi^{2}-h\bar{\psi}\psi\chi~{}.$$
(46)
To avoid confusion, we drop the prime on $\phi$ remembering that $\phi<0$
after inflation has ended.
It should be noticed that the $\chi$ field has no bare mass,
its effective mass being determined by the field $\phi$ and the value of the coupling
constant $g$ ($m_{\chi}=g|\phi|$).
The production of $\chi$ particles commences as soon as $m_{\chi}$ begins changing
non-adiabatically FKL
$$|\dot{m_{\chi}}|\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}m_{\chi}^{2}~{}%
~{}~{}or~{}~{}~{}~{}|\dot{\phi}|\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{%
}g\phi^{2}~{}.$$
(47)
The condition for particle production (47) is satisfied when
$$|\phi|\lower 3.87pt\hbox{ $\buildrel<\over{\sim}$}~{}|\phi_{prod}|=\sqrt{\frac%
{|\dot{\phi}_{end}|}{g}}=\sqrt{{V_{end}^{1/2}\over{\sqrt{3}g}}}~{}.$$
(48)
From (16) we find that $\phi_{prod}\ll M_{p}$ for $g\gg 10^{-9}$.
The production time for $\chi$ particles can
be estimated to be
$$\Delta t_{prod}\sim{|\phi|\over{|\dot{\phi}|}}\sim\frac{1}{\sqrt{gV_{end}^{1/2%
}}}~{}.$$
(49)
The uncertainty relation provides an estimate for the momentum of $\chi$ particles
created non-adiabatically: $k_{prod}\simeq(\Delta t_{prod})^{-1}\sim g^{1/2}V_{end}^{1/4}$. Proceeding as in FKL we can show that
the occupation number of $\chi$ particles jumps sharply from zero to
$$n_{k}\simeq\exp(-{\pi k^{2}/{gV_{end}}^{1/2}})~{},$$
(50)
during the time interval $\Delta t_{prod}$.
The $\chi$-particle number density is estimated to be
$$n_{\chi}={1\over{2\pi^{3}}}\int_{0}^{\infty}{k^{2}n_{k}}dk\simeq{(gV_{end}^{1/%
2})^{3/2}\over{8\pi^{3}}}~{}.$$
(51)
Quanta of the $\chi$-field are created during the time interval $\Delta t_{prod}$
that the field $\phi$ spends in the vicinity of $\phi=0$. Thereafter the mass
of the $\chi$-particle begins to grow since $m_{\chi}=g|\phi(t)|$, and
the energy density of particles of the $\chi$-field created in this manner is
given by
$$\rho_{\chi}=m_{\chi}n_{\chi}\left({a_{end}\over a}\right)^{3}={(gV^{1/2}_{end}%
)^{3/2}\over{8\pi^{3}}}{{g|\phi(t)|}}\left({a_{end}\over a}\right)^{3}~{}.$$
(52)
where the $({a_{end}/a})^{3}$ term accounts for the cosmological
dilution of the energy density with time.
As shown above, the process of $\chi$ particle-production
takes place immediately after inflation has ended, provided $g\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}10^{-9}$.
In what follows we will show that the $\chi$-field can rapidly decay into fermions.
It is easy to show that if the quanta of the $\chi$-field were converted (thermalized)
into radiation instantaneously, the radiation energy density would become
$$\rho_{r}\simeq\rho_{\chi}\sim{(gV^{1/2}_{end})^{3/2}\over{8\pi^{3}}}g\phi_{%
prod}\sim 10^{-2}g^{2}V_{end}~{}.$$
(53)
From equation (53) follows the important result
$$\left({\rho_{\phi}\over\rho_{r}}\right)_{end}\sim\left(\frac{10}{g}\right)^{2}%
~{}.$$
(54)
Comparing (54) with (33) we find that, in order for
relic gravity waves to respect the nucleosynthesis constraint, we should
have $g\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}4\times 10^{-3}$.
(The energy density created by instant preheating
$\left(\rho_{r}/\rho_{\phi}\right)\simeq(g/10)^{2}$ can clearly be
much larger than the energy density
produced by quantum particle production, for which
$\left(\rho_{r}/\rho_{\phi}\right)\simeq 10^{-16}g_{p}$.)
The constraint
$g\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}4\times 10^{-3}$, implies that the particle production time-scale
(49) is much smaller than the Hubble time since
$$\frac{1}{\Delta t_{prod}H_{end}}\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{%
}300\alpha^{2},~{}~{}\alpha\gg 1~{}.$$
(55)
Thus the effects of expansion can safely be neglected
during the very short time interval in which
‘instant preheating’ takes place.
We also find, from equation (48), that $|\phi_{prod}|/M_{p}\lower 3.87pt\hbox{ $\buildrel<\over{\sim}$}~{}10^{-3}$
implying that particle production takes place in a very narrow band around
$\phi=0$.
Figure 4 demonstrates
the violation of the adiabaticity condition (at the end of inflation) which is a
necessary prerequisite
for particle production to take place.
For the range of $g$
allowed by
the nucleosynthesis constraint, the particle production turns out to be almost instantaneous.
We now briefly mention about the back-reaction of created $\chi$-particles
on the background. As shown in Refsamiv , for any generic value of the coupling $g\lower 3.87pt\hbox{ $\buildrel<\over{\sim}$}~{}0.3$, the back-reaction
of $\chi$ particles in the evolution equation is negligible during the time
scale $\sim H_{kin}^{-1}$ ( Here $\sim H_{kin}^{-1}$ characterizes the epoch the kinetic regime commences. $H_{kin}=H_{end}(0.085-0.688/\alpha^{2})$vst ).
We now turn to the matter of reheating which occurs through the
decay of $\chi$ particles to fermions, as a consequence of the interaction term in the
Lagrangian (46).
The decay rate of $\chi$ particles is given by
$\Gamma_{\bar{\psi}\psi}=h^{2}m_{\chi}/8\pi$, where $m_{\chi}=g|\phi|$.
Clearly the decay rate is faster for larger values of $|\phi|$.
For $\Gamma_{\bar{\psi}\psi}>H_{kin}$, the decay process will
be completed within the time that back-reaction effects (of $\chi$ particles)
remain small. Using the expression for ${Hkin}$ this requirement translates into
$$h^{2}>{{8\pi\alpha}\over{\sqrt{3}g\phi}}{V_{end}^{1/2}\over M_{p}}F(\alpha)~{}.$$
(56)
For reheating to be completed by $\phi/M_{p}\lower 3.87pt\hbox{ $\buildrel<\over{\sim}$}~{}1$,
we find from equations (33) and
(56) that $h\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}10^{-4}g^{-1/2}$($g\lower 3.87pt\hbox{ $\buildrel>\over{\sim}$}~{}4\times 10^{-3}$) for $\alpha\simeq 5$. This along with the
constraint imposed by the back reaction defines the allowed region in the parameter space (g, h).
We observe that there is a wide region in the parameter space for which (i) reheating is rapid and
(ii) the relic gravity background in non-oscillatory braneworld models of
quintessential inflation
is consistent with nucleo-synthesis constraints. However, this is not the complete story. One should
further subject the model to the recent WMAP observations. The measurement of CMB anisotropies
places fairly strong constraints on inflationary models
spergel03 ; tegmark03 .
It appears that the tensor perturbations are not adequately suppressed in the models of steep brane world inflation
and as a result these models are on the verge of being ruled out.
As indicated by Lidsey and Nunes, inflation in a Gauss-Bonnet braneworld could appear to be
in better agreement with observations than inflation in a RS II scenario lidsey .
In the following section, we briefly discuss the prospects of brane world inflation with the
Gauss-Bonnet correction term in the bulk.
VI Gauss-Bonnet Brane Worlds
Though we are trying to motivate the GB term in the bulk having a specific
application in mind, the Gauss-Bonnet
correction is interesting in its own right and has a deep meaning. Let us
begin at the very beginning and ask for the compelling physical motivation
for general relativity (GR). It is the interaction of zero mass particle
with gravitation. Zero mass particle has the universal constant speed
which can not change yet it must feel gravity. This can only happen if
gravitational field curves space. Since space and time are already bound
together by incorporation of zero mass particle in mechanics,
gravitational field thus curves spacetime. In other words it can truly be
described by curvature of spacetime and it thus becomes a property of
spacetime - no longer an external field n1
From the physical standpoint the new feature that GR has to incorporate is
that gravitational field itself has energy and hence like any other energy
it must also link to gravity. That is, field has gravitational charge and
hence it is self interacting. Field energy density will go as square of
first derivative of the metric and it must be included in the Einstein
field equation. It is indeed included for the Riemann curvature involves
the second derivative and square of the first derivative. However in the
specific case of field of an isolated body, we obtain $1/r$ potential, the
same as in the Newtonian case. Where has the square of $\nabla\Phi$ ($\Phi$ denotes the gravitational potential) gone?
It turns out that its contribution has gone into curving the space,
$g_{rr}$ component of the metric being different from $1$.
The main point is that gravitational field equation should follow from the
curvature of spacetime and they should be second order quasilinear
differential equations (quasilinear means the highest order of derivative
must occur linearly so that the equation admits a unique solution).
Riemann curvature through the Bianchi identities leads to the Einstein
equation with the $\Lambda$ term. We should emphasize here that $\Lambda$
enters here as naturally as the stress energy tensor. It is indeed a true
new constant of the Einsteinian gravity n2 .
It is a pertinent question to
ask, is this the most general second order quasilinear equation one can
obtain from curvature of spacetime? The answer is No. There exists a
remarkable combination of square of Riemann tensor and its contractions,
which when added to the action gives a second order quasilinear equation
involving second and fourth power of the first derivative. This is what is
the famous Gauss-Bonnet (GB) term. Thus GB term too appears naturally and
should have some non-trivial meaning.
However GB term is topological in $D<5$ and hence has no dynamics. It
attains dynamics in $D>4$. Note that gravity does not have its full
dynamics in $D<4$ and hence the minimum number of dimensions required for
complete description of gravitation is $4$. This self interaction of
gravity arises through square of first derivative of the metric. Self
interaction should however be iterative and hence higher order terms
should also be included. It turns out that there exists generalization of the
GB term in higher dimensions in terms of the Lovelock Lagrangian
which is a polynomial in the Riemann curvature. That again yields the
quasilinear second order equation with higher powers of the first derivative.
Thus GB and Lovelock Lagrangian represent higher order loop corrections to
the Einstein gravity.
They do however make non-trivial contribution
classically only for $D>4$ dimensions. This is rather important. If GB term
had made a non-trivial
contribution in $4$-dimensions, it would have conflicted with the $1/r$
character of the potential because of the presence of $(\nabla\Phi)^{4}$
terms in the equation. The square terms (to account for contribution of gravitational field energy) were taken care of by the space curvature ($g_{rr}$ in
the metric) and now nothing more is left to accommodate the fourth (and higher) power term. However we can not tamper with the inverse square law (i.e. $1/r$ potential) which is
independently required by the Gauss law of conservation of flux. That can
not be defied at any cost. Thus it is not for nothing that the GB and its
Lovelock generalization term
makes no contribution for $D=4$. It further carries an important message
that gravitational field cannot be kept confined to $4$-dimensions. It is
indeed a higher dimensional interaction where the higher order iterations
attain meaning and dynamics. This is the most profound message
the GB term indicates. This is yet another independent and new motivation
for higher dimensional gravity n2 .
Self interaction is always to be evaluated iteratively. For gravity
iteration is on the curvature of spacetime. It is then not surprising that GB
term arises naturally from the one loop correction to classical gravity.
String theory should however encompass whatever is obtained by iterative
the iterative process. GB term is therefore strongly motivated by
string theoretic considerations as well.
Further GB is topological in $4$-D but in quantum considerations it
defines new vacuum state. It is quantum mechanically non-trivial. In
higher dimensions, it attains dynamics even at classical level. In the
simplest case in higher dimension it should have a classical analogue of
$4$-D quantum case. That is what indeed happens. Space of constant
curvature or equivalently conformally flat Einstein space solves the
equation with GB term with redefined vacuum. This is a general result for
all $D>4$. It is interesting to see quantum in lower dimension becoming
classical in higher dimension.
In the context of the brane bulk system we should therefore include GB
term in the bulk and see its effects on the dynamics on the brane. The
brane world gravity should thus be studied with GB term not necessarily as
correction but in its own right. It is a true description of high energy
gravity. However, for the purpose of following discussion, we shall treat
GB as a correction term.
The Einstein-Gauss-Bonnet action for five dimensional bulk containg a 4D brane is
$$\displaystyle S$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2\kappa_{5}^{2}}\int d^{5}x\sqrt{-g}\big{\{}{\cal R}-2%
\Lambda_{5}+\alpha_{\rm GB}[{\cal R}^{2}-4{\cal R}_{AB}{\cal R}^{AB}$$
(57)
$$\displaystyle+$$
$$\displaystyle{\cal R}_{ABCD}{\cal R}^{ABCD}]\big{\}}+\int d^{4}x\sqrt{-h}({%
\cal L}_{m}-\lambda_{b})~{},$$
${\cal R},R$ refer to the Ricci scalars in the bulk metric $g_{AB}$ and the induced metric on the brane
$h_{AB}$; $\alpha_{rmGB}$ has dimensions of (length)${}^{2}$ and
is the Gauss-Bonnet coupling, while $\lambda_{b}$ is the brane tension
and $\Lambda_{5}\,(<0)$ is the bulk
cosmological constant. The constant $\kappa_{5}$ contains the $M_{5}$, the 5D fundamental energy scale ($\kappa_{5}^{2}=M_{5}^{-3}$).
The
modified Friedman equation on the (spatially flat) brane
may be written as D ; T ; lidsey (see also Refg )
$$\displaystyle H^{2}$$
$$\displaystyle=$$
$$\displaystyle{1\over 4\alpha_{\rm GB}}\left[(1-4\alpha_{\rm GB}\mu^{2})\cosh%
\left({2\chi\over 3}\right)-1\right]\,,$$
(58)
$$\displaystyle\kappa_{5}^{2}(\rho+\lambda_{b})$$
$$\displaystyle=$$
$$\displaystyle\left[{{2(1-4\alpha_{\rm GB}\mu^{2})^{3}}\over{\alpha}_{\rm GB}}%
\right]^{1/2}\sinh\chi\,,$$
(59)
where $\chi$ is a dimensionless measure of the energy-density.
In order to regain general relativity at low energies, the
effective 4D Newton constant is defined by T
$$\kappa_{4}^{2}\equiv{1\over M_{p}^{2}}={\kappa_{5}^{4}\lambda_{b}\over 6(1-4%
\alpha_{\rm GB}\Lambda_{5}/9)}\,.$$
(60)
When $\alpha_{\rm GB}=0$, we recover the RS expression. We can fine-tune
the brane tension to achieve zero cosmological constant on the
brane T :
$$\kappa_{5}^{4}\lambda_{b}^{2}=-4\Lambda_{5}+{1\over\alpha_{\rm GB}}\left[1-%
\left(1+{4\over 3}\alpha_{\rm GB}\Lambda_{5}\right)^{\!3/2}\right].$$
(61)
The modified Friedman equation (58), together with
Eq. (59), shows that there is a characteristic GB energy
scale $M_{\rm GB}$DR such that,
$$\displaystyle\rho\gg M_{\rm GB}^{4}$$
$$\displaystyle\Rightarrow$$
$$\displaystyle~{}H^{2}\approx\left[{\kappa_{5}^{2}\over 16\alpha_{\rm GB}}\,%
\rho\right]^{2/3}\,,$$
(62)
$$\displaystyle M_{\rm GB}^{4}\gg\rho\gg\lambda_{b}$$
$$\displaystyle\Rightarrow$$
$$\displaystyle~{}H^{2}\approx{\kappa_{4}^{2}\over 6\lambda_{b}}\,\rho^{2}\,,$$
(63)
$$\displaystyle\rho\ll\lambda_{b}$$
$$\displaystyle\Rightarrow$$
$$\displaystyle~{}H^{2}\approx{\kappa_{4}^{2}\over 3}\,\rho\,.$$
(64)
It should be noted that Hubble law acquires an unusual form for energies higher that than the
GB scale. Interestingly, for an exponential potential, the modified Eq.(62) leads to
exactly scale invariant spectrum for primordial density perturbations. Inflation continues below GB scale
and terminates in the RS regime leading to the spectral index very close to one. This is amazing
that it happens without tuning the slope of the potential. The Gauss-Bonnet inflation has
interesting consequences for steep brane world inflation (see Fig. 5 and the discussion in the next section).
VII Summary
In this paper we have reviewed the recent work on unification of inflation with quintessence
in the frame work of brane worlds. These models belong
to the class of non-oscillatory models in which the underlying alternative reheating mechanism
plays a crucial role. The popular reheating alternative via quantum mechanical production of
particle during inflation leads to an unacceptable relic gravity wave background which
violates the nucleo-synthesis constraint at the commencement of radiative regime. We have mentioned other alternatives to conventional (p)reheating and have shown that ’instant preheating’ discovered by
Felder, Linde and Kofman is superior and best suited to brane world models of quintessential inflation. The recent measurement of CMB anisotropies
by WMAP, appears to heavily constraint these models.
The steep brane world inflation seems to be excluded by observation in
RS scenariosuji04 . As shown in Ref.ssr , the inclusion of GB term in the bulk effects the constraints on the
inflationary
potentials and can rescue the steep exponential potential allowing it to be compatible
with observations for a range of energy scales. The GB term leads to an increase of the spectral index $n_{S}$
and decrease of tensor to scalar ratio of perturbations $R$ in the intermediate region between RS and GB.
As seen from Fig.5, there is an intermediate region
where the steep inflation driven by exponential potential lies within $2\sigma$ contour for ${\cal N}=70$ .
Thus, the steep inflation in a Gauss-Bonnet braneworld appears to be
in agreement with observations.
VIII Acknowledgments
We thank N. Mavromatos, S. Odintsov, V. Sahni and Shinji Tsujikawa for useful comments.
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Sensor Switching Control Under Attacks Detectable by Finite Sample Dynamic Watermarking Tests
Pedro Hespanhol, Matthew Porter, Ram Vasudevan, and Anil Aswani
This work was supported by the UC Berkeley Center for Long-Term Cybersecurity, and by a grant from Ford Motor Company via the Ford-UM Alliance under award N022977.Pedro Hespanhol and Anil Aswani are with the Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA
[email protected], [email protected] Porter and Ram Vasudevan are with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
[email protected], [email protected]
Abstract
Control system security is enhanced by the ability
to detect malicious attacks on sensor measurements. Dynamic
watermarking can detect such attacks on linear time-invariant
(LTI) systems. However, existing theory focuses on attack detection
and not on the use of watermarking in conjunction with
attack mitigation strategies. In this paper, we study the problem
of switching between two sets of sensors: One set of sensors has
high accuracy but is vulnerable to attack, while the second set of
sensors has low accuracy but cannot be attacked. The problem is
to design a sensor switching strategy based on attack detection
by dynamic watermarking. This requires new theory because
existing results are not adequate to control or bound the behavior
of sensor switching strategies that use finite data. To overcome
this, we develop new finite sample hypothesis tests for dynamic
watermarking in the case of bounded disturbances, using the
modern theory of concentration of measure for random matrices.
Our resulting switching strategy is validated with a simulation
analysis in an autonomous driving setting, which demonstrates
the strong performance of our proposed policy.
Dynamic watermarking, observer switching control, finite sample tests
I Introduction
The secure and resilient control of cyber-physical systems (CPS) requires safe operation in the face of malicious attacks that can occur on either the physical layer (e.g., sensors and actuators) or the cyber layer (e.g., communication and computation capabilities)[1]. Real-life incidents like the Maroochy-Shire incident [2], the Stuxnet worm [3], and others [4] illustrate the importance of concerns about CPS security. One approach to secure control has been to focus on cybersecurity of CPS [5, 6, 7, 8], but this does not fully exploit the physical aspects of CPS. An alternative is attack identification and detection considering the interplay between the cyber and physical parts of CPS [9, 4, 10, 11]. Many of these techniques are static (i.e., do not consider system dynamics) [12] or passive (i.e., do not actively control system to identify malicious nodes and sensors) [13, 14, 15].
In contrast, dynamic watermarking is an active defense technique that injects perturbations into the system control in order to detect attacks [16, 17, 18, 19]. More specifically, this method applies a private excitation to the system, which is a disturbance only known to the controller. Then it uses consistency tests to detect attacks by checking for correlation between sensor measurements and the private excitation. The goal is to be able to detect all sensor attacks whose magnitude exceeds some prespecified amount.
I-A Asymptotic Results for Dynamic Watermarking
Research on dynamic watermarking can be divided into two main areas of contribution: The first is the development of statistical hypothesis testing that tries to detect corrupted measurements by observing correlations between sensor outputs and the dynamic watermark [17, 20, 18, 16, 21]. This set of techniques apply to general LTI systems, but cannot ensure the zero-average-power property for general attack models. The second line of work [22, 23] considers general attack models and develop tests able to ensure that only attacks which add a zero-average-power signal to the sensor measurements can remain undetected, but constrain their analysis to LTI systems with specific structure on their dynamics.
More recently, The work done in [24] and [25] attempts to bridge this gap by providing statistical guarantees for complex types of attacks for general LTI systems.
While both papers address a general MIMO LTI system, the set of assumptions are somewhat different: the former assumes open-loop stability of the LTI system, and the latter restricts the attack form. In particular, in [25], the tests provided are able to detect if a general MIMO LTI system is under a fairly general type of attack. In particular, it considers additive attacks that can dampen/amplify the system measurements, can replay the system from a different initial condition, or can do both. This form of attack, while arguably simple, encompasses many of the types of attacks reported in real-life incidents (e.g., replay attacks [3]) as well as compensate for external disturbances not accounted by the system model (e.g., wind when represented via internal model principle [25]).
We proceed to briefly summarize the results of [25], as it is the foundation for this current paper. Consider a MIMO LTI system with partial observations
$$\displaystyle x_{n+1}=Ax_{n}+Bu_{n}+w_{n}$$
(1)
$$\displaystyle y_{n}=Cx_{n}+z_{n}+v_{n}$$
for some measurement noise $z_{n}$, system disturbance $w_{n}$, and attack vector $v_{n}$. Suppose $(A,B)$ is stabilizable, $(A,C)$ is detectable. Typically, dynamic watermarking approaches will add an additive signal to the control input $u_{n}=Kx_{n}+e_{n}$, where $K$ is some feedback matrix and $e_{n}$ is our watermarking signal that is unknown to the attacker. Now let $k^{\prime}=\min\{k\geq 0\ |\ C(A+BK)^{k}B\neq 0\}$. If we define the test vectors
$$\psi^{\top}_{n}=\begin{bmatrix}(C\hat{x}_{n}-y_{n})^{\top}&e_{n-k^{\prime}-1}^%
{\top}\end{bmatrix},$$
(2)
and the following holds [25]:
$$\textstyle\operatorname{as-lim}_{N}\frac{1}{N}\sum_{n=0}^{N-1}\psi_{n}\psi_{n}%
^{\top}=\begin{bmatrix}C\Sigma_{\Delta}C^{\top}+\Sigma_{Z}&0\\
0&\Sigma_{E}\end{bmatrix}$$
(3)
for some specific matrices $\Sigma_{\Delta},\Sigma_{E},\Sigma_{Z}$, then all attack vectors $v_{n}$ following a particular model [25] are constrained in power
$$\textstyle\operatorname{as-lim}_{N}\frac{1}{N}\sum_{n=0}^{N-1}v_{n}^{\top}v_{n%
}=0.$$
(4)
Though these tests only provide asymptotic guarantees, that is enough to construct a statistical version of the test, similar to [22] where a hypothesis test is constructed by thresholding the negative log-likelihood. It follows that under a Gaussianity assumption for process and sensor noise, the matrix in (3) follows a well-behaved Wishart distribution. While that approach allows us to construct hypothesis tests using known distributions, the dependency of subsequent samples make finite sums display more complex behavior.
Then it is up to the designer of the watermark to specify a threshold that controls the false error rate. In this framework a rejection of the hypothesis test corresponds to detection of an attack, while an acceptance corresponds to the lack of detection of an attack. This notation emphasizes the fact that achieving a specified false error rate requires changing the threshold.
I-B Intelligent Transportation Systems and Observer Switching
Though the design and analysis of intelligent transportation systems (ITS) has drawn renewed interest [23, 26, 27, 28, 29, 30, 31], there has been less work on secure control of ITRS. One recent work considered the use of dynamic watermarking to detect sensor attacks in a network of autonomous vehicles coordinated by a supervisory controller[23], while [32] considered a platoon of vehicles where attacks happen not only on the sensors but also on the communication channel.
A particular feature of ITS is the possibility of redundancy in sensing. For instance, one can use a highly accurate satellite-based sensor (susceptible to external attack) and an on-board infrared sensor (not susceptible to external attack) in order to obtain spatial data. Then, one way of safeguarding a system susceptible to attacks is to switch from the high accuracy sensor to the on-board sensor when an attack is detected [33]. This approach naturally leads to systems with distributed observers with dynamic switching decision rules [34, 35]. In this scenario, it is crucial to design hypothesis tests that are able to detect attacks while having a decision rule that correctly selects which observer is to be used. Because control switching occurs at finite instances in time, the previous asymptotic results of dynamic watermarking cannot be used for this purpose. The reason, which is subtle, is that hypothesis tests based on characterization of asymptotic distributions will not have the correct theoretical properties in order to ensure proper control of the false alarm rate. Consequently, new finite sample hypothesis tests need to be constructed.
The first contribution of this paper it to provide finite-time guarantees on attack detection via dynamic watermarking, which to the best of our knowledge has not been done before. Namely, we provide statistical tests that provide finite-time guarantees on attack detection, instead of relying of asymptotic behavior of sums of random matrices.
We also relate the magnitude of an attack to our test power, by describing the inherent trade-off between the test capability of triggering true detection, and the magnitude of the attacks that are allowed to remain undetected in the long run. The finite sample analysis of dynamic watermarking requires the use of random matrix concentration inequalities, which are useful in analyzing the matrices involved in the evolution of LTI system dynamics. The second major contribution of this paper is to provide finite sample concentration-based tests, which allow us to detect attacks and allow switching decisions based on such tests to correctly report attack detections infinitely often. Namely, if there is no attack, we develop a finite sample test that falsely reports attacks only a finite number of times. This is a crucial feature because it also implies that in the long-run the switching rule based on such a test is correctly selecting which observer is active infinitely often.
I-C Outline
In Sect. II, we define our notation for this paper. We also present the random matrix concentration inequalities that we use to perform our finite sample analysis. Next, in Sect. III we present our general LTI framework with switching observers. Then, we apply the concentration inequalities to the LTI setting in Sect. IV in order to obtain appropriate concentration for the matrices involved. In Sect. V and Sect. VI, we present the finite sample consistency tests and a simple threshold that relates the attack magnitude to the power of our test. Next, in Sect. VI we provide some numerical results demonstrating our approach on an autonomous vehicle application.
II Preliminaries
In this section, we define all relevant notation concerning the random matrix analysis done throughout the paper. We also define the key concepts of Stein’s Method [36, 37] applied to matrices and the relevant matrix concentration inequalities that will be used. This method turns out to be key to our finite sample analysis of dynamic watermarking, as it involves analyzing sums of inter-temporal dependent matrices.
II-A Notation
We use the symbol $\left\lVert\cdot\right\rVert$ for the spectral norm of a matrix, which is the largest singular value of a general matrix. The space of $d\times d$ Hermitian real-valued matrices is denoted by $\mathcal{H}{{}^{d}}$. Moreover, the symbols $\lambda_{\max}(A),\lambda_{\min}(A)$ are respectively the maximum and the minimum eigenvalues of an Hermitian matrix $A\in\mathcal{H}^{d}$. The symbol $\preceq$ refers to the semidefinite partial order, namely $A\preceq B$ if and only if $B-A$ is positive semi-definite (p.s.d). For a matrix $A$, we let $(A)_{ij}$ denote the $(ij)$-th element of $A$. We let $\text{tr}(\cdot)$ do the denote the trace operator.
We also define a master probability space $(\Omega,\mathcal{F},P)$ and a filtration $\{\mathcal{F}_{k}\}$ contained in the master sigma algebra:
$$\mathcal{F}_{k}\subset\mathcal{F}_{k+1}\text{ and }\mathcal{F}_{k}\subset%
\mathcal{F},\forall k\geq 0.$$
(5)
Given such filtration we also define the conditional expectation $\mathbb{E}_{k}[\cdot]$. We also let $\epsilon$ denote a Radamacher random variable, that takes values in $\{-1,1\}$ with equal probability. The random matrix concentration inequalities involved in this work are derived based on the method of exchangeable pairs based on the Stein’s Method [36]. Let $Z$ and $Z^{\prime}$ be random vectors taking values in a space $\mathbb{R}^{d}$. We say that $(Z,Z^{\prime})$ is an exchangeable pair if it has the same distribution as $(Z^{\prime},Z)$. Next, we define a matrix Stein pair:
Definition 1.
Let $Z$ and $Z^{\prime}$ be an exchangeable pair of random vectors taking values in a space $\mathcal{Z}$, and let $\psi:\mathcal{Z}\rightarrow\mathcal{H}^{d}$ be a measurable function. Define the random Hermitian matrices
$$X=\psi(Z)\textit{ and }X^{\prime}=\psi(Z^{\prime}).$$
(6)
We say that $(X,X^{\prime})$ is a matrix Stein pair if there is a constant $\beta\in(0,1]$ for which $\mathbb{E}[X-X^{\prime}|Z]=\beta X\textit{ a.s.}$.
Note it follows from the above definition that $\mathbb{E}[X]=0$. Also, $\beta$ is called the scale factor of the pair $(X,X^{\prime})$.
Lastly, we present the concept of dilations, which are used to derive our results. A symmetric dilation of a real-valued rectangular matrix $B$ is
$$\mathcal{D}(B)=\begin{bmatrix}0&B\\
B^{\top}&0\end{bmatrix}$$
(7)
Note that $\mathcal{D}(B)$ is always symmetric, and it satisfies the following useful property:
$$\mathcal{D}(B)^{2}=\begin{bmatrix}0&BB^{\top}\\
B^{\top}B&0\end{bmatrix}$$
(8)
Moreover, observe that the norm of the symmetric dilation has a useful relationship with the norm of the original matrix $\lambda_{\max}(\mathcal{D}(B))=\|\mathcal{D}(B)\|=\|B\|$. We will construct bounds for symmetric matrices and then we will extend those bounds to non-symmetric matrices by using dilations.
II-B Matrix Concentration Inequalities
In order for us to develop finite sample tests we require matrix concentration inequalities. The random matrices involved in this paper are not independent in the general case. We first present a version of matrix Hoeffding inequality for conditionally independent sums of random matrices, that is random matrices that become independent after conditioning on another matrix. This theorem, and the following theorems about concentrations, were first introduced by [37], as generalizations of the (respective) independent cases.
Proposition 1.
[37] Consider a finite sequence $(Y_{k})_{(k\geq 1)}$ of random matrices in $\mathcal{H}^{d}$ that are conditionally independent given an auxiliary random matrix $Z$ and finite sequences $(P_{k})_{k\geq 1}$ and $(Q_{k})_{k\geq 1}$ of deterministic matrices in $\mathcal{H}^{d}$. Assume that
$$\mathbb{E}[Y_{k}|Z]=0\textit{, }Y^{2}_{k}\preceq P^{2}_{k}\textit{, }\mathbb{E%
}[Y^{2}_{k}|(Y_{j})_{j\neq k}]\preceq Q^{2}_{k}\textit{ a.s.}\forall k,$$
(9)
then for all $t\geq 0$ we have
$$\mathrm{P}\left(\lambda_{\max}\left(\sum_{k=0}Y_{k}\right)\geq t\right)\leq d%
\cdot e^{-t^{2}/2\sigma^{2}}$$
(10)
where $\sigma^{2}=\frac{1}{2}\|\sum_{k}P^{2}_{k}+Q^{2}_{k})\|$.
Next we present a version of the McDiarmid inequality for self-reproducing random matrices.
Proposition 2.
[37] Let $z=(Z_{1},...,Z_{n})$ be a random vector taking values in a space $\mathcal{Z}$, and, for each index k, let $Z^{\prime}_{k}$ and $Z_{k}$ be conditionally i.i.d. given $(Z_{j})_{j\neq k}$. Suppose that $H:\mathcal{Z}\rightarrow\mathcal{H}^{d}$ is a function that satisfies the self-reproducing property
$$\sum_{k=1}^{n}(H(z)-\mathbb{E}[H(z)|(Z_{j})_{j\neq k}])=s\cdot(H(z)-\mathbb{E}%
[H(z)])\text{ a.s. }$$
(11)
for a parameter $s>0$, as well as the bounded difference property
$$\mathbb{E}[(H(z)-H(Z_{1},...,Z_{k}^{{}^{\prime}},...,Z_{n}))^{2}|z]\preceq P^{%
2}_{k}$$
(12)
for each index k a.s., where $P_{k}$ is a deterministic matrix in $\mathcal{H}^{d}$. Then, for all $t\geq 0$,
$$\mathrm{P}\{\lambda_{\max}(H(z)-\mathbb{E}[H(z)])\geq t\}\leq d\cdot e^{-st^{2%
}/L}$$
(13)
for $L=\left\lVert\sum_{k=1}^{n}P^{2}_{k}\right\rVert$.
Now we provide an essential property that is called symmetrization, which is a generalization for summation of the symmetrization property presented in [37] for a single matrix:
Lemma 1.
Let $\{X_{i}\}_{i=1}^{n}$ be a sequence of random Hermitian matrices with $\mathbb{E}[X_{i}]=0$. Then
$$\mathbb{E}\left[\textup{tr}\left(\textup{e}^{\sum_{i=1}^{n}X_{i}}\right)\right%
]\leq\mathbb{E}\left[\textup{tr}\left(\textup{e}^{2\sum_{i=1}^{n}\epsilon_{i}X%
_{i}}\right)\right]$$
(14)
where $\{\epsilon_{i}\}_{i=1}^{n}$ are i.i.d. Radamacher random variables.
Proof.
First, we construct a sequence of copies $\{X^{\prime}_{i}\}_{i=1}^{n}$ independent from $\{X_{i}\}_{i=1}^{n}$, and let $\mathbb{E}^{\prime}$ denote the expectation with respect to $\{X^{\prime}_{i}\}_{i=1}^{n}$. So we have
$$\displaystyle\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i=1}^{n}X_{i}}%
\right)\right]=\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i=1}^{n}X_{i}-%
\mathbb{E}^{\prime}[X^{\prime}_{i}]}\right)\right]\leq\\
\displaystyle\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i=1}^{n}X_{i}-X^%
{\prime}_{i}}\right)\right]=\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i%
=1}^{n}\epsilon_{i}(X_{i}-X^{\prime}_{i})}\right)\right]$$
(15)
where we have sequentially used Jensen’s inequality and then the symmetry of $(X_{i}-X^{\prime}_{i})$. Now we finish the proof by noting
$$\displaystyle\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i=1}^{n}X_{i}}%
\right)\right]\leq\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i=1}^{n}%
\epsilon_{i}(X_{i}-X^{\prime}_{i})}\right)\right]\leq\\
\displaystyle\mathbb{E}\left[\text{tr}\left(\text{e}^{\sum_{i=1}^{n}\epsilon_{%
i}X_{i}}\text{e}^{-\sum_{i=1}^{n}\epsilon_{i}X^{\prime}_{i}}\right)\right]\leq%
\\
\displaystyle\mathbb{E}\left[\text{tr}\left(\text{e}^{2\sum_{i=1}^{n}\epsilon_%
{i}X_{i}}\right)^{1/2}\text{tr}\left(\text{e}^{-2\sum_{i=1}^{n}\epsilon_{i}X^{%
\prime}_{i}}\right)^{1/2}\right]=\\
\displaystyle\mathbb{E}\left[\text{tr}\left(\textup{e}^{2\sum_{i=1}^{n}%
\epsilon_{i}X_{i}}\right)\right]$$
(16)
where we have sequentially used the Golden-Thompson inequality, the Cauchy-Schwartz inequality two times, and the fact that both factors are identically distributed. (See [38] for the definition of those properties.)
∎
III LTI System with Switching
We consider a MIMO LTI system that allows the controller to switch between two sets of sensors, and we will assume that both the measurement and process noise have stochastic distributions with a bounded support. Namely, we will assume that the noise vectors have bounded norm almost surely.
III-A LTI Formulation
Consider a MIMO LTI system with partial observations and switching in the sensing
$$\displaystyle x_{n+1}$$
$$\displaystyle=Ax_{n}+Bu_{n}+w_{n}$$
(17)
$$\displaystyle y_{n}$$
$$\displaystyle=C(\alpha_{n})x_{n}+z_{n}(\alpha_{n})+\alpha_{n}v_{n}$$
where $x\in\mathbb{R}^{p}$, $u\in\mathbb{R}^{q}$, $y,z,v\in\mathbb{R}^{m}$, and $\alpha_{n}\in\{0,1\}$. The $w_{n}$ represents zero mean i.i.d. process noise with covariance $\Sigma_{W}$. Moreover, we have
$$\displaystyle C_{n}=C(\alpha_{n})=\alpha_{n}C_{1}+(1-\alpha_{n})C_{2}$$
(18)
$$\displaystyle z_{n}(\alpha_{n})=\alpha_{n}\zeta_{n}+(1-\alpha_{n})\eta_{n}$$
where $\zeta_{n}$ and $\eta_{n}$ represent zero mean i.i.d. measurement noise with covariance matrices $\Sigma_{\zeta}\preceq\Sigma_{\eta}$, respectively. Note that $\alpha_{n}\in\{0,1\}$ should be interpreted as the switching control action that selects between the observability matrices $C_{1}$ or $C_{2}$. The $v_{n}$ is as an additive measurement disturbance added by an attacker, which can only affect the observations made when the mode $\alpha=1$ is selected. The idea of this model is that $C_{1}$ corresponds to a more accurate set of sensors than $C_{2}$, but conversely that some subset of sensors within $C_{1}$ are susceptible to an attack whereas the set of all sensors within $C_{2}$ are not susceptible to an attack. We further assume the process noise is independent of the measurement noise, that is $w_{n}$ for $n\geq 0$ is independent of $\zeta_{n},\eta_{n}$ for $n\geq 0$. Lastly we assume both measurement and disturbance noises are bounded in magnitude. Namely, we assume that both measurement noise and systems disturbances are given by i.i.d. bounded random vectors: $\left\lVert w_{k}\right\rVert\leq K_{w}$ and $\left\lVert z_{k}\right\rVert\leq K_{z},\forall k\geq 0$.
If $(A,B)$ is stabilizable and both $(A,C_{1})$ and $(A,C_{2})$ are detectable, then an output-feedback controller can be designed when $v_{n}\equiv 0$ using an observer and the separation principle. Let $K$ be a constant state-feedback gain matrix such that $A+BK$ is Schur stable, and let $L_{i}$ be a constant observer gain matrix such that $A+L_{i}C_{i}$ is Schur stable for $i\in\{1,2\}$. The idea of dynamic watermarking in this context will be to superimpose a private (and random) excitation signal $e_{n}$ known in value to the controller but unknown in value to the attacker. As a result, we will apply the control input $u_{n}=Kx^{\prime}_{n}+e_{n}$, where $x^{\prime}_{n}$ is the observer-estimated state and $e_{n}$ are i.i.d. random vectors on a bounded support, such that $\left\lVert e_{k}\right\rVert\leq K_{e},\forall k\geq 0$, with zero mean and constant variance $\Sigma_{E}$ fixed by the controller. Let
$$\displaystyle L(\alpha)=\alpha L_{1}+(1-\alpha)L_{2}$$
(19)
$$\displaystyle L_{n}=L(\alpha_{n})$$
$$\displaystyle\underline{L(\alpha)}^{\top}=\begin{bmatrix}0&-L(\alpha)^{\top}%
\end{bmatrix}$$
Moreover, let $\tilde{x}^{\top}=\begin{bmatrix}x^{\top}&x^{\prime\top}\end{bmatrix}$, and define:
$$\displaystyle\underline{B}^{\top}$$
$$\displaystyle=\begin{bmatrix}B^{\top}&B^{\top}\end{bmatrix}\text{, }\underline%
{D}^{\top}=\begin{bmatrix}\mathbb{I}&0\end{bmatrix}\text{, and}$$
(20)
$$\displaystyle\underline{A}(\alpha)$$
$$\displaystyle=\begin{bmatrix}A&BK\\
-L(\alpha)C(\alpha)&A+BK+L(\alpha)C(\alpha)\end{bmatrix}.$$
Then the closed-loop system with private excitation is given by:
$$\tilde{x}_{n+1}=\underline{A}(\alpha_{n})\tilde{x}_{n}+\underline{B}e_{n}+%
\underline{D}w_{n}+\underline{L}(\alpha_{n})(z_{n}(\alpha_{n})+\alpha_{n}v_{n}).$$
(21)
If we define the observation error $\delta^{\prime}=x^{\prime}-x$, then with the change of variables $\check{x}^{\top}=\begin{bmatrix}x^{\top}&\delta^{\prime\top}\end{bmatrix}$ we have the dynamics
$$\check{x}_{n+1}=\underline{\underline{A}}(\alpha_{n})\check{x}_{n}+\underline{%
\underline{B}}e_{n}+\underline{\underline{D}}w_{n}+\underline{\underline{L}}(%
\alpha_{n})(z_{n}(\alpha)+\alpha v_{n})$$
(22)
where we further define the following matrices
$$\displaystyle\underline{\underline{B}}^{\top}=\begin{bmatrix}B^{\top}&0\end{%
bmatrix}\text{, }\underline{\underline{D}}^{\top}=\begin{bmatrix}\mathbb{I}&-%
\mathbb{I}\end{bmatrix}\text{, }\underline{\underline{L}}(\alpha)=\underline{L%
}(\alpha),$$
(23)
$$\displaystyle\text{and }\underline{\underline{A}}(\alpha)=\begin{bmatrix}A+BK&%
BK\\
0&A+L(\alpha)C(\alpha)\end{bmatrix}.$$
Recall that $\underline{\underline{A}}(\alpha)$ is Schur stable whenever $A+BK$ and $A+L(\alpha)C(\alpha)$ are both Schur stable.
There is one technical point that needs to be addressed before proceeding: Since there is switching between observers, the closed-loop system will not necessarily be stable even though $A+BK$ and $A+L(\alpha)C(\alpha)$ are both Schur stable. One approach to resolving this issue is limiting the rate of switching, as follows:
Proposition 3.
let $P$ be the solution of the Lyapunov equation
$$\underline{\underline{A}}(1)P\underline{\underline{A}}(1)^{\top}-P=-\mathbb{I},$$
(24)
where $\mathbb{I}$ is the identity matrix. Let $\tau$ be the smallest positive integer such that
$$\underline{\underline{A}}(0)^{\tau}P(\underline{\underline{A}}(0)^{\tau})^{%
\top}-P\leq-\mathbb{I}.$$
(25)
Then the the closed-loop system is stable under switching policies where: whenever we switch from $\alpha=1$ to $\alpha=0$ we maintain $\alpha=0$ for at least $\tau$ time steps before any possible switching occurs to $\alpha=1$.
Lastly, we note that such a $\tau$ exists because $\underline{\underline{A}}(0)$ is Schur stable.
IV Matrix Inequalities for General LTI Systems
We will now apply the abstract concentration inequalities presented in Sect. II to our LTI setting with switching. We will begin our analysis consider that the system is under no attack. Under no attack we would like to keep using the most accurate sensor – that is keeping our switching control $\alpha_{n}\equiv 1$ for all $n\geq 0$. However, as it is usually observed for any kind of tests based on random quantities, we are susceptible to commit what is commonly known as false positive or type I errors. Hence our goal is to provide finite sample tests based on matrix concentration of measure such that type I errors happen only a finite number of times throughout the evolution of the system. This would imply that those tests report correctly that there is no attack infinitely often. To that end, we will utilize two observers: The first observer obtain system measurements from the switched system, using $C(\alpha_{n})$; The second observer never switches and keeps measuring the system using the vulnerable sensor, using $C_{1}$. The finite-time statistical tests and the concentration inequalities analysis presented in this section are referring to quantities associated with the second observer. For ease of notation and presentation we drop the subscript of the analysis define $C=C_{1}$ and $L=L_{1}$. Moreover, for the second observer we define: $\hat{x}_{n}$ and $\delta_{n}$ to denote the estimate state and observation error:
$$\hat{x}_{n+1}=(A+BK)\hat{x}_{n}+LC(\hat{x}_{n}-x_{n})+Be_{n}-Lz_{n}$$
(26)
and $\delta_{n}=\hat{x}_{n}-x_{n}$. Then by the same type of variable substitution:
$$\delta_{n+1}=(A+LC)\delta_{n}-w_{n}+-Lz_{n}$$
(27)
We will start by bounding the vector $C\delta_{n}-z_{n}$:
Theorem 1.
Let $\delta_{n}=\hat{x}_{n}-x_{n}$. Assume that both measurement noise and systems disturbances are given by i.i.d. bounded random vectors: $\left\lVert w_{k}\right\rVert\leq K_{w}\text{ and }\left\lVert z_{k}\right%
\rVert\leq K_{z},\forall k\geq 0$. Then when $v_{n}\equiv 0$ for all $n\geq 0$ we have
$$\|C\delta_{n}-z_{n}\|\leq\bar{K}_{n}$$
(28)
where $\bar{K}_{n}=K_{z}+\sum_{k=0}^{n-1}\left\lVert C\bar{D}_{k}\right\rVert K_{w}+%
\left\lVert C\bar{L}_{k}\right\rVert K_{z}$
and
$$(C\delta_{n}-z_{n})(C\delta_{n}-z_{n})^{\top}\preceq\bar{K}^{2}_{n}\mathbb{I}.$$
(29)
Moreover, it follows that
$$\displaystyle\mathbb{E}[(C\delta_{n}$$
$$\displaystyle-z_{n})(C\delta_{n}-z_{n})^{\top}]=$$
(30)
$$\displaystyle C\left(\sum_{k=0}^{n-1}\bar{D}_{k}\Sigma_{w}\bar{D}^{\top}_{k}+%
\bar{L}_{k}\Sigma_{z}\bar{L}^{\top}_{k}\right)C^{\top}+\Sigma_{z}$$
where
$$\displaystyle\bar{D}_{k}$$
$$\displaystyle=-(A+LC)^{n-1-k}$$
(31)
$$\displaystyle\bar{L}_{k}$$
$$\displaystyle=-(A+LC)^{n-1-k}L^{\top}.$$
Proof.
Recall our definition of $\delta_{n}$ (Eq. 27) we can write:
$$\delta_{n}=(A+LC)^{n}\delta_{0}-\sum_{k=0}^{n-1}(A+LC)^{n-1-k}(\mathbb{I}w_{k}%
+L^{\top}z_{k}).$$
(32)
Assuming $\delta_{0}=0$, we have that
$$\delta_{n}=\sum_{k=0}^{n-1}\bar{D}_{k}w_{k}+\bar{L}_{k}z_{k}.$$
(33)
Now, we can define the following:
$$\displaystyle C\delta_{n}\delta_{n}^{\top}C^{\top}=$$
(34)
$$\displaystyle C\left(\sum_{k=0}^{n-1}\bar{D}_{k}w_{k}+\bar{L}_{k}z_{k}\right)%
\left(\sum_{k=0}^{n-1}\bar{D}_{k}w_{k}+\bar{L}_{k}z_{k}\right)^{\top}C^{\top},$$
and obtain the expectation directly:
$$\displaystyle\mathbb{E}[(C\delta_{n}$$
$$\displaystyle-z_{n})(C\delta_{n}-z_{n})^{\top}]=$$
(35)
$$\displaystyle C\left(\sum_{k=0}^{n-1}\bar{D}_{k}\Sigma_{w}\bar{D}^{\top}_{k}+%
\bar{L}_{k}\Sigma_{z}\bar{L}^{\top}_{k}\right)C^{\top}+\Sigma_{z}.$$
Since $z_{n}$ and $\delta_{n}$ are independent for all $n$. Moreover, both system disturbances and measurement noise are independent. Under our key assumption that both measurement noise and systems disturbances are given by i.i.d. bounded random vectors we have that
$$\|\delta_{n}\|\leq\sum_{k=0}^{n-1}\left\lVert\bar{D}_{k}\right\rVert K_{w}+%
\left\lVert\bar{L}_{k}\right\rVert K_{z},$$
(36)
and that
$$\displaystyle\|C\delta_{n}-z_{n}\|\leq\\
\displaystyle K_{z}+\sum_{k=0}^{n-1}\left\lVert C\bar{D}_{k}\right\rVert K_{w}%
+\left\lVert C\bar{L}_{k}\right\rVert K_{z}=\bar{K}_{n}.$$
(37)
So we have $(C\delta_{n}-z_{n})(C\delta_{n}-z_{n})^{\top}\preceq\bar{K}^{2}_{n}\mathbb{I}.$
∎
Now consider the matrix (3) that was used in the introduction to define the asymptotic tests. But now, instead of letting $n$ go to infinity, we keep it finite and then analyze the finite summation of matrices. Let $k^{\prime}=\min\{k\geq 0\ |\ C(A+BK)^{k}B\neq 0\}$. The existence of such $k^{\prime}$ is guaranteed (see [25]). Moreover, define
$$\psi^{\top}_{n}=\begin{bmatrix}(C\hat{x}_{n}-y_{n})^{\top}&e^{\top}_{n-k^{%
\prime}-1}\end{bmatrix}.$$
(38)
Then we have:
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\psi_{n}\psi_{n}^{\top}=$$
$$\displaystyle\frac{1}{N}\scalebox{0.9}{$\begin{bmatrix}\sum_{n=0}^{N-1}(C\hat{%
x}_{n}-y_{n})(C\hat{x}_{n}-y_{n})^{\top}&\sum_{n=0}^{N-1}(C\hat{x}_{n}-y_{n})e%
^{\top}_{n-k^{\prime}-1}\\
\sum_{n=0}^{N-1}e_{n-k^{\prime}-1}(C\hat{x}_{n}-y_{n})^{\top}&\sum_{n=0}^{N-1}%
e_{n-k^{\prime}-1}e^{\top}_{n-k^{\prime}-1}\end{bmatrix}$}$$
(39)
It suits our purposes to make sure that the above matrix is centered (that is have zero expected value). In order to achieve this, we construct the matrix
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\Psi_{n}$$
$$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}\psi_{n}\psi_{n}^{\top}-$$
(40)
$$\displaystyle\frac{1}{N}\scalebox{0.9}{$\begin{bmatrix}\sum_{n=0}^{N-1}\mathbb%
{E}[(C\delta_{n}-z_{n})(C\delta_{n}-z_{n})^{\top}]&0\\
0&N\Sigma_{e}\end{bmatrix}$}$$
Note that it follows that: $\mathbb{E}[\Psi_{n}]=0,\forall n\geq 0$, since $C\hat{x}_{n}-y_{n}=C\delta_{n}-z_{n}$. We wish to control the singular values of the above matrix. We will do so by analyzing each individual block. To ease the notation we define
$$\Phi_{N}=\frac{1}{N}\sum_{n=0}^{N-1}\Psi_{n}$$
(41)
and we define each submatrix
$$\displaystyle\Phi^{(1)}_{N}$$
$$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}(C\hat{x}_{n}-y_{n})(C\hat{x}_{n}-y_{%
n})^{\top}-$$
$$\displaystyle\quad\quad\quad\frac{1}{N}\sum_{n=0}^{N-1}\mathbb{E}[(C\hat{x}_{n%
}-y_{n})(C\hat{x}_{n}-y_{n})^{\top}]$$
(42)
$$\displaystyle\Phi^{(2)}_{N}$$
$$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}(C\hat{x}_{n}-y_{n})e^{\top}_{n-k^{%
\prime}-1}$$
(43)
$$\displaystyle\Phi^{(3)}_{N}$$
$$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}(e_{n-k^{\prime}-1}e^{\top}_{n-k^{%
\prime}-1}-\Sigma_{e})$$
(44)
such that
$$\Phi_{N}=\begin{bmatrix}\Phi^{(1)}_{N}&\Phi^{(2)}_{N}\\
(\Phi^{(2)}_{N})^{\top}&\Phi^{(3)}_{N}\end{bmatrix}.$$
(45)
Our next step is to bound the norm of $\Phi^{(1)}_{N}$.
Theorem 2.
If $v_{n}\equiv 0$ for all $n\geq 0$, then the following concentration inequality holds for all $N\geq 1$ and all $t$:
$$\mathrm{P}\bigg{(}\left\lVert\Phi^{(1)}_{N}\right\rVert\geq t\Bigg{)}\leq m%
\cdot e^{-N^{2}t^{2}/c^{(1)}_{N}}$$
(46)
where $c^{(1)}_{N}=8\left\lVert\sum_{k=0}^{N-1}\left(\bar{K}^{4}_{k}\mathbb{I}\right)\right\rVert$.
Proof.
We start by defining the matrix $Y_{n}$ as
$$Y_{n}=(C\delta_{n}-z_{n})(C\delta_{n}-z_{n})^{\top}-\mathbb{E}[(C\delta_{n}-z_%
{n})(C\delta_{n}-z_{n})^{\top}].$$
(47)
Now define a vector of independent i.i.d. Radamacher random variables $\{\epsilon_{n}\}_{n=0}^{N-1}$. We use the symmetrization property to write
$$\left\lVert\Phi^{(1)}_{N}\right\rVert\leq\left\lVert\frac{1}{N}\sum_{n=0}^{N-1%
}Y_{n}\right\rVert\leq\left\lVert\frac{2}{N}\sum_{n=0}^{N-1}\epsilon_{n}Y_{n}%
\right\rVert.$$
(48)
Now we define a filtration $Z=(Y_{n})_{n\geq 1}$ where $W_{n}=\epsilon_{n}Y_{n},n\geq 1$. Then we see that each summand $W_{n}$ is conditionally independent given $Z$, because the Radamacher random variables are all i.i.d. This allows us to use the Hoeffding Bound for conditionally independent sums to obtain
$$\displaystyle\mathrm{P}\left(\left\lVert\frac{1}{N}\sum_{n=0}^{N-1}Y_{n}\right%
\rVert\geq t\right)\leq\\
\displaystyle\mathrm{P}\left(\left\lVert\frac{2}{N}\sum_{n=0}^{N-1}\epsilon_{n%
}Y_{n}\right\rVert\geq t\right)\leq d\cdot e^{-N^{2}t^{2}/8\sigma^{2}}$$
(49)
for $\sigma^{2}=\left\lVert\sum_{k=0}^{N-1}\left(\bar{K}^{4}_{k}\mathbb{I}\right)\right\rVert$.
The first inequality follows from applying the Laplace transform method and using the property
$$\mathbb{E}\left[\text{tr}\left(\textup{e}^{\sum_{i=1}^{n}X_{i}}\right)\right]%
\leq\mathbb{E}\left[\text{tr}\left(\textup{e}^{2\sum_{i=1}^{n}\epsilon_{i}X_{i%
}}\right)\right].$$
(50)
We also used the fact $W^{2}_{k}\preceq\bar{K}^{4}_{k}\mathbb{I}$ for all $k$, and the fact that
$$\frac{1}{2}\left\lVert\sum_{k}\bar{K}^{4}_{k}\mathbb{I}+E[W^{2}_{k}|(W_{j})_{j%
\neq k}]\right\rVert\leq\left\lVert\sum_{k}\left(\bar{K}^{4}_{k}\mathbb{I}%
\right)\right\rVert$$
(51)
since $\mathbb{E}[W^{2}_{k}|(W_{j})_{j\neq k}]=\mathbb{E}[Y^{2}_{k}|(W_{j})_{j\neq k}%
]\preceq\bar{K}^{4}_{k}\mathbb{I}$.
∎
Next, we provide a bound on the norm of $\Phi^{(2)}_{N}$. But before that we need the following proposition:
Proposition 4.
Let $e=(e_{1},...,e_{k},...,e_{n})$ be a sequence of random vectors taking values in a space $\mathcal{Z}$. Now construct an exchangeable pair $e^{\prime}=(e_{1},...,e^{\prime}_{k},...,e_{n})$ where $e_{k}$ and $e^{\prime}_{k}$ are conditionally i.i.d. given $(e_{j})_{j\neq k}$ and $k$ is an independent coordinate drawn uniformly from $\{1,...,n\}$. We define
$$H(e)=\begin{bmatrix}0&\sum_{n=0}^{N-1}(d_{n})e^{\top}_{n-k^{\prime}-1}\\
\sum_{n=0}^{N-1}e_{n-k^{\prime}-1}(d_{n})^{\top}&0\end{bmatrix}$$
(52)
where $d_{n}=(C\delta_{n}-z_{n})$. If $v_{n}\equiv 0$ for all $n\geq 0$, then the function $H(e)$ satisfies the bounded differences property
$$\displaystyle\mathbb{E}[(H(e)-H(e^{\prime}))^{2}|e]\preceq\bar{P}^{2}_{n}$$
(53)
for $\bar{P}^{2}_{n}=\max\{P^{2}_{n},P^{{}^{\prime}2}_{n}\}\mathbb{I}$ with positive constants $P^{2}_{n},P^{{}^{\prime}2}_{n}$:
$$P^{{}^{\prime}2}_{n}=\left\lVert\mathbb{E}[Q^{\prime}_{n}|e]\right\rVert\leq%
\bar{K}^{2}_{n}(K^{2}_{e}+\left\lVert\Sigma_{E}\right\rVert)$$
(54)
$$\displaystyle P^{2}_{n}=\left(K^{2}_{e}+\textup{tr}(\Sigma_{E}\right)\times$$
(55)
$$\displaystyle\left\lVert C\left(\sum_{k=1}^{n-1}\bar{D}_{k}\Sigma_{w}\bar{D}^{%
\top}_{k}+\bar{L}_{k}\Sigma_{z}\bar{L}^{\top}_{k}\right)C^{\top}+\Sigma_{z}\right\rVert$$
(56)
Proof.
Let $q_{n}=d_{n}e^{\top}_{n-k^{\prime}-1}-d_{n}e^{{}^{\prime}\top}_{n-k^{\prime}-1}$ and observe that
$$\displaystyle\mathbb{E}[(H(e)-H(e^{\prime}))^{2}|e]=\mathbb{E}\bigg{[}\begin{%
bmatrix}0&q_{n}\\
q_{n}^{\top}&0\end{bmatrix}^{2}|e\bigg{]}=\\
\displaystyle\mathbb{E}\bigg{[}\begin{bmatrix}Q_{n}&0\\
0&Q^{\prime}_{n}\end{bmatrix}|e\bigg{]}$$
(57)
where we have defined
$$\displaystyle Q_{n}=d_{n}e^{\top}_{n-k^{\prime}-1}e_{n-k^{\prime}-1}d^{\top}_{%
n}+d_{n}e^{{}^{\prime}\top}_{n-k^{\prime}-1}e^{{}^{\prime}}_{n-k^{\prime}-1}d^%
{\top}_{n}$$
(58)
$$\displaystyle Q^{\prime}_{n}=e_{n-k^{\prime}-1}d^{\top}_{n}d_{n}e^{\top}_{n-k^%
{\prime}-1}+e^{{}^{\prime}}_{n-k^{\prime}-1}d^{\top}_{n}d_{n}e^{{}^{\prime}%
\top}_{n-k^{\prime}-1}$$
Now we have
$$\displaystyle\mathbb{E}[Q_{n}|e]=\\
\displaystyle\mathbb{E}[d_{n}e^{\top}_{n-k^{\prime}-1}e_{n-k^{\prime}-1}d^{%
\top}_{n}+d_{n}e^{{}^{\prime}\top}_{n-k^{\prime}-1}e^{{}^{\prime}}_{n-k^{%
\prime}-1}d^{\top}_{n}|e]=\\
\displaystyle(e^{\top}_{n-k^{\prime}-1}e_{n-k^{\prime}-1})\mathbb{E}[d_{n}d^{%
\top}_{n}|e]+\\
\displaystyle\mathbb{E}[e^{{}^{\prime}\top}_{n-k^{\prime}-1}e^{{}^{\prime}}_{n%
-k^{\prime}-1}|e]\mathbb{E}[d_{n}d^{\top}_{n}|e]$$
(59)
Recalling that $\left\lVert e_{k}\right\rVert\leq K_{e}~{}\forall k\geq 0$ and (35), it follows that
$$\displaystyle\left\lVert\mathbb{E}[Q_{n}|e]\right\rVert\leq\left(K^{2}_{e}+%
\text{tr}(\Sigma_{E}\right)\times\\
\displaystyle\left\lVert C\left(\sum_{k=1}^{n-1}\bar{D}_{k}\Sigma_{w}\bar{D}^{%
\top}_{k}+\bar{L}_{k}\Sigma_{z}\bar{L}^{\top}_{k}\right)C^{\top}+\Sigma_{z}%
\right\rVert=P^{2}_{n}.$$
(60)
Moreover, it follows that
$$\displaystyle\left\lVert\mathbb{E}[Q^{\prime}_{n}|e]\right\rVert=\\
\displaystyle e_{n-k^{\prime}-1}d^{\top}_{n}d_{n}e^{\top}_{n-k^{\prime}-1}+e^{%
{}^{\prime}}_{n-k^{\prime}-1}d^{\top}_{n}d_{n}e^{{}^{\prime}\top}_{n-k^{\prime%
}-1}\\
\displaystyle=(\mathbb{E}[(d^{\top}_{n}d_{n})|e])e_{n-k^{\prime}-1}e^{\top}_{n%
-k^{\prime}-1}+\\
\displaystyle(\mathbb{E}[d^{\top}_{n}d_{n}|e])\mathbb{E}[e^{{}^{\prime}}_{n-k^%
{\prime}-1}e^{{}^{\prime}\top}_{n-k^{\prime}-1}|e]$$
(61)
So we get
$$\left\lVert\mathbb{E}[Q^{\prime}_{n}|e]\right\rVert\leq\bar{K}^{2}_{n}(K^{2}_{%
e}+\left\lVert\Sigma_{E}\right\rVert)=P^{{}^{\prime}2}_{n}$$
(62)
Hence it follows that
$$\displaystyle\left\lVert\mathbb{E}[(H(e)-H(e^{\prime}))^{2}|e]\right\rVert\leq%
\max\{P^{2}_{n},P^{{}^{\prime}2}_{n}\}$$
(63)
So it follows that
$$\displaystyle\mathbb{E}[(H(e)-H(e^{\prime}))^{2}|e]\preceq\bar{P}^{2}_{n}$$
(64)
where $\bar{P}^{2}_{n}=\max\{P^{2}_{n},P^{{}^{\prime}2}_{n}\}\mathbb{I}$.
∎
Now we are ready to provide our theorem.
Theorem 3.
If $v_{n}\equiv 0$ for all $n\geq 0$, then the following concentration inequality holds for all $N\geq 1$ and all $t$:
$$\mathrm{P}\bigg{(}\left\lVert\Phi^{(2)}_{N}\right\rVert\geq t\Bigg{)}\leq(m+p)%
\cdot e^{-N^{2}t^{2}/c^{(2)}_{N}}$$
(65)
where $c^{(2)}_{N}=\left\lVert\sum_{k=0}^{N-1}\bar{P}^{2}_{k}\right\rVert$ for $\bar{P}^{2}_{k}=\max\{P^{2}_{k},P^{{}^{\prime}2}_{k}\}\mathbb{I}$, where
$$\displaystyle P^{2}_{k}$$
$$\displaystyle=\left(K^{2}_{e}+\textup{tr}(\Sigma_{E}\right)\times$$
(66)
$$\displaystyle\quad\quad\left\lVert C\left(\sum_{k=1}^{n-1}\bar{D}_{k}\Sigma_{w%
}\bar{D}^{\top}_{k}+\bar{L}_{k}\Sigma_{z}\bar{L}^{\top}_{k}\right)C^{\top}+%
\Sigma_{z}\right\rVert$$
$$\displaystyle P^{\prime 2}_{k}$$
$$\displaystyle=\bar{K}^{2}_{n}(K^{2}_{e}+\left\lVert\Sigma_{E}\right\rVert).$$
Proof.
We wish to provide bounds on the operator norm of
$$\Phi^{(2)}_{N}=\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_{n}-z_{n})e^{\top}_{n-k^{%
\prime}-1}$$
(67)
To achieve that, we will use the concept of matrix Stein pairs as defined previously. Let $E=(e_{1},...,e_{k},...,e_{n})$ be a sequence of random vectors taking values in a space $\mathcal{Z}$. Now construct an exchangeable pair $E^{\prime}=(e_{1},...,e^{\prime}_{k},...,e_{n})$ where $e_{k}$ and $e^{\prime}_{k}$ are conditionally i.i.d. given $(e_{j})_{j\neq k}$ and $k$ is an independent coordinate drawn uniformly from $\{1,...,n\}$. We define $H(e)$ as in Proposition 4:
$$H(e)=\begin{bmatrix}0&\sum_{n=0}^{N-1}b_{n}e^{\top}_{n-k^{\prime}-1}\\
\sum_{n=0}^{N-1}e_{n-k^{\prime}-1}b_{n}^{\top}&0\end{bmatrix}$$
(68)
where $b_{n}=C\delta_{n}-z_{n}$. Since $\mathbb{E}(H(e))=0$, this means $H(e)$ satisfies the self-reproducing property
$$\sum_{n=1}^{N}H(e)-\mathbb{E}[H(e)|(e_{j})_{j\neq(n-k^{\prime}-1)}]=H(e)$$
(69)
for the choice of parameter $s=1$ (see (11) for the definition of $s$), since for all $n\in\{1,...,N\}$ we have
$$\displaystyle H(e)-\mathbb{E}[H(e)|(e_{j})_{j\neq(n-k^{\prime}-1)}]=\\
\displaystyle\begin{bmatrix}0&(C\delta_{n}-z_{n})e^{\top}_{n-k^{\prime}-1}\\
e_{n-k^{\prime}-1}(C\delta_{n}-z_{n})^{\top}&0\end{bmatrix}$$
(70)
Next, we use Proposition 4 to state that $H(e)$ also satisfies the bounded differences property. So we have
$$\mathbb{E}[(H(e)-H(e^{\prime})^{2}|e]\preceq\bar{P}^{2}_{n}$$
(71)
for $\bar{P}^{2}_{n}=\max\{P^{2}_{n},P^{{}^{\prime}2}_{n}\}\mathbb{I}$. Hence, we apply the McDiarmid inequality to the dilation $H(e)\in\mathcal{H}^{m+p}$ to obtain
$$\displaystyle\mathrm{P}\Bigg{(}\left\lVert\frac{1}{N}H(e)\right\rVert\geq t%
\Bigg{)}=\\
\displaystyle\mathrm{P}\Bigg{(}\left\lVert\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_%
{n}-z_{n})e^{\top}_{n-k^{\prime}-1}\right\rVert\geq t\Bigg{)}\leq\\
\displaystyle(m+p)\cdot e^{-N^{2}t^{2}/L}$$
(72)
for $L=\left\lVert\sum_{k=0}^{N-1}\bar{P}^{2}_{k}\right\rVert$.
∎
Now we focus on bounding the last submatrix $\Phi^{(3)}_{N}$.
Theorem 4.
The following concentration inequality holds for all $N\geq 1$ and all $t$:
$$\mathrm{P}\bigg{(}\left\lVert\Phi^{(3)}_{N}\right\rVert\geq t\Bigg{)}\leq 2q%
\cdot e^{-N^{2}t^{2}/c^{(3)}_{N}}$$
(73)
where $c^{(3)}_{N}=\left\lVert\sum_{k=0}^{N-1}(\bar{K}^{2}_{e}\mathbb{I}-\Sigma_{e})^%
{2}+\mathbb{E}[(e_{n}e^{\top}_{n})^{4}]-\Sigma^{2}_{e}\right\rVert$.
Proof.
We wish to provide a bound on the norm of
$$\Phi^{(3)}_{N}=\frac{1}{N}\sum_{n=0}^{N-1}(e_{n-k^{\prime}-1}e^{\top}_{n-k^{%
\prime}-1}-\Sigma_{e})$$
(74)
Define $\bar{E}_{n}=e_{n-k^{\prime}-1}e^{\top}_{n-k^{\prime}-1}-\Sigma_{e}$. We apply the Hoeffding bound for the independent sum to obtain
$$\mathrm{P}\left(\left\lVert\frac{1}{N}\sum_{n=0}^{N-1}\bar{E}_{n}\right\rVert%
\geq t\right)\leq d\cdot e^{-N^{2}t^{2}/2\sigma^{2}}$$
(75)
for $\sigma^{2}=\frac{1}{2}\left\lVert\sum_{k=0}^{N-1}\left(\bar{K}^{2}_{e}\mathbb{%
I}-\Sigma_{e}\right)^{2}+\mathbb{E}[(e_{n}e^{\top}_{n})^{4}]-\Sigma^{2}_{e}\right\rVert$, since
$$\bar{E}_{n}^{2}\preceq\sum_{k=0}^{N-1}\left(\bar{K}^{2}_{e}\mathbb{I}-\Sigma_{%
e}\right)^{2}$$
(76)
and by the definition of expectation we have that $\mathbb{E}\left[\bar{E}_{n}^{2}\right]=\mathbb{E}[(e_{n-k^{\prime}-1}e^{\top}_%
{n-k^{\prime}-1})^{4}]-\Sigma^{2}_{e}$.
∎
V Finite Sample Tests for General LTI Systems
In this section, we provide our finite sample tests based on dynamic watermarking for general LTI Systems with switching. In the previous section, we obtained concentration inequalities for each of the submatrices of $\Phi_{N}$ (45). Note $\Phi_{3}$ is the private excitation matrix we get to design, and so it is in our power to choose the dynamic watermark to display a desired concentration behavior.
We are now ready to state the main theorem of this work, which basically characterizes the behavior of a switching rule based on the finite-time concentration inequalities. Our switching rule is constructed by thresholding the block submatrices $\Phi^{(1)}_{N}$ and $\Phi^{(2)}_{N}$ using the measurements of the second observer ((26) and (27)) and applying the switch on the first observer once those thresholds are violated, and the we switch back on violations disappear. Let $S$ be a positive constant such that $\max\{c^{(1)}_{N},c^{(2)}_{N},c^{(3)}_{N}\}\leq NS$; such an $S$ exists when $(A+BK)$ and $(A+L_{n}C_{n})$ are Schur stable provided that the switching rule satisfies the condition specified in Proposition 3.
Theorem 5.
Recall the closed-loop MIMO LTI system (17) with $\alpha_{n}$ being our switching control action that chooses between two different observation matrices. Define the threshold $t_{N}=\sqrt{(1+\rho)S\log N/N}$, where $\rho>0$. Let $\Phi^{(1)}_{N}$ and $\Phi^{(2)}_{N}$ be defined using the measurements from Eq. 26 and Eq. 27. Let $\alpha_{N}$ be the switching decision rule with
•
we choose the switching input $\alpha_{N}=0$ when we have $\left\lVert\Phi^{(1)}_{N}\right\rVert<t_{N}$ or $\left\lVert\Phi^{(2)}_{N}\right\rVert<t_{N}$
•
we switch from $\alpha_{N-1}=0$ to $\alpha_{N}=1$ when $\alpha_{N-i}=0$ for $i\in\{1,\ldots,\tau\}$ and $\left\lVert\Phi^{(1)}_{N}\right\rVert\geq t_{N}$ and $\left\lVert\Phi^{(2)}_{N}\right\rVert\geq t_{N}$.
Moreover, let $E_{N}$ for all $n\geq 1$ denote the event
$$E_{N}=\Bigg{[}\left\lVert\Phi^{(1)}_{N}\right\rVert>t_{N}\bigcup\left\lVert%
\Phi^{(2)}_{N}\right\rVert>t_{N}\Bigg{]}$$
(77)
Then if $v_{N}\equiv 0$ for all $N\geq 0$, we have that
$$\mathrm{P}(\limsup_{N\rightarrow\infty}E_{N})=0.$$
(78)
That is, under no attacks our switching rule incorrectly switches the system only a finite number of times.
Proof.
Recall that we previously proved the following matrix concentration inequalities for each submatrix:
$$\displaystyle\mathrm{P}\bigg{(}\left\lVert\Phi^{(1)}_{N}\right\rVert\geq t_{N}%
\Bigg{)}$$
$$\displaystyle\leq m\cdot e^{-N^{2}t_{N}^{2}/c^{(1)}_{N}}$$
(79)
$$\displaystyle\mathrm{P}\Bigg{(}\left\lVert\Phi^{(2)}_{N}\right\rVert\geq t_{N}%
\Bigg{)}$$
$$\displaystyle\leq(m+p)\cdot e^{-N^{2}t_{N}^{2}/c^{(2)}_{N}}$$
(80)
for the constants $c^{(1)}_{N}$ and $c^{(2)}_{N}$. Summing over all $N$, we have
$$\sum_{k=1}^{\infty}\mathrm{P}\bigg{(}\left\lVert\Phi^{(j)}_{k}\right\rVert\geq
t%
_{k}\Bigg{)}\leq(m+p)\int_{1}^{\infty}\frac{1}{k^{1+\rho}}dk<\infty.$$
(81)
Hence the Borel-Cantelli Lemma implies that for the event
$$E^{(j)}_{N}=\Bigg{[}\left\lVert\Phi^{(j)}_{N}\right\rVert\geq t_{N}\Bigg{]}$$
(82)
we have
$$\mathrm{P}(\limsup_{N\rightarrow\infty}E^{(j)}_{N})=0,~{}\forall j=\{1,2,3\}.$$
(83)
Now, if we define the event
$$E_{N}=\Bigg{[}\left\lVert\Phi^{(1)}_{n}\right\rVert>t_{n}\bigcup\left\lVert%
\Phi^{(2)}_{n}\right\rVert>t_{n}\Bigg{]},~{}N\geq 1,$$
(84)
then it follows that
$$\mathrm{P}(E_{N})\leq\mathrm{P}\left(\bigcup_{j=1}^{2}E^{(j)}_{N}\right)\leq%
\sum_{j=1}^{2}\mathrm{P}\left(E^{(j)}_{N}\right),~{}N\geq 1.$$
(85)
So summing once more for all $N$ gives
$$\displaystyle\sum_{k=1}^{\infty}\mathrm{P}(E_{k})\leq\sum_{k=1}^{\infty}\sum_{%
j=1}^{2}\mathrm{P}\left(E^{(j)}_{N}\right)<\\
\displaystyle 2(m+p)\int_{1}^{\infty}\frac{1}{k^{1+\rho}}dk<\infty.$$
(86)
We obtain by applying Borel-Cantelli lemma that
$$\mathrm{P}\left(\limsup_{N\rightarrow\infty}E_{N}\right)=0,$$
(87)
which is the desired result.
∎
The result of this theorem implies that if there is no attack to the system, the operator norm of the matrices involved can have “large” deviations only a finite number of times, hence we obtain that a switching rule based on tests derived from the concentration inequalities defined previously will only trigger attack alerts only a finite number of times. In addition, we note that the having a second observer to compute the finite tests is the key to ensure that the concentration inequalities are consistent with the obtained measurements. While the first observer measurements with switching plays the role in the control synthesis. Lastly, we observe that we do not need to enforce the test on $\left\lVert\Phi^{(3)}_{3}\right\rVert$ since this submatrix is only composed of the watermaking signal, and the attacks do not have the power to affect the watermarking imposed by the controller.
VI Attack Magnitude Thresholding
The previous section gives a finite sample test that works properly when there is no attack.
Our goal here is to determine the trade-off between our test’s statistical power and the attack magnitude.
Namely, we are interested in how the right-hand side of our finite sample tests are related to the magnitude of the attack vectors. To do so, we consider the first observation matrix under an attack $y_{n}=Cx_{n}+z_{n}+v_{n}$, where $v_{n}$ is an additive attack. We first consider attacks that are small perturbations and then consider more complex attacks of the form explored in [25] .
Note we have again omitted the subscript of the observation matrix for clarity. Namely, in the next two subsections we will consider two attack forms.
VI-A Perturbation Attacks
The first attack we analyze is when $v_{n}$ consists of a small perturbation that could be determinstic and/or stochastic. To begin our analysis, let $\bar{\delta}_{n}$ be the measurement error when the system is under attack, and observe that
$$\bar{\delta}_{n+1}=(A+LC)\bar{\delta}_{n}-Dw_{n}-L^{\top}z_{n}-L^{\top}v_{n}.$$
(88)
Expanding this expression gives that
$$\bar{\delta}_{n}=(A+LC)^{n}\bar{\delta}_{0}-\sum_{k=0}^{n-1}(A+LC)^{n-1-k}(%
\mathbb{I}w_{k}-L^{\top}z_{k}-L^{\top}v_{k})$$
(89)
where $\bar{\delta}_{0}=\delta_{0}=0$. So we can rewrite the above as
$$\bar{\delta}_{n}=\sum_{k=0}^{n-1}\bar{D}_{k}w_{k}+\bar{L}_{k}z_{k}+\bar{L}_{k}%
v_{k}=\delta_{n}+\sum_{k=0}^{n-1}\bar{L}_{k}v_{k}.$$
(90)
Next we define $V_{n}=C\sum_{k=0}^{n-1}\bar{L}_{k}v_{k}-v_{n}$, and observe that the quantity $V_{n}$ is determined by the attacker since it depends upon the values of $v_{k}$. Qualitatively, we note that the magnitude of $V_{n}$ is related to the attack magnitude, since if there is no attack then $V_{n}\equiv 0$ for all $n$.
Theorem 6.
Consider the closed-loop MIMO LTI system (17) with $\alpha_{n},t_{N},E_{n}$ as defined in Theorem 5, and suppose the attacker chooses the perturbation attack described above. If the attack values $v_{k}$ are such that there exists a positive constant $G$ with
$$\frac{1}{N}\sum_{k=0}^{N-1}\left\lVert V_{k}\right\rVert\leq\frac{G}{N}.$$
(91)
then we have that $\mathrm{P}(\limsup_{n\rightarrow\infty}E_{n})=0$. That is, under a perturbation attack with the above specifications the attack is detected only a finite number of times.
Proof.
We begin by considering
$$\displaystyle\Phi^{(1)}_{N}=\frac{1}{N}\sum_{n=0}^{N-1}(C\hat{x}_{n}-y_{n})(C%
\hat{x}_{n}-y_{n})^{\top}-$$
(92)
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\mathbb{E}[(C\hat{x}_{n}-y_{n})(C\hat{%
x}_{n}-y_{n})^{\top}]=$$
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}(C\bar{\delta}_{n}-z_{n}-v_{n})(C\bar{%
\delta}_{n}-z_{n}-v_{n})^{\top}-$$
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\mathbb{E}[(C\hat{x}_{n}-y_{n})(C\hat{%
x}_{n}-y_{n})^{\top}]=$$
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_{n}-z_{n})(C\delta_{n}-z_{n})%
^{\top}+D_{n}+D^{\top}_{n}+M_{n}-$$
$$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\mathbb{E}[(C\hat{x}_{n}-y_{n})(C\hat{%
x}_{n}-y_{n})^{\top}].$$
where $\delta_{n}$ is the measurement error under no attack, and
$$\displaystyle D_{n}$$
$$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_{n}-z_{n})V_{n}^{\top}$$
(93)
$$\displaystyle M_{n}$$
$$\displaystyle=\frac{1}{N}\sum_{n=0}^{N-1}V_{n}V_{n}^{\top}.$$
Now using Theorem 2, we have that
$$\displaystyle\mathrm{P}(\left\lVert\Phi^{(1)}_{N}\right\rVert\geq t_{N})\leq\\
\displaystyle\mathrm{P}\bigg{(}\|\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_{n}-z_{n}%
)(C\delta_{n}-z_{n})^{\top}-\\
\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}\mathbb{E}[(C\hat{x}_{n}-y_{n})(C\hat{%
x}_{n}-y_{n})^{\top}]\|\geq t_{N}+\\
\displaystyle-2\left\lVert{D}_{N}\right\rVert-\left\lVert M_{N}\right\rVert%
\bigg{)}\leq\\
\displaystyle m\text{ }e^{\frac{-N^{2}\left(t_{N}-2\left\lVert{D}_{N}\right%
\rVert-\left\lVert M_{N}\right\rVert\right)^{2}}{c^{(1)}_{N}}}.$$
(94)
Next observe that
$$\displaystyle 2\left\lVert\bar{D}_{N}\right\rVert+\left\lVert M_{N}\right%
\rVert\leq\frac{2\bar{K}_{N}}{N}\sum_{n=0}^{N-1}\left\lVert V_{n}\right\rVert+%
\frac{1}{N}\sum_{n=0}^{N-1}\left\lVert V_{n}\right\rVert^{2}\leq\\
\displaystyle\frac{2\bar{K}_{N}G}{N}+\frac{G^{2}}{N}.$$
(95)
Since $(A+LC)$ is Schur stable, then from the definition of $\bar{K}_{N}$ we immediately get that there exists a positive constant $\bar{S}$ such that $\bar{K}_{N}\leq\bar{S}$ for all $N\geq 1$. Combining this with the above implies that
$$\sum_{k=1}^{\infty}\mathrm{P}(\left\lVert\Phi^{(1)}_{k}\right\rVert\geq t^{(1)%
}_{k})<\infty,$$
(96)
and so the Borel-Cantelli lemma implies that $\left\lVert\Phi^{(1)}_{N}\right\rVert\geq t^{(1)}_{N}$ only finitely many times. Our next step considers
$$\displaystyle\Phi^{(2)}_{N}=\frac{1}{N}\sum_{n=0}^{N-1}(C\hat{x}_{n}-y_{n})e^{%
\top}_{n-k^{\prime}-1}=\\
\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}(C\bar{\delta}_{n}-z_{n}-v_{n})e^{\top%
}_{n-k^{\prime}-1}=\\
\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_{n}-z_{n})e^{\top}_{n-k^{%
\prime}-1}+H_{n}$$
(97)
where
$$H_{N}=\frac{1}{N}\sum_{n=0}^{N-1}V_{n}e^{\top}_{n-k^{\prime}-1}.$$
(98)
Now using Theorem 3, we have that
$$\displaystyle\mathrm{P}(\left\lVert\Phi^{(2)}_{N}\right\rVert\geq t_{N})\leq\\
\displaystyle\mathrm{P}\bigg{(}\|\frac{1}{N}\sum_{n=0}^{N-1}(C\delta_{n}-z_{n}%
)e^{\top}_{n-k^{\prime}-1}\|\geq t_{N}-\left\lVert H_{N}\right\rVert\bigg{)}%
\leq\\
\displaystyle 2m\text{ }e^{\frac{-N^{2}\left(t_{N}-\left\lVert H_{N}\right%
\rVert\right)^{2}}{c^{(2)}_{N}}}.$$
(99)
Next observe that
$$\left\lVert H_{N}\right\rVert\leq\frac{K_{e}}{N}\sum_{k=0}^{N-1}\left\lVert V_%
{N}\right\rVert\leq\frac{K_{e}G}{N}.$$
(100)
Combining this with the above implies that
$$\sum_{k=1}^{\infty}\mathrm{P}(\left\lVert\Phi^{(2)}_{k}\right\rVert\geq t_{k})%
<\infty,$$
(101)
and so the Borel-Cantelli lemma implies that $\left\lVert\Phi^{(2)}_{N}\right\rVert\geq t_{N}$ only finitely many times. The remainder of the proof follows similarly to that of the last steps of Theorem 5.
∎
Our analysis in this subsection is capable of only providing a simple relation between the power of our detection scheme and the magnitude of $V_{n}$. An analysis that translates to the bounds of each individual $v_{n}$ is more involved because it depends explicitly on the structure/behavior of the matrix $(A+LC)$.
VI-B Replay Attacks
The second attack we analyze is when
$$v_{n}=C\xi_{n}+\zeta_{n}-(Cx_{n}+z_{n})$$
(102)
where $\xi_{n+1}=(A+BK)\xi_{n}+\omega_{n}$ and $\omega_{n}$ is a bounded disturbance. This is a replay attack [3], since it subtracts the real sensor measurements and substitutes these with a replay of the dynamics starting from a different initial condition. In fact, we will perform our analysis for a more general attack
$$v_{n}=C\xi_{n}+\zeta_{n}-\gamma\cdot(Cx_{n}+z_{n}),$$
(103)
where $\gamma\in\mathbb{R}$. This attack also allows for dampening or amplifying the true sensor measurements $(Cx_{n}+z_{n})$.
Theorem 7.
Consider the closed-loop MIMO LTI system (17) with $\alpha_{n},t_{N},E_{n}$ as defined in Theorem 5, and suppose the attacker chooses the attack (103). If the attack is not trivial (i.e., a trivial attack has $v_{n}\equiv 0$ for all $n\geq 0$), then we have that $\mathrm{P}(\limsup_{n\rightarrow\infty}\neg E_{n})=0$. That is, under the attack with the above specifications the attack is not detected only a finite number of times.
Proof.
Suppose $\gamma\neq 0$. Then the proof of Theorem 1 in [25] shows that $\lim_{N\rightarrow\infty}\Phi^{(2)}_{N}$ exists almost surely and is not equal to 0. This means that $\mathrm{P}(\limsup_{n\rightarrow\infty}\neg E_{N}^{(2)})=0$. Now consider the case $\gamma=0$. Then the proof of Theorem 1 in [25] shows that $\lim_{N\rightarrow\infty}\Phi^{(1)}_{N}$ exists almost surely and is not equal to 0. This means that $\mathrm{P}(\limsup_{n\rightarrow\infty}\neg E_{N}^{(1)})=0$. The remainder of the proof by repeating the last steps of Theorem 5 for the two cases, after noting that $\neg E_{N}=\neg E_{N}^{(1)}\vee\neg E_{N}^{(2)}$ by De Morgan’s laws.
∎
This result is stronger than Theorem 6 in that it says all replay attacks, and more generally attacks of the form (103), will not be detected by the finite sample tests only a finite number of times. In fact, this result is analagous to the zero-average-power results (4) of past work on dynamic watermarking for LTI systems with general structure [25], since this result says that only (trivial) replay attacks with zero-average-power cannot be detected.
VII Experimental Results
To further demonstrate the effectiveness of this method, we return to the lane keeping example used in [25] which is based off of the standard model for lane keeping and speed control [39].
In this model the state vector takes the form $x^{T}=[\psi~{}y~{}s~{}\gamma~{}v]$ and input vector $u^{T}=[r~{}a]$, where $\psi$ is heading error, $y$ is lateral error, $s$ is trajectory distance, $\gamma$ is vehicle angle, $v$ is vehicle velocity, $r$ is steering, and $a$ is acceleration.
Linearizing about a straight trajectory at a velocity of 10 m/s and step size of 0.05 seconds gives us an LTI system:
$$A=\begin{bmatrix}1&0&0&\frac{1}{10}&0\\
\frac{1}{2}&1&0&\frac{1}{40}&0\\
0&0&1&0&\frac{1}{2}\\
0&0&0&1&0\\
0&0&0&0&1\end{bmatrix}B=\begin{bmatrix}\frac{1}{400}&0\\
\frac{1}{2400}&0\\
0&\frac{1}{800}\\
\frac{1}{20}&0\\
0&\frac{1}{20}\end{bmatrix}$$
(104)
with $C_{1}=C_{2}=[I,0]\in\mathbb{R}^{3\times 5}$. The process noise and watermark take the form of uniform random variables such that $w\in[-2.5\times 10^{-4},~{}2.5\times 10^{-4}]^{5}$ and $e\in[-2,~{}2]^{2}$. Similarly the measurement noise for each sensor is also estimated as uniform random variables where $\zeta\in[-1\times 10^{-2},~{}1\times 10^{-2}]^{3}$ and $\eta\in[-2\times 10^{-2},~{}2\times 10^{-2}]^{3}$. For this example we can think of the $\zeta$ measurements as localization using visual or lidar based localization with high definition mapping, and $\eta$ as GPS localization. Finally controller and observer gains $K$ and $L_{1}=L_{2}$ were chosen to stabilize the closed loop system.
For this system it was found that
$$\displaystyle c_{n}^{(1)}$$
$$\displaystyle\leq 6.7502\times 10^{-5}n$$
(105)
$$\displaystyle c_{n}^{(2)}$$
$$\displaystyle\leq 0.0968n.$$
(106)
Using the threshold structure defined in Theorem 5 results in
$$\displaystyle\tau_{n}^{(1)}$$
$$\displaystyle=\sqrt{(1+\rho^{(1)})(6.7502\times 10^{-5})\log(n)/n}$$
(107)
$$\displaystyle\tau_{n}^{(2)}$$
$$\displaystyle=\sqrt{(1+\rho^{(2)})(0.968)\log(n)/n}.$$
(108)
While the finite switching guarantee given by Theorem 5 only applies for $\rho^{(1)},\rho^{(2)}>0$, due to the conservative nature of the bounds in (105)-(106) in addition to the desire to also maintain a sufficiently quick detection we instead heuristically tune these values to find the desired balance.
For our analysis of this system, we once again consider the two forms of attack discussed in Section VI.
The perturbation attack takes the form of random noise pulled from a uniform distribution such that $v_{n}\in[-0.150.15]^{3}$
The replay attack is described in (102) where $\xi_{0}=0$ and $\zeta$ and $\omega$ are uniformly distributed such that $\zeta\in[-2.5\times 10^{-4},~{}2.5\times 10^{-4}]^{3}$ and $\omega\in[-2.5\times 10^{-4},~{}2.5\times 10^{-4}]^{5}$.
Each attacked system, along with an un-attacked system were simulated 1000 times for 10,000 discrete time steps.
While the perturbation attack is detected and switching occurs almost immediately for $\rho^{(1)},\rho^{(2)}<1$, the replay attack can take a much longer time to be detected.
Figure 1 shows the average time to detection for each of our switching conditions in addition to the number of trials that result in switching for the un-attacked case plotted against the corresponding value of $\rho^{(1)}$ or $\rho^{(2)}$.
While the number of switching simulations for the un-attacked system under switching condition 2 appear to be quite large even when the average time to detect is relatively large, it is important to note that many of the unwanted switches occur in the first four discrete steps which can be mitigated in practice by ignoring the first four values.
Choosing values of $\rho^{(1)}=\rho^{(2)}=-0.98$, each attack was again simulated this time for 1000 discrete time steps both with and without the switching policy.
Figure 2 shows the value of $\|\Phi_{n}^{(1)}\|$ and $\|\Phi_{n}^{(2)}\|$ for both normal operation and under each of the attacks when the switching policy is not being used. The plot shows that for both attack 1 and attack 2 the switching policy will result in an almost immediate and consistent transfer from the attacked sensor to the protected sensor. Furthermore, when the system is un-attacked the values of $\|\Phi_{n}^{(1)}\|$ and $\|\Phi_{n}^{(2)}\|$ remain below the switching threshold.
Figure 3 compares the performance of the lane keeping algorithm for each attack with respect to the un-attacked performance both with and without the switching policy. This plot shows that for both attacks the switching policy is able to transfer to the protected sensor before significant deviation can occur. This switch allows the vehicles performance to gracefully degrade while avoiding total failure.
VIII Conclusion
This paper constructed a dynamic watermarking approach for detecting malicious sensor attacks for general LTI systems, and the two main contributions were: to extend dynamic watermarking to general LTI systems under a specific attack model that is more general than replay attacks, and to show that modeling is important for designing watermarking techniques by demonstrating how persistent disturbances can negatively affect the accuracy of dynamic watermarking. Our approach to resolve this issue was to incorporate a model of the persistent disturbance via the internal model principle. Future work includes generalizing the attack models that can be detected by our approach. An additional direction for future work is to study the problem of robust controller design in the regime of when an attack is detected.
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ON TURBULENT RECONNECTION
EUN-JIN KIM and P. H. DIAMOND
Department of Physics, University of California
at San Diego, La Jolla, CA 92093, USA
Abstract
We examine the dynamics of turbulent reconnection in
2D and 3D reduced MHD by calculating the effective
dissipation due to coupling between small–scale fluctuations
and large–scale magnetic fields. Sweet–Parker type
balance relations are then used to calculate the global
reconnection rate. Two approaches are employed — quasi–linear
closure and an eddy-damped fluid model. Results indicate
that despite the presence of turbulence,
the reconnection rate remains inversely proportional
to $\sqrt{R_{m}}$, as in the Sweet–Parker analysis.
In 2D, the global reconnection rate is shown to be enhanced
over the Sweet–Parker result by a factor of magnetic Mach
number. These results are the consequences of the constraint
imposed on the global reconnection rate by the requirement
of mean square magnetic potential balance. The incompatibility
of turbulent fluid–magnetic energy equipartition and
stationarity of mean square magnetic potential is demonstrated.
Subject headings: MHD — magnetic fields — turbulence
1 INTRODUCTION
Magnetic reconnection is the process whereby large scale magnetic
field energy is dissipated and magnetic topology is altered in
MHD fluids and plasmas (for instance, see, Vasyliunas 1975;
Parker 1979; Forbes & Priest 1984; Biskamp 1993;
Wang, Ma, & Bhattacharjee 1996 and references therein).
Reconnection is often invoked as the
explanation of large scale magnetic energy release in space,
astrophysical, and laboratory plasmas. Specifically, magnetic
reconnection is thought to play an integral role in the dynamics of
the magnetotail, the solar dynamo, solar coronal heating, and
in the major disruption in tokamaks. For these reasons, magnetic
reconnection has been extensively studied in the context of
MHD, two-fluid and kinetic models, via theory,
numerical simulations and laboratory experiments.
The basic paradigm for magnetic reconnection is the Sweet–Parker
(called SP hereafter) problem (Parker 1957; Sweet 1958),
in which a steady inflow velocity
advects oppositely directed magnetic field lines ($\pm{\bf B}$)
together, resulting in current sheet formation and, thus, reconnection
(see Fig. 1). The current sheet has thickness $\Delta$ and length
$L$, so that imposition of continuity ($v_{r}L=v_{0}\Delta$),
momentum balance ($v_{0}=v_{A}$) and magnetic energy balance
($v_{r}B=\eta B/\Delta$) constrains the inflow, or
“reconnection”, velocity to be
$v_{r}=v_{A}/\sqrt{S}\propto v_{A}/\sqrt{R_{m}}$.
Here $v_{0}$ is the outflow velocity;
$v_{A}$ is the Alfvén speed associated with ${\bf B}$;
$S\equiv v_{A}L/\eta$ is the Lundquist number;
$R_{m}=ul/\eta$ is the magnetic Reynolds number, with $u$
and $l$ being the characteristic amplitude and length scale of
the velocity — $S$ is called the magnetic Reynolds number
$R_{m}$ in some literatures.
Note that the SP process forms strangely anisotropic current sheets
since $\Delta/L=\sqrt{S}$ and $S\gg 1$. Note also the link
between sheet anisotropy and the reconnection speed $v_{r}$, i.e.,
$v_{r}/v_{A}=\Delta/L=1/\sqrt{S}$.
Finally, it should be
noted that $v_{r}$ is a measure of the global reconnection
rate, in that it parameterizes the mean inflow velocity to the
layer.
The SP Picture is intrinsically appealing, on account of its
simplicity and dependence only upon conservation laws.
Moreover, the SP prediction has been verified by laboratory
experiments (Ji, Yamada, & Kulsrud 1998).
However, since $R_{m}$ is extremely
large in most astrophysical applications of interest
(i.e. $R_{m}\sim 10^{13}$ in the solar corona), the SP
reconnection speed is pathetically slow. Hence, there
have been many attempts to develop models of fast
reconnection. For example, in 1964 Petschek proposed a
fast reconnection model involving shock formation near the
reconnection layer, which predicted $v_{r}=v_{A}/\ln{S}$.
Unfortunately, subsequent numerical (Biskamp 1986)
and theoretical (Kulsrud 2000) study has indicated that
Petschek’s model is internally inconsistent. While
research on fast, laminar reconnection continues today (i.e.,
Kleva, Drake, & Waelbroeck 1995) in the context of
two-fluid models, the failure
of the Petschek scenario has sparked
increased interest in turbulent reconnection (Matthaeus & Lamkin 1986)
in which turbulent transport coefficients (which can be
large for large Reynolds number) act as effective
dissipation coefficients, and so are thought to
facilitate fast reconnection (i.e., Diamond et al. 1984; Strauss 1988).
Interest in turbulent reconnection has also been stimulated by
the fact that many instances of reconnection occur in systems
where turbulence is ubiquitous, i.e., coronal heating
of turbulent accretion disks, the dynamo in the sun’s
convection zone, and turbulent tokamak plasmas during
disruptions.
Recently, Lazarian and Vishniac (1999) (referred to hereafter
as LV) presented a detailed discussion of turbulent
reconnection. LV took a rather novel approach to the
problem by considering the interaction of two slabs of
oppositely directed, chaotic magnetic fields when
advected together. LV modeled the effects of
turbulence by treating the slabs’ surfaces as
rough, where the roughness was symptomatic of a
chaotic turbulent magnetic field structure. This
‘rough surface’ model naturally led LV to decompose the
reconnection process into an ensemble of local,
‘micro’-reconnection events, which interact to form a
net ‘global’ reconnection process. LV argue that micro-reconnection
events occur in small scale ‘layers’, with dimensions set by
the structure of the underlying Alfvénic MHD turbulence
(i.e., the $k_{\perp}^{-1}$ and $k_{\parallel}^{-1}$,
as set by the Goldreich–Sridhar model). The upper
bound for the micro-reconnection rate obtained by LV
is $v_{r}=v_{A}(u/v_{A})^{2}=v_{A}(b/B_{H})^{2}$, where $B_{H}$
is the mean, reconnection field, and $u$ and $b$ are
small–scale velocity and magnetic field. While the LV arguments
concerning micro-reconnection are at least plausible,
their assertion that the global reconnection rate
can be obtained by effectively superposing micro–reconnection
events is unsubstantiated and rather dubious, in that it
neglects dynamical interactions between micro-layers.
Such interactions are particularly important for enforcing
topological conservation laws. Since the process of turbulent
reconnection is intimately related to the rate of flux
dissipation, and the latter is severely constrained by
mean square magnetic potential conservation, it stands
to reason that such a topological conservation law will also
constrain the rate of global magnetic reconnection.
In particular, for a mean $B$-field with strength in excess
of $B_{{\rm crit}}\sim\sqrt{{\langle}u^{2}{\rangle}/R_{m}}$, the flux
2D was shown to be suppressed by a factor
$${1\over 1+R_{m}{\langle B\rangle^{2}/\langle u^{2}\rangle}}\,,$$
(1)
where $\langle B\rangle$ is the large–scale magnetic field
and $\langle u^{2}\rangle$ the turbulent kinetic energy
(Cattaneo & Vainshtein 1991; Gruzinov & Diamond 1994);
The above expression implies that even a weak magnetic
field (i.e, one far below the equipartition value ${\langle}b^{2}{\rangle}\sim{\langle}u^{2}{\rangle}$) is potentially important. The origin
of this suppression is ultimately linked to the conservation
of mean square potential (see Das & Diamond 2000 for flux
diffusion in EMHD). Hence, it is natural to investigate
the effect of such constraints on reconnection, as well.
In turbulent reconnection, fluctuating magnetic fields are
dynamically coupled to a large–scale magnetic field so that
a similar suppression of energy transfer is expected to occur.
In other words, fluctuating magnetic fields
will inhibit the energy transfer from large–scale to small–scale
magnetic fields (responsible for turbulent diffusion), even when
the latter is far
below equipartition value. This link between small and large scale
magnetic field dynamics is indeed the very feature that is missing
in LV, where a global reconnection rate is considered to be a simple sum of
local reconnection events, without depending on either $\langle B\rangle$
or $R_{m}$. That is, even if one local reconnection event
may proceeds fast, the energy transfer from large–scale to small–scale
is suppressed inversely with $R_{m}$, preventing many local
reconnection events for a large $R_{m}$ and fixed large–scale field
strength. Thus, the global reconnection rate is very likely to be reduced for
large $R_{m}$.
The purpose of this paper is to determine the global reconnection rate
by treating the dynamics of large and small–scale magnetic fields
in a consistent way. The key idea is to compute the effective
dissipation rate of a large–scale magnetic field (turbulent
diffusivity) by taking into account small–scale field backreaction
and then to use
Sweek-Parker type balance relations to obtain the global reconnection rate.
Since magnetic fields across current sheets are not
always strictly antiparallel in real systems, we assume that only one
component of the magnetic field (e.g., poloidal or horizontal field) changes
its sign across the current sheet (see Fig. 2). The other component
(e.g., axial field) is assumed to be very strong compared to the
poloidal component. A strong axial magnetic field avoids the
null point problem inherent in SP slab model, justifying the
assumption of incompressibility of the flow in the poloidal (horizontal)
plane. Such a magnetic configuration is ideal for the application
of so–called 3D reduced MHD (3D RMHD) (Strauss 1976).
In 3D RMHD, the conservation of the mean square
potential is linearly broken due to the propagation of Alfvén waves
along an axial field, but preserved by the nonlinearity.
As we shall show later, the latter effect introduces additional
suppression in the effective dissipation of a large–scale
magnetic field compared to 2D MHD.
We also discuss the 2D MHD case which can be recovered from our results
simply by taking the limit $B_{0}\to 0$, where $B_{0}$ is a axial magnetic
field.
To be able to obtain analytic results, we adopt the following two methods.
The first is a quasi–linear closure together using $\tau$ approximation by
assuming the same correlation time for fluctuating velocity and magnetic fields
employing unity magnetic Prandtl number.
The second is an eddy-damped fluid model, based on large viscosity (Kim 1999),
which may have relevance in Galaxy where $\nu\gg\eta$.
In this model, the nonlinear
backreaction can be incorporated consistently, without
having to invoking the presence of fully developed MHD turbulence,
or assumptions such as a quasi–linear closure or $\tau$ approximation.
In both models, the isotropy and homogeneity of
turbulence is assumed in the horizontal (poloidal) plane since the
reduction
in effective dissipation of a large–scale poloidal magnetic field is
likely to occur when its strength is far below the equipartition value.
The effect of hyper-resistivity is incorporated in our analysis. This
can potentially accelerate the dissipation of a large–scale poloidal
magnetic field.
The paper is organized in the following way.
In §2, we set up our problem in 3D RMHD and provide the quasi–linear
closure using $\tau$ approximation where the flux is
estimated in a stationary case. Section 3 contains a similar analysis
for an eddy-damped fluid model. The global reconnection rate for both
models is presented in §4. Our main conclusion and discussion
is found in §5.
2 QUASI–LINEAR MEAN FIELD EQUATIONS
We assume that a strong constant axial magnetic field $B_{0}$ is aligned
in the $z$ direction and that a poloidal (horizontal) magnetic field ${\bf B}_{H}$
lies in the horizontal $x$-$y$ plane, as shown in Fig. 2. The subscript $H$
denotes horizontal direction. The total magnetic field is then expressed
as ${\bf B}=B_{0}{\hat{z}}+{\bf B}_{H}=B_{0}{\hat{z}}+{\nabla\times}\psi{\hat{z}}$, in terms
of a parallel component of the vector potential $\psi$
(i.e., ${\bf B}_{H}={\nabla\times}\psi{\hat{z}}$). According to the
RMHD ordering, the flow in the horizontal plane ${\bf u}$
is incompressible and therefore can be written using a scalar
potential $\phi$ as ${\bf u}={\nabla\times}\phi{\hat{z}}$.
Then, the equations governing 3D RMHD are (see Strauss 1976):
$$\displaystyle{\partial}_{t}\psi+{\bf u}\cdot\nabla\psi$$
$$\displaystyle=$$
$$\displaystyle\eta{\nabla^{2}}\psi+B_{0}{\partial}_{z}\psi\,,$$
(2)
$$\displaystyle{\partial}_{t}{\nabla^{2}}\phi+{\bf u}\cdot\nabla{\nabla^{2}}\phi$$
$$\displaystyle=$$
$$\displaystyle\nu{\nabla^{2}}{\nabla^{2}}\phi+{\bf B}\cdot\nabla{\nabla^{2}}%
\psi\,,$$
(3)
where $\eta$ and $\nu$ are Ohmic diffusivity and viscosity, respectively.
For the quasi–linear closure, unity magnetic Prandtl number
($\eta=\nu$) will implicitly be assumed.
In comparison with 2D MHD, the equation for the vector potential
contains an additional term $B_{0}{\partial}_{z}\phi$, which reflects the propagation
of Alfvén wave along the axial magnetic field $B_{0}{\hat{z}}$. Due
to this additional term, the conservation of the mean square
potential is broken in 3D RMHD, albeit only linearly. In other words,
the nonlinear term in equation (2) conserves ${\langle}\psi^{2}{\rangle}$
since $\langle{\bf u}\cdot\nabla\psi^{2}\rangle=\nabla\cdot\langle{\bf u}\psi^{2}{%
\rangle}=0$, assuming that boundary terms vanish (cf. Blackman
& Field 2000). Similarly, the momentum
equation contains an additional term $B_{0}{\partial}_{z}{\nabla^{2}}\psi$. These
additional terms
are proportional to the wavenumber $k_{z}$ along $B_{0}{\hat{z}}$. Thus, the
2D case can be recovered by taking $k_{z}\to 0$ or $B_{0}\to 0$.
Note that due to a strong axial field $B_{0}{\hat{z}}$, the vertical wavenumber
$k_{z}$ is much smaller than horizontal wavenumber ${\bf k}_{H}=k_{x}{\hat{x}}+k_{y}{\hat{z}}$; specifically, the 3D RMHD ordering implies that
$k_{z}/k_{H}\sim B_{H}/B_{0}\sim\epsilon\ll 1$.
We envisage a situation where large–scale magnetic fields
with a horizontal component ${\bf B}_{H}=\langle{\bf B}_{H}\rangle={\nabla\times}{\langle}\psi{\rangle}{%
\hat{z}}$ are embedded in a turbulent background.
The turbulence can be generated by an external forcing, for instance.
The horizontal component of a large–scale magnetic field
${\langle}{\bf B}_{H}{\rangle}$ flows to form a current sheet of
thickness $\Delta$ in the horizontal plane, so ${\langle}{\bf B}_{H}{\rangle}$
changes sign across the current sheet. As reconnection proceeds,
small–scale flows as well as magnetic fields are generated within
the current sheet. It is reasonable to model the physical processes
within a current sheet as well as the background turbulence
by an (approximately) isotropic and homogeneous
turbulence with fluctuating velocity ${\bf u}$ and magnetic field
${\bf b}={\nabla\times}\psi^{\prime}{\hat{z}}$.
Here the assumption of isotropy is justified since
${\langle}B_{H}{\rangle}^{2}\ll{\langle}u^{2}{\rangle}$, i.e. the reconnecting field
is taken to be weak.
Outside the reconnection region, there are large–scale inflow
and outflow in addition to the background turbulence. Thus,
to obtain SP-like balance relations, small–scale flow as well
as large–scale flow should be incorporated. However, since small–scale
velocity is assumed to be homogeneous and isotropic, there is no net
contribution from the fluctuating velocity to mass continuity.
Effectively, the small–scale velocity does not appear
in the momentum balance either.
However, Ohm’s law (magnetic energy balance) now contains an
additional term due to the correlation between fluctuating fields
${\langle}{\bf u}\times{\bf b}{\rangle}$,
leading to turbulent diffusitivy (effective dissipation
rate), which then effectively changes the Ohmic diffusivity to the sum
of Ohmic diffusivity and turbulent diffusivity inside current sheet.
Therefore, similar balance relations to the original SP
hold in our case
as long as the Ohmic diffusivity is replaced by the total diffusivity.
To recapitulate, homogeneous and isotropic turbulence is assumed
to be present with magnetic fields ${\bf B}_{H}=\langle{\bf B}_{H}\rangle+{\bf b}$ (${\langle}{\bf b}{\rangle}=0$) and small–scale velocity ${\bf u}$ (${\langle}{\bf u}{\rangle}={\langle}\phi{\rangle}=0$).
Once the effective dissipation rate of ${\langle}{\bf B}_{H}{\rangle}$ within the
reconnection zone is computed, it will be used to determine the
reconnection velocity $v_{r}$ through SP balance relations
by using the total diffusivity in place of Ohmic diffusivity.
2.1 Mean Field Equation
The evolution equation for $\psi$ is obtained by taking the average
of the above equation as:
$$\displaystyle{\partial}_{t}\langle\psi\rangle+\langle{\bf u}\cdot\nabla\psi^{%
\prime}\rangle$$
$$\displaystyle=$$
$$\displaystyle\eta{\nabla^{2}}\langle\psi\rangle\,.$$
(4)
Note that although equation (4) does not exhibit an explicit dependence
on $B_{0}$, it does depend on $B_{0}$ through the flux $\Gamma_{i}\equiv\langle u_{i}\psi^{\prime}\rangle$. To compute the flux $\Gamma_{i}$, we first do a
quasi–linear closure of $\langle{\bf u}\cdot\nabla\psi^{\prime}\rangle$.
The effect of the backreaction can be incorporated in the flux
$\Gamma_{i}$ by considering the change in flux $\Gamma_{i}$ to be due to the
change in the velocity as well as the fluctuating magnetic field. That is,
we can rewrite the flux as
$$\displaystyle\Gamma_{i}$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}\langle{\partial}_{j}\phi\psi^{\prime}\rangle=%
\epsilon_{ij3}\langle{\partial}_{j}\phi\delta\psi^{\prime}-\delta\phi{\partial%
}_{j}\psi^{\prime}\rangle\,,$$
(5)
where unity magnetic Prandtl number is assumed for the
equal splitting between $\langle{\partial}_{j}\phi\delta\psi^{\prime}\rangle$
and $\langle\delta\phi{\partial}_{j}\psi^{\prime}\rangle$; the latter
essentially takes the backreaction to be as important as the kinematic
contribution.
2.2 Fluctuations
¿From equations (2) and (3), we can write the equation for the
fluctuations in the following form.
$$\displaystyle({\partial}_{t}+{\bf u}\cdot\nabla)\psi^{\prime}-{\langle}{\bf u}%
\cdot\nabla\psi^{\prime}{\rangle}$$
$$\displaystyle=$$
$$\displaystyle-{\bf u}\cdot\nabla{\langle}\psi{\rangle}+\eta{\nabla^{2}}\psi^{%
\prime}+B_{0}{\partial}_{z}\psi^{\prime}\,,$$
$$\displaystyle({\partial}_{t}+{\bf u}\cdot\nabla){\nabla^{2}}\phi-{\langle}{\bf
u%
}\cdot\nabla{\nabla^{2}}\phi{\rangle}$$
$$\displaystyle=$$
$$\displaystyle\nu{\nabla^{2}}{\nabla^{2}}\phi+B_{0}{\partial}_{z}{\nabla^{2}}%
\psi^{\prime}+{\langle}B_{H}{\rangle}\cdot\nabla_{H}{\nabla^{2}}\psi^{\prime}+%
{\bf b}\cdot\nabla_{H}{\nabla^{2}}{\langle}\psi{\rangle}\,.$$
Here we have assumed that there is no large–scale flow in the current
sheet.
To estimate $\delta\phi$ and $\delta\psi^{\prime}$ in equation (5), we
introduce a correlation time $\tau$ that represents the overall
effect of inertial and advection terms on the left hand side
of the above equations. That is, we approximate
$({\partial}_{t}+{\bf u}\cdot\nabla)\psi^{\prime}-\langle{\bf u}\cdot\nabla\psi%
^{\prime}\rangle\equiv\tau^{-1}\psi^{\prime}$, and
$({\partial}_{t}+{\bf u}\cdot\nabla){\nabla^{2}}\phi-\langle{\bf u}\cdot\nabla{%
\nabla^{2}}\phi{\rangle}\equiv\tau^{-1}{\nabla^{2}}\phi$, where the same correlation time
$\tau$ is assumed
for both the fluctuating flow and magnetic field due to unity magnetic
Prandtl number. Then, $\delta\phi$ and $\delta\psi^{\prime}$ in
equation (5) can be estimated from the above equations as follows:
$$\displaystyle\delta\psi^{\prime}$$
$$\displaystyle=$$
$$\displaystyle\tau\left[B_{0}{\partial}_{z}\phi^{\prime}-\epsilon_{ij3}{%
\partial}_{j}\phi^{\prime}{\partial}_{i}\langle\psi\rangle\right]\,,$$
(6)
$$\displaystyle\delta{\nabla^{2}}\phi$$
$$\displaystyle=$$
$$\displaystyle\tau\left[B_{0}{\partial}_{z}{\nabla^{2}}\psi^{\prime}+\epsilon_{%
ij3}{\partial}_{j}\langle\psi\rangle{\partial}_{i}{\nabla^{2}}\psi^{\prime}+%
\epsilon_{ij3}{\partial}_{j}\psi^{\prime}{\partial}_{i}{\nabla^{2}}\langle\psi%
\rangle\right]\,.$$
(7)
In Fourier space, the above equations take the following form:
$$\displaystyle\delta\psi^{\prime}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle\tau\bigl{[}B_{0}ik_{z}\phi({\bf k})+\epsilon_{ij3}\int d^{3}k^{%
\prime}k^{\prime}_{j}\phi({\bf k}^{\prime})(k-k^{\prime})_{i}\langle\psi({\bf k%
}-{\bf k}^{\prime})\rangle\bigr{]}\,,$$
(8)
$$\displaystyle\delta\phi({\bf k})$$
$$\displaystyle=$$
$$\displaystyle i\tau\biggl{[}B_{0}k_{z}\psi^{\prime}({\bf k})+i\epsilon_{ij3}{1%
\over k^{2}}\int d^{3}k^{\prime}\left[(k-k^{\prime})_{j}k^{\prime}_{i}k^{%
\prime 2}+k^{\prime}_{j}(k-k^{\prime})_{j}({\bf k}-{\bf k}^{\prime})^{2}\right%
]\psi^{\prime}({\bf k}^{\prime})$$
(9)
$$\displaystyle \times\langle%
\psi({\bf k}-{\bf k}^{\prime})\rangle\biggr{]}\,.$$
Note that in principle, the correlation time can be a function of
the spatial scale,
or the wavenumber, i.e., $\tau=\tau_{{{\bf k}}}$. Nevertheless, for
the notational simplicity, we have taken $\tau$ to be
a constant by assuming that the variation of $\tau_{{\bf k}}$ in ${{\bf k}}$ is
small or that the small–scale fields possess a characteristic
scale with a small spread in ${\bf k}$. Our final result will not fundamentally
change when the scale dependence of $\tau$ is incorporated.
The flux $\Gamma_{i}$ can readily be computed once the statistics of
small–scale magnetic field and the velocity are specified. As mentioned
earlier, the statistics of both fluctuations are assumed to be homogeneous and
isotropic in the $x$-$y$ plane. We further assume that the former
is homogeneous and reflectionally symmetric in the $z$ direction
with no cross correlation between horizontal and vertical components,
thereby eliminating a helicity term. The absence of helicity terms
rules out a possibility of a mean field dynamo in our model.
Note that due to the presence of a strong axial field $B_{0}{\hat{z}}$,
the correlation functions cannot be everywhere isotropic.
Specifically, the correlation functions at equal time
$t$ are taken to have the form:
$$\displaystyle\langle\psi^{\prime}({\bf k}_{1},t)\psi^{\prime}({\bf k}_{2},t)\rangle$$
$$\displaystyle=$$
$$\displaystyle\delta({\bf k}_{1}-{\bf k}_{2}){\overline{\psi}}(k_{1H},k_{1z})\,,$$
(10)
$$\displaystyle\langle\phi({\bf k}_{1},t)\phi({\bf k}_{2},t)\rangle$$
$$\displaystyle=$$
$$\displaystyle\delta({\bf k}_{1}-{\bf k}_{2}){\overline{\phi}}(k_{1H},k_{1z})\,,$$
(11)
where ${\overline{\psi}}(k_{1H},k_{1z})$ and ${\overline{\phi}}(k_{1H},k_{1z})$
are the power spectra of $\psi^{\prime}$ and $\phi$, respectively. These depend
on only the magnitude of horizontal wavenumber $k_{1H}=\sqrt{k_{1x}^{2}+k_{1y}^{2}}$ and vertical wavenumber $k_{1z}$. Finally, we assume
that ${\langle}\phi\psi^{\prime}{\rangle}=0$, which can be shown to be equivalent
to excluding the generation of a large–scale flow by the Lorentz force.
Straightforward but tedious algebra using equations (8)–(11)
in equation (5) leads to the following expression for the flux
(the details are given in Appendix A):
$$\displaystyle\Gamma_{i}$$
$$\displaystyle=$$
$$\displaystyle-{\tau\over 2}\left[(\langle u^{2}\rangle-\langle b^{2}\rangle){%
\partial}_{i}\langle\psi\rangle-\langle\psi^{\prime 2}\rangle{\partial}_{i}{%
\nabla^{2}}\langle\psi\rangle\right]\,,$$
(12)
where $\langle u^{2}\rangle=\int d^{3}kk^{2}{\overline{\phi}}({\bf k})$,
$\langle\psi^{\prime 2}\rangle=\int d^{3}k{\overline{\psi}}({\bf k})$, and
$\langle b^{2}\rangle=\int d^{3}kk^{2}{\overline{\psi}}({\bf k})$.
The first term on the right hand side of equation (12) represents the
kinematic turbulent diffusion by fluid advection of the flux;
the second represents the flux coalescence due to the backreaction
of small–scale magnetic fields with the (negative) diffusion coefficient
proportional to the small–scale magnetic energy ${\langle}b^{2}{\rangle}$.
The third term is the hyper-resistivity, reflecting the contribution
to $\Gamma_{i}$ due to
the gradient of a large–scale current ${\langle}J{\rangle}=-{\nabla^{2}}{\langle}\psi{\rangle}$.
($J{\hat{z}}={\nabla\times}{{\bf B}}_{H}$).
Note that the value of hyper-resistivity, being proportional to mean square
potential, is related to the small–scale magnetic energy
as ${\langle}\psi^{\prime 2}{\rangle}=L_{bH}^{2}{\langle}b^{2}{\rangle}$, where
$L_{bH}$ is the typical horizontal scale of ${\bf b}$.
Thus, the negative magnetic diffusion (second) term and hyper-resistivity
(third) term are closely linked through the small–scale magnetic
energy ${\langle}b^{2}{\rangle}$. Indeed, the negative diffusivity and
hyper-resistivity together conserve total ${\langle}\psi^{\prime 2}{\rangle}$, while
shuffling the ${\langle}\psi^{\prime 2}{\rangle}$ spectrum toward large scales.
We now put equation (12) in the following form:
$$\displaystyle{\langle}b^{2}{\rangle}$$
$$\displaystyle=$$
$$\displaystyle{2\Gamma_{i}/\tau+{\langle}u^{2}{\rangle}{\partial}_{i}{\langle}%
\psi{\rangle}\over{\partial}_{i}{\langle}\psi{\rangle}+L_{bH}^{2}{\partial}_{i%
}{\nabla^{2}}{\langle}\psi{\rangle}}\,,$$
(13)
where no summation over the index $i$ occurs.
2.3 Stationary Case: ${\partial}_{t}\langle\psi^{\prime 2}\rangle=0$
To compute the flux $\Gamma_{i}$, we need an additional relation
between ${\langle}b^{2}{\rangle}$ and $\Gamma_{i}$ besides equation (13). This can
be attained by imposing a stationarity condition on ${\langle}\psi^{\prime 2}{\rangle}$.
The stationarity of
fluctuations is achieved in a situation where the energy transfer
from large–scale fields balances the dissipation of fluctuations
locally, as is usually the case in the presence of an external forcing
and dissipation. To obtain this relation, we multiply the equation
for $\psi^{\prime}$ by $\psi^{\prime}$ and then take the average
$$\displaystyle{1\over 2}{\partial}_{t}\langle\psi^{\prime 2}\rangle+\epsilon_{%
ij3}\langle{\partial}_{j}\phi\psi^{\prime}\rangle{\partial}_{i}\langle\psi\rangle$$
$$\displaystyle=$$
$$\displaystyle-\eta\langle({\partial}_{i}\psi^{\prime})^{2}\rangle+B_{0}\langle%
\psi^{\prime}{\partial}_{z}\phi\rangle\,.$$
(14)
Here, the integration by parts was used
assuming that there are no boundary terms. We note that
either when the stationarity
condition is not satisfied or when boundary terms do not
vanish, there will be a correction to our results
(Blackman & Field 2000).
When $\langle\psi^{\prime 2}\rangle$ is stationary, the first term
on the left hand side of equation (14) vanishes, simplifying
the equation that relates $\langle b^{2}\rangle$ to
$\Gamma_{i}={\langle}u_{i}\psi^{\prime}{\rangle}=\epsilon_{ij3}{\langle}{%
\partial}_{i}\phi\psi^{\prime}{\rangle}$ to
the form:
$$\displaystyle\langle({\partial}_{i}\psi^{\prime})^{2}\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle b^{2}\rangle={1\over\eta}\left[-\Gamma_{i}{\partial}_{i}%
\langle\psi\rangle+B_{0}{\langle}\psi^{\prime}{\partial}_{z}\phi{\rangle}%
\right]\,.$$
(15)
Note that in 2D MHD ($B_{0}=0$), the flux is proportional to $\eta{\langle}b^{2}{\rangle}$. This balance reflects the conservation of ${\langle}\psi^{2}{\rangle}$,
which is damped only by Ohmic diffusion.
The second term on the right hand side of equation (15) can be
evaluated in a similar way as for $\Gamma_{i}$, i.e.,
by writing
$$\displaystyle{\langle}\psi^{\prime}{\partial}_{z}\phi{\rangle}$$
$$\displaystyle=$$
$$\displaystyle{\langle}\delta\psi^{\prime}{\partial}_{z}\phi-{\partial}_{z}\psi%
^{\prime}\delta\phi{\rangle}\,,$$
(16)
and then by using equations (8)–(11). Omitting the intermediate
steps (see Appendix A for details), the final result is
$$\displaystyle{\langle}\psi^{\prime}{\partial}_{z}\phi{\rangle}$$
$$\displaystyle=$$
$$\displaystyle\tau B_{0}[{\xi_{v}}{\langle}u^{2}{\rangle}-{\xi_{b}}{\langle}b^{%
2}{\rangle}]\,.$$
(17)
Here
$$\displaystyle{\xi_{v}}$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{3}kk_{z}^{2}{\overline{\phi}}({\bf k})/\int d^{3}kk_{H}^{%
2}{\overline{\phi}}({\bf k})\,,$$
(18)
$$\displaystyle{\xi_{b}}$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{3}kk_{z}^{2}{\overline{\psi}}({\bf k})/\int d^{3}kk_{H}^{%
2}{\overline{\psi}}({\bf k})\,,$$
(19)
and $k_{H}^{2}=k_{x}^{2}+k_{y}^{2}$. If the characteristic
horizontal and vertical scales of ${\bf u}$ are $L_{vH}$ and $L_{vz}$, and if
those of ${\bf b}$ are $L_{bH}$ and $L_{bz}$, then ${\xi_{v}}$ and ${\xi_{b}}$ can
be expressed in terms of these characteristic scales as:
$$\displaystyle{\xi_{v}}={L_{vH}^{2}\over L_{vz}^{2}}\,,$$
$$\displaystyle{\xi_{b}}={L_{bH}^{2}\over L_{bz}^{2}}\,.$$
(20)
Insertion of equation (17) into (15) gives us
$$\displaystyle\langle b^{2}\rangle$$
$$\displaystyle=$$
$$\displaystyle{1\over\eta}\left[-\Gamma_{i}{\partial}_{i}\langle\psi\rangle+%
\tau{\xi_{v}}B_{0}^{2}{\langle}u^{2}{\rangle}\right]/\left(1+{\tau{\xi_{b}}%
\over\eta}B_{0}^{2}\right)\,.$$
(21)
Thus, from equations (13) and (21), we obtain
$$\displaystyle\Gamma_{i}$$
$$\displaystyle=$$
$$\displaystyle-{\tau\over 2}{\langle}u^{2}{\rangle}{1+{\tau\over\eta}B_{0}^{2}(%
{\xi_{b}}-{\xi_{v}})+{\tau L_{bH}^{2}\over\eta}{\xi_{v}}B_{0}^{2}|{{\partial}_%
{i}{\nabla^{2}}{\langle}\psi{\rangle}\over{\partial}_{i}{\langle}\psi{\rangle}%
}|\over 1+{\tau\over\eta}\left[{1\over 2}{\langle}B_{H}{\rangle}^{2}+{\xi_{b}}%
B_{0}^{2}-{L_{bH}^{2}\over 2}{\langle}J{\rangle}^{2}\right]}{\partial}_{i}{%
\langle}\psi{\rangle}\,,$$
(22)
where ${J}{\hat{z}}={\nabla\times}{\bf B}_{H}$ and the integration by part is
used to express ${\partial}_{i}{\langle}\psi{\rangle}{\partial}_{i}{\nabla^{2}}{\langle}\psi{%
\rangle}=-({\nabla^{2}}{\langle}\psi{\rangle})^{2}=-{\langle}J{\rangle}^{2}<0$.
Note the last term in the numerator and denominator
in equation (22) comes from the hyper-resistivity.
Equation (22) is the flux in 3D RMHD, which generalizes the 2D MHD
result (Cattaneo & Vainshtein 1991; Gruzinov & Diamond 1994).
Several aspects of this result are of interest.
First, in the limit
as ${\bf B}_{0}\to 0$ and ${\langle}{\bf B}_{H}{\rangle}\to 0$ (${\langle}J{\rangle}\to 0$), the flux
reduces to the kinematic value $\Gamma_{i}=-\eta_{k}{\partial}_{i}{\langle}\psi{\rangle}$,
with the kinematic turbulent diffusivity $\eta_{k}=\tau{\langle}u^{2}{\rangle}/2$.
This corresponds to the 2D hydrodynamic result where the effect of
the Lorentz force is neglected. The full 2D MHD result can
be obtained by taking the limit ${\bf B}_{0}\to 0$ in equation (22),
which will reproduce equation (1). This agrees
with the well–known result on the suppression of flux
diffusion in 2D (Cattaneo & Vainshtein 1991; Gruzinov & Diamond 1994).
Another interesting case may be the limit ${\langle}{\bf B}_{H}{\rangle}\to 0$.
In fact, this limit can be shown to be consistent with the ordering
of 3D RMHD as follows.
First, note that 3D RMHD ordering
($k_{z}/k_{H}\sim B_{H}/B_{0}\sim\epsilon<1$) requires
${\xi_{b}}B_{0}^{2}\sim{\langle}B_{H}^{2}{\rangle}$.
Since ${\langle}B_{H}{\rangle}^{2}\ll{\langle}B_{H}^{2}{\rangle}\sim{\langle}b^{2}{\rangle}$,
we expect that ${\xi_{b}}B_{0}^{2}\sim{\langle}b^{2}{\rangle}\gg{\langle}B_{H}{\rangle}^{2}$.
Furthermore, $L_{bH}^{2}{\langle}J{\rangle}^{2}\sim(L_{bH}/L_{BH})^{2}{\langle}B_{H}{\rangle%
}^{2}<{\langle}B_{H}{\rangle}^{2}$, where $L_{BH}$ is the characteristic scale of
${\langle}B_{H}{\rangle}$. Thus, the dominant term in the square brackets in
the denominator of equation (22) is ${\xi_{b}}B_{0}^{2}\sim{\langle}b^{2}{\rangle}$.
That is, the effect of $B_{0}$ seems to be stronger
than that of ${\langle}{\bf B}_{H}{\rangle}$ in 3D RMHD.
Finally, to determine whether ${\bf B}_{0}$ enhances the flux or not,
we note that ${\xi_{v}}-{\xi_{b}}$ in equation (22) can be taken to be zero,
since the scales for ${\bf b}$ and ${\bf u}$ are likely to be comparable
in this model, which employs unity magnetic Prandtl number.
Then, we estimate the last term in the numerator, due to
hyper–resistivity, to be
$\tau{\langle}b^{2}{\rangle}L_{bH}^{2}/(\eta L_{BH}^{2})\sim(L_{bH}/L_{BH})^{2}%
R_{m}$ where
${\xi_{b}}B_{0}^{2}\sim{\langle}b^{2}{\rangle}$ and
${\langle}b^{2}{\rangle}\sim{\langle}u^{2}{\rangle}$ are used.
If $(L_{bH}/L_{BH})^{2}\sim R_{m}^{-1}$, this term will be of
order unity.
Note $R_{m}=ul/\eta$ is the magnetic Reynolds number, with $u$ and
$l$ being the characteristic amplitude and length scale of the
velocity.
Therefore, equation (22) indicates that the flux is reduced on account of
the strong axial magnetic field $B_{0}$ as well as the horizontal
reconnecting field ${\langle}{\bf B}_{H}{\rangle}$.
The above analyses will be used
in §4.1 in order to estimate the effective dissipation
and global reconnection rate.
3 EDDY-DAMPED FLUID MODEL
The analysis performed in the previous section introduced an arbitrary
correlation time $\tau$ that is assumed to be the same for both small–scale
velocity and small–scale magnetic fields. Moreover, the quasi–linear
closure is valid
strictly only when the small–scale fields remain weaker than the
large–scale
fields. In order to compensate for these shortcomings, we now consider
an eddy-damped fluid model which is based a large viscosity
(Kim 1999). In this model, the fluid motion is
self–consistently generated by a forcing with a prescribed statistics
as well as by the Lorentz force, without having to assume the presence
of fully developed MHD turbulence, to invoke a quasi–linear closure,
or to introduce an arbitrary correlation time for the fluctuating
fields. This is the simplest model within which the nonlinear
effect of the back–reaction can rigorously be treated. Even though
this model has limited applicability to a system with a large viscosity,
it could be quite relevant to small scale fields in Galaxy where
$\nu\gg\eta$.
As shall be shown later, this model gives rise to an effective correlation
time for the fluctuating magnetic fields that is given by the viscous
time $\tau_{\nu}=l_{bH}^{2}/\nu$, where $l_{bH}$ is the typical scale of
the magnetic fluctuations in the horizontal plane (cf eqs. [22] and [32]).
Thus, in comparison with the $\tau$ approximation in the previous
section, this model is equivalent to replacing $\tau$ by $\tau_{\nu}$
despite the fact that some of detailed results for the two models are not the
same.
3.1 Splitting of Velocity
In a high viscosity limit with the fluid kinetic Reynolds number
$Re=ul/\nu<1$, the nonlinear advection term as well
as inertial term in the momentum equation can be neglected. Then, the
linearity of the remaining terms in the
momentum equation enables us to split the velocity into two components; the
first — random velocity — is solely governed by the random forcing, and
the second — induced velocity — is governed by the Lorentz force only.
Specifically, we express the total velocity ${\bf u}$ as ${\bf u}={\bf v}+{\bf v}^{\prime}$,
where ${\bf v}$ and ${\bf v}^{\prime}$ are the random and induced velocity, respectively,
and introduce velocity potential $\phi_{0}$ and $\phi_{I}$ as ${\bf v}={\nabla\times}\phi_{0}{\hat{z}}$ and ${\bf v}^{\prime}={\nabla\times}\phi_{I}{\hat{z}}$. Then, the
equations for these potentials are:
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle\nu{\nabla^{2}}\phi_{0}+F\,,$$
(23)
$$\displaystyle 0$$
$$\displaystyle=$$
$$\displaystyle\nu{\nabla^{2}}\phi_{I}+{\bf B}\cdot\nabla{\nabla^{2}}\psi\,,$$
(24)
where the nonlinear advection term as well as the inertial term
is neglected since $Re<1$ is assumed.
In equation (23), $F$ is a prescribed forcing with known statistics.
Instead of solving equation (23) for $\phi_{0}$, we can
equivalently prescribe the statistics of the random velocity $\phi_{0}$
(or ${\bf v}$). Therefore, we assume that the statistics of random component
satisfies homogeneity and isotropy in the horizontal plane and
homogeneity and reflectional symmetry in the
$z$ direction, respectively. Furthermore, we assume that it is delta
correlated in time. The correlation function is then given by:
$$\displaystyle\langle\phi_{0}({\bf k}_{1},t_{1})\phi_{0}({\bf k}_{2},t_{2})\rangle$$
$$\displaystyle=$$
$$\displaystyle\delta({\bf k}_{1}-{\bf k}_{2})\delta(t_{1}-t_{2}){\overline{\phi%
}}_{0}(k_{1H},k_{1z})\,,$$
(25)
where ${\overline{\phi}}_{0}(k_{1H},k_{1z})$ is the power spectrum of $\phi_{0}$.
Note that $\tau_{0}{\langle}\phi_{0}^{2}{\rangle}=\int d^{3}k{\overline{\phi}}({\bf k})$
and $\tau_{0}{\langle}v^{2}{\rangle}=\int d^{3}kk^{2}{\overline{\phi}}({\bf k})$,
where $\tau_{0}$ is the correlation time of ${\bf v}$ that is assumed to
be short.
On the other hand, the induced velocity can be constructed by solving
equation (24)
for $\phi_{I}$ in terms of ${\bf B}$. This can easily be done in Fourier
space as:
$$\displaystyle\phi_{I}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle{i\over\nu k^{2}k_{H}^{2}}\left[B_{0}k^{2}k_{H}^{2}+i\epsilon_{ij%
3}\int d^{3}k^{\prime}(k-k^{\prime})_{j}k_{Hi}^{\prime}k_{H}^{\prime 2}\psi({%
\bf k}-{\bf k}^{\prime})\psi({\bf k}^{\prime})\right]\,,$$
(26)
where $B_{Hi}({\bf k})=i\epsilon_{ij3}k_{j}\psi({\bf k})$ is used.
Note that the $\psi$ in the above equation contains both mean
and fluctuating parts.
3.2 Magnetic Field
Both random and induced velocities are to be substituted in
equation (2) to solve for the magnetic field. Notice that
equation (2) then has a cubic nonlinearity, since the induced
velocity is quadratic in ${\bf B}$. We again assume that the magnetic field
in the horizontal plane consists of mean and fluctuating components,
i.e., $\psi={\langle}\psi{\rangle}+\psi^{\prime}$ and that the fluctuation is
homogeneous and isotropic in the $x$-$y$ plane and homogeneous and
reflectionally symmetric in the $z$ direction, satisfying the same
correlation function as equation (10).
To obtain equations for ${\langle}\psi{\rangle}$ and ${\langle}\psi^{2}{\rangle}$, we
utilize the delta–correlation in time of ${\bf v}$ and
iterate equation (2) for small time intervals $\delta t$.
Specifically, we use ${\langle}v_{i}(t_{1})B(t)_{j}{\rangle}=0$ for $t_{1}>t$
and $v\sim O((\delta t)^{-1/2})$ since
${\langle}v_{i}(t_{1})v_{j}(t_{2}){\rangle}\propto\delta(t_{1}-t_{2})\sim 1/{%
\delta t}$, where
${\delta t}=t_{1}-t_{2}$. Then, for $\delta t\ll 1$, equation (2)
can be iterated up to order $O(\delta t)$ as:
$$\displaystyle\psi(t+{\delta t})$$
$$\displaystyle=\psi(t)+{\delta t}\eta{\nabla^{2}}\psi(t)+\int_{t}^{t+{\delta t}%
}dt_{1}\left[\epsilon_{ij3}{\partial}_{j}\psi(t){\partial}_{i}\phi(t_{1})+B_{0%
}{\partial}_{z}\psi(t_{1})\right]$$
$$\displaystyle+{1\over 2}\epsilon_{ij3}\int_{t}^{t+{\delta t}}dt_{1}dt_{2}\left%
[\epsilon_{lm3}{\partial}_{i}\phi(t_{1}){\partial}_{j}[{\partial}_{m}\psi(t){%
\partial}_{l}\phi(t_{2})]+B_{0}{\partial}_{i}\phi(t_{1}){\partial}_{jz}\phi(t_%
{2})\right]+O({\delta t}^{3/2})\,,$$
(27)
where $\psi$ and $\phi$ are to be evaluated at the same spatial
position ${\bf x}$.
The mean field equation is obtained by substituting equation (26)
in (27), by taking the average with the help of equations (10)
and (25), and then by taking the limit ${\delta t}\to 0$.
The derivation is tedious and is outlined
in Appendix B. Here, we give the final result
$$\displaystyle{\partial}_{t}{\langle}\psi{\rangle}$$
$$\displaystyle=$$
$$\displaystyle\eta{\nabla^{2}}{\langle}\psi{\rangle}+\left[{\tau_{0}\over 4}{%
\langle}v^{2}{\rangle}-{1\over 2\nu}G\right]{\nabla^{2}}{\langle}\psi{\rangle}%
-{F\over\nu}{\nabla^{2}}{\nabla^{2}}{\langle}\psi{\rangle}$$
(28)
$$\displaystyle=$$
$$\displaystyle(\eta+\eta_{M}){\nabla^{2}}{\langle}\psi{\rangle}-\mu{\nabla^{2}}%
{\nabla^{2}}{\langle}\psi{\rangle}\,.$$
Here $\tau_{0}$ is the short correlation time of random velocity ${\bf v}$
and
$$\displaystyle\eta_{M}$$
$$\displaystyle\equiv$$
$$\displaystyle{\tau_{0}\over 4}{\langle}v^{2}{\rangle}-{1\over 2\nu}G\equiv\eta%
_{k}-{1\over 2\nu}G\,,$$
$$\displaystyle\mu$$
$$\displaystyle\equiv$$
$$\displaystyle{F\over\nu}\,,$$
$$\displaystyle G$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{3}k{k_{H}^{2}\over k^{2}}{\overline{\psi}}({\bf k})\simeq%
{\langle}\psi^{\prime 2}{\rangle}\equiv\kappa{\langle}b^{2}{\rangle}\,,$$
$$\displaystyle F$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{3}k{k_{H}^{2}k_{z}^{2}\over k^{6}}{\overline{\psi}}({\bf k%
})\simeq{L_{bH}^{4}\over L_{bz}^{2}}G\equiv\gamma G\,,$$
where $\eta_{k}=\tau_{0}{\langle}v^{2}{\rangle}/4$ is the kinematic diffusivity;
$\kappa\equiv L_{bH}^{2}$ and $\gamma\equiv L_{bH}^{4}/L_{bz}^{2}=\kappa{\xi_{b}}$.
The above equation implies that the flux $\Gamma_{i}={\langle}u_{i}\psi^{\prime}{\rangle}$ is given by
$$\displaystyle\Gamma_{i}$$
$$\displaystyle=$$
$$\displaystyle-\eta_{M}{\partial}_{i}{\langle}\psi{\rangle}+\mu{\partial}_{i}{%
\nabla^{2}}{\langle}\psi{\rangle}\,.$$
(29)
Again, the two terms in $\eta_{M}$ are due to the kinematic
turbulent diffusivity and backreaction. Note that the
kinematic diffusivity $\eta_{k}=\tau_{0}{\langle}v^{2}{\rangle}/4$ now comes
only from the random velocity, with $\tau_{0}$ being its correlation
time that can be prescribed. The backreaction term is proportional
to ${\langle}\psi^{\prime 2}{\rangle}$, not ${\langle}b^{2}{\rangle}$ (cf. eq. [11]) and
inversely proportional to the viscosity $\nu$. It is because the cutoff
scale of the magnetic field $l_{\eta}$ is smaller than that of the
velocity $l_{\nu}$ in this model so that for a larger $\nu$, there are
magnetic modes over a larger interval of scale $l$ between $l_{\eta}$
and $l_{\nu}$ (i.e. $l_{\eta}<l<l_{\nu}$) where the velocity is absent
due to viscous damping. That is, the induced velocity (Lorentz force)
cannot be generated on this scale ($l_{\eta}<l<l_{\nu}$) due to viscous
damping, thereby weakening the overall effect of backreaction
(see eq. [48]).
Now, the last term in equation (29) is the contribution from
the hyper-resistivity $\mu$. It is interesting to see that $\mu$
is inversely proportional to $L_{bz}^{2}$ and thus
vanishes as $L_{bz}\to\infty$ (or $\gamma\to 0$) which corresponds
to the 2D limit. Therefore,
in this eddy-damped fluid model, the hyper-resistivity
term vanishes in two dimensions. It should be contrasted to the case
considered in the previous section where the hyper-resistivity,
being proportional ${\langle}\psi^{\prime 2}{\rangle}$, survives in 2D MHD limit
(see eq. [12]).
For use later, we solve equation (29) for ${\langle}b^{2}{\rangle}$ yielding
$$\displaystyle{\langle}b^{2}{\rangle}$$
$$\displaystyle=$$
$$\displaystyle{\Gamma_{i}+\eta_{k}{\partial}_{i}{\langle}\psi{\rangle}\over{%
\kappa\over 2\nu}{\partial}_{i}{\langle}\psi{\rangle}+{\kappa\gamma\over\nu}{%
\partial}_{i}{\nabla^{2}}{\langle}\psi{\rangle}}\,,$$
(30)
where again the summation over the index $i$ is not implied.
3.3 Stationary Case: ${\partial}_{t}{\langle}\psi^{\prime 2}{\rangle}=0$
The additional relation between the flux $\Gamma_{i}$ and
magnetic energy ${\langle}b^{2}{\rangle}$ is obtained for the case of
stationary $\langle\psi^{\prime 2}\rangle$. To derive an equation
for ${\langle}\psi^{2}{\rangle}$, we
multiply equation (27) by itself, take average, and then
take the limit of $\delta t\to 0$. After considerable
algebra (see Appendix B), we obtain the following equation
$$\displaystyle{\partial}_{t}{\langle}\psi^{\prime 2}{\rangle}+{\partial}_{t}{%
\langle}\psi{\rangle}^{2}-2\eta\left[-{\langle}({\partial}_{i}\psi)^{2}{%
\rangle}+{\langle}\psi{\rangle}{\nabla^{2}}{\langle}\psi{\rangle}\right]$$
$$\displaystyle=$$
$$\displaystyle B_{0}^{2}\left[{\xi_{v}}{\langle}v^{2}{\rangle}-{2\over\nu}{%
\overline{G}}\right]\,,$$
(31)
where
$$\displaystyle{\overline{G}}$$
$$\displaystyle\equiv$$
$$\displaystyle\int d^{3}k{k_{z}^{2}\over k^{2}}{\overline{\psi}}({\bf k})\sim{L%
_{bz}^{2}\over L_{bH}^{2}}G={\xi_{b}}G\,,$$
In a stationary case, equations (28), (30), and (31) lead us to
the following expression for the flux:
$$\displaystyle\Gamma_{i}$$
$$\displaystyle=$$
$$\displaystyle-{\tau_{0}\over 4}{\langle}v^{2}{\rangle}{1+{\kappa\over\eta\nu}B%
_{0}^{2}({\xi_{b}}-{\xi_{v}})+{2\kappa\gamma\over\eta\nu}{\xi_{v}}B_{0}^{2}%
\left|{\partial}_{i}{\nabla^{2}}{\langle}\psi{\rangle}\over{\partial}_{i}{%
\langle}\psi{\rangle}\right|\over 1+{\kappa\over\eta\nu}\left[{\xi_{b}}B_{0}^{%
2}+{1\over 2}{\langle}B_{H}{\rangle}^{2}-\gamma{\langle}J{\rangle}^{2}\right]}%
{\partial}_{i}{\langle}\psi{\rangle}\,,$$
(32)
where ${J}{\hat{z}}={\nabla\times}{\bf B}_{H}$, and
${\partial}_{i}{\nabla^{2}}{\langle}\psi{\rangle}{\partial}_{i}{\langle}\psi{%
\rangle}=-({\nabla^{2}}{\langle}\psi{\rangle})^{2}=-{\langle}J{\rangle}^{2}<0$ is used. When the characteristic
scales of fluctuating velocity and magnetic field are
comparable, or when only the ratios of vertical to horizontal scales
of the fluctuating velocity and magnetic fields are comparable,
${\xi_{v}}$ can be taken to be equal
to ${\xi_{b}}$, simplifying the above expression.
It is worth considering a few interesting limits of equation (32).
First, in the limit $B_{0}\to 0$ and $B_{H}\to 0$, equation (32)
again recovers the 2D hydrodynamic result with the kinematic
diffusivity $\eta_{k}=\tau_{0}{\langle}v^{2}{\rangle}/4$. The limit
$B_{0}\to 0$ leads to 2D MHD case where the suppression of
the turbulent diffusion arises from ${\langle}{\bf B}_{H}{\rangle}$.
In 3D RMHD, the dominant suppression in the flux comes from
$B_{0}$ when ${\xi_{v}}={\xi_{b}}$, as discussed in §2.3.
We note that the last term in the numerator and denominator
is due to the hyper-resistivity,
which comes with a multiplicative factor $\gamma=L_{bH}^{2}/L_{bz}^{2}\ll 1$. Therefore, the effect of hyper-resistivity
can be neglected as compared to other terms in equation (32).
Since $\gamma\to 0$ in 2D MHD, there is no contribution
from the hyper-resistivity to the flux in 2D in this model.
The estimate of the effective
dissipation in this model is provided in §4.2.
It is very interesting to compare equation (32) with (22). We recall
that in order to derive equation (22), the same correlation time $\tau$
was assumed for both fluctuating magnetic field and velocity, which
appears in front of the mean magnetic fields $B_{0}$ and ${\langle}\psi{\rangle}$
in equation (22). In contrast, $\tau_{0}$ in equation (32)
is the correlation time of the random component of the velocity,
which can be arbitrarily prescribed. Moreover, $\tau$ in front of
mean magnetic fields in equation (22) is now replaced by viscous
time scale $\tau_{\nu}=\kappa/\nu=L_{bH}^{2}/\nu$ in equation (32).
The latter
represents the viscous time scale across the typical horizontal scale
of fluctuating magnetic fields. Thus, as noted at the beginning
of this section, this viscous time $\tau_{\nu}$ replaces $\tau$ in
the quasi–linear closure, which was assumed to be a parameter.
4 RECONNECTION RATE
In previous sections, the flux $\Gamma_{i}$ was derived by using a
quasi–linear closure and an eddy-damped fluid model. Assuming the
flux $\Gamma_{i}$ has a form proportional to ${\partial}_{i}{\langle}\psi{\rangle}$ in both
cases (see eqs. [22] and [32]), it can be expressed in terms of
the effective dissipation rate (or, turbulent
diffusivity) $\eta_{eff}$ as follows:
$$\displaystyle\Gamma_{i}$$
$$\displaystyle=$$
$$\displaystyle-\eta_{eff}{\partial}_{i}{\langle}\psi{\rangle}\,.$$
(33)
Upon using equation (33), the mean field equation (4) then becomes
$$\displaystyle{\partial}_{t}{\langle}\psi{\rangle}$$
$$\displaystyle=$$
$$\displaystyle(\eta+\eta_{eff}){\nabla^{2}}{\langle}\psi{\rangle}\equiv\eta_{T}%
{\nabla^{2}}{\langle}\psi{\rangle}\,.$$
(34)
where $\eta_{T}\equiv\eta+\eta_{eff}$ is the total dissipation rate of
the mean field. The effective dissipation rate is the quantity that
represents the overall decay rate of a large–scale magnetic field
due to both small–scale motions and magnetic fluctuations.
That is, the dynamical system consisting of both small and
large scale fields can be represented by the evolution
of a large–scale field only when the effect of small–scale
fields is absorbed in this turbulent coefficient.
In order to determine a global reconnection rate, we now invoke
the original SP type balance equations and use the total dissipation rate
in place of the Ohmic diffusivity (see §2):
$$\displaystyle v_{r}$$
$$\displaystyle=$$
$$\displaystyle{v_{A}\over\sqrt{v_{A}L/\eta_{T}}}\,.$$
(35)
Note that we have neglected a multiplicative correction factor to
the reconnection rate in the eddy-damped model since its
dependence on $\nu$ is weak with $1/4$ power
(for instance, see, Biskamp 1993).
In the following subsections, we assume ${\xi_{v}}={\xi_{b}}$ for
simplicity and estimate the reconnection rate via
equation (35). Then, we briefly comment on the implication
for reconnection assuming ‘Alfvénic turbulence’, as Lazarian
and Vishniac (1999) did.
4.1 Using the Quasi-linear Result
The effective dissipation rate follows from equations (22) and
(32):
$$\displaystyle\eta_{eff}$$
$$\displaystyle\simeq$$
$$\displaystyle{\tau\over 2}{\langle}u^{2}{\rangle}{1+{\tau L_{bH}^{2}\over\eta}%
{\xi_{v}}B_{0}^{2}|{{\partial}_{i}{\nabla^{2}}{\langle}\psi{\rangle}\over{%
\partial}_{i}{\langle}\psi{\rangle}}|\over 1+{\tau\over\eta}\left[{1\over 2}{%
\langle}B_{H}{\rangle}^{2}+{\xi_{b}}B_{0}^{2}-{L_{bH}^{2}\over 2}{\langle}J{%
\rangle}^{2}\right]}\,,$$
(36)
after using ${\xi_{v}}={\xi_{b}}$.
As shown in §2.3, the dominant term in the square brackets in
the denominator of equation (36) is ${\xi_{b}}B_{0}^{2}\sim{\langle}b^{2}{\rangle}$,
and the second term in the numerator is of order unity
if $L_{BH}^{2}/L_{bH}^{2}\sim R_{m}$. In that case,
$\eta_{eff}$ is roughly given by
$$\displaystyle\eta_{eff}$$
$$\displaystyle\sim$$
$$\displaystyle\eta_{k}{1\over 1+\tau{\langle}b^{2}{\rangle}/\eta}\sim\eta_{k}{1%
\over 1+2R_{m}{\langle}b^{2}{\rangle}/{\langle}u^{2}{\rangle}}\,,$$
(37)
where $\eta_{k}=\tau{\langle}u^{2}{\rangle}/2$ is the kinematic value
of turbulent diffusivity in 2D and $R_{m}=\eta_{k}/\eta$.
In contrast to the 2D MHD result
(eq. [1])), the equation (37) reveals that
the effective diffusivity in 3D RMHD is more severely reduced
as ${\langle}b^{2}{\rangle}\gg{\langle}B_{H}{\rangle}^{2}$ ($={\langle}B{\rangle}^{2}$).
To determine the leading order contribution in equation (37),
we need to estimate ${\langle}b^{2}{\rangle}$. To do so, we
substitute equations (33) and (37) in (13) and use $L_{bH}<L_{BH}$
to obtain:
$$\displaystyle{\langle}b^{2}{\rangle}$$
$$\displaystyle\sim$$
$$\displaystyle{\langle}u^{2}{\rangle}-{\eta\over\tau}\sim{\langle}u^{2}{\rangle%
}\left[1-{1\over 2R_{m}}\right]\,,$$
(38)
where $R_{m}=\eta_{k}/\eta=\tau{\langle}u^{2}{\rangle}/2\eta$
is used.
We note that ${\langle}b^{2}{\rangle}>0$ is guaranteed since
${\langle}b^{2}{\rangle}>{\langle}B_{H}{\rangle}^{2}$ (implying $R_{m}>1$) was assumed to
derive the above equation.
Thus,
$$\displaystyle{\tau{\langle}b^{2}{\rangle}\over\eta}$$
$$\displaystyle\sim$$
$$\displaystyle 2R_{m}-{1}\,.$$
That is, for $R_{m}\gg 1$, $\tau{\langle}b^{2}{\rangle}/\eta\gg 1$. Insertion
of the above equation in (37) then gives us
$$\displaystyle\eta_{eff}$$
$$\displaystyle\sim$$
$$\displaystyle\eta_{k}{1\over 2R_{m}}\sim{\eta\over 2}\,.$$
(39)
In other words, to leading order, the effective dissipation rate is
just that given by Ohmic diffusivity! Therefore, by inserting equation (39)
into (35) with $\eta_{T}=\eta+\eta_{eff}$,
the reconnection rate is found to have the original
SP scaling with $\eta$, i.e.
$$\displaystyle v_{r}$$
$$\displaystyle\sim$$
$$\displaystyle{v_{A}\over\sqrt{v_{A}L/\eta}}\,.$$
(40)
It is interesting to contrast this result to the 2D case where
$B_{0}=0$. In that case, the dominant term in equation (36)
is ${\langle}B_{H}{\rangle}^{2}$, with $\eta_{eff}\sim\eta_{k}{\langle}u^{2}{\rangle}/R_{m}{\langle}B_{H}{\rangle}^{2}%
\sim\eta{\langle}u^{2}{\rangle}/{\langle}B_{H}{\rangle}^{2}\sim\eta u^{2}/v_{A%
}^{2}>\eta$, where $u$ is the typical velocity.
Therefore, in 2D, the global reconnection rate becomes
$$\displaystyle v_{r}$$
$$\displaystyle\sim$$
$$\displaystyle{v_{A}\over\sqrt{v_{A}L/\eta}}{u\over v_{A}}\,,$$
(41)
which is larger than SP by a factor of magnetic Mach number
$M_{A}=u/v_{A}$.
Note that the reduction in the effective dissipation
of a large–scale magnetic field is more severe
in 3D RMHD than in 2D MHD by a factor of ${\langle}u^{2}{\rangle}/{\langle}B_{H}{\rangle}^{2}\sim{\langle}u^{2}{\rangle}%
/v_{A}^{2}$.
4.2 Using the Eddy-Damped Fluid Model Result
For an eddy-damped fluid model, equation (32) yields:
$$\displaystyle\eta_{eff}$$
$$\displaystyle=$$
$$\displaystyle{\tau_{0}\over 4}{\langle}v^{2}{\rangle}{1+{2\kappa\gamma\over%
\eta\nu}{\xi_{v}}B_{0}^{2}\left|{\partial}_{i}{\nabla^{2}}{\langle}\psi{%
\rangle}\over{\partial}_{i}{\langle}\psi{\rangle}\right|\over 1+{\kappa\over%
\eta\nu}\left[{\xi_{b}}B_{0}^{2}+{1\over 2}{\langle}B_{H}{\rangle}^{2}-\gamma{%
\langle}J{\rangle}^{2}\right]}\,,$$
(42)
after assuming ${\xi_{v}}={\xi_{b}}$. We recall that the contribution from
the hyper-resistivity comes with a multiplicative factor $\gamma=L_{bH}^{2}/L_{bz}^{2}\ll 1$ (vanishing in the 2D MHD limit) and thus can be
neglected as compared to other terms in equation (42). Then, a similar
estimation as in §4.1 simplifies equation (42) to
$$\displaystyle\eta_{eff}$$
$$\displaystyle\sim$$
$$\displaystyle\eta_{k}{1\over 1+{\kappa\over\nu\eta}{\langle}b^{2}{\rangle}}\,,$$
(43)
where $\eta_{k}=\tau_{0}{\langle}v^{2}{\rangle}/4$ is the kinematic value of the turbulent
diffusivity in 2D and $\kappa=L_{bH}^{2}$. To obtain the leading order
behavior of equation (43), we estimate ${\langle}b^{2}{\rangle}$ with the help of
equation (30) to be
$$\displaystyle{\langle}b^{2}{\rangle}$$
$$\displaystyle\sim$$
$$\displaystyle{\eta\nu\over\kappa}(2R_{m}-1)\,,$$
(44)
where $R_{m}=\eta_{k}/\eta$.
By inserting equation (44) in (43), we obtain
$$\displaystyle\eta_{eff}$$
$$\displaystyle\sim$$
$$\displaystyle{\eta_{k}\over 2R_{m}}\sim{\eta\over 2}\,.$$
(45)
Thus, the reconnection rate is again given by
$$\displaystyle v_{r}$$
$$\displaystyle\sim$$
$$\displaystyle{v_{A}\over\sqrt{v_{A}L/\eta}}\,,$$
(46)
i.e., SP scaling with $\eta$ persists!
It is interesting to estimate ${\langle}b^{2}{\rangle}$ in equation (44)
by using
$$\displaystyle{\eta\nu\over\kappa}$$
$$\displaystyle=$$
$$\displaystyle{\langle}v^{2}{\rangle}{\eta\over\sqrt{{\langle}v^{2}{\rangle}}L_%
{bH}}{\nu\over\sqrt{{\langle}v^{2}{\rangle}}L_{bH}}\sim{\langle}v^{2}{\rangle}%
{1\over R_{m}R_{e}}\,,$$
(47)
where $R_{e}=\sqrt{{\langle}v^{2}{\rangle}}L_{bH}/\nu$ is the fluid Reynolds number.
Thus, equation (44) becomes
$$\displaystyle{\langle}b^{2}{\rangle}$$
$$\displaystyle\sim$$
$$\displaystyle{\langle}v^{2}{\rangle}{1\over R_{e}}\left(2-{1\over R_{m}}\right%
)\,.$$
(48)
The above equation clearly demonstrates that ${\langle}b^{2}{\rangle}>{\langle}v^{2}{\rangle}$
for our model ($R_{e}<1$) when $R_{m}>1$, as pointed out near the
end of §3.2.
Finally, we note that in 2D limit with $B_{0}\to 0$,
the dominant term in the square brackets in the denominator of
equation (42) is ${\langle}B_{H}{\rangle}^{2}$.
Thus, $\eta_{eff}\sim\eta_{k}{\langle}v^{2}{\rangle}/R_{e}R_{m}{\langle}B_{H}{\rangle%
}^{2}\sim\eta{\langle}v^{2}{\rangle}/R_{e}{\langle}B_{H}{\rangle}^{2}\sim\eta u%
^{2}/R_{e}v_{A}^{2}>\eta$, where $u$ is the typical velocity.
Therefore, in 2D, the global reconnection rate becomes
$$\displaystyle v_{r}$$
$$\displaystyle\sim$$
$$\displaystyle{1\over\sqrt{R_{e}}}{v_{A}\over\sqrt{v_{A}L/\eta}}{u\over v_{A}}\,,$$
(49)
where $u/v_{A}=M_{A}$ is the magnetic Mach number.
In comparison with equation (41), the global reconnection rate
in this model is thus larger in the 2D limit (recall $R_{e}<1$).
4.3 Alfvénic Turbulence
In Alfvénic turbulence (Goldreich & Sridhar 1994; 1995; 1997),
the equipartition between ${\langle}b^{2}{\rangle}$
and ${\langle}u^{2}{\rangle}$ is assumed from the start. It is to be contrasted to
the present analysis in which the relation between ${\langle}b^{2}{\rangle}$ and
${\langle}u^{2}{\rangle}$ i.e., equations (38) and (49), follows from the condition
of stationarity of ${\langle}\psi^{\prime 2}{\rangle}$ in the presence of ${\bf B}_{0}$
and ${\langle}{\bf B}_{H}{\rangle}$. As can be seen from equation (38),
in the quasi–linear closure with unity magnetic Prandtl number,
exact equipartition is possible only for $\eta=0$. In the
eddy-damped fluid model, exact equipartition can never be satisfied
since the assumption $R_{e}<1$ implies ${\langle}b^{2}{\rangle}>{\langle}v^{2}{\rangle}$
when $R_{m}>1$ (see eq. [48])!
Therefore, in general, stationarity of ${\langle}\psi^{\prime 2}{\rangle}$ and exact
Alfvénic equipartition cannot be simultaneously achieved. In other
words, if Alfvénic turbulence is assumed, ${\langle}\psi^{\prime 2}{\rangle}$
cannot be stationary; if ${\langle}\psi^{\prime 2}{\rangle}$ is stationary,
the turbulence cannot be in a state of Alfvénic equipartition.
We easily confirm this in 2D MHD by quasi–linear closure. The exact
equipartition (${\langle}u^{2}-b^{2}{\rangle}=0$) implies that the flux $\Gamma_{i}$
in equation (12) is given by hyper-resistivity only:
$\Gamma_{i}=-\tau{\langle}\psi^{\prime 2}{\rangle}{\partial}_{i}{\nabla^{2}}{%
\langle}\psi{\rangle}/2$.
Then, if we were to impose the stationarity of ${\langle}\psi^{\prime 2}{\rangle}$,
equation (15) would indicate ${\langle}\psi^{\prime 2}{\rangle}\tau{\partial}{\langle}J{\rangle}{\langle}B_{%
H}{\rangle}=\eta{\langle}b^{2}{\rangle}$. Thus,
$$\displaystyle{{\langle}B_{H}{\rangle}^{2}\over{\langle}b^{2}{\rangle}}R_{m}$$
$$\displaystyle\sim$$
$$\displaystyle\left({l_{B}\over l_{b}}\right)^{2}\,,$$
(50)
where $l_{B}$ and $l_{b}$ are the characteristic scales of ${\langle}{\bf B}_{H}{\rangle}$
and ${\bf b}$, respectively. Since ${\langle}B_{H}{\rangle}^{2}/{\langle}u^{2}{\rangle}\sim 1/R_{m}$
(with ${\langle}b^{2}{\rangle}\sim{\langle}u^{2}{\rangle}$)
and $(l_{B}/l_{b})^{2}\sim 1/R_{m}$ in 2D MHD, the relation (49)
(for stationarity) cannot be satisfied.
5 CONCLUSION AND DISCUSSIONS
In view of the ubiquity of turbulence in space and astrophysical plasmas,
magnetic reconnection will likely occur in an environments with
turbulence. On the other hand, the reconnection itself generates
small–scale fluctuation, feeding back the turbulence. Thus, it is
important to treat these two processes consistently, accounting for
the back reaction. Although LV argued that the local reconnection
rate can be fast, they basically neglected the dynamic coupling between
small and large scale fields, therefore leaving the issue of the global
reconnection rate unresolved. The coupling between global and local
reconnection rates should be treated self consistently.
The aim of the present work was to shed some light on this issue
by taking the simplest approach that is analytically tractable.
Our main strategy was to self–consistently compute the effective
dissipation rate of a large–scale magnetic field within the current sheet
by using stationarity of ${\langle}\psi^{\prime 2}{\rangle}$ and then
use the effective dissipation rate in SP type balance relations to
obtain the global reconnection rate.
To avoid the null point problem associated with a 2D slab model,
we considered 3D RMHD, within which we can solidly justify the
incompressibility
of the fluid in the horizontal plane. To facilitate analysis, two
models (methods) were employed, one being a quasi–linear closure
with $\tau$ approximation and the other eddy-damped fluid model.
The effective dissipation rate $\eta_{eff}$ that we obtained
generalizes the 2D MHD result (Cattaneo & Vainshtein 1991;
Gruzinov & Diamond 1994).
The quasi–linear closure predicted $\eta_{eff}\sim\eta_{k}/(1+2R_{m}{\langle}b^{2}{\rangle}/{\langle}u^{2}{\rangle%
})\sim\eta/2$
(see eqs. [37]–[39]).
A similar result was obtained in the eddy-damped fluid model
with $\eta_{eff}\sim\eta_{k}/(1+R_{m}R_{e}{\langle}b^{2}{\rangle}/{\langle}u^{2}{%
\rangle})\sim\eta/2$ (see eqs. [43]–[45] and [47]).
The 2D result can simply be recovered from our results on the flux
by taking the limit $B_{0}\to 0$. In that limit,
$\eta_{eff}\sim\eta_{k}/(1+R_{m}{\langle}B_{H}{\rangle}^{2}/{\langle}u^{2}{%
\rangle})$
according to the quasi–linear closure,
consistent with previous work.
In the eddy-damped fluid model,
$\eta_{eff}\sim\eta_{k}/(1+R_{m}R_{e}{\langle}B_{H}{\rangle}^{2}/{\langle}u^{2}%
{\rangle})$.
Since the effective dissipation rate $\eta_{eff}$ was found
to be the same in both models (in 3D RMHD), the global reconnection,
obtained by invoking SP balance relations, was also the same
with the value $v_{r}\sim v_{A}/\sqrt{v_{A}L/\eta}$ in both models. This
result indicates that
the global reconnection rate is suppressed for large $R_{m}$
as an inverse power of $R_{m}^{1/2}$ such that the original SP
scaling with $\eta$ persists. Again, this persistent $\eta$ scaling
results from the reduction in the effective
dissipation rate of a large–scale magnetic field for large $R_{m}$
mainly due to a strong axial magnetic field, with the effective
dissipation rate $\eta_{eff}\sim\eta$.
Furthermore, in the 2D limit, the quasi–linear closure
yielded the global reconnection rate $v_{r}\sim(v_{A}/\sqrt{v_{A}L/\eta})(u/v_{A})$, which is enhanced over SP by a factor
of $M_{A}=u/v_{A}$ (note that $M_{A}$ can be large).
In contrast, the eddy-damped fluid model gave
$v_{r}\sim\sqrt{R_{e}}^{-1}(v_{A}/\sqrt{v_{A}L/\eta})(u/v_{A})$.
The implication of these results for the LV scenario is that
no matter how fast local reconnection
events proceed, there is not enough energy transfer from large–scale
to small–scale magnetic fields to allow fast global reconnection. Therefore,
global reconnection cannot be given by a simple sum of the
local reconnection events as LV suggested. We emphasize again
that the ${\langle}\psi^{\prime 2}{\rangle}$ balance played the crucial role
in determining the global reconnection rate consistently.
Alternatively, an accurate calculation of the global reconnection
rates requires that (global) topological conservation laws
be enforced.
The reduction in the effective dissipation in 2D is closely linked to
the conservation of mean square magnetic potential. In 3D RMHD,
the mean square of parallel component of potential is no longer an
ideal invariant due to the propagation of Alfvén waves along a
strong axial magnetic field. Nevertheless, the conservation
of mean magnetic potential is broken only linearly, which
turned out to introduce additional suppression factors, as
compared to 2D. The
interesting question is then how relevant these results would be in
3D. The mean square potential is not an invariant of 3D MHD.
However, its conservation is broken nonlinearly, unlike 3D RMHD. Therefore,
the effective dissipation in 3D MHD may be very
different from that in 3D RMHD, with the possibility that
the former may not be reduced, at least,
in the weak magnetic field limit (Gruzinov & Diamond 1994; Kim 1999).
Moreover, in 3D, there is a possibility of a dynamo,
which brings in an additional transport coefficient (the $\alpha$ effect)
into the problem. Some insights into the problem
of effective dissipation of a large–scale field in the presence
of a dynamo process might be obtained by considering a simple extension
of the present 3D RMHD model by allowing a large–scale
dynamo in the horizontal plane. Recall that this possibility was
ruled out in the present paper by assuming isotropy in the horizontal
plane and reflectional symmetry in the axial direction, with no helicity
term (i.e., no correlation between horizontal and vertical component
of fluctuations).
Considering some of limitations of the two models that were analyzed
in the paper, such as the $\tau$ approximation, quasi–linear closure,
low kinetic Reynolds number limit, etc,
it will be very interesting to investigate our predictions via
numerical computation. The stationarity of ${\langle}\psi^{\prime 2}{\rangle}$
can be maintained as long as there is an energy source in the
system, such as an external forcing. By incorporating the proper
ordering required for 3D RMHD, one can measure the decay rate of
${\langle}{\bf B}_{H}{\rangle}$ to check our predictions for $\eta_{eff}\sim\eta$
(see eqs. [40] and [46]).
Ultimately, a numerical simulation with a simple reconnection
configuration should be performed to measure a global reconnection rate
as a function of $R_{m}$ as well as $B_{0}$ and ${\langle}B_{H}{\rangle}$.
It will also be interesting to investigate non–stationary
states such as plasmoid formation (Forbes & Priest 1983;
Priest 1984; Matthaeus & Lamkin 1986).
We thank E. Zweibel for bringing this problem to our attention
and for many interesting discussions. We also thank E.T. Vishniac
and A.S. Ware for stimulating conversations.
This research was supported by U.S. DOE FG03-88ER 53275.
P.H. Diamond also acknowledges partial support from the National
Science Foundation under Grant No. PHY99-07949 to the Institute
for Theoretical Physics at U.C.S.B., where part of this work
was performed. E. Kim acknowledges partial support from
HAO/NCAR where part of this work was completed.
Appendix A
In this appendix, we provide some of steps leading to equations (12)
and (17). First, to derive equation (12), we
let $\Gamma_{i}=\Gamma_{i}^{(1)}-\Gamma_{i}^{(2)}$, where
$\Gamma_{i}^{(1)}=\epsilon_{ij3}\langle{\partial}_{j}\phi\psi^{\prime}\rangle$ and
$\Gamma_{i}^{(2)}=\epsilon_{ij3}\langle\phi{\partial}_{i}\psi^{\prime}\rangle$,
and begin with $\Gamma_{i}^{(1)}$.
$$\displaystyle\Gamma_{i}^{(1)}$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}\langle{\partial}_{j}\phi\psi^{\prime}\rangle$$
(A.1)
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}\int d^{3}k_{1}d^{3}k_{2}ik_{1j}{\langle}\phi({\bf k%
}_{1})\delta\psi^{\prime}({\bf k}_{2}){\rangle}\exp{\{i({\bf k}_{1}+{\bf k}_{2%
})\cdot{\bf x}\}}\,.$$
After inserting equation (8) in (A1) and using equation (11),
we can easily obtain
$$\displaystyle\Gamma_{i}^{(1)}$$
$$\displaystyle=$$
$$\displaystyle-i\tau\epsilon_{ij3}\epsilon_{lm3}\int d^{3}k_{1}d^{3}kk_{1j}k_{%
im}k_{l}{\overline{\phi}}({\bf k}_{1}){\langle}\psi({\bf k}){\rangle}e^{i{\bf k%
}\cdot{\bf x}}+\tau\epsilon_{ij3}\int d^{3}kk_{1j}k_{1z}B_{0}{\overline{\phi}}%
({\bf k}_{1})$$
(A.2)
$$\displaystyle=$$
$$\displaystyle-{\tau\over 2}{\partial}_{l}{\langle}\psi{\rangle}\delta_{il}\int
d%
^{3}k_{1}k_{1}^{2}{\overline{\phi}}({\bf k}_{1})=-{\tau\over 2}{\langle}u^{2}{%
\rangle}{\partial}_{i}{\langle}\psi{\rangle}\,.$$
where ${\langle}u^{2}{\rangle}=\int d^{3}k_{1}k_{1}^{2}{\overline{\phi}}({\bf k}_{1})$.
To obtain the last line in equation (A2), we use the following
relations
$$\displaystyle\int d^{3}kk_{j}k_{m}{\overline{\phi}}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}\delta_{jm}\int d^{3}kk^{2}{\overline{\phi}}({\bf k})\,,$$
$$\displaystyle\int d^{3}kk_{j}k_{z}{\overline{\phi}}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle 0\,,$$
(A.3)
which follows from the isotropy of $\phi$ in the $x$-$y$ plane,
and reflectional symmetry in the $z$ direction.
The second part, $\Gamma_{i}^{(2)}$, is calculated in a similar way.
$$\displaystyle\Gamma_{i}^{(2)}$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}\langle\phi{\partial}_{j}\psi^{\prime}\rangle$$
(A.4)
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}\int d^{3}k_{1}d^{3}k_{2}ik_{2j}{\langle}\delta\phi%
({\bf k}_{1})\psi^{\prime}({\bf k}_{2}){\rangle}\exp{\{i({\bf k}_{1}+{\bf k}_{%
2})\cdot{\bf x}\}}\,.$$
We insert equation (9) in (A4) and use (10) to obtain
$$\displaystyle\Gamma_{i}^{(2)}$$
$$\displaystyle=$$
$$\displaystyle i\tau\epsilon_{ij3}\biggl{[}-iB_{0}\int d^{3}k_{1}k_{1z}k_{1j}{%
\overline{\psi}}({\bf k}_{1})$$
(A.5)
$$\displaystyle+\epsilon_{lm3}\int d^{3}k_{2}d^{3}ke^{i{\bf k}\cdot{\bf x}}{1%
\over({\bf k}+{\bf k}_{2})^{2}}\left[k_{m}k_{2l}k_{2}^{2}+k_{2m}k_{l}k^{2}%
\right]k_{2j}{\overline{\psi}}({\bf k}_{2}){\langle}\psi({\bf k}){\rangle}%
\biggr{]}$$
Since ${\langle}\psi{\rangle}$ has a scale much larger than $\psi^{\prime}$,
$k_{2}\gg k$ in the second integral on the right hand side. We thus expand
the integrand of this second term and use the following isotropy relations:
$$\displaystyle\int d^{3}kk_{j}k_{m}{\overline{\psi}}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}\delta_{jm}\int d^{3}kk^{2}{\overline{\psi}}({\bf k})\,,$$
$$\displaystyle\int d^{3}kk_{i}k_{j}k_{l}k_{m}{\overline{\psi}}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle{1\over 8}\left(\delta_{ij}\delta_{lm}+\delta_{il}\delta_{jm}+%
\delta_{im}\delta_{jl}\right)\int d^{3}kk^{4}{\overline{\psi}}(k)\,,$$
$$\displaystyle\int d^{3}kk_{i}k_{z}{\overline{\psi}}({\bf k})$$
$$\displaystyle=$$
$$\displaystyle 0\,.$$
(A.6)
A bit of algebra then gives us
$$\displaystyle\Gamma_{i}^{(2)}$$
$$\displaystyle=$$
$$\displaystyle{\tau\over 2}\left[-{\langle}b^{2}{\rangle}{\partial}_{i}{\langle%
}\psi({\bf x}){\rangle}-{\langle}\psi^{\prime 2}{\rangle}{\partial}_{i}{\nabla%
^{2}}{\langle}\psi({\bf x}){\rangle}\right]\,.$$
(A.7)
Thus, from equations (A3) and (A7), we obtain equation (12)
in the main text.
Second, to derive equation (17), we again compute the correlation
function on the right hand side of equation (16) in Fourier
space. The first term can be rewritten as:
$$\displaystyle\langle\delta\psi^{\prime}{\partial}_{z}\phi\rangle$$
$$\displaystyle=$$
$$\displaystyle\int d^{3}k_{1}d^{3}k_{2}ik_{1z}{\langle}\phi({\bf k}_{1})\delta%
\psi^{\prime}({\bf k}_{2}){\rangle}\exp{\{i({\bf k}_{1}+{\bf k}_{2})\cdot{\bf x%
}\}}\,.$$
(A.8)
Then, inserting equation (8) in (A8) and using equation (11) give
us
$$\displaystyle\langle\delta\psi^{\prime}{\partial}_{z}\phi\rangle$$
$$\displaystyle=$$
$$\displaystyle\tau\biggl{[}\int d^{3}k_{1}k_{1z}k_{1z}B_{0}{\overline{\phi}}({%
\bf k}_{1})-\epsilon_{lm3}\int d^{3}k_{1}d^{3}kk_{1z}k_{im}k_{l}{\overline{%
\phi}}({\bf k}_{1}){\langle}\psi({\bf k}){\rangle}e^{i{\bf k}\cdot{\bf x}}%
\biggr{]}$$
(A.9)
$$\displaystyle=$$
$$\displaystyle\tau B_{0}\int d^{3}k_{1}k_{1z}^{2}{\overline{\phi}}({\bf k}_{1})%
=\tau B_{0}{\xi_{v}}{\langle}u^{2}{\rangle}\,,$$
where the isotropy and equation (18) were used to obtain the last line.
Similarly, the second term on the right side of equation (16) is
easily calculated (in Fourier space) by using the isotropy condition.
The result is
$$\displaystyle\langle{\partial}_{z}\psi^{\prime}\delta\phi\rangle$$
$$\displaystyle=$$
$$\displaystyle\tau B_{0}\int d^{3}k_{1}k_{1z}^{2}{\overline{\psi}}({\bf k}_{1})%
=\tau B_{0}{\xi_{b}}{\langle}b^{2}{\rangle}\,.$$
(A.10)
Thus, equations (16), (A9), and (A10) yield equation (17), in the main
text.
Appendix B
In this Appendix, we provide some of intermediate steps used to
obtain equations (28) and (31).
For the mean field equation (28), we first take the average
of equation (27)
$$\displaystyle{\langle}\psi(t+{\delta t}){\rangle}-{\langle}\psi(t){\rangle}-{%
\delta t}\eta{\nabla^{2}}{\langle}\psi(t){\rangle}$$
$$\displaystyle=$$
$$\displaystyle I_{1}+I_{2}+I_{3}$$
(B.1)
where
$$\displaystyle I_{1}$$
$$\displaystyle=$$
$$\displaystyle\int_{t}^{t+{\delta t}}dt_{1}\left[\epsilon_{ij3}{\partial}_{j}%
\psi(t){\partial}_{i}\phi_{I}(t_{1})\right]\simeq{\delta t}\epsilon_{ij3}{%
\partial}_{i}{\langle}{\partial}_{j}\psi(t)\phi_{I}(t){\rangle}\equiv{\delta t%
}{\partial}_{i}\Delta_{i}\,,$$
$$\displaystyle I_{2}$$
$$\displaystyle=$$
$$\displaystyle\int_{t}^{t+{\delta t}}dt_{1}B_{0}{\partial}_{z}{\langle}\psi_{I}%
(t_{1}){\rangle}\simeq{\delta t}B_{0}{\partial}_{z}{\langle}\phi_{I}(t){%
\rangle}\,,$$
$$\displaystyle I_{3}$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}\epsilon_{ij3}\int_{t}^{t+{\delta t}}dt_{1}dt_{2}{%
\langle}\epsilon_{lm3}{\partial}_{i}\phi_{0}(t_{1})[{\partial}_{jm}\psi(t){%
\partial}_{l}\phi_{0}(t_{2})+{\partial}_{m}\psi(t){\partial}_{jl}\phi(t_{2})]$$
(B.2)
$$\displaystyle +B_{0}{\partial}_{i}\phi_{0}(t_{1}){%
\partial}_{jz}\phi_{0}(t_{2}){\rangle}\,,$$
where $\Delta_{i}\equiv\epsilon_{ij3}{\langle}{\partial}_{j}\psi(t)\phi_{I}(t){\rangle}$
and the smooth variation of the induced velocity $\phi_{I}$ in time was used
to approximate the time integrals in $I_{1}$ and $I_{2}$.
To compute the averages, it is convenient to express the
correlation function (24) in terms of ${\bf v}$ in real space as:
$$\displaystyle{\langle}v_{i}({\bf x},t_{1})v_{j}({\bf y},t_{2}){\rangle}$$
$$\displaystyle=$$
$$\displaystyle\delta(t_{1}-t_{2})\left[T_{L}({\bf r}_{H},r_{z})\delta_{ij}+r_{H%
}{{\partial}T_{L}\over{\partial}r_{H}}\left(\delta_{ij}-{r_{Hi}r_{Hj}\over r_{%
H}^{2}}\right)\right]\,,$$
(B.3)
where ${\bf r}\equiv{\bf y}-{\bf x}$ and ${\bf r}_{H}$ is the horizontal component.
Note that the above relation implies
that at ${\bf r}=0$, ${\langle}v_{i}({\bf x},t_{1})v_{j}({\bf x},t_{2}){\rangle}=\delta(t_{1}-t_{2})%
\delta_{ij}T_{L}(r=0)$ so that $T_{L}(0)=\tau_{0}{\langle}v^{2}{\rangle}/2=2\eta_{k}$.
Here $\tau_{0}$ is the short correlation time of ${\bf v}$ and $\eta_{k}=\tau_{0}{\langle}v^{2}{\rangle}/4$ is the kinematic diffusivity.
${\langle}v_{i}({\bf x})v_{j}({\bf x}){\rangle}$ is obviously related
to $\phi_{0}$ by ${\langle}{\partial}_{i}\phi_{0}({\bf x},t_{1}){\partial}_{l}\phi_{0}({\bf y},t%
_{2}){\rangle}=\delta_{il}{\langle}v_{j}({\bf x},t_{1})v_{j}({\bf y},t_{2}){%
\rangle}+{\langle}v_{i}({\bf x},t_{1})v_{l}({\bf y},t_{2}){\rangle}$.
By using ${\bf v}_{j}=-\epsilon_{ij3}{\partial}_{i}\phi_{0}$ and ${\langle}\phi_{0}(t_{1})\psi(t){\rangle}=0$, $I_{3}$ is determined to be:
$$\displaystyle I_{3}$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}{\delta t}T_{L}(0){\nabla^{2}}{\langle}\psi{\rangle}\,.$$
(B.4)
$I_{3}$ represents the kinematic turbulent
diffusivity. Next, to compute $I_{2}$, we take the inverse Fourier
transform of equation (26) and then take the average. Upon neglecting
${\partial}_{z}{\langle}\psi{\rangle}\sim 0$, one can easily show that $I_{2}=0$.
Finally, $I_{1}$ contains the backreaction as well as hyper-resistivity.
To evaluate this term, we insert equation (26) in $\Delta_{i}$
to obtain
$$\displaystyle\Delta_{i}$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}{\langle}{\partial}_{j}\psi(t)\phi_{I}(t){\rangle}$$
(B.5)
$$\displaystyle=$$
$$\displaystyle-{i\over\nu}\epsilon_{ij3}\epsilon_{lm3}\int d^{3}k_{2}d^{3}k^{%
\prime}e^{i{\bf k}^{\prime}\cdot{\bf x}}{1\over({\bf k}+{\bf k}^{\prime})^{2}(%
{\bf k}_{H}+{\bf k}_{H}^{\prime})^{2}}{\overline{\psi}}(-{\bf k})P_{jlm}{%
\langle}\phi({\bf k}^{\prime}){\rangle}\,,$$
here
$$\displaystyle P_{jlm}$$
$$\displaystyle\equiv$$
$$\displaystyle-k_{j}\left[k_{m}k_{l}^{\prime}k_{H}^{\prime 2}+k_{l}k_{m}^{%
\prime}k_{H}^{2}\right]\,.$$
For notational convenience, we introduce ${\bf q}={\bf k}_{H}$ so that
$q_{3}=0$.
Since the characteristic scale of ${\langle}\psi{\rangle}$ is much larger
than that of $\psi^{\prime}$, $k^{\prime}\ll k$ in equation (B5). Thus, we
expand the integrand of equation (B5) to second order in $(k^{\prime}/k)$ and
exploit the isotropy and homogeneity of $\psi^{\prime}$ in the $x-y$ plane.
The latter implies equation (A5) (recall ${\bf q}={\bf k}_{H}$) and
also the following relations
$$\displaystyle\int d^{3}kq_{j}q_{l}q_{r}k_{n}$$
$$\displaystyle=$$
$$\displaystyle\int d^{3}kq_{j}q_{l}q_{r}q_{n}\,,$$
$$\displaystyle\int d^{3}kq_{j}q_{l}k_{z}k_{z}$$
$$\displaystyle=$$
$$\displaystyle{1\over 2}\delta_{jl}\int d^{3}kq^{2}k_{z}^{2}\,.$$
(B.6)
Then, a fair amount of algebra reduces equation (B5) to
$$\displaystyle\Delta_{i}$$
$$\displaystyle=$$
$$\displaystyle-{1\over 2\nu}{\partial}_{i}{\langle}\psi{\rangle}\int d^{3}k{k_{%
H}^{2}\over k^{2}}{\overline{\psi}}({\bf k})-{1\over\nu}{\partial}_{i}{\nabla^%
{2}}{\langle}\psi{\rangle}\int d^{3}k{k_{H}^{2}k_{z}^{2}\over k^{6}}{\overline%
{\psi}}({\bf k})$$
(B.7)
$$\displaystyle=$$
$$\displaystyle-{G\over 2\nu}{\partial}_{i}{\langle}\psi{\rangle}-{F\over\nu}{%
\partial}_{i}{\nabla^{2}}{\langle}\psi{\rangle}\,.$$
Note that there is no contribution from the first order term.
By inserting equation (B7) into (B1), by dividing both
sides by ${\delta t}$, and then by taking the limit of ${\delta t}\to 0$,
we obtain equation (28).
Next, to derive equation (31), we multiply equation (27) by $\psi$
and then take average to obtain the following equation:
$$\displaystyle{\langle}\psi^{2}(t+{\delta t}){\rangle}-{\langle}\psi^{2}(t){%
\rangle}-2\eta{\delta t}{\langle}\psi(t){\nabla^{2}}\psi(t){\rangle}$$
$$\displaystyle=$$
$$\displaystyle J_{1}+J_{2}+2J_{3}\,,$$
(B.8)
where
$$\displaystyle J_{1}$$
$$\displaystyle\equiv$$
$$\displaystyle\int_{t}^{t+{\delta t}}dt_{1}dt_{2}\biggl{\{}\epsilon_{ij3}%
\epsilon_{lm3}{\langle}{\partial}_{j}\psi(t){\partial}_{m}(t){\partial}_{i}%
\phi_{I}(t_{1}){\partial}_{l}\phi_{I}(t_{2}){\rangle}+2B_{0}\epsilon_{ij3}{%
\langle}{\partial}_{j}\psi(t){\partial}_{i}\phi_{I}(t_{1}){\partial}_{x}\phi_{%
I}(t_{2}){\rangle}\biggr{\}}\,,$$
$$\displaystyle J_{2}$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{ij3}\int_{t}^{t+{\delta t}}dt_{1}dt_{2}{\langle}\psi(t)%
\left[{\partial}_{i}\phi_{0}(t_{1})\epsilon_{lm3}{\partial}_{j}\left[{\partial%
}_{m}\psi(t){\partial}_{l}\phi_{0}(t_{2})\right]+B_{0}{\partial}_{i}\phi_{0}(t%
_{1}){\partial}_{jz}\phi_{0}(t_{2})\right]{\rangle}\,,$$
$$\displaystyle J_{3}$$
$$\displaystyle=$$
$$\displaystyle\int_{t}^{t+{\delta t}}dt_{1}{\langle}\psi(t)\left[\epsilon_{ij3}%
{\partial}_{j}\psi(t){\partial}_{i}\phi_{I}(t_{1})+B_{0}{\partial}_{z}\phi_{I}%
(t_{1})\right]{\rangle}\equiv{\delta t}(J_{31}+J_{32})\,,$$
(B.9)
where $J_{31}\equiv\epsilon_{ij3}{\langle}\psi(t){\partial}_{j}\psi(t){\partial}_{i}%
\phi_{I}(t){\rangle}$ and $J_{32}\equiv B_{0}{\langle}\psi(t){\partial}_{z}\phi_{I}(t){\rangle}$.
First, $J_{1}$ can easily be computed by using the correlation functions
as
$$\displaystyle J_{1}$$
$$\displaystyle=$$
$$\displaystyle{\delta t}\left[T_{L}(0)\left[{\langle}b^{2}{\rangle}+{\langle}B_%
{H}{\rangle}^{2}\right]-B_{0}^{2}\int d^{3}k_{z}^{2}{\overline{\phi}}({\bf k})%
\right]\,.$$
(B.10)
Next, $J_{2}$ can be computed upon substituting
equation (26) and then splitting average by
using ${\langle}\psi(t)\phi(t_{1}){\rangle}=0$, with the result
$$\displaystyle J_{2}$$
$$\displaystyle=$$
$$\displaystyle\delta tT_{L}(0)\left[-{\langle}b^{2}{\rangle}+{\langle}\psi{%
\rangle}{\nabla^{2}}{\langle}\psi{\rangle}\right]\,.$$
(B.11)
For $J_{3}$, one can first show $J_{31}=0$ due to isotropy.
To compute $J_{32}$, we substitute equation (26) and use
${\langle}\phi_{I}{\rangle}=0$ to obtain
$$\displaystyle J_{32}$$
$$\displaystyle=$$
$$\displaystyle-B_{0}\int d^{3}k_{1}d^{3}k\exp{\{i({\bf k}_{1}+{\bf k}_{2})\cdot%
{\bf x}\}}{k_{z}\over\nu k_{H}^{2}k^{2}}$$
(B.12)
$$\displaystyle\times{\langle}\psi^{\prime}({\bf k}_{1})\left[B_{0}k_{z}k_{H}^{2%
}\psi^{\prime}({\bf k})+i\epsilon_{ij3}\int d^{3}k^{\prime}\psi({\bf k}-{\bf k%
}^{\prime})(k-k^{\prime})_{j}k^{\prime}_{i}k_{H}^{\prime 2}\psi({\bf k}^{%
\prime})\right]{\rangle}$$
$$\displaystyle=$$
$$\displaystyle-{B_{0}\over\nu}\biggl{[}B_{0}\int d^{3}k_{1}{k_{1z}^{2}\over k_{%
1}^{2}}{\overline{\psi}}({\bf k}_{1})$$
$$\displaystyle+i\epsilon_{ij3}\int d^{3}kd^{3}k_{1}\exp{\{i({\bf k}_{1}+{\bf k}%
_{2})\cdot{\bf x}\}}{k_{z}\over k_{H}^{2}k^{2}}Q_{ij}{\overline{\psi}}({\bf k}%
_{1}){\langle}\psi({\bf k}+{\bf k}_{1}){\rangle}\biggr{]}$$
where $Q_{ij}\equiv-k_{1j}(k+k_{1})_{i}({\bf k}_{H}+{\bf k}_{1H})^{2}-k_{1i}(k+k_{1})%
_{j}k_{1H}^{2}$. By using the definition of ${\overline{G}}$
(see immediately after eq. [31]) and $\epsilon_{ij3}Q_{ij}=-\epsilon_{ij3}(k_{H}^{2}+2{\bf k}_{H}\cdot{\bf k}_{H})k_%
{i}k_{1j}$, equation (B12)
becomes
$$\displaystyle J_{32}$$
$$\displaystyle=$$
$$\displaystyle-{B_{0}\over\nu}\left[B_{0}{\overline{G}}-i\epsilon_{ij3}\int d^{%
3}k^{\prime}d^{3}ke^{i{\bf k}^{\prime}\cdot{\bf x}}{k_{z}\over k_{H}^{2}k^{2}}%
\left[k_{H}^{2}+2k_{l}(k^{\prime}-k)_{l}\right]k_{i}(k^{\prime}-k)_{j}{%
\overline{\psi}}(-{\bf k}+{\bf k}^{\prime}){\langle}\psi({\bf k}^{\prime}){%
\rangle}\right]$$
(B.13)
$$\displaystyle=$$
$$\displaystyle-{B_{0}\over\nu}\left[B_{0}{\overline{G}}-\epsilon_{ij3}{\partial%
}_{j}d^{3}k^{\prime}\int d^{3}ke^{i{\bf k}^{\prime}\cdot{\bf x}}{k_{z}k_{i}%
\over k^{2}}\left[-1+{2k_{l}k_{l}^{\prime}\over k_{H}^{2}}\right]{\overline{%
\psi}}(-{\bf k}+{\bf k}^{\prime}){\langle}\psi({\bf k}^{\prime}){\rangle}%
\right]\,.$$
Now, since $k^{\prime}\ll k$, we expand the integrand of equation (B13)
to second order in $k^{\prime}/k$, in order to show that there is no
contribution from the second term in equation (B13) (to this order).
Therefore, $J_{32}=-B_{0}^{2}{\overline{G}}/\nu$.
Inserting $J_{1}$, $J_{2}$, and $J_{3}$ in equation (B8), dividing by
${\delta t}$, and then taking the limit ${\delta t}\to 0$ finally yields
equation (31).
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The Hopf algebra of ($q$)multiple polylogarithms
with non-positive arguments
Kurusch Ebrahimi-Fard
ICMAT,
C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain.
On leave from Univ. de Haute Alsace, Mulhouse, France
[email protected], [email protected]
www.icmat.es/kurusch
,
Dominique Manchon
Univ. Blaise Pascal,
C.N.R.S.-UMR 6620,
63177 Aubière, France
[email protected]
http://math.univ-bpclermont.fr/$\sim$manchon/
and
Johannes Singer
Department Mathematik,
Friedrich–Alexander–Universität Erlangen–Nürnberg,
Cauerstraße 11,
91058 Erlangen, Germany
[email protected]
http://math.fau.de/singer
(Date:: January 14, 2021)
Abstract.
We consider multiple polylogarithms in a single variable at non-positive integers. Defining a connected graded Hopf algebra, we apply Connes’ and Kreimer’s algebraic Birkhoff decomposition to renormalize multiple polylogarithms at non-positive integer arguments, which satisfy the shuffle relation. The $q$-analogue of this result is as well presented, and compared to the classical case.
Key words and phrases:multiple polylogarithms, multiple zeta values, Rota-Baxter algebra, renormalization, Hopf algebra, $q$-analogues
2010 Mathematics Subject Classification: 11M32,16T05
Contents
1 Introduction
2 Meromorphic Continuation of MZVs
3 Algebraic Framework
3.1 Rota–Baxter Algebra and multiple zeta values
3.2 $q$-multiple zeta values
3.3 General Word Algebraic Part
3.3.1 The algebra $\mathcal{H}_{\lambda},\,\lambda\neq 0$
3.3.2 The coproduct $\overline{\Delta}_{\lambda},\,\lambda\in\mathbb{Q}$
3.3.3 Compatibility properties of the coproduct ($\lambda\neq 0$ case)
3.3.4 The Hopf algebra $\mathcal{H}_{\lambda},\,\lambda\neq 0$
3.3.5 Compatibility between the product and the coproduct ($\lambda=0$ case)
3.3.6 The Hopf algebra $\mathcal{H}_{0}$
3.3.7 Shuffle factorization
3.3.8 A combinatorial description of the coproduct $\Delta_{\lambda}$
4 Renormalization of regularized MZVs
4.1 Connes–Kreimer Renormalization in a Nutshell
4.2 Renormalization of MZVs
4.3 Renormalization of $q$MZVs
1. Introduction
Let $n,k_{1},\ldots,k_{n}$ be positive integers. Multiple polylogarithms (MPLs) in a single variable are defined by
$$\displaystyle\operatorname{Li}_{k_{1},\ldots,k_{n}}(z):=\sum_{m_{1}>\cdots>m_{%
n}>0}\frac{z^{m_{1}}}{m_{1}^{k_{1}}\cdots m_{n}^{k_{n}}}$$
when $z$ is a complex number. The function $\operatorname{Li}_{k_{1},\ldots,k_{n}}(z)$ is of depth $\operatorname{dpt}(\mathbf{k}):=n\geq 1$ and weight $\operatorname{wt}(\mathbf{k}):=k_{1}+\cdots+k_{n}$, for $\mathbf{k}:=(k_{1},\ldots,k_{n})$. It is analytic in the open unit disk and, in the case $k_{1}>1$, continuous on the closed unit disk. In this case we observe the connection of MPLs and multiple zeta values (MZVs)
(1)
$$\displaystyle\zeta(k_{1},\ldots,k_{n}):=\sum_{m_{1}>\cdots>m_{n}>0}\frac{1}{m_%
{1}^{k_{1}}\cdots m_{n}^{k_{n}}}=\operatorname{Li}_{k_{1},\ldots,k_{n}}(1).$$
Equivalently we can define MPLs by induction on the weight $\operatorname{wt}(\mathbf{k})$ as follows:
(2)
$$\displaystyle z\frac{d}{dz}\operatorname{Li}_{k_{1},\ldots,k_{n}}(z)=%
\operatorname{Li}_{k_{1}-1,k_{2},\ldots,k_{n}}(z)\hskip 8.535827pt\text{if}%
\hskip 8.535827ptk_{1}>1,$$
(3)
$$\displaystyle(1-z)\frac{d}{dz}\operatorname{Li}_{1,k_{2},\ldots,k_{n}}(z)=%
\operatorname{Li}_{k_{2},\ldots,k_{n}}(z)\hskip 8.535827pt\text{if}\hskip 8.53%
5827ptn>1,$$
(4)
$$\displaystyle\operatorname{Li}_{k_{1},\ldots,k_{n}}(0)=0.$$
Therefore one can observe an integral formula for MPLs using iterated Chen integrals. Indeed, let $\varphi_{1},\ldots,\varphi_{p}$ be complex-valued differential $1$-forms defined on a compact interval. Then we define inductively for real numbers $x$ and $y$
$$\displaystyle\int_{x}^{y}\varphi_{1}\cdots\varphi_{p}:=\int_{x}^{y}\varphi_{1}%
(t)\int_{x}^{t}\varphi_{2}\cdots\varphi_{p}.$$
Now we set
$$\displaystyle\omega_{k_{1},\ldots,k_{n}}:=\omega_{0}^{k_{1}-1}\omega_{1}\cdots%
\omega_{0}^{k_{n}-1}\omega_{1},$$
where
$$\displaystyle\omega_{0}(t):=\frac{dt}{t}\hskip 11.381102pt\text{and}\hskip 11.%
381102pt\omega_{1}(t):=\frac{dt}{1-t}.$$
Using the differential equations (2), (3) and the initial conditions (4) we obtain
(5)
$$\displaystyle\operatorname{Li}_{k_{1},\ldots,k_{n}}(z)=\int_{0}^{z}\omega_{k_{%
1},\ldots,k_{n}}$$
using the convention $\operatorname{Li}_{\emptyset}(z)=1$.
This representation gives rise to the well known shuffle products of MPLs and MZVs (see e.g. [Wal02, Wal11]).
Recall that the $\mathbb{Q}$-vector space spanned by MZVs forms an algebra equipped with two products. The quasi-shuffle product is obtained when one multiplies series (1) directly, which yields a linear combination of MZVs due to the product rule for sums. The aforementioned shuffle product between MZVs derives from integration by parts for iterated integrals. The resulting so-called double shuffle relations among MZVs arise from the interplay between these two products. An alternative characterization of MPLs can be given by the following formula
(6)
$$\displaystyle\operatorname{Li}_{k_{1},\ldots,k_{n}}(z)=J^{k_{1}}[yJ^{k_{2}}[y%
\cdots J^{k_{n}}[y]\cdots]](z),$$
where $y(z):=\frac{z}{1-z}$ and $J[f](z):=\int_{0}^{z}\frac{f(t)}{t}\,dt$ (see Lemma 3.2). Since $J$ is a Rota–Baxter operator of weight zero, iterations of the operator $J$ induce a product that coincides with the usual shuffle product for MPLs (see Lemma 3.7). For $|z|<1$ Equation (6) is valid for all $k_{1},\ldots,k_{n}\in\mathbb{Z}$. The inverse of $J$ is given by $J^{-1}[f](t)=\delta[f](t):=t\frac{\partial f}{\partial t}(t)$ (Proposition 3.1).
We study the $\mathbb{Q}$-vector space
$$\displaystyle\mathcal{MP}:=\langle z\mapsto\operatorname{Li}_{-k_{1},\ldots,-k%
_{n}}(z)\colon k_{1},\ldots,k_{n}\in\mathbb{N}_{0},n\in\mathbb{N}\rangle_{%
\mathbb{Q}},$$
which is indeed an algebra, where the product is induced by Equation (6) (see Lemma 3.5).
The algebra $\mathcal{MP}$ admits also an interpretation for MZVs at non-positive integers. Indeed, let $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$. It is easily seen that $\operatorname{Li}_{-k_{1},\ldots,-k_{n}}(z)$ is convergent for $|z|<1$ and divergent for $z=1$. Nevertheless we can perceive the product induced for $|z|<1$ as an analogue for the shuffle product of MZVs at non-positive integers. In order to make this connection more precise we have to establish a renormalization procedure. This permits us to extract explicit numbers for MZVs with non-positive arguments in a consistent way, such that they satisfy the shuffle product relations induced by the algebra structure of $\mathcal{MP}$.
We should keep in mind that a characterization of the shuffle product at non-positive integers – in contrast to the quasi-shuffle product – is a crucial point. Since the quasi-shuffle product is induced by the series representation of MZVs, the combinatorics is essentially the same as for positive arguments. On the other hand the shuffle product for positive indices is induced by the integral representation (5). The combinatorics behind this product comes from shuffling of integration variables. It could be illustrated by the shuffling of two decks of cards, say a deck of red and blue cards, each consecutively numbered such that the internal numbering of red and blue cards is preserved. In this approach, however, it is not clear how to handle non-positive arguments, which corresponds to a non-positive number of cards.
Extracting finite numbers for MZVs at non-positive integers is accomplished by the process of renormalization, which involves two steps:
(I)
introduction of a regularization scheme,
(II)
applying a subtraction method.
In step (I) we consider divergent MZVs $\zeta(-k_{1},\ldots,-k_{n})$ with $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$, and introduce a so-called regularization parameter $z$, which systematically deforms the divergent MZV in order to obtain a meromorphic function in $z$ with the only singularity in $z=0$.
Step (II) involves a systematic procedure to eliminate singularities in terms of recursively defined subtractions. A rather natural way to achieve such eliminations is widely known as minimal subtraction scheme. The renormalization process is an integral part of perturbative quantum field theory (QFT). See e.g. [Col84]. The “right” choice of the regularization scheme in QFT is essential in the light of constraints coming from physics. In our context those constraints are of mathematical nature: The deformation of divergent MZVs has to be established in such a way that the regularized MZVs coincide with the meromorphic continuation of (M)ZVs. The recursively defined subtractions in step (II) involve combinatorial structures, which are concisely captured by the Connes–Kreimer Hopf algebraic approach to renormalization [CK00, CK01, Man04].
One of the key points in our approach is based on providing an adequate Hopf algebra together with an algebra morphism from that Hopf algebra into the space $\mathcal{MP}$ (Theorem 3.18), which permits to define a consistent renormalization process. Regarding regularization schemes, we will use the fact, that MPLs may be considered as regularized classical MZVs. However, we also consider a specific $q$-analogue of MZVs [OOZ12], where the variable $q$ takes the role of a natural regulator.
Remark 1.1.
Renormalization of MZVs at non-positive integers appeared already in [GZ08] and [MP10]. The authors applied regularization schemes together with well-chosen subtraction methods suitable for preserving the quasi-shuffle product for renormalized MZVs – at non-positive arguments. Using Ecalle’s Mould calculus, Bouillot proposes in his work [Bou14] a unifying picture of MZVs at non-positive arguments respecting the quasi-shuffle product. The common point of our approach with those presented in the aforementioned references is the use of the Connes–Kreimer Hopf algebraic approach to renormalization, and the corresponding algebraic Birkhoff decomposition, which encodes the subtraction procedure for singularities. However, we should emphasize that in our work it is the shuffle product, in a naturally extended sense, which is satisfied by renormalized MZVs at non-positive arguments.
In [CEM14b] the authors indicated that the $q$-parameter appearing in a specific $q$-analogue of MZVs [OOZ12] (see Equation (8) below) may be considered as a regularization parameter for MZVs at non-positive arguments. The approach presented in our paper can be consistently extended to this $q$-analogue of MZVs ($q$MZVs). Indeed, we would like to demonstrate that under the $q$-parameter regularization [CEM14a, CEM14b, OOZ12] a proper renormalization of MZVs can be defined. The $q$-multiple polylogarithm ($q$MPL) in one variable is defined as
(7)
$$\displaystyle\operatorname{Li}^{q}_{k_{1},\ldots,k_{n}}(z):=\sum_{m_{1}>\cdots%
>m_{n}>0}\frac{z^{m_{1}}}{[m_{1}]_{q}^{k_{1}}\cdots[m_{n}]_{q}^{k_{n}}}$$
with $[m]_{q}:=\frac{1-q^{m}}{1-q}$. It turns out that for $|z|<1$ the series in (7) is convergent for $k_{1},\ldots,k_{n}\in\mathbb{Z}$, and especially for $|q|<1$ we obtain the formal power series
(8)
$$\displaystyle\mathfrak{z}_{q}(k_{1},\ldots,k_{n}):=\operatorname{Li}^{q}_{k_{1%
},\ldots,k_{n}}(q)=\sum_{m_{1}>\cdots>m_{n}>0}\frac{q^{m_{1}}}{[m_{1}]_{q}^{k_%
{1}}\cdots[m_{n}]_{q}^{k_{n}}}\in\mathbb{Z}[[q]].$$
These $q$MZVs were introduced by Ohno, Okuda and Zudilin in [OOZ12], and further studied in [CEM14a, CEM14b], see also [Sin15, Sin14, Zha14]. For $k_{1}>1$ and $k_{2},\ldots,k_{n}\geq 1$ we see that
(9)
$$\displaystyle\lim_{q\nearrow 1}\mathfrak{z}_{q}(k_{1},\ldots,k_{n})=\zeta(k_{1%
},\ldots,k_{n}),$$
where $q\nearrow 1$ means $q\to 1$ inside an angular sector $-\frac{\pi}{2}+\varepsilon\leq\operatorname{Arg}(1-q)\leq\frac{\pi}{2}-\varepsilon$ with $\varepsilon>0$ sufficiently small. It will be convenient for technical reasons to consider the modified $q$MZVs introduced in [OOZ12]
(10)
$$\displaystyle\overline{\mathfrak{z}}_{q}(k_{1},\ldots,k_{n}):=(1-q)^{-(k_{1}+%
\cdots+k_{n})}\mathfrak{z}(k_{1},\ldots,k_{n}).$$
They are used to establish a Hopf algebra structure on the space of modified $q$MPLs. The modification has to be reversed after renormalization, in order to relate the renormalized $q$MZVs via (9) to renormalized MZVs.
The paper is organized as follows. In Section 2 we recall the basic results on the meromorphic continuation of MZVs. Section 3 contains the main result, i.e., the detailed construction of a Hopf algebra for MPLs at non-positive integers. A generalization of this finding to the $q$-analogue of MZVs defined by Ohno, Okuda and Zudilin is presented as well. In Section 4 we recall the Hopf algebra approach to perturbative renormalization by Connes and Kreimer, and apply one of its main theorems to the renormalization of MPLs at non-positive integer arguments. The $q$-analogue of this result is as well presented, and compared to the classical case.
Acknowledgement: The first author is supported by a Ramón y Cajal research grant from the Spanish government. The second and third authors gratefully acknowledge support by ICMAT and the Severo Ochoa Excellence Program. The second author is supported by Agence Nationale de la Recherche (projet CARMA).
2. Meromorphic Continuation of MZVs
In this section we review some well-known facts about the meromorphic continuation of MZVs. For $n\in\mathbb{N}$ we consider the function
(11)
$$\displaystyle\zeta_{n}\colon\mathbb{C}^{n}\to\mathbb{C},\hskip 28.452756pt%
\zeta_{n}(s_{1},\ldots,s_{n}):=\sum_{m_{1}>\cdots>m_{n}>0}\frac{1}{m_{1}^{s_{1%
}}\cdots m_{n}^{s_{n}}}.$$
Proposition 2.1 ([Zha00]).
The infinite sum (11) converges absolutely for $\operatorname{Re}(s_{1})>1$ and $\sum_{j=1}^{k}\operatorname{Re}(s_{j})>k$, $k=1,\ldots,n$. In this domain $\zeta_{n}$ defines an analytic function in $n$ variables.
Theorem 2.2 ([AET01, AT01, Zha00]).
The function $\zeta_{n}(s_{1},\ldots,s_{n})$ admits a meromorphic extension to $\mathbb{C}^{n}$. The subvariety $\mathcal{S}_{n}$ of singularities is given by
$$\displaystyle\mathcal{S}_{n}=\left\{(s_{1},\ldots,s_{n})\in\mathbb{C}^{n}%
\colon s_{1}=1;s_{1}+s_{2}=2,1,0,-2,-4,\ldots;\sum_{i=1}^{j}s_{i}\in\mathbb{Z}%
_{\leq j}~{}(j=3,4,\ldots,n)\right\}.$$
In the subsequent sections $\zeta_{n}$ always denotes the meromorphic continuation of MZVs.
Remark 2.3.
In this paper we discuss $\zeta_{n}$ restricted to the set $(\mathbb{Z}_{\leq 0})^{n}$. The Bernoulli numbers are defined by the following generating series
$$\displaystyle\frac{te^{t}}{e^{t}-1}=\sum_{m\geq 0}\frac{B_{m}}{m!}t^{m}.$$
The first few values are $B_{0}=1,B_{1}=\frac{1}{2},B_{2}=\frac{1}{6},B_{3}=0,B_{4}=-\frac{1}{30},B_{5}=0,$ etc., especially $B_{2l+1}=0$ for $l\in\mathbb{N}$.
Therefore we have the following cases for $\zeta_{n}$ restricted to non-positive arguments:
•
Case $n=1$: For $l\in\mathbb{N}_{0}$ we have the well known formula
$$\displaystyle\zeta_{1}(-l)=-\frac{B_{l+1}}{l+1}.$$
•
Case $n=2$: In the light of Theorem 2.2 we assume the sum $k_{1}+k_{2}$ to be odd. Therefore we obtain from [AET01] that
$$\displaystyle\zeta_{2}(-k_{1},-k_{2})=\frac{1}{2}\left(1+\delta_{0}(k_{2})%
\right)\frac{B_{k_{1}+k_{2}+1}}{k_{1}+k_{2}+1}.$$
•
Case $n\geq 3$: From Theorem 2.2 we deduce that
$$\displaystyle(\mathbb{Z}_{\leq 0})^{n}\subseteq\mathcal{S}_{n}.$$
Therefore we obtain no information from the meromorphic continuation.
3. Algebraic Framework
We briefly introduce Rota–Baxter algebras, since they conveniently relate to shuffle-type products on word algebras. Two such shuffle products are presented, which encode products of MPLs and $q$MPLs at integer arguments. The main result of this section is the construction of a graded connected commutative and cocommutative shuffle Hopf algebra for ($q$)MPLs at non-positive integer arguments.
3.1. Rota–Baxter Algebra and multiple zeta values
Let $k$ be a ring, $\lambda\in k$ and $\mathcal{A}$ a $k$-algebra. A Rota–Baxter operator (RBO) of weight $\lambda$ on $\mathcal{A}$ over $k$ is a $k$-module endomorphism $L$ of $\mathcal{A}$ such that
$$\displaystyle L(x)L(y)=L(xL(y))+L(L(x)y)+\lambda L(xy)$$
for any $x,y\in\mathcal{A}$. A Rota–Baxter $k$-algebra (RBA) of weight $\lambda$ is a pair $(\mathcal{A},L)$ with a $k$-algebra $\mathcal{A}$ and a Rota–Baxter operator $L$ of weight $\lambda$ on $\mathcal{A}$ over $k$. On the algebra of continuous functions $C(\mathbb{R})$ the integration operator
$$\displaystyle R\colon C(\mathbb{R})\to C(\mathbb{R}),\hskip 14.226378ptR[f](z)%
:=\int_{0}^{z}f(x)\,dx$$
is a RBO of weight zero, which is an immediate consequence of the integration by parts formula. We consider the $\mathbb{C}$-algebra of power series
$$\displaystyle\mathcal{P}_{\geq 1}:=\bigg{\{}f(t):=\sum_{k\geq 1}a_{k}t^{k}%
\colon R_{f}\geq 1\bigg{\}}\subseteq t\mathbb{C}[[t]]$$
without a term of degree zero in $t$, and radius of convergence, $R_{f}$, of at least $1$. We define the operator
$$\displaystyle J\colon\mathcal{P}_{\geq 1}\to\mathcal{P}_{\geq 1},\hskip 14.226%
378ptJ[f](t):=\int_{0}^{t}f(z)\frac{dz}{z}.$$
Further the Euler derivation $\delta$ is given by
$$\displaystyle\delta\colon\mathcal{P}_{\geq 1}\to\mathcal{P}_{\geq 1},\hskip 14%
.226378pt\delta[f](t):=t\frac{\partial f}{\partial t}(t).$$
Proposition 3.1.
(i)
The pair $(\mathcal{P}_{\geq 1},J)$ is a RBA of weight $\lambda=0$.
(ii)
The operator $\delta$ is a derivation, i.e., $\delta[fg]=\delta[f]g+f\delta[g]$, for any $f,g\in\mathcal{P}_{\geq 1}$.
(iii)
The operators $J$ and $\delta$ are mutually inverse, i.e., $J\circ\delta=\delta\circ J=\operatorname{Id}$.
Proof.
Statement (i) follows form integration by part. The second claim is straightforward to show. Finally, item (iii) is an immediate consequence of the fundamental theorem of calculus together with the fact that $f(0)=0$ for any $f\in\mathcal{P}_{\geq 1}$.
∎
Lemma 3.2.
Let $k_{1},\ldots,k_{n}$ be integers. Then $\operatorname{Li}_{k_{1},\ldots,k_{n}}(t)\in\mathcal{P}_{\geq 1}$, explicitly
$$\displaystyle\operatorname{Li}_{k_{1},\ldots,k_{n}}(t)=J^{k_{1}}[yJ^{k_{2}}[y%
\cdots J^{k_{n}}[y]\cdots]](t),$$
where $y(t):=\frac{t}{1-t}\in\mathcal{P}_{\geq 1}$.
Proof.
Using the fact that $J^{-1}=\delta$ we prove the claim for $\mathbf{k}:=(k_{1},\ldots,k_{n})\in\mathbb{Z}^{n}$ by induction on its depth, $\operatorname{dpt}(\mathbf{k})=n$. For $\operatorname{dpt}(\mathbf{k})=1$ we easily compute
$$\displaystyle J^{k}[y](t)$$
$$\displaystyle=\left.\begin{cases}\sum_{m\geq 1}\frac{t^{m}}{m^{k}},&\text{for~%
{}}k\geq 0\\
\sum_{m\geq 1}m^{|k|}{t^{m}}=\sum_{m\geq 1}\frac{t^{m}}{m^{k}},&\text{for~{}}k%
<0\end{cases}\right\}=\operatorname{Li}_{k}(t)$$
for any $k\in\mathbb{Z}$. In the inductive step we get
$$\displaystyle J^{k_{1}}[yJ^{k_{2}}[y\cdots J^{k_{n}}[y]\cdots]](t)$$
$$\displaystyle=J^{k_{1}}\left[\sum_{m>0}t^{m}\sum_{m_{2}>\cdots>m_{n}>0}\frac{t%
^{m_{2}}}{m_{2}^{k_{2}}\cdots m_{n}^{k_{n}}}\right]$$
$$\displaystyle=J^{k_{1}}\left[\sum_{m_{1}>m_{2}>\cdots>m_{n}>0}\frac{t^{m_{1}}}%
{m_{2}^{k_{2}}\cdots m_{n}^{k_{n}}}\right]$$
$$\displaystyle=\sum_{m_{1}>m_{2}>\cdots>m_{n}>0}\frac{t^{m_{1}}}{m_{1}^{k_{1}}m%
_{2}^{k_{2}}\cdots m_{n}^{k_{n}}}$$
$$\displaystyle=\operatorname{Li}_{k_{1},\ldots,k_{n}}(t)$$
using the induction hypothesis.
∎
This lemma gives rise to the following algebraic formalism. Let $X_{0}:=\{j,d,y\}$, and $W_{0}$ denotes the set of words on the alphabet $X_{0}$, subject to the rule $jd=dj=\mathbf{1}$, where $\mathbf{1}$ denotes the empty word. Therefore any word $w\in W_{0}$ can be uniquely written in the canonical form
$$\displaystyle w=j^{k_{1}}yj^{k_{2}}y\cdots j^{k_{n-1}}yj^{k_{n}}$$
for $k_{1},\ldots,k_{n}\in\mathbb{Z}$ using the notation $j^{-1}=d$ and $j^{0}=\mathbf{1}$. The length of the word $w$ above is $|w|=k_{1}+\cdots+k_{n}+n-1$.
Further, $\mathcal{A}_{0}$ denotes the vector space $\mathcal{A}_{0}:=\langle W_{0}\rangle_{\mathbb{Q}}$ spanned by the words in $W_{0}$.
Next we define the product $\,\shuffle_{0}\,\colon\mathcal{A}_{0}\otimes\mathcal{A}_{0}\to\mathcal{A}_{0}$ by $\mathbf{1}\,\shuffle_{0}\,w:=w\,\shuffle_{0}\,\mathbf{1}:=w$ for any word $w\in W_{0}$, and recursively with respect to the sum of the length of two words in $W_{0}$:
(i)
$yu\,\shuffle_{0}\,v:=u\,\shuffle_{0}\,yv:=y(u\,\shuffle_{0}\,v)$,
(ii)
$ju\,\shuffle_{0}\,jv:=j(u\,\shuffle_{0}\,jv)+j(ju\,\shuffle_{0}\,v)$,
(iii)
$du\,\shuffle_{0}\,dv:=d(u\,\shuffle_{0}\,dv)-u\,\shuffle_{0}\,d^{2}v$,
(iv)
$du\,\shuffle_{0}\,jv:=d(u\,\shuffle_{0}\,jv)-u\,\shuffle_{0}\,v$,
(v)
$ju\,\shuffle_{0}\,dv:=d(ju\,\shuffle_{0}\,v)-u\,\shuffle_{0}\,v$.
Remark 3.3.
• Note that (iv) can be deduced from (iii) by replacing $v$ by $j^{2}v$.
• (iii) does not really define $du\,\shuffle_{0}\,dv$ by induction on the sum of lengths of two words, because $|du|+|dv|=|u|+|d^{2}v|$. Using (i) and writing $u^{\prime}=du=d^{k}yw$ for some $k\geq 1$, we can however get a recursive definition by iterating (iii) as follows:
$$\displaystyle d^{k}yw\,\shuffle_{0}\,dv=$$
$$\displaystyle~{}d\big{(}d^{k-1}yw\,\shuffle_{0}\,dv-d^{k-2}yw\,\shuffle_{0}\,d%
^{2}v+\cdots$$
$$\displaystyle+(-1)^{k-1}yw\,\shuffle_{0}\,d^{k}v\big{)}+(-1)^{k}y(w\,\shuffle_%
{0}\,d^{k+1}v).$$
Lemma 3.4.
The $\mathbb{Q}$-vector space
$$\displaystyle\mathcal{T}:=\langle j^{k_{1}}yj^{k_{2}}y\cdots j^{k_{n-1}}yj^{k_%
{n}}\in W_{0}\colon k_{n}\neq 0,n\in\mathbb{N}\rangle_{\mathbb{Q}}$$
is a two sided ideal of $(\mathcal{A}_{0},\,\shuffle_{0}\,)$.
Proof.
Let $a\in\{j,d\}$ and $u:=u^{\prime}a\in W_{0}$ and $v\in W_{0}$. We prove $u\,\shuffle_{0}\,v\in\mathcal{T}$ by induction on $r:=|u|+|v|$. The base cases are true because we observe for $r=1$
that $d\,\shuffle_{0}\,\mathbf{1}=d,j\,\shuffle_{0}\,\mathbf{1}=j$ and for $r=2$ that
$$\displaystyle d\,\shuffle_{0}\,y=yd,\hskip 22.762205ptj\,\shuffle_{0}\,y=yj,$$
$$\displaystyle d\,\shuffle_{0}\,d=0,\hskip 11.381102ptj\,\shuffle_{0}\,j=2j^{2}%
,\hskip 11.381102ptj\,\shuffle_{0}\,d=d\,\shuffle_{0}\,j=0.$$
For the inductive step we have several cases:
• 1st case: $u=y\tilde{u}d$ or $v=y\tilde{v}$. This is an immediate consequence of (i) and the induction hypothesis.
• 2nd case: $u=j\tilde{u}d$ and $v=j\tilde{v}$. We observe using (ii) and the induction hypothesis that
$$\displaystyle j\tilde{u}d\,\shuffle_{0}\,j\tilde{v}=j(\tilde{u}d\,\shuffle_{0}%
\,j\tilde{v}+j\tilde{u}d\,\shuffle_{0}\,\tilde{v})\in\mathcal{T}.$$
• 3rd case: $u=d\tilde{u}d$ and $v=j\tilde{v}$. We observe using (iv) and the induction hypothesis that
$$\displaystyle d\tilde{u}d\,\shuffle_{0}\,j\tilde{v}=d(\tilde{u}d\,\shuffle_{0}%
\,j\tilde{v})-\tilde{u}d\,\shuffle_{0}\,\tilde{v}\in\mathcal{T}.$$
• 4th case: $u=j\tilde{u}d$ and $v=d\tilde{v}$. We observe using (v) and the induction hypothesis that
$$\displaystyle j\tilde{u}d\,\shuffle_{0}\,d\tilde{v}=d(j\tilde{u}d\,\shuffle_{0%
}\,\tilde{v})-\tilde{u}d\,\shuffle_{0}\,\tilde{v}\in\mathcal{T}.$$
• 5th case: $u=d\tilde{u}d$ and $v=d\tilde{v}$. We observe using (iii) that
$$\displaystyle d\tilde{u}d\,\shuffle_{0}\,d\tilde{v}=d(\tilde{u}d\,\shuffle_{0}%
\,d\tilde{v})-\tilde{u}d\,\shuffle_{0}\,d^{2}\tilde{v}.$$
By induction hypothesis the first term is an element of $\mathcal{T}$. If $\tilde{u}$ starts with a $j$ or $y$ we are in one of the above cases. Therefore we only consider the case, where $\tilde{u}$ is a word consisting purely of $d$, i.e., $\tilde{v}=d^{n}$ for $n\in\mathbb{N}$. Now we prove $d^{n}\,\shuffle_{0}\,d^{m}w=0$ for any $w\in W_{0}$ and $m\in\mathbb{N}$. For $n=1$ we have $d\,\shuffle_{0}\,d^{m}\tilde{v}=d(\mathbf{1}\,\shuffle_{0}\,d^{m}\tilde{v})-%
\mathbf{1}\,\shuffle_{0}\,d^{m+1}\tilde{v}=0$. Therefore we obtain by induction hypothesis that
$$\displaystyle d^{n+1}\,\shuffle_{0}\,d^{m}\tilde{v}=d(d^{n}\,\shuffle_{0}\,d^{%
m}\tilde{v})-d^{n}\,\shuffle_{0}\,d^{m+1}\tilde{v}=0.$$
All in all we have shown that $\,\shuffle_{0}\,(\mathcal{T}\otimes\mathcal{A}_{0})\subseteq\mathcal{T}$.
Since $\,\shuffle_{0}\,$ is not commutative we also have to prove $v\,\shuffle_{0}\,u\in\mathcal{T}$ by induction on $r:=|v|+|u|$. The base cases are true. The first four cases are completely analogous to the first four cases above. We only discuss the following case: $v=d\tilde{v}$ and $u=d\tilde{u}d$.
We observe using (iii) that
$$\displaystyle d\tilde{v}\,\shuffle_{0}\,d\tilde{u}d=d(\tilde{v}\,\shuffle_{0}%
\,d\tilde{u}d)-\tilde{v}\,\shuffle_{0}\,d^{2}\tilde{u}d.$$
By induction hypothesis the first term is an element of $\mathcal{T}$. If $\tilde{v}$ starts with $j$ or $y$ we are in one of the above cases. Again only the case $\tilde{v}=d^{n}$ for $n\in\mathbb{N}$ has to be considered. By the same induction as in the previous case we obtain that the last term is zero. This proves $\,\shuffle_{0}\,(\mathcal{A}_{0}\otimes\mathcal{T})\subseteq\mathcal{T}$. The proof is now complete.
∎
Let $Y_{0}:=\{\mathbf{1}\}\cup W_{0}y$ be the set of admissible words, i.e. words which do not end up with a $j$ or a $d$. It is easily seen that $\mathcal{A}^{\prime}_{0}:=\langle Y_{0}\rangle_{\mathbb{Q}}$ is a subalgebra of $(\mathcal{A}_{0},\,\shuffle_{0}\,)$ isomorphic to $\mathcal{A}_{0}/\mathcal{T}$. A priori, the product $\,\shuffle_{0}\,$ on $\mathcal{A}_{0}$ is neither commutative nor associative. Now let $\mathcal{L}$ (resp. $\mathcal{L}^{\prime}$) be the ideal of $\mathcal{A}_{0}$ (resp. $\mathcal{A}^{\prime}_{0}$) generated by
$$\{j^{k}\big{(}d(u\,\shuffle_{0}\,v)-du\,\shuffle_{0}\,v-u\,\shuffle_{0}\,dv%
\big{)},\,u,v\in W_{0}y,\,k\in\mathbb{Z}\}.$$
Let $\mathcal{B}_{0}$ (resp. $\mathcal{B}^{\prime}_{0}$) be the quotient algebra $\mathcal{A}_{0}/\mathcal{L}$ (resp. $\mathcal{A}^{\prime}_{0}/\mathcal{L}^{\prime}$). We obviously have the isomorphism:
$$\mathcal{B}^{\prime}_{0}\sim\mathcal{A}_{0}/(\mathcal{T}+\mathcal{L}).$$
Proposition 3.5.
The pair $(\mathcal{B}_{0},\,\shuffle_{0}\,)$ is a commutative, associative and unital algebra.
Proof.
We first prove commutativity $u^{\prime}\,\shuffle_{0}\,v^{\prime}-v^{\prime}\,\shuffle_{0}\,u^{\prime}\in%
\mathcal{L}$ by induction on $r=|u^{\prime}|+|v^{\prime}|$. The cases $r=0$ and $r=1$ are immediate. Several cases must be considered:
• 1st case: $u^{\prime}=yu$ or $v^{\prime}=yv$. The induction hypothesis immediately applies, using (i).
• 2nd case: $u^{\prime}=ju$ and $v^{\prime}=jv$. Then we have by induction hypothesis:
$$\displaystyle ju\,\shuffle_{0}\,jv-jv\,\shuffle_{0}\,ju=j(ju\,\shuffle_{0}\,v+%
u\,\shuffle_{0}\,jv-jv\,\shuffle_{0}\,u-v\,\shuffle_{0}\,ju)\in\mathcal{L}.$$
• 3rd case: $u^{\prime}=du$ and $v^{\prime}=jv$ or vice-versa. We have then:
$$\displaystyle du\,\shuffle_{0}\,jv-jv\,\shuffle_{0}\,du$$
$$\displaystyle=d(u\,\shuffle_{0}\,jv)-u\,\shuffle_{0}\,v-d(jv\,\shuffle_{0}\,u)%
+v\,\shuffle_{0}\,u$$
$$\displaystyle=d(u\,\shuffle_{0}\,jv-jv\,\shuffle_{0}\,u)-(u\,\shuffle_{0}\,v-v%
\,\shuffle_{0}\,u),$$
which belongs to $\mathcal{L}$ by induction hypothesis.
• 4th case: $u^{\prime}=du$ and $v^{\prime}=dv$. Then $dv\,\shuffle_{0}\,du-d(dv\,\shuffle_{0}\,u)+d^{2}v\,\shuffle_{0}\,u\in\mathcal%
{L}$, hence:
$$\displaystyle du\,\shuffle_{0}\,dv-dv\,\shuffle_{0}\,du$$
$$\displaystyle=d(u\,\shuffle_{0}\,dv)-u\,\shuffle_{0}\,d^{2}v-d(dv\,\shuffle_{0%
}\,u)+d^{2}v\,\shuffle_{0}\,u\mod\mathcal{L}$$
$$\displaystyle=d(u\,\shuffle_{0}\,dv-dv\,\shuffle_{0}\,u)-(u\,\shuffle_{0}\,d^{%
2}v-d^{2}v\,\shuffle_{0}\,u)\mod\mathcal{L}.$$
The first term belongs to $\mathcal{L}$ by induction hypothesis. We further suppose that $u^{\prime}$ is written $d^{k}yw$ for some $k\geq 1$ and $w\in Y_{0}$. Iterating the process using (iii) we finally get $du\,\shuffle_{0}\,dv-dv\,\shuffle_{0}\,du=(-1)^{k}(yw\,\shuffle_{0}\,d^{k+1}v-%
d^{k+1}v\,\shuffle_{0}\,yw)\mod\mathcal{L}$. We are then back to the first case.
Associativity follows by showing $u^{\prime}\,\shuffle_{0}\,(v^{\prime}\,\shuffle_{0}\,w^{\prime})=(u^{\prime}\,%
\shuffle_{0}\,v^{\prime})\,\shuffle_{0}\,w^{\prime}$ via induction on the sum $|u^{\prime}|+|v^{\prime}|+|w^{\prime}|$. If one of the words is the empty one nothing is to show. Now let $u^{\prime}=au$, $v^{\prime}=bv$ and $w^{\prime}=cw$ with $a,b,c\in\{d,j,y\}$.
• 1st case: one of the letters is $y$, for example $u^{\prime}=yu$. Using the induction hypothesis, we obtain
$$\displaystyle(yu\,\shuffle_{0}\,v^{\prime})\,\shuffle_{0}\,w^{\prime}=(y(u\,%
\shuffle_{0}\,v^{\prime}))\,\shuffle_{0}\,w^{\prime}$$
$$\displaystyle=y((u\,\shuffle_{0}\,v^{\prime})\,\shuffle_{0}\,w^{\prime})$$
$$\displaystyle=y(u\,\shuffle_{0}\,(v^{\prime}\,\shuffle_{0}\,w^{\prime}))\mod%
\mathcal{L}$$
$$\displaystyle=yu\,\shuffle_{0}\,(v^{\prime}\,\shuffle_{0}\,w^{\prime})\mod%
\mathcal{L}.$$
Note that the other cases $v^{\prime}=yv$ or $w^{\prime}=yw$ are similar, and the arguments are completely analogous.
• 2nd case: $a=b=c=j$. On the one hand we have
$$\displaystyle(ju\,\shuffle_{0}\,jv)\,\shuffle_{0}\,jw=$$
$$\displaystyle~{}j((u\,\shuffle_{0}\,jv)\,\shuffle_{0}\,jw)+j((ju\,\shuffle_{0}%
\,v)\,\shuffle_{0}\,jw)$$
$$\displaystyle+j(j(u\,\shuffle_{0}\,jv)\,\shuffle_{0}\,w)+j(j(ju\,\shuffle_{0}%
\,v)\,\shuffle_{0}\,w)$$
$$\displaystyle=$$
$$\displaystyle~{}j((u\,\shuffle_{0}\,jv)\,\shuffle_{0}\,jw)+j((ju\,\shuffle_{0}%
\,v)\,\shuffle_{0}\,jw)+j((ju\,\shuffle_{0}\,jv)\,\shuffle_{0}\,w),$$
on the other hand
$$\displaystyle ju\,\shuffle_{0}\,(jv\,\shuffle_{0}\,jw)=$$
$$\displaystyle~{}j(u\,\shuffle_{0}\,j(v\,\shuffle_{0}\,jw))+j(u\,\shuffle_{0}\,%
j(jv\,\shuffle_{0}\,w))$$
$$\displaystyle+j(ju\,\shuffle_{0}\,(v\,\shuffle_{0}\,jw))+j(ju\,\shuffle_{0}\,(%
jv\,\shuffle_{0}\,w))$$
$$\displaystyle=$$
$$\displaystyle~{}j(u\,\shuffle_{0}\,(jv\,\shuffle_{0}\,jw))+j(ju\,\shuffle_{0}%
\,(v\,\shuffle_{0}\,jw))+j(ju\,\shuffle_{0}\,(jv\,\shuffle_{0}\,w)).$$
Hence $(ju\,\shuffle_{0}\,jv)\,\shuffle_{0}\,jw=ju\,\shuffle_{0}\,(jv\,\shuffle_{0}\,%
jw)\mod\mathcal{L}$.
• 3rd case: two $j$’s and one $d$. On the one hand we have
$$\displaystyle(ju\,\shuffle_{0}\,jv)\,\shuffle_{0}\,dw=$$
$$\displaystyle~{}d(j(u\,\shuffle_{0}\,jv)\,\shuffle_{0}\,w)-(u\,\shuffle_{0}\,%
jv)\,\shuffle_{0}\,w$$
$$\displaystyle+d(j(ju\,\shuffle_{0}\,v)\,\shuffle_{0}\,w)-(ju\,\shuffle_{0}\,v)%
\,\shuffle_{0}\,w$$
$$\displaystyle=$$
$$\displaystyle~{}d((ju\,\shuffle_{0}\,jv)\,\shuffle_{0}\,w)-(u\,\shuffle_{0}\,%
jv)\,\shuffle_{0}\,w-(ju\,\shuffle_{0}\,v)\,\shuffle_{0}\,w,$$
on the other hand
$$\displaystyle ju\,\shuffle_{0}\,(jv\,\shuffle_{0}\,dw)$$
$$\displaystyle=ju\,\shuffle_{0}\,d(jv\,\shuffle_{0}\,w)-ju\,\shuffle_{0}\,(v\,%
\shuffle_{0}\,w)$$
$$\displaystyle=d(ju\,\shuffle_{0}\,(jv\,\shuffle_{0}\,w))-u\,\shuffle_{0}\,(jv%
\,\shuffle_{0}\,w)-ju\,\shuffle_{0}\,(v\,\shuffle_{0}\,w).$$
then $(ju\,\shuffle_{0}\,jv)\,\shuffle_{0}\,dw=ju\,\shuffle_{0}\,(jv\,\shuffle_{0}\,%
dw)\mod\mathcal{L}$.
• 4th case: two $d$’s and one $j$. We have to prove
$$\displaystyle(du\,\shuffle_{0}\,dv)\,\shuffle_{0}\,jw=du\,\shuffle_{0}\,(dv\,%
\shuffle_{0}\,jw)\mod\mathcal{L}.$$
It suffices to show that
$$\displaystyle(d^{k}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,jw=d^{k}yu\,\shuffle_{%
0}\,(dv\,\shuffle_{0}\,jw)\mod\mathcal{L}$$
with $u\in W_{0}$ and $k\in\mathbb{N}$. Using (iii) we observe
$$\displaystyle(d^{k}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,jw=d(d^{k-1}yu\,%
\shuffle_{0}\,dv)\,\shuffle_{0}\,jw-(d^{k-1}yu\,\shuffle_{0}\,d^{2}v)\,%
\shuffle_{0}\,jw$$
$$\displaystyle=d\big{(}(d^{k-1}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,jw\big{)}-(%
d^{k-1}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,w-(d^{k-1}yu\,\shuffle_{0}\,d^{2}v%
)\,\shuffle_{0}\,jw$$
$$\displaystyle=d\big{(}d^{k-1}yu\,\shuffle_{0}\,(dv\,\shuffle_{0}\,jw)\big{)}-d%
^{k-1}yu\,\shuffle_{0}\,(dv\,\shuffle_{0}\,w)-(d^{k-1}yu\,\shuffle_{0}\,d^{2}v%
)\,\shuffle_{0}\,jw\mod\mathcal{L}$$
$$\displaystyle=d^{k}yu\,\shuffle_{0}\,(dv\,\shuffle_{0}\,jw)+d^{k-1}yu\,%
\shuffle_{0}\,(d^{2}v\,\shuffle_{0}\,jw)-(d^{k-1}yu\,\shuffle_{0}\,d^{2}v)\,%
\shuffle_{0}\,jw\mod\mathcal{L}.$$
For the difference of the two terms in the previous line to belong to $\mathcal{L}$, it suffices to prove
$$d^{k-1}yu\,\shuffle_{0}\,(d^{2}v\,\shuffle_{0}\,jw)=(d^{k-1}yu\,\shuffle_{0}\,%
d^{2}v)\,\shuffle_{0}\,jw\mod\mathcal{L}.$$
Applying the above procedure iteratively this could be reduced to
$$\displaystyle yu\,\shuffle_{0}\,(d^{k}v\,\shuffle_{0}\,jw)=(yu\,\shuffle_{0}\,%
d^{k}v)\,\shuffle_{0}\,jw\mod\mathcal{L},$$
which is true by using (i) and the induction hypothesis.
• 5th case: $a=b=c=d$. We have to prove
$$\displaystyle(du\,\shuffle_{0}\,dv)\,\shuffle_{0}\,dw=du\,\shuffle_{0}\,(dv\,%
\shuffle_{0}\,dw)\mod\mathcal{L}.$$
It suffices to show that
$$\displaystyle(d^{k}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,dw=d^{k}yu\,\shuffle_{%
0}\,(dv\,\shuffle_{0}\,dw)\mod\mathcal{L},$$
with $u\in W_{0}$ and $k\in\mathbb{N}$. Using (iii) we observe
$$\displaystyle(d^{k}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,dw=d(d^{k-1}yu\,%
\shuffle_{0}\,dv)\,\shuffle_{0}\,dw-(d^{k-1}yu\,\shuffle_{0}\,d^{2}v)\,%
\shuffle_{0}\,dw$$
$$\displaystyle=d\big{(}(d^{k-1}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,dw\big{)}-(%
d^{k-1}yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,d^{2}w-(d^{k-1}yu\,\shuffle_{0}\,d%
^{2}v)\,\shuffle_{0}\,dw$$
$$\displaystyle=d^{k}yu\,\shuffle_{0}\,(dv\,\shuffle_{0}\,dw)+d^{k-1}yu\,%
\shuffle_{0}\,(d^{2}v\,\shuffle_{0}\,dw)+d^{k-1}yu\,\shuffle_{0}\,(dv\,%
\shuffle_{0}\,d^{2}w)$$
$$\displaystyle~{}-(d^{k-1}yu\,\shuffle_{0}\,d^{2}v)\,\shuffle_{0}\,dw-(d^{k-1}%
yu\,\shuffle_{0}\,dv)\,\shuffle_{0}\,d^{2}w\mod\mathcal{L}.$$
Iteratively applying this procedure leads – as in the 4th case – to the claim using (i) and the induction hypothesis. Proposition 3.5 is thus proven.
∎
Now we define the map $\zeta_{t}^{\shuffle}\colon\mathcal{B}^{\prime}_{0}\to\mathbb{Q}[[t]]$ by $\zeta_{t}^{\shuffle}(\mathbf{1}):=1$, and for any $k_{1},\ldots,k_{n}\in\mathbb{Z}$,
$$\displaystyle j^{k_{1}}y\cdots j^{k_{n}}y\mapsto\zeta_{t}^{\shuffle}(j^{k_{1}}%
y\cdots j^{k_{n}}y):=\operatorname{Li}_{k_{1},\ldots,k_{n}}(t).$$
Lemma 3.6.
The map $\zeta_{t}^{\shuffle}$ is multiplicative, i.e. is an algebra morphism.
Proof.
From Proposition 3.1 (ii) and (iii) we obtain for any $f,g\in\mathcal{P}_{\geq 1}$
(12)
$$\displaystyle\delta[J[f]g]=J[f]\delta[g]+fg.$$
Therefore the definition of $\,\shuffle_{0}\,$ and Proposition 3.1 (i), (ii), (iii) and (12) imply that
$$\displaystyle\zeta_{t}^{\,\shuffle\,}\colon\mathcal{B}^{\prime}_{0}\to\mathbb{%
Q}[[t]],\hskip 14.226378ptj^{k_{1}}y\cdots j^{k_{n}}y\mapsto J^{k_{1}}[y\cdots
J%
^{k_{n}}[y]\cdots](t)$$
with $k_{1},\ldots,k_{n}\in\mathbb{Z}$ is an algebra morphism.
∎
Next we show that if we restrict the shuffle product $\,\shuffle_{0}\,$ to admissible words corresponding to positive arguments we obtain the ordinary shuffle product. Let $\mathcal{C}:=\mathbb{Q}\mathbf{1}\oplus j\mathbb{Q}\langle j,y\rangle y$ and $\mathcal{D}:=\mathbb{Q}\mathbf{1}\oplus x_{0}\mathbb{Q}\langle x_{0},x_{1}%
\rangle x_{1}$.
Lemma 3.7.
The algebras $(\mathcal{C},\,\shuffle_{0}\,)$ and $(\mathcal{D},\,\shuffle\,)$ are isomorphic, where $\,\shuffle\,$ denotes the ordinary shuffle product.
Proof.
It is easily seen that $\Phi\colon(\mathcal{D},\,\shuffle\,)\to(\mathcal{C},\,\shuffle_{0}\,)$ given by $\mathbf{1}\mapsto\mathbf{1}$ and
$$\displaystyle x_{0}^{k_{1}-1}x_{1}x_{0}^{k_{2}-1}x_{1}\cdots x_{0}^{k_{n}-1}x_%
{1}\mapsto j^{k_{1}}yj^{k_{2}}y\cdots j^{k_{n}}y$$
is an algebra morphism, for $k_{1},\ldots,k_{n}\in\mathbb{N}$, with $k_{1}>1$, $n\in\mathbb{N}$. Since $\Phi$ is bijective the proof is complete.
∎
3.2. $q$-multiple zeta values
For a formal power series $f\in\mathbb{Q}[[t]]$ we define the $q$-dilation operator as $E_{q}[f](t):=f(qt).$ Let $\mathcal{A}:=t\mathbb{Q}[[t,q]]$ be the space of formal power series in the variables $t$ and $q$, without a term of degree zero in $t$. We can interpret $\mathcal{A}$ as the $\mathbb{Q}[[q]]$-algebra $t\mathbb{Q}[[t]]$. Then the $\mathbb{Q}[[q]]$-linear map $P_{q}\colon\mathcal{A}\to\mathcal{A}$ is defined by
(13)
$$\displaystyle P_{q}[f](t):=\sum_{n\geq 0}E_{q}^{n}[f](t).$$
Furthermore, the $q$-difference operator $D_{q}\colon\mathcal{A}\to\mathcal{A}$ is defined as $D_{q}:=\operatorname{Id}-E_{q}.$ We have the following known result:
Proposition 3.8 ([CEM14b]).
(i)
The pair $(\mathcal{A},P_{q})$ is a RBA of weight $\lambda=-1$.
(ii)
For any $f,g\in\mathcal{P}_{\geq 1}$ the operator $D_{q}$ satisfies the generalized Leibniz rule, i.e.
$$\displaystyle D_{q}[fg]=D_{q}[f]g+fD_{q}[g]-D_{q}[f]D_{q}[g].$$
(iii)
The operators $P_{q}$ and $D_{q}$ are mutually inverse, i.e. $D_{q}\circ P_{q}=P_{q}\circ D_{q}=\operatorname{Id}$.
Remark 3.9.
Recall that the Jackson integral
$$\mathcal{J}[f](x):=\int_{0}^{x}f(y)d_{q}y=(1-q)\>\sum_{n\geq 0}f(q^{n}x)q^{n}x$$
is the $q$-analogue of the classical indefinite Riemann integral $R$. For functions $\frac{f(x)}{x}$ – where the Jackson integral is well defined – it reduces to
$$(1-q)\>\sum_{n\geq 0}f(q^{n}x)=\int_{0}^{x}\frac{f(y)}{y}d_{q}y=(1-q)P_{q}[f](%
x),$$
which is the $q$-analogue of the integral operator $J$. Correspondingly, the $q$-analogue of the Euler derivation $\delta$ reduces to $(\operatorname{Id}-E_{q})$.
Lemma 3.10 ([CEM14b]).
Let $k_{1},\ldots,k_{n}\in\mathbb{Z}$. Then we have
$$\displaystyle\overline{\mathfrak{z}}_{q}(k_{1},\ldots,k_{n})=P_{q}^{k_{1}}[yP_%
{q}^{k_{2}}[y\cdots P_{q}^{k_{n}}[y]\cdots]](q).$$
Surprisingly enough, the algebraic formalism for $q$MZVs is simpler than in the classical case. Let $X_{-1}:=\{p,d,y\}$. By $W_{-1}$ we denote the set of words on the alphabet $X_{-1}$, subject to the rule $pd=dp=\mathbf{1}$, where $\mathbf{1}$ denotes the empty word. Again, $\mathcal{A}_{-1}$ denotes the algebra spanned by the words in $W_{-1}$, i.e., $\mathcal{A}_{-1}:=\langle W_{-1}\rangle_{\mathbb{Q}}$.
Then we define the product $\,\shuffle_{-1}\,\colon\mathcal{A}_{-1}\otimes\mathcal{A}_{-1}\to\mathcal{A}_{%
-1}$ by $\mathbf{1}\,\shuffle_{-1}\,w:=w\,\shuffle_{-1}\,\mathbf{1}:=w$ for any $w\in W_{-1}$, and for any words $u,v\in W_{-1}$
(i)
$yu\,\shuffle_{-1}\,v:=u\,\shuffle_{-1}\,yv:=y(u\,\shuffle_{-1}\,v)$,
(ii)
$pu\,\shuffle_{-1}\,pv:=p(u\,\shuffle_{-1}\,pv)+p(pu\,\shuffle_{-1}\,v)-p(u\,%
\shuffle_{-1}\,v)$,
(iii)
$du\,\shuffle_{-1}\,dv:=u\,\shuffle_{-1}\,dv+du\,\shuffle_{-1}\,v-d(u\,\shuffle%
_{-1}\,v)$,
(iv)
$du\,\shuffle_{-1}\,pv=pv\,\shuffle_{-1}\,du:=d(u\,\shuffle_{-1}\,pv)+du\,%
\shuffle_{-1}\,v-u\,\shuffle_{-1}\,v$.
Remark 3.11.
We can deduce (iv) from (iii).
Lemma 3.12.
The pair $(\mathcal{A}_{-1},\,\shuffle_{-1}\,)$ is a commutative, associative and unital algebra.
Proof.
The proof is similar to that of [CEM14b, Theorem 7], and left to the reader.
∎
Next we introduce the set of words ending in the letter $y$ and containing the empty word
$$\displaystyle Y_{-1}:=W_{-1}y\cup\{\mathbf{1}\}\subseteq W_{-1},$$
subject to the rule $pd=dp=\mathbf{1}$. Note that $(\langle Y_{-1}\rangle_{\mathbb{Q}},\,\shuffle_{-1}\,)$ is a subalgebra of $(\mathcal{A}_{-1},\,\shuffle_{-1}\,)$. Moreover we introduce the map $\overline{\mathfrak{z}}_{q}^{\shuffle}\colon\langle Y_{-1}\rangle_{\mathbb{Q}}%
\to\mathbb{Q}[[q]]$ by
$$\displaystyle p^{k_{1}}y\cdots p^{k_{n}}y\mapsto\overline{\mathfrak{z}}_{q}^{%
\shuffle}(p^{k_{1}}y\cdots p^{k_{n}}y):=\overline{\mathfrak{z}}_{q}(k_{1},%
\ldots,k_{n})$$
for any integers $k_{1},\ldots,k_{n}$.
Lemma 3.13 ([CEM14b]).
The map $\overline{\mathfrak{z}}_{q}^{\shuffle}$ is an algebra morphism.
3.3. General Word Algebraic Part
In this section we explore the algebraic structure that is related to non-positive arguments for MZVs and $q$MZVs. For this reason we introduce a parameter $\lambda\in\mathbb{Q}$. The case $\lambda=0$ corresponds to MZVs and the case $\lambda=-1$ to (modified) $q$MZVs.
Let $L:=\{d,y\}$ be an alphabet of two letters. The free monoid of $L$ with empty word $\mathbf{1}$ is denoted by $L^{\ast}$. We denote the free algebra of $L$ by $\mathbb{Q}\langle L\rangle$ and define the subspace of words ending in $d$ by
$$\displaystyle\mathcal{T}_{-}:=\mathcal{T}\cap\mathbb{Q}\langle L\rangle=%
\langle\left\{wd\colon w\in L^{\ast}\right\}\rangle_{\mathbb{Q}}\subseteq%
\mathbb{Q}\langle L\rangle,$$
with $\mathcal{T}$ defined in Lemma 3.4.
The set of admissible words is defined as
$$\displaystyle Y:=L^{\ast}y\cup\{\mathbf{1}\},$$
and the $\mathbb{Q}$-vector space spanned by $Y$ is notated as $\mathcal{H}:=\langle Y\rangle_{\mathbb{Q}}$. It is isomorphic to the quotient $\mathbb{Q}\langle L\rangle/\mathcal{T}_{-}.$ The weight $\operatorname{wt}(w)$ of a word $w\in Y$ is given by the number of letters of $w$, and we use the convention $\operatorname{wt}(\mathbf{1}):=0$.
Furthermore, the depth $\operatorname{dpt}(w)$ of a word $w\in Y$ is given by the number of $y$ in $w$. The $\mathbb{Q}$-vector space $\mathcal{H}$ is graded by depth, i.e.
$$\displaystyle\mathcal{H}=\bigoplus_{n\geq 0}\mathcal{H}_{(n)}$$
with $\mathcal{H}_{(}n):=\langle w\in Y\colon\operatorname{dpt}(w)=n\rangle_{\mathbb%
{Q}}$.
3.3.1. The algebra $\mathcal{H}_{\lambda},\,\lambda\neq 0$
Let $\lambda\in\mathbb{Q}\setminus\{0\}$. We define the product
$$\displaystyle\,\shuffle_{\lambda}\,\colon\mathbb{Q}\langle L\rangle\otimes%
\mathbb{Q}\langle L\rangle\to\mathbb{Q}\langle L\rangle$$
iteratively by
(P1)
$\mathbf{1}\,\shuffle_{\lambda}\,w:=w\,\shuffle_{\lambda}\,\mathbf{1}:=w$ for any $w\in L^{\ast}$;
(P2)
$yu\,\shuffle_{\lambda}\,v:=u\,\shuffle_{\lambda}\,yv:=y(u\,\shuffle_{\lambda}%
\,v)$ for any $u,v\in L^{\ast}$;
(P3)
$du\,\shuffle_{\lambda}\,dv:=\frac{1}{\lambda}\big{[}d(u\,\shuffle_{\lambda}\,v%
)-du\,\shuffle_{\lambda}\,v-u\,\shuffle_{\lambda}\,dv\big{]}$ for any $u,v\in L^{\ast}$.
Furthermore, we define the unit map $\eta\colon\mathbb{Q}\to\mathbb{Q}\langle L\rangle$, $1\mapsto\mathbf{1}$.
Proposition 3.14.
For $\lambda\in\mathbb{Q}$, the triple $(\mathbb{Q}\langle L\rangle,\,\shuffle_{\lambda}\,,\eta)$ is a commutative, associative, and unital $\mathbb{Q}$-algebra. The subspace $\mathcal{T}_{-}$ is a two-sided ideal of $\mathbb{Q}\langle L\rangle$.
Proof.
In the case $\lambda=-1$ the proof is a consequence of Lemma 3.12. We give a proof for any $\lambda\neq 0$ for completeness, although it could be derived from the case $\lambda=-1$ by appropriate rescaling. Commutativity is clear from the definition. We only have to verify associativity if all words begin with a letter $d$. We apply induction on the sum of the lengths of the words. The base case is trivial. For the inductive step we observe for $a,b,c\in L^{\ast}$ that
$$\displaystyle(da\,\shuffle_{\lambda}\,db)\,\shuffle_{\lambda}\,dc=$$
$$\displaystyle~{}\frac{1}{\lambda}\left[d(a\,\shuffle_{\lambda}\,b)-da\,%
\shuffle_{\lambda}\,b-a\,\shuffle_{\lambda}\,db\right]\,\shuffle_{\lambda}\,dc$$
$$\displaystyle=$$
$$\displaystyle~{}\frac{1}{\lambda^{2}}\left[d((a\,\shuffle_{\lambda}\,b)\,%
\shuffle_{\lambda}\,c)-d(a\,\shuffle_{\lambda}\,b)\,\shuffle_{\lambda}\,c-(a\,%
\shuffle_{\lambda}\,b)\,\shuffle_{\lambda}\,dc\right]$$
$$\displaystyle-\frac{1}{\lambda}\left[(da\,\shuffle_{\lambda}\,b)\,\shuffle_{%
\lambda}\,dc+(a\,\shuffle_{\lambda}\,db)\,\shuffle_{\lambda}\,dc\right]$$
$$\displaystyle=$$
$$\displaystyle~{}\frac{1}{\lambda^{2}}\left[d(a\,\shuffle_{\lambda}\,b\,%
\shuffle_{\lambda}\,c)-a\,\shuffle_{\lambda}\,b\,\shuffle_{\lambda}\,dc-da\,%
\shuffle_{\lambda}\,b\,\shuffle_{\lambda}\,c-a\,\shuffle_{\lambda}\,db\,%
\shuffle_{\lambda}\,c\right]$$
$$\displaystyle-\frac{1}{\lambda}\left[da\,\shuffle_{\lambda}\,db\,\shuffle_{%
\lambda}\,c+da\,\shuffle_{\lambda}\,b\,\shuffle_{\lambda}\,dc+a\,\shuffle_{%
\lambda}\,db\,\shuffle_{\lambda}\,dc\right]$$
and
$$\displaystyle da\,\shuffle_{\lambda}\,(db\,\shuffle_{\lambda}\,dc)=$$
$$\displaystyle~{}\frac{1}{\lambda}da\,\shuffle_{\lambda}\,\left[d(b\,\shuffle_{%
\lambda}\,c)-db\,\shuffle_{\lambda}\,c-b\,\shuffle_{\lambda}\,dc\right]$$
$$\displaystyle=$$
$$\displaystyle~{}\frac{1}{\lambda^{2}}\left[d(a\,\shuffle_{\lambda}\,(b\,%
\shuffle_{\lambda}\,c))-da\,\shuffle_{\lambda}\,(b\,\shuffle_{\lambda}\,c)-a\,%
\shuffle_{\lambda}\,d(b\,\shuffle_{\lambda}\,c)\right]$$
$$\displaystyle-\frac{1}{\lambda}\left[da\,\shuffle_{\lambda}\,(db\,\shuffle_{%
\lambda}\,c)+da\,\shuffle_{\lambda}\,(b\,\shuffle_{\lambda}\,dc)\right]$$
$$\displaystyle=$$
$$\displaystyle~{}\frac{1}{\lambda^{2}}\left[d(a\,\shuffle_{\lambda}\,b\,%
\shuffle_{\lambda}\,c)-da\,\shuffle_{\lambda}\,b\,\shuffle_{\lambda}\,c-a\,%
\shuffle_{\lambda}\,db\,\shuffle_{\lambda}\,c-a\,\shuffle_{\lambda}\,b\,%
\shuffle_{\lambda}\,dc\right]$$
$$\displaystyle-\frac{1}{\lambda}\left[a\,\shuffle_{\lambda}\,db\,\shuffle_{%
\lambda}\,dc+da\,\shuffle_{\lambda}\,db\,\shuffle_{\lambda}\,c+da\,\shuffle_{%
\lambda}\,b\,\shuffle_{\lambda}\,dc\right],$$
which shows associativity. From definition we obtain that $\,\shuffle_{\lambda}\,(\mathcal{T}_{-}\otimes\mathbb{Q}\langle L\rangle)=\,%
\shuffle_{\lambda}\,(\mathbb{Q}\langle L\rangle\ \otimes\mathcal{T}_{-})%
\subseteq\mathcal{T}_{-}$ and therefore $\mathcal{T}_{-}$ is a two-sided ideal of $\mathbb{Q}\langle L\rangle$.
∎
3.3.2. The coproduct $\overline{\Delta}_{\lambda},\,\lambda\in\mathbb{Q}$
Now we define the coproduct
$$\displaystyle\overline{\Delta}_{\lambda}\colon\mathbb{Q}\langle L\rangle\to%
\mathbb{Q}\langle L\rangle\otimes\mathbb{Q}\langle L\rangle$$
by
(C1)
$\overline{\Delta}_{\lambda}(y):=\mathbf{1}\otimes y+y\otimes\mathbf{1}$,
(C2)
$\overline{\Delta}_{\lambda}(d):=\mathbf{1}\otimes d+d\otimes\mathbf{1}+\lambda
d\otimes
d$,
which extends uniquely to an algebra morphism (with respect to concatenation) on the free algebra $\mathbb{Q}\langle L\rangle$. The counit map $\varepsilon\colon\mathbb{Q}\langle L\rangle\to\mathbb{Q}$ is given by $\varepsilon(\mathbf{1})=1$ and $\varepsilon(w)=0$ for any word $w\in L^{*}\setminus\{\mathbf{1}\}$.
Example 3.15.
We have
$$\displaystyle\overline{\Delta}_{\lambda}(dy)=$$
$$\displaystyle~{}\mathbf{1}\otimes dy+dy\otimes\mathbf{1}+d\otimes y+y\otimes d%
+\lambda dy\otimes d+\lambda d\otimes dy.$$
Proposition 3.16.
For $\lambda\in\mathbb{Q}$ the triple $(\mathbb{Q}\langle L\rangle,\overline{\Delta}_{\lambda},\varepsilon)$ is a cocommmutative and counital coalgebra, and $\mathcal{T}_{-}$ is a coideal of $\mathbb{Q}\langle L\rangle$.
Proof.
Cocommutativity is clear by definitions (C1) and (C2). The counit axiom is not hard to verify. Finally we have to check coassociativity. We have
$$\displaystyle(\operatorname{Id}\otimes\overline{\Delta}_{\lambda})\overline{%
\Delta}_{\lambda}(d)=(\operatorname{Id}\otimes\overline{\Delta}_{\lambda})(%
\mathbf{1}\otimes d+d\otimes\mathbf{1}+\lambda d\otimes d)$$
$$\displaystyle=$$
$$\displaystyle~{}\mathbf{1}\otimes\mathbf{1}\otimes d+\mathbf{1}\otimes d%
\otimes\mathbf{1}+\lambda\mathbf{1}\otimes d\otimes d+d\otimes\mathbf{1}%
\otimes\mathbf{1}+\lambda d\otimes\mathbf{1}\otimes d+\lambda d\otimes d%
\otimes\mathbf{1}+\lambda^{2}d\otimes d\otimes d$$
and
$$\displaystyle(\overline{\Delta}_{\lambda}\otimes\operatorname{Id})\overline{%
\Delta}_{\lambda}(d)=(\overline{\Delta}_{\lambda}\otimes\operatorname{Id})(%
\mathbf{1}\otimes d+d\otimes\mathbf{1}+\lambda d\otimes d)$$
$$\displaystyle=$$
$$\displaystyle~{}\mathbf{1}\otimes\mathbf{1}\otimes d+\mathbf{1}\otimes d%
\otimes\mathbf{1}+d\otimes\mathbf{1}\otimes\mathbf{1}+\lambda d\otimes d%
\otimes\mathbf{1}+\lambda\mathbf{1}\otimes d\otimes d+\lambda d\otimes\mathbf{%
1}\otimes d+\lambda^{2}d\otimes d\otimes d.$$
The case $(\operatorname{Id}\otimes\overline{\Delta}_{\lambda})\overline{\Delta}_{%
\lambda}(y)=(\overline{\Delta}_{\lambda}\otimes\operatorname{Id})\overline{%
\Delta}_{\lambda}(y)$ is easy to see. We immediately obtain
$$\displaystyle\overline{\Delta}_{\lambda}(\mathcal{T}_{-})\subseteq\mathcal{T}_%
{-}\otimes\mathbb{Q}\langle L\rangle+\mathbb{Q}\langle L\rangle\otimes\mathcal%
{T}_{-},$$
which concludes the proof.
∎
3.3.3. Compatibility properties of the coproduct ($\lambda\neq 0$ case)
Lemma 3.17.
For words $u,v\in L^{\ast}$ we have
(14)
$$\displaystyle\overline{\Delta}_{\lambda}(y)[\overline{\Delta}_{\lambda}(u)\,%
\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(v)]=\overline{\Delta}_{\lambda%
}(yu)\,\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(v)=\overline{\Delta}_{%
\lambda}(u)\,\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(yv)$$
and
(15)
$$\displaystyle\begin{split}\displaystyle\overline{\Delta}_{\lambda}(d)[%
\overline{\Delta}_{\lambda}(u)\,\shuffle_{\lambda}\,\overline{\Delta}_{\lambda%
}(v)]=&\displaystyle\overline{\Delta}_{\lambda}(du)\,\shuffle_{\lambda}\,%
\overline{\Delta}_{\lambda}(v)+\overline{\Delta}_{\lambda}(u)\,\shuffle_{%
\lambda}\,\overline{\Delta}_{\lambda}(dv)\\
&\displaystyle+\lambda[\overline{\Delta}_{\lambda}(du)\,\shuffle_{\lambda}\,%
\overline{\Delta}_{\lambda}(dv)].\end{split}$$
Proof.
Using (P2) and Sweedler’s notation, $\overline{\Delta}_{\lambda}(u)=\sum_{(u)}u_{1}\otimes u_{2}$, we obtain for the first equality of (14)
$$\displaystyle\overline{\Delta}_{\lambda}(y)\left[\overline{\Delta}_{\lambda}(u%
)\,\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(v)\right]$$
$$\displaystyle=\overline{\Delta}_{\lambda}(y)\left[\sum_{(u),(v)}(u_{1}\,%
\shuffle_{\lambda}\,v_{1})\otimes(u_{2}\,\shuffle_{\lambda}\,v_{2})\right]$$
$$\displaystyle=\sum_{(u),(v)}\left[(u_{1}\,\shuffle_{\lambda}\,v_{1})\otimes y(%
u_{2}\,\shuffle_{\lambda}\,v_{2})+y(u_{1}\,\shuffle_{\lambda}\,v_{1})\otimes(u%
_{2}\,\shuffle_{\lambda}\,v_{2})\right]$$
$$\displaystyle=\sum_{(u),(v)}\left[(u_{1}\,\shuffle_{\lambda}\,v_{1})\otimes(yu%
_{2}\,\shuffle_{\lambda}\,v_{2})+(yu_{1}\,\shuffle_{\lambda}\,v_{1})\otimes(u_%
{2}\,\shuffle_{\lambda}\,v_{2})\right]$$
$$\displaystyle=\overline{\Delta}_{\lambda}(yu)\,\shuffle_{\lambda}\,\overline{%
\Delta}_{\lambda}(v).$$
The second equality follows completly analogously. For (15) we observe
$$\displaystyle\overline{\Delta}_{\lambda}(d)[\overline{\Delta}_{\lambda}(u)\,%
\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(v)]$$
$$\displaystyle=$$
$$\displaystyle\overline{\Delta}_{\lambda}(d)\left[\sum_{(u),(v)}(u_{1}\,%
\shuffle_{\lambda}\,v_{1})\otimes(u_{2}\,\shuffle_{\lambda}\,v_{2})\right]$$
$$\displaystyle=$$
$$\displaystyle\sum_{(u),(v)}\left[d(u_{1}\,\shuffle_{\lambda}\,v_{1})\otimes(u_%
{2}\,\shuffle_{\lambda}\,v_{2})+(u_{1}\,\shuffle_{\lambda}\,v_{1})\otimes d(u_%
{2}\,\shuffle_{\lambda}\,v_{2})+\lambda d(u_{1}\,\shuffle_{\lambda}\,v_{1})%
\otimes d(u_{2}\,\shuffle_{\lambda}\,v_{2})\right]$$
$$\displaystyle=$$
$$\displaystyle\sum_{(u),(v)}\left[(\mathbf{1}\otimes d+d\otimes\mathbf{1}+%
\lambda d\otimes d)(u_{1}\otimes u_{2})\,\shuffle_{\lambda}\,(v_{1}\otimes v_{%
2})\right]$$
$$\displaystyle+\sum_{(u),(v)}\left[(u_{1}\otimes u_{2})\,\shuffle_{\lambda}\,(%
\mathbf{1}\otimes d+d\otimes\mathbf{1}+\lambda d\otimes d)(v_{1}\otimes v_{2})\right]$$
$$\displaystyle+\lambda\sum_{(u),(v)}\left[(\mathbf{1}\otimes d+d\otimes\mathbf{%
1}+\lambda d\otimes d)(u_{1}\otimes u_{2})\,\shuffle_{\lambda}\,(\mathbf{1}%
\otimes d+d\otimes\mathbf{1}+\lambda d\otimes d)(v_{1}\otimes v_{2})\right]$$
$$\displaystyle=$$
$$\displaystyle\overline{\Delta}_{\lambda}(du)\,\shuffle_{\lambda}\,\overline{%
\Delta}_{\lambda}(v)+\overline{\Delta}_{\lambda}(u)\,\shuffle_{\lambda}\,%
\overline{\Delta}_{\lambda}(dv)+\lambda\left[\overline{\Delta}_{\lambda}(du)\,%
\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(dv)\right],$$
which yields the claim.
∎
3.3.4. The Hopf algebra $\mathcal{H}_{\lambda},\,\lambda\neq 0$
Theorem 3.18.
The quintuple $\mathcal{H}_{\lambda}=(\mathcal{H},\,\shuffle_{\lambda}\,,\eta,\Delta_{\lambda%
},\varepsilon)$ is a Hopf algebra with
$$\displaystyle\Delta_{\lambda}(w)$$
$$\displaystyle:=\overline{\Delta}_{\lambda}(w)\mod(\mathcal{T}_{-}\otimes%
\mathbb{Q}\langle L\rangle+\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-})$$
for any word $w\in Y$, where $\mathcal{H}=\mathbb{Q}\langle L\rangle/\mathcal{T}_{-}$ is always identified with $\langle Y\rangle_{\mathbb{Q}}$.
Proof.
On the one hand $\mathcal{H}$ is the quotient by a two-sided ideal $\mathcal{T}_{-}$ and therefore $(\mathcal{H},\,\shuffle_{\lambda}\,,\eta)$ is an algebra. On the other hand $\mathcal{T}_{-}$ is a coideal. Hence, $(\mathcal{H},\Delta_{\lambda},\varepsilon)$ is a coalgebra. Since $\mathcal{H}$ is connected it suffices to prove that $(\mathcal{H},\,\shuffle_{\lambda}\,,\eta,\Delta_{\lambda},\varepsilon)$ is a bialgebra.
We show that
$$\overline{\Delta}_{\lambda}(u^{\prime}\,\shuffle_{\lambda}\,v^{\prime})=%
\overline{\Delta}_{\lambda}(u^{\prime})\,\shuffle_{\lambda}\,\overline{\Delta}%
_{\lambda}(v^{\prime})\mod(\mathcal{T}_{-}\otimes\mathbb{Q}\langle L\rangle+%
\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-})$$
by induction on the sum of weights $\operatorname{wt}(u)+\operatorname{wt}(v)$, of the words $u,v\in Y$. The base cases are straightforward.
• 1st case: $u^{\prime}=yu$ or $v^{\prime}=yv$. We have with Lemma 3.17
$$\displaystyle\overline{\Delta}_{\lambda}(yu\,\shuffle_{\lambda}\,v^{\prime})$$
$$\displaystyle=\overline{\Delta}_{\lambda}(y(u\,\shuffle_{\lambda}\,v^{\prime})%
)=\overline{\Delta}_{\lambda}(y)\overline{\Delta}_{\lambda}(u\,\shuffle_{%
\lambda}\,v^{\prime})$$
$$\displaystyle=\overline{\Delta}_{\lambda}(y)(\overline{\Delta}_{\lambda}(u)\,%
\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(v))\mod(\mathcal{T}_{-}\otimes%
\mathbb{Q}\langle L\rangle+\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-})$$
$$\displaystyle=\overline{\Delta}_{\lambda}(yu)\,\shuffle_{\lambda}\,\overline{%
\Delta}_{\lambda}(v)\mod(\mathcal{T}_{-}\otimes\mathbb{Q}\langle L\rangle+%
\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-}).$$
• 2nd case: $u^{\prime}=du$ and $v^{\prime}=dv$. We have with Lemma 3.17 and the induction hypothesis
$$\displaystyle\overline{\Delta}_{\lambda}(du\,\shuffle_{\lambda}\,dv)=\frac{1}{%
\lambda}\left[\overline{\Delta}_{\lambda}\left(d(u\,\shuffle_{\lambda}\,v)-du%
\,\shuffle_{\lambda}\,v-u\,\shuffle_{\lambda}\,dv\right)\right]$$
$$\displaystyle=\frac{1}{\lambda}\left[\overline{\Delta}_{\lambda}(d)(\overline{%
\Delta}_{\lambda}(u)\,\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(v))-%
\overline{\Delta}_{\lambda}(du)\,\shuffle_{\lambda}\,\overline{\Delta}_{%
\lambda}(v)\right.$$
$$\displaystyle\left.-\overline{\Delta}_{\lambda}(u)\,\shuffle_{\lambda}\,%
\overline{\Delta}_{\lambda}(dv)\right]\mod(\mathcal{T}_{-}\otimes\mathbb{Q}%
\langle L\rangle+\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-})$$
$$\displaystyle=\frac{1}{\lambda}\left[\overline{\Delta}_{\lambda}(du)\,\shuffle%
_{\lambda}\,\overline{\Delta}_{\lambda}(v)+\overline{\Delta}_{\lambda}(u)\,%
\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(dv)+\lambda(\overline{\Delta}_%
{\lambda}(du)\,\shuffle_{\lambda}\,\overline{\Delta}_{\lambda}(dv))\right.$$
$$\displaystyle\left.-\overline{\Delta}_{\lambda}(du)\,\shuffle_{\lambda}\,%
\overline{\Delta}_{\lambda}(v)-\overline{\Delta}_{\lambda}(u)\,\shuffle_{%
\lambda}\,\overline{\Delta}_{\lambda}(dv)\right]\mod(\mathcal{T}_{-}\otimes%
\mathbb{Q}\langle L\rangle+\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-})$$
$$\displaystyle=\overline{\Delta}_{\lambda}(du)\,\shuffle_{\lambda}\,\overline{%
\Delta}_{\lambda}(dv)\mod(\mathcal{T}_{-}\otimes\mathbb{Q}\langle L\rangle+%
\mathbb{Q}\langle L\rangle\otimes\mathcal{T}_{-}),$$
which concludes the proof.
∎
Example 3.19.
For $n\in\mathbb{N}$ we have $\Delta_{\lambda}(y^{n})=\sum_{l=0}^{n}\binom{n}{l}y^{l}\otimes y^{n-l};$, and
$$\displaystyle\Delta_{\lambda}(d^{n}y)=$$
$$\displaystyle~{}\mathbf{1}\otimes d^{n}y+d^{n}y\otimes\mathbf{1};$$
$$\displaystyle\Delta_{\lambda}(yd^{n}y)=$$
$$\displaystyle~{}\mathbf{1}\otimes yd^{n}y+y\otimes d^{n}y+d^{n}y\otimes y+yd^{%
n}y\otimes\mathbf{1};$$
$$\displaystyle\Delta_{\lambda}(dyd^{n}y)=$$
$$\displaystyle~{}\mathbf{1}\otimes dyd^{n}y+y\otimes d^{n+1}y+d^{n}y\otimes dy+%
dy\otimes d^{n}y+d^{n+1}y\otimes y+dyd^{n}y\otimes\mathbf{1}$$
$$\displaystyle+\lambda dy\otimes d^{n+1}y+\lambda d^{n+1}y\otimes dy.$$
3.3.5. Compatibility between the product and the coproduct ($\lambda=0$ case)
Let us now focus on the case $\lambda=0$. Recall from Paragraph 3.1 that $\mathcal{L}$ is the two-sided ideal of the (noncommutative and nonassociative) algebra $(\mathcal{A}_{0},\,\shuffle_{0}\,)$ generated by the elements
$$\displaystyle j^{k}\big{(}d(u\,\shuffle_{0}\,v)-du\,\shuffle_{0}\,v-u\,%
\shuffle_{0}\,dv\big{)},\,k\in\mathbb{Z},\,u,v\in W_{0}y.$$
Now let $\mathcal{L}_{-}$ be the two-sided ideal of the (noncommutative and nonassociative) subalgebra $(\mathbb{Q}\langle L\rangle,\,\shuffle_{0}\,)$ generated by the elements
$$\displaystyle d^{k}\big{(}d(u\,\shuffle_{0}\,v)-du\,\shuffle_{0}\,v-u\,%
\shuffle_{0}\,dv\big{)},\,k\in\mathbb{N}_{0},\,u,v\in L^{\ast}.$$
Further let $\mathcal{L}_{-}^{(2)}:=\mathcal{L}_{-}\otimes\mathbb{Q}\langle L\rangle+%
\mathbb{Q}\langle L\rangle\otimes\mathcal{L}_{-}$.
Proposition 3.20.
For any $u^{\prime},v^{\prime}\in L^{\ast}$ we have:
(16)
$$\overline{\Delta}_{0}(u^{\prime}\,\shuffle_{0}\,v^{\prime})=\overline{\Delta}_%
{0}(u^{\prime})\,\shuffle_{0}\,\overline{\Delta}_{0}(v^{\prime})\mod\mathcal{L%
}_{-}^{(2)}.$$
Proof.
We use induction on $r:=|u^{\prime}|+|v^{\prime}|$. The cases $r=0$ and $r=1$ being immediate. The case $u^{\prime}=yu$ or $v^{\prime}=yv$ is easy and left to the reader. In the case $u^{\prime}=du$ and $v^{\prime}=dv$ we compute:
$$\displaystyle\overline{\Delta}_{0}(du\,\shuffle_{0}\,dv)=\overline{\Delta}_{0}%
\big{(}d(u\,\shuffle_{0}\,dv)-u\,\shuffle_{0}\,d^{2}v)\big{)}$$
$$\displaystyle=$$
$$\displaystyle~{}\overline{\Delta}_{0}(d)\overline{\Delta}_{0}(u\,\shuffle_{0}%
\,dv)-\overline{\Delta}_{0}(u\,\shuffle_{0}\,d^{2}v)$$
$$\displaystyle=$$
$$\displaystyle~{}\overline{\Delta}_{0}(d)\big{(}\overline{\Delta}_{0}(u)\,%
\shuffle_{0}\,\overline{\Delta}_{0}(dv)\big{)}-\overline{\Delta}_{0}(u\,%
\shuffle_{0}\,d^{2}v)\mod\mathcal{L}_{-}^{(2)}$$
$$\displaystyle=$$
$$\displaystyle~{}\overline{\Delta}_{0}(d)\overline{\Delta}_{0}(u)\,\shuffle_{0}%
\,\overline{\Delta}_{0}(dv)+\overline{\Delta}_{0}(u)\,\shuffle_{0}\,\overline{%
\Delta}_{0}(d)\overline{\Delta}_{0}(d)\overline{\Delta}_{0}(v)-\overline{%
\Delta}_{0}(u\,\shuffle_{0}\,d^{2}v)\mod\mathcal{L}_{-}^{(2)},$$
hence we get:
(17)
$$\overline{\Delta}_{0}(du\,\shuffle_{0}\,dv)-\overline{\Delta}_{0}(du)\,%
\shuffle_{0}\,\overline{\Delta}_{0}(dv)=-\big{(}\overline{\Delta}_{0}(u\,%
\shuffle_{0}\,d^{2}v)-\overline{\Delta}_{0}(u)\,\shuffle_{0}\,\overline{\Delta%
}_{0}(d^{2}v)\big{)}\mod\mathcal{L}_{-}^{(2)}.$$
Iterating this process we return to the case when one of the arguments starts with a $y$.
∎
3.3.6. The Hopf algebra $\mathcal{H}_{0}$
Let $\mathcal{H}_{0}:=\mathbb{Q}\langle L\rangle/(\mathcal{L}_{-}+\mathcal{T}_{-})$.
Proposition 3.21.
The ideal $\mathcal{L}_{-}$ is a coideal of $(\mathbb{Q}\langle L\rangle,\overline{\Delta}_{0})$, where $\overline{\Delta}_{0}$ is defined by (C1) and (C2) with $\lambda=0$.
Proof.
Using Proposition 3.20 we compute
$$\displaystyle\overline{\Delta}_{0}\big{(}d(u\,\shuffle_{0}\,v)-du\,\shuffle_{0%
}\,v-u\,\shuffle_{0}\,dv\big{)}$$
$$\displaystyle=$$
$$\displaystyle~{}\overline{\Delta}_{0}(d)\big{(}\overline{\Delta}_{0}(u)\,%
\shuffle_{0}\,\overline{\Delta}_{0}(v)\big{)}-\overline{\Delta}_{0}(du)\,%
\shuffle_{0}\,\overline{\Delta}_{0}(v)-\overline{\Delta}_{0}(u)\,\shuffle_{0}%
\,\overline{\Delta}_{0}(dv)\mod\mathcal{L}_{-}^{(2)}$$
$$\displaystyle=$$
$$\displaystyle~{}\overline{\Delta}_{0}(d)\big{(}\overline{\Delta}_{0}(u)\,%
\shuffle_{0}\,\overline{\Delta}_{0}(v)\big{)}-\overline{\Delta}_{0}(d)%
\overline{\Delta}_{0}(u)\,\shuffle_{0}\,\overline{\Delta}_{0}(v)-\overline{%
\Delta}_{0}(u)\,\shuffle_{0}\,\overline{\Delta}_{0}(d)\overline{\Delta}_{0}(v)%
\mod\mathcal{L}_{-}^{(2)}.$$
Hence, $\overline{\Delta}_{0}\big{(}d(u\,\shuffle_{0}\,v)-du\,\shuffle_{0}\,v-u\,%
\shuffle_{0}\,dv\big{)}\in\mathcal{L}_{-}^{(2)}$.
∎
Corollary 3.22.
$\mathcal{H}_{0}$ is a commutative Hopf algebra.
Proof.
From Proposition 3.16 and Proposition 3.21 we get that the ideal $\mathcal{T}_{-}+\mathcal{L}_{-}$ is also a coideal of $\mathbb{Q}\langle L\rangle$. Hence the quotient $\mathcal{H}_{0}$ is a bialgebra. It is graded by the depth (and weight), hence connected, thus $\mathcal{H}_{0}$ is a Hopf algebra.
∎
We will denote by $\Delta_{0}$ the coproduct on $\mathcal{H}_{0}$. By a slight abuse of notation the coproduct on $\mathbb{Q}\langle L\rangle/\mathcal{T}$, which was deduced from $\overline{\Delta}_{0}$, will also be denoted $\Delta_{0}$. Note that the ideal $\mathcal{T}_{-}+\mathcal{L}_{-}$ is also graded by depth. One then gets a grading on the quotient $\mathcal{H}_{0}$, which we still denote by $\operatorname{dpt}$.
3.3.7. Shuffle factorization
Let $\lambda\in\mathbb{Q}$, including the case $\lambda=0$. From connectedness we can always write
$$\displaystyle\Delta_{\lambda}([w])=\mathbf{1}\otimes[w]+[w]\otimes\mathbf{1}+%
\tilde{\Delta}_{\lambda}([w])\hskip 14.226378pt\text{with}\hskip 14.226378pt%
\tilde{\Delta}_{\lambda}([w])\in\bigoplus_{p+q=n\atop p\neq 0,q\neq 0}\mathcal%
{H}_{(p)}\otimes\mathcal{H}_{(q)}.$$
Therefore in the following we use two variants of Sweedler’s notation
$$\displaystyle\Delta_{\lambda}([w])=\sum_{([w])}[w]_{1}\otimes[w]_{2}\hskip 14.%
226378pt\text{and}\hskip 14.226378pt\tilde{\Delta}_{\lambda}([w])=\sum_{([w])}%
[w]^{\prime}\otimes[w]^{\prime\prime}.$$
The following theorem, valid for any $\lambda$, including $\lambda=0$, provides a nice example of the theory outlined in [Pat93].
Theorem 3.23.
Let $\lambda\in\mathbb{Q}$. Then for all $w\in L^{*}$ we have
$$\displaystyle\,\shuffle_{\lambda}\,\circ\Delta_{\lambda}([w])=2^{\operatorname%
{dpt}([w])}[w],$$
where $[w]$ stands for the class of $w$ modulo $\mathcal{T}_{-}$ in the case $\lambda\neq 0$ (resp. modulo $\mathcal{T}_{-}+\mathcal{L}_{-}$ in the case $\lambda=0$).
Proof.
We prove this by induction on the weight of $[w]$. For $\operatorname{wt}([w])=0$ we have $[w]=\mathbf{1}$ and obtain$\,\shuffle_{\lambda}\,\circ\Delta_{\lambda}(\mathbf{1})=\,\shuffle_{\lambda}\,%
(\mathbf{1}\otimes\mathbf{1})=\mathbf{1}.$ For the inductive step we consider two cases:
• 1st case: $w=yv$ with $v\in L^{*}$
We have $\operatorname{dpt}([w])=\operatorname{dpt}([v])+1$ and obtain
$$\displaystyle\,\shuffle_{\lambda}\,\circ\Delta_{\lambda}([yv])$$
$$\displaystyle=\,\shuffle_{\lambda}\,\circ(\Delta_{\lambda}([y])\Delta_{\lambda%
}([v]))=\,\shuffle_{\lambda}\,\left(\sum_{([v])}[v]_{1}\otimes[yv]_{2}+\sum_{(%
[v])}[yv]_{1}\otimes[v]_{2}\right)$$
$$\displaystyle=[y]\left(\sum_{([v])}\,\shuffle_{\lambda}\,([v]_{1}\otimes[v]_{2%
})+\sum_{([v])}\,\shuffle_{\lambda}\,([v]_{1}\otimes[v]_{2})\right)=2[y](\,%
\shuffle_{\lambda}\,\circ\Delta_{\lambda}([v]))$$
$$\displaystyle=2^{\operatorname{dpt}([v])+1}[yv]=2^{\operatorname{dpt}([w])}[w].$$
• 2nd case: $w=dv$ with $v\in L^{*}$
Since $\operatorname{dpt}([w])=\operatorname{dpt}([v])$ we observe
$$\displaystyle\,\shuffle_{\lambda}\,\circ\Delta_{\lambda}([dv])$$
$$\displaystyle=\,\shuffle_{\lambda}\,\circ(\Delta_{\lambda}([d])\Delta_{\lambda%
}([v]))$$
$$\displaystyle=\,\shuffle_{\lambda}\,\left(\sum_{([v])}[dv]_{1}\otimes[v]_{2}+[%
v]_{1}\otimes[dv]_{2}+\lambda([dv]_{1}\otimes[dv]_{2})\right)$$
$$\displaystyle=[d]\left(\sum_{([v])}\,\shuffle_{\lambda}\,([v]_{1}\otimes[v]_{2%
})\right)=[d](\,\shuffle_{\lambda}\,\circ\Delta_{\lambda}([v]))$$
$$\displaystyle=2^{\operatorname{dpt}([v])}[dv]=2^{\operatorname{dpt}([w])}[w].$$
∎
Corollary 3.24.
Let $\lambda\in\mathbb{Q}$.
Then for all $w\in L^{*}$, we have
$$\displaystyle(2^{\operatorname{dpt}([w])}-2)[w]$$
$$\displaystyle=\,\shuffle_{\lambda}\,\circ\tilde{\Delta}_{\lambda}([w])$$
$$\displaystyle=\sum_{([w])}[w]^{\prime}\,\shuffle_{\lambda}\,[w]^{\prime\prime}%
=K\star K([w]).$$
The linear map $K:=\operatorname{Id}-\eta\circ\varepsilon\in{\mathrm{End}}_{\mathbb{Q}}(%
\mathcal{H}_{\lambda})$ is a projector to the augmentation ideal $\mathcal{H}^{\prime}:=\bigoplus_{n>0}\mathcal{H}_{n}$, and $f\star g:=\,\shuffle_{\lambda}\,\circ(f\otimes g)\circ\Delta_{\lambda}$, $f,g\in{\mathrm{End}}_{\mathbb{Q}}(\mathcal{H}_{\lambda})$.
3.3.8. A combinatorial description of the coproduct $\Delta_{\lambda}$
In the following we give a combinatorial description of the coproduct $\Delta_{\lambda}$. However, note that we consider the construction only on an admissible representative $w\in Y$ of a given equivalence class in $\mathbb{Q}\langle L\rangle/\mathcal{T}_{-}$ for $\lambda\neq 0$ (resp. in $\mathbb{Q}\langle L\rangle/(\mathcal{L}_{-}+\mathcal{T}_{-})$ for $\lambda=0$).
Let $w:=d^{n_{1}-1}y\cdots d^{n_{k-1}-1}yd^{n_{k}-1}y\in Y$ be a word, with $n:=\sum_{i=1}^{k}n_{i}$, and define for $1\leq m\leq k$, $N^{m}_{w}:=\{n_{1},n_{1}+n_{2},\ldots,n_{1}+\cdots+n_{m}\}$. The coproduct $\Delta_{\lambda}(w)$ can be calculated as follows. Let $S:=\{s_{1}<\cdots<s_{l}\}\subseteq[n]:=\{1,\ldots,n\}$ and $\bar{S}:=[n]\backslash S=\{\bar{s}_{1}<\cdots<\bar{s}_{n-l}\}$. Define the words $w_{S}:=w_{s_{1}}\cdots w_{s_{l}}$ and $w_{\bar{S}}:=w_{\bar{s}_{1}}\cdots w_{\bar{s}_{n-l}}$. The set $S$ is called admissible if both $w_{S}$ and $w_{\bar{S}}$ are in $Y$, i.e., if $s_{l},\bar{s}_{n-l}\in N^{k}_{w}$ with $w\in Y$. The coproduct is then given by
(18)
$$\displaystyle\Delta_{\lambda}(w)=\sum_{S\subseteq[n]\atop S\ {\rm{adm}}}w_{S}%
\otimes w_{\bar{S}}+\sum_{S\subseteq[n]\atop S\ {\rm{adm}}}\sum_{J=\{j_{1}<%
\cdots<j_{p}\}\subset S\atop{J\neq\emptyset,\ J\cap N^{k}_{w}=\emptyset\atop j%
_{p}<n_{1}+\cdots+n_{k-1}}}\lambda^{|J|}w_{S}\otimes w_{[n]\backslash(S%
\backslash J)}.$$
For $\lambda=0$ this reduces to the coproduct corresponding to MPLs at non-positive integer arguments
$$\Delta_{0}(w):=\sum_{S\subseteq[n]\atop S\ {\rm{adm}}}w_{S}\otimes w_{\bar{S}}.$$
We introduce now a graphical notation, which should make the above more transparent. The set of vertices $V:=\{\leavevmode\hbox to8.4pt{\vbox to8.4pt{\pgfpicture\makeatletter\hbox to 0%
.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{%
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{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }%
\nullfont\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}}{}{}{}{}{}{}{}%
{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{%
pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }%
\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{%
rgb}{0,0,0}{}\pgfsys@moveto{2.0pt}{0.0pt}\pgfsys@curveto{2.0pt}{1.104569pt}{1.%
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{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke%
\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to8.4pt{\vbox to8.4pt{%
\pgfpicture\makeatletter\hbox to 0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }%
\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}%
\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }%
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to 0.0pt{%
\pgfsys@beginscope\pgfsys@invoke{ }{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope%
\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}%
\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }{}\pgfsys@moveto{2.0pt}{0.0pt}%
\pgfsys@curveto{2.0pt}{1.104569pt}{1.104569pt}{2.0pt}{0.0pt}{2.0pt}%
\pgfsys@curveto{-1.104569pt}{2.0pt}{-2.0pt}{1.104569pt}{-2.0pt}{0.0pt}%
\pgfsys@curveto{-2.0pt}{-1.104569pt}{-1.104569pt}{-2.0pt}{0.0pt}{-2.0pt}%
\pgfsys@curveto{1.104569pt}{-2.0pt}{2.0pt}{-1.104569pt}{2.0pt}{0.0pt}%
\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{%
}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}%
\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}%
\lxSVG@closescope\endpgfpicture}}\}$ is used to define a polygon. The black vertex $\leavevmode\hbox to8.4pt{\vbox to8.4pt{\pgfpicture\makeatletter\hbox to 0.0pt{%
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\lxSVG@closescope\endpgfpicture}}\sim d$ and the white one $\leavevmode\hbox to8.4pt{\vbox to8.4pt{\pgfpicture\makeatletter\hbox to 0.0pt{%
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\lxSVG@closescope\endpgfpicture}}\sim y$. To each admissible word $w=d^{n_{1}-1}y\cdots d^{n_{k-1}-1}yd^{n_{k}-1}y\in Y$ corresponds an polygon with clockwise oriented edges, and the vertices colored clockwise according to the word $w$. For instance, the word $w=ddydy$ corresponds to
An admissible subset $S\subseteq[n]$ corresponds to a sub polygon. The admissible subsets for the above example are as follows:
$\{1,3\},\{2,3\},\{4,5\}$ correspond respectively to
$\{1,2,3\},\{2,4,5\}$, $\{1,4,5\}$ correspond respectively to
$\{1,2,4,5\}$ and $\{3\}$ correspond respectively to
$$\scalebox{0.7}{
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In the last polygon we have marked the single vertex subpolygon by a circle around the white vertex.
The coproduct
$$\displaystyle\Delta_{0}(ddydy)$$
$$\displaystyle=\sum_{S\subseteq[n]\atop S\ {\rm{adm}}}w_{S}\otimes w_{\bar{S}}$$
$$\displaystyle=ddydy\otimes\mathbf{1}+\mathbf{1}\otimes ddydy+3dy\otimes ddy+3%
ddy\otimes dy+dddy\otimes y+y\otimes dddy.$$
The first two terms on the right-hand side correspond to $S=[n]$ and $S=\emptyset$, respectively.
The pictorial description of the weight-$\lambda$ coproduct (18) is captured as follows. The second term on the right-hand side of the coproduct (18) reflects the term $\lambda d\otimes d$ in the coproduct
$$\overline{\Delta}_{\lambda}(d):=\mathbf{1}\otimes d+d\otimes\mathbf{1}+\lambda
d\otimes
d$$
defined further above. It amounts to a certain doubling of those black vertices in an admissible word $w=d^{n_{1}-1}y\cdots d^{n_{k-1}-1}yd^{n_{k}-1}y\in Y$, which appear before the $y$ at position $n_{1}+\cdots+n_{k-1}$. Algebraically this means that extracting a subpolygon corresponding to the admissible subset $S\subseteq[n]$ leads to a splitting of the word $w$ into $w_{S}$ and $w_{\bar{S}^{\prime}}$, where the augmented complement sets ${\bar{S}^{\prime}}$ contain $\bar{S}$, i.e., $\bar{S}\subset{\bar{S}^{\prime}}$. This is due to not eliminating several black vertices, i.e., $d$’s that appear before the $y$ at position $n_{1}+\cdots+n_{k-1}$. Pictorially we denoted this by doubling black vertices. Returning to the example above. We find for the admissible word $w=ddydy$
which correspond to the admissible subsets and related augmented complement sets: $\{1,3\}$, ${\bar{S}^{\prime}}=\{1,2,4,5\}$, and $\{2,3\}$, ${\bar{S}^{\prime}}=\{1,2,4,5\}$, respectively. Next we consider
where we have $\{2,4,5\}$, ${\bar{S}^{\prime}}=\{1,2,3\}$, and $\{1,4,5\}$, ${\bar{S}^{\prime}}=\{1,2,3\}$, respectively. For
we have $\{1,2,3\}$, ${\bar{S}^{\prime}}=\{1,4,5\}$, and $\{1,2,3\}$, ${\bar{S}^{\prime}}=\{2,4,5\}$, and $\{1,2,3\}$, ${\bar{S}^{\prime}}=\{1,2,4,5\}$, respectively. Finally
where we have $\{1,2,4,5\}$, ${\bar{S}^{\prime}}=\{1,3\}$, and $\{1,2,4,5\}$, ${\bar{S}^{\prime}}=\{2,3\}$, and $\{1,2,4,5\}$, ${\bar{S}^{\prime}}=\{1,2,3\}$, respectively.
The coproduct
$$\displaystyle\Delta_{\lambda}(ddydy)=\sum_{S\subseteq[5]\atop S\ {\rm{adm}}}w_%
{S}\otimes w_{\bar{S}}+\sum_{S\subseteq[5]\atop S\ {\rm{adm}}}\sum_{J=\{j_{1}<%
\cdots<j_{p}\}\subset S\atop{J\neq\emptyset,\ J\cap N^{2}_{ddydy}=\emptyset%
\atop j_{p}<3}}\lambda^{|J|}w_{S}\otimes w_{[5]\backslash(S\backslash J)}$$
$$\displaystyle=ddydy\otimes\mathbf{1}+\mathbf{1}\otimes ddydy+3dy\otimes ddy+3%
ddy\otimes dy+dddy\otimes y+y\otimes dddy$$
$$\displaystyle\quad+2\lambda dy\otimes dddy+4\lambda ddy\otimes ddy+2\lambda
dddy%
\otimes dy+\lambda^{2}ddy\otimes dddy+\lambda^{2}dddy\otimes ddy.$$
Again, the first two terms on the right-hand side correspond to $S=[n]$ and $S=\emptyset$, respectively.
4. Renormalization of regularized MZVs
Alain Connes and Dirk Kreimer discovered a Hopf algebraic approach to the BPHZ renormalization method in perturbative quantum field theory [CK00, CK01]. See [Man04] for a review. One of the fundamental results of these seminal works is the formulation of the process of perturbative renormalization in terms of a factorization theorem for regularized Hopf algebra characters. We briefly recall this theorem, and apply it in the context of the Hopf algebra introduced on $t$- and $q$-regularized MPLs, when considering them at non-positive arguments.
4.1. Connes–Kreimer Renormalization in a Nutshell
We regard the commutative algebra $\mathcal{A}:=\mathbb{Q}[z^{-1},z]]$ with the renormalization scheme
$\mathcal{A}=\mathcal{A}_{-}\oplus\mathcal{A}_{+},$ where $\mathcal{A}_{-}:=z^{-1}\mathbb{Q}[z^{-1}]$ and $\mathcal{A}_{+}:=\mathbb{Q}[[z]]$.
On $\mathcal{A}$ we define the corresponding projector $\pi:\mathcal{A}\to\mathcal{A}_{-}$ by
$$\displaystyle\pi\left(\sum_{n=-k}^{\infty}a_{n}z^{n}\right):=\sum_{n=-k}^{-1}a%
_{n}z^{n}$$
with the common convention that the sum over the empty set is zero. Then $\pi$ and $\operatorname{Id}-\pi:\mathcal{A}\to\mathcal{A}_{+}$ are Rota–Baxter operators of weight $-1$. See e. g. [CK00, Ebr02, EG07].
Let $(\mathcal{H},m_{\mathcal{H}},\Delta)$ be a bialgebra and $(\mathcal{A},m_{\mathcal{A}})$ an algebra. Then we define the convolution product $\star\colon\operatorname{Hom}(\mathcal{H},\mathcal{A})\otimes\operatorname{Hom%
}(\mathcal{H},\mathcal{A})\to\operatorname{Hom}(\mathcal{H},\mathcal{A})$ by the composition
$$\displaystyle\mathcal{H}\stackrel{{\scriptstyle\Delta}}{{\longrightarrow}}%
\mathcal{H}\otimes\mathcal{H}\stackrel{{\scriptstyle\varphi\otimes\psi}}{{%
\longrightarrow}}\mathcal{A}\otimes\mathcal{A}\stackrel{{\scriptstyle m_{%
\mathcal{A}}}}{{\longrightarrow}}\mathcal{A}$$
for $\varphi,\psi\in\operatorname{Hom}(\mathcal{H},\mathcal{A})$, or in Sweedler’s notation
$$\displaystyle(\varphi\star\psi)(x):=m_{\mathcal{A}}(\varphi\otimes\psi)\Delta(%
x)=\sum_{(x)}\varphi(x_{1})\psi(x_{2}).$$
The Connes–Kreimer Hopf algebra approach unveiled a beautiful encoding of one of the key concepts of the renormalization process, i.e., Bogoliubov’s counter term recursion, in terms of an algebraic Birkhoff decomposition:
Theorem 4.1 ([CK00],[CK01],[Man04],[EGP07]).
Let $(\mathcal{H},m_{\mathcal{H}},\Delta)$ be a connected filtered Hopf algebra and $\mathcal{A}$ a commutative unital algebra equipped with a renormalization scheme $\mathcal{A}=\mathcal{A}_{-}\oplus\mathcal{A}_{+}$ and corresponding idempotent Rota–Baxter operator $\pi$, where $\mathcal{A}_{-}=\pi(\mathcal{A})$ and $\mathcal{A}_{+}=(\operatorname{Id}-\pi)(\mathcal{A})$. Further let $\phi\colon\mathcal{H}\to\mathcal{A}$ be a Hopf algebra character. Then:
a)
The character $\phi$ admits a unique decomposition
(19)
$$\displaystyle\phi=\phi_{-}^{\star{(-1)}}\star\phi_{+}$$
called algebraic Birkhoff decomposition, in which $\phi_{-}\colon\mathcal{H}\to\mathbb{Q}\oplus\mathcal{A}_{-}$ and $\phi_{+}\colon\mathcal{H}\to\mathcal{A}_{+}$ are characters.
b)
The maps $\phi_{-}$ and $\phi_{+}$ are recursively given fixed point equations
(20)
$$\displaystyle\phi_{-}$$
$$\displaystyle=e-\pi\left(\phi_{-}\star(\phi-e)\right),$$
(21)
$$\displaystyle\phi_{+}$$
$$\displaystyle=e+(\operatorname{Id}-\pi)\left(\phi_{-}\star(\phi-e\right)),$$
where the unit for the convolution algebra product is $e=\eta_{\mathcal{A}}\circ\varepsilon$, and $\eta_{\mathcal{A}}:\mathbb{Q}\to\mathcal{A}$ is the unit map of the algebra $\mathcal{A}$.
4.2. Renormalization of MZVs
An important remark is in order. To improve readableness we skip brackets in the notation of classes of words, that is, in the following a word $w$ stands for the class $[w]$.
Let $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$. Then we define a map $\phi\colon\mathcal{H}_{0}\to\mathbb{Q}[z^{-1},z]]$ by
(22)
$$\displaystyle d^{k_{1}}y\cdots d^{k_{n}}y\mapsto\phi(d^{k_{1}}y\cdots d^{k_{n}%
}y)(z):=\partial_{z}^{k_{1}}[x\partial_{z}^{k_{2}}[x\cdots\partial_{z}^{k_{n}}%
[x]]\cdots](z),$$
where $x(z):=\frac{e^{z}}{1-e^{z}}$.
Lemma 4.2.
The map $\phi\colon(\mathcal{H}_{0},\,\shuffle_{0}\,)\to(\mathbb{Q}[z^{-1},z]],\cdot)$ is a Hopf algebra character. Furthermore, the following diagram commutes:
{xy}
(0,20)*+(H_0, \shuffle_0 )=”a”; (40,20)*+(Q[[t]],⋅)=”b”;
(40,0)*+(Q[z^-1,z]],⋅)=”d”;
\ar”a”;”b”?*!/_4mm/ζ_t^\shuffle;
\ar”b”;”d”?*!/_8mm/t↦e^z;
\ar”a”;”d”;?*!/_2mm/ϕ;
Proof.
From the chain and product rule of differentiation we easily obtain that $\phi=\zeta^{\,\shuffle\,}_{e^{z}}$. Furthermore, the evaluation map $t\mapsto e^{z}$ and $\zeta_{t}^{\shuffle}$ are both algebra morphisms (see Lemma 3.6). Therefore $\phi$ is – as a composition of multiplicative maps – itself a character.
∎
Next we apply Theorem 4.1 to the character $\phi$ (Lemma 4.2). Then we define renormalized MZVs $\zeta_{+}$ – using the character $\phi_{+}$ with image in $\mathbb{Q}[[z]]$ in the Birkhoff decomposition (19) of $\phi$ – by
$$\displaystyle\zeta_{+}(-k_{1},\ldots,-k_{n}):=\lim_{z\to 0}\phi_{+}(d^{k_{1}}y%
\cdots d^{k_{n}}y)(z)$$
for $k_{1},\ldots,k_{n}\in\mathbb{N}_{0},n\in\mathbb{N}$. The first values of $\zeta_{+}$ in depth two are given in Table 1 (for an explicit calculation example see Example 4.5).
Note that $\zeta_{+}$ respects the shuffle product $\,\shuffle_{0}\,$ as $\phi_{+}$ is a character with respect to the algebra $(\mathcal{H}_{0},\,\shuffle_{0}\,)$. However, note that the quasi-shuffle relations are not verified because it would require $\zeta_{+}(0,0)=\frac{3}{8}$.
Next we show that the renormalized MZVs coincide with the meromorphic continuation of MZVs discussed in Section 2:
Theorem 4.3.
The renormalization procedure is compatible with the meromorphic continuation of MZVs, i.e. for $k\in\mathbb{N}_{0}$
(23)
$$\displaystyle\zeta_{+}(-k)=\zeta_{1}(-k)$$
and for $a,b\geq 0$ with $a+b$ odd
(24)
$$\displaystyle\zeta_{+}(-a,-b)=\zeta_{2}(-a,-b).$$
Note that for $\operatorname{dpt}(w)>2$ there is no information form the meromorphic continuation (see Remark 2.3).
Proof.
We begin with the proof of (23). From Equation (22) we obtain
$$\displaystyle\phi(d^{k}y)(z)$$
$$\displaystyle=\partial_{z}^{k}\left(\frac{e^{z}}{1-e^{z}}\right)$$
$$\displaystyle=-\partial_{z}^{k}\left(\frac{1}{z}\frac{ze^{z}}{e^{z}-1}\right)$$
$$\displaystyle=-\partial_{z}^{k}\left(\frac{B_{0}}{z}+\sum_{n\geq 0}\frac{B_{n+%
1}}{(n+1)!}z^{n}\right)$$
$$\displaystyle=-\left(\frac{(-1)^{k}k!B_{0}}{z^{k+1}}+\sum_{n\geq 0}\frac{B_{n+%
k+1}}{n+k+1}\frac{1}{n!}z^{n}\right).$$
Since $d^{k}y\in Y$ is a primitive element for the coproduct $\Delta_{0}$ we obtain with Remark 2.3 that
$$\displaystyle\phi_{+}(d^{k}y)(z)=(\operatorname{Id}-\pi)\phi(z)=-\frac{B_{k+1}%
}{k+1}+O(z)=\zeta_{1}(-k)+O(z).$$
For Equation (24) we calculate for $a+b$ odd with $a,b\geq 0$ (see Remark 2.3)
$$\displaystyle\phi(d^{a}yd^{b}y)(z)=$$
$$\displaystyle~{}\partial_{z}^{a}\left[x(z)\partial_{z}^{b}\left[x(z)\right]\right]$$
$$\displaystyle=$$
$$\displaystyle~{}\partial_{z}^{a}\left[\left(\frac{B_{0}}{z}+\sum_{m\geq 0}%
\frac{B_{m+1}}{(m+1)!}z^{m}\right)\left(\frac{(-1)^{b}b!B_{0}}{z^{b+1}}+\sum_{%
n\geq 0}\frac{B_{n+b+1}}{n+b+1}\frac{1}{n!}z^{n}\right)\right]$$
$$\displaystyle=$$
$$\displaystyle~{}\partial_{z}^{a}\left[\text{pole part~{}}+\sum_{n\geq 0}\frac{%
B_{0}B_{n+b+2}}{n+b+2}\frac{z^{n}}{(n+1)!}+\sum_{m\geq 0}\frac{(-1)^{b}b!B_{0}%
B_{m+b+2}}{(m+b+2)!}z^{m}\right.$$
$$\displaystyle~{}\left.+\sum_{l\geq 0}\sum_{\begin{smallmatrix}n+m=l\\
n,m\geq 0\end{smallmatrix}}\frac{B_{m+1}B_{n+b+1}}{n+b+1}\frac{z^{l}}{n!(m+1)!%
}\right]$$
$$\displaystyle=$$
$$\displaystyle~{}\text{pole part~{}}+\frac{B_{0}B_{a+b+2}}{(a+b+2)(a+1)}+(-1)^{%
b}\frac{a!b!B_{0}B_{a+b+2}}{(a+b+2)!}$$
$$\displaystyle~{}+\sum_{\begin{smallmatrix}n+m=a\\
n,m\geq 0\end{smallmatrix}}\frac{B_{m+1}B_{n+b+1}}{(m+1)!(n+b+1)}\frac{a!}{n!}%
+O(z).$$
The second and third summand are zero since $a+b+2\geq 3$ is an odd number and therefore $B_{a+b+2}=0$. We have three possibilities for the last sum to be different from zero:
•
Case 1: $m+1$ and $n+b+1$ are even numbers. Then we have $m+n+b+2=a+b+2$ even, which contradicts that $a+b$ is odd.
•
Case 2: $m=0$. The last summand is equal to
$$\displaystyle\frac{B_{1}B_{a+b+1}}{a+b+1}.$$
•
Case 3: $n+b=0$. Then $n=b=0$ and we have for the last summand
$$\displaystyle\frac{B_{1}B_{a+1}}{a+1}.$$
Therefore we obtain together with Remark 2.3 that
$$\displaystyle\phi(d^{a}yd^{b}y)(z)$$
$$\displaystyle=\text{pole part~{}}+\frac{1}{2}\left(1+\delta_{0}(b)\right)\frac%
{B_{a+b+1}}{a+b+1}+O(z)$$
$$\displaystyle=\text{pole part~{}}+\zeta_{2}(-a,-b)+O(z).$$
Let $s:=a+b$ be an odd number and $c,d\geq 0$ with $c+d=s$. Then we observe using the above calculation that
$$\displaystyle\phi_{-}(p^{c}y)(z)\phi(p^{d}y)(z)$$
$$\displaystyle=\frac{(-1)^{c}c!B_{0}}{z^{c+1}}\left(\frac{(-1)^{d}d!B_{0}}{z^{d%
+1}}+\sum_{n\geq 0}\frac{B_{n+d+1}}{n+d+1}\frac{1}{n!}z^{n}\right)$$
$$\displaystyle=\text{pole part~{}}+(-1)^{c}\frac{B_{0}B_{c+d+2}}{(c+1)(c+d+2)}+%
O(z)$$
Since $c+d+2=a+b+2\geq 3$ is an odd number $B_{c+d+2}=0$ and the constant term is zero.
Hence, we obtain $\phi_{+}(d^{a}yd^{b}y)(z)=\zeta_{2}(-a,-b)+O(z).$ Here we have used that $\tilde{\Delta}_{0}$ respects the weight graduation.
∎
In the light of Theorem 3.23 and Corollary 3.24 we deduce a simple way to calculate the renormalized MPL, which is presented in
Corollary 4.4.
For $w\in Y$, $\operatorname{dpt}(w)>1$
(25)
$$\displaystyle\phi_{+}(w)=\frac{1}{2^{\operatorname{dpt}(w)}-2}\sum_{(w)}\phi_{%
+}(w^{\prime})\phi_{+}(w^{\prime\prime}).$$
Note that both $\operatorname{dpt}(w^{\prime})$ and $\operatorname{dpt}(w^{\prime\prime})$ are strictly smaller than $\operatorname{dpt}(w)$. On the right-hand side of (25) one can continue to apply Theorem 3.23 to the words $w^{\prime},w^{\prime\prime}$, until $w$ has been fully decomposed into primitive elements. The renormalization of $w$ respectively the corresponding MZV reduce to the simple renormalization of single MPLs at non-positive arguments corresponding to primitive words in $\mathcal{H}$. For example
$$\phi_{+}(dyd^{n}y)=\phi_{+}(y)\phi_{+}(d^{n+1}y)+\phi_{+}(dy)\phi_{+}(d^{n}y).$$
Proof of Corollary 4.4.
Statement (25) follows directly from Theorem 3.23, since $\phi_{+}$ is a character by construction. See (21) in Theorem 4.1. Observe that (25) it is compatible with (21), since ${(\operatorname{Id}-\pi)\phi_{+}=\phi_{+}}$.
∎
Example 4.5.
Let us calculate the renormalized MZVs $\zeta_{+}(0,-2)$ and $\zeta_{+}(-1,-1)$.
From Example 3.19 we find
$$\displaystyle\tilde{\Delta}_{0}(yd^{2}y)=y\otimes d^{2}y+d^{2}y\otimes y\hskip
1%
4.226378pt\text{and}\hskip 14.226378pt\tilde{\Delta}_{0}(dydy)=y\otimes d^{2}y%
+d^{2}y\otimes y+2dy\otimes dy.$$
Therefore we obtain form the iterative formulas (20) and (21) of Theorem 4.1
$$\displaystyle\phi_{+}(yd^{2}y)$$
$$\displaystyle=(\operatorname{Id}-\pi)\left[\phi(yd^{2}y)-\left(\phi_{-}(y)\phi%
(d^{2}y)+\phi_{-}(d^{2}y)\phi(y)\right)\right],$$
$$\displaystyle\phi_{+}(dydy)$$
$$\displaystyle=(\operatorname{Id}-\pi)\left[\phi(dydy)-\left(\phi_{-}(y)\phi(d^%
{2}y)+\phi_{-}(d^{2}y)\phi(y)+2\phi_{-}(dy)\phi(dy)\right)\right].$$
Using
$$\displaystyle\phi(y)(z)$$
$$\displaystyle=-{z}^{-1}-{\frac{1}{2}}-{\frac{1}{12}}z+{\frac{1}{720}}{z}^{3}+O%
\!\left({z}^{4}\right),$$
$$\displaystyle\phi(dy)(z)$$
$$\displaystyle={z}^{-2}-{\frac{1}{12}}+{\frac{1}{240}}{z}^{2}+O\!\left({z}^{4}%
\right),$$
$$\displaystyle\phi(d^{2}y)(z)$$
$$\displaystyle=-2\,{z}^{-3}+{\frac{1}{120}}z-{\frac{1}{1512}}{z}^{3}+O\!\left({%
z}^{4}\right)$$
and
$$\displaystyle\phi(yd^{2}y)(z)$$
$$\displaystyle=2\,{z}^{-4}+{z}^{-3}+\frac{1}{6}\,{z}^{-2}-{\frac{1}{90}}-{\frac%
{1}{240}}\,z+{\frac{1}{30240}}\,{z}^{2}+{\frac{1}{3024}}\,{z}^{3}+O\left({z}^{%
4}\right),$$
$$\displaystyle\phi(dydy)(z)$$
$$\displaystyle=3\,{z}^{-4}+{z}^{-3}+{\frac{1}{240}}-{\frac{1}{240}}z-{\frac{1}{%
1008}}{z}^{2}+{\frac{1}{3024}}{z}^{3}+O\!\left({z}^{4}\right),$$
we observe that
$$\displaystyle(\operatorname{Id}-\pi)\left[\phi_{-}(y)\phi(d^{2}y)+\phi_{-}(d^{%
2}y)\phi(y)\right](z)$$
$$\displaystyle=-\frac{1}{90}+O(z)$$
and
$$\displaystyle(\operatorname{Id}-\pi)\left[\phi_{-}(y)\phi(d^{2}y)+\phi_{-}(d^{%
2}y)\phi(y)+2\phi_{-}(dy)\phi(dy)\right](z)$$
$$\displaystyle=-\frac{1}{360}+O(z).$$
Hence,
$$\displaystyle\phi_{+}(yd^{2}y)(z)=O(z)\hskip 28.452756pt\text{and}\hskip 28.45%
2756pt\phi_{+}(dydy)(z)=\frac{1}{240}+\frac{1}{360}+O(z)=\frac{1}{144}+O(z),$$
which results in $\zeta_{+}(0,-2)=0$ and $\zeta_{+}(-1,-1)=\frac{1}{144}$.
Alternatively, we can use the shuffle product to calculate $\zeta_{+}(0,-2)$ and $\zeta_{+}(-1,-1)$. Note that since $y\,\shuffle_{0}\,dy=ydy$ we have $\zeta_{+}(0,-2)=\zeta_{+}(0)\zeta_{+}(-2)=0$.
Because of $dy\,\shuffle_{0}\,dy=dydy-yd^{2}y$ we see that $\zeta_{+}(-1,-1)=\zeta_{+}(0,-2)+\zeta_{+}(-1)^{2}=\frac{1}{144}$.
A third way to calculate, say, $\zeta_{+}(-1,-1)$, is based on Corollary 3.24, and described in Corollary 4.4:
$$\displaystyle\zeta_{+}(-1,-1)=\frac{1}{2}\left(\zeta_{+}(0)\zeta_{+}(-2)+\zeta%
_{+}(-2)\zeta_{+}(0)+2\zeta_{+}(-1)\zeta_{+}(-1)\right)=\frac{1}{144}.$$
4.3. Renormalization of $q$MZVs
The Hopf algebra $(\mathcal{H},\,\shuffle_{-1}\,,\Delta_{-1})$ is related to the modified $q$-analogue $\overline{\mathfrak{z}}_{q}$ whereas the relation between MZVs and $q$MZVs (see Equation (9)) relies on a limit process involving the non-modified $q$MZVs $\mathfrak{z}_{q}$. Therefore the renormalization related to the $q$MZV deformation is more involved than the renormalization described in the previous section. First of all we apply Theorem 4.1 in the framework of modified $q$MZVs. We define the map $\psi\colon(\mathcal{H},\,\shuffle_{-1}\,)\to\mathbb{Q}[z^{-1},z]]$ by
$$\displaystyle d^{k_{1}}y\cdots d^{k_{n}}y\mapsto\psi(d^{k_{1}}y\cdots d^{k_{n}%
}y)(z):=\sum_{m_{1},\ldots,m_{n}\geq 0}\frac{B_{m_{1}}}{m_{1}!}\cdots\frac{B_{%
m_{n}}}{m_{n}!}\cdot C^{k_{1},\ldots,k_{n}}_{m_{1},\ldots,m_{n}}z^{m_{1}+%
\cdots+m_{n}-n}$$
for $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$, where
$$\displaystyle C^{k_{1},\ldots,k_{n}}_{m_{1},\ldots,m_{n}}:=\sum_{l_{i}=0\atop i%
=1,\ldots,n}^{k_{i}}\left(\prod_{i=1}^{n}\binom{k_{i}}{l_{i}}(-1)^{l_{i}+1}(l_%
{1}+\cdots+l_{i}+1)^{m_{i}-1}\right).$$
Lemma 4.6.
The map $\psi\colon(\mathcal{H},\,\shuffle_{-1}\,)\to(\mathbb{Q}[z^{-1},z]],\cdot)$ is a character. Furthermore, the following diagram commutes:
{xy}
(0,20)*+(H, \shuffle_-1 )=”a”; (40,20)*+(Q[[q]],⋅)=”b”;
(40,0)*+(Q[z^-1,z]],⋅)=”d”;
\ar”a”;”b”?*!/_4mm/¯z_q^\shuffle;
\ar”b”;”d”?*!/_8mm/q↦e^z;
\ar”a”;”d”;?*!/_2mm/ψ;
Proof.
Let $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$. First we observe that
$$\displaystyle\overline{\mathfrak{z}}_{q}^{\,\shuffle\,}(d^{k_{1}}y\cdots d^{k_%
{n}}y)$$
$$\displaystyle=$$
$$\displaystyle\sum_{m_{1}>\cdots>m_{n}>0}q^{m_{1}}(1-q^{m_{1}})^{k_{1}}(1-q^{m_%
{2}})^{k_{2}}\cdots(1-q^{m_{n}})^{k_{n}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{m_{1},\ldots,m_{n}>0}q^{m_{1}+\cdots+m_{n}}(1-q^{m_{1}+%
\cdots+m_{n}})^{k_{1}}(1-q^{m_{2}+\cdots+m_{n}})^{k_{2}}\cdots(1-q^{m_{n}})^{k%
_{n}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{l_{1}=0}^{k_{1}}\cdots\sum_{l_{n}=0}^{k_{n}}(-1)^{l_{1}+%
\cdots+l_{n}}\binom{k_{1}}{l_{1}}\cdots\binom{k_{n}}{l_{n}}\sum_{m_{1},\ldots,%
m_{n}>0}q^{m_{1}(l_{1}+1)}q^{m_{2}(l_{1}+l_{2}+1)}\cdots q^{m_{n}(l_{1}+\cdots%
+l_{n}+1)}$$
$$\displaystyle=$$
$$\displaystyle\sum_{l_{1}=0}^{k_{1}}\cdots\sum_{l_{n}=0}^{k_{n}}(-1)^{l_{1}+%
\cdots+l_{n}+n}\binom{k_{1}}{l_{1}}\cdots\binom{k_{n}}{l_{n}}\frac{q^{l_{1}+1}%
}{q^{l_{1}+1}-1}\cdots\frac{q^{l_{1}+\cdots+l_{n}+1}}{q^{l_{1}+\cdots+l_{n}+1}%
-1}.$$
This leads to
$$\displaystyle\overline{\mathfrak{z}}_{q}^{\,\shuffle\,}(d^{k_{1}}y\cdots d^{k_%
{n}}y)\stackrel{{\scriptstyle q\mapsto e^{z}}}{{\longmapsto}}$$
$$\displaystyle\sum_{l_{i}=0\atop i=1,\ldots,n}^{k_{i}}\left(\prod_{j=1}^{n}(-1)%
^{l_{j}+1}\binom{k_{j}}{l_{j}}\frac{e^{z(l_{1}+\cdots+l_{j}+1)}}{e^{z(l_{1}+%
\cdots+l_{j}+1)}-1}\right)$$
$$\displaystyle=$$
$$\displaystyle\sum_{l_{i}=0\atop i=1,\ldots,n}^{k_{i}}\left(\prod_{j=1}^{n}(-1)%
^{l_{j}+1}\binom{k_{j}}{l_{j}}\sum_{m_{j}\geq 0}\frac{B_{m_{j}}}{m_{j}!}(z(l_{%
1}+\cdots+l_{j}+1))^{m_{j}-1}\right)$$
$$\displaystyle=$$
$$\displaystyle\sum_{m_{1},\ldots,m_{n}\geq 0}\frac{B_{m_{1}}}{m_{1}!}\cdots%
\frac{B_{m_{n}}}{m_{n}!}\cdot C^{k_{1},\ldots,k_{n}}_{m_{1},\ldots,m_{n}}z^{m_%
{1}+\cdots+m_{n}-n}.$$
The map $\psi$ is a character since it is a composition of algebra morphisms.
∎
Next we reverse the modification process applied in Equation (10). Therefore we apply Theorem 4.1 to the character $\psi$ and define the renormalized $q$MZVs $\mathfrak{z}_{+}$ by
(26)
$$\displaystyle\mathfrak{z}_{+}(-k_{1},\ldots,-k_{n}):=\lim_{z\to 0}\frac{(-1)^{%
k_{1}+\cdots+k_{n}}}{z^{k_{1}+\cdots+k_{n}}}\psi_{+}(d^{k_{1}}y\cdots d^{k_{n}%
}y)(z)$$
for $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$.
Theorem 4.7.
Let $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$. Then $\mathfrak{z}_{+}(-k_{1},\ldots,-k_{n})$ is well defined, and we have
$$\displaystyle\mathfrak{z}_{+}(-k_{1},\ldots,-k_{n})=\zeta_{+}(-k_{1},\ldots,-k%
_{n}).$$
Especially, the renormalized $q$MZVs $\mathfrak{z}_{+}$ respect the shuffle product $\,\shuffle_{0}\,$.
For the proof of this theorem we need an auxiliary result:
Lemma 4.8.
We have $\psi_{+}(d^{k}y)(z)=(-1)^{k}\zeta_{+}(-k)z^{k}+O(z^{k+1})$ for all $k\in\mathbb{N}_{0}$.
Proof.
Since $d^{k}y$ is primitive with respect to the coproduct $\Delta_{-1}$, we obtain from Lemma 4.6 that
$$\displaystyle\psi_{+}(d^{k}y)(z)=(\operatorname{Id}-\pi)\psi(d^{k}y)(z)=(%
\operatorname{Id}-\pi)\left(\sum_{m\geq 0}\frac{B_{m}}{m!}\cdot C_{m}^{k}z^{m-%
1}\right)=\sum_{m>0}\frac{B_{m}}{m!}\cdot C_{m}^{k}z^{m-1}$$
with $C_{m}^{k}=\sum_{l=0}^{k}\binom{k}{l}(-1)^{l+1}(l+1)^{m-1}$. We have
$$\displaystyle C_{m}^{k}$$
$$\displaystyle=\sum_{l=0}^{k}\binom{k}{l}(-1)^{l+1}(l+1)^{m-1}=\frac{1}{k+1}%
\sum_{l=0}^{k+1}\binom{k+1}{l}(-1)^{l}l^{m}$$
$$\displaystyle=\left.\frac{1}{k+1}\delta_{z}^{m}\left(\sum_{l=0}^{k+1}\binom{k+%
1}{l}(-z)^{l}\right)\right|_{z=1}=\left.\frac{1}{k+1}\delta_{z}^{m}(1-z)^{k+1}%
\right|_{z=1}$$
This shows that $C_{m}^{k}=0$ for $m=1,\ldots,k$. Furthermore, we observe that $C_{k+1}^{k}=(-1)^{k+1}\frac{(k+1)!}{k+1}$, which completes the proof.
∎
Proof of Theorem 4.7.
Let $w:=d^{k_{1}}y\cdots d^{k_{n}}y$ with $k_{1},\ldots,k_{n}\in\mathbb{N}_{0}$. In order to prove that $\mathfrak{z}_{+}$ is well defined we show $\psi_{+}(w)\in O(z^{\operatorname{wt}(w)-n})$. We split up the coproduct $\Delta_{-1}(w)$ into two parts
(27)
$$\displaystyle\Delta_{-1}(w)=\Delta_{0}(w)+\left(\Delta_{-1}(w)-\Delta_{0}(w)%
\right).$$
From Corollary 4.4 we deduce that $\Delta_{-1}(w)$ induces a $\mathbb{Q}$-linear combination of products with $n$ factors of $\psi_{+}$ in primitive elements of $\mathcal{H}$. The part of $\psi_{+}$ corresponding to $\Delta_{0}(w)$ in the coproduct factorization is homogeneous in weight $\operatorname{wt}(w)$ and the one related to $\Delta_{-1}(w)-\Delta_{0}(w)$ has weight greater than $\operatorname{wt}(w)$. Therefore Lemma 4.8 implies that $\psi_{+}(w)\in O(z^{\operatorname{wt}(w)-n})$. Hence, the limit in (26) exists. On the one hand we can apply Corollary 4.4 to $\phi_{+}(w)$, defined in the previous section, which corresponds to the factorization induced by $\Delta_{0}(w)$. On the other hand we can do the same with $\psi_{+}(w)$. However, this factorization is related to $\Delta_{-1}(w)$. After dividing by $z^{\operatorname{wt}(w)-n}$ and taking the limit $z\to 0$ only the first part in the decomposition (27) of $\Delta_{-1}(w)$ makes a contribution in the factorization of $\psi_{+}(w)$. Using the fact that the leading factor of $\psi_{+}(d^{k}y)(z)$ equals $(-1)^{k}\zeta_{+}(-k)z^{k}$ concludes the proof.
∎
Example 4.9.
Let us calculate the renormalized $q$MZV $\mathfrak{z}_{+}(-1,-1)$. From Example 3.19 we obtain
$$\displaystyle\tilde{\Delta}_{0}(dydy)=y\otimes d^{2}y+d^{2}y\otimes y+2dy%
\otimes dy-dy\otimes d^{2}y-d^{2}y\otimes dy,$$
which gives
$$\displaystyle\psi_{+}(dydy)=(\operatorname{Id}-\pi)$$
$$\displaystyle\left[\psi(dydy)-\left(\psi_{-}(y)\psi(d^{2}y)+\psi_{-}(d^{2}y)%
\psi(y)+2\psi_{-}(dy)\psi(dy)\right.\right.$$
$$\displaystyle\left.\left.-\psi_{-}(dy)\psi(d^{2}y)-\psi_{-}(d^{2}y)\psi(dy)%
\right)\right]$$
Using
$$\displaystyle\psi(y)(z)$$
$$\displaystyle=-{z}^{-1}-\frac{1}{2}-\frac{1}{12}\,z+{\frac{1}{720}}\,{z}^{3}-{%
\frac{1}{30240}}\,{z}^{5}+O\!\left({z}^{7}\right),$$
$$\displaystyle\psi(dy)(z)$$
$$\displaystyle=-\frac{1}{2}\,{z}^{-1}+\frac{1}{12}\,z-{\frac{7}{720}}\,{z}^{3}+%
{\frac{31}{30240}}\,{z}^{5}+O\!\left({z}^{7}\right),$$
$$\displaystyle\psi(d^{2}y)(z)$$
$$\displaystyle=-\frac{1}{3}\,{z}^{-1}+{\frac{1}{60}}\,{z}^{3}-{\frac{1}{168}}\,%
{z}^{5}+O\!\left({z}^{7}\right)$$
and
$$\displaystyle\psi(dydy)(z)={\frac{5}{12}}\,{z}^{-2}+\frac{1}{6}\,{z}^{-1}-%
\frac{1}{36}+{\frac{1}{216}}\,{z}^{2}-{\frac{1}{120}}\,{z}^{3}-{\frac{19}{9072%
}}\,{z}^{4}+O\!\left({z}^{5}\right),$$
we observe that
$$\displaystyle(\operatorname{Id}-\pi)\left[\psi_{-}(y)\psi(d^{2}y)+\psi_{-}(d^{%
2}y)\psi(y)+2\psi_{-}(dy)\psi(dy)\right](z)=-\frac{1}{18}-\frac{1}{135}z^{2}+O%
(z^{3})$$
and
$$\displaystyle(\operatorname{Id}-\pi)\left[-\psi_{-}(dy)\psi(d^{2}y)-\psi_{-}(d%
^{2}y)\psi(dy)\right](z)=\frac{1}{36}+\frac{11}{2160}z^{2}+O(z^{3}).$$
Therefore we have $\psi_{+}(dydy)(z)=\frac{1}{144}z^{2}+O(z^{3})$ and consequently $\mathfrak{z}_{+}(-1,-1)=\frac{1}{144}$, which coincides with $\zeta_{+}(-1,-1)$.
Remark 4.10.
The crucial point in the renormalization of $q$MZVs is the fact that $\psi_{+}(w)(z)\in O(z^{\operatorname{wt}(w)-n})$. We used a corollary of Theorem 3.23 to prove this. However, in the light of Theorem 4.1 the previous example shows that this is obtained by non-trivial cancellations.
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Characterizing divergence and thickness in right-angled Coxeter groups
Ivan Levcovitz
(Date:: )
Abstract.
We completely classify the possible divergence functions for right-angled Coxeter groups (RACGs). In particular, we show that the divergence of any such group is either polynomial, exponential or infinite.
We prove that a RACG is strongly thick of order $k$ if and only if its divergence function is a polynomial of degree $k+1$. Moreover, we show that the exact divergence function of a RACG can easily be computed from its defining graph by an invariant we call the hypergraph index.
The author was supported in part by a Technion fellowship.
1. Introduction
Given a finite simplicial graph $\Gamma$ with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$,
the corresponding right-angled Coxeter group (RACG for short) is given by the presentation:
$$\langle s\in V(\Gamma)~{}|~{}s^{2}=1~{}\text{ for all }s\in V(\Gamma)\text{ %
and }st=ts\text{ for all }(s,t)\in E(\Gamma)\rangle$$
In this article, we provide an explicit method to compute the divergence and order of (strong) thickness of a RACG, and we prove that these quasi-isometry invariants are in fact equivalent in this setting.
There are very few other explicitly computable quasi-isometry invariants available for non-relatively hyperbolic RACGs; consequently, we can distinguish many more RACGs up to quasi-isometry than was previously possible.
Additionally, this result establishes an exact connection between
the largest rate that a pair of geodesic rays can diverge in (the Cayley graph of) a RACG and the coarse complexity of an optimal decomposition of this group into subsets which do not exhibit non-positive curvature (i.e. whose asymptotic cones do not contain cutpoints).
Given a bi-infinite geodesic $\gamma:\mathbb{R}\to X$ in a metric space $X$, its geodesic divergence is the function $\text{Div}^{\gamma}(r)$ whose value is the infimum over the length of $\alpha$, where $\alpha$ is a path from $\gamma(-r)$ to $\gamma(r)$ which does not intersect the ball based at $\gamma(0)$ of radius $r$. If no such path exists, then we say that $\text{Div}^{\gamma}(r)$ is infinite.
There is a corresponding notion of the divergence function of a metric space which, for $r>0$, roughly takes value the supremum over the lengths of all minimal length paths, which avoid a ball of radius proportional to $r$ and which connect two points that are distance proportional to $r$ apart.
The divergence function of a finitely generated group is defined to be the divergence function of one of its Cayley graphs, and
it is a quasi-isometry invariant of finitely generated groups up to a usual equivalence of functions used in geometric group theory.
The geodesic divergence of a bi-infinite geodesic in the Cayley graph of a group gives a lower bound on the group’s divergence.
The second quasi-isometry invariant treated in this article is strong thickness (which we often simply refer to as “thickness”).
A group is thick of order $0$ if and only if all of its asymptotic cones do not contain cutpoints.
Roughly, a group is thick of order $k$, if it is not thick of order $k-1$, and its Cayley graph coarsely decomposes into thick pieces of order strictly less than $k$. Moreover, given any two such pieces $P$ and $P^{\prime}$ in this decomposition, there exists a sequence of pieces $P=P_{1},\dots,P_{m}=P^{\prime}$ in the decomposition such that $P_{i}$ has infinite-diameter coarse intersection with $P_{i+1}$ for $1\leq i<m$.
Finally, we will need a third notion: the hypergraph index.
Given a simplicial graph, one can explicitly compute its hypergraph index, which takes value either a non-negative integer or $\infty$.
Our main theorem, stated below, characterizes divergence and thickness in RACGs in terms of the hypergraph index of its defining graph. This gives an easy and explicit method of computing thickness and divergence of any RACG.
Theorem A.
Let $W_{\Gamma}$ be a RACG and $k\geq 0$ an integer. Then the following are equivalent:
(1)
The hypergraph index of $\Gamma$ is $k$.
(2)
The divergence of $W_{\Gamma}$ is $r^{k+1}$, and the Cayley graph of $W_{\Gamma}$ contains a periodic geodesic with geodesic divergence $r^{k+1}$.
(3)
The group $W_{\Gamma}$ is strongly thick of order $k$.
As we later discuss, there were already some known bounds between divergence, thickness and the hypergraph index [BD14, Lev19].
Moreover, the above theorem was known in the special cases where $k=0$ and $k=1$ [BHS17, DT15, Lev18] and was conjectured in [Lev19].
However, the proof of those two known cases resisted a generalization, and new, significantly more refined methods had to be developed in order to prove our result in its full generality.
Additionally, given an arbitrary RACG which is not thick of order $0$ or $1$, its exact divergence function and order thickness was not previously known except in very specialized cases such as those treated in [DT15] and [Lev18] which involve RACGs whose associated CAT(0) cube complexes contain hyperplanes with very well-behaved separation properties.
As a corollary, we obtain a complete classification of divergence functions in RACGs:
Corollary B.
The divergence of a RACG is either polynomial, exponential or infinite.
These large gaps exhibited in the divergence function spectrum of RACGs do not exist in arbitrary finitely generated groups: there are groups with “intermediate” divergence functions.
For instance, Olshanskii–Osin–Sapir show there are lacunary hyperbolic groups exhibiting divergence functions which are strictly between linear and quadratic [OOS09]. Additional groups with exotic divergence functions were found by Gruber–Sisto [GS18]. More recently, Brady–Tran amazingly construct finitely-presented groups with divergence functions $r^{\alpha}$ for a dense set of $\alpha\in[2,\infty]$ [BT].
Theorem A can also be utilized in the study of random RACGs, i.e., RACGs defined by a random graph in the Erdős–Rényi model.
Behrstock–Hagen–Sisto compute an explicit threshold function for when a random RACG is thick or relatively hyperbolic [BHS17].
Building on this work, Behrstock–Falgas-Ravry–Hagen–Susse give threshold functions for the transition between thick of order $0$, $1$ and $2$ in RACGs [BFRHS18].
These authors demonstrate an interesting interplay between random graphs and the coarse geometry of RACGs. Theorem A should make it possible for this analysis to extend to RACGs with higher orders of thickness.
We now discuss some background on divergence, thickness and the hypergraph index.
Gromov expected that one-ended groups which act geometrically on a CAT(0) space should exhibit either linear or exponential divergence [Gro93]. However, it turns out that many important classes of groups (including CAT(0) ones) do not fall into this dichotomy. For instance, Gersten showed that the divergence of a $3$–manifold group is linear, quadratic or exponential [Ger94]. Additionally, Behrstock–Charney prove that the divergence of a right-angled Artin group is, similarly, linear, quadratic or infinite (when not one-ended) [BC12]. Interestingly, most mapping class groups also exhibit quadratic divergence [Beh06, DR09].
There are also CAT(0) groups with divergence function a polynomial of any degree, and such groups were first constructed by Macura [Mac13] and independently also by Behrstock-Druţu [BD14].
For each non-negative integer, Dani–Thomas give an example of a RACG whose divergence is a polynomial of this degree [DT15].
Furthermore, these authors give a graph-theoretic characterization of $2$–dimensional RACGs (i.e., whose defining graph does not contain $3$–cycles) with linear and quadratic divergence. These characterizations were later generalized to RACGs of arbitrary dimension, with the linear case being done by Behrstock–Hagen–Sisto [BHS17] and the quadratic case done by the author [Lev18].
Thick spaces were first defined by Behrstock–Druţu–Mosher in [BDM09] where it is shown that the order of thickness is a quasi-isometry invariant and that thick groups are non-relatively hyperbolic. Behrstock–Druţu later define a slightly stronger, more quantified, version of thickness, known as strong thickness which appears to be becoming the standard definition.
Strong thickness is still a quasi-isometry invariant, and all groups known to be thick, are strongly thick of the same order.
Furthermore, these authors show that a group which is strongly thick of order $k$ has divergence function bound above by a polynomial of degree $k+1$ [BD14].
Many well-studied non-relatively hyperbolic groups are strongly thick, and there is often a dichotomy where a group, in a given class of groups, is either strongly thick or hyperbolic relative to strongly thick peripheral subgroups (see for instance [BHS17] and [Hag19]).
The hypergraph index was introduced by the author in [Lev19].
The hypergraph index of a RACG is defined to be the hypergraph index of its defining graph, and it was previously known to give some measure of the RACG’s coarse complexity.
For instance, the hypergraph index of a RACG is $\infty$ if and only if the RACG is relatively hyperbolic.
Moreover, if a given RACG is quasi-isometric to a right-angled Artin group then its hypergraph index is either $0$, $1$ or $\infty$.
The hypergraph index was previously only known to be quasi-isometry invariant within the class of $2$–dimensional RACGs, and the proof of this used the structure of quasi-flats.
Finally, the hypergraph index is known to give an upper bound on thickness: a RACG of hypergraph index $k\neq\infty$ is thick of order at most $k$.
Given these known bounds between divergence, thickness and the hypergraph index,
in order to prove Theorem A, a lower bound on the divergence function of a RACG in terms of the hypergraph index must be established.
This is the content of the following theorem:
Theorem C.
Let $\Gamma$ be a simplicial graph with hypergraph index $k\neq\infty$. Then the Cayley graph of the RACG $W_{\Gamma}$ contains a periodic geodesic with geodesic divergence a polynomial of degree $k+1$.
The proof of Theorem C involves a careful analysis of disk diagrams.
We first define $L$–fences in a disk diagram over a RACG in Section 3.
These inductively defined objects consist of a set of dual curves whose intersection pattern naturally corresponds to a subgraph of hypergraph index $L$ in the RACG’s defining graph.
Additionally, dual curves which “cross” an $L$–fence are forced to intersect it in a way that is compatible with the associated hypergraph index structure.
In Section 4, we define sequences of “structured” dual curves in a disk diagram. These sequences have desirable properties, and we show how to find large enough such sequences.
In Section 5 we define disk diagram surgeries which allow us to insert a disk diagram, which contains a well-behaved path, into another disk diagram.
After establishing these necessary concepts and proving some essential properties about them,
in Section 6 we simultaneously prove two technical propositions which are at the heart of the proof of Theorem C.
These propositions give lower bounds on the lengths of certain paths, and their hypotheses are designed to be weak enough to allow for the inductive argument to work.
The arguments in this section involve a careful analysis of $L$–fences in disk diagrams and the paths which they connect.
We are also required to perform a series of disk diagram surgeries.
A technical challenge to surgeries is that pathologies such as bigons and nongons are introduced into the resulting diagram, and they cannot be easily removed without possibly destroying $L$–fence structures already found.
We often then need to “let bigons be bigons” and to work around these pathologies.
Finally, in Section 7, we utilize the technical work from the previous section to prove the results from the introduction.
Acknowledgements
I am thankful to Jason Behrstock for helpful comments.
2. Preliminaries
We establish some of the definitions and notation used throughout the article and refer the reader to references for a more extensive background.
Let $X$ be a metric space. Given a point $x\in X$ and a constant $R\geq 0$, we always denote the $R$–ball about $x$ by $B_{x}(R)$. Furthermore, given a subspace $Y\subset X$, we denote the $R$–neighborhood of $Y$ by $N_{R}(Y)$.
2.1. Divergence
We review the definitions of divergence of a metric space and of a geodesic.
We refer the reader to [DMS10] for further background and proofs that various notions of divergence are equivalent under mild hypotheses (such as the metric space being the Cayley graph of a finitely generated group).
Let $(X,d)$ be a metric space, and let $0<\delta\leq 1$ and $\lambda\geq 0$ be constants.
Given points $x,y,b\in X$ such that $\min\{d(b,x),d(b,y)\}=r>0$, we define $\text{div}_{\delta,\lambda}(x,y,b)$ to be the infimum over the lengths of paths from $x$ to $y$ which do not intersect the ball $B_{b}(\delta r-\lambda)$. If there is no such path, we set $\text{div}_{\delta,\lambda}(x,y,b)=\infty$.
The divergence of $X$ is the function $\text{Div}^{X}_{\delta,\lambda}(r)$ which, for each $r\geq 0$, takes value the supremum of $\text{div}_{\delta,\lambda}(x,y,b)$ over all $x,y,b\in X$ such that $d(x,y)\leq r$.
Given a pair of non-decreasing functions $f,g:\mathbb{R}_{+}\to\mathbb{R}_{+}$, we write $f\preceq g$ if for some constant $C\geq 1$ we have that
$$f(r)\leq Cg(Cr+C)+Cr+C$$
for all $r\in\mathbb{R}_{+}$. We write $f\asymp g$ if $f\preceq g$ and $g\preceq f$.
Up to the equivalence relation $\asymp$ and for $\delta\leq\frac{1}{2}$ and $\lambda\geq 2$, the divergence function $\text{Div}^{X}_{\delta,\lambda}(r)$ is a quasi-isometry invariant [DMS10][Corollary 3.12] when $X$ is restricted to metric spaces which are the Cayley graph of a finitely generated group.
In light of this, we can define the divergence of a finitely generated group to be the divergence (with $\delta\leq\frac{1}{2}$ and $\lambda\geq 2$) of a Cayley graph of the group with respect to a finite generating set, up to the equivalence relation $\asymp$.
We remark that the divergence of a group is equivalent to $\infty$ if and only if the group is not one-ended.
We now describe the notion of the divergence of a geodesic.
Let $\gamma:\mathbb{R}\to X$ be a bi-infinite geodesic with basepoint $b$ in the metric space $X$. The geodesic divergence of $\gamma$ is the the function $\text{Div}^{\gamma}_{\delta,\lambda}(r)=\text{div}_{\delta,\lambda}(\gamma(r),%
\gamma(-r),b)$. It is immediate from the definitions that given a bi-infinite geodesic $\gamma$ in a metric space $X$, we have that $\text{Div}^{\gamma}_{\delta,\lambda}(r)\leq\text{Div}^{X}_{\delta,\lambda}(r)$. Thus, geodesic divergence gives a lower bound on the divergence of a space. Furthermore, it is not difficult to show, that $\text{Div}^{\gamma}_{\delta,\lambda}(r)\asymp\text{Div}^{\gamma}_{1,0}(r)$. Thus, when computing geodesic divergence, we can always assume that $\delta=1$ and $\lambda=0$.
2.2. Strongly thick metric spaces
In this article, we will not directly use the definition of strongly thick spaces, as we are able to apply known results giving relationships between thickness, divergence and the hypergraph index (see Theorem 2.1 and Theorem 2.2). For completeness, we still include the definition here.
We refer the reader to [BD14] for a more detailed background.
Let $C,L>0$ be constants. A subset $Y$ of a metric space $X$ is $(C,L)$–quasi-convex if given any $y,y^{\prime}\in Y$ there exists an $(L,L)$–quasi-geodesic contained in $N_{C}(Y)$ from $y$ to $y^{\prime}$.
A metric space is strongly $(C,L)$–thick of order $0$ if the following two conditions hold: (1) No asymptotic cone of $X$ contains a cutpoint (equivalently, $\text{Div}_{\delta,\lambda}^{X}(r)$ is a linear function for every $0<\delta<\frac{1}{54}$ and $\lambda\geq 0$ [DMS10]);
(2) For each $x\in X$, there exists a bi-infinite $(L,L)$–quasi-geodesic in $X$ which intersects the ball $B_{x}(C)$.
A metric space that is strongly $(C,L)$–thick of order $0$ for some $C$ and $L$, is also called wide.
For each integer $k\geq 1$, a metric space $X$ is strongly $(C,L)$–thick of order at most $k$ if there is a collection $\mathcal{Y}$ of $(C,L)$–quasi-convex subsets of $X$
which are each strongly $(C,L)$–thick of order at most $k-1$ with respect to each of their induced metrics.
Moreover, we have that $X=\bigcup_{Y\in\mathcal{Y}}N_{C}(Y)$, and, additionally, for every $Y,Y^{\prime}\in\mathcal{Y}$ and every $x\in X$ such that $B_{x}(3C)\cap Y\neq\emptyset$ and $B_{x}(3C)\cap Y^{\prime}\neq\emptyset$, it follows that there exists a sequence $Y=Y_{1},\dots,Y_{n}=Y^{\prime}$ of subspaces in $\mathcal{Y}$, with $n\leq L$,
such that for all $1\leq i<n$, $N_{C}(Y_{i})\cap N_{C}(Y_{i+1})$ has infinite diameter, $N_{C}(Y_{i})\cap N_{C}(Y_{i+1})\cap B_{x}(L)\neq\emptyset$ and $N_{L}(N_{C}(Y_{i})\cap N_{C}(Y_{i+1}))$ is path connected.
We say that a metric space is strongly thick of order $k$ if it is strongly $(C,L)$–thick of order at most $k$ for some $C,L>0$ and is not strongly $(C^{\prime},L^{\prime})$–thick of order $k-1$ for any choices of $C^{\prime},L^{\prime}>0$ and any choice of subspaces. The order of strong thickness is a quasi-isometry invariant (see [BD14] and [BDM09]). Behrstock-Druţu also show that the order of strong thickness gives an upper bound on divergence:
Theorem 2.1 (Corollary 4.17 of [BD14]).
Let $X$ be a metric space which is strongly thick of order at most $k$, then $\text{Div}^{X}_{\delta,\lambda}(r)\preceq r^{k+1}$ for all $0<\delta<\frac{1}{54}$ and all $\lambda\geq 0$.
2.3. Right-angled Coxeter groups
We refer the reader to [Dav08] for the general theory of Coxeter groups and to [Dan18] for a survey on RACG results.
Given a RACG $W_{\Gamma}$ and $w=s_{1}\dots s_{n}$, with each $s_{i}\in V(\Gamma)$, we say that $w$ is a word in $W_{\Gamma}$.
We say that the word $w^{\prime}$ is an expression for the word $w$ if $w$ and $w^{\prime}$ are equal as group elements of $W_{\Gamma}$.
Given a word $w=s_{1}\dots s_{n}$, its length $|w|$ is $n$.
We say that a word $w$ is reduced if $|w|$ is minimal out of all possible expressions for $w$.
A RACG $W_{\Gamma}$ acts geometrically on a CAT(0) cube complex $\Sigma_{\Gamma}$ known as the Davis complex.
The $1$-skeleton of $\Sigma_{\Gamma}$ is the Cayley graph of $W_{\Gamma}$ (with the standard generating set) where bigons are collapsed to single edges.
The edges of $\Sigma_{\Gamma}$ are labeled by the generators $V(\Gamma)$.
Moreover, for $n\geq 2$, there is an $n$–cube in $\Sigma_{\Gamma}$ spanning any set of $2^{n}$ edges which is (label-preserving) isomorphic to the Cayley graph of $W_{K}$ where $K$ is a subclique of $\Gamma$.
We refer the reader to [Dav08] and [Wis12] for further background on the Davis complex and CAT(0) cube complexes respectively. We only directly utilize CAT(0) cube complexes in the proof of Theorem C given in the final section.
2.4. Disk Diagrams
A disk diagram over a RACG $W_{\Gamma}$ is square complex $D$, with a fixed planar embedding, whose edges are labeled by vertices of $\Gamma$.
Moreover, given a square in $D$, the label of its edges, read in cyclic order, is $stst$ where $s$ and $t$ are a pair of adjacent vertices of $\Gamma$.
All disk diagrams in this article are over RACGs. We refer the reader [Sag95] and [Wis12] for the general theory of disk diagrams over CAT(0) cube complexes.
A square $[0,1]\times[0,1]$ in the disk diagram $D$ contains two midcubes: $\{\frac{1}{2}\}\times[0,1]$ and $[0,1]\times\{\frac{1}{2}\}$.
A dual curve $H$ in $D$ is a minimal, non-empty, connected collection of midcubes in $D$ such that given any pair of midcubes $m$ and $m^{\prime}$ in $D$, whose intersection is contained in an edge of $D$, it follows that $m\in H$ if and only if $m^{\prime}\in H$.
We say that an edge of $D$ is dual to $H$ if $H$ intersects this edge.
The carrier $N(H)$ of a dual curve $H$ is the set of all cells in $D$ which the dual curve intersects.
As opposite sides of squares in $D$ have the same label, every edge dual to a given dual curve has this same label which we call the type of the dual curve.
It readily follows by how squares are labeled in $D$ that the types of a pair of intersecting dual curves consist of a pair of distinct adjacent vertices in $V(\Gamma)$.
We can also deduce that no dual curve contains both mid-cubes of a given square.
We will frequently use these facts throughout.
A path in $D$ is a sequence $e_{1},\dots,e_{n}$ of pairwise incident edges in the $1$–skeleton of $D$, and its label is $s_{1}\dots s_{n}$ where $s_{i}$ is the label of $e_{i}$.
In particular, paths have a natural orientation given by the ordering on its edges.
We say that a path is reduced if its label is a reduced word in the corresponding RACG.
A dual curve is dual to a path, if it is dual to an edge contained in the path.
A dual curve is dual to at most one edge of a reduced path.
When we say that $\gamma=\gamma_{1}\gamma_{2}$ is a path, it is understood that $\gamma_{1}$ and $\gamma_{2}$ are paths and $\gamma$ is the concatenation of $\gamma_{1}$ and $\gamma_{2}$.
We say that the disk diagram $D$ has boundary path $\gamma$ if $\gamma$ is a path in $D$ containing every edge on the boundary of $D$ (with respect to the given planar embedding) and is minimal length out of such possible paths.
The basepoint of a disk diagram with boundary path $\gamma$ is defined to be the starting vertex of $\gamma$.
Given any word $w$ in the RACG $W_{\Gamma}$ which represents the identity element of $W_{\Gamma}$, it follows from van Kampen’s lemma that there is a disk diagram with boundary path labeled by $w$.
Finally, given a closed path $\eta$ in a disk diagram $D$, the subdiagram $D^{\prime}\subset D$ with boundary path $\eta$, is the largest subcomplex of $D$ that is contained in the closure of the bounded component of $\mathbb{R}^{2}\setminus\eta$ (recall that, by its planar embedding, $D$ is a subset of $\mathbb{R}^{2}$).
2.5. Hypergraph index
We denote the vertex set and edge set of a graph $\Gamma$ respectively by $V(\Gamma)$ and $E(\Gamma)$.
Let $T\subset V(\Gamma)$ be a subset of vertices of the graph $\Gamma$. The subgraph of $\Gamma$ induced by $T$ is the subgraph whose vertex set is $T$ and whose edges consist of all edges in $\Gamma$ connecting a pair of vertices in $T$. We say that the graph $\Delta$ is a join if $\Delta$ contains two subgraphs $\Delta_{1}$ and $\Delta_{2}$ such that $V(\Delta)=V(\Delta_{1})\cup V(\Delta_{2})$ and every vertex of $\Delta_{1}$ is adjacent to every vertex of $\Delta_{2}$. We denote such a join graph by $\Delta=\Delta_{1}\star\Delta_{2}$.
Let $\Delta$ be an induced subgraph of $\Gamma$ which decomposes as the join $\Delta=\Delta_{1}\star\Delta_{2}$. We say that $\Delta$ is a wide subgraph if, for each $i\in\{1,2\}$, $\Delta_{i}$ contains two non-adjacent vertices. Furthermore, we say that $\Delta=\Delta_{1}\star\Delta_{2}$ is a strip subgraph if $\Delta_{1}$ consists of exactly two non-adjacent vertices and $\Delta_{2}$ is a clique.
We note that the RACG $W_{\Delta}$ is a wide group if $\Delta$ is wide, and $W_{\Delta}$ is isomorphic to $D_{\infty}\times\mathbb{Z}_{2}^{k}$ (which is quasi-isometric to $\mathbb{Z}$) if $\Delta$ is a strip subgraph.
Recall that a hypergraph $\Lambda$ is a set of vertices $V(\Lambda)$ and a set of hyperedges $\mathcal{E}(\Lambda)$, where a hyperedge is a non-empty subset of $V(\Lambda)$. In particular, a graph is just a hypergraph whose hyperedges each contain exactly two vertices.
Fix now a simplicial graph $\Gamma$.
Let $\Omega$ be the set of all maximal wide subgraphs of $\Gamma$, and let $\Psi$ be the set of all maximal strip subgraphs of $\Gamma$.
Let $\Lambda_{0}=\Lambda_{0}(\Gamma)$ be the hypergraph with vertex set $V(\Gamma)$ and hyperedge set $\{V(\Delta)~{}|~{}\Delta\in\Omega\cup\Psi\}$.
We now define hypergraphs $\Lambda_{n}=\Lambda_{n}(\Gamma)$ inductively for integers $n>0$.
Suppose that the hypergraph $\Lambda_{i}$ is defined for some $i$.
First, we define an equivalence class $\equiv_{i}$ on the hyperedges of $\Lambda_{i}$: given hyperedges $E,E^{\prime}\in\mathcal{E}(\Lambda_{i})$, $E\equiv_{i}E^{\prime}$ if there exist a sequence $E=E_{1},\dots,E_{n}=E^{\prime}$ of hyperedges in $\mathcal{E}(\Lambda_{i})$ such that for each $1\leq i<n$, $E_{i}\cap E_{i+1}$ contains a pair of distinct vertices which are not adjacent in $\Gamma$.
We now define $\Lambda_{i+1}$. The vertex set of $\Lambda_{i+1}$ is equal to $V(\Gamma)$.
Furthermore, $E$ is a hyperedge of $\Lambda_{i+1}$ if and only if $E=E_{1}\cup\dots\cup E_{m}$ where $\{E_{1},\dots,E_{m}\}$ is a maximal collection of $\equiv_{i}$–equivalent hyperedges of $\Lambda_{i}$.
For each $0\leq i<\infty$, we say that $\Lambda_{i}$ is the $i$’th hypergraph associated to $\Gamma$.
We now define the hypegraph index of $\Gamma$.
Suppose first that $\Omega\neq\emptyset$. Then the hypergraph index of $\Gamma$ is defined to be the smallest integer $k\geq 0$ such that the $k$’th hypergraph, $\Lambda_{k}$, associated to $\Gamma$ contains a hyperedge $E$ such that $E=V(\Gamma)$.
If no such $k$ exists, we set the hypergraph index of $\Gamma$ to be $\infty$.
Additionally, if $\Omega=\emptyset$ we also set the hypergraph index of $\Gamma$ to be $\infty$.
We refer the reader to [Lev19][Figure 1] for an explicit example of the computation of the hypergraph index.
The hypergraph index of $\Gamma$ is $\infty$ if and only if the RACG $W_{\Gamma}$ is relatively hyperbolic. The hypergraph index also gives an upper bound on the order of strong thickness:
Theorem 2.2 (Theorem B from [Lev19]).
Let $W_{\Gamma}$ be a RACG with hypergraph index $k\neq\infty$, then $W_{\Gamma}$ is strongly thick of order at most $k$.
3. Fences in disk diagrams
In this section we introduce the notion of $L$–fences in disk diagrams.
We show how these objects relate to the hypergraph index in Proposition 3.7, and in Proposition 3.10 we show that, in some sense, $L$–fences separate a disk diagram.
We also define $L$–splitting points, which roughly measure the largest “height” of an $L$–fence connecting two given paths.
3.1. $L$–fences
Before defining $L$–fences, we first define spokes below which will serve as the building blocks of an $L$–fence.
Definition 3.1 (Spoke).
A spoke $\mathcal{S}$ in a disk diagram $D$ is a pair $\mathcal{S}=\{H,K\}$ where $H$ and $K$
are dual curves in $D$ whose types are distinct non-adjacent vertices of $\Gamma$. The type of the spoke
$\{H,K\}$ is the pair $\{s,t\}$ where $s$ and $t$ are the types of $H$ and
$K$ respectively.
Remark 3.2.
We remark that given a spoke $\{H,K\}$ in the disk diagram $D$, it follows that $H$ and $K$ do not intersect in $D$ (as their types are non-adjacent vertices of $\Gamma$).
We say that a dual curve $Q$ (resp. a path $\gamma$) intersects a spoke $\mathcal{S}=\{H,K\}$, if both $H\cap Q\neq\emptyset$ and $K\cap Q\neq\emptyset$ (resp. both $H\cap\gamma\neq\emptyset$ and $K\cap\gamma\neq\emptyset$). A spoke $\mathcal{S}=\{H,K\}$ intersects the spoke $\mathcal{S}^{\prime}=\{H^{\prime},K^{\prime}\}$ if both $H$ intersects $\mathcal{S}^{\prime}$ and $K$ intersects $\mathcal{S}^{\prime}$.
An $L$–fence, inductively defined below, is a collection of spokes in a disk diagram satisfying a certain intersection pattern.
Definition 3.3 ($L$-fence).
Let $D$ be a disk diagram.
A $0$–fence in $D$ is a spoke in $D$. Moreover, the spoke of a $0$–fence is just the $0$–fence itself.
For integers $L\geq 1$, an $L$-fence $\mathcal{F}$ in $D$ is a sequence $(\mathcal{F}_{1},\mathcal{F}_{2},\dots,\mathcal{F}_{n})$
of $(L-1)$–fences in $D$ satisfying the following:
for each $1\leq i<n$,
either there exists a spoke $\{H,K\}$ of $\mathcal{F}_{i}$
such that $H$ intersects a spoke of $\mathcal{F}_{i+1}$ and $K$ intersects a
(possibly different) spoke of $\mathcal{F}_{i+1}$, or, alternatively,
there exists a spoke $\{H^{\prime},K^{\prime}\}$ of $F_{i+1}$
such that $H^{\prime}$ intersects a spoke of $F_{i}$ and $K^{\prime}$ intersects a (possibly different)
spoke of $F_{i}$.
A spoke of $\mathcal{F}$ (equivalently, a spoke contained in $\mathcal{F}$) is defined to be a spoke of $\mathcal{F}_{i}$ for some
$1\leq i\leq n$.
Remark 3.4.
By definition, an $L$–fence is also an $L^{\prime}$–fence for all $L^{\prime}\geq L$.
We say that a dual curve intersects an $L$–fence, if the dual curve intersects a spoke of the $L$–fence.
A dual curve is contained in the $L$–fence $\mathcal{F}$ if it is
contained in some spoke of $\mathcal{F}$.
One needs to take care when considering which spokes are contained an $L$–fence:
Remark 3.5.
Let $\mathcal{F}$ be an $L$-fence in a disk diagram $D$ and $\mathcal{S}=\{H,K\}$ a spoke in $D$ such that
$H$ and $K$ are dual curves contained in $\mathcal{F}$. Then $\mathcal{S}$ may or may not be a spoke of $\mathcal{F}$.
The next lemma gives a way of combining two $L$–fences into a larger $L$–fence.
Lemma 3.6.
Let $L\geq 0$ be an integer, and let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be $L$–fences in a disk diagram. Let $\mathcal{R}$ and $\mathcal{R}^{\prime}$ be the sets of all spokes of $\mathcal{F}$ and $\mathcal{F}^{\prime}$ respectively. If $\mathcal{R}\cap\mathcal{R}^{\prime}\neq\emptyset$, then there exists an $L$–fence $\mathcal{F}^{\prime\prime}$ such that the set of all spokes of $\mathcal{F}^{\prime\prime}$ is $\mathcal{R}\cup\mathcal{R}^{\prime}$.
Proof.
We prove the claim by induction on $L$. If $L=0$, then $\mathcal{R}=\mathcal{R}^{\prime}=\{\mathcal{S}\}$ for some spoke $\mathcal{S}$. The claim then follows by taking $\mathcal{F}^{\prime\prime}=\mathcal{S}$.
Suppose now that $L>0$ and the claim is true for $L-1$.
Let $\mathcal{F}=(\mathcal{F}_{1},\dots,\mathcal{F}_{n})$ and $\mathcal{F}^{\prime}=(\mathcal{F}_{1}^{\prime},\dots,\mathcal{F}_{n^{\prime}}^%
{\prime})$ be as in the definition of an $L$–fence, where $\mathcal{F}_{1},\dots,\mathcal{F}_{n},\mathcal{F}_{1}^{\prime},\dots,\mathcal{%
F}_{n^{\prime}}^{\prime}$ are each $(L-1)$–fences.
Let $\mathcal{S}\in\mathcal{R}\cap\mathcal{R}^{\prime}$, and let $1\leq j\leq n$ and $1\leq j^{\prime}\leq n^{\prime}$ be such that $\mathcal{S}$ is contained in $\mathcal{F}_{j}$ and in $\mathcal{F}_{j^{\prime}}^{\prime}$.
By the induction hypothesis, there exists an $(L-1)$–fence $\mathcal{T}$ such that the spokes contained in $\mathcal{T}$ consist of all the spokes contained in $\mathcal{F}_{j}$ and all the spokes contained in $\mathcal{F}_{j^{\prime}}^{\prime}$.
It now follows that
$$(\mathcal{F}_{1},\dots,\mathcal{F}_{j-1},\mathcal{T},\mathcal{F}_{j^{\prime}+1%
}^{\prime},\dots,\mathcal{F}_{n^{\prime}}^{\prime},\dots\mathcal{F}_{j^{\prime%
}+1}^{\prime},\mathcal{T},\mathcal{F}_{j^{\prime}-1}^{\prime},\dots,\mathcal{F%
}_{1}^{\prime},\dots,\mathcal{F}_{j^{\prime}-1}^{\prime},\mathcal{T},\mathcal{%
F}_{j+1},\dots,\mathcal{F}_{n})$$
is an $L$–fence whose spokes are exactly $\mathcal{R}\cup\mathcal{R}^{\prime}$.
∎
We say that an $L$–fence $\mathcal{F}$ in a disk diagram $D$ is maximal, if any $L$–fence which contains every spoke of $\mathcal{F}$ is equal to $\mathcal{F}$.
By the previous lemma, it follows that given a spoke $\mathcal{S}$, there is a unique maximal $L$–fence containing $\mathcal{S}$.
Given an $L$–fence $\mathcal{F}$, we define $V(\mathcal{F})\subset V(\Gamma)$ to be the set of
all vertices $s\in V(\Gamma)$ which are the type of some dual curve in $\mathcal{F}$.
An $L$–fence, together with a collection of dual curves intersecting it, naturally corresponds to a subgraph of $\Gamma$ of hypergraph index at most $L$:
Proposition 3.7.
Let $D$ be a disk diagram over the RACG $W_{\Gamma}$.
Let $\mathcal{F}$ be an $L$–fence in $D$.
Let $Q_{1},\dots,Q_{m}$ be a sequence of dual curves in $D$
of types respectively $q_{1},\dots,q_{m}$ such that $Q_{i}$ intersects $\mathcal{F}$ for each $1\leq i\leq m$.
Then the subgraph of $\Gamma$ induced by $V(\mathcal{F})\cup q_{1}\cup\dots\cup q_{m}$ is either a strip subgraph or has hypergraph index at most $L$.
Proof.
We prove the lemma by induction on $L$.
We first suppose that $L=0$.
In this case, $\mathcal{F}$ is a spoke of type $\{s,t\}$ for some non-adjacent vertices $s,t\in\Gamma$.
It follows that the vertices $q_{1},\dots,q_{m}$ are each adjacent to both $s$ and $t$ in $\Gamma$. Thus, the vertices $\{q_{1},\dots,q_{m},s,t\}$ induce a join subgraph of $\Gamma$ which is either a wide subgraph (and so has hypergraph index $0$) or a strip subgraph,
depending on whether the vertices $q_{1},\dots,q_{m}$ all lie in a common clique of $\Gamma$.
This completes the base case.
We now fix $L\geq 1$ and assume by induction that the claim is true for all
$L^{\prime}<L$.
Let $\mathcal{F}=(\mathcal{F}_{1}\dots,\mathcal{F}_{n})$ be an $L$-fence in $D$.
For each $1\leq i\leq n$, let $V_{i}$ be the vertices of $\Gamma$ consisting of
$V(\mathcal{F}_{i})$, the set of all $q_{j}\in\{q_{1},\dots,q_{m}\}$ such that $Q_{j}$ intersects $\mathcal{F}_{i}$ and, additionally, all vertices $s\in\Gamma$ such that there exists a dual curve $Q$ of type $s$ in $\mathcal{F}$ which intersects $\mathcal{F}_{i}$.
By the induction hypothesis, $V_{i}$ is either a strip subgraph or has hypergraph index at most $L-1$.
Furthermore, we have that $V(\mathcal{F})=\bigcup_{i=1}^{n}V_{i}$.
Fix $1\leq i<n$.
Suppose first that there exists a spoke $\mathcal{S}=\{H,K\}$, of type $\{h,k\}$, in
$\mathcal{F}_{i}$ such that both $H$ and $K$ intersect $\mathcal{F}_{i+1}$.
It follows that $h$ and $k$ is a pair of non-adjacent vertices of $\Gamma$ contained in $V_{i}\cap V_{i+1}$.
On the other hand, if there is no such spoke $\mathcal{S}$, then by the definition of an $L$-fence there must exist
a spoke of $\mathcal{S}^{\prime}=\{H^{\prime},K^{\prime}\}$, of type $\{h^{\prime},k^{\prime}\}$, of $\mathcal{F}_{i+1}$ such that both $H^{\prime}$ and $K^{\prime}$ intersect $\mathcal{F}_{i}$.
In this case, $h^{\prime}$ and $k^{\prime}$ are non-adjacent vertices of $\Gamma$ both in $V_{i}\cap V_{i+1}$.
Thus, $V(\mathcal{F})=\bigcup_{i=1}^{n}V_{i}$
has hypergraph at most $L$.
∎
Definition 3.8 ($L$-fence connected paths).
Let $D$ be a disk diagram containing an $L$–fence $\mathcal{F}$ and paths $\gamma$ and $\gamma^{\prime}$.
We say that $\mathcal{F}$ is connected to $\gamma$, if some spoke of $\mathcal{F}$ intersects $\gamma$.
We say that the path $\gamma^{\prime}$ is $L$–fence connected to the path $\gamma$ if some $L$–fence $\mathcal{F}^{\prime}$ intersects both $\gamma$ and $\gamma^{\prime}$. In this case, we also say that $\mathcal{F}^{\prime}$ connects $\gamma$ and $\gamma^{\prime}$.
It will often be the case that an $L$–fence $\mathcal{F}$ connects two paths, both of which are on the boundary of a disk diagram.
Moreover, in some sense, $\mathcal{F}$ separates this disk diagram and any dual curve “crossing” $\mathcal{F}$ must intersect a spoke of $\mathcal{F}$. This is made precise in Proposition 3.10. Before proving that proposition, we prove a lemma showing that $L$–fences exhibit a certain connectivity property.
Lemma 3.9.
Let $\mathcal{F}$ be an $L$–fence in a disk diagram $D$, and let $\mathcal{R}=\{\mathcal{S}_{1},\dots,\mathcal{S}_{n}\}$ be the set of all spokes of $\mathcal{F}$. Let $Y=\{H_{1},\dots,H_{n}\}$ be a collection of dual curves such that for each $1\leq i\leq n$, $H_{i}$ is a dual curve in $\mathcal{S}_{i}$. Then $Y$ is a connected subset of $D$.
Proof.
The proof will be by induction on $L$. If $L=0$, then $Y$ consists of a single dual curve and so is connected.
Suppose now that $L>0$ and that the statement is true for all $L^{\prime}<L$.
Write $\mathcal{F}=(\mathcal{F}_{1},\dots,\mathcal{F}_{n})$
as in the definition of an $L$-fence.
For each $1\leq i\leq n$, let $\mathcal{R}_{i}=\{\mathcal{S}_{i_{1}}\dots\mathcal{S}_{i_{m}}\}\subset\mathcal%
{R}$ be the set of spokes of $\mathcal{F}_{i}$, and let $Y_{i}=\{H_{i_{1}},\dots,H_{i_{m}}\}$. By the induction hypothesis, $Y_{i}$ is connected.
Now, fix $1\leq i<n$.
Suppose that some spoke $\mathcal{S}=\{H,K\}\in\mathcal{R}_{i}$ is such that $H$ intersects the spoke $\{H^{\prime},K^{\prime}\}\in\mathcal{R}_{i+1}$ and $K$ intersects a, possibly different, spoke of $\mathcal{F}_{i+1}$.
Up to relabeling, we suppose that $H\in Y_{i}$ and $H^{\prime}\in Y_{i+1}$.
As $H\cap H^{\prime}\neq\emptyset$, it follows that $Y_{i}\cap Y_{i+1}\neq\emptyset$.
On the other hand, if such a spoke $\mathcal{S}$ does not exists, then by the definition of an $L$–fence it follows that some spoke of $\mathcal{F}_{i+1}$ has each of its dual curves intersecting $\mathcal{F}_{i}$, and we similarly deduce that $Y_{i}\cap Y_{i+1}\neq\emptyset$.
Thus, we conclude that $Y=Y_{1}\cup\dots\cup Y_{n}$ is connected, as $Y_{i}\cap Y_{i+1}\neq\emptyset$ for all $1\leq i<n$.
∎
Proposition 3.10.
Let $D$ be a disk diagram with boundary path $\gamma\eta\gamma^{\prime}\eta^{\prime}$ such that $\gamma\cap\eta$, $\gamma^{\prime}\cap\eta$, $\gamma\cap\eta^{\prime}$ and $\gamma^{\prime}\cap\eta^{\prime}$ all consist of a single vertex.
Let $\mathcal{F}$ be an $L$–fence connecting $\gamma$ and $\gamma^{\prime}$.
Then any dual curve that is dual to both $\eta$ and $\eta^{\prime}$ intersects a spoke of $\mathcal{F}$.
Proof.
The proof will be by contradiction. Let $Q$ be a dual curve, dual to both $\eta$ and $\eta^{\prime}$.
Let $\mathcal{R}$ be the set of all spokes of $\mathcal{F}$.
We assume, for a contradiction, that each spoke in $\mathcal{R}$ contains a dual curve which does not intersect $Q$.
Let $Y$ be the collection of all such dual curves. Note that $Q$ does not intersect $Y$, and $Y$ is connected by Lemma 3.9.
As $\mathcal{F}$ connects $\gamma$ to $\gamma^{\prime}$, there are spokes $\{H,K\}$ and $\{H^{\prime},K^{\prime}\}$ of $\mathcal{F}$ which intersect $\gamma$ and $\gamma^{\prime}$ respectively.
Thus, $Y$ contains a point $p$ of $\gamma$ and a point $p^{\prime}$ of $\gamma^{\prime}$.
Let $\zeta$ be a path in $Y$ from $p$ to $p^{\prime}$.
By the structure of dual curves in a disk diagram, $\zeta$ can be chosen to not intersect $\eta$ or $\eta^{\prime}$.
As $\zeta$ separates $\eta$ from $\eta^{\prime}$ in $D$, it follows that
$Q$ intersects $\zeta\subset Y$, a contradiction.
∎
3.2. Splitting Points
In this subsection we define $L$–splitting points. Intuitively, an $L$–splitting point is the first point along an oriented path $\gamma_{1}$ such that the subpath of $\gamma_{1}$ after this point is not $L$–fence connected to another given path $\gamma_{2}$ in a specified disk diagram.
Definition 3.11 ($L$–splitting point).
Let $D$ be a disk diagram over a RACG.
Let $\alpha$ be a proper subpath of a boundary path of $D$, let $\gamma$ be a reduced path in $D$, and let
$\gamma_{1}$ and $\gamma_{2}$ be reduced paths each with their starting point on $\gamma$ and endpoint on $\alpha$.
Furthermore, suppose that $\gamma_{1}\cap\gamma$ and $\gamma_{2}\cap\gamma$ are each a single vertex, $v_{1}$ and $v_{2}$ respectively, and that $\gamma_{1}\cap\gamma_{2}=\emptyset$.
Let $\gamma^{\prime}$ be the smallest subpath of $\gamma$ from $v_{2}$ to $v_{1}$, and let $\alpha^{\prime}$ be the smallest subpath of $\alpha$ from the endpoint of $\gamma_{1}$ to the endpoint of $\gamma_{2}$.
Let $D^{\prime}$ be the subdiagram of $D$ bounded by the path $\gamma_{1}\alpha^{\prime}\gamma_{2}^{-1}\gamma^{\prime}$.
Let $L\geq 0$ be an integer.
Consider the (possibly empty) set
$$\mathcal{R}=\{\{H_{1},K_{1}\},\dots,\{H_{n},K_{n}\}\}$$
of all spokes in $D^{\prime}$ such that each spoke $\{H_{i},K_{i}\}\in\mathcal{R}$ intersects $\gamma_{1}$ and is contained in an $L$–fence in $D^{\prime}$ connecting $\gamma_{1}$ and $\gamma_{2}$.
By possibly relabeling, we can suppose that
for each $1\leq i\leq n$, $H_{i}\cap\gamma_{1}$ occurs before $K_{i}\cap\gamma_{1}$ along the orientation of $\gamma_{1}$ and that
for each $1\leq i<n$, either $H_{i}=H_{i+1}$ or $H_{i}\cap\gamma_{1}$ occurs before $H_{i+1}\cap\gamma_{1}$ with respect to the orientation of $\gamma_{1}$.
We define the $L$–splitting point of $(\gamma_{1},\gamma_{2};\gamma,\alpha)$ to be the point $H_{n}\cap\gamma_{1}$ if $\mathcal{R}$ is not empty and to be the starting point of $\gamma_{1}$ if $\mathcal{R}$ is empty.
Definition 3.12 (Initial and terminal paths with respect to an $L$–splitting point).
Fix the notation as in the previous definition.
Let $x$ be the $L$–splitting point of $(\gamma_{1},\gamma_{2};\gamma,\alpha)$. Suppose first that $x$ is not equal to the starting point of $\gamma_{1}$.
Let $e$ be the edge of $\gamma_{1}$ whose midpoint is $x$.
Let $\gamma_{1}^{\prime}$ be the initial subpath of $\gamma_{1}$ from the starting point of $\gamma_{1}$ up to, and not including, $e$.
Let $\gamma_{1}^{\prime\prime}$ be the subpath of $\gamma_{1}$ from $e$ to the endpoint of $\gamma_{1}$ which does not include $e$.
On the other hand, if $x$ is the starting point of $\gamma_{1}$, then we define $\gamma_{1}^{\prime}=x$ (a length $0$ path) and $\gamma_{1}^{\prime\prime}=\gamma_{1}$.
In either case, we say that $\gamma_{1}^{\prime}$ and $\gamma_{1}^{\prime\prime}$ are, respectively, the initial and terminal paths of $\gamma_{1}$ with respect to the $L$–splitting point $x$.
Remark 3.13.
With the notation as in the previous two definitions, it is immediate that no $L$–fence in $D^{\prime}$ connects $\gamma_{1}^{\prime\prime}$ and $\gamma_{2}$.
Additionally, if $x$ is not equal to the starting point of $\gamma_{1}$, then there exists an $L$–fence in $D^{\prime}$ connecting $\gamma_{1}\setminus\gamma_{1}^{\prime}=\gamma_{1}^{\prime\prime}\cup e$ and $\gamma_{2}$.
4. Structured sequences of dual curves
Given a path $\gamma$ in a disk diagram, we will often need to find a sequence of dual curves, intersecting $\gamma$ and satisfying certain desirable properties. For instance, we will want these dual curves to be pairwise non-intersecting and to naturally correspond to spokes whose dual curves intersect $\gamma$ close to one another.
In this section, we define such sequences of dual curves and prove we can find them in different settings.
Definition 4.1 (Structured sequence).
Let $D$ be a disk diagram over the RACG $W_{\Gamma}$.
Let $\gamma$ be an oriented path in $D$.
We say that a sequence of dual curves $H_{1},K_{1},\dots,H_{n},K_{n}$, each intersecting $\gamma$, is structured with respect to $\gamma$ if:
(1)
The dual curves $H_{1},K_{1},\dots,H_{n},K_{n}$ are ordered with respect to the orientation of $\gamma$. More precisely, for all $1\leq i\leq n$, $H_{i}\cap\gamma$ occurs before $K_{i}\cap\gamma$ with respect to the orientation of $\gamma$ and for all $1\leq i<n$, $K_{i}\cap\gamma$ occurs before $H_{i+1}\cap\gamma$ with respect to the orientation of $\gamma$.
(2)
There are non-adjacent vertices $s,t\in\Gamma$ such that, for all $1\leq i\leq n$, $H_{i}$ and $K_{i}$ are of types $s$ and $t$ respectively.
Similarly, we say that a sequence of spokes $\{H_{1},K_{1}\},\dots,\{H_{n},K_{n}\}$ is structured with respect to $\gamma$ if the corresponding sequence of dual curves $H_{1},K_{1},\dots,H_{n},K_{n}$ is structured with respect to $\gamma$.
Note that a sequence of dual curves is structured with respect to $\gamma$ if and only if the corresponding sequence of spokes is structured with respect to $\gamma$.
If $\alpha$ is another path in $D$, then the sequence $H_{1},K_{1},\dots,H_{n},K_{n}$ (resp. $\{H_{1},K_{1}\},\dots,\{H_{n},K_{n}\}$)
is structured with respect to $(\gamma,\alpha)$
if it is structured with respect to $\gamma$ and, moreover, both $H_{i}$ and $K_{i}$ intersect $\alpha$ for all $1\leq i\leq n$.
We say that a sequence $H_{1},K_{1},\dots,H_{n},K_{n}$ (resp. $\{H_{1},K_{1}\},\dots,\{H_{n},K_{n}\}$) structured with respect to $\gamma$ is tight if for each $1\leq i\leq n$, the smallest subpath of $\gamma$ containing both $H_{i}\cap\gamma$ and $K_{i}\cap\gamma$ is of length at most $|V(\Gamma)|$.
By (2) in the definition above, it follows that the dual curves, in a sequence of dual curves structured with respect to a path, are pairwise non-intersecting.
We will use this observation freely throughout.
Given a reduced path $\gamma$, the next lemma guarantees we can always find a tight sequence of dual curves structured with respect to $\gamma$ of size proportional to $|\gamma|$.
Lemma 4.2.
Let $\Gamma$ be a non-clique graph.
Let $\gamma$ be a reduced path in the disk diagram $D$ over the RACG $W_{\Gamma}$.
Then there is a tight sequence $H_{1},K_{1},\dots,H_{n},K_{n}$ of dual curves structured with respect to $\gamma$ such that $n\geq\frac{1}{|V(\Gamma)|^{2}}\Big{\lfloor}\frac{|\gamma|}{|V(\Gamma)|}\Big{\rfloor}$.
Proof.
Set $M:=|V(\Gamma)|$. We partition $\gamma=\gamma_{1}\dots\gamma_{m+1}$ such that, for each $1\leq i\leq m$, $|\gamma_{i}|=M$ and $|\gamma_{m+1}|<M$.
Note that $m=\big{\lfloor}\frac{|\gamma|}{M}\big{\rfloor}$.
For each $1\leq i\leq m$, let $w_{i}=s_{i_{1}}\dots s_{i_{M}}$ be the label of $\gamma_{i}$.
We claim that, for each $1\leq i\leq m$, $\{s_{i_{1}},\dots,s_{i_{M}}\}$ contains a pair of distinct non-adjacent vertices of $\Gamma$.
For, suppose otherwise, that $s_{i_{1}},\dots,s_{i_{M}}$ are all vertices in a common clique of $\Gamma$.
As $|V(\Gamma)|=M$ and as $\Gamma$ is not a clique, it follows that for some $1\leq j<j^{\prime}\leq M$ we have that $s_{i_{j}}$ and $s_{i_{j^{\prime}}}$ are equal as vertices of $\Gamma$.
However, it also then follows that $w_{i}$ is not reduced, contradicting the fact that $\gamma$ is a reduced path.
This shows our claim.
Thus, there exists a sequence of dual curves $P_{1},Q_{1},\dots,P_{m},Q_{m}$ such that for each $1\leq i\leq m$, $P_{i}$ and $Q_{i}$ both intersect $\gamma_{i}$ and the types $s_{i}$ and $t_{i}$, of $P_{i}$ and $Q_{i}$ respectively, are not adjacent in $\Gamma$.
Moreover, the smallest subpath of $\gamma$ containing both $P_{i}\cap\gamma$ and $Q_{i}\cap\gamma$ has length at most $|V(\Gamma)|$ as both $P_{i}$ and $Q_{i}$ intersect $\gamma_{i}$.
As there are at most $M(M-1)\leq M^{2}$ possible pairs of non-adjacent vertices of $\Gamma$, by the pigeonhole principle, there exist non-adjacent vertices $s,t\in\Gamma$ such that at least $\frac{m}{M^{2}}$ of the spokes $\{P_{i},Q_{i}\}$ are of type $(s,t)$. Thus, there exists a subsequence $H_{1}=P_{i_{1}},K_{1}=Q_{i_{1}},\dots,H_{n}=P_{i_{n}},K_{n}=Q_{i_{n}}$ which is structured with respect to $\gamma$ such that $n\geq\frac{1}{M^{2}}\big{\lfloor}\frac{|\gamma|}{M}\big{\rfloor}$ and which is tight.
∎
We will often need for “enough” spokes, each satisfying some property, to intersect a path. The following definition makes this notion precise.
Definition 4.3 ($M$–adequate sets).
Let $\gamma$ be a path in a disk diagram $D$, and let $\mathcal{R}$ be a set of spokes (resp. dual curves) which intersect $\gamma$. For each integer $M\geq 0$, we say that $\mathcal{R}$ is $M$–adequate with respect to $\gamma$ if given any sequence of spokes (resp. dual curves) structured with respect to $\gamma$, it follows that all but possibly $M$ of these spokes (resp. dual curves) are in $\mathcal{R}$.
When $\gamma$ is implicit, we simply say that $\mathcal{R}$ is an $M$–adequate set.
The next lemma, which will be heavily used in Section 6, guarantees that, under the right hypotheses, we can find a tight sequence of dual curves structured with respect to $(\gamma,\alpha)$.
Lemma 4.4.
Let $\Gamma$ be a non-clique graph.
Let $D$ be a disk diagram over the RACG $W_{\Gamma}$ with boundary path $\psi\mu_{0}\dots\mu_{k}\theta\alpha\eta\beta$ such that $\psi\mu_{0}\dots\mu_{k}$ is a reduced path and such that no dual curve intersects both $\psi\mu_{0}\dots\mu_{k}$ and $\theta$.
Let $M\geq\max{\{|V(\Gamma)|,|\beta|\}}$ be an integer.
Suppose that for each $0\leq i\leq k$, there is an $M$–adequate set $\mathcal{R}(\mu_{i})$ of spokes intersecting $\mu_{i}$ such that no spoke
in this set intersects $\eta$.
Then there exists a tight sequence of spokes $\{H_{1},K_{1}\},\dots,\{H_{n},K_{n}\}$ structured with respect to $(\mu_{0}\dots\mu_{k},\alpha)$ such that:
(1)
$n\geq\frac{1}{M^{2}}\big{\lfloor}\frac{|\mu_{0}\dots\mu_{k}|}{M}\big{\rfloor}-%
3(k+1)M$
(2)
For each $1\leq i\leq n$, $\{H_{i},K_{i}\}\in\mathcal{R}(\mu_{j})$ for some $0\leq j\leq k$.
Proof.
By Lemma 4.2, there is a tight sequence of spokes $\{P_{1},Q_{1}\},\dots,\{P_{r},Q_{r}\}$
structured with respect to $\mu_{0}\dots\mu_{k}$ and such that $r\geq\frac{1}{M^{2}}\big{\lfloor}\frac{|\mu_{0}\dots\mu_{k}|}{M}\big{\rfloor}$.
As the dual curves $P_{1},Q_{1},\dots,P_{r},Q_{r}$ are pairwise non-intersecting (as they are structured), it follows that for all but possibly $k$ values of $i\in\{1,\dots,r\}$ we have that $\{P_{i},Q_{i}\}$ intersects $\mu_{j}$ for a distinct $0\leq j\leq k$.
Additionally, each of the dual curves $P_{1},Q_{1},\dots,P_{r},Q_{r}$ also intersects $\alpha\eta\beta$ as they cannot be dual to two edges of $\psi\mu_{0}\dots\mu_{k}$ (as it is reduced) and they cannot intersect $\theta$ by hypothesis.
Thus, for all but possibly $|\beta|+1$ values of $i\in\{1,\dots,r\}$ we have that $\{H_{i},K_{i}\}$ intersects either $\alpha$ or $\eta$.
There then exists a subsequence $\{P_{i_{1}},Q_{i_{1}}\},\dots,\{P_{i_{r^{\prime}}},Q_{i_{r^{\prime}}}\}$ of length $r^{\prime}\geq r-k-|\beta|-1$ such that, for each $1\leq l\leq r^{\prime}$, $\{P_{i_{l}},Q_{i_{l}}\}$ intersects $\mu_{j}$ for some distinct $0\leq j\leq k$ and intersects either $\alpha$ or $\eta$.
By hypothesis, for each $0\leq j\leq k$, all but possibly $M$ of the spokes
which intersect $\mu_{j}$ are in $\mathcal{R}(\mu_{j})$ and in particular do not intersect $\eta$.
It follows that there exists a subsequence $\{H_{1},K_{1}\},\dots,\{H_{n},K_{n}\}$ of $\{P_{i_{1}},Q_{i_{1}}\},\dots,\{P_{i_{r^{\prime}}},Q_{i_{r^{\prime}}}\}$ of length $n\geq r^{\prime}-(k+1)M$
which is structured with respect to $(\mu_{0}\dots\mu_{k},\alpha)$ and satisfies (2) above. Furthermore, we get that:
$$n\geq r^{\prime}-(k+1)M\geq r-k-|\beta|-1-(k+1)M\geq\frac{1}{M^{2}}\Big{%
\lfloor}\frac{|\mu_{0}\dots\mu_{k}|}{M}\Big{\rfloor}-3(k+1)M$$
∎
5. Disk diagram surgery
In this section we discuss disk diagram surgery, an operation which allows us to “insert” a path into a disk diagram in place of another. We prove two lemmas which will allow us to insert well-behaved paths into a disk diagram.
We say that a path in a disk diagram is simple if it is topologically a simple path.
Definition 5.1 (Disk diagram surgery).
Let $D$ be a disk diagram, and let $\gamma$ a simple path in $D$ with label $w$. Let $w^{\prime}$ be an expression for $w$.
Let $D^{\prime}$ be a disk diagram with boundary path $\gamma^{\prime}\eta^{-1}$ such that the labels of $\gamma^{\prime}$ and $\eta$ are $w^{\prime}$ and $w$ respectively.
Let $D^{\prime\prime}$ be the disk diagram consisting of a copy of $D^{\prime}$ and another reflected copy of $D^{\prime}$ glued together along the path $\gamma^{\prime}$
(see Figure 2). Note that the boundary path of $D^{\prime\prime}$ has label $ww^{-1}$.
We first slightly thicken $\gamma$ in $D$ and then cut along this path to produce an annular diagram $A$, one of whose boundary paths has label $ww^{-1}$ and the other has label the same as that of a boundary path of $D$.
We then attach $D^{\prime\prime}$ along its boundary to the boundary of $A$ with label $ww^{-1}$.
Let $E$ be this resulting disk diagram.
We can naturally think of $\gamma^{\prime}$ as a path in $E$ with label $w^{\prime}$.
We say that the resulting diagram is obtained from $D$ by surgery to insert $\gamma^{\prime}$ in place of $\gamma$ and that $E$ is obtained from $D$ by surgery.
Note that $E$ contains two (possibly equal) paths, labeled by $w$, which naturally correspond to $\gamma$ along the boundary of the inserted disk $D^{\prime\prime}$. We say that these paths are copies of $\gamma$ in $E$. By a slight abuse of notation, we will often refer to $\gamma$ when we mean a copy of $\gamma$.
Fix the notation from the previous definition.
There is a natural map $\Psi:E\to D$ collapsing the disk which was inserted into $D$.
We need to take great care when performing surgeries.
For instance, given a dual curve $H$ of $E$, it could be that $\Psi(H)$ is contained in two distinct dual curves of $D$.
Moreover,
bigons and nongons (as described in [Wis12]) can be introduced after surgery, even if they are not present in $D$ or $D^{\prime\prime}$.
However, some things can be seen to be preserved by surgery.
For instance, the boundary path of $E$ is canonically identified with a boundary path of $D$ (even if $\gamma$ contains edges of $\partial D$).
Moreover,
given a subdiagram $G\subset D$ whose interior does not intersect $\gamma$, it follows that $G$ is naturally a subdiagram of $E$.
The next convention will from now on be used as a book-keeping device for paths which track other paths (as in the definition below).
Convention 5.2.
As a convention, we will always use hat notation as follows: the word $\hat{w}$ will always be understood be equal to a word $w$ with some letters deleted. Similarly, the path $\hat{\gamma}$ will always be understood to track the path $\gamma$ (as defined below).
Definition 5.3 (Track).
Let $\gamma$ and $\gamma^{\prime}$ be oriented paths in a disk diagram, and suppose that $\gamma^{\prime}$ is reduced. Let $H_{1},\dots,H_{n}$ be the set of all dual curves which intersect $\gamma^{\prime}$ ordered by the orientation of $\gamma^{\prime}$, i.e. $H_{i}\cap\gamma^{\prime}$ occurs prior to $H_{i+1}\cap\gamma^{\prime}$ along the orientation of $\gamma^{\prime}$ for all $1\leq i<n$.
We say that $\gamma^{\prime}$ tracks $\gamma$ if the dual curves $H_{1},\dots,H_{n}$ each also intersect $\gamma$ and are ordered along the orientation of $\gamma$. In particular, if $w$ is the label of $\gamma$, then the label of $\gamma^{\prime}$ is a word $\hat{w}$ (as in Convention 5.2).
By definition, if $\hat{\gamma}$ tracks $\gamma$ and $\hat{\hat{\gamma}}$ tracks $\hat{\gamma}$, then $\hat{\hat{\gamma}}$ tracks $\gamma$.
Additionally, if $\hat{\gamma}$ tracks $\gamma$, then a sequence of dual curves (resp. spokes) structured with respect to $\hat{\gamma}$ is also structured with respect to $\gamma$.
These observations will be frequently used without mention.
The next two lemmas guarantee we can use disk diagram surgery to insert paths with certain desirable properties into a disk diagram.
Lemma 5.4.
Let $\gamma$ be a simple path with label $w$ in a disk diagram $D$. We can obtain a disk diagram $E$ by applying surgery to $D$ to insert a path $\hat{\gamma}$ in place of $\gamma$, such that $\hat{\gamma}$ tracks each copy of $\gamma$ in $E$ and has label a reduced expression $\hat{w}$ for $w$.
Proof.
By [Wis11][Lemma 2.6 and Corollary 2.7], there exists a disk diagram $D^{\prime}$ with boundary path $\eta(\gamma^{\prime})^{-1}$ where $\eta$ and $\gamma^{\prime}$ have labels $w$ and $w^{\prime}$ respectively, where $w^{\prime}$ is a reduced expression for $w$.
Additionally, $D^{\prime}$ is such that there is no intersection between two dual curves emanating from distinct edges of $\gamma^{\prime}$. As no dual curve intersects a reduced path twice, every dual curve dual to $\gamma^{\prime}$ intersects $\eta$ in $D^{\prime}$. Consequently, $\gamma^{\prime}$ tracks $\eta$ in $D^{\prime}$. We apply surgery to $D$ as in Definition 5.1 to obtain the disk diagram $E$ by, as in this definition, inserting two copies of $D^{\prime}$. It follows that $\gamma^{\prime}$ tracks the copies of $\gamma$ in $E$. Thus, we can set $\hat{\gamma}=\gamma^{\prime}$ and $\hat{w}=w^{\prime}$.
∎
Lemma 5.5.
Let $D$ be a disk diagram over the RACG $W_{\Gamma}$.
Let $\gamma$ be a simple path in $D$ with label $w=s_{1}\dots s_{n}$.
Then, we can apply surgery to $D$ to insert a path $\gamma^{\prime}$ in place of $\gamma$ such that the label $w^{\prime}=s_{1}^{\prime}\dots s_{n}^{\prime}$ of $\gamma^{\prime}$ is a reduced expression for $w$. Moreover, the following occurs in the resulting disk diagram.
For some $1\leq j\leq n$, $s_{j}^{\prime}=s_{n}$ and, for all $j<i\leq n$, $s_{j}^{\prime}$ and $s_{i}^{\prime}$ are adjacent vertices of $\Gamma$. The initial subpath of $\gamma^{\prime}$ with label $s_{1}^{\prime}\dots s_{j}^{\prime}$ tracks each copy of $\gamma$. Finally, for each $1\leq k<j$, the dual curve dual to the edge of $\gamma^{\prime}$ labeled by $s_{k}^{\prime}$ does not intersect the dual curve dual to the edge of $\gamma^{\prime}$ labeled by $s_{j}^{\prime}$.
Proof.
Let $w^{\prime}=s_{1}^{\prime}\dots s_{n}^{\prime}$ be an expression for $w$ and $1\leq j\leq n$ be such that $s_{j}^{\prime}=s_{n}$ and, for all $j<i\leq n$, $s_{j}$ and $s_{i}$ are adjacent vertices of $\Gamma$.
We additionally choose $w^{\prime}$ and $j$ so that $j$ is minimal out of all such possible choices.
Note that such an expression exists as we can take $w^{\prime}=w$ and $j=n$ (where $j$ is not necessarily minimal).
Let $h_{1}=s_{1}^{\prime}\dots s_{j-1}^{\prime}$ and $h_{2}=s_{j+1}^{\prime}\dots s_{n}^{\prime}$ so that $w^{\prime}=h_{1}s_{j}^{\prime}h_{2}$.
It follows from [Wis11][Lemma 2.6 and Corollary 2.7] that, by
possibly replacing $h_{1}$ in $w^{\prime}$ with another expression for $h_{1}$,
there is a disk diagram $D^{\prime}$ with boundary path labeled $ww^{\prime-1}$
such that there are no intersection between pairs of dual curves dual to the subpath of $\partial D^{\prime}$ corresponding to $h_{1}$.
In particular, as dual curves intersect a reduced path at most once, the subpath of $\partial D^{\prime}$ labeled by $s_{1}^{\prime}\dots s_{j}^{\prime}$ tracks the subpath labeled by $w$.
We now apply surgery to $D$ to insert two glued copies of $D^{\prime}$ into $D$ as in Definition 5.1.
Let $E$ be the resulting disk diagram, and let
$\gamma^{\prime}$ be the corresponding path labeled by $w^{\prime}$ in $E$.
We claim that, for each $1\leq k<j$, the dual curve dual to the edge of $\gamma^{\prime}$ labeled by $s_{k}^{\prime}$ does not intersect the dual curve dual to the edge labeled by $s_{j}^{\prime}$.
For suppose, otherwise, and take $k$ maximal with this property.
It then follows by the commuting relations imposed by intersections of dual curves in $E$ that $s_{1}^{\prime}\dots s_{k-1}^{\prime}s_{k+1}^{\prime}\dots s_{j}^{\prime}s_{k}^%
{\prime}s_{j+1}^{\prime}\dots s_{n}^{\prime}$ is an expression for $w$ and that $s_{k}^{\prime}$ is adjacent to $s_{j}^{\prime}$ in $\Gamma$, contradicting our choice of $j$ being minimal.
The lemma now follows.
∎
6. Divergence Bounds
We wish prove Theorem C by induction; however, for the induction to work, we need to first prove the more technical Proposition 6.2 and Proposition 6.3. These two propositions are the crux of the argument, and they are the focus of this section.
We first establish some terminology.
Given a disk diagram $D$ with boundary path $\gamma$ and a subpath $\alpha\subset\gamma$, there is a closed path $\gamma^{\prime}$ in the $1$–skeleton of the Davis complex $\Sigma_{\Gamma}$ (i.e., the Cayley graph of $W_{\Gamma}$ with standard generators) based at the vertex representing the identity and with the same label as that of $\gamma$. Furthermore, there is a subpath $\alpha^{\prime}\subset\gamma^{\prime}$ naturally corresponding to $\alpha$. We say that $\alpha$ is $R$–avoidant with respect to $D$ if the corresponding path $\alpha^{\prime}$ in $\Sigma_{\Gamma}$ does not intersect the ball of radius $R$ about the vertex representing the identity element.
We would like to define a spoke to be “$L$–fence separated” from a path $\eta$ if no $L$–fence containing this spoke intersects $\eta$.
However, we need a spoke to remain $L$–fence separated even after surgeries have been performed and new $L$–fences have possibly been created. This motivates the actual definition, below, of an $L$–fence separated spoke.
Definition 6.1 ($L$–fence separated).
Let $D$ be a disk diagram with a fixed boundary path $\gamma$.
Let $\mathcal{S}=\{H,K\}$ be a spoke in $D$ which intersects $\gamma$, and let
$e_{1}$ and $e_{2}$ be the edges of $\gamma$ dual to $H$ and $K$ respectively.
Let $\eta$ be a subpath of $\gamma$.
We say that $\mathcal{S}$ is $L$–fence separated from $\eta$ with respect to $D$ if the following holds. Given any disk diagram $E$ obtained from $D$ by a series of surgeries and dual curves $H^{\prime}$ and $K^{\prime}$ in $E$ dual respectively to (the images of) $e_{1}$ and $e_{2}$, it follows that the spoke $\{H^{\prime},K^{\prime}\}$ is not contained in an $L$–fence in $E$ which intersects $\eta$.
We define the function $f_{L}^{M}(R):=\frac{R}{M^{100L+50}}$. This function will be used in the next two propositions. When $M$ is implicit, we denote this function by $f_{L}(R)$.
Proposition 6.2.
Let $D$ be a disk diagram with boundary path $\gamma_{1}\gamma_{2}\gamma_{3}\alpha\eta\beta$ and basepoint $b$. Let $M>\max\{|V(\Gamma)|+1,|\beta|\}$, $L\geq 0$ and $R>0$ be integers. Furthermore, suppose that:
(A1)
The path $\alpha$ is $R$–avoidant with respect to $D$.
(A2)
The path $\gamma_{1}\gamma_{2}$ is reduced, no dual curve is dual to both $\gamma_{1}\gamma_{2}$ and $\gamma_{3}$,
$|\gamma_{1}|\leq f_{L}(R)$ and $|\gamma_{2}|\geq f_{L}(R)$.
(A3)
There is an $M$–adequate set $\mathcal{R}(\gamma_{2})$ of spokes intersecting $\gamma_{2}$ which are each $L$–fence separated from $\eta$.
Then, for $R$ large enough (depending only on $M$ and $L$), we can apply a sequence of surgeries to $D$ to obtain a disk diagram $E$ containing a path $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta\pi$ from $b$ to $\alpha$. Let $E^{\prime}\subset E$ be the subdiagram with boundary path
$\gamma_{1}\gamma_{2}\gamma_{3}\alpha^{\prime}(\gamma_{1}^{\prime}\gamma_{2}^{%
\prime}\zeta\pi)^{-1}$ where $\alpha^{\prime}$ is the subpath of $\alpha$ between the endpoint of $\gamma_{3}$ and the endpoint of $\pi$. We additionally have that:
(B1)
The path $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta$ is reduced, no dual curve is dual to both $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta$
and $\pi$, $|\gamma_{2}^{\prime}\zeta|\geq 32f_{L+1}(R)$, $\gamma_{1}^{\prime}$ tracks $\gamma_{1}$ and $\gamma_{2}^{\prime}$ tracks $\gamma_{2}$.
(B2)
In $E^{\prime}$, the set of spokes intersecting $\zeta$ which are contained in an $(L+1)$–fence that contains a spoke of $\mathcal{R}(\gamma_{2})$ is an $M$–adequate set.
(B3)
In $E^{\prime}$, no dual curve intersects both $\zeta$ and $\gamma_{1}$.
(B4)
The path $\alpha^{\prime}$ has length at least $C_{L}R^{L+1}$ where $C_{L}$ depends only on $L$ and $M$.
Proposition 6.3.
Let $D$ be a disk diagram with boundary path $\gamma\alpha\eta\beta$ and basepoint $b$.
Let $M>\max\{|V(\Gamma)|+1,|\beta|\}$, $L\geq 1$, $n\leq\frac{f_{L}(R)}{50M^{4}}-1$ and $R>0$ be integers.
Let $\psi\mu_{0}\dots\mu_{n}\theta$ be a simple path
in $D$ from $b$ to $\alpha$.
Let $D_{1}\subset D$ and $D_{2}\subset D$ be the subdiagrams with boundary paths $\gamma\kappa_{1}(\psi\mu_{0}\dots\mu_{n}\theta)^{-1}$ and $\psi\mu_{0}\dots\mu_{n}\theta\kappa_{2}\eta\beta$ respectively, where $\alpha=\kappa_{1}\kappa_{2}$ and the endpoint of $\kappa_{1}$ is the endpoint of $\theta$.
Suppose that in $D_{1}$ we have an $L$–fence $\mathcal{F}$ containing a spoke which is $L$–fence separated from $\eta$ with respect to $D$.
Additionally, suppose that:
(X1)
The path $\alpha$ is $R$–avoidant with respect to $D$.
(X2)
The path $\psi\mu_{0}\dots\mu_{n}$ is reduced, no dual curve intersects both $\psi\mu_{0}\dots\mu_{n}$ and $\theta$, $|\psi|\leq f_{L}(R)$ and $|\mu_{0}\dots\mu_{n}|\geq f_{L}(R)$.
(X3)
For each $1\leq i\leq n$, the set $\mathcal{R}(\mu_{i})$ of spokes intersecting $\mu_{i}$ which are contained in $\mathcal{F}$ is $M$–adequate.
(X4)
The set of dual curves intersecting $\mu_{1}\dots\mu_{n}$ which also intersect $\mathcal{F}$ is $M$–adequate.
(X5)
The set $\mathcal{R}(\mu_{0})$ of spokes intersecting $\mu_{0}$ which are $L$–fence separated from $\eta$ with respect to $D_{2}$ is $M$–adequate.
Then, for $R$ large enough (depending only on $M$ and $L$), we can apply a sequence of surgeries to $D$, none of which involves a path that intersects the interior of $D_{1}\subset D$, to obtain a disk diagram $E$. Moreover, $E$ contains a simple path $\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}\theta^{\prime}$ from $b$ to $\kappa_{2}$ that does not intersect the interior of the image of $D_{1}$ in $E$. Additionally, we have that:
(Y1)
Let $E^{\prime}\subset E$ be the subdiagram with boundary path $\gamma\kappa_{3}(\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}\theta^{%
\prime})^{-1}$ where $\kappa_{3}$ is the initial subpath of $\alpha$ up to the endpoint of $\theta^{\prime}$.
There exists an $L$–fence $\mathcal{F}^{\prime}$ in $E^{\prime}$ which either contains a spoke of $\mathcal{F}$, or alternatively contains a spoke of $\mathcal{R}(\mu_{0})$.
(Y2)
The path $\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}$ is reduced, no dual curve intersects both $\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}$ and $\theta^{\prime}$, the paths $\psi^{\prime},\mu_{0}^{\prime},\dots,\mu_{n}^{\prime}$ each track $\psi,\mu_{0},\dots,\mu_{n}$ respectively, and no dual curve intersects both $\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}$ and $\psi$.
(Y3)
For all $1\leq i\leq n+2$, the set of spokes intersecting $\mu_{i}^{\prime}$
which are in $\mathcal{F}^{\prime}$ is $M$–adequate.
(Y4)
The set of dual curves intersecting $\mu_{1}^{\prime}\dots\mu_{n+2}^{\prime}$ which intersect $\mathcal{F}^{\prime}$ is $M$–adequate.
(Y5)
Let $\alpha^{\prime}$ be the subpath of $\alpha$ from the endpoint of $\theta$ to the endpoint of $\theta^{\prime}$. Then one of the two possibilities holds:
(a)
$|\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}|\geq 32f_{L+1}(R)$ and $|\alpha^{\prime}|\geq C_{L}R^{L+1}$ where $C_{L}$ depends only on $M$ and $L$.
(b)
$|\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}|\geq f_{L}(R)$ and $|\alpha^{\prime}|\geq C_{L}^{\prime}R^{L}$ where $C_{L}^{\prime}$ depends only on $M$ and $L$.
Our strategy in proving the above two propositions is the following. We first show that Proposition 6.2 holds when $L=0$. Next, we show that
Proposition 6.3 holds for $L=\ell\geq 1$ given that
Proposition 6.2 holds for $L=\ell-1$.
Finally, we show that if Proposition 6.3 holds for $L=\ell\geq 1$ then Proposition 6.2 also holds for $L=\ell$.
Proof of Proposition 6.2 for $L=0$..
Let $D$ be a disk diagram as in Proposition 6.2 with $L=0$.
Set $c:=f_{1}(R)$.
By Lemma 4.4 (where in that lemma we set $k=0$, $\psi=\gamma_{1}$, $\mu_{0}=\gamma_{2}$ and $\theta=\gamma_{3}$)
there exists a sequence of spokes $\{H_{1},K_{1}\},\dots,\{H_{t},K_{t}\}$ in $\mathcal{R}(\gamma_{2})$ and structured with respect to $(\gamma_{2},\alpha)$ such that
$$t\geq\frac{1}{M^{2}}\Big{\lfloor}\frac{|\gamma_{2}|}{M}\Big{\rfloor}-3M\geq%
\frac{1}{M^{2}}\Big{\lfloor}\frac{f_{0}(R)}{M}\Big{\rfloor}-3M\geq\frac{f_{0}(%
R)}{2M^{3}}\geq 34f_{1}(R)=34c$$
where the second and third inequality follow respectively from (A2) and $R$ being large enough.
Let $e$ be the edge of $\gamma_{2}$ dual to $H_{33c}$.
Let $\rho$ be the initial subpath of $\gamma_{1}\gamma_{2}$ up to and including $e$, and
let $w=s_{1}\dots s_{n}$ be its label.
As $\{H_{1},K_{1}\},\dots,\{H_{t},K_{t}\}$ is a structured sequence, there exist non-adjacent vertices $s$ and $t$ of $\Gamma$ such that $H_{i}$ and $K_{i}$ are of type $s$ and $t$ respectively for all $1\leq i\leq t$.
In particular, $s_{n}=s$.
Using Lemma 5.5, we apply a disk diagram surgery to replace $\rho$ with a path $\rho^{\prime}$, with label $w^{\prime}=s_{1}^{\prime}\dots s_{n}^{\prime}$ satisfying the properties of that lemma.
As in that lemma, let $1\leq j\leq n$ be such that $s_{j}^{\prime}$ is of type $s$ and, for all $i>j$, $s_{i}$ and $s_{j}$ are adjacent vertices of $\Gamma$.
As both $\rho$ and $\rho^{\prime}$ are reduced, it follows that $H_{33c}$ (in this resulting diagram) intersects the edge $e^{\prime}$ labeled by $s_{j}^{\prime}$.
Let $\rho^{\prime\prime}$ be the initial subpath of $\rho^{\prime}$
up to, and not including, $e^{\prime}$.
We have that $\rho^{\prime\prime}=\hat{\gamma}_{1}\hat{\gamma}_{2}$ (with the notation as in Convention 5.2).
Furthermore, as structured dual curves are pairwise non-intersecting, for all $1\leq i<33c$, $H_{i}$ intersects $\hat{\gamma}_{2}$ and in particular $|\hat{\gamma}_{2}|>32c$.
We define $\gamma_{1}^{\prime}=\hat{\gamma}_{1}$ and $\gamma_{2}^{\prime}=\hat{\gamma}_{2}$.
We set $\zeta$ to be the endpoint of $\gamma_{2}^{\prime}$ (i.e., a length $0$ path).
Finally, we set $\pi$ to be a simple path in the carrier $N(H_{33c})$ from the endpoint of $\gamma_{2}^{\prime}$ to $\alpha$.
We now check that the conclusions of Proposition 6.2 are satisfied with these choices of $\gamma_{1}^{\prime}$, $\gamma_{2}^{\prime}$, $\zeta$ and $\pi$.
By our application of Lemma 5.5, $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta$ is reduced (as $\hat{\gamma}_{1}\hat{\gamma}_{2}$ is reduced), $\gamma_{1}^{\prime}$ tracks $\gamma_{1}$, $\gamma_{2}^{\prime}$ tracks $\gamma_{2}$, any dual curve dual to $\gamma_{1}^{\prime}\gamma_{2}^{\prime}$ does not intersect $\pi$ (as it does not intersect $H_{33c}$) and $|\gamma_{2}^{\prime}\zeta|\geq 32c=32f_{1}(R)$. Thus, (B1) follows.
Conclusions (B2) and (B3) hold trivially as $|\zeta|=0$.
To see (B4), note that since the dual curves $H_{1},K_{1},\dots,H_{t},K_{t}$ are pairwise non-intersecting, it follows that, for $33c\leq i\leq 34c$, $H_{i}$ intersects the subpath of $\alpha$ between the endpoint of $\gamma_{3}$ and the endpoint of $\pi$.
In particular, this subpath has length at least $c=f_{1}(R)\geq C_{0}R$ where $C_{0}$ depends only on $M$.
Thus, the conclusions of Proposition 6.2 hold.
∎
Proof of Proposition 6.3 for $L=\ell\geq 1$, assuming Proposition 6.2 for $L=\ell-1$.
We suppose now we have a disk diagram $D$ satisfying the hypotheses of Proposition 6.3 with $L=\ell$ and $R=r$.
We also suppose that the statement of Proposition 6.2 is true for $L=\ell-1$.
Set $c:=f_{\ell+1}(r)$.
Let $\nu$ be the initial subpath of $\mu_{0}\dots\mu_{n}$ of length $\lceil f_{\ell}(r)\rceil$, which exists by (X2).
By Lemma 4.4, (X2), (X3) and (X5) there exists a tight sequence of spokes $\{H_{1},K_{1}\},\dots,\{H_{t},K_{t}\}$ in $D_{2}$ structured with respect to $(\nu,\kappa_{2})$ such that
$$t\geq\frac{1}{M^{2}}\Big{\lfloor}\frac{|\nu|}{M}\Big{\rfloor}-3(n+1)M\geq\frac%
{1}{M^{2}}\Big{\lfloor}\frac{f_{\ell}(r)}{M}\Big{\rfloor}-\frac{3f_{\ell}(r)}{%
50M^{3}}\geq\frac{f_{\ell}(r)}{2M^{3}}\geq 34f_{\ell+1}(r)=34c$$
and for each $1\leq i\leq t$, $\{H_{i},K_{i}\}$ is in the set $\mathcal{R}(\mu_{j})$ for some distinct $0\leq j\leq n$ .
For each $33c<i\leq 34c$,
Let $h_{i}$ (resp. $k_{i}$) be a simple path in the carrier $N(H_{i})\subset D_{2}$ (resp. $N(K_{i})\subset D_{2}$) starting from the edge of $\mu_{0}\dots\mu_{n}$ dual to $H_{i}$ (resp. $K_{i}$) and up to $\kappa_{2}$ such that this path not intersect $H_{i}$ (resp. $K_{i}$).
As the dual curves $H_{1},K_{1},\dots,H_{t},K_{t}$ are pairwise non-intersecting, these paths can be chosen to also be pairwise non-intersecting.
By applying a series of surgeries to insert reduced paths in place of the $h_{i}$ and $k_{i}$ using Lemma 5.4, we assume that $h_{i}$ and $k_{i}$ are reduced for each $i$. Note that, even after such a replacements, we still have that these paths are pairwise non-intersecting and that any dual curve that intersects $h_{i}$ (resp. $k_{i}$) must also intersect $H_{i}$ (resp. $K_{i}$).
By a slight abuse of notation, we still denote this resulting diagram by $D$.
Let $z_{i}$ be the subpath of $\mu_{0}\dots\mu_{n}$ from the starting point of $h_{i}$ to that of $k_{i}$. By construction, $z_{i}\subset\mu_{j}$ for some distinct $0\leq j\leq n$, $|z_{i}|\leq|V(\Gamma)|$ (as our structured sequence of dual curves is tight) and the path $h_{i}^{-1}z_{i}k_{i}$ is a simple path with endpoints on $\kappa_{2}$.
For each $33c<i\leq 34c$, let $D_{i}\subset D$ be the subdiagram with boundary path $h_{i}\alpha_{i}k_{i}^{-1}z_{i}^{-1}$, where $\alpha_{i}$
is the subpath of $\alpha$ between the endpoint of $h_{i}$ and the endpoint of $k_{i}$.
Note that $\alpha_{i}\cap\alpha_{j}=\emptyset$ for all $i\neq j$.
Let $x_{i}$ be the $(\ell-1)$–splitting point of $(h_{i},k_{i};z_{i},\alpha_{i})$.
Up to applying surgeries to $D_{i}$, we can assume that no diagram obtained from $D_{i}$ by a sequence of surgeries is such that the $(\ell-1)$–splitting point of $(h_{i},k_{i};z_{i},\alpha_{i})$ in this diagram occurs after $x_{i}$ along the orientation of $h_{i}$.
In other words, up to surgeries on $D_{i}$, $x_{i}$ is a far along $h_{i}$ as possible.
Let $h_{i}^{\prime}$ and $h_{i}^{\prime\prime}$ be the initial and terminal paths of $h_{i}$ with respect to $x_{i}$.
There are now two main cases to consider, depending on whether or not $|h_{i}^{\prime}|$ is small for all $33c<i\leq 34c$. We will prove that Proposition 6.3 holds with (Y5a) in the first case and with (Y5b) in the second.
Case 1:
Suppose first that $|h_{i}^{\prime}|\leq\frac{f_{\ell-1}(r)}{8}$ for all $33c<i\leq 34c$.
Let $e$ be the edge of $\mu_{0}\dots\mu_{n}$ which is dual to $H_{33c}$.
Let $0\leq m\leq n$ be such that $e$ lies on $\mu_{m}$.
Let $\rho$ be the initial subpath of $\psi\mu_{0}\dots\mu_{m}$ up to and including $e$.
We apply Lemma 5.5 to replace $\rho$ with a path $\rho^{\prime}$ such that the conclusions of that lemma are satisfied.
By a slight abuse of notation, we denote by $H_{33c}$ the dual curve in this resulting diagram dual to a copy of the edge $e$.
Let $\rho^{\prime\prime}$ be the initial subpath of $\rho^{\prime}$ up to, and not including, the edge dual to $H_{33c}$.
As in the proof above of Proposition 6.2 for the case $L=0$, we have that $\rho^{\prime\prime}=\hat{\psi}\hat{\mu}_{0}\dots\hat{\mu}_{m}$ and that $|\hat{\mu}_{0}\dots\hat{\mu}_{m}|\geq 32c$.
Furthermore, no dual curve dual to $\rho^{\prime\prime}$ intersects $H_{33c}$.
We would like to show that the conclusions of Proposition 6.3 hold by setting $\psi^{\prime}=\hat{\psi}$,
setting $\mu_{i}^{\prime}=\hat{\mu}_{i}$ for $1\leq i\leq m$,
setting $\mu_{i}^{\prime}$ to be the endpoint of $\hat{\mu}_{m}$ for $m<i\leq n+2$,
setting $\mathcal{F}^{\prime}=\mathcal{F}$, and defining $\theta^{\prime}$ to be a simple subpath of $N(H_{33c})$ with starting point the endpoint of $\hat{\mu}_{m}$ and endpoint on $\kappa_{2}$.
Indeed, conclusions (Y1) and (Y2) immediately follows from our choices, and (Y3) holds from hypothesis (X3) as $\mu_{i}^{\prime}$ either tracks $\mu_{i}$ or has length $0$.
Conclusion (Y4) holds from hypothesis (X4), as $\mu_{1}^{\prime}\dots\mu_{n+2}^{\prime}$ tracks $\mu_{1}\dots\mu_{n}$
Additionally, we have that $|\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}|=|\hat{\mu}_{0}\dots\hat{\mu}_{m}|%
\geq 32c=32f_{\ell+1}(R)$, so the first part of (Y5a) holds.
The remainder of this case consists of showing the bound from (Y5a) on the subpath $\alpha^{\prime}$ of $\alpha$.
For $33c<i\leq 34c$, set $\gamma_{1}^{i}$ to be the initial subpath of $h_{i}$ up to the starting point of $h_{i}^{\prime\prime}$.
By the definition of splitting points, $\gamma_{1}^{i}$ is just $h_{i}^{\prime}$ with possibly the addition of an edge.
Set $\gamma_{2}^{i}=h_{i}^{\prime\prime}$ and $\gamma_{3}^{i}$ to be the endpoint of $h_{i}^{\prime\prime}$ (i.e., a length $0$ path).
We get the following equations for $r$ large enough (depending only on $M$ and $\ell$):
(1)
$$\displaystyle|\gamma_{2}^{i}|=|h_{i}^{\prime\prime}|\geq r-|\psi|-|\nu|-|h_{i}%
^{\prime}|-1\geq r-2f_{\ell}(r)-\frac{f_{\ell-1}(r)}{8}-1\geq f_{\ell-1}\big{(%
}\frac{r}{4}\big{)}$$
(2)
$$\displaystyle r-|\psi|-|\nu|\geq r-2f_{\ell}(r)\geq\frac{r}{4}$$
(3)
$$\displaystyle|\gamma_{i}^{1}|\leq|h_{i}^{\prime}|+1\leq\frac{f_{\ell-1}(r)}{8}%
+1\leq f_{\ell-1}(\frac{r}{4})$$
We now show that the hypotheses of Proposition 6.2 hold for the disk diagram $D_{i}$ with boundary path $\gamma_{i}^{1}\gamma_{i}^{2}\gamma_{i}^{3}\alpha_{i}k_{i}^{-1}z_{i}^{-1}$ for $L=\ell-1$ and $R=\frac{r}{4}$ (where in that proposition we set $\gamma_{1}=\gamma_{i}^{1}$, $\gamma_{2}=\gamma_{i}^{2}$, $\gamma_{3}=\gamma_{i}^{3}$, $\alpha=\alpha_{i}$, $\eta=k_{i}^{-1}$ and $\beta=z_{i}^{-1}$).
First note that as $|z_{i}|\leq|V(\Gamma)|$, the same $M$ can be used in that proposition as the one used in this proof.
Hypothesis (A1) of that proposition holds, as by equation (2), the path $\alpha_{i}$ is $\frac{r}{4}$–avoidant with respect to $D_{i}$.
We now check (A2).
The path $\gamma_{1}^{i}\gamma_{2}^{i}$ is reduced as it is equal to the reduced path $h_{i}$.
No dual curve intersects $\gamma_{3}^{i}$ as it has length $0$.
That $|\gamma_{2}^{i}|\geq f_{\ell-1}(\frac{r}{4})$ and $|\gamma_{1}^{i}|\leq f_{\ell-1}(\frac{r}{4})$ follow respectively from equations (1) and (3).
By Remark 3.13, no $(\ell-1)$–fence connects $h_{i}^{\prime\prime}$ to $k_{i}$ in $D_{i}$.
Furthermore, by our choice of $D_{i}$ (with $x_{i}$ as furthest as possible along $h_{i}$) this is still true after performing surgeries to $D_{i}$.
Thus, every spoke which intersects $\gamma_{2}^{i}$ is $L$–fence separated from $k_{i}^{-1}$, and hypothesis (A3) follows.
As all the required hypotheses hold, we apply Proposition 6.2 and deduce, from conclusion (B4) of that proposition, that $|\alpha_{i}|\geq C_{\ell-1}(\frac{r}{4})^{\ell}$ for each $33c<i\leq 34c$.
As the paths $\{\alpha_{i}\}$ are disjoint, we have that
$$|\alpha^{\prime}|\geq\sum_{i={33c+1}}^{34c}|\alpha_{i}|\geq c\Big{(}C_{\ell-1}%
\big{(}\frac{r}{4}\big{)}^{\ell}\Big{)}\geq f_{\ell+1}(r)\Big{(}C_{\ell-1}(%
\frac{r}{4})^{\ell}\Big{)}\geq C_{\ell}r^{\ell+1}$$
for some constant $C_{\ell}$ depending only on $M$ and $\ell$.
Thus (Y5b) holds, and we are done in this case.
Case 2:
By the previous case, we may assume that $|h_{j}^{\prime}|>\frac{f_{\ell-1}(r)}{8}$ for some $33c<j\leq 34c$.
We fix such a $j$, and we fix $0\leq m\leq n$ such that $\{H_{j},K_{j}\}$ intersects $\mu_{m}$.
Let $v$ be the starting point of $h_{j}$.
Let $\omega$ be the initial subpath of $\psi\mu_{0}\dots\mu_{m}$ up to $v$.
By Lemma 5.4, we can apply surgery to $\omega h_{j}$ to obtain a reduced path $\hat{\omega}\hat{h}_{j}=\hat{\psi}\hat{\mu}_{0}\dots\hat{\mu}_{m}\hat{h}_{j}$ which tracks $\omega h_{j}$ and such that $\hat{h}_{j}=\hat{h}_{j}^{\prime}\hat{a}\hat{h}_{j}^{\prime\prime}$ where $a$ is edge between the endpoint of $h_{j}^{\prime}$ and the starting point of $h_{j}^{\prime\prime}$ as in the definition of splitting points.
As $\omega$ and $h_{j}^{\prime}$ are reduced paths, by the triangle inequality
we have that
(4)
$$\displaystyle|\hat{h}_{j}^{\prime}|\geq|h_{j}^{\prime}|-|\psi|-|\nu|>\frac{f_{%
\ell-1}(r)}{8}-2f_{\ell}(r)\geq f_{\ell-1}\big{(}\frac{r}{16}\big{)}$$
Additionally, we have:
(5)
$$\displaystyle|\hat{\psi}\hat{\mu}_{1}\dots\hat{\mu}_{m}|$$
$$\displaystyle\leq|\psi|+|\nu|\leq 2f_{\ell}(r)\leq f_{\ell-1}\big{(}\frac{r}{1%
6}\big{)}$$
Set $\gamma_{1}=\hat{\psi}\hat{\mu}_{0}\dots\hat{\mu}_{m}$.
Let $\gamma_{2}$ be the initial subpath of $\hat{h}_{j}^{\prime}$ of length $\lceil f_{\ell-1}(\frac{r}{16})\rceil$, which exists by equation (4) above.
Let $\gamma_{3}$ be the subpath of $\hat{h}_{j}$ from the endpoint of $\gamma_{2}$ to the endpoint of $\hat{h}_{j}$.
Let $\alpha^{\prime\prime}$ be the subpath of $\alpha$ from the endpoint of $\gamma_{3}$ to the endpoint of $\alpha$.
We would now like to apply Proposition 6.2 to the disk diagram $D^{\prime}\subset D$ with boundary path $\gamma_{1}\gamma_{2}\gamma_{3}\alpha^{\prime\prime}\eta\beta$ with $L=\ell-1$ and $R=\frac{r}{16}$.
We first check that the hypotheses of that proposition hold.
The path $\alpha^{\prime\prime}$ is $\frac{r}{16}$–avoidant with respect to $D^{\prime}$, as it is a subpath of $\alpha$ which is $r$–avoidant with respect to $D$. Thus, (A1) holds.
We now check (A2).
As $\gamma_{1}\gamma_{2}\gamma_{3}=\hat{\omega}\hat{h}_{j}$ is reduced, so is the path $\gamma_{1}\gamma_{2}$, and consequently no dual curve intersects both $\gamma_{1}\gamma_{2}$ and $\gamma_{3}$.
The bounds on $\gamma_{1}$ and $\gamma_{2}$ follow respectively from equation (5) and our choice of $\gamma_{2}$.
We now turn to showing (A3).
To do so, we first define an $\ell$–fence $\mathcal{F^{\prime}}$ in
the subdiagram $D\setminus D^{\prime}$.
If $m>0$, we define $\mathcal{F}^{\prime}$ be the maximal $\ell$–fence in $D\setminus D^{\prime}$ which contains every spoke of $\mathcal{F}$.
As $\mathcal{F}$ contains a spoke which is $L$–fence separated from $\eta$ with respect to $D$, so does $\mathcal{F}^{\prime}$.
Otherwise, if $m=0$, we define $\mathcal{F}^{\prime}$ to be the maximal $\ell$–fence in $D\setminus D^{\prime}$ which contains $\{H_{j},K_{j}\}$.
In this case, $\mathcal{F}^{\prime}$ is $\ell$–fence separated from $\eta$ with respect to $D_{2}$ as $\{H_{j},K_{j}\}\in\mathcal{R}(\mu_{0})$.
Also note that, in either case, $\{H_{j},K_{j}\}$ is contained in $\mathcal{F}^{\prime}$.
Hypothesis (A3) now immediately follows from the second statement in the following claim:
Claim 6.4.
A dual curve intersecting $\gamma_{2}$ must either intersect $\mathcal{F}^{\prime}$ or intersect $z_{j}$.
Additionally, the set $\mathcal{R}(\gamma_{2})$ of spokes intersecting $\gamma_{2}$ which are contained in $\mathcal{F}^{\prime}$ is an $M$–adequate set.
Proof.
By Remark 3.13, there exists an $(\ell-1)$–fence $\mathcal{G}$ connecting $h_{j}\setminus h_{j}^{\prime}$ to $k_{j}$.
As every dual curve which intersects $h_{j}$ (resp. $k_{j}$) must intersect $H_{j}$ (resp. $K_{j}$), it follows that $(\{H_{j},K_{j}\},\mathcal{G})$ is an $\ell$–fence.
By Lemma 3.6 the spokes of $\mathcal{G}$ and the spoke $\{H_{j},K_{j}\}$ are all contained in $\mathcal{F}^{\prime}$.
To show the first claim, consider a dual curve $Q$ intersecting $\gamma_{2}$.
As $\gamma_{2}\subset\hat{h}_{j}^{\prime}$ and as $\hat{h}_{j}^{\prime}$ tracks $h_{j}^{\prime}$, it follows that $Q$ intersects $h_{j}^{\prime}$.
Furthermore, as $h_{j}$ is a reduced path, by considering $Q$ as a dual curve in $D_{j}$, one sees that $Q$ must intersect either $k_{j}$, $z_{j}$ or $\alpha_{j}$.
If $Q$ intersects $k_{j}$, then it intersects $\{H_{j},K_{j}\}$ and so intersects $\mathcal{F}^{\prime}$. On the other hand, if $Q$ intersects $\alpha_{j}$, then by Proposition 3.10, it intersects $\mathcal{G}$ and so intersects $\mathcal{F}^{\prime}$.
The first claim now follows.
To prove the second claim, consider a sequence $\{X_{1},Y_{1}\},\dots,\{X_{q},Y_{q}\}$ of spokes structured with respect to $\gamma_{2}$.
We must show that all but possibly $M$ of these spokes are contained in $\mathcal{F}^{\prime}$.
As $\hat{h}_{j}^{\prime}$ tracks $h_{j}^{\prime}$, these spokes are structured with respect to $h_{j}^{\prime}$.
As $|z_{j}|\leq|V(\Gamma)|$ and as $M\geq|V(\Gamma)|+1$, all but possibly $M$ of the spokes $\{X_{i},Y_{i}\}$ intersect either $\alpha_{j}$ or $k_{j}$.
If $\{X_{i},Y_{i}\}$ intersects $k_{j}$, then it intersects $\{H_{j},K_{j}\}$.
In this case, $(\{X_{i},Y_{i}\},\{H_{j},K_{j}\})$ is a $1$–fence which contains a spoke of $\mathcal{F}^{\prime}$ and so $\{X_{i},Y_{i}\}$ is in $\mathcal{F}^{\prime}$ by Lemma 3.6.
On the other hand, if $\{X_{i},Y_{i}\}$ intersects $\alpha_{j}$, then it intersects $\mathcal{G}$ by Proposition 3.10. Thus, $(\{X_{i},Y_{i}\},\mathcal{G})$ is an $\ell$–fence containing a spoke of $\mathcal{F}^{\prime}$ and again by Lemma 3.6 we deduce that $\{X_{i},Y_{i}\}$ is contained in $\mathcal{F}^{\prime}$.
∎
As the appropriate hypotheses are satisfied, we can now apply Proposition 6.2 to obtain a path $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta\pi$ in $D^{\prime}$ from $b$ to $\alpha^{\prime\prime}$ satisfying the conclusions of that proposition for $L=\ell-1$ and $R=\frac{r}{16}$. As $\gamma_{1}^{\prime}$ tracks $\gamma_{1}$ by (B1), we have that $\gamma_{1}^{\prime}=\hat{\hat{\psi}}\hat{\hat{\mu}}_{0}\dots\hat{\hat{\mu}}_{m}$, and as $\gamma_{2}^{\prime}$ tracks $\gamma_{2}$, we have that $\gamma_{2}^{\prime}=\hat{\hat{h}}_{j}^{\prime}$.
We now set $\psi^{\prime}=\hat{\hat{\psi}}$ and $\mu_{i}^{\prime}=\hat{\hat{\mu}}_{i}$ for each $0\leq i\leq m$.
For all $m<i\leq n$ we set $\mu_{i}^{\prime}$ to be the endpoint of $\mu_{m}^{\prime}$.
Let $\sigma$ be the initial subpath of $\gamma_{2}^{\prime}\zeta$ of length $\lceil f_{\ell}(r)-|\mu_{0}^{\prime}\dots\mu_{n}^{\prime}|\rceil$ which exists by (B1) as $|\gamma_{2}^{\prime}\zeta|\geq 32f_{\ell}(\frac{r}{16})=2f_{\ell}(r)$.
If the endpoint of $\gamma_{2}^{\prime}$ does not lie on $\sigma$, we set $\mu_{n+1}^{\prime}$ to be equal to $\sigma$ and $\mu_{n+2}^{\prime}$ to be the endpoint of $\sigma$. Otherwise, we set $\mu_{n+1}^{\prime}$ to be equal to $\gamma_{2}^{\prime}$ and for $\mu_{n+2}^{\prime}$ to be the subpath of $\zeta$ from the endpoint of $\gamma_{2}^{\prime}$ to the endpoint of $\sigma$.
Finally, we set $\theta^{\prime}$ to be the subpath of $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta\pi$ from the endpoint of $\mu_{n+2}^{\prime}$ to the endpoint of $\pi$.
To conclude the proof, we now check that the conclusions of Proposition 6.3 are satisfied with these choices.
Conclusion (Y1) immediately follows by our choice of $\mathcal{F}^{\prime}$.
We now check (Y2).
The path $\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}$ is reduced as it is a subpath of $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta$ which is reduced by (B1).
No dual curve intersects both $\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}$ and $\theta^{\prime}$ by (B1).
The paths $\psi^{\prime},\mu_{0}^{\prime}\dots,\psi_{n}^{\prime}$ each track $\psi,\mu_{0},\dots,\mu_{n}$ respectively by construction.
Let $Q$ be a dual curve intersecting $\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}$. If $Q$ intersects $\mu_{0}^{\prime}\dots\mu_{n}^{\prime}$, then it intersects $\mu_{0}\dots\mu_{n}$ and so does not intersect $\psi$ as $\psi\mu_{0}\dots\mu_{n}$ is reduced.
If $Q$ intersects $\mu_{n+1}^{\prime}$, then it intersects $\hat{h}_{j}$ and consequently intersects $h_{j}$.
By our choice of $h_{j}$, $Q$ must intersect $H_{33c}$ and so cannot intersect $\psi$.
Finally, if $Q$ intersects $\mu_{n+2}^{\prime}$, then it intersects $\zeta$ and
by (B3) $Q$ either intersects $\hat{\mu}_{0}\dots\hat{\mu}_{m}$ (and consequently also intersects $\mu_{0}\dots\mu_{m}$), it intersects $\alpha$ or it intersects $\hat{h}_{j}$ and consequently $h_{j}$. As before, $Q$ cannot intersect $\psi$. Conclusion (Y2) follows.
We now check conclusion (Y3).
Given $1\leq i\leq m$, (Y3) holds for the spokes intersecting $\mu_{i}^{\prime}$ by (X3), as this path tracks $\mu_{i}$.
Given $m<i\leq n$, (Y3) trivially holds for spokes intersecting $\mu_{i}^{\prime}$ as this is a length $0$ path.
Conclusion (Y3) holds for $\mu_{n+1}^{\prime}=\gamma_{2}^{\prime}$ by the second part of Claim 6.4 and as $\gamma_{2}^{\prime}$ tracks $\gamma_{2}$.
Finally, suppose that we have a sequence of spokes structured with respect to $\mu_{n+2}^{\prime}\subset\zeta$.
By (B2), all but possibly $M$ of these spokes are contained in an $\ell$–fence which contains a spoke of $\mathcal{R}(\gamma_{2})$ (as defined in Claim 6.4).
Thus, by Lemma 3.6 and as $\mathcal{F}^{\prime}$ is maximal, all but possibly $M$ of these spokes are contained in $\mathcal{F}^{\prime}$.
Thus, (Y3) holds.
Before checking (Y4), we first prove that any dual curve intersecting $\mu_{n+2}^{\prime}$, either intersects $\mathcal{F}^{\prime}$ or intersects $z_{j}$.
Let $\zeta^{\prime}$ be the subpath of $\zeta$ from the endpoint of $\mu_{n+2}^{\prime}$ to the endpoint of $\zeta$.
As $|\mu_{n+1}^{\prime}\mu_{n+2}^{\prime}|\leq\lceil f_{\ell}(r)\rceil$, $\gamma_{2}^{\prime}\zeta\geq 2f_{\ell}(r)$, and $\mu_{n+1}^{\prime}\mu_{n+2}^{\prime}$ is an initial subpath of $\gamma_{2}^{\prime}\zeta$, it follows that $|\zeta^{\prime}|\geq f_{\ell}(r)-1$.
By Lemma 4.2 and for $r$ large enough, there is a sequence of $M+1$ spokes structured with respect to $\zeta^{\prime}$.
By conclusion (B2) of Proposition 6.2, one of these spokes is contained in an $\ell$–fence $\mathcal{G}^{\prime}$ which contains a spoke of $\mathcal{R}(\gamma_{2})$.
In particular, by $\mathcal{F}^{\prime}$ being maximal and Lemma 3.6, $\mathcal{G}^{\prime}$ is contained in $\mathcal{F}^{\prime}$.
Now, by conclusion (B3) of Proposition 6.2, any dual curve which intersects $\mu_{n+2}^{\prime}$ must either intersect $\gamma_{2}$ or intersect $\gamma_{3}\cup\alpha$.
In the first case, it intersects $\mathcal{F}^{\prime}$ or $z_{j}$ by the first part of Claim 6.4, and in the second case it intersects $\mathcal{G}^{\prime}$ by Proposition 3.10 and therefore intersects $\mathcal{F}^{\prime}$.
Thus, the claim follows.
We are now ready to check (Y4).
Let $A_{1},B_{1},\dots,A_{p},B_{p}$ be a sequence of dual curves structured with respect to $\mu_{1}^{\prime}\dots\mu_{n+2}^{\prime}$.
By the first part of Claim 6.4, the previous paragraph and as $\mu_{i}^{\prime}$ tracks $\mu_{i}$ for $1\leq i\leq n$, it follows that each $A_{i}$ (resp. $B_{i}$) either intersects $\mathcal{F}^{\prime}$, intersects $\mu_{1}\dots\mu_{n}$ or intersects $z_{j}$.
If $m>0$, as $z_{j}\subset\mu_{m}$, it follows that these dual curves either intersect $\mathcal{F}^{\prime}$ or intersect $\mu_{1}\dots\mu_{n}$ and the claim follows from (X4).
On the other hand, if $m=0$, then $\mu_{1}^{\prime}\dots\mu_{n+2}^{\prime}=\mu_{n+1}^{\prime}\mu_{n+2}^{\prime}$, and it follows from the first part of Claim 6.4 and the previous paragraph that a dual curve intersecting $\mu_{n+1}^{\prime}\mu_{n+2}^{\prime}$ either intersects $\mathcal{F}^{\prime}$ or intersects $z_{j}\subset\mu_{0}$.
As $|z_{j}|\leq V(\Gamma)<M$, the claim also follows in this case.
We now check that (Y5b) holds. The first part of this claim follows as $|\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}|=\lceil f_{\ell}(r)\rceil$ by our choices. The subpath of $\alpha$ between the endpoint of $\gamma_{3}$ and the endpoint of $\pi$ (which is the same as the endpoint of $\theta^{\prime}$) has length $C_{\ell-1}(\frac{r}{16})^{\ell}$ by conclusion (B4) of Proposition 6.2. Thus, (Y5b) follows. This completes the proof for this case.
∎
Before proving the final step, we show how Proposition 6.3 can naturally be iterated.
Lemma 6.5.
Suppose we have integers $L\geq 1$, $M\geq 2$, $R\geq 0$ and a path $\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{m+2}^{\prime}\theta^{\prime}$, with $m+2\leq\frac{f_{L}(R)}{50M^{3}}-1$, and an $L$–fence $\mathcal{F}^{\prime}$ satisfying conclusions (Y1)-(Y4) and (Y5b) of Proposition 6.3 (by setting $n=m$ in that proposition). Then, by setting $\mathcal{F}=\mathcal{F}^{\prime}$, $\psi=\psi^{\prime}$, $\theta=\theta^{\prime}$ and for $1\leq i\leq m+2$ setting $\mu_{i}=\mu_{i}^{\prime}$, it follows that hypotheses (X2)–(X5) of Proposition 6.3 hold for the path $\psi\mu_{0}\dots\mu_{m+2}\theta$ (by setting $n=m+2$ in that proposition).
Proof.
First note that $\mathcal{F}$ contains a spoke which is $L$–fence separated from $\eta$ by (Y1).
We first check (X2). From (Y2) we see that $\psi\mu_{0}\dots\mu_{m+2}$ is reduced and no dual curve intersects both $\psi\mu_{0}\dots\mu_{m+2}$ and $\theta$. That $|\psi|\leq f_{L}(R)$ follows by (Y2) as $\psi$ tracks a path of length at most $f_{L}(R)$. Finally, $|\mu_{0}\dots\mu_{m+2}|\geq f_{L}(R)$ by (Y5b). Thus, (X2) holds.
The claims (X3) and (X4) each immediately follow from (Y3) and (Y4) respectively. Finally, (X5) follows as, by (Y2), $\mu_{0}$ tracks a path satisfying (X5).
∎
Proof of Proposition 6.2 for $L\geq 1$ assuming Proposition 6.3 for $L$..
Let $D$ be a disk diagram with boundary path $\gamma_{1}\gamma_{2}\gamma_{3}\alpha\eta\beta$ satisfying the hypotheses of Proposition 6.2 for some $L\geq 1$, and suppose that Proposition 6.3 holds for $L$.
We show that the conclusions of Proposition 6.2 hold for $D$.
Our strategy will be to use Lemma 6.5 to iteratively apply Proposition 6.3.
At the $n$’th iteration, we show that we have a path satisfying conclusions (B1)–(B3) of Proposition 6.2. Furthermore, we additionally show that either conclusion (B4) holds (and we are done) or, alternatively, the subpath of $\alpha$, as in conclusion (B4), has length at least $C_{L}^{\prime}\frac{n}{2}R^{L}$ where $C_{L}^{\prime}$ is as in Proposition 6.3.
This is enough to prove the proposition, as when $n\geq\frac{f_{L}(R)}{50M^{4}}$, we get that $|\alpha|\geq C_{L}^{\prime}\frac{f_{L}(R)}{100M^{4}}R^{L}\geq CR^{L+1}$, where $C$ depends only on $M$ and $L$.
First we show there exists path $\psi\mu_{0}\theta$ in $D$ satisfying the hypotheses of Proposition 6.3 with $n=0$. This is seen by setting $\psi=\gamma_{1}$, $\mu_{0}=\gamma_{2}$ and $\theta=\gamma_{3}$.
We also take $\mathcal{F}$ to be any spoke intersecting $\gamma_{2}$ which exists for $R$ large enough.
The hypotheses of (X1), (X2) and (X5) of Proposition 6.3 follow respectively from hypotheses (A1), (A2) and (A3) of Proposition 6.2.
Furthermore, hypotheses (X3) and (X4) follow trivially.
Additionally, note that by setting $\gamma_{1}^{\prime}=\psi=\gamma_{1}$, $\gamma_{2}^{\prime}=\mu_{0}=\gamma_{2}$, $\zeta$ to be the endpoint of $\gamma_{2}^{\prime}$ and $\pi=\theta=\gamma_{3}$, we have that (B2) and (B3) are trivially satisfied as $\zeta$ has length $0$. Additionally, (B1) is satisfied by our choices and as $f_{L}(R)\geq 32f_{L+1}(R)$.
Next, suppose that we have a path $\psi\mu_{0}\dots\mu_{n}\theta$ (where $\psi$, $\mu_{0}$ and $\theta$ are possibly different than as in the previous paragraph) in $D$ satisfying the hypotheses of Proposition 6.3.
Furthermore, we suppose that conclusions (B1)–(B3) of Proposition 6.2 hold by setting $\gamma_{1}^{\prime}=\psi$, $\gamma_{2}^{\prime}=\mu_{0}$, $\zeta=\mu_{1}\dots\mu_{n}$ and $\pi=\theta$.
We additionally suppose that the subpath of $\alpha$ from the endpoint of $\gamma_{3}$ to the endpoint of $\theta$ has length $C_{L}^{\prime}\frac{n}{2}R^{L}$ where $C^{\prime}$ depends only on $M$ and $L$.
We apply Proposition 6.3 to obtain a new path $\psi^{\prime}\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}\theta^{\prime}$ satisfying the conclusions of that proposition.
We now show that conclusions (B1)–(B3) of Proposition 6.2 hold by setting $\gamma_{1}^{\prime}=\psi^{\prime}$, $\gamma_{2}^{\prime}=\mu_{0}^{\prime}$, $\zeta=\mu_{1}^{\prime}\dots\mu_{n+2}^{\prime}$ and $\pi=\theta^{\prime}$.
We first check that (B1) holds.
By (Y2) we get that $\gamma_{1}^{\prime}\gamma_{2}^{\prime}\zeta$ is reduced, no dual curve intersects both $\psi\mu_{0}\dots\mu_{n}$ and $\theta$,
$\gamma_{1}^{\prime}$ tracks $\gamma_{1}$ and $\gamma_{2}^{\prime}$ tracks $\gamma_{2}$.
If (Y5a) holds then we immediately get that $|\gamma_{2}^{\prime}\zeta|\geq 32f_{L+1}(R)$.
On the other hand, if (Y5b) holds, then $|\gamma_{2}^{\prime}\zeta|=|\mu_{0}^{\prime}\dots\mu_{n+2}^{\prime}|\geq f_{L}%
(R)\geq 32f_{L+1}(R)$.
Conclusion (B1) now follows.
Furthermore, conclusion (B2) follows from (Y4) and (Y1).
Finally, conclusion (B3) follows directly from the last claim of (Y2).
Now, if (Y5a) holds, then (B4) follows and we are done.
On the other hand, if (Y5b) holds, it then follows that the subpath of $\alpha$ between the endpoint of $\theta$ and the endpoint of $\theta^{\prime}$ has length $C_{L}^{\prime}R^{L}$. Thus, the subpath of $\alpha$ between the endpoint of $\gamma_{3}$ and the endpoint of $\theta^{\prime}$ has length $C_{L}^{\prime}\frac{n}{2}R^{L}+C_{L}^{\prime}R^{L}=C_{L}^{\prime}\frac{n+2}{2}%
R^{L}$.
Additionally, if $n+2\leq\frac{f_{L}(R)}{50M^{4}}-1$, then by Lemma 6.5 we may iterate and apply Proposition 6.3 again.
This completes the proof.
∎
7. Main theorems
In this section, we use Proposition 6.2 from the previous section to prove Theorem C, and the other results from the introduction.
Before proving Theorem C, we first define $\Gamma$-complete words.
These words were first defined in [DT15] and are also utilized in [Lev18].
The periodic geodesic we construct to prove Theorem C will have label the concatenation of $\Gamma$–complete words.
Definition 7.1 ($\Gamma$-complete word).
Let $\Gamma$ be a graph which is not a join.
As $\Gamma$ is not a join, the
complement graph $\Gamma^{c}$ is connected, and it follows that we can choose
a sequence of vertices $s_{0},\dots,s_{n}$ of $\Gamma$ such that
(1)
For every vertex $s\in\Gamma$, $s_{i}=s$ for some $0\leq i\leq n$.
(2)
The vertices $s_{i}$ and $s_{i+1}$ are distinct, non-adjacent vertices of
$\Gamma$ for all $0\leq i\leq n$ (taken mod $n+1$).
Note that it could be
that $s_{i}=s_{j}$ for some $i\neq j$.
We say that $w=s_{0}\dots s_{n}$ is
a $\Gamma$-complete word.
We remark that by Tits’ solution to the word problem (see [Dav08] for instance), $w^{n}$ is a reduced word for all integers $n$.
We now deduce the following lemma from Proposition 3.7 and Proposition 3.10, which will be needed in the proof of Theorem C.
Lemma 7.2.
Suppose that the graph $\Gamma$ has integer hypergraph index $L>0$ and is not a join.
Let $D$ be a disk diagram with boundary path $\gamma\alpha\gamma^{\prime}\beta$ such that $\gamma\cap\alpha$, $\gamma\cap\beta$, $\alpha\cap\gamma^{\prime}$ and $\gamma^{\prime}\cap\beta$ all consist of a single vertex.
Suppose that the label of $\beta$ is a $\Gamma$-complete word, and that every dual curve dual to $\beta$ is also dual to
$\alpha$.
Then $\gamma$ and $\gamma^{\prime}$ are not connected by an $(L-1)$–fence.
Proof.
Suppose for a contradiction that there is an $(L-1)$–fence $\mathcal{F}$ connecting
$\gamma$ and $\gamma^{\prime}$.
Let $w=s_{0}\dots s_{n}$ be the $\Gamma$–complete word which is the label of $\beta$.
By Proposition 3.10, every dual curve dual to $\beta$
intersects a spoke of $\mathcal{F}$.
By Proposition 3.7, the set of vertices $V(\mathcal{F})\cup\{s_{0},,\dots,s_{n}\}$ either induces a strip subgraph of $\Gamma$ or induces a subgraph of hypergraph index at most $L-1$.
However, as $w$ is a $\Gamma$–complete word, every $s\in V(\Gamma)$ is equal to $s_{i}$
for some $i$.
This implies that $V(\Gamma)=V(\mathcal{F})\cup\{s_{0},,\dots,s_{n}\}$. Thus, $\Gamma$ is either a strip subgraph (and has hypergraph index $\infty$) or has hypergraph index at most $L-1$.
In either case we get a contradiction.
∎
We can now prove the theorems from the introduction.
Proof of Theorem C.
First note that we may assume that $\Gamma$ is not a clique, as otherwise $W_{\Gamma}$ is a finite group and has hypergraph index $\infty$. Suppose first that $\Gamma=\Gamma_{1}\star\Gamma_{2}$ is a join.
If $\Gamma_{2}$ is a clique, then $W_{\Gamma}=W_{\Gamma_{1}}\times W_{\Gamma_{2}}$ where $W_{\Gamma_{2}}$ is finite.
In this case, it can readily deduced that $\Gamma_{1}$ and $\Gamma$ have the same hypergraph index
and that the geodesic divergence of a bi-infinite geodesic in the Cayley graph of $W_{\Gamma_{1}}$ is equivalent, under the $\asymp$ equivalence of functions, to the geodesic divergence of this geodesic when considered as a geodesic in the Cayley graph of $W_{\Gamma}$.
Thus, we may assume that $\Gamma_{1}$ and $\Gamma_{2}$ each contain a pair of non-adjacent vertices.
In particular, $\Gamma$ has hypergraph index $0$.
Furthermore, $W_{\Gamma}$ is strongly thick of order $0$ and has linear divergence [BFRHS18][Proposition 2.11].
In particular, any periodic geodesic in $W_{\Gamma}$ has geodesic divergence a linear function, and the claim follows in this case.
By the previous paragraph, we may assume that $\Gamma$ is not a join graph, and we form a $\Gamma$–complete word $w=s_{0}\dots s_{n}$.
Let $\Sigma_{\Gamma}$ be the Davis complex of the RACG $W_{\Gamma}$.
Let $\sigma$ be the
bi-infinite geodesic based at the identity vertex $b\in\Sigma_{\Gamma}$ which has one of its infinite rays emanating from $b$ with label
$www\dots$ and the other ray emanating from $b$ with label $w^{-1}w^{-1}w^{-1}\dots$.
For $i\in\mathbb{Z}$, let $p_{i}$ be the vertex of $\sigma$ which is the endpoint of the subpath of $\sigma$ with starting point $b$ and label $w^{i}$.
Fix an integer $r>0$. Let $B$ be a ball of radius $|w|r$ based at $b\in\Sigma_{\Gamma}$, and let $\nu$ be a path in $\Sigma_{\Gamma}\setminus B$ from $p_{r}$ to $p_{-r}$.
The path $\nu$ exists as $W_{\Gamma}$ is one-ended (since it has integer hypergraph index and consequently is not relatively hyperbolic).
To prove the theorem, it is enough to show that, for $r$ large enough, the length of $\nu$ is bound below by a function $Cr^{k+1}$ for some constant $C$.
For each $0\leq i<r$, let $H_{i}$ be the hyperplane dual to the edge of $\sigma$ which is adjacent to $p_{i}$ and is labeled by $s_{0}$.
As hyperplanes separate $\Sigma_{\Gamma}$ and do not intersect geodesics twice, it follows that $H_{i}$ intersects $\nu$ for each $i$.
Let $\rho_{i}$ be a minimal length geodesic in the carrier $N(H_{i})$ with starting point $p_{i}$ and endpoint on $\nu$.
Let $\nu_{i}$ be the subpath of $\nu$ between $\rho_{i}\cap\nu$ and $\rho_{i+1}\cap\nu$.
As $w$ is a $\Gamma$–complete word, it readily follows that no pair of hyperplanes dual to $\sigma$ intersect.
In particular, we have that $\nu_{i}\cap\nu_{j}=\emptyset$ for all $i\neq j$.
Let $D_{i}$ be the disk diagram with boundary path $\rho_{i}\nu_{i}\rho_{i+1}^{-1}\sigma_{i}^{-1}$ where $\sigma_{i}$ has label $w$.
For each $0\leq i\leq\frac{r}{2}$, we will apply Proposition 6.2 to $D_{i}$ by setting, in that proposition, $\gamma_{1}=p_{i}$ (i.e., a length $0$ path), $\gamma_{2}=\rho_{i}$, $\gamma_{3}$ to be the endpoint of $\rho_{i}$ (also a length $0$ path), $\alpha=\nu_{i}$, $\eta=\rho_{i+1}^{-1}$, $\beta=\sigma_{i}^{-1}$ and $R=|w|(r-i)$.
Hypothesis (A1) holds as $\nu$ does not intersect the ball $B_{p_{0}}(|w|r)$ and so $\nu_{i}$ does not intersect the ball $B_{p_{i}}(|w|(r-i))=B_{p_{i}}(R)$.
Hypothesis (A2) holds as $\gamma_{1}\gamma_{2}\gamma_{3}=\rho_{i}$ is reduced, $|\gamma_{1}|=0$ and $|\gamma_{2}|\geq R$.
By Lemma 7.2, no $(k-1)$–fence connects $\rho_{i}$ to $\rho_{i+1}^{-1}$ in any disk diagram with boundary path $\rho_{i}\nu_{i}\rho_{i+1}^{-1}\sigma_{i}^{-1}$.
Thus, any spoke intersecting $\gamma_{2}$ is $L$–fence separated from $\eta$, and
hypothesis (A3) follows.
Thus, for $r$ large enough, by Proposition 6.2, $|\nu_{i}|\geq C^{\prime}(|w|(r-i))^{k}$ where $C^{\prime}$ depends only on $|V(\Gamma)|$ and $k$. As the $\{\nu_{i}\}$ are disjoint, for $r$ large enough we get:
$$|\nu|\geq\sum_{i=1}^{\big{\lfloor}\frac{r}{2}\big{\rfloor}}|\nu_{i}|\geq\Big{(%
}\frac{r}{2}-1\Big{)}C^{\prime}\Big{(}|w|\big{(}r-\frac{r}{2}\big{)}\Big{)}^{k%
}\geq Cr^{k+1}$$
where $C$ depends only on $|V(\Gamma)|$ and $k$.
∎
Proof of Theorem A.
Suppose first that the hypergraph index of $\Gamma$ is $\infty$.
It then follows that $W_{\Gamma}$ is relatively hyperbolic (see [Lev19]). Thus, $W_{\Gamma}$ has either exponential divergence or infinite divergence [Sis][Theorem 1.3], and $W_{\Gamma}$ is not strongly thick [BDM09].
Thus, we may suppose that $W_{\Gamma}$ has hypergraph index a non-negative integer $k$.
It then follows that $W_{\Gamma}$ is strongly thick of order at most $k$ by Theorem 2.2, and
has divergence function bound above by the function $r^{k+1}$ by Theorem 2.1.
By Theorem C, $W_{\Gamma}$ contains a periodic geodesic with geodesic divergence the function $r^{k+1}$ and, consequently, $W_{\Gamma}$ has divergence function bound below by $r^{k+1}$. Thus, $W_{\Gamma}$ has divergence exactly $r^{k+1}$.
Applying Theorem 2.1 once more, we see that $W_{\Gamma}$ is strongly thick of order exactly $k$.
The theorem now follows.
∎
Proof of Corollary B.
Let $W_{\Gamma}$ be a RACG. If the hypergraph index of $\Gamma$ is $\infty$, then $W_{\Gamma}$ is relatively hyperbolic (see [Lev19]) and has divergence an exponential function if one-ended [Sis][Theorem 1.3] and has infinite divergence if it is finite, infinite-ended or two-ended. Otherwise, the hypergraph index of $\Gamma$ is an integer and $W_{\Gamma}$ has divergence a polynomial function by Theorem A.
∎
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Twisted quasiperiodic textures of biaxial nematics.
V.L. Golo
Department of Mechanics and Mathematics, Lomonosov Moscow
State University
Moscow, Russia, and
National Research University Higher School of Economics
Moscow, Russia
E.I. Kats
Landau Institute for Theoretical Physics
Chernogolvka, Moscow region, Russia
[email protected]
A.A. Sevenyuk
Department of Mechanics and Mathematics, Lomonosov Moscow
State University
Moscow, Russia
D.O. Sinitsyn
Semenov Institute of Chemical Physics
Moscow, Russia
Аннотация
Textures (i.e., smooth space non-uniform distributions of the order parameter) in biaxial nematics turned out to be much more
difficult and interesting than expected. Scanning the literature we find only a very few publications on this topic.
Thus, the immediate motivation of the present paper is to develop a systematic procedure to study, classify and visualize
possible textures in biaxial nematics. Based on the elastic energy of a biaxial nematic (written in the most simple form
that involves the least number of phenomenological parameters) we derive and solve numerically
the Lagrange equations of the first kind. It allows one to visualize the solutions
and offers a deep insight into their geometrical and topological features. Performing Fourier analysis we find some particular
textures possessing two
or more characteristic space periods (we term such solutions quasiperiodic ones because the periods are not necessarily commensurate).
The problem is not only of intellectual interest but also of relevance to
optical characteristics of the liquid-crystalline textures.
biaxial liquid crystals, textures, gyroscopes.
pacs: 61.30.G, 78.66, 45.40
I Introduction.
The combination of orientational order (leading to non-trivial optical response) and
relatively soft elasticity (leading to high sensitivity to boundary conditions) makes liquid crystals very interesting
systems possessing rich variety of textures (topologically stable
smooth configurations of the order parameter).
Although textures and related optical properties of uniaxial nematics are well studied and results of the investigations have numerous applications,
there are still only a few publications on textures in biaxial nematics, i.e. when molecular orientations are arranged regularly
in two directions (see e.g. CP05 , KV09 ). In this work we investigate geometrical and topological
features of textures in biaxial liquid crystals. We show that there are some particular configurations with quasiperiodic
(i.e., possessing two or more generally speaking incommensurate periods) space distribution of the order parameter.
In what follows we always have in mind a liquid crystalline slab placed between two polarizers.
The nematic order parameter (director ${\bf n}$ for uniaxial nematics or three mutually orthogonal
unit vectors ${\bf n}$, ${\bf l}$, and ${\bf m}={\bf n}\times{\bf l}$ for biaxial nematics) is uniform in-plane (independent of
$x$ and $y$) GP93 , PI91 .
However a question of how to identify biaxial nematics is still a problem. One evident approach - optical microscopy- would be to examine the optical properties of the
textures between crossed polarizers.
Dealing with optical properties we assume that the light beam is propagating along the normal to the slab ($z$-axis). In such a situation for uniaxial
nematics one can learn almost everything about coarse grained features of the textures from polarizing microscopy
investigations which give average 2D information integrated
over the path of the light (or to get full 3D information more sophisticated and expensive confocal
polarizing microscopy methods SL10 should be used). Indeed the pattern of transmitted light intensity depends on the difference
of times for ordinary and extraordinary beams to pass through the slab. If the uniaxial director is everywhere parallel to $z$-axis, there is
no transmitted light at all in the crossed polarizers. Therefore non-zero intensity indicates deviations from uniform
alignment (${\bf n}=(0\,,0\,,1)$).
Behavior is so simple because for the uniaxial nematics the optical axis is uniquely
defined by the director (either parallel to the director for prolate permeability tensor, or perpendicular to it
for the oblate tensor). Even more, all physical property tensors contain elements expressed in terms of laboratory axes,
and the axes are simply related to crystallographic Cartesian axes.
Everything is not so simple for biaxial nematics, where all three eigenvalues
of the order parameter tensor (i.e., dielectric permeability tensor) are different.
Although the system of Maxwell’s equations irrespective to the liquid crystal symmetry (uniaxial or biaxial)
can be always described in terms of only two normal modes, the transmitted intensity for biaxial textures can be
zero only for two special configurations, corresponding to the light propagating along so-called biradials LL63 (see more details
in MR93 ).
In their turn the biradial directions depend on the biaxiality and on three unit vectors ${\bf n}$, ${\bf l}$, ${\bf m}$.
These quantities for a generic biaxial nematic texture are
non-trivially oriented with respect to the light propagation direction and distributed over $z$.
We conclude that in biaxial nematics there are intrinsic relations between geometry and topology of the order parameter textures
and optical properties expressed in terms of geometry and topology of the wave or ray surfaces (determined
by Fresnel’s equations LL63 ). It must be frankly admitted that we know practically nothing about optical properties of
biaxial nematic textures, and the first step to get insight on optical properties is to calculate
and then visualize the biaxial order parameter textures.
We believe that not only the study of biaxial nematic textures is an interesting problem in its own right but as well
the basic ideas inspiring our work can be applied to a large variety
of other interesting problems of textures in liquid crystals.
The phenomenological model and the order parameters on which our description is based are introduced in the next section
II. Technical details of our variational approach and computation method are presented in section III.
The main results of our work are summarized in section V. In the conclusion section VI we discuss significance and limitations
of our findings.
II Biaxial nematic: Background.
Biaxial nematics ($N_{b}$) possess three soft (Goldstone) orientational degrees of freedom: two corresponding to the direction
of the long axis of a molecule and one degree for the rotations of the molecule around this axis.
Thus, they have the same freedom of orientations in space as a solid body. Consequently, they should have three optical axes,
and, in comparison with uniaxial nematics, an additional pair of diffuse (liquid-like) X-ray diffraction peaks.
It should be noted that biaxial nematics have been experimentally identified, see saupe - chan3 and also more recent
publications DM06 , VP08 , PD12 ;
an important tool for their investigation being the X-ray scattering, feiser , windle .
The symmetry structure of biaxial nematics requires the use of a matrix order parameter within the framework of the
Landau-de Gennes theory, GP93 - PI91 .
The guidelines to the effect are due to the theory
of uniaxial nematics ($N_{u}$). It was de Gennes who suggested that the order parameter for $N_{u}$ phase may be cast in the form
$$Q_{ij}=S_{u}\cdot\left(n_{i}n_{j}-\frac{1}{3}\delta_{ij}\right)$$
(1)
where $S_{u}$ is the module of the order parameter, whereas $n_{i}$ are coordinates of the director,
that is a vector indicating the average orientation of molecules.
Similarly, the orientation in biaxial nematics relies on the use of a frame structure,
that is the order parameter of the form, which is a generalization of that given by Eq.(1),
$$Q_{ij}=S_{u}(n_{i}n_{j}-\frac{1}{3}\delta_{ij})+S_{b}(m_{i}m_{j}-l_{i}l_{j}),$$
(2)
where $S_{b}$ describes the system’s biaxiality.
Biaxial nematics are studied by employing this frame.
In this paper we shall use a method that is equivalent to the above approach, but provides useful tools for studying spatial inhomogeneities
of the orientational order, or textures. It is to be noted that Eq.(2) indicates that the matrix of the order parameter
${\hat{Q}}$
may be cast in the form
$${\hat{Q}}={\hat{R}}^{-1}{\hat{Q}}_{0}{\hat{R}}$$
(3)
where ${\hat{R}}$ is a rotational $3\times 3$ matrix and matrix ${\hat{Q}}_{0}$ reads
$${\hat{Q}}_{0}=\left(\begin{array}[]{ccc}\lambda_{1}&0&0\\
0&\lambda_{2}&0\\
0&0&-(\lambda_{1}+\lambda_{2})\\
\end{array}\right)$$
(4)
The representation (3) of the order parameter
opens a convenient way for the application of topology to various problems
of field theory and condensed matter, monst2 . Recently, Monastyrskii and Sasorov, monst3 ,
employed the method for studying singularities in biaxial nematics.
It is important that values of the order parameter may change from one region of the volume to another.
In this sense they are local characteristics of the system’s state.
The key point about the study of the phenomena is to employ
the continuum theory by considering the order parameter as a variable that describes the intrinsic structure of the liquid, GP93 .
It can be incorporated in the free energy through appropriate terms that may also describe its spatial variations,
and one can employ the minimization of the free energy for writing down the equation of textures.
To that end we shall write down the part of the free energy related to spatial inhomogeneities in the form
$${\cal F}_{\nabla}=\int F_{\nabla}d^{3}x$$
where the density of the free energy reads
$$F_{\nabla}=K_{1}\partial_{i}Q_{ki}\partial_{j}Q_{kj}+K_{2}\partial_{i}Q_{kj}%
\partial_{i}Q_{kj}+K_{3}\partial_{i}Q_{kj}\partial_{j}Q_{ki}$$
(5)
It is worth noting that we have little knowledge as to the relative size of the coefficients $K_{1},K_{2},K_{3}$ for the biaxial nematics.
It is equally important that, applying integration by parts to the bulk free energy, we may substantially change the form of $F_{\nabla}$
through taking into account appropriate boundary conditions.
The minimization equations for $F_{\nabla}$ given by Eq.(5) are extremely hard to solve analytically.
Therefore we consider a special but very important case of one dimensional textures.
Thus we assume that the order parameter depends only on one spatial coordinate, say $z$.
This approximation can describe textures arising in the commonly used experimental configuration
in which the liquid crystal is placed in a thin layer between two planar uniform glass substrates.
III Variational principle: Technical details.
The gradient part of the free energy for one dimensional textures follows from Eq.(5) by neglecting terms with derivatives in $x_{1}=x,x_{2}=y$,
and preserving only terms in $x_{3}=z$. In what follows we shall use the notation
$$\frac{d}{dz}f=\dot{f}$$
for any function $f$ of z. Thus we obtain the following expression for the gradient energy
$$F_{\nabla}=K_{2}Tr\left({\hat{\dot{Q}}}\cdot{\hat{\dot{Q}}}\right)+(K_{1}+K_{3%
})\left({\hat{\dot{Q}}}\cdot{\hat{\dot{Q}}}\right)_{33}$$
(6)
We have to minimize the functional of the gradient part of the free energy
$${\cal F}=\int\limits_{0}^{z}\,F_{\nabla}\,dz$$
(7)
under the constraint that the symmetric matrix
of the order parameter have fixed eigenvalues, see Eq.(3), and trace zero.
This requirement may be accommodated by imposing the constraints
$$Tr({\hat{Q}})=0;\quad Tr({\hat{Q}}\cdot{\hat{Q}})=const_{1};\quad Tr({\hat{Q}}%
\cdot{\hat{Q}}\cdot{\hat{Q}})=const_{2}$$
(8)
The minimization problem may be solved either by the use of the Lagrangian equations of the 1-st kind, that is using Lagrangian multipliers,
or by resolving the above constraints and employing the Lagrangian equations of the 2-nd kind. To that end we shall use the representation
given by Eq.(3).
The Lagrangian equations of the 1-st kind are more appropriate for numerical simulation. Generally, it is a hard problem to resolve constraints.
It is equally important that the constraints being resolved, the obtained equations depend on the choice of local parameters, that is generalized coordinates,
and therefore subject to some non-evident tricks. For that reason we shall use the equations of the 1-st kind for our numerics, and introduce the effective
Lagrangian
$${\cal L}=F_{\nabla}-\Lambda_{1}\;Tr({\hat{Q}})-\Lambda_{2}\;Tr({\hat{Q}}\cdot{%
\hat{Q}})-\Lambda_{3}\;Tr({\hat{Q}}\cdot{\hat{Q}}\cdot{\hat{Q}})$$
(9)
and solve the minimization problem in accord with the usual rules. It should be noted that we find only extremal solutions, which, generally speaking, minimize the functional only for sufficiently short intervals in $z$.
To describe the change of position of a molecule in space we may employ the skew-symmetric matrix
$${\hat{\Omega}}={\hat{R}}^{-1}\dot{\hat{R}}$$
(10)
where ${\hat{R}}$ is the rotation matrix defining the value of the order parameter, see Eq.(3). Matrix ${\hat{\Omega}}$ is intimately related
to the usual vector of angular velocity, $\bm{\omega}$, through the equation
$$\Omega_{ij}=-\epsilon_{ijk}\,\omega_{k}$$
or in the matrix form
$${\hat{\Omega}}=\sum\limits_{k=1}^{3}\omega_{k}\,f^{k}$$
where the $f^{k}$ are generators of rotations about axes $\hat{x}_{k},\;k=1,2,3$,
and the components $\omega_{k}$ of the angular velocity vector can be expressed in terms of the Euler angles
$\theta,\varphi,\psi$ for the rotation matrix ${\hat{R}}$ by the standard formulas, see LL78 :
$$\begin{array}[]{lcl}\omega_{1}=\dot{\varphi}\sin{\theta}\sin{\psi}+\dot{\theta%
}\cos{\psi},\\
\omega_{2}=\dot{\varphi}\sin{\theta}\cos{\psi}-\dot{\theta}\sin{\psi},\\
\omega_{3}=\dot{\varphi}\cos{\theta}+\dot{\psi}.\end{array}$$
IV Equations of the textures
As explained above, for the numerical analysis, we employ the Lagrange equations of the first kind
derived from the Lagrangian (9),(6), which have the general form:
$$\frac{d}{dt}\frac{\partial\cal L}{\partial\dot{Q}_{ij}}=\frac{\partial\cal L}{%
\partial Q_{ij}}.$$
These equations contain the
Lagrange multipliers $\Lambda_{1},\Lambda_{2},\Lambda_{3}$, which we express in terms of the
elements of ${\hat{Q}}\,,{\hat{\dot{Q}}}$ using the constraints (8)
according to the standard procedure of Lagrange multipliers exclusion (see e.g. Gantm ).
The expressions used along the way get quite involved very quickly,
so an essential component of success in the work with systems of such analytical complexity
is the automation of the work with formulas provided by means of symbolic computation.
After the explicit equations are obtained, approximate solutions can be found numerically.
For describing the possible textures of the system we numerically solve the Cauchy
problem for the Lagrange equations with
a diagonal initial order parameter matrix ${\hat{Q}}(0)$ given by (4)
and a given initial derivative matrix ${\hat{\dot{Q}}}(0)$,
which we compute using the representation (3)
and choosing an arbitrary initial angular velocity vector $\bm{\omega}(0)$.
This guarantees the tangency of the initial velocity to the manifold defined by
the constraints (8).
From here on we use the following notation for the parameters of the Lagrangian:
$$k_{1}=K_{2},\quad k_{2}=K_{1}+K_{3}.$$
It is also worthwhile to note that the terms ‘‘evolution’’, ‘‘velocity’’,
the ‘‘dot’’ derivatives etc. refer in our context to the change of the variables along the Z axis.
V Results of the simulations
V.1 Homogeneous phase
The simplest and trivial arrangement of the orientational order parameters arises in the case of zero initial velocity, in which the
order parameter is constant (for any values of $k_{1},k_{2}$), and the structure is homogeneous.
Correspondingly in this case all biaxial molecules (which can be visualized as identical bricks) have the same orientation.
V.2 Simple Helicoidal phase
The Lagrange equations for the order parameter (for any values of $k_{1},k_{2}$) admit of a partial exact solution of the form:
$${\hat{Q}}(z)={\hat{R}}_{3}^{-1}(\omega z){\hat{Q}}(0){\hat{R}_{3}(\omega z)},$$
(11)
where ${\hat{R}}_{3}(\omega z)$ is the matrix of the three-dimensional rotation about the Z axis through the angle $\omega z$, and ${\hat{Q}}(0)$ is given by (4).
The fact that (11) satisfies the Lagrange equations was checked using
symbolic computation. The angular velocity for this solution does not depend on z:
$${\bm{\omega}}=(0,0,\omega).$$
The eigenvectors of the matrix ${\hat{Q}}(z)$ for this solution have the form:
$$\begin{array}[]{lcl}{\bf l}=(-\cos{\omega z},\sin{\omega z},0),\\
{\bf m}=(\sin{\omega z},\cos{\omega z},0),\\
{\bf n}=(0,0,1).\end{array}$$
Thus, from layer to layer, the directions of the molecule’s axes perform uniform rotation in the XY plane
with the angular velocity $\omega$.
The arrangement of molecules corresponding to this solution is shown
in Fig. 1, A. This phase shares some characteristics with the cholesteric structure,
the evolution of the molecule orientation from layer to layer consisting in
the rotation about the Z axis with the angular velocity $\omega$.
The period of this structure in z (also known as the pitch) has the value
$T={\pi}/{\omega}$
($\pi$ and not $2\pi$ because in non-polar biaxial nematics ${\bf n}$, ${\bf m}$, and ${\bf l}$ are
physically equivalent to ${-\bf n}$, ${-\bf m}$, and ${-\bf l}$ respectively, thus
a rotation through $\pi$ radians transforms an order parameter matrix into itself).
This similarity is a natural consequence of a close connection between chirality (mirror symmetry breaking)
and biaxiality (i.e., additional to uniaxial director ${\bf n}$ ordering in the perpendicular to ${\bf n}$ plane).
In our case chirality is produced externally by applied on the boundary twist ${\hat{\Omega}}$ and biaxiality
is an intrinsic property of the liquid crystalline material (therefore can be arbitrarily large). In conventional cholesterics
(see e.g. HK99 ) chirality is a material property, whereas biaxiality is a geometrical consequence of simple
spiral twist structure and is typically very small.
V.3 Generic quasiperiodic textures
When the initial angular velocity direction differs from the Z axis, a more complicated
solution arises. The molecular arrangement of this type is shown in
Fig. 1, B. One can see that the angular velocity is changing
from layer to layer, and the molecular orientation planes are not parallel.
To study this structure in more detail, we draw the trajectories of the eigenvectors
$\bf l\,,\bf m\,,\bf n$ of the order parameter matrix ${\hat{Q}}$, which correspond to the directions
of the molecules’ axes, see Fig. 2. In the case illustrated
the vector $\bf m$ corresponds to the eigenvalue with the largest magnitute,
depicted in Fig. 1, B by the longest
axis of the parallelepiped representing the molecule.
Its trajectory is shown in Fig. 2, B. One can see the rotation
of this vector about the Z axis (as in the helicoidal phase) combined with an oscillatory motion in the vertical direction.
Thus, in contrast to the helicoidal phase, this texture has more than one characteristic
frequency, exhibiting quasiperiodic structure.
The third eigenvector $\bf n$ corresponds in Fig. 1 to
the shortest axis of a parallelepiped, or, in other words, the normal to the plane of the molecule.
The evolution of this normal is shown in Fig. 2, C.
In the helicoidal structure this vector is directed along the Z axis,
whereas in the quasiperiodic structure the Z direction is only the average
position of $\bf n$, while its momentary values oscillate around this mean orientation.
To throw more light upon the change of the orientational order along $Z$ axis, we also
investigate the evolution of the angular velocity vector for the quasiperiodic solution.
Its trajectory is shown in Fig. 3, A. In the particular case considered,
the 3rd component of the angular velocity doesn’t change its sign, while the other two
components oscillate around zero. This agrees with Fig. 1,
where we see the same main pattern as in the helicoidal phase, that is the rotation of molecules
about the Z axis from layer to layer, but with an additional pattern governing the
precession of the molecular planes. Another picture illustrating this structure is shown in
Fig. 4, where the ends of the angular velocity vectors are depicted, when each
vector is drawn from the location of the corresponding molecule,
i.e. the figure shows the vector $(0\,,0\,,z)+{\bm{\omega}}(z)$.
A segment of this curve resembles the trefoil knot. This confirms that the structure has
more than one characteristic frequency.
A similar curve appears also in the case $k_{2}=0$, Fig. 3, B, Fig. 4, B.
In this case there are additional first integrals corresponding to the rotational symmetry
of the Lagrangian. This property affects the shape of the solution, which nonetheless retains
its quasiperiodic character.
V.4 Spectral properties of the textures.
The characteristic frequencies (in $z$) of the orientation patterns
and the corresponding (generalized) pitches,
i.e. the lengths at which the structure approximately repeats itself, may indicate the wavelengths
of light that correspond to special optical properties of these textures.
It is thus important to study the spectral characteristics of the solutions,
sampling the functions of $z$ with a sufficiently small step in an appropriate interval
and computing the discrete Fourier transform (the spectrum).
As a characteristic quantity we consider the X component $n_{x}$ of the third eigenvector ${\bf n}$ of
the order parameter matrix. Fig. 5 shows this function and a plot of the magnitude of its discrete
Fourier transform.
One can see three major peaks in the spectrum corresponding to three characteristic frequencies
of the texture. This confirms the quasiperiodic nature of the texture under consideration
and indicates the frequencies of light at which special optical properties of the texture
may be expected.
The general structure of the set of characteristic frequencies and their dependence
on the parameters of the system needs further exploration.
The observed peaks may correspond to
linear combinations of a number of basic frequencies,
as is the case in the spectra of quasiperiodic functions.
One of the situations in which quasiperiodic solutions can appear
is the case of integrable, or close to integrable, Lagrange equations.
Thus, the nature of the characteristic frequencies
in this system may be clarified by a further analysis of its dynamical
properties, including the study of the first integrals and possible dimensionality reductions.
VI Conclusions.
In this paper we have presented a theory for orientational textures in biaxial nematic liquid crystals.
The theory combines two parts: the Landau - de Gennes free energy expansion over the gradients of the tensor order
parameter, and numeric solution of the Lagrange equations of the first kind.
The deceptively simple general main message of our work is that description of biaxial nematic textures involves new symmetry
and topology features, admit the Hamiltonian formulation, and could result in new optical phenomena. In particular,
in this paper we develop a systematic numeric procedure to study, classify and visualize
possible textures in biaxial nematics. Based on the elastic energy of a biaxial nematic written in the most simple form
we derive and solve numerically the Lagrange equations of the first kind.
Performing Fourier analysis for the biaxial order parameter texture we find some particular
textures possessing three characteristic space periods (which are not necessarily commensurate). Such a multi-periodic twisted
structure can be considered as a sort of artificial photonic crystal. One can expect very unusual optical characteristics,
e.g., two or three band gaps (unlike one band gap for single periodic cholesteric photonic crystals KA71 , BD79 ).
Our results can be verified either in particle-resolved computer simulations or in optic experiments.
Future work should extend the present study to three spatial dimensions of the textures, which would require
more heavy and sophisticated numerical investigations but promises a reacher plethora of structures
interesting for optical applications.
The results of the present paper allow us to bring forward some hypotheses concerning other physical properties of biaxial nematics.
The fact is that the three directors ${\bf n}\,,{\bf m}\,,{\bf l}$ – the main axes of the orientational
order parameter – are not necessarily the principal axes for all macroscopic second order tensorial physical properties. Such requirement is
neither warranted by a theory, nor imposed by experimental data. A variety of different symmetries are possible depending on the additional symmetry operations for the material under consideration.
Our guess relies mainly on multiperiodical structure of their textures, which was discussed above.
For example the multi-periodic textures found in our work should result in additional pairs of X-ray diffraction peaks.
As a note of caution we should also mention that for the optical or X-ray methods
with a response integrated over the sample thickness, conventional uniaxial nematic textures (where the single optic axis varies in the three-dimensional space) can be confused with a response from a genuine biaxial nematic with two optical axes.
The methods of this paper could be extended to other systems, for example, DNA solutions.
Under appropriate conditions there exist DNA phases that are believed to be cholesteric liquid crystals.
The phases are prepared from segments of the DNA of a size approximately equal to the persistence length, or $50\,nm$. It is important
that the segments do not have the rotational symmetry about their axes owing to the structure of the double helix.
Therefore, one may expect their being strongly biaxial (unlike only weakly biaxial conventional cholesterics
formed by molecules of low molecular mass), see JETPlett .
The theory presented in our work can also have implications not only for optical or X-ray experiment qualitative rationalization.
One might think about various electro-optic and magneto-optic applications of biaxial nematic liquid crystals.
For example the response of the ’’biaxial’’ directors ${\bf m}\,,{\bf l}$ can be much faster than that of the uniaxial
director ${\bf n}$.
Благодарности.V.L.G. acknowledges the support of the Program Progress of Basic Research,
National Research University Higher School of Economics.
E.K. acknowledges the support of the RFBR grant No 13-02-00120 and hospitality of the Issac Newton Institute
for Mathematical Science.
D.S. acknowledges the support of the Government of the Russian Federation grant
for support of research projects implemented by leading scientists at Lomonosov Moscow State University
under the agreement No. 11.G34.31.0054.
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Spectral gaps for hyperbounded operators
Jochen Glück
Jochen Glück, Institute of Applied Analysis, Ulm University, 89069 Ulm, Germany
[email protected]
(Date:: January 14, 2021)
Abstract.
We consider a positive and contractive linear operator $T$ on $L^{p}$ over a finite measure space and prove that, if $TL^{p}\subseteq L^{q}$ for some $q>p$, then the essential spectral radius of $T$ is strictly smaller than $1$. As a special case, we obtain a recent result of Miclo who proved this assertion for self-adjoint operators in the case $p=2$ (under a few additional assumptions). Moreover, we also prove a version of our main theorem on non-finite measure spaces.
Our methods are qualitative in nature. They rely on an ultra power argument and on the fact that an infinite dimensional $L^{p}$-space cannot by isomorphic to an $L^{q}$-space for $q\not=p$.
Key words and phrases:Essential spectral radius; quasi-compactness of positive operators; convergence of semigroups; functions spaces; ultra power techniques; geometry of Banach lattices
2010 Mathematics Subject Classification: Primary 47A10; Secondary: 47B65, 47B38, 47D06, 46E30, 46B08
1. Introduction
Let us consider a linear operator semigroup $(T_{t})_{t\in(0,\infty)}$ on a Banach space. The asymptotic behaviour of the operators $T_{t}$ as the time $t$ tends to $\infty$ is one of the essential questions in the study of such a semigroup. In the present paper we are particularly interested in the question whether the operators $T_{t}$ converge uniformly (i.e. with respect to the operator norm) as $t\to\infty$. In order to prove this kind of behaviour for a given operator semigroup, one always needs some kind of spectral gap property of the operators $T_{t}$. For instance, it is very helpful to know that one of the operators $T_{t}$ is quasi-compact, i.e. that its essential spectral radius is strictly smaller than $1$ (see for instance [8, Section V.3]). One is thus interested in good criteria which ensure quasi-compactness of a linear operator. The aim of the present article is to prove the the Theorems 1.1 and 1.2 below which both give such a criterion for an important class of operators. The first theorem is a special case of the second one.
Theorem 1.1.
Let $1\leq p<q\leq\infty$ and let $(\Omega,\mu)$ be a finite measure space. Let $T$ be a positive linear operator on $L^{p}(\Omega,\mu)$ of norm $\lVert T\rVert\leq 1$. If $TL^{p}(\Omega,\mu)\subseteq L^{q}(\Omega,\mu)$, then the essential spectral radius of $T$ is strictly smaller than $1$.
Operators on $L^{p}$ whose range is contained in $L^{q}$ for some $q>p$ are sometimes called hyperbounded. This explains the title of the paper.
A few notes on the terminology used in the theorem are in order. We call an operator on $L^{p}(\Omega,\mu)$ positive if $Tf\geq 0$ for every function $f\geq 0$. Moreover, recall that the essential spectral radius of a bounded linear operator $T$ on a Banach space $E$ is defined to be the spectral radius of $T$ in the Calkin algebra $\mathcal{L}(E)/\mathcal{K}(E)$, where $\mathcal{L}(E)$ denotes the Banach algebra of all bounded linear operators on $E$ and $\mathcal{K}(E)$ denotes the ideal of all compact operators in $\mathcal{L}(E)$. The essential spectral radius of $T$ is strictly smaller than $1$ if and only if every spectral value of $T$ which has modulus at least $1$ is a pole of the resolvent of $T$ and the corresponding spectral projection has finite-dimensional range. Another equivalent condition is that there exists an integer $n\in\mathbb{N}$ and a compact operator $K$ on $E$ such that $\lVert T^{n}-K\rVert<\infty$; this is why operators with essential spectral radius $<1$ are sometimes called quasi-compact.
Our second theorem is a bit more general version of the first one; here we consider $L^{p}$ and $L^{q}$ over different, possibly infinite measure spaces and assume that $L^{q}$ is embedded as a sublattice in $L^{p}$. We also allow the case $q<p$ in this general situation.
Theorem 1.2.
Let $p,q\in[1,\infty]$ be two distinct numbers and let $(\Omega_{1},\mu_{1})$ and $(\Omega_{2},\mu_{2})$ be two arbitrary measure spaces. Moreover, let $j:L^{q}:=L^{q}(\Omega_{2},\mu_{2})\to L^{p}:=L^{p}(\Omega_{1},\mu_{1})$ be an injective lattice homomorphism (i.e. $\lvert j(f)\rvert=j(\lvert f\rvert)$ for all $f\in L^{q}$).
If $T$ is a positive linear operator on $L^{p}$ of norm $\lVert T\rVert\leq 1$ and if $TL^{p}\subseteq j(L^{q})$, then the essential spectral radius of $T$ is strictly smaller than $1$.
We point out that the lattice homomorphism $j$ is of course positive, and thus automatically continuous [29, Theorem II.5.3]. The generality of Theorem 1.2 is perhaps best illustrated by the following corollary for (unbounded) subsets of $\mathbb{R}^{d}$:
Corollary 1.3.
Let $\Omega$ be a Borel measurable subset of $\mathbb{R}^{d}$, endowed with the $d$-dimensional Lebesgue measure $\lambda$, and let $1\leq p<\infty$. Let $T$ be a positive linear operator on $L^{p}(\Omega,\lambda)$ of norm $\lVert T\rVert\leq 1$ and assume that there exist numbers $\varepsilon_{1},\varepsilon_{2}>0$ such that
$$\displaystyle\int_{\Omega}\lvert Tf(x)\rvert^{p}(1+\lvert x\rvert)^{%
\varepsilon_{2}}\;\mathrm{d}\lambda(x)<\infty$$
and
$$\displaystyle\int_{\Omega}\lvert Tf(x)\rvert^{p+\varepsilon_{1}}(1+\lvert x%
\rvert)^{\varepsilon_{2}}\;\mathrm{d}\lambda(x)<\infty$$
for each $f\in L^{p}(\Omega,\lambda)$. Then the essential spectral radius of $T$ is strictly smaller than $1$.
We demonstrate at the end of the introduction how Corollary 1.3 can be derived from Theorem 1.2.
Assume for a moment that the Borel measurable set $\Omega\subseteq\mathbb{R}^{d}$ is bounded; then Corollary 1.3 becomes a special case of Theorem 1.1; conversely, it is not difficult to deduce Theorem 1.1 from Corollary 1.3 in case that the space $\Omega$ in the theorem is a bounded and Borel measurable subset of $\mathbb{R}^{d}$ and $\mu$ is the Lebesgue measure.
Obviously, one can also apply Theorem 1.2 to operators on $\ell^{p}$ whose range is contained in $\ell^{q}$ for some $q<p$. In this case, however, the assertion of the theorem is an immediate consequence of a much more general of result of Pitt which asserts that, for $1\leq q<p<\infty$, every bounded linear operator from $\ell^{p}$ to $\ell^{q}$ is compact (the case $p=\infty$ is not included in Pitt’s theorem, but this case is rather simple anyway; compare Subsection 2.1). For a thorough discussion of Pitt’s theorem and possible generalisations we refer the reader to the Appendix of [28].
Historical remarks and related literature
For self-adjoint operators on $L^{2}$ Theorem 1.1 was recently proven – under a few additional assumptions – by Miclo [21, Theorem 1]; this solved a long open conjecture of Simon and Høegh-Krohn [30]; the case of self-adjoint operators on $L^{2}$ is, of course, of special importance in mathematical physics. Of all the related articles that appeared during the last decades, let us mention the works [1, 16, 17, 13] which impose a strengthened positivity assumption on the operator, the paper [33] which assumes explicit numerical bounds for the operator norm of $T$ from $L^{p}$ to $L^{q}$ and the recent article [34] where the technique used by Miclo is further developed.
For the general asymptotic theory of operator semigroups we refer to [8, Chapter V], [3, Chapter 5] and to the monograph [32]. For the long term behaviour of, in particular, positive semigroups we refer to the [2, 7] and to the recent monograph [4]. In addition to the references mentioned above, some recent contributions to the asymptotic theory of positive semigroups include the article [22] which has its focus on spectral theory and growth fragmentation equations, the paper [10] on so-called lower bound methods for semigroups on $L^{1}$, the articles [25, 11, 27, 26, 9] which all deal with semigroups that dominate integral operators, and the work [23] which considers perturbed semigroups on $L^{1}$-spaces and which is related to the aforementioned series of articles.
Preliminaries
We assume the reader to be familiar with the basic theory of real and complex Banach lattices; standard references for this theory are, for instance, the monographs [29, 36, 20]. Here we only recall the basic terminology that a linear operator $T$ on a Banach lattice $E$ is called positive if $Tf\geq 0$ for each $0\leq f\in E$. The reader is also assumed to be familiar with standard spectral theory for linear operators on Banach spaces; for a detailed treatment we refer, for instance, to the spectral theory chapters in the monographs [31, 18, 35]. If $T$ is a bounded linear operator on a complex Banach space $E$ and if $\lambda\in\mathbb{C}$ is not a spectral value of $T$, then we denote the resolvent of $T$ at $\lambda$ by $\mathcal{R}(\lambda,T):=(\lambda-T)^{-1}$.
We make extensive use of ultra power arguments; the most important facts about the construction of ultra powers are briefly recalled at the beginning of Subsection 2.3; for a detailed treatment of ultra powers and ultra products of Banach spaces we refer the reader to the survey article [15].
Throughout the paper, all occurring Banach spaces and Banach lattices are assumed to be defined over the complex scalar field. All measure spaces in the paper are allowed to be non-$\sigma$-finite unless otherwise stated.
Organisation of the article
In the remaining part of the introduction we prove Corollary 1.3. In Subsection 2.1 we first demonstrate that our main theorems are not particularly surprising if one of the numbers $p$ and $q$ is either $1$ or $\infty$. The rest of Section 2 is then devoted to the proofs of our main results in the non-trivial case $p,q\in(1,\infty)$. We briefly discuss the consequences of our results for the long term behaviour of positive operator semigroups in Section 3, and we conclude the paper with an open problem in Section 4.
A proof of Corollary 1.3
On this subsection we show how Corollary 1.3 can be derived from Theorem 1.2.
Proof of Corollary 1.3.
Let us define $\delta:\Omega\to\mathbb{R}$ by $\delta(x)=1+\lvert x\rvert$ for all $x\in\Omega$. Choose $q\in(p,\infty)$ sufficiently close to $p$ such that $q\leq p+\varepsilon_{1}$ and $d(\frac{q}{p}-1)<\frac{\varepsilon_{2}}{2}$. From $p<q\leq p+\varepsilon_{1}$ it follows by interpolation that
$$\displaystyle\int_{\Omega}\lvert Tf\rvert^{q}\,\delta^{\varepsilon_{2}}\;%
\mathrm{d}\lambda<\infty$$
for each $f\in L^{p}(\Omega,\lambda)$. Now we choose a real number $\alpha$ which strictly larger than $d(1-\frac{p}{q})$ but strictly smaller than $d(1-\frac{p}{q})+\frac{\varepsilon_{2}}{2}\frac{p}{q}$. Then we have
$$\displaystyle d(1-\frac{p}{q})<\alpha\qquad\text{and}\qquad\frac{q}{p}\alpha<%
\varepsilon_{2},$$
hence, $\int_{\Omega}\lvert Tf\rvert^{q}\,\delta^{\frac{q}{p}\alpha}\;\mathrm{d}%
\lambda<\infty$ for each $f\in L^{p}(\Omega,\lambda)$.
Set $r:=q/p\in(1,\infty)$ and choose $r^{\prime}\in(1,\infty)$ such that $1/r+1/r^{\prime}=1$ (i.e. $r^{\prime}=\frac{q}{q-p}$); moreover, let $\mu$ denote the measure on the Borel $\sigma$-algebra over $\Omega$ which has density $\delta^{\alpha r}$ with respect to the Lebesgue measure, i.e. $\;\mathrm{d}\mu=\delta^{\alpha r}\;\mathrm{d}\lambda$.
For every function $f\in L^{q}(\Omega,\mu)$ it follows from Hölder’s inequality that
$$\displaystyle\int_{\Omega}\lvert f\rvert^{p}\;\mathrm{d}\lambda=\int_{\Omega}%
\lvert f\rvert^{p}\,\delta^{\alpha}\,\frac{1}{\delta^{\alpha}}\;\mathrm{d}%
\lambda\leq\left(\int_{\Omega}\lvert f\rvert^{pr}\delta^{\alpha r}\;\mathrm{d}%
\lambda\right)^{1/r}\;\left(\int_{\Omega}\delta^{-\alpha r^{\prime}}\;\mathrm{%
d}\lambda\right)^{1/r^{\prime}},$$
and hence
$$\displaystyle\lVert f\rVert_{L^{p}(\Omega,\lambda)}\leq\lVert f\rVert_{L^{q}(%
\Omega,\mu)}\;\left(\int_{\Omega}\delta^{-\alpha r^{\prime}}\;\mathrm{d}%
\lambda\right)^{\frac{1}{r^{\prime}p}}$$
We chose $\alpha$ to be strictly larger than $d\frac{q-p}{q}$, hence we have $\alpha r^{\prime}>d$. This implies that $\int_{\Omega}\delta^{-\alpha r^{\prime}}\;\mathrm{d}\lambda<\infty$, so we have checked that $L^{q}(\Omega,\mu)$ continuously embeds into $L^{p}(\Omega,\lambda)$ (and obviously, the embedding is a lattice homomorphism). Moreover, we have already noted above that $\int_{\Omega}\lvert Tf\rvert^{q}\,\delta^{\alpha r}\;\mathrm{d}\lambda<\infty$ for all $f\in L^{p}(\Omega,\lambda)$, so the range of $T$ is contained in $L^{q}(\Omega,\mu)$. Thus, the assertion follows from Theorem 1.2.
∎
2. Main arguments
2.1. The end points of the $L^{p}$-scale
We first consider the cases $p\in\{1,\infty\}$ and $q\in\{1,\infty\}$ in Theorem 1.2. In this cases, the theorems are much simpler since we can show that a power of $T$ is compact under the given assumptions (and in fact, we do not even need the contractivity nor the positivity $T$ for that).
We need the following observations from Dunford–Pettis theory: let $T$ be a bounded linear operator between two Banach spaces $E$ and $F$. Recall that $T$ is called weakly compact if it maps the closed unit ball in $E$ to a relatively weakly compact subset of $E$. Moreover, $T$ is said to be a Dunford–Pettis operator or to be completely continuous if, for every sequence $(x_{n})$ in $E$ which converges weakly to a vector $x\in E$, the sequence $(Tx_{n})$ in $F$ converges in norm to the vector $Tx$; equivalently, $T$ maps weakly compact subsets of $E$ to norm-compact (equivalently: relatively norm-compact) subsets of $F$. Every compact operator between $E$ and $F$ is a Dunford–Pettis operator, and the converse is true if $E$ is reflexive; thus, Dunford–Pettis operators are particularly interesting on non-reflexive Banach spaces. We point out that, if $E,F,G$ are Banach spaces, $T:E\to F$ is weakly compact and $S:F\to G$ is a Dunford–Pettis operator, then $ST:E\to G$ is compact. We will make repeated use of this simple observation in the proofs of the two subsequent propositions.
Now, let $(\Omega,\mu)$ be an arbitrary measure space. Then the spaces $E=L^{1}(\Omega,\mu)$ and $E=L^{\infty}(\Omega,\mu)$ are so-called Dunford–Pettis spaces, i.e. every weakly compact linear operator from $E$ to any Banach space $F$ is a Dunford–Pettis operator; see [20, Proposition 3.7.9].
The following proposition proves Theorem 1.2 (and much more) if $p\in\{1,\infty\}$ or $q\in\{1,\infty\}$. Although the result follows from rather standard arguments from Dunford–Pettis theory, we include the proof for the convenience of the reader.
Proposition 2.1.
Let $p,q\in[1,\infty]$ and let $(\Omega_{1},\mu_{1})$ and $(\Omega_{2},\mu_{2})$ be two arbitrary measure spaces. Moreover, let $j:L^{q}:=L^{q}(\Omega_{2},\mu_{2})\to L^{p}:=L^{p}(\Omega_{1},\mu_{1})$ be an injective lattice homomorphism
Let $T$ be a bounded linear operator on $L^{p}$ and assume that $TL^{p}\subseteq j(L^{q})$.
(a)
If $q\in(1,\infty]$ and $p=1$, then $T^{2}$ is compact.
(b)
If $q\in[1,\infty)$ and $p=\infty$, then $T^{2}$ is compact.
(c)
If $p\in(1,\infty)$ and $q\in\{1,\infty\}$, then $T$ is compact.
Proof.
We first observe that, in any case, $j^{-1}T:L^{p}\to L^{q}$ is continuous due to the closed graph theorem.
(a) Let us first show that $T:L^{1}\to L^{1}$ is weakly compact. Consider the case $q\not=\infty$ first. Then $j^{-1}T$ is even weakly compact since $L^{q}$ is reflexive. Since the embedding $j$ is continuous, we conclude that $T=jj^{-1}T$ is weakly compact. Now consider the case $q=\infty$. Then $j$ maps the unit ball of $L^{q}=L^{\infty}$ into a subset of $L^{1}$ of the form $J+iJ$, where $J$ is an order interval in $L^{1}$. But order intervals in $J$ are weakly compact as $L^{1}$ has order continuous norm (see [20, Theorem 2.4.2(i) and (vi)]), so $j$ is weakly compact and hence, so is $T=jj^{-1}T$.
We have thus proved that $T$ is weakly compact. In particular, $T$ is a Dunford–Pettis operator as $L^{1}$ is a Dunford–Pettis space and hence, $T^{2}$ is compact (as a composition of a weakly compact operator with a Dunford–Pettis operator).
(b) The mapping $j^{-1}T:L^{\infty}\to L^{q}$ maps the unit ball of $L^{\infty}$ into a subset of $L^{q}$ of the form $J+iJ$, where $J$ is an order interval in $L^{q}$. Since $L^{q}$ has order continuous norm, we again conclude that order intervals in $L^{q}$ are weakly compact [20, Theorem 2.4.2(i) and (vi)]. Hence, $j^{-1}T$ is weakly compact and thus, so is $T=jj^{-1}T$. As $L^{\infty}$ is a Dunford–Pettis space, we conclude that $T$ is a Dunford–Pettis operator, so $T^{2}$ is compact (as a composition of a weakly compact operator with a Dunford–Pettis operators).
(c) As $L^{q}$ is reflexive, the mapping $j^{-1}T:L^{p}\to L^{q}$ is weakly compact and hence a Dunford–Pettis operator, as $L^{p}$ is a Dunford–Pettis space (since $p\in\{1,\infty\}$). Moreover, the reflexivity of $L^{q}$ also implies that the embedding $j:L^{q}\to L^{p}$ is weakly compact. Hence, $T=jj^{-1}T$ is compact.
∎
The proof of assertion (c) in the above proposition is actually a special case of the more general (and well-known) observation that a bounded linear operator on a Dunford–Pettis space which factorises through a reflexive space is compact. The arguments from Dunford–Pettis theory used in the above proofs can be put in a more general context if one considers so-called principle ideals in Banach lattices; this is explained in detail in [6, Section 2] and in [5, Section 2].
2.2. Dimension of the fixed space
Our first ingredient to the proof of Theorem 1.2 in the case $p,q\in(1,\infty)$ is the following proposition. It says that, in the situation of the theorem, the fixed space of $T$ is finite dimensional; by the fixed space of $T$ we mean the closed vector subspace of $E$ given by $\operatorname{fix}T:=\ker(1-T)$.
Proposition 2.2.
Let $p,q\in(1,\infty)$ be two distinct numbers and let $(\Omega_{1},\mu_{1})$ and $(\Omega_{2},\mu_{2})$ be two arbitrary measure spaces. Moreover, let $j:L^{q}:=L^{q}(\Omega_{2},\mu_{2})\to L^{p}:=L^{p}(\Omega_{1},\mu_{1})$ be an injective lattice homomorphism.
If $T$ is a positive linear operator on $L^{p}$ of norm $\lVert T\rVert\leq 1$ and if $TL^{p}\subseteq j(L^{q})$, then $\operatorname{fix}T$ is finite dimensional.
For the case of self-adjoint operators on $L^{2}$, a version of Proposition 2.2 (under the assumption that $(\Omega_{1},\mu_{1})=(\Omega_{2},\mu_{2})$ is a finite measure space) has already been proved by Gross as a part Theorem 1 in [14]; in this reference, an explicit bound of the dimension of $\operatorname{fix}T$ in terms of the operator norm of $T$ as an operator from $L^{2}$ to $L^{q}$ is given.
For the proof of Proposition 2.2 we need the following simple lemma. We call a vector subspace $F$ of a Banach lattice $E$ a sublattice of $E$ if $\lvert f\rvert\in F$ for each $f\in F$.
Lemma 2.3.
Let $1<p<\infty$, let $(\Omega,\mu)$ be an arbitrary measure space and let $T$ be a positive linear operator on $L^{p}(\Omega,\mu)$ of norm at most $1$. Then $\operatorname{fix}T$ is a sublattice of $L^{p}(\Omega,\mu)$.
Proof.
Let $f\in\operatorname{fix}T$. Then we have $T\lvert f\rvert\geq\lvert Tf\rvert=\lvert f\rvert$. On the other hand, the norm of $T\lvert f\rvert$ is not larger than the norm of $\lvert f\rvert$ since $T$ is contractive. Thus, $T\lvert f\rvert=\lvert f\rvert$.
∎
At the end of the above argument we used the fact that the norm on $L^{p}$ is strictly monotone, i.e. if we have $0\leq f\leq g$ for two distinct vectors $f$ and $g$ in $L^{p}$, then $\lVert f\rVert<\lVert g\rVert$. This is also true for $p=1$, so the above lemma remains true on $L^{1}$ (but not on $L^{\infty}$). Now we can prove Proposition 2.2.
Proof of Proposition 2.2.
First note that $j^{-1}T:L^{p}\to L^{q}$ is continuous by the closed graph theorem. In particular, the operator $S:=j^{-1}Tj:L^{q}\to L^{q}$ is continuous. Moreover, a vector $f\in L^{q}$ is in the fixed space of $S$ if and only if $j(f)$ is in the fixed space of $T$, i.e. $\operatorname{fix}S=j^{-1}(\operatorname{fix}T)$.
The spaces $\operatorname{fix}T$ and $\operatorname{fix}S$ are closed in $L^{p}$ and $L^{q}$, respectively. Moreover, it follows from Lemma 2.3 that $\operatorname{fix}T$ is a sublattice of $L^{p}$. Since $j$ is a lattice homomorphism, this implies that $\operatorname{fix}S$ is a sublattice of $L^{q}$. Now it follows from Kakutani’s representation theorem for abstract $L^{p}$-spaces [20, Theorem 2.7.1] that $\operatorname{fix}T$, with the norm induced by $L^{p}$, is itself isometrically lattice isomorphic to an $L^{p}$-space over some measure space, and likewise it follows that $\operatorname{fix}S$, with the norm induced by $L^{q}$, is isometrically lattice isomorphic to an $L^{q}$-space over some measure space.
Yet, the mapping $j|_{\operatorname{fix}S}:\operatorname{fix}S\to\operatorname{fix}T$ is bijective and a lattice homomorphism, hence a lattice isomorphism. As $p\not=q$, Proposition A.1 in the Appendix shows that this can only be true of $\operatorname{fix}T$ is finite dimensional.
∎
2.3. Ultra powers
Let us briefly recall the concept of an ultra power of a Banach space $E$. Fix a free ultra filter $\mathcal{U}$ on $\mathbb{N}$, endow the $E$-valued sequence space $\ell^{\infty}(E)$ with its canonical norm $\lVert z\rVert:=\sup_{n\in\mathbb{N}}\lVert z_{n}\rVert_{E}$ for $z=(z_{n})_{n\in\mathbb{N}}\in\ell^{\infty}(E)$, and define
$$\displaystyle c_{0,\mathcal{U}}(E):=\{z\in\ell^{\infty}(E):\;\lim_{n\to%
\mathcal{U}}\lVert z_{n}\rVert_{E}=0\}.$$
Then $\ell^{\infty}(E)$ is a Banach space and $c_{0,\mathcal{U}}(E)$ is a vector subspace of it. The quotient space
$$\displaystyle E^{\mathcal{U}}:=\ell^{\infty}(E)/c_{0,\mathcal{U}}(E)$$
is called the ultra power of $E$ with respect to the ultra filter $\mathcal{U}$. For each $z=(z_{n})_{n\in\mathbb{N}}\in\ell^{\infty}(E)$ we use the notation $z^{\mathcal{U}}$ for the equivalence class of $z$ in $E^{\mathcal{U}}$. Moreover, for $x\in E$ we use the notation $x^{\mathcal{U}}$ for the equivalence class of the constant sequence $(x)_{n\in\mathbb{N}}\in\ell^{\infty}(E)$ in $E^{\mathcal{U}}$. The mapping $E\ni x\mapsto x^{\mathcal{U}}\in E^{\mathcal{U}}$ is isometric, and via this mapping we may consider $E$ as a closed subspace of $E^{\mathcal{U}}$. For every $z\in\ell^{\infty}(E)$ we can compute the norm of $z^{\mathcal{U}}$ in $E^{\mathcal{U}}$ by means of the formula $\lVert z^{\mathcal{U}}\rVert=\lim_{n\to\mathcal{U}}\lVert z_{n}\rVert_{E}$.
If $E$ is a Banach lattice, then so is $\ell^{\infty}(E)$, and then the space $c_{0,\mathcal{U}}(E)$ is a closed ideal in $\ell^{\infty}(E)$. Thus, the ultra power $E^{\mathcal{U}}$ is a Banach lattice, too, and the embedding $E\ni x\mapsto x^{\mathcal{U}}\in E^{\mathcal{U}}$ is an isometric lattice homomorphism in this case. The formula $\lVert z^{\mathcal{U}}\rVert=\lim_{n\to\mathcal{U}}\lVert z_{n}\rVert_{E}$ for $z\in\ell^{\infty}(E)$ implies that, if $E$ is an $L^{p}$-space over some measure space for $p\in[1,\infty)$, then $E^{\mathcal{U}}$ is an abstract $L^{p}$-space and thus isometrically lattice isomorphic to a concrete $L^{p}$-space by means of Kakutani’s representation theorem [20, Theorem 2.7.1].
Let $E,F$ be Banach spaces. Every bounded linear operator $T:E\to F$ can be canonically extended to a bounded linear operator $T^{\mathcal{U}}:E^{\mathcal{U}}\to F^{\mathcal{U}}$ which is given by $T^{\mathcal{U}}z^{\mathcal{U}}=(Tz_{n})^{\mathcal{U}}$ for each $z=(z_{n})\in\ell^{\infty}(E)$. If $E=F$, then the mapping $T\mapsto T^{\mathcal{U}}$ is an isometric and unital Banach algebra homomorphism from the space of all bounded linear operators on $E$ into the space of all bounded linear operators on $E^{\mathcal{U}}$. If $E$ and $F$ are Banach lattices, then $T$ is positive if and only if $T^{\mathcal{U}}$ is positive; likewise, $T$ is a lattice homomorphism if and only if $T^{\mathcal{U}}$ is a lattice homomorphism.
Ultra products are an important tool in operator theory. One of their most useful properties is that that lifting $T\mapsto T^{\mathcal{U}}$ improves the behaviour of certain parts of the spectrum of $T$ without changing the spectrum as a whole; see for instance [20, Theorem 4.1.6]. Another useful property is that information about the operator $T^{\mathcal{U}}$ can sometimes be used to deduce stronger information about the original operator $T$. Here is an example of this phenomenon:
Proposition 2.4.
Let $E$ be a reflexive Banach space, let $T$ be a power-bounded linear operator on $E$ (i.e. $\sup_{n\in\mathbb{N}_{0}}\lVert T^{n}\rVert<\infty$), and let $\mathcal{U}$ be a free ultra filter on $\mathbb{N}$. Assume that $\operatorname{fix}(T^{\mathcal{U}})$ is finite dimensional. Then $\operatorname{fix}T$ is finite dimensional, too, and the number $1$ is a pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$ of order at most $1$.
Here we use the convention that a number $\lambda_{0}\in\mathbb{C}$ is a pole of $\mathcal{R}(\mathord{\,\cdot\,},T)$ of order $0$ iff $\lambda_{0}$ is not contained in the spectrum of $T$.
Proof of Proposition 2.4.
Clearly, every fixed vector of $T$ is mapped to a fixed vector of $T^{\mathcal{U}}$ by the injective mapping $E\ni x\mapsto x^{\mathcal{U}}\in E^{\mathcal{U}}$, so $\operatorname{fix}T$ is finite dimensional, too.
As $E$ is reflexive and $T$ is power bounded, the Cesàro means of the powers of $T$ converge strongly to a projection $P$ on $E$ which commutes with $T$ and whose range coincides with the fixed space of $T$. In particular, $T$ leaves the range and the kernel of $P$ invariant, and it acts as the identity mapping on the range of $P$. In order to show that the number $1$ is a zero or first order pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},T)$, it thus suffices to show that $1$ is not a spectral value of the restricted operator $T|_{\ker P}$. So assume for a contradiction that $1$ is a spectral value of $T|_{\ker P}$.
We first note that $1$ is not an eigenvalue of $T|_{\ker P}$. However, as $T|_{\ker P}$ is power bounded, its spectral radius cannot be larger than $1$, so $1$ is contained in the topological boundary of the spectrum of $T|_{\ker P}$. In particular, $1$ is an approximate eigenvalue of $T|_{\ker P}$, i.e. there exists a sequence $(x_{n})_{n\in\mathbb{N}}$ in $\ker P$ such that $\lVert x_{n}\rVert=1$ for all indices $n$ and such that $(1-T|_{\ker P})x_{n}=(1-T)x_{n}\to 0$ as $n\to\infty$. Now we use an argument taken from [2, Lemma C-III-3.10]:
If a subsequence of $(x_{n})_{n\in\mathbb{N}}$ converges to a vector $x$, then $x$ is obviously an eigenvector of $T|_{\ker P}$ for the eigenvalue $1$; hence, the sequence $(x_{n})_{n\in\mathbb{N}}$ has no convergent subsequence. In particular, the set $\{x_{n}:\;\in\mathbb{N}\}\subseteq\ker P$ is not pre-compact, so there exists an $\varepsilon>0$ such that this set cannot be covered by finitely many balls of radius $\varepsilon$. Therefore, we can find a subsequence $(y_{n})_{n\in\mathbb{N}}$ of $(x_{n})_{n\in\mathbb{N}}$ such that $\lVert y_{n}-y_{m}\rVert_{E}\geq\varepsilon$ for all distinct $m,n\in\mathbb{N}$.
For each $k\in\mathbb{N}_{0}$ we define $y^{(k)}:=\big{(}(y_{n+k})_{n\in\mathbb{N}}\big{)}^{\mathcal{U}}\in E^{\mathcal%
{U}}$. Then $y^{(k)}$ is a fixed vector of $T^{\mathcal{U}}$ and has norm $\lVert y^{(k)}\rVert=1$. However, for $j\not=k$ we obtain
$$\displaystyle\lVert y^{(k)}-y^{(j)}\rVert=\lim_{n\to\mathcal{U}}\lVert y_{n+k}%
-y_{n+j}\rVert_{E}\geq\varepsilon.$$
Thus, the sequence $(y^{k)})_{k\in\mathbb{N}_{0}}$ in the unit ball of $\operatorname{fix}(T^{\mathcal{U}})$ has no convergent subsequence, so $\operatorname{fix}(T^{\mathcal{U}})$ is infinite dimensional.
∎
2.4. Proofs of the main results
Now we can prove Theorem 1.2 (and thus also Theorem 1.1). The case where at least one of the numbers $p$ and $q$ is $1$ or $\infty$ has already been treated in Subsection 2.1, so we are only left to deal with the case $p,q\in(1,\infty)$ here.
Proof of Theorem 1.2 in the case $p,q\in(1,\infty)$.
If the spectral radius of $T$ is strictly smaller than $1$, there is nothing to prove, so we assume throughout the proof that $T$ has spectral radius $1$. In particular, $1$ is a spectral value of $T$ since $T$ is positive [29, Proposition V.4.1].
Fix a free ultra filter $\mathcal{U}$ on $\mathbb{N}$. We first show that the fixed space of the operator $T^{\mathcal{U}}$ on $(L^{p})^{\mathcal{U}}$ is finite dimensional. To this end, we are going to employ Proposition 2.2. The space $(L^{p})^{\mathcal{U}}$ is itself (isometrically lattice isomorphic to) an $L^{p}$-space over some measure space, and we have $\lVert T^{\mathcal{U}}\rVert=\lVert T\rVert\leq 1$, so we have to find an appropriate $L^{q}$-space to make the proposition work.
The space $(L^{q})^{\mathcal{U}}$ is (isometrically lattice isomorphic to) an $L^{q}$-space over some measure space, and the mapping $j^{\mathcal{U}}:(L^{q})^{\mathcal{U}}\to(L^{p})^{\mathcal{U}}$ is a lattice homomorphism, but it is not necessarily injective. However, its kernel $\ker(j^{\mathcal{U}})$ is a closed ideal in $(L^{q})^{\mathcal{U}}$ and thus, as $(L^{q})^{\mathcal{U}}$ has order continuous norm, even a band in $(L^{q})^{\mathcal{U}}$. The quotient space $(L^{q})^{\mathcal{U}}/\ker(j^{\mathcal{U}})$ is isometrically lattice isomorphic to the complementary band and thus to an $L^{q}$ -space. Moreover, $j^{\mathcal{U}}$ induces an injective lattice homomorphism $J:(L^{q})^{\mathcal{U}}/\ker(j^{\mathcal{U}})\to(L^{p})^{\mathcal{U}}$ which has the same range as $j^{\mathcal{U}}$.
Next we note that the range of $T^{\mathcal{U}}$ is contained in the range of $j^{\mathcal{U}}$ (and thus in the range of $J$). Indeed, let $f^{\mathcal{U}}\in(L^{p})^{\mathcal{U}}$ where $f=(f_{n})_{n\in\mathbb{N}}\in\ell^{\infty}(L^{p})$. It follows from the closed graph theorem that the mapping $j^{-1}T:L^{p}\to L^{q}$ is continuous, so the sequence $(j^{-1}Tf_{n})_{n\in\mathbb{N}}$ is an element of $\ell^{\infty}(L^{q})$. We thus obtain
$$\displaystyle T^{\mathcal{U}}f^{\mathcal{U}}=\big{(}(Tf_{n})_{n\in\mathbb{N}}%
\big{)}^{\mathcal{U}}=\big{(}(jj^{-1}Tf_{n})_{n\in\mathbb{N}}\big{)}^{\mathcal%
{U}}=j^{\mathcal{U}}\,\big{(}(j^{-1}Tf_{n})_{n\in\mathbb{N}}\big{)}^{\mathcal{%
U}},$$
so $T^{\mathcal{U}}f^{\mathcal{U}}$ is in the range of $j^{\mathcal{U}}$ (and thus in the range of $J$). Therefore, the assumptions of Proposition 2.2 are fulfilled, and we conclude that the fixed space of $T^{\mathcal{U}}$ is finite dimensional.
Now we can apply Proposition 2.4 which tells us that the fixed space of $T$ is also finite dimensional and that the number $1$ is a pole of the resolvent of $T$ of order $1$. Note that the range of the corresponding spectral projection $Q$ coincides with the fixed space of $T$ since the order of the pole equals $1$.
Next we employ a theorem which goes originally back to Niiro and Sawashima [24, Theorem 9.2] and which can, in the version that we use here, be found in [29, Theorem V.5.5]. The theorem says that, as the spectral radius of our operator $T$ is a pole of the resolvent and as the corresponding spectral projection has finite-dimensional range, every spectral value of $T$ of maximal modulus is a pole of the resolvent. Hence, $T$ has only finitely many spectral values on the unit circle, and each such spectral value is a pole of the resolvent. Moreover, it readily follows from the Neumann series representation of the resolvent that the order of each such pole is dominated by the order of the pole $1$; hence, all unimodular spectral values of $T$ are first order poles of $\mathcal{R}(\mathord{\,\cdot\,},T)$.
It only remains to show that the eigenspace of each unimodular spectral value of $T$ is finite dimensional. This follows, for instance, from the dimension estimate in [12, Theorem 5.5] which asserts that $\dim\ker(\lambda-T)\leq\dim\ker(\lambda^{n}-T)$ for each number $\lambda$ on the complex unit circle and for each integer $n\in\mathbb{Z}$; plugging in $n=0$ we obtain $\dim\ker(\lambda-T)\leq\dim\ker(1-T)$ (note that the assumptions of [12, Theorem 5.5] are fulfilled here since every power bounded operator on a reflexive Banach space is weakly almost periodic).
∎
3. Operator semigroups
Now we briefly explain how our main results can be applied to obtain operator norm convergence of positive semigroups. Let $E$ be a Banach lattice. An operator semigroup on $E$ is a family $(T_{t})_{t\in(0,\infty)}$ of bounded linear operators on $E$ such that the so-called semigroup law $T_{s+t}=T_{s}T_{t}$ is fulfilled for all $s,t\in(0,\infty)$. For what follows, we do not need to impose any regularity assumption with respect to the time parameter $t$ on the semigroup. The semigroup $(T_{t})_{t\in(0,\infty)}$ is called bounded if $\sup_{t\in(0,\infty)}\lVert T_{t}\rVert<\infty$ and it is called contractive if $\lVert T_{t}\rVert\leq 1$ for all $t\in(0,\infty)$; it is called positive if $T_{t}$ is a positive operator for each $t\in(0,\infty)$.
Let $(T_{t})_{t\in(0,\infty)}$ be a positive and bounded operator semigroup on a Banach lattice $E$. It was proved by Lotz in [19, Theorem 4 on p. 153] that, if for some $t_{0}\in(0,\infty)$ the essential spectral radius of $T_{t_{0}}$ is strictly smaller than $1$, then $T_{t}$ converges with respect to the operator norm as $t\to\infty$. Hence, we obtain the following corollary of Theorem 1.2.
Corollary 3.1.
Let $p,q\in[1,\infty]$ be distinct numbers and let $(\Omega_{1},\mu_{1})$ and $(\Omega_{2},\mu_{2})$ be arbitrary measure spaces. Moreover, let $j:L^{q}:=L^{q}(\Omega_{2},\mu_{2})\to L^{p}:=L^{p}(\Omega_{1},\mu_{1})$ be an injective lattice homomorphism.
Let $(T_{t})_{t\in(0,\infty)}$ be a positive and contractive operator semigroup on $L^{p}$ and assume that $T_{t_{0}}L^{p}\subseteq j(L^{q})$ for at least one time $t_{0}\in(0,\infty)$. Then $T_{t}$ converges with respect to the operator norm as $t\to\infty$.
Note that the corollary remains valid if we replace the assumption that the operator semigroup be contractive with the assumption that the operator semigroup is bounded and that merely the operator $T_{t_{0}}$ is contractive.
4. Concluding remarks
We conclude the paper with the following open problem:
Open Problem 4.1.
Do Theorems 1.1 and 1.2 remain valid if we replace the assumption $\lVert T\rVert\leq 1$ with the weaker assumption that $T$ be power-bounded (i.e. $\sup_{n\in\mathbb{N}_{0}}\lVert T^{n}\rVert<\infty$)?
If we merely assume that $T$ is power-bounded instead of contractive, there is only one point where our proof of Theorem 1.2 fails: we can no longer use the argument from Lemma 2.3 to conclude that the fixed space of $T$ is a sublattice of $L^{p}(\Omega_{1},\mu_{1})$.
However, it is still possible to show that $\operatorname{fix}T$ is a lattice subspace of $L^{p}(\Omega_{1},\mu_{1})$, i.e. a vector lattice in its own right with respect to the order inherited from $L^{p}(\Omega_{1},\mu_{1})$ (but with possibly different lattice operations). Indeed, if $f\in\operatorname{fix}T$, then the limit $g:=\lim_{n\to\infty}T^{n}\lvert f\rvert$ exists with respect to the norm on $L^{p}(\Omega_{1},\mu_{1})$ and yields the modulus of $f$ in the space $\operatorname{fix}T$ (this argument is taken from the proof of [2, Corollary C-III-4.3(a)]).
Still, since the lattice operations in a lattice subspace can differ from the lattice operations in $L^{p}(\Omega_{1},\mu_{1})$ itself, we cannot simply conclude that the norm on $\operatorname{fix}T$ is $p$-additive, and this is where our argument fails.
Acknowledgements
I am indebted to Delio Mugnolo for several very helpful discussions and comments; he told me about L. Miclo’s article [21], which was the motivation for writing the present paper, he brought Pitt’s theorem (mentioned after Corollary 1.3) to my attention and he suggested to generalise the result in Theorem 1.1 to Theorem 1.2.
Appendix A Lattice isomorphisms between $L^{p}$- and $L^{q}$-spaces
The fact that, for instance, $L^{p}([0,1])$ and $L^{q}([0,1])$ are not isomorphic as Banach spaces for $p\not=q$ is usually shown by techniques from the geometric theory of Banach spaces. For our purposes, though, we only need the much simpler fact that an infinite dimensional $L^{p}$-space is never lattice isomorphic to an $L^{q}$-space for $p\not=q$. In the following proposition we give an elementary proof of this fact.
Proposition A.1.
Let $p,q\in[1,\infty)$ be two distinct numbers, let $(\Omega_{1},\mu_{1})$ and $(\Omega_{2},\mu_{2})$ be arbitrary measure spaces and assume that $L^{p}:=L^{p}(\Omega_{1},\mu_{1})$ and $L^{q}:=L^{q}(\Omega_{2},\mu_{2})$ are isomorphic as Banach lattices (i.e. there exists a lattice isomorphism $L^{q}\to L^{p}$). Then $L^{q}$ (and hence $L^{p}$) has finite dimension.
For the proof we need the simple observation that, in every infinite dimensional Banach lattice $E$, there exists a sequence $(x_{k})_{k\in\mathbb{N}}\subseteq E_{+}$ of normalised and pairwise disjoint vectors (i.e. $\lVert x_{k}\rVert=1$ for all indices $k$ and $x_{j}\land x_{k}=0$ whenever $j\not=k$).
Proof of Proposition A.1.
We may assume that $p<q$. Assume for a contradiction that $L^{q}$ is infinite-dimensional. Then there exists a sequence $(f_{k})_{k\in\mathbb{N}}$ of normalised and pairwise disjoint vectors $0\leq f_{k}\in L^{q}$. The series $\sum_{k=1}^{\infty}\frac{f_{k}}{k^{1/p}}$ converges in $L^{q}$ since, for $1\leq m\leq n$, we have
$$\displaystyle\Big{\lVert}\sum_{k=m}^{n}\frac{f_{k}}{k^{1/p}}\Big{\rVert}_{q}^{%
q}=\sum_{k=m}^{n}\frac{\lVert f_{k}\rVert_{q}^{q}}{k^{q/p}}=\sum_{k=m}^{n}%
\frac{1}{k^{q/p}}\to 0\quad\text{as }m,n\to\infty.$$
Now, let $J:L^{q}\to L^{p}$ be a lattice isomorphism. Then the vectors $g_{k}:=Jf_{k}\in L^{p}$ are also pairwise disjoint. As $J$ is continuous, the series $\sum_{k=1}^{\infty}\frac{g_{k}}{k^{p}}$ converges in $L^{p}$. However, the mapping $J^{-1}$ is continuous, too, so we have $\lVert g_{k}\rVert_{p}\geq c\lVert f_{k}\rVert_{q}=c$ for a constant $c>0$ and all indices $k$. This implies that, for $1\leq m\leq n$,
$$\displaystyle\Big{\lVert}\sum_{k=m}^{n}\frac{g_{k}}{k^{1/p}}\Big{\rVert}_{p}^{%
p}=\sum_{k=m}^{n}\frac{\lVert g_{k}\rVert_{p}^{p}}{k}\geq\sum_{k=m}^{n}\frac{c%
^{p}}{k}\not\to 0\quad\text{as }m,n\to\infty.$$
This is a contradiction.
∎
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Inflation in de Sitter spacetime and CMB large scales anomaly
Dong Zhao${}^{1:1)}$ Ming-Hua Li${}^{2}$
Ping Wang${}^{1}$
Zhe Chang${}^{1}$
[email protected]
${}^{1}$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
${}^{2}$ School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China
Abstract
The influence of cosmological constant type dark energy in the early universe is investigated. This is accommodated by a new dispersion relation in de Sitter spacetime. We perform a global fitting to explore the cosmological parameters space by using the CosmoMC package with the recently released Planck TT and WMAP Polarization datasets. Using the results from global fitting, we compute a new CMB temperature-temperature spectrum. The obtained TT spectrum has lower power compared with the one based on $\Lambda$CDM model at large scales.
d
pacs: 9
{CJK*}
GBKsong
e Sitter spacetime, dark energy, inflation, anisotropy
8.80.Cq
1 Introduction
The anisotropy of Cosmic Microwave Background radiation (CMBR) was first discovered by NASA’s Cosmic Background Explorer (COBE) satellite in the 1990s[2]. The results were later confirmed by the balloon experiments and the Wilkinson Microwave Anisotropy Probe (WMAP) satellite[3]. The observation data can be well fitted by the $\Lambda$CDM (cold dark matter plus dark energy in a form of the cosmological constant $\Lambda$) model. With the help of the Planck data [4], one can now place an unprecedentedly precise constraint on six cosmological parameters (with an accuracy down to $10\%$ level)[5].
However, for the power spectrum of CMBR, the observed values of $C_{\ell}$ for low $\ell$, especially for the quadrupole component $\ell=2$, are smaller than that predicted by the standard cosmological model. H. Liu and T.-P. Li [6][7] proposed that the CMB quadrupole in the WMAP data is artificial and the corresponding values should actually be near zero. Contrarily, C. Bennett et al. [8] reexamined the WMAP 7-year data carefully and reported that the quadrupole amplitude value is consistent with that predicted by the the $\Lambda$CDM model at a $95\%$ confidence level and shows no anomalies.
In a previous paper, we proposed inflation in a de Sitter spacetime that could possibly alleviate this controversy[9]. In fact, a de Sitter Universe is a cosmological solution of Einstein’s field equations in general relativity with a positive cosmological constant $\Lambda$. We considered the possible effects of dark energy in form of a cosmological constant during the inflationary period. In this scenario, the cosmological constant type dark energy was once predominant in the early universe. The dynamics of the universe is accommodated by a new dispersion relation—the dispersion relation in de Sitter spacetime. We found that for certain cosmological parameter values, the modified inflation model gives a CMB TT power spectrum with lower power at large scales, alleviating the low-$\ell$ multipole issue.
In this paper, we would use the Markov Chain Monte Carlo sampler (CosmoMC)[10] to explore the cosmological parameters space. The recently released Planck TT[4] and WMAP Polarization[3] datasets will be used in our global fitting. More stringent constraints are presented on the cosmological parameters. We obtain values of the cosmological parameters from global fitting and compute a new TT spectrum with the Code for Anisotropies in the Microwave Background (CAMB)[11].
The rest of the paper is organized as follows. Section 2 is devoted to setup of the inflation in de Sitter spacetime. In section 3, we present a global fitting with combined datasets and obtain numerical results. Discussions and conclusions are given in section 4.
2 Inflation in de Sitter Spacetime
De Sitter spacetime is a vacuum solution of the Einstein’s field equations with a positive cosmological constant. It can be realized as a four-dimensional pseudo-sphere imbedded in a five dimensional Minkowski flat space(with coordinates $\xi_{\mu}$, $\mu=0,1,2,3,4$) [12]
$$\begin{array}[]{l}\displaystyle-\xi_{0}^{2}+\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2%
}+\xi_{4}^{2}=\frac{1}{K}=R^{2}\ ,\\
ds^{2}=-d\xi_{0}^{2}+d\xi_{1}^{2}+d\xi_{2}^{2}+d\xi_{3}^{2}+d\xi_{4}^{2}\ ,%
\end{array}$$
(1)
where $K$ is the Riemannian curvature and $R$ is the radius of de Sitter spacetime.
For a free particle with mass $m_{0}$, the five dimensional angular momentum $M_{\mu\nu}$ is defined as
$$M_{\mu\nu}\equiv m_{0}\left(\xi_{\mu}\frac{d\xi_{\nu}}{ds}-\xi_{\nu}\frac{d\xi%
_{\mu}}{ds}\right)\ ,~{}~{}~{}~{}~{}\mu,\nu=0,1,2,3,4\ ,$$
(2)
where $s$ is the affine parameter. For the rest of the article, the Latin indices (i.e. $i,j,k$, etc.) run from 1 to 3 and the Greek indices (i.e. $\mu,\nu,\alpha,\beta$, etc.) run from 0 to 3.
The momentum of a free particle in de Sitter spacetime is defined as
$$P_{\mu}\equiv R^{-1}M_{4\mu}\ .$$
(3)
The angular momentum $J_{\mu\nu}$ in de Sitter spacetime is defined as
$$J_{\mu\nu}\equiv M_{\mu\nu}=x_{\mu}P_{\nu}-x_{\nu}P_{\mu}\ .$$
(4)
An invariant can be constructed in terms of the angular momentum $J^{\mu\nu}$,
$$\begin{array}[]{c}m^{2}_{0}=\displaystyle\frac{K}{2}J_{\mu\nu}J^{\mu\nu}=E^{2}%
-{\bf P}^{2}+\frac{K}{2}J_{ij}J^{ij}~{},\\
E=P_{0}~{},~{}~{}~{}~{}{\bf P}=(P_{1},~{}P_{2},~{}P_{3})~{}.\end{array}$$
(5)
The generators $\hat{M}_{\mu\nu}$ of de Sitter group $SO(1,4)$ are of the form
$$\hat{M}_{\mu\nu}\equiv-i\left(\xi_{\mu}\frac{\partial}{\partial\xi^{\nu}}-\xi_%
{\nu}\frac{\partial}{\partial\xi^{\mu}}\right)\ .$$
(6)
The Klein-Gordon equation for a free scalar can be written as
$$\left(\frac{K}{2}\hat{M}_{\mu\nu}\hat{M}^{\mu\nu}-m^{2}_{0}\right)\phi(\xi^{%
\mu})=0~{}.$$
(7)
The equation (7) can be solved analytically [12]. The dispersion relation for a free scalar in de Sitter spacetime is
$$E^{2}=m_{0}^{2}+k^{2}+K(2n+l)(2n+l+2)\ ,$$
(8)
where $n$ and $l$ respectively denotes the radial and the angular quantum number of the system. For massless particles like photons, the above dispersion relation becomes
$$\begin{array}[]{c}\omega^{2}=k^{2}+\varepsilon^{*2}_{\gamma}\ ,\\
\varepsilon^{*}_{\gamma}\equiv\sqrt{K(2n_{\gamma}+l_{\gamma})(2n_{\gamma}+l_{%
\gamma}+2)}\ ,\end{array}$$
(9)
where $w$ and $k$ are frequency and wavenumber of the photon.
Now, we calculate the primordial power spectrum ${\cal P}_{\delta\phi}(k)$ in de Sitter spacetime.
The inflation field is denoted by $\delta\phi({\bf x},t)$.
In the momentum representation, the evolution equation of the primordial perturbation is given by [13]
$$\delta\sigma^{\prime\prime}_{\bf k}+\left(\omega^{2}-\frac{2}{\tau^{2}}\right)%
\delta\sigma_{\bf k}=0\ ,~{}~{}~{}~{}~{}\omega^{2}\equiv k^{2}+\varepsilon^{*2%
}_{\gamma}~{},$$
(10)
where
$$\delta\sigma_{\bf k}\equiv a\delta\phi_{\mathbf{k}}\ .$$
(11)
A prime represents differentiation with respect to the conformal time $\tau$ [14].
The above equation has an exact particular solution
$$\displaystyle\delta\sigma_{\bf k}$$
$$\displaystyle=$$
$$\displaystyle\frac{e^{-i\omega\tau}}{\sqrt{2\omega}}\left(1+\frac{i}{\omega%
\tau}\right)$$
(12)
$$\displaystyle=$$
$$\displaystyle\frac{e^{-i\sqrt{k^{2}+\varepsilon^{*2}_{\gamma}}~{}\tau}}{\sqrt{%
2}\left(k^{2}+\varepsilon^{*2}_{\gamma}\right)^{1/4}}\left(1+\frac{i}{\sqrt{k^%
{2}+\varepsilon^{*2}_{\gamma}}~{}\tau}\right)\ .$$
The power spectrum ${\cal P}_{\delta\phi}(k)$ is defined as [13]
$${\cal P}_{\delta\phi}(k)\equiv\frac{k^{3}}{2\pi^{2}}\,|\frac{1}{a}\,\delta\phi%
_{\textbf{k}}|^{2}\ .$$
(13)
Considering the super-horizon criterion, one has
$$-k\tau=\frac{k}{aH}\ll 1\ ,$$
(14)
where $a=-1/H\tau$ ($H$ is the Hubble parameter) [13].
By making use of (12), (13) and (14), we obtain the primordial power spectrum[9]
$$\displaystyle{\cal P}_{\delta\phi}(k)=\frac{H^{2}}{4\pi^{2}}\cdot\frac{k^{3}}{%
\left({k^{2}+\varepsilon^{*2}_{\gamma}}\right)^{3/2}}\ .$$
(15)
For perturbations on small scales, we obtain the usual scale-invariant primordial power spectrum
$${\cal P}_{\delta\phi}(k)~{}\simeq~{}k^{0}.$$
(16)
For large scales, we have
$${\cal P}_{\delta\phi}(k)\simeq\frac{H^{2}}{4\pi^{2}}\cdot k^{3}.$$
(17)
The power spectrum of the comoving curvature perturbation $\mathcal{R}$ is usually parameterized as [14]
$$\mathcal{P}_{\mathcal{R}}(k)=\frac{H^{2}}{\dot{\phi}^{2}}~{}{\cal P}_{\delta%
\phi}(k)\equiv A^{2}_{s}\left(\frac{k}{k_{p}}\right)^{n_{s}-1}\cdot\frac{(k/k_%
{p})^{3}}{[{\left(k/k_{p}\right)^{2}+\varepsilon^{*2}}]^{3/2}}\ ,$$
(18)
where $k_{p}=0.05~{}\rm{Mpc^{-1}}$ and $\varepsilon^{*}=\varepsilon^{*}_{\gamma}/k_{p}$. $A_{s}$ is the power amplitude and $n_{s}$ is the scalar spectral index.
We plot the primordial spectrum (18) in Figure 2.
\figcaption
The primordial power spectrum $P_{\delta\phi}(k)$. The black curve stands for the $P_{\delta\phi}(k)$ in standard $\Lambda$CDM model and the red curve represents the $P_{\delta\phi}(k)$ in de Sitter spacetime. One can see that there is a cutoff at large scales. It should lead to a suppressed TT spectrum when $\ell$ is small.
3 The Numerical Results
We use the Markov Chain Monte Carlo sampler (CosmoMC) to perform a global fitting of cosmological parameters. The Planck TT[4] and WMAP polarization[3] datasets will be used in the global fitting. There are six parameters originated in the $\Lambda$CDM model and one extra parameter $\varepsilon^{*}$. The results are shown in Table 1 and Figure 1.
Parameter
$$\Lambda$$CDM model
de Sitter
$$\Omega_{b}h^{2}$$
0.02205$$\pm$$0.00028
0.02203$$\pm$$0.00028
$$\Omega_{c}h^{2}$$
0.1199$$\pm$$0.0027
0.1200$$\pm$$0.0026
$$100\theta_{MC}$$
1.04131$$\pm$$0.00063
1.04121$$\pm$$0.00063
$$\tau$$
$$0.089_{-0.014}^{+0.012}$$
0.098$$\pm$$0.015
$$n_{s}$$
0.9603$$\pm$$0.0073
0.9585$$\pm$$0.0073
$$ln(10^{10}A^{s})$$
$$3.089_{-0.027}^{+0.024}$$
3.106$$\pm$$0.030
$$100\varepsilon^{*}$$
-
0.4266$$\pm$$0.1945
\tabcaption
The 68$\%$ limits for the cosmological parameters originated in the $\Lambda$CDM and de Sitter models with data combination Planck+WP.
\figcaption
Contour plots and the likelihood distributions of the seven cosmological parameters with data combination Planck+WP in de Sitter spacetime.
In Table 3, one can see that the values of the six basic parameters in de Sitter spacetime are similar with the ones based on standard cosmological model[4]. We are more interested in the new parameter $\varepsilon^{*}_{\gamma}$. With $\varepsilon^{*}_{\gamma}=2.13\times 10^{-4}~{}\rm{Mpc^{-1}}$, we can calculate the wavelength related to $\varepsilon^{*}_{\gamma}$, i.e. $\lambda=4.69\times 10^{3}~{}\rm{Mpc}$. The wavelength is comparable to the hubble horizon. From Figure 3, one can see that the combined datasets favour the new parameter $\varepsilon^{*}_{\gamma}$ is not zero at around 1$\sigma$ confidence level.
Finally, we use the modified version of CAMB to compute the new CMB temperature-temperature spectrum in de Sitter spacetime with the global fitting results. The obtained TT spectrum has been suppressed obviously at large scales.
\figcaption
The TT spectra with data combination Planck+WP. The black curve represents TT spectrum based on the standard $\Lambda$CDM model and the red one indicates TT spectrum in de Sitter spacetime.
In Figure 3, the black curve indicates the best-fitting CMB TT spectrum in standard $\Lambda$CDM model and the red curve represents the TT spectrum in de Sitter spacetime. Compared to the TT spectrum in the $\Lambda$CDM model, the spectrum in de Sitter spacetime has been suppressed obviously when $\ell$ $<$ 20. Especially at quadrupole, the spectrum drops to the lowest point, which is almost half of the standard TT spectrum.
4 Conclusions
In this paper, we analyzed an inflation model in de Sitter spacetime and got modified primordial spectrum. Possible effects of cosmological constant type dark energy were considered during the inflationary period. We got a lower TT spectrum at large scales which refers to low-$\ell$ multipole anomaly. The new scenario has a new parameter, namely $\varepsilon^{*}_{\gamma}$. To constrain the cosmological parameters, we used the CosmoMC package to perform a global fitting with Planck TT and WMAP Polarization datasets. We found that the results of global fitting in de Sitter spacetime are similar to ones based on standard $\Lambda$CDM model. With $\varepsilon^{*}_{\gamma}=2.13\times 10^{-4}~{}\rm{Mpc^{-1}}$, we calculated the wavelength related to $\varepsilon^{*}_{\gamma}$, i.e. $\lambda=4.69\times 10^{3}~{}\rm{Mpc}$. The wavelength is comparable to the hubble horizon. The combined datasets favour that the new parameter $\varepsilon^{*}_{\gamma}$ is not zero at about 1$\sigma$ confidence level. The new TT spectrum shows lower energy at large scales, which just as we expected.
We are grateful to Dr. Hai-Nan Lin, Si-Yu Li, Sai Wang and Yu Sang for useful discussion. This work has been funded by the National Natural Science Fund of China under Grant no. 11375203.
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Detailed Abundances in the Ultra-Faint Magellanic Satellites Carina II and III111This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile.
A. P. Ji
Hubble Fellow
Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
T. S. Li
NHFP Einstein Fellow
Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
J. D. Simon
Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
J. Marshall
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
A. K. Vivas
Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile
A. B. Pace
Mitchell Astronomy Fellow
McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
K. Bechtol
Physics Department, University of Wisconsin-Madison, 1150 University Avenue Madison, WI 53706, USA
A. Drlica-Wagner
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
Department of Astronomy and Astrophysics, University of Chicago, Chicago IL 60637, USA
S. E. Koposov
McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
T. T. Hansen
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
S. Allam
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
R. A. Gruendl
Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, IL 61801, USA
National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA
M. D. Johnson
National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA
M. McNanna
Physics Department, University of Wisconsin-Madison, 1150 University Avenue Madison, WI 53706, USA
N. E. D. Noël
Department of Physics, University of Surrey, Guildford, GU2 7XH, UK
D. L. Tucker
Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA
A. R. Walker
Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared Astronomy Research Laboratory, Casilla 603, La Serena, Chile
Abstract
We present the first detailed elemental abundances in the ultra-faint Magellanic satellite galaxies Carina II (Car II) and Carina III (Car III).
With high-resolution Magellan/MIKE spectroscopy, we determined abundances of nine stars in Car II including the first abundances of an RR Lyrae star in an ultra-faint dwarf galaxy; and two stars in Car III.
The chemical abundances demonstrate that both systems are clearly galaxies and not globular clusters.
The stars in these galaxies mostly display abundance trends matching those of other similarly faint dwarf galaxies:
enhanced but declining [$\alpha$/Fe] ratios, iron-peak elements matching the stellar halo, and unusually low neutron-capture element abundances. One star displays a low outlying [Sc/Fe]$=-1.0$.
We detect a large Ba scatter in Car II, likely due to inhomogeneous enrichment by low-mass AGB star winds.
The most striking abundance trend is for [Mg/Ca] in Car II, which decreases from $+0.4$ to $-0.4$ and indicates clear variation in the initial progenitor masses of enriching core-collapse supernovae.
So far, the only ultra-faint dwarf galaxies displaying a similar [Mg/Ca] trend are likely satellites of the Large Magellanic Cloud.
We find two stars with $\mbox{[Fe/H]}\leq-3.5$, whose abundances likely trace the first generation of metal-free Population III stars and are well-fit by Population III core-collapse supernova yields.
An appendix describes our new abundance uncertainty analysis that propagates line-by-line stellar parameter uncertainties.
stars: abundances — galaxies: dwarf — Local Group
††journal: ApJ††facilities: Magellan-Clay (MIKE)††software: MOOG (Sneden, 1973; Sobeck et al., 2011), SMHR (Casey, 2014), numpy (van der Walt et al., 2011),
scipy (Jones et al., 2001),
matplotlib (Hunter, 2007),
pandas (Mckinney, 2010),
seaborn, (Waskom et al., 2016),
astropy (Astropy Collaboration et al., 2013)
1 Introduction
Ultra-faint dwarf galaxies (UFDs) are the luminous counterparts to the least massive star-forming dark matter halos, likely forming stars during the first $\sim$1 Gyr before being quenched by reionization (e.g., Bullock et al., 2000; Benson et al., 2002; Simon & Geha, 2007; Brown et al., 2014; Simon, 2019).
As a result, the chemical abundances of stars in UFDs preserve a clean snapshot of chemical enrichment from the earliest stages of galaxy formation and reionization, providing a window to the most metal-poor stellar populations and their nucleosynthetic output (Kirby et al., 2008; Frebel & Bromm, 2012; Geha et al., 2013; Weisz et al., 2014; Wise et al., 2014; Ji et al., 2015).
Dozens of UFDs have now been discovered in deep, wide, and uniform photometric surveys such as the Sloan Digital Sky Survey, Pan-STARRS, and the Dark Energy Survey (DES) (e.g., Willman et al., 2005; Belokurov et al., 2007; Laevens et al., 2015; Bechtol et al., 2015; Koposov et al., 2015a; Drlica-Wagner et al., 2015).
The large number of UFDs provides a large population of local objects that retain signatures of high-redshift star and galaxy formation.
Until recently, these UFDs have generally been assumed to be satellites of the Milky Way. However, the two most massive dwarfs orbiting the Milky Way, the Large and Small Magellanic Clouds (LMC and SMC), should have had their own satellite UFDs (e.g., D’Onghia & Lake, 2008; Koposov et al., 2015b; Drlica-Wagner et al., 2015; Jethwa et al., 2016; Dooley et al., 2017; Sales et al., 2017). Since the LMC and SMC are likely on their first infall into the Milky Way (Besla et al., 2007; Busha et al., 2011; Kallivayalil et al., 2013; Simon, 2018; Fritz et al., 2019; Pace & Li, 2019), any dwarfs that were previously Magellanic satellites could now be in the process of accretion into the Milky Way. Gaia proper motion measurements have revealed that several UFDs are kinematically associated with the LMC/SMC system (Kallivayalil et al., 2018; Erkal & Belokurov, 2019).
Two of these LMC satellites are Carina II (Car II, $M_{V}=-4.5$, $L/L_{\odot}\sim 10^{3.7}$) and Carina III (Car III, $M_{V}=-2.4$, $L/L_{\odot}\sim 10^{2.9}$), discovered in the Magellanic Satellites Survey (MagLiteS, Drlica-Wagner et al. 2016; Torrealba et al. 2018) with the Dark Energy Camera (DECam, Flaugher et al. 2015) on the Blanco telescope.
Li et al. (2018) spectroscopically confirmed Car II to be a dwarf galaxy, and Li et al. (in prep) have now confirmed Car III as a dwarf galaxy as well.
These UFDs are only ${\sim}20$ kpc away from the LMC, and are also close to the Sun (37.4 and 27.8 kpc for Car II and III, respectively). Thus, they have a relatively large number of bright stars amenable for high-resolution spectroscopic followup and chemical abundance measurements.
In this paper, we present a comprehensive chemical abundance analysis of Magellan/MIKE spectroscopy of 9 stars in Car II and 2 stars in Car III.
Along with Horologium I (Nagasawa et al., 2018), these are currently the only ultra-faint LMC satellites with high-resolution abundance measurements.
Section 2 explains the observations, data reduction, and velocity measurements.
Section 3 details our abundance analysis.
We discuss the formation history of these galaxies in Section 4, highlighting the interesting $\alpha$-element abundance trends in Section 4.3.
We focus on potential signatures of metal-free Pop III stars in Section 5,
then summarize and conclude in Section 6.
2 Observations, Data Reduction, Radial Velocities
Our Carina II and III targets were selected to be the brightest radial velocity members from Magellan/IMACS, AAT/AAO, and VLT/FLAMES moderate resolution spectra, including five bright member stars from Li et al. (2018) and five new bright member stars from Li et al. (in prep). In addition, we include one RR Lyrae member in Carina II identified in Torrealba et al. (2018).
We observed these stars with Magellan/MIKE (Bernstein et al., 2003) over four separate runs (Tables 1 and 2).
Slits of width 0$\farcs$5, 0$\farcs$7, and 1$\farcs$0 were used depending on the seeing, resulting in typical resolutions of $R\sim$50k/40k, 35k/28k, and 28k/22k on the blue/red arms of MIKE, respectively.
We used 2x2 binning for the 0$\farcs$7 and 1$\farcs$0 slits, and 2x1 binning for the 0$\farcs$5 slit.
The MIKE data were reduced with CarPy (Kelson, 2003).
We used the code SMHR (Casey, 2014)222https://github.com/andycasey/smhr, first described in Casey 2014 to coadd, normalize, stitch orders, and Doppler correct the reduced spectra for abundance analysis.
Data from multiple runs were combined by coadding order-by-order, using a common set of spline knot locations and line masks after adjusting for observed radial velocity.
The signal-to-noise at the order center closest to rest wavelengths of 4500Å, 5300Å, and 6500Å is given in Table 1.
The total integrated time spent on these stars is 34 hours.
Note there is significant reddening towards Car II and III (E($B-V$) $\sim 0.2$ mag).
Figure 1 shows our spectra around the C-H G band, the strongest barium line, and the Mg b triplet.
In general, we reduced all MIKE data from a given observing run together before measuring the radial velocity.
The exception is the RR Lyrae (RRL) star CarII-V3, which experiences large radial velocity variations on a short timescale.
Using the known pulsation phases (Torrealba et al., 2018),
we observed CarII-V3 across phases $0.40-0.55$ with five consecutive 30-min exposures.
Over this phase range, the star has fairly consistent stellar parameters (For et al., 2011), so individual exposures can be coadded after correcting for a velocity offset.
We reduced each exposure separately, measured radial velocities for each observation separately using the Mg b triplet, corrected each order to rest frame, and coadded order-by-order before stitching orders in SMHR.
Radial velocities are given in Table 2. For the velocity measurements, we re-reduced each exposure individually with CarPy. We measured radial velocities of the 40 orders from 3900Å to 6800Å (order numbers $51-90$). Of these, we masked the telluric lines around 6300Å, discarded three orders from 5820-6020Å because of interstellar Na D absorption, and discarded the bluest order on the red side due to uniformly low S/N. We cross-correlated individual orders of our MIKE spectra against a normalized high-S/N MIKE spectrum of HD122563. To remove outliers, we iteratively sigma clip orders with velocities that are more than 5 biweight scales away from the biweight average.
The final number of orders for each spectrum is given by $N_{\text{ord}}$ in Table 2.
Statistical errors for each order were then found by calculating the $\chi^{2}$ at different velocities and taking $\Delta\chi^{2}=1$ away from the minimum.
Naively, we could combine these measurements by taking a weighted average of all orders to get a final average velocity and in principle reaching an extremely high velocity precision of ${\sim}0.1$ km/s.
However, systematic effects dominate both the velocity measurement and error.
For example, MIKE is not attached to the instrument rotator and until recently did not have an atmospheric dispersion compensator. At high airmasses, atmospheric refraction in the narrow slit direction causes systematic velocity offsets as a function of wavelength that can be as large as $2-3$ km/s.
We will correct for these effects in later work, but such velocity differences do not impact the abundance analyses that are the focus of this paper.
Thus, for now in Table 2 we provide the radial velocity of each individual spectrum computed by an inverse-variance weighted average of all $N_{\text{ord}}$ orders. The systematic error is the weighted standard deviation of those orders and dominates over the ${\sim}0.1$ km/s statistical uncertainty.
3 Abundance Analysis
3.1 Abundance Analysis Details
We performed a standard 1D-LTE analysis using the 2017 version of the 1D LTE radiative transfer code MOOG (Sneden, 1973; Sobeck et al., 2011)333https://github.com/alexji/moog17scat and the Castelli & Kurucz (2004) (ATLAS) model atmospheres.
We used SMHR to measure equivalent widths, interpolate model atmospheres, and run MOOG.
For the red giant branch (RGB) stars, stellar parameters were derived spectroscopically. Briefly, we start assuming $\alpha$-enhanced $[\alpha/\text{Fe}]=+0.4$ model atmospheres. The effective temperature, surface gravity, and microturbulence ($T_{\rm eff}$, $\log g$, $\nu_{t}$) were determined by balancing excitation, ionization, and line strength for Fe lines, respectively.
We then applied the $T_{\rm eff}$ correction from Frebel et al. (2013) to place the measurements on a photometric temperature scale and redetermined $\log g$ and $\nu_{t}$.
After this initial determination, if the star turned out to have low Mg abundances, we switched to $[\alpha/\text{Fe}]=0$ atmospheres and redetermined the stellar parameters.
Statistical stellar parameter uncertainties are found following Ji et al. (2019a), and we adopt systematic uncertainties of 150 K for $T_{\rm eff}$, 0.3 dex for $\log g$, and $0.2\,\text{km}\,\text{s}^{-1}$ for $\nu_{t}$ due to uncertainties in the Frebel et al. (2013) temperature calibration.
The statistical and systematic uncertainties were added in quadrature to obtain the total stellar parameter uncertainties in Table 3.
We used a combination of equivalent widths and spectral syntheses to measure the abundances of individual lines.
We also determined statistical and systematic abundance uncertainties for each individual feature.
For lines measured using equivalent widths, we propagated the $1\sigma$ equivalent width uncertainty into a $1\sigma$ statistical abundance uncertainty.
For lines measured using syntheses, we increased the element abundance until $\Delta\chi^{2}=1$, also corresponding to a $1\sigma$ statistical uncertainty.
These uncertainties account for continuum placement uncertainty (see Appendix A for details).
For the systematic uncertainties, we varied each stellar parameter ($T_{\rm eff}$, $\log g$, $\nu_{t}$, [M/H]) individually by its error and remeasured the abundance.
The total systematic uncertainty is the quadrature sum of the individual stellar parameter uncertainties.
Finally, the total abundance uncertainty for an individual line is the quadrature sum of the statistical and systematic uncertainty.
Individual line measurements and uncertainties are found in Table 4.
We use inverse-variance weighted averages to combine lines into a final abundance.
Because we have included a detailed account of line-by-line uncertainties, this automatically downweights lines in regions of low spectral S/N; saturated lines that are sensitive to small equivalent width variations; and lines that are particularly sensitive to stellar parameters.
We verified that the weighted averages are usually only a few hundredths of a dex different from the unweighted averages. The exception is elements with few measurable lines like Si and Al, where some lines are much lower quality than others.
See Appendix A for detailed equations.
[X/Fe] ratios are derived by taking ratios of common ionization states (e.g., [Mg I/Fe I], [Ti II/Fe II]).
This mostly (though not always) results in smaller [X/Fe] errors than [X/H] errors, since some stellar parameter differences cancel out.
We also consistently propagate stellar parameter uncertainties for [X/Y] ratios, such as [Mg/Ca].
Upper limits were derived by spectrum synthesis. For a given feature, we fit a synthetic spectrum that well-matched the observed spectrum to determine a reference $\chi^{2}$ and local spectrum smoothing.
Then holding the continuum and smoothing fixed, we increased the abundance until $\Delta\chi^{2}=25$. This is formally a $5\sigma$ upper limit but does not include uncertainties in continuum placement.
3.2 Abundance corrections
Various systematics can affect 1D-LTE abundances of red giants. We tabulate several abundance corrections in Table 5, which are the average of line-by-line corrections. These corrections have been applied in all figures but not in Tables 4 or 6.
Carbon is systematically converted to nitrogen in evolved red giants due to CN cycling. We estimate the natal carbon abundances of these stars with the corrections from Placco et al. (2014)444http://vplacco.pythonanywhere.com/.
Hotter stars have no correction, while for cooler/more evolved stars the correction can be as large as $+0.75$ dex.
We use the default correction grid assuming [N/Fe]$=0$, but changing [N/Fe] makes minimal difference.
Note that we assume all our stars are on the RGB, but if we had red clump or AGB stars in our sample they would have larger carbon corrections than applied here.
Only the Na D lines are available for sodium abundances, and these can have fairly large negative NLTE corrections.
We apply Na corrections from Lind et al. (2011)555www.inspect-stars.com, which range from $-0.13$ to $-0.48$ dex.
For CarII-6544 and CarII-7872, and CarII-5664 we set $\log g=1$ to avoid the edge of the corrections grid.
Mg is marginally affected by NLTE effects in our stars. However, since Mg will be a very important element later, we tabulate the NLTE corrections just to show they are only affected by $<0.04$ dex (Osorio et al., 2015; Osorio & Barklem, 2016).
For several stars (CarII-6544, CarII-4704, CarII-0064, CarII-5664, CarII-7872) we set $\log g=1.5$ to avoid the edge of the corrections grid.
Note that we have used the two high-equivalent width Mg b lines in all our Mg abundances, but removing these two lines everywhere does not significantly affect our RGB star abundances.
Other elements that are known to have significant NLTE corrections include Al, Mn, K, and Fe.
For these elements we do not calculate star-by-star corrections, but instead just estimate the magnitude and direction of a typical correction.
If desired, the effect of these corrections can be approximated by adding the correction to the relevant abundance, as well as adding the total correction in quadrature to the total abundance error; but we do not do so here.
For aluminum, we measured the 3944Å and 3961Å lines, which are heavily affected by NLTE in cool metal-poor stars as well as being in the wings of strong lines, so we only estimate the abundance corrections.
We examined the corrections grid from Nordlander & Lind (2017)666https://www.mso.anu.edu.au/~thomasn/NLTE/ for these lines. Half of our stars are cooler and have lower $\log g$ than the grid range.
The abundance corrections for the 3961Å line tend to be large and positive, from $+0.7$ to $+1.5$ dex.
The corrections for 3944Å are more moderate, from $+0.0$ to $+0.5$ dex.
The corrections for these lines tend to go in opposite directions, such that averaging corrections for these lines in the warmer stars ($T_{\rm eff}\gtrsim 4800$ K) gives corrections in a smaller range from $+0.5$ to $+0.7$ dex.
However, this also tends to make the individual 3944Å and 3961Å abundances more discrepant.
Given these uncertainties, we caution against overinterpretation of our Al abundances or trends.
For manganese, we always use the resonant triplet near 4030Å, as well as redder lines (e.g. 4754Å, 4783Å) when detected. Bergemann et al. (2019) have recently published grids of Mn corrections, showing overall corrections of about $+0.4$ to $+0.6$ dex, though the corrections are likely larger for cooler and metal-poor stars.
As our Mn abundances just fall within the overall halo trend (which are also not corrected for NLTE), we will not discuss this further.
For potassium, we can measure the 7699Å line in all stars. The 7665Å line was also clear of telluric lines for a few stars, and when measurable is always consistent with the 7699Å line. K has negative NLTE corrections that could be as large as $-0.9$ dex (Ivanova & Shimanskiĭ, 2000), although Reggiani et al. (2019) have recently calculated grids of corrections that are more typically $-0.0$ to $-0.4$ dex in our stellar parameter range.
Fe I abundances are affected by NLTE effects, with corrections typically $+0.2$ to $+0.3$ dex in our parameter range (e.g., Bergemann et al., 2012; Mashonkina et al., 2016; Ezzeddine et al., 2017).
Our temperature correction procedure partially accounts for these effects, though not completely (Frebel et al., 2013; Ji et al., 2016b).
We have decided not to apply Fe corrections so as to be able to compare our Fe measurements to literature values, which are essentially all done in LTE.
Finally, we note that Ca can be affected by NLTE as well (Mashonkina et al., 2016).
The available grids do not span our whole stellar parameter space777http://spectrum.inasan.ru/nLTE/, but the available corrections are about ${+}0.1$ dex for our stars. We have not applied this correction.
3.3 RRL Abundance analysis
Stellar parameters for the RRL star CarII-V3 were determined by examining the phase-parameter relations in For et al. (2011).
As our observations are between phases 0.40 to 0.55, stellar parameters are expected to be fairly stable over all exposures.
We adopted initial stellar parameters of $T_{\rm eff}=6000\pm 100$ K, $\log g=1.80\pm 0.2$ dex, $\nu_{t}=3.00\pm 0.20\,\text{km}\,\text{s}^{-1}$ where the error bars are adopted systematic uncertainties based on scatter in the For et al. (2011) values.
Then, we measured equivalent widths by fitting Gaussian profiles to the line list from For & Sneden (2010) (rather than our usual line list, which is optimized for red giants).
To slightly improve Fe excitation, ionization, and line strength balance from 28 Fe I lines and 10 Fe II lines, we adjusted the stellar parameters to $T_{\rm eff}=6150$ K, $\log g=1.75$ dex, $\nu_{t}=3.15\,\text{km}\,\text{s}^{-1}$, resulting in $\mbox{[M/H]}=-2.70$.
Total stellar parameter and abundance uncertainties were then determined the same way as the RGB stars.
We do not apply any abundance corrections for this star, as the correction grids are computed for cool giants.
CarII-V3 is one of the most metal-poor RRLs ever studied spectroscopically, with similar [Fe/H] as X Ari and the most Fe-poor RRLs in the LMC (For et al., 2011; Haschke et al., 2012; Nemec et al., 2013).
3.4 Abundance Summary
Our full abundance results are tabulated in Table 6 (Appendix B) and Figures 2 and 4.
We compare the results to halo stars in small grey points (Abohalima & Frebel, 2018), and to other UFD measurements in the literature.
The UFD literature compilation includes Bootes I (Feltzing et al., 2009; Norris et al., 2010; Gilmore et al., 2013; Ishigaki et al., 2014; Frebel et al., 2016), Bootes II (Ji et al., 2016a), Canes Venatici II (François et al., 2016), Coma Berenices (Frebel et al., 2010), Grus I (Ji et al., 2019a), Hercules (Koch et al., 2008, 2013), Horologium I (Nagasawa et al., 2018), Leo IV (Simon et al., 2010; François et al., 2016), Pisces II (Spite et al., 2018), Reticulum II (Ji et al., 2016c; Roederer et al., 2016), Segue 1 (Frebel et al., 2014), Segue 2 (Roederer & Kirby, 2014), Triangulum II (Ji et al., 2019a; Kirby et al., 2017; Venn et al., 2017), Tucana II (Ji et al., 2016b; Chiti et al., 2018a), Tucana III (Hansen et al., 2017; Marshall et al., 2018), and Ursa Major II (Frebel et al., 2010).
We reiterate that throughout this paper, the error bars for Car II and III include full propagation of the line-by-line statistical and stellar parameter uncertainties.
The RRL star CarII-V3 generally has consistent abundances with the RGB stars, although there are fewer lines and only moderate S/N so the abundance uncertainties for this star are fairly large.
The main outlier is the Si abundance, which is unusually low but has large uncertainty as it is measured only from the 3905Å line.
Given the abundance similarities to other stars in Car II, we will treat this star’s abundances on the same footing as RGB stars when lines are detected.
C, N, O. Carbon abundances are derived from synthesizing the CH bands at ${\sim}4300-4325$Å. CO molecular equilibrium affects CH abundances, and we always assume the MOOG default of [O/Fe]$=0$ even when O is measured independently.
Literature measurements suggest [O/Fe] is typically $>0.5$ (e.g., Brown et al., 2014). If we used [O/Fe]$=+1.0$ instead, [C/Fe] would typically increase by ${+}0.08$ dex with star-to-star scatter of 0.08 dex, but we keep the MOOG default for consistency with previously analyzed literature stars.
Nitrogen is derived from fitting CN bands at ${\sim}3850$Å after fixing the CH abundance.
In two relatively cool and metal-rich stars, we detect the two forbidden oxygen lines at ${\sim}6300$Å. These can only be measured when the O abundance is very high, so are probably a biased sample of measurements.
The stronger 6300Å line was deblended from telluric absorption, and the weaker 6363Å line can be affected by a wide calcium ionization feature (e.g., Barbuy et al., 2015). However in both cases, the two different lines give very close abundances.
We include oxygen upper limits for all stars (including the two detections) in the machine-readable version of Table 4 from the 6300Å line.
$\alpha$-elements: Mg, Si, Ca. The $\alpha$-element abundances are determined from equivalent widths in all stars. Magnesium is determined from 5-7 lines including the Mg b lines in all stars (except CarII-V3, where only the Mg b lines can be measured). The Mg b lines are quite strong and saturated but give similar abundances as the weaker lines for all stars.
Si is measured from both the 3905Å and 4102Å lines, but these are both rather poor-quality lines. The 3905Å line is fairly saturated, and the 4102Å line is in a Balmer wing.
Ca is usually measured from 10-20 lines with three exceptions: the warmer and more Fe-poor stars CarII-4928 and CarIII-1120 have only 2 and 1 Ca lines, respectively; and only the strong 4226Å line is detected in the RRL CarII-V3. We do not use the 4226Å line in any of the RGB stars due to large and uncertain NLTE corrections (e.g., Sitnova et al., 2019).
Odd-Z elements: Na, Al, K, Sc.
We use equivalent widths to measure sodium abundances from the two Na D lines, which have been corrected for NLTE effects.
We synthesize the 3944Å and 3961Å Al lines, which are both very strong and subject to NLTE effects so our Al abundances are very uncertain.
K abundances are mostly from the 7699Å line, although occasionally the 7665Å is not blended with tellurics.
Sc abundances are mostly measured with spectral synthesis from five lines at $4246<\lambda<4415$Å, though the redder line abundances (e.g. 5031Å, 5526Å) agree.
CarII-0064 is a significant low Sc outlier in Car II with [Sc/Fe] $\approx-1$ (Figure 2). We plot two Sc line spectrum in Figure 3, along with its synthetic fit and two other stars that have higher Sc abundances. The Sc abundance is clearly lower in CarII-0064, though visually not as much as would be expected from Figure 2.
This is because each individual line difference is significant at $\lesssim 2\sigma$, but they are all consistent and the combination of $5-6$ Sc lines reduces the uncertainty. Also note the [Sc/Fe] abundance error is smaller, due to correlated uncertainties in stellar parameters.
Such low Sc abundances have previously been seen in “iron-rich” stars (those with overall low [X/Fe] ratios, e.g., Cohen & Huang 2010; Cohen et al. 2013; Yong et al. 2013).
However, this cannot explain CarII-0064 because it is an outlier from the overall Car II trend only in [Sc/Fe].
Similarly Sc-deficient stars have been found in the bulge where it has been argued that this signature may indicate unusually old stars (Casey & Schlaufman, 2015), but we see no sign of this in the more Fe-poor stars in Car II.
It is unclear to us how to interpret this star’s extreme Sc abundance.
Fe-peak elements: Ti, Cr, Mn, Co, Ni, Zn.
We use equivalent widths to measure abundances for both ionization states of titanium, but we adopt the Ti II abundances everywhere as our default; it is measured in all our stars, has more and stronger lines, and is less susceptible to NLTE effects.
The Fe-peak elements closely follow the halo trends within their abundance uncertainties.
There are minor deviations that are all significant at $<2\sigma$, so we do not concern ourselves with these further, other than to comment that Zn could be moderately enhanced in Car III and moderately deficient in Car II.
Neutron-capture elements: Sr, Ba. These elements have low abundances or upper limits, similar to most other UFDs.
The nucleosynthetic origin of these very low Sr and Ba abundances remains unknown (it is in general not even clear if they are from the $r$- or $s-$processes, see Ji et al. (2019a) for an extensive discussion), but it appears to be unique to UFDs and occasional halo stars that are presumably stripped from UFDs.
Given the low abundance of neutron-capture elements, no other neutron-capture elements could be detected, so we place [Eu/Fe] upper limits and show [Ba/Eu] in Figure 4.
There are two stars in Car II with relatively high $\mbox{[Ba/Fe]}\gtrsim-1$ compared to the other Car II stars.
One of these relatively Ba-rich stars, CarII-7872, also has a low Eu upper limit that results in $\mbox{[Ba/Eu]}\gtrsim 0$, suggesting its Ba is predominantly from the $s$-process (e.g., Sneden et al., 2008).
We discuss this large barium scatter in Section 4.4.
4 Formation history of Carina II and III
4.1 Carina II and III are Dwarf Galaxies
Low luminosity stellar systems are classified as either dwarf galaxies or star clusters. Dwarf galaxies are generally more spatially extended than clusters, with velocity dispersions implying significant dark matter content and nonzero metallicity (or more specifically, iron-peak abundance) dispersions (Willman & Strader, 2012).
Faint dwarf galaxies also tend to display very low abundances of neutron-capture elements (e.g., Ji et al., 2019a), while globular clusters have light element anticorrelations associated with hot bottom burning (e.g., Bastian & Lardo, 2018).
Both Carina II and III are clearly dwarf galaxies and not globular clusters.
Their half-light radii and luminosities place them within the dwarf galaxy morphological locus (Torrealba et al., 2018).
Carina II displays both a significant velocity and metallicity dispersion from medium-resolution data (Li et al., 2018).
Our two Carina III stars have [Fe/H] values that differ by almost 2 dex, definitively establishing a significant metallicity dispersion. We have also now resolved the velocity dispersion (Li et al., in prep).
The neutron-capture elements Sr and Ba are low in both systems, like nearly every other UFD (Figure 4).
These criteria alone already show that Car II and III are galaxies, but as a final confirmation we show there are no light element anticorrelations.
Figure 5 shows these relations for our stars.
In the top two panels, we show Na-Mg and Al-Mg for our UFD stars (symbols as in Figure 2) and globular cluster stars as purple circles (from references Carretta et al., 2007, 2009; Gratton et al., 2006; Cohen & Kirby, 2012).
Most globular clusters do not show significant dispersion in [Mg/Fe], but those that do always display an anti-correlation in Na-Mg and Al-Mg.
In contrast, there is very clearly a positive correlation for these elements in both Car II and III.
Note that Na and Mg have NLTE corrections applied, while the Al corrections should on average provide an offset and are unlikely to turn a strong positive Mg-Al correlation into an anticorrelation.
The bottom panel of Figure 5 shows the Mg-K anticorrelation found in NGC 2419 (Mucciarelli et al., 2012), which is not present in Car II. However, our two stars in Car III (including one K upper limit) do not rule out an Mg-K anticorrelation in this system.
4.2 Car II and III are consistent with being accreted along with the LMC/SMC
Li et al. (2018) showed that the positions and radial velocities of both Car II and Car III were consistent with having accreted with the LMC, according to the Jethwa et al. (2016) model.
Kallivayalil et al. (2018) then added proper motion data from Gaia DR2 (Gaia Collaboration et al., 2018, 2016), finding that Car II and Car III are also likely LMC satellites based on LMC analogues in the Aquarius simulations (Springel et al. 2008, also see Sales et al. 2017; Simon 2018; Erkal & Belokurov 2019).
Kinematically, it thus appears likely that both Car II and Car III entered the Milky Way with the LMC/SMC system, although Car II is towards the edge of the likely region due to its high velocity.
Kallivayalil et al. (2018) also associate Hyi I and Hor I with the LMC.
Thus, Car II and III, along with Hor I (Nagasawa et al., 2018), can be studied in contrast to other UFDs to see if abundance ratios have any environmental dependence.
Nagasawa et al. (2018) point out that the three stars in Hor I have unusually low Mg and Ca, with one possible explanation being that LMC satellites might have typically different enrichment histories compared to Milky Way UFDs.
Figure 2 does not suggest that Car II or Car III obviously deviate from the typical abundance scatter of other UFDs, including for Mg and Ca. The unusually low Mg and Ca in Hor I thus likely has some other origin.
4.3 $\alpha$-element evolution: time delay scenario or initial mass function variations?
4.3.1 $\alpha$-element abundance ratios in Car II and Car III
The $\alpha$-elements (O, Mg, Si, Ca)
are primarily produced in core-collapse supernovae (CCSNe) and thus tend to be enhanced at low [Fe/H].
After a delay of $100-1000$ Myr (Maoz et al., 2014), Type Ia supernovae (SNe1a) begin to add Fe peak elements, causing a “knee” in [$\alpha$/Fe] vs [Fe/H] (Tinsley, 1979).
In this time delay scenario, the location of the knee can be interpreted as an overall star formation timescale for a galaxy (e.g., Tolstoy et al., 2009; Kirby et al., 2011).
Figure 2 shows clear downward trends in [Mg/Fe] and [Ca/Fe] vs. [Fe/H] for both Car II and III, with a possible knee at $\mbox{[Fe/H]}\sim-2.8$ for Car II that would indicate very slow chemical evolution in this low mass galaxy.
However, there is a striking difference in the size of the trend for [Mg/Fe] and [Ca/Fe]: [Mg/Fe] declines by over 1 dex, while [Ca/Fe] declines by only about 0.4 dex.
We will focus primarily on Car II, because Car III has only two stars and the more Fe-poor star has only one Ca line.
To clarify the Mg and Ca difference, in the top panel of Figure 6 we plot [Mg/Ca] vs [Fe/H], where [Mg/Ca] declines from about $+0.4$ to $-0.4$ as [Fe/H] increases from $-3.5$ to $-2.2$.
These extreme [Mg/Ca] ratios are often interpreted as variations in the high mass end of the initial mass function.
Stars with high [Mg/Ca] ratios are typically associated with enrichment by very massive stars with $M>20-30M_{\odot}$ (e.g., Norris et al., 2000; Cohen et al., 2007; Koch et al., 2008, also see Section 5.2).
Stars with $\mbox{[Mg/Ca]}<0$ form out of gas enriched by lower mass CCSN progenitors with $M\lesssim 15M_{\odot}$ (e.g., Tolstoy et al., 2003; McWilliam et al., 2013).
The variable [Mg/Ca] ratios in Car II may thus indicate that the $\alpha$-elements in this galaxy is tracing changes in the high-mass end of the initial mass function (IMF).
Indeed, the low-mass end of the IMF in UFDs has previously been shown to vary between different UFDs (Geha et al., 2013; Gennaro et al., 2018), which tantalizingly hints that the high-mass end of the IMF might vary as well (although the low-mass IMF varies from galaxy to galaxy, while here we consider time variations within a single galaxy, so the mechanisms may not be related).
In the bottom panels of Figure 6, we plot [Mg/H] and [Ca/H] vs [Fe/H], which shows that there may actually be two phases of [Mg/Ca] evolution: from $\mbox{[Fe/H]}=-3.6$ to $-3.0$ this is primarily driven by a smaller increase in [Mg/H] than [Ca/H]; while from $\mbox{[Fe/H]}=-3.0$ to $-2.2$, [Mg/H] stays mostly flat while [Ca/H] increases.
The first phase unambiguously shows that Car II has been enriched by at least two different masses of CCSNe: the most Fe-poor star in Car II has high [Mg/Ca] ratios suggesting enrichment by high mass stars, but it has lower [Mg/H] than the higher metallicity stars. Since SNe1a produce negligible Mg, this means that CCSNe with $\mbox{[Mg/Ca]}\sim 0$ must have enriched Car II after the formation of the most Fe-poor star.
This could potentially be evidence of a transition from very massive Pop III stars to regular mass Pop II CCSNe.
The second phase of evolution could be attributed to either IMF variation or SN1a enrichment.
To illustrate this, we show an extremely simple chemical evolution track in Figure 6. First, we set an initial [Mg/H], [Ca/H], and [Fe/H] that matches the [Mg/Ca] ratio at $\mbox{[Fe/H]}=-3$ (black square).
Then, we assume a fixed [Ca/Fe] yield and negligible Mg yield for SNe1a (Kirby et al., 2019), and compute the evolution of Mg, Ca, and Fe assuming no more CCSNe and no gas accretion/expulsion.
Kirby et al. (2019) have recently made an empirical measurement of the SN1a [Ca/Fe] yield in larger dSph galaxies, finding values that range between $-0.5<\mbox{[Ca/Fe]}<0.0$.
We thus apply our simple model with SN1a yields of [Ca/Fe] $=0.0$ and $-0.5$, which are shown as black solid and dotted lines respectively in Figure 6 and reasonably match the observed Mg and Ca ratios.
This would be quite an extreme situation: if most of the metal enrichment in Car II is due to SNe1a and not CCSNe, but stars still formed to sample the SN1a yields, that implies an extremely top-light IMF where no massive stars formed.
However, this is definitely not a unique model, and specifically the flat [Mg/H] trend does not rule out contributions from additional CCSNe because gas accretion can increase the hydrogen reservoir (e.g., Ji et al., 2016a).
Detailed chemical evolution modeling of more elements might help clarify the picture but is beyond the scope of this paper.
Furthermore, stochastic sampling of individual SN explosions may dominate the observed trends (e.g., Koch et al., 2008, 2013; Revaz et al., 2016; Applebaum et al., 2018), especially given that Car II produced only ${\sim}100$ CCSNe in total (assuming a Salpeter initial mass function and present-day mass-to-light ratio of 2.2, Ji et al. 2016a). Car III is even more susceptible to stochastic enrichment, having been enriched by only ${\sim}15$ supernovae. We thus caution against over-interpreting the available data.
4.3.2 [Mg/Ca] abundances across the UFD population
Some more insight can be derived by comparing the [Mg/Ca] vs [Fe/H] trends of Car II to the trends in other UFDs.
It turns out that few other UFDs have similarly negative [Mg/Ca] vs [Fe/H] slopes.
To quantify this result, we fit lines to the [Mg/Ca] vs [Fe/H] evolution of every UFD individually, and consider the slope angle (i.e., $0^{\circ}$ corresponds to a flat line, and negative slope angles indicate declining [Mg/Ca] as [Fe/H] increases).
We then calculate the slopes and slope uncertainties by assuming that data points are drawn from a thin line with multivariate Gaussian uncertainties (see section 7 of Hogg et al., 2010).
We take a uniform prior in slope angle (as opposed to slope) for $\theta\in[-90^{\circ},+90^{\circ})$ and a flat prior for the intercept, then use emcee to sample the posterior (Foreman-Mackey et al., 2013).
We take the posterior median as the point estimate and the 16th-84th percentile range as the 68% credible interval.
We remove the four UFDs that have unconstrained posteriors (since their stars have essentially the same [Fe/H]).
Note that the literature UFD stars have inhomogenously determined uncertainties, so we instead assume independent error bars of 0.2 dex for both [Fe/H] and [Mg/Ca]; but use our actual abundance uncertainties for Car II and Car III.
The [Mg/Ca] vs [Fe/H] slopes for all UFDs where ${\geq}2$ stars have detailed abundance measurements are shown in Figure 7.
The top panel of Figure 7 shows the UFD [Mg/Ca] slopes vs luminosity (luminosities from the Simon 2019 compilation, including data from Bechtol et al. 2015; Muñoz et al. 2018; Torrealba et al. 2018; Mutlu-Pakdil et al. 2018).
There is not an obvious relation between slope angle and luminosity.
The bottom panel shows a histogram of the slope angle point estimates from the top panel.
Many UFDs have too few stars to place a useful slope constraint, so we shade each UFD in the histogram by the number of stars used to calculate the slope, with darker colors indicating more stars. The UFDs with the most confident measurements (i.e., ${\geq}7$ stars with detailed abundances) are Car II (this work), Ret II (Ji et al., 2016c), Bootes I (Frebel et al., 2016), Segue 1 (Frebel et al., 2014), and Tuc II (Chiti et al., 2018a).
We also highlight the slope of Car II and Car III as a vertical solid red line and vertical orange dashed line, respectively.
Of the other UFDs, only Ret II exhibits a declining [Mg/Ca] slope that deviates from zero by $\gtrsim 1\sigma$.
4.3.3 Effect of environment on [Mg/Ca] abundances
The results above raise an interesting question about the role of environment in determining abundance trends:
Car II and III are LMC satellites, and Ret II is also a candidate LMC satellite
(Kallivayalil et al., 2018; Erkal & Belokurov, 2019)888Hor I (Nagasawa et al., 2018) also is an LMC satellite, but all three currently observed stars have $\mbox{[Fe/H]}\sim-2.6$ within uncertainties and thus no useful constraint on its [Mg/Ca] vs [Fe/H] trend. The three Hor I stars all have $\mbox{[Mg/Fe]}\approx\mbox{[Ca/Fe]}\approx 0$..
In the bottom panel of Figure 7, we show the [Mg/Ca] vs [Fe/H] slope angles from grouping all LMC UFD stars and all MW UFD stars.
It is very obvious that the LMC satellite UFD stars have a significant negative slope, while the MW satellite UFD stars have a flat slope; though we note that the LMC trend is mostly driven by Car II and should await additional abundances in LMC satellite UFDs to clarify this suggestion.
However, we speculate briefly on how the large scale environment could possibly affect chemical evolution in UFDs.
At first glance, UFDs should not display significant environment dependence.
UFDs form most of their stars by $z\sim 6$ (Brown et al., 2014), and in simulations the closest more massive galaxy at $z>6$ is typically 400 physical kpc away (Wetzel et al. 2015).
Even generously sized galactic superbubbles reach only tens of kpc (Griffen et al., 2018), so external enrichment or directly affecting UFD gas with ram pressure stripping is unlikely (Wetzel et al., 2015).
However, radiation (both ionizing and Lyman-Werner) can span these distances, though there are limited ways we can imagine this would affect stellar populations.
At the metal-rich end, one possibility is the integrated galactic IMF theory (IGIMF, e.g., Weidner et al., 2013; McWilliam et al., 2013), which suggests that as galaxies become gas-poor they cannot form the most massive stars. If LMC UFDs formed later and thus reionized earlier in their evolution, they would form more of their stars in this phase.
At the metal-poor end, delaying Pop III star formation with Lyman-Werner feedback may increase susceptibility of UFD progenitors to external enrichment (e.g., Magg et al., 2018).
Also, metal-free gas with relatively high ionization fractions can form HD molecules during collapse, which may (or may not) affect the Pop III initial mass function (Glover, 2013).
A final note is that the distance scales from Wetzel et al. (2015) assume that UFDs reside in dark matter halos of $M_{\rm peak}\sim 10^{9}M_{\odot}$ (Wetzel et al., 2015).
If instead UFDs reside in smaller dark matter halos of $M_{\rm peak}\sim 10^{7-8}M_{\odot}$ (e.g., Jeon et al. 2014; Ji et al. 2015; Jethwa et al. 2018; Graus et al. 2019), then separation distances would become smaller and environmental effects could be more important.
4.4 Inhomogenous metal mixing of AGB winds in Car II
There is real scatter in [Ba/Fe] at $\mbox{[Fe/H]}\sim-2.5$ in Car II, with some stars having relatively high Ba abundances and others having low Ba abundances (Figure 4). The extent of the scatter in Ba is ${\sim}1$ dex, much larger than the scatter in any other abundance ratio.
A plausible explanation for the Ba scatter is inhomogeneous mixing of AGB wind ejecta into the galaxy’s ISM. Unlike supernova ejecta, which mix rapidly upon entering the hot phase of the ISM, AGB winds mix into relatively cool ISM phases and can thus stay quite inhomogeneous (Emerick et al., 2018, 2019).
Since Ba is produced by the $s$-process and released in AGB winds, this mechanism could explain the large Ba scatter.
This scenario is supported by the fact that one of the high-Ba stars (CarII-7872) has $\mbox{[Ba/Eu]}\gtrsim 0$ (Figure 4), suggesting its Ba is predominantly from the $s$-process.
Since most barium comes from AGB stars with initial mass $M\leq 4M_{\odot}$ and lifetimes $\geq 10^{8}$ years, the presence of AGB enrichment requires that Car II formed stars for at least ${\sim}100$ Myr (Lugaro et al., 2012; Karakas & Lugaro, 2016).
Note that the nucleosynthetic origin of the low Sr and Ba floor in UFDs remains unknown (see Ji et al., 2019a, for more discussion).
One might also expect a correlation between Ba and other AGB elements like C. We find a moderate but not statistically significant correlation between stars that have both Ba and C detected in Car II (correlation of $0.48$ with a $p$-value of $0.34$ from scipy.stats.pearsonr).
5 Population III Star Signatures
5.1 Carbon-enhanced fraction in UFDs
Carbon-Enhanced Metal-Poor (CEMP) stars are stars with high [C/Fe] ratios (Beers & Christlieb, 2005).
Below $\mbox{[Fe/H]}\sim-3$, about half the stars in the Milky Way halo are CEMP stars (i.e., $\mbox{[C/Fe]}\gtrsim+0.7$, Aoki et al. 2007).
It is generally thought that a specific subclass (CEMP-no stars; Beers & Christlieb 2005)999The “no” is short for “no strong enhancement of neutron-capture elements”. of the CEMP stars traces unique nucleosynthesis in Pop III stars (e.g., Norris et al., 2013; Frebel & Norris, 2015; Placco et al., 2016).
If so, the observed CEMP fraction provides a window to the distribution of some Pop III star properties, such as initial mass, explosion energy, or stellar rotation (e.g., Cooke & Madau, 2014; Ji et al., 2015).
In Figure 8, we show the fraction of carbon-enhanced stars below a given [Fe/H] in our ${\gtrsim}80$ star UFD literature sample and the halo star compilation by Placco et al. (2014).
Both samples have included the Placco et al. (2014) evolutionary carbon corrections.
For the UFD sample, we show 68% Wilson confidence intervals on the CEMP fraction.
Figure 8 shows that the carbon-enhanced fraction in UFDs is essentially identical to halo stars at all levels of carbon enhancement.
For comparison, the CEMP fraction in larger dwarf galaxies like Sculptor has been studied in some detail (e.g., Skúladóttir et al., 2015; Salvadori et al., 2015; Chiti et al., 2018b), but it is still debated whether the CEMP fraction in those galaxies is consistent with the halo.
If we are after pure Pop III signatures, it also makes sense to look at entire UFDs as either C-rich or C-normal (Ji et al., 2015).
Seven UFDs have stars with $\mbox{[Fe/H]}<-3$. The most metal-poor stars in five of these UFDs are C-rich (Car III, Segue 1, Boo I, Tuc II, UMa II), while the other two are C-normal (Ret II, Car II).
This suggests that the fraction of Pop III stars producing carbon-enhanced abundances is $0.71_{-0.19}^{+0.13}$, following the simple models in Ji et al. (2015).
A more stringent cut of $\mbox{[Fe/H]}<-3.5$ results in three C-enhanced galaxies out of five, or a carbon-enhanced rate of $0.60^{+0.34}_{-0.39}$.
More to the point, the existence of carbon-normal stars with $\mbox{[Fe/H]}\lesssim-3.5$ in Ret II and Car II is evidence against the hypothesis that 100% of Pop III stars produce carbon-enhanced signatures, as is often assumed in theoretical models and simulations (e.g., Salvadori et al., 2015; Jeon et al., 2017).
5.2 Full fits to individual UFD stars
The two stars CarII-5664 and CarIII-1120 have low enough [Fe/H] that they are plausibly enriched only by Pop III stars (e.g., Frebel & Norris, 2015).
Under this assumption, we fit models from Heger & Woosley (2010) to the data to estimate the initial progenitor mass, explosion energy, internal mixing, and gas dilution mass for these stars.
To summarize the fitting procedure, we find the optimum dilution mass for all 16800 models in the Heger & Woosley (2010) grid, reject all models inconsistent with our upper limits, then weight each remaining model by using its deviation from the best-fit $\chi^{2}$ as input to a $\chi^{2}$ survival function with 4 degrees of freedom.
The detailed fitting procedure and parameter description is described in Frebel et al. (2019)101010Code at https://github.com/alexji/alexmods/blob/master/alexmods/alex_starfit.py.
Here, we exclude the elements Al, K, and Mn due to the uncertain size of NLTE corrections; and the elements Sc, Cr, Cu, and Zn due to model calculation uncertainties (Heger & Woosley, 2010).
Abundance corrections to C, Na, and Mg have been included (Table 5).
We note that the Heger & Woosley (2010) models do not include stellar rotation. However, rotation can substantially influence stellar evolution and the resulting nucleosynthesis (e.g., Maeder et al., 2015) and should be considered in future analyses.
The results are shown in Figure 9. We plot all models within $2\sigma$ contours of $\chi^{2}$ (i.e., models with weight $\gtrsim 0.05$).
In the top panel for each star, we show the data as filled red squares with error bars and upper limits as downward pointing arrows. Unused measurements and upper limits are indicated as open squares and downward pointing triangles, respectively.
The best-fit model is shown as a solid blue line, while other models within $2\sigma$ are shown as black lines.
For visualization purposes, models with worse $\chi^{2}$ are plotted as thinner transparent lines.
The bottom left panel for each star shows the weighted histogram for the resulting progenitor masses of the full fit.
The bottom right panel shows the best-fit energy and dilution masses, where again models with worse $\chi^{2}$ are displayed as smaller and more transparent points. The best fit model is again shown as a solid blue point.
In general, satisfactory fits were found for these two stars with $\mbox{[Fe/H]}<-3.5$.
CarIII-1120 is most consistent with a relatively low mass progenitor between $10-20M_{\odot}$ with a typical ${\sim}1\times 10^{51}$ erg explosion energy. Note that CarIII-1120 is a Group 2 CEMP-no star according to Yoon et al. (2016).
CarII-5664 is also best fit by a similar low-mass progenitor, but most of the best-fit models actually prefer a higher mass progenitor of $25-35M_{\odot}$ with slightly higher explosion energy.
The combination of explosion energy and dilution mass introduces another consistency check.
A supernova with explosion energy $E$ will produce a supernova remnant that sweeps up a certain amount of mass before merging with the ISM (e.g., Cioffi et al., 1988). This is the minimum dilution mass allowable for that explosion energy (assuming no rare interactions such as colliding supernova blastwaves).
In the bottom right panels of Figure 9 we show the approximate swept-up mass of a supernova remnant expanding into an efficiently cooling ISM $M_{\rm dil,H}=0.75\times 10^{4.5}M_{\odot}(E/10^{51}\,\text{erg})^{0.95}$ as a dotted red line (Cioffi et al., 1988; Ryan et al., 1996).
Models below this line are inconsistent with the explosion energy (though could be explained with enrichment by multiple supernovae), while models above the line are diluted beyond the supernova remnant due to turbulent mixing.
Applying this constraint tends to prefer higher explosion energies and higher masses.
In general, the best-fit dilution masses satisfying this constraint are ${\sim}10^{5}M_{\odot}$, suggesting that recollapsed gas within a minihalo is the most likely explanation for the origin of these stars rather than external pollution, as externally polluted halos have higher effective dilution masses (e.g., Cooke & Madau, 2014; Ji et al., 2015; Smith et al., 2015; Griffen et al., 2018).
6 Conclusion
We present a comprehensive abundance analysis of the Magellanic satellite galaxies Carina II and Carina III using high-resolution Magellan/MIKE data, including the first abundances of an RR Lyrae star in any UFD.
The abundance results are shown in Figures 2 and 4.
The stars in these two dwarf galaxies clearly do not show light element anticorrelations associated with globular clusters (Figure 5).
The most notable chemical evolution trend is the variations in different $\alpha$-element ratios.
Car II clearly shows different trends in [Mg/Fe] and [Ca/Fe] (Figure 6).
The origin of this evolution could be differences in core collapse and/or Type Ia supernova yields, and it is not yet clear which.
However, there are obvious differences in the [Mg/Ca] trends between different UFDs (Figure 7), and we tentatively suggest this could be an environment-dependent abundance signature as LMC satellite UFDs have a different trend than MW satellite UFDs.
This suggestion will require studying the abundances of additional LMC satellites to confirm.
The most metal-poor stars in UFDs may contain signatures of the first metal-free Population III stars.
Studying the whole population of Fe-poor UFD stars, we find that the carbon-enhanced fraction of UFD stars is essentially the same as the Milky Way halo (Figure 8).
But, not all of the most Fe-poor stars in UFDs are carbon-enhanced: the most Fe-poor star in Car II is clearly carbon-normal.
We also found two new stars with $\mbox{[Fe/H]}\leq-3.5$, bringing the total number of such stars in UFDs up to 8. The abundances of these stars are well-fit by Pop III core-collapse supernova yields (Figure 9).
Our analysis of Car II and III, along with the past decade of observations, brings the total number of UFD stars with high-resolution abundances up to ${\sim}85$ stars across $16$ different UFDs, of which now 5 UFDs have a “large” (${\geq}7$) number of stars studied (see references in Section 3.4).
While these data have already provided key insights into early nucleosynthesis and galaxy formation and pointed to many interesting abundance trends and signatures, the numbers of stars are still relatively small.
These sample sizes are currently dictated by the limits of current large telescopes, but 30m class telescopes will allow high-resolution spectroscopic abundances for 10s$-$100s of stars per UFD out to the virial radius of the Milky Way (Ji et al., 2019b), transforming our ability to unravel the detailed history of these first galaxy relics.
We thank Andy McWilliam, Evan Kirby, Ian Thompson, George Preston, Dan Kelson, Andrew Emerick, and Thomas Nordlander, and Chris Sneden for fruitful discussions; and Eduardo Bañados for saving our RRL observations from certain doom.
APJ and TSL are supported by NASA through Hubble Fellowship grant HST-HF2-51393.001 and HST-HF2-51439.001 respectively, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.
JDS is supported by the National Science Foundation under grant AST-1714873.
SK is partially supported by NSF awards AST-1813881 and AST-1909584.
The work of ABP is supported by NSF grant AST-1813881.
This research has made use of NASA’s Astrophysics Data System Bibliographic Services.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.
This project used data obtained with the Dark Energy Camera (DECam),
which was constructed by the Dark Energy Survey (DES) collaboration.
Funding for the DES Projects has been provided by
the U.S. Department of Energy,
the U.S. National Science Foundation,
the Ministry of Science and Education of Spain,
the Science and Technology Facilities Council of the United Kingdom,
the Higher Education Funding Council for England,
the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign,
the Kavli Institute of Cosmological Physics at the University of Chicago,
the Center for Cosmology and Astro-Particle Physics at the Ohio State University,
the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University,
Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovacão,
the Deutsche Forschungsgemeinschaft,
and the Collaborating Institutions in the Dark Energy Survey.
The Collaborating Institutions are
Argonne National Laboratory,
the University of California at Santa Cruz,
the University of Cambridge,
Centro de Investigaciones Enérgeticas, Medioambientales y Tecnológicas-Madrid,
the University of Chicago,
University College London,
the DES-Brazil Consortium,
the University of Edinburgh,
the Eidgenössische Technische Hochschule (ETH) Zürich,
Fermi National Accelerator Laboratory,
the University of Illinois at Urbana-Champaign,
the Institut de Ciències de l’Espai (IEEC/CSIC),
the Institut de Física d’Altes Energies,
Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe,
the University of Michigan,
the National Optical Astronomy Observatory,
the University of Nottingham,
the Ohio State University,
the OzDES Membership Consortium
the University of Pennsylvania,
the University of Portsmouth,
SLAC National Accelerator Laboratory,
Stanford University,
the University of Sussex,
and Texas A&M University.
Based on observations at Cerro Tololo Inter-American Observatory, National Optical
Astronomy Observatory (NOAO Prop. ID 2016A-0366 and PI Keith Bechtol), which is operated by the Association of
Universities for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation.
This project is partially supported by the NASA Fermi Guest Investigator Program Cycle 9 No. 91201.
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Appendix A Abundance Error Analysis Formalism
Here we explicitly list the equations used for our error analysis.
For element $X$, with lines indexed by $i$ that have abundances $A_{i}$, statistical error $\sigma_{i,\text{stat}}$, and systematic abundance offsets $\delta_{i,T_{\rm eff}}$, $\delta_{i,\log g}$, $\delta_{i,\nu_{t}}$ and $\delta_{i,\text{[M/H]}}$ (note that the systematic abundance offsets retain their sign so we refer to them as $\delta_{i}$):
$$\displaystyle\sigma_{i,\text{sys}}^{2}$$
$$\displaystyle=\delta_{i,T_{\rm eff}}^{2}+\delta_{i,\log g}^{2}+\delta_{i,\nu_{%
t}}^{2}+\delta_{i,\text{[M/H]}}^{2}$$
(A1)
$$\displaystyle\equiv\sum_{SP}\delta_{i,SP}^{2}$$
(A2)
$$\displaystyle\sigma_{i}^{2}$$
$$\displaystyle=\sigma_{i,\text{stat}}^{2}+\sigma_{i,\text{sys}}^{2}$$
(A3)
The statistical error $\sigma_{i,\text{stat}}$ quantifies the spectrum noise, either through the $1\sigma$ equivalent width uncertainty or $\chi^{2}$ uncertainty for synthesis.
Our equivalent width and synthesis fits allow the local continuum to vary by a linear function, using $\chi^{2}$ minimization to find the continuum level. Our quoted statistical uncertainties $\sigma_{i,\text{stat}}$ propagate these continuum uncertainties, and they match those inferred from simpler formulas based on the line FWHM within 5% (e.g., Battaglia et al., 2008; Frebel et al., 2006).
It is in principle possible that our local spectrum models are not accurate, and
the most impactful systematic would be misplacing the overall continuum level. As an extra conservative error bar, we include an additional column $\sigma_{\rm cont}$ in Table 4, which is the uncertainty from systematically changing the overall continuum by the local 1$\sigma$ spectrum noise (i.e., the abundance difference after multiplying each equivalent width by $1\pm 1/\text{SNR}$).
For synthesis measurements, we estimate this uncertainty by calculating the equivalent width of the synthetic feature without any other elements, then treating it as an equivalent width measurement. We thus did not estimate the continuum error for the molecular features.
A very conservative error estimate would also add this error in quadrature as part of equation A3.
However, we are confident that our continuum placement procedure uncertainties are accurately reflected in the statistical error bar, so we do not include $\sigma_{\rm cont}$ in our abundance uncertainties.
We then assign each line a weight $w_{i}$
$$w_{i}=\sigma_{i}^{-2}$$
(A4)
We adopt the weighted average of the lines as the final abundance, with statistical and systematic uncertainties:
$$\displaystyle A(X)$$
$$\displaystyle=\frac{\sum_{i}w_{i}A_{i}}{\sum_{i}w_{i}}$$
(A5)
$$\displaystyle\sigma_{\text{stat}}^{2}(X)$$
$$\displaystyle=\frac{\sum_{i}w_{i}(A_{i}-A(X))^{2}}{\sum_{i}w_{i}}+\frac{1}{%
\sum_{i}w_{i}}$$
(A6)
$$\displaystyle\delta_{\text{sys},SP}(X)$$
$$\displaystyle=\frac{\sum_{i}w_{i}(A_{i}+\delta_{i,SP})}{\sum_{i}w_{i}}-A(X)$$
(A7)
$$\displaystyle=\frac{\sum_{i}w_{i}\delta_{i,SP}}{\sum_{i}w_{i}}$$
(A8)
The total statistical uncertainty accounts for both noise in individual lines as well as the weighted standard error of different lines.
Here we adopt just the first order Taylor expansion for the stellar parameter uncertainty, neglecting covariance between stellar parameters (see McWilliam et al. 2013).
Finally, the total abundance error for [X/H] and element ratios [X/Y] combines the statistical and systematic uncertainties in quadrature:
$$\displaystyle\sigma_{\text{[X/H]}}^{2}$$
$$\displaystyle=\sigma_{\text{stat}}^{2}+\sum_{SP}\delta_{\text{sys},SP}^{2}$$
(A9)
$$\displaystyle\sigma_{\text{[X/Y]}}^{2}$$
$$\displaystyle=\sigma_{X,\text{stat}}^{2}+\sigma_{Y,\text{stat}}^{2}+\sum_{SP}%
\left(\delta_{X,SP}-\delta_{Y,SP}\right)^{2}$$
(A10)
Note that for an element ratio of X and Y, we only allow covariance between $X$ and $Y$ through the stellar parameters.
Appendix B Abundance Tables
\startlongtable |
Confined Vortices in Topologically Massive U(1)$\times$U(1) Theory
Mohamed M. Anber
[email protected]
Yannis Burnier
[email protected]
Eray Sabancilar
[email protected]
Mikhail Shaposhnikov
[email protected]
Institut de Théorie des Phénomènes Physiques, EPFL,
CH-1015 Lausanne, Switzerland.
(November 21, 2020)
Abstract
We report on a new topological vortex solution in U(1)$\times$U(1) Maxwell-Chern-Simons theory.
The existence of the vortex is envisaged by analytical means, and a numerical solution is obtained by integrating the equations of motion. These vortices have a long-range force because one of the U(1)s remains unbroken in the infrared, which is guarded by the Coleman-Hill theorem. The sum of the winding numbers of an ensemble of vortices has to vanish; otherwise the system would have a logarithmically divergent energy. In turn, these vortices exhibit classical confinement. We investigate the rich parameter space of the solutions, and show that one recovers the Abrikosov-Nielsen-Olesen, U(1) Maxwell-Chern-Simons, U(1) pure Chern-Simons and global vortices as various limiting cases. Unlike these limiting cases, the higher winding solutions of our vortices carry non-integer charges under the broken U(1). This is the first vortex solution exhibiting such behavior.
I Introduction
Vortices are topological defects that were first discussed in the context of type-II superconductors by Abrikosov Abrikosov (1957), where the core of a vortex is in the normal fluid phase whereas outside the core is in the superfluid phase. The relativistic generalization of vortices was given by Nielsen and Olesen Nielsen and Olesen (1973) for the Abelian Higgs model. Vortices arise in field theories with degenerate vacuum manifolds, whose first homotopy group is non-trivial, $\pi_{1}[\mathcal{M}]\neq I$. According to Kibble’s classification Kibble (1976), e.g., the degenerate vacuum of a spontaneously broken $\rm{U(1)}$ theory has $\pi_{1}[{\rm U(1)}]=\mathbb{Z}$ (see e.g., Refs. Vilenkin and Shellard (1994); Hindmarsh and Kibble (1995) for reviews).
The Abrikosov-Nielsen-Olesen (ANO) vortex has no electric charge, but has quantized magnetic flux, $\Phi_{B}=2\pi n/e$, where $e$ is the gauge coupling constant and $n\in{\mathbb{Z}}$ is the winding number of the Higgs field corresponding to different topological sectors classified by $\pi_{1}[{\rm U(1)}]$. As both the gauge and scalar fields are short range, they do not exhibit long range interactions.
Interesting vortex solutions accompany the addition of a Chern-Simons term Deser et al. (1982a, b) $\int d^{3}x~{}\mu~{}\epsilon^{\alpha\beta\gamma}A_{\alpha}F_{\beta\gamma}$, which breaks the $P$ and $T$ invariance of the theory and gives a mass to the photon.
It was shown in Ref. Paul and Khare (1986) that if a Chern-Simons term is added to the Abelian Higgs model, the vortices carry both a quantized magnetic flux $\Phi_{B}=2\pi n/e$ and charge $Q=\mu\Phi_{B}$, where $\mu$ is the Chern-Simons coefficient. Similar to the ANO vortex, the interaction is short range and the charge is screened as the gauge field is higgsed (see, e.g., Refs. Dunne (1998); Horvathy and Zhang (2009) for a review of various applications of Chern-Simons vortices).
Generally, Chern-Simons terms will appear in the context of finite temperature four-dimensional gauge theories such as the standard electroweak theory Laine and Shaposhnikov (1999). Upon dimensionally reducing from four to three dimensions and integrating out the fermions, non-zero Matsubara modes of the gauge bosons and the zero Matsubara mode of the temporal component of the gauge fields one obtains Chern-Simons terms Redlich and Wijewardhana (1985). They are also used as effective field theory models to study the quantum Hall effect Frohlich and Kerler (1991); Frohlich and Zee (1991). Here, we specifically consider $\rm{U(1)}_{\scriptscriptstyle Z}\times\rm{U(1)}_{\scriptscriptstyle A}$ theory with a Chern-Simons mixing term, as given by the action (4). In fact, the Chern-Simons mixing term, $\mu_{1}\epsilon^{\mu\nu\alpha}\mathcal{F}_{\mu\nu}\mathcal{Z}_{\alpha}$, in (4) is the $2+1$ dimensional version of the BF theory Horowitz (1989).
In this work, we report on a new class of vortex solutions in $\rm{U(1)}_{\scriptscriptstyle Z}\times\rm{U(1)}_{\scriptscriptstyle A}$ Maxwell-Chern-Simons theory. One of ${\rm U(1)}$s is spontaneously broken by a complex scalar field, whereas the other remains unbroken. As a result, the new vortex is charged under the unbroken $\rm{U(1)}_{\scriptscriptstyle A}$, in addition of being charged under the broken $\rm{U(1)}_{\scriptscriptstyle Z}$, and it mediates a long-range force. Therefore, an ensemble of vortices and antivortices will be confined to minimize the energy of the system. This is the dynamical realization of the classical confinement that was pointed out by Cornalba and Wilczek Cornalba and Wilczek (1997) and de Wild Propitius de Wild Propitius (1997). Since our vortices carry magnetic fluxes, they will also exhibit non-trivial statistics in the infrared. Thus, a collection of these vortices will behave like anyons with long-range fields. We also show that the parameter space of these vortices is vast and includes the limiting cases of various known vortex solutions: ANO, U(1) Maxwell-Chern-Simons, U(1) pure Chern-Simons, and global vortices Vilenkin and Everett (1982). Interestingly enough, we find that unlike these limiting cases, the $\rm{U(1)}_{\scriptscriptstyle Z}$ charge and the $\rm{U(1)}_{\scriptscriptstyle A}$ magnetic flux of the higher winding solutions of our vortices are not integers times the charge and flux of the lowest winding solution. This is the first vortex solution exhibiting this behavior.
It is crucial that the model we consider does not have a self Chern-Simons term, $\mu\epsilon^{\mu\nu\beta}A_{\mu}F_{\nu\beta}$, for the $\rm{U(1)}_{\scriptscriptstyle A}$ gauge field which would otherwise spoil its long-range behavior. Then, one wonders if quantum corrections can generate such a term that destroys the nice long-range property of the vortices. Fortunately enough, if this term is absent on the tree and one-loop level, which is the case at hand, then the Coleman-Hill theorem Coleman and Hill (1985) guarantees that this term will not be generated at any higher loop level (see also Ref. Laine and Shaposhnikov (1999)).
The plan of this paper is as follows. In Sec. II, we introduce the Chern-Simons theory with the Chern-Simons mixing term for both the mixed and unmixed basis, and then discuss their basic properties. In Sec. III, we give a proof of existence for topological vorticies that are charged under the long-range $\rm{U(1)}_{\scriptscriptstyle A}$, and then present our results for the numerical solutions. In Sec. IV, we calculate the flux, charge and energy of the vortex solution and of a vortex-antivortex pair. We then show that the energy of the vortex-antivortex system is finite whereas the single vortex energy is logarithmically divergent, hence the vortices are classically confined. We conclude with a summary of our results and discussion in Sec. V.
II Topologically Massive U(1)$\times$U(1) Theory
We consider two topologically massive Abelian gauge fields $\mathcal{Y}_{\mu}$ and $\mathcal{W}_{\mu}$ with corresponding gauge groups $\rm{U(1)}_{\scriptscriptstyle W}\times\rm{U(1)}_{\scriptscriptstyle Y}$, and a complex scalar field $\varphi$ that is coupled to a linear combination of $\mathcal{Y}_{\mu}$ and $\mathcal{W}_{\mu}$:
$$\displaystyle\begin{aligned} \displaystyle S&\displaystyle=\int d^{3}x\biggl{[%
}-\frac{1}{4}\mathcal{Y}_{\mu\nu}\mathcal{Y}^{\mu\nu}-\frac{1}{4}\mathcal{W}_{%
\mu\nu}\mathcal{W}^{\mu\nu}+\mu_{\rm\scriptscriptstyle Y}\epsilon^{\mu\nu%
\alpha}\mathcal{Y}_{\mu\nu}\mathcal{Y}_{\alpha}\\
&\displaystyle\hskip 14.226378pt-\mu_{\rm\scriptscriptstyle W}\epsilon^{\mu\nu%
\alpha}\mathcal{W}_{\mu\nu}\mathcal{W}_{\alpha}+|(\partial_{\mu}-ig_{1}%
\mathcal{Y}_{\mu}-ig_{2}\mathcal{W}_{\mu})\varphi|^{2}\\
&\displaystyle\hskip 14.226378pt-\frac{\lambda}{4}\left(|\varphi|^{2}-v^{2}%
\right)^{2}\biggr{]}\,,\end{aligned}$$
(1)
where $\mathcal{Y}_{\mu\nu}=\partial_{\mu}\mathcal{Y}_{\nu}-\partial_{\nu}\mathcal{Y}%
_{\mu}$ and $\mathcal{W}_{\mu\nu}=\partial_{\mu}\mathcal{W}_{\nu}-\partial_{\nu}\mathcal{W}%
_{\mu}$. For generic values of $\mu_{\rm\scriptscriptstyle Y}$ and $\mu_{\rm\scriptscriptstyle W}$, each of the fields $\mathcal{Y}_{\mu}$ and $\mathcal{W}_{\mu}$ has a single degree of freedom which is screened in the infrared, thanks to the topological masses. Adding the two degrees of freedom of the complex scalar, our system has four degrees of freedom in total. In the following it will be useful to go to the new basis ${\cal A}_{\mu}$ and ${\cal Z}_{\mu}$:
$$\displaystyle\begin{aligned} \displaystyle\mathcal{Y}_{\mu}&\displaystyle=\cos%
\theta\mathcal{A}_{\mu}+\sin\theta\mathcal{Z}_{\mu}\,,\\
\displaystyle\mathcal{W}_{\mu}&\displaystyle=-\sin\theta\mathcal{A}_{\mu}+\cos%
\theta\mathcal{Z}_{\mu}\,,\end{aligned}$$
(2)
where $\tan\theta=g_{1}/g_{2}$. Now, we fix
$$\displaystyle\mu_{\rm\scriptscriptstyle Y}=\mu_{\rm\scriptscriptstyle W}\tan^{%
2}\theta$$
(3)
to obtain the action for the corresponding $\rm{U(1)}_{\scriptscriptstyle Z}\times\rm{U(1)}_{\scriptscriptstyle A}$ theory
$$\displaystyle\begin{aligned} \displaystyle S&\displaystyle=\int d^{3}x\biggl{[%
}-\frac{1}{4}\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}-\frac{1}{4}\mathcal{Z}_{%
\mu\nu}\mathcal{Z}^{\mu\nu}+\mu_{1}\epsilon^{\mu\nu\alpha}\mathcal{F}_{\mu\nu}%
\mathcal{Z}_{\alpha}\\
&\displaystyle\hskip 14.226378pt+\frac{\mu_{2}}{2}\epsilon^{\mu\nu\alpha}%
\mathcal{Z}_{\mu\nu}\mathcal{Z}_{\alpha}+|D_{\mu}\varphi|^{2}-\frac{\lambda}{4%
}\left(|\varphi|^{2}-v^{2}\right)^{2}\biggr{]}\,,\end{aligned}$$
(4)
where $\mathcal{F}_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}%
_{\mu}$, $\mathcal{Z}_{\mu\nu}=\partial_{\mu}\mathcal{Z}_{\nu}-\partial_{\nu}\mathcal{Z}%
_{\mu}$, and $D_{\mu}=\partial_{\mu}-ie\mathcal{Z}_{\mu}$. The parameters of the $\rm{U(1)}_{\scriptscriptstyle Z}\times\rm{U(1)}_{\scriptscriptstyle A}$ theory are related to the ones in the $\rm{U(1)}_{\scriptscriptstyle W}\times\rm{U(1)}_{\scriptscriptstyle Y}$ theory as follows: $e=\sqrt{g_{1}^{2}+g_{2}^{2}}$, $\mu_{1}=2\mu_{\rm\scriptscriptstyle W}\tan\theta$, and $\mu_{2}=2\mu_{\rm\scriptscriptstyle W}(\tan^{2}\theta-1)$. The Chern-Simons coefficients $\mu_{1}$ and $\mu_{2}$ as well as the parameter $\lambda$ have mass dimension $M$, while the coupling constant $e$ and the vacuum expectation value $v$ have mass dimension $M^{1/2}$. We set $c=1$, $\hbar=1$, $\epsilon^{012}=1$, and use the metric $\eta_{\mu\nu}={\rm diag}(1,-1,-1)$ in what follows.
It is a simple exercise to study the fluctuations about the vacuum $|\varphi|=v$ in Eq. (4). First, the gauge field ${\cal A}_{\mu}$ carries a single massless degree of freedom, thanks to the unbroken $\rm{U(1)}_{\scriptscriptstyle A}$. Writing the complex field $\varphi$ as $\varphi=(v+h)e^{i\Pi}$, we find that there is a single massive radial field $h$ in the infrared. In addition, the would-be Goldstone boson, $\Pi$, is eaten by the massive ${\cal Z}_{\mu}$ field. In fact, the mass of ${\cal Z}_{\mu}$ receives contributions from three sources: the self Chern-Simons term $\mu_{2}$, the Chern-Simons mixing term $\mu_{1}$, and the Higgs vacuum expectation value. This will be clear from our vortex solution, as is evident from Eq. (17) below. Thus, the field ${\cal Z}_{\mu}$ has two degrees of freedom, and we recover the total sum of the four degrees of freedom we started with.
One wonders whether the condition (3) and hence the spectrum described above, especially the massless $\rm{U(1)}_{\scriptscriptstyle A}$ field, are not spoiled by quantum effects. In fact, a one-loop calculation in the theory described by (1) does not yield any corrections to the Chern-Simons terms Laine and Shaposhnikov (1999). Besides, according to the Coleman-Hill theorem Coleman and Hill (1985), there are no more corrections to these topological terms other than the one-loop contribution. Therefore, the massless $\rm{U(1)}_{\scriptscriptstyle A}$ gauge field is protected against quantum effects.
III Charged Vortex Solution
Throughout this work, we seek cylindrically symmetric vortex solutions of the theory given by the action (4).
By varying the action [Eq. (4)], we obtain the field equations
$$\displaystyle\begin{aligned} &\displaystyle\partial_{\beta}\mathcal{F}^{\beta%
\sigma}+\mu_{1}\epsilon^{\beta\alpha\sigma}\mathcal{Z}_{\beta\alpha}=0\,,\\
&\displaystyle\partial_{\beta}\mathcal{Z}^{\beta\sigma}+\mu_{1}\epsilon^{\beta%
\alpha\sigma}\mathcal{F}_{\beta\alpha}+\mu_{2}\epsilon^{\beta\alpha\sigma}%
\mathcal{Z}_{\beta\alpha}+j^{\sigma}=0\,,\\
&\displaystyle D_{\beta}D^{\beta}\varphi+\frac{\lambda}{2}\left(|\varphi|^{2}-%
v^{2}\right)\varphi=0\,,\end{aligned}$$
(5)
where we defined the current as
$$j^{\sigma}=ie\bigl{[}\varphi^{*}D^{\sigma}\varphi-(D^{\sigma}\varphi)^{*}%
\varphi\bigr{]}\,.$$
(6)
To this end, we take the cylindrically symmetric Nielsen-Olesen like Ansätze, namely,
$$\displaystyle\varphi$$
$$\displaystyle=$$
$$\displaystyle vf(r)e^{in\theta}\,,\quad\mathcal{Z}_{i}=-\epsilon^{ij}x_{j}%
\frac{Z(r)}{er^{2}}\,,$$
$$\displaystyle\mathcal{Z}_{0}$$
$$\displaystyle=$$
$$\displaystyle eZ_{0}(r)\,,\quad~{}~{}~{}\mathcal{A}_{i}=-\epsilon^{ij}x_{j}%
\frac{A(r)}{er^{2}}\,,$$
(7)
$$\displaystyle\mathcal{A}_{0}$$
$$\displaystyle=$$
$$\displaystyle eA_{0}(r)\,,$$
where the profile functions $f(r),Z(r),Z_{0}(r),A(r)$, and $A_{0}(r)$ are dimensionless. Using these Ansätze in the equations of motion (5), we obtain
$$\displaystyle f^{\prime\prime}+\frac{f^{\prime}}{r}-(n-Z)^{2}\frac{f}{r^{2}}+e%
^{4}Z_{0}^{2}f-\frac{\lambda v^{2}}{2}(f^{2}-1)f=0\,,$$
$$\displaystyle Z^{\prime\prime}-\frac{Z^{\prime}}{r}+2e^{2}v^{2}f^{2}(n-Z)-2e^{%
2}r(\mu_{1}A_{0}^{\prime}+\mu_{2}Z_{0}^{\prime})=0\,,$$
$$\displaystyle Z_{0}^{\prime\prime}+\frac{Z_{0}^{\prime}}{r}-2e^{2}v^{2}f^{2}Z_%
{0}-\frac{2}{e^{2}r}(\mu_{1}A^{\prime}+\mu_{2}Z^{\prime})=0\,,$$
(8)
$$\displaystyle A^{\prime\prime}-\frac{A^{\prime}}{r}-2\mu_{1}e^{2}rZ_{0}^{%
\prime}=0\,,$$
$$\displaystyle A_{0}^{\prime\prime}+\frac{A_{0}^{\prime}}{r}-\frac{2\mu_{1}}{e^%
{2}r}Z^{\prime}=0\,.$$
Note that in the limit $\mu_{1}=0$ the two U(1) sectors decouple and we obtain the equation of motion for the normal $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex Paul and Khare (1986). The last two equations in Eq. (III) can be integrated to find
$$\displaystyle A^{\prime}=2\mu_{1}e^{2}rZ_{0}+{\cal D}_{1}r\,,\quad A_{0}^{%
\prime}=\frac{2\mu_{1}}{e^{2}r}Z+\frac{{\cal D}_{2}}{r}\,,$$
(9)
where ${\cal D}_{1}$ and ${\cal D}_{2}$ are integration constants. In order to determine the constants ${\cal D}_{1}$ and ${\cal D}_{2}$, we examine the near core and large $r$ behavior of the system. As we shall show in Sec. III.1, the behavior of $Z(r)$ near the core goes like $r^{2}$, and hence, one has to set ${\cal D}_{2}=0$ in order to have a regular solution of the electric field $eA_{0}^{\prime}(r)$ at $r=0$. Besides, a regular solution for $A^{\prime}(r)$ at large $r$ demands that ${\cal D}_{1}=-2\mu_{1}e^{2}Z_{0}(\infty)$. However, since nonzero $Z_{0}(\infty)$ leads to a quadratically divergent energy [see Eq. (IV.2)], it has to be set to zero, so does ${\cal D}_{1}$.
Before delving into the detailed vortex solution, one can read the physics of the vortex solution from the second equation in (9). This relation states that starting with a single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex, i.e. setting $\mu_{1}=0$, which has an asymptotic $Z$ solution of the form $Z(\infty)=n\,,n\in{\mathbb{Z}}$, and turning on a small $\mu_{1}$ will cause the vortex to acquire a long-range electric field proportional to $\mu_{1}$:
$$\displaystyle E_{{\cal A}}=eA_{0}^{\prime}\cong\frac{2\mu_{1}n}{er}\,.$$
(10)
Therefore, our vortices will carry a long-range field, thanks to the unbroken $\rm{U(1)}_{\scriptscriptstyle A}$. This physics will be confirmed by detailed analytical as well as numerical checks, as we show below.
III.1 Boundary Conditions
In the usual Nielsen-Olesen vortex solution, the asymptotic behaviors of the fields are determined easily by their regularity at the core and the finiteness of the energy; namely, the profile functions vanish at the core, Higgs goes to its expectation value, whereas the gauge field goes to a pure gauge value determined by the vanishing of the kinetic energy of the Higgs field at infinity. In our model, the boundary conditions of the profile functions at $r\to\infty$ are obtained by substituting Eq. (9) into Eq. (III), setting $f=1$, and neglecting the derivative and ${\cal O}(1/r^{2})$ terms:
$$\displaystyle\begin{aligned} \displaystyle\quad Z(\infty)&\displaystyle=\frac{%
e^{2}v^{2}n}{e^{2}v^{2}+2\mu_{1}^{2}}\,,\quad Z_{0}(\infty)=0\,,\quad f(\infty%
)=1\,,\\
\displaystyle A(\infty)&\displaystyle=n{\cal C}_{2}\,,\quad A_{0}(\infty)=%
\frac{2\mu_{1}v^{2}n}{e^{2}v^{2}+2\mu_{1}^{2}}\ln\frac{e^{2}r}{{\cal C}_{1}}\,%
,\end{aligned}$$
(11)
where the constants ${\cal C}_{1,2}$ are determined numerically. Unlike the single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex, the ${\mathcal{Z}}$ winding is not an integer. This peculiar behavior will have its dramatic consequences on the single as well as the multi-vortex solutions as we discuss later on. In the limit $\mu_{1}\to 0$, we find $Z(\infty)=n$ as expected for the single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex.
Near the core, $r\rightarrow 0$, the fields can be expanded in a Taylor series with arbitrary parameters. Requiring that the physical fields are continuous at the origin and forcing the expansion to fulfill the second order field equations (III), we can fix all but the five parameters $a_{00},z_{00},z_{2},a_{2},f_{1}$:
$$\displaystyle f(r)$$
$$\displaystyle=$$
$$\displaystyle f_{1}r^{|n|}+\mathcal{O}(r^{3})\,,$$
$$\displaystyle Z(r)$$
$$\displaystyle=$$
$$\displaystyle ez_{2}r^{2}+\mathcal{O}(r^{4})\,,$$
$$\displaystyle Z_{0}(r)$$
$$\displaystyle=$$
$$\displaystyle\frac{z_{00}+r^{2}(a_{2}\mu_{1}+z_{2}\mu_{2})}{e}+\mathcal{O}(r^{%
4})\,,$$
(12)
$$\displaystyle A(r)$$
$$\displaystyle=$$
$$\displaystyle ea_{2}r^{2}+\mathcal{O}(r^{4})\,,$$
$$\displaystyle A_{0}(r)$$
$$\displaystyle=$$
$$\displaystyle\frac{a_{00}+z_{2}\mu_{1}r^{2}}{e}+\mathcal{O}(r^{4})\,.$$
Note that the value of $a_{00}$ is purely a gauge choice, which doesn’t contribute to the equations of motion (III). Below we will arbitrarily set $a_{00}=0$.
III.2 Existence of the Solution
Before moving to the numerical solution of the field equations, in this section we sketch a proof of existence of the vortex solution. To this end, we analyze the system of equations (III) at asymptotic infinity, $r\rightarrow\infty$, taking into account the first order equations (9). Since our vortices carry a long-range U(1) field, it is expected that the far-field will follow a power-law behavior.
The asymptotic power law behavior can be obtained by expanding the profile functions as $\Psi(r)=\Psi(\infty)+\sum_{m=1}^{\infty}\psi_{m}/r^{m}$ and solving for $\psi_{m}$ using Eq. (III). Upon performing this expansion, all constants $\psi_{m}$ can be fixed and we get for the asymptotic profile functions $\Psi(r)\to\Psi^{\infty}(r)$ to leading order:
$$\displaystyle\begin{aligned} \displaystyle f^{\infty}(r)&\displaystyle=1-\frac%
{4n^{2}\mu_{1}^{4}}{\lambda v^{2}(e^{2}v^{2}+2\mu_{1}^{2})^{2}~{}r^{2}}+{\cal O%
}(1/r^{4})\,,\\
\displaystyle Z^{\infty}(r)&\displaystyle=Z(\infty)-\frac{16e^{2}n^{3}\mu_{1}^%
{6}}{\lambda(e^{2}v^{2}+2\mu_{1}^{2})^{4}~{}r^{2}}+{\cal O}(1/r^{4})\,,\\
\displaystyle Z_{0}^{\infty}(r)&\displaystyle=-\frac{32n^{3}\mu_{1}^{6}\mu_{2}%
}{\lambda(e^{2}v^{2}+2\mu_{1}^{2})^{5}~{}r^{4}}+{\cal O}(1/r^{6})\,,\\
\displaystyle A_{0}^{\infty}(r)&\displaystyle=A_{0}(\infty)+\frac{16n^{3}\mu_{%
1}^{7}}{\lambda(e^{2}v^{2}+2\mu_{1}^{2})^{4}~{}r^{2}}+{\cal O}(1/r^{4})\,,\\
\displaystyle A^{\infty}(r)&\displaystyle=n{\cal C}_{2}+\frac{32e^{2}n^{3}\mu_%
{1}^{7}\mu_{2}}{\lambda(e^{2}v^{2}+2\mu_{1}^{2})^{5}~{}r^{2}}+{\cal O}(1/r^{4}%
)\,.\end{aligned}$$
(13)
Note that in the $\mu_{2}=0$ limit, $Z_{0}(r)=A(r)=0$ to all orders in the large $r$ expansion. In addition, in the $\mu_{1}\to\infty$ limit, all the profile functions except $f(r)$ vanish identically, hence we recover the U(1) global vortex solution Vilenkin and Everett (1982).
Now we expand the profile functions around the large $r$ limit as
$$\displaystyle f$$
$$\displaystyle=$$
$$\displaystyle f^{\infty}+\delta f,~{}~{}Z=Z^{\infty}+\delta Z,~{}Z_{0}=Z_{0}^{%
\infty}+\delta Z_{0}\,,$$
(14)
and plug these Ansätze back into the equations for the profile functions (III). To the leading order in the fluctuations $\delta f,\delta Z,\delta Z_{0}$ (and neglecting terms of the form $(\delta f,\delta Z_{0},\delta Z)/r^{2}$), we get
$$\displaystyle\delta f^{\prime\prime}+\frac{\delta f^{\prime}}{r}-\lambda v^{2}%
\delta f\approx 0\,,$$
$$\displaystyle\delta Z^{\prime\prime}-\frac{\delta Z^{\prime}}{r}-(2e^{2}v^{2}+%
4\mu_{1}^{2})\delta Z-2e^{2}r\mu_{2}\delta Z_{0}^{\prime}\approx 0\,,$$
(15)
$$\displaystyle\delta Z_{0}^{\prime\prime}+\frac{\delta Z_{0}^{\prime}}{r}-(2e^{%
2}v^{2}+4\mu_{1}^{2})\delta Z_{0}-\frac{2}{e^{2}r}\mu_{2}\delta Z^{\prime}%
\approx 0\,.$$
By first writing $\delta Z(r)=e\sqrt{r}z(r)$ and $\delta Z_{0}(r)=z_{0}(r)/(e\sqrt{r})$, then solving for $z(r)$ and $z_{0}(r)$ at large $r$, we find following regular solutions to the homogenous equations Eq. (III.2)
$$\displaystyle\delta f(r)\simeq\frac{{\cal C}_{5}}{e\sqrt{r}}e^{-\sqrt{\lambda}%
vr}\,,$$
$$\displaystyle\delta Z(r)\simeq{\cal C}_{3}e\sqrt{r}e^{-{\cal M}_{Z}r}\,,$$
(16)
$$\displaystyle\delta Z_{0}(r)\simeq\frac{{\cal C}_{4}}{e\sqrt{r}}e^{-{\cal M}_{%
Z}r}\,,$$
where ${\cal M}_{Z}$ is the ${\mathcal{Z}}_{\mu}$ mass:
$$\displaystyle{\cal M}_{Z}=-|\mu_{2}|+\sqrt{4\mu_{1}^{2}+2e^{2}v^{2}+\mu_{2}^{2%
}}\,.$$
(17)
The solution (III.2) describes the intermediate region $\frac{\log(e^{2}v^{2}/2\mu_{1}^{2})}{\sqrt{\lambda v}}\gtrsim r$, where the equality sign results by equating $f$ from the power-law behavior (13) to the exponential one (III.2).
Assuming that a vortex solution with radius $r_{\rm c}$ exists, then the profile functions and their derivatives have to be continuous at $r_{\rm c}$.
Thus, we match the profile functions and their derivatives in the small $r$ limit given by Eq. (III.1) with the large $r$ limit in Eq. (14). If the solution exists, then we should have the right number of free parameters.
Notice that if we use the first order equations (9) instead of the second order ones (III), we obtain a non-trivial relation between $a_{2}$ and $z_{00}$ :
$$\displaystyle a_{2}=\mu_{1}z_{00}\,,$$
(18)
which reduces the number of the free parameters by one. We are then left with three free parameters at the core $z_{00},z_{2},f_{1}$, and five free constants at infinity ${\cal C}_{1,2,3,4,5}$. Now matching $f$ and its derivative across $r_{\rm c}$ gives $f_{1}$ and ${\cal C}_{5}$, $Z$ and its derivative across $r_{\rm c}$ gives $z_{2}$ and ${\cal C}_{3}$, and $Z_{0}$ and its derivative across $r_{\rm c}$ gives $z_{00}$ and ${\cal C}_{4}$. Matching $A_{0}^{\prime}$ and $A^{\prime}$ across $r_{\rm c}$ does not give new information since both of these functions are dependent on $Z$ and $Z_{0}$, as is clear from Eq. (9). Finally, we can solve for ${\cal C}_{1,2}$ by matching $A$ and $A_{0}$ across $r_{\rm c}$.
The explicit expressions of ${\cal C}_{1,3,4,5}$ are cumbersome and not very illuminating, and we refrain from giving them here.
It will turn out that the value of ${\cal C}_{2}$ determines the magnetic flux of ${\cal A}_{\mu}$ and partially the electric charge of ${\cal Z}_{\mu}$.
Now two comments are in order. First, one can read the masses of the particle spectrum from Eq. (III.2): the ${\cal A}_{\mu}$ field is massless, the Higgs mass is $\sqrt{\lambda}v$, while the ${\mathcal{Z}}_{\mu}$ mass is given by Eq. (17). The $\mathcal{Z}_{\mu}$ mass gets contribution from the Higgs vacuum expectation value, after eating the would-be Goldstone boson, and from the topological Chern-Simons terms. Thus, as stated before, the ${\cal Z}_{\mu}$ field has two degrees of freedom. Second, we note that in the limit $\mu_{2}=0$, the coefficients $z_{00}$, ${\cal C}_{2}$ and ${\cal C}_{4}$ are identically zero for all values of $\mu_{1}$ as we checked numerically. In this limit, the Chern-Simons vortex degenerates to Abrikosov-Nielsen Olesen vortex. The vanishing of ${\cal C}_{2}$ for $\mu_{2}=0$ means the absence of the ${\cal A}_{\mu}$ magnetic flux and ${\cal Z}_{\mu}$ electric charge as we detail below.
III.3 Numerical Results
The full numerical solution of the second order equations (III) is obtained via a shooting method as implemented in Burnier and Shaposhnikov (2005). Starting from a small but non-vanishing radius $r_{\rm min}=10^{-5}$, in units of $e^{2}$, and using the small $r$ expansion (III.1) to sixth order, the shooting method finds the value of the four free parameters that lead to a solution satisfying the first four boundary conditions at large distance $r_{\rm max}$ as given in Eq. (11). Here, to reach the boundary conditions to an absolute precision of $10^{-8}$ at $r_{\rm max}=27$, we use 128 digits of precision to solve the set of non-linear differential equations (III). Note that the last boundary condition at infinity in Eq. (11) is satisfied automatically. The profile functions, electric and magnetic fields, charges and energy density for the specific case $v=1$, $\lambda=1$, $\mu_{1}=\mu_{2}=1/4$ are shown in Figs. 1, 2, 3, 4 and 5 for different winding numbers, $n$, as a function of $r$, which is in units of $e^{2}$.
The existence of a well defined solution is ensured by checking that it follows the expansion at small $r$ given in Eq. (III.1) and merges smoothly to the asymptotic behavior given in Eq. (13). In Fig. 3, we show how the profile function $f(r)$ converges towards $1$ at large distance and see that it satisfies the large $r$ expansion [Eq. (13)]. As a further test we checked that the solution does not depend on the value of $r_{\rm min}$ and $r_{\rm max}$ as long as they remain small and large enough, respectively.
IV Physical Properties of Charged Vortices
IV.1 Magnetic Fluxes and Charges
The electric and magnetic fields as well as the kinetic term for the Higgs field can be related to the profile functions using Eq. (III):
$$\displaystyle E_{\mathcal{Z}}$$
$$\displaystyle=$$
$$\displaystyle eZ_{0}^{\prime}\,,\qquad B_{\mathcal{Z}}=\frac{1}{2}\epsilon^{0%
ij}Z_{ij}=\frac{Z^{\prime}}{er}\,,$$
$$\displaystyle E_{\mathcal{A}}$$
$$\displaystyle=$$
$$\displaystyle eA_{0}^{\prime}\,,\qquad B_{\mathcal{A}}=\frac{1}{2}\epsilon^{0%
ij}F_{ij}=\frac{A^{\prime}}{er}\,,$$
(19)
$$\displaystyle\mathcal{Z}_{0}$$
$$\displaystyle=$$
$$\displaystyle eZ_{0}\,,\qquad|{\bf D}\varphi|^{2}=v^{2}f^{\prime 2}+v^{2}(n-Z)%
^{2}\frac{f^{2}}{r^{2}}\,.$$
The magnetic flux of $\mathcal{Z}_{\mu}$ and $\mathcal{A}_{\mu}$ fields for the charged vortex solution are
$$\displaystyle\begin{aligned} \displaystyle\Phi_{B_{\mathcal{Z}}}&\displaystyle%
=&\displaystyle\oint_{S^{1}_{\infty}}{\bf\mathcal{Z}}\cdot{\bf d\ell}=\oint_{S%
^{1}_{\infty}}\frac{Z(r)}{er}rd\theta=\frac{2\pi}{e}Z(\infty)\,,\\
\displaystyle\Phi_{B_{\mathcal{A}}}&\displaystyle=&\displaystyle\oint_{S^{1}_{%
\infty}}{\bf\mathcal{A}}\cdot{\bf d\ell}=\oint_{S^{1}_{\infty}}\frac{A(r)}{er}%
rd\theta=\frac{2\pi}{e}A(\infty)\,,\end{aligned}$$
(20)
where $S^{1}_{\infty}$ is a circle enclosing the vortex at infinity.
Using Eq. (11), we find
$$\displaystyle\Phi_{B_{\mathcal{Z}}}=\frac{2\pi n}{e}\frac{e^{2}v^{2}}{e^{2}v^{%
2}+2\mu_{1}^{2}}\,,\quad\Phi_{B_{\mathcal{A}}}=\frac{2\pi n}{e}{\cal C}_{2}\,.$$
(21)
Although we have used the asymptotic values of the fields to calculate the fluxes, it should be clear that these fluxes originate from the near-core region of the vortex since both ${\cal Z}_{\mu}$ and ${\cal A}_{\mu}$ are screened outside the core, as is clear from Eqs. (III.2) and (13) and Fig. 1. As we discussed before, ${\cal C}_{2}$ and hence $\Phi_{B_{\mathcal{A}}}$ vanish identically at $\mu_{2}=0$ for all values of $\mu_{1}$.
Note that in the $\mu_{1}\to 0$ limit Eq. (21) reduces to $\Phi_{B_{\mathcal{Z}}}=2\pi n/e$ as expected for a single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex Paul and Khare (1986). Unlike the single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex, the flux in our case is not an integer times $2\pi/e$, which is attributed to the mismatch between the ${\cal Z}_{\mu}$ and $\varphi$ windings. More on this point will be discussed in Sec. IV.2.
The charge of the vortex under the $\mathcal{Z}_{\mu}$ field can be obtained from Eq. (6) as (this is the Noether’s charge)
$$Q_{\mathcal{Z}}=\int d^{2}xj^{0}=2e^{3}v^{2}\int_{0}^{2\pi}d\alpha\int_{0}^{%
\infty}dr~{}rf^{2}Z_{0}\,.$$
(22)
Using the third equation in (III), we have
$$\displaystyle\begin{aligned} \displaystyle Q_{\mathcal{Z}}&\displaystyle=2\pi e%
\int_{0}^{\infty}dr\left[(rZ_{0}^{\prime})^{\prime}-\frac{2}{e^{2}}(\mu_{1}A^{%
\prime}+\mu_{2}Z^{\prime})\right]\\
&\displaystyle=2\pi e\left[(rZ_{0}^{\prime})-\frac{2}{e^{2}}(\mu_{1}A+\mu_{2}Z%
)\right]_{0}^{\infty}\,,\end{aligned}$$
(23)
and then, using the boundary conditions in Eq. (11), $Q_{\mathcal{Z}}$ reduces to
$$Q_{\mathcal{Z}}=-\frac{4\pi n}{e}\left[\mu_{1}{\cal C}_{2}+\mu_{2}\frac{e^{2}v%
^{2}}{e^{2}v^{2}+2\mu_{1}^{2}}\right]\,.$$
(24)
Remembering that ${\cal C}_{2}=0$ at $\mu_{2}=0$, we see right away that the Noether charge of the vortex under the ${\cal Z}_{\mu}$ field vanishes in this limit. In fact, the absence of the ${\cal Z}_{\mu}$ electric charge and ${\cal A}_{\mu}$ magnetic flux at $\mu_{2}=0$ is not a coincidence. We can understand this observation as follows. For $\mu_{1}=\mu_{2}=0$, we recover the normal Abrikosov-Nielsen-Olesen vortex which carries only ${\cal Z}_{\mu}$ magnetic flux. Turning on a non-zero value for $\mu_{2}$, keeping $\mu_{1}=0$, the Chern-Simons term will induce an electric charge for the ${\cal Z}_{\mu}$ field. Recalling Eq. (22) —which determines the electric charge as a function of $Z_{0}$— and the discussion after Eq. (III.2), the values of the coefficients $z_{00}$ and ${\cal C}_{4}$ are determined upon matching the near-core and the far-region values of the profile function $Z_{0}$ and its derivative across the vortex wall $r_{\rm c}$. For $\mu_{2}\neq 0$ both ${\cal C}_{2}$ and $z_{00}$ are non-vanishing, and hence we obtain non-zero values for $Q_{\mathcal{Z}}$ as easily seen by taking the $\mu_{1}\to 0$ limit in Eq. (24), which reduces $Q_{\mathcal{Z}}$ to $-4\pi n\mu_{2}/e$ as expected for the single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex.
Now, let us turn on a non-zero value for $\mu_{1}$ keeping
$\mu_{2}\neq 0$. The first equation in (9) relates $A^{\prime}$ to $Z_{0}$ at all values of $r$. At the core, non-zero values of $Z_{0}$, which are expected for $\mu_{2}\neq 0$, will induce non-zero value for the $A$ profile which will induce magnetic field $B_{{\cal A}}$, and hence magnetic flux $\Phi_{B_{\cal A}}$. Thus, we see that the vanishing of $\mu_{2}$ means the vanishing of $Z_{0}$, and hence $Q_{\mathcal{Z}}$, and in sequence the vanishing of the flux $\Phi_{B_{\cal A}}$.
The interesting feature of $Q_{\mathcal{Z}}$ is that it is not quantized. Although the charge neutrality condition is always satisfied, i.e., $Q_{\mathcal{Z}}(-n)=-Q_{\mathcal{Z}}(n)$, we find that $Q_{\mathcal{Z}}(n)\neq nQ_{\mathcal{Z}}(n=1)$ if $\mu_{1}\neq 0$ and $\mu_{2}\neq 0$. If $\mu_{2}=0$ then $Q_{\mathcal{Z}}$ vanishes, and if $\mu_{1}=0$ the charge is quantized as is obvious from formula (24). In fact, the second term of Eq. (24) is directly proportional to $n$ but the first one is not as the asymptotic value ${\cal C}_{2}$ depends non-trivially on $|n|$. This is the first example of a vortex with such behavior.
The $\mathcal{Z}$ charge of the vortices with winding $n=1,2,3$ are given in Table 1 for $\mu_{2}=1/4,e=v=\lambda=1$ and several values of $\mu_{1}$.
Far in the infrared, the $\mathcal{Z}_{\mu}$ field is screened while the ${\cal A}_{0}$ field is long-range, thanks to the unbroken $\rm{U(1)}_{\scriptscriptstyle A}$. The charge of the vortex under $\rm{U(1)}_{\scriptscriptstyle A}$ can be obtained by integrating the electric field over the $\mathbb{R}^{2}$ plane and using Stokes’ theorem.
Substituting the asymptotic value of $\mathcal{A}_{0}$ in Eq. (11) we obtain
$$Q_{\mathcal{A}}=\oint_{S^{1}_{\infty}}{\bf E_{\mathcal{A}}}\cdot d{\bm{\ell}}=%
e\oint_{S^{1}_{\infty}}A_{0}^{\prime}rd\theta=\frac{4\pi nev^{2}\mu_{1}}{e^{2}%
v^{2}+2\mu_{1}^{2}}\,.$$
(25)
This is the electric charge of the vortex under the $\rm{U(1)}_{\scriptscriptstyle A}$ field as defined from Gauss’s law [note that $Q_{\mathcal{A}}=2\mu_{1}\Phi_{B_{\cal Z}}$, where $\Phi_{B_{\cal Z}}$ given by Eq. (21)]. Therefore, an external probe with test charge $Q_{\rm test}$ will experience a force111The interaction between vortices will be elucidated in Sec. IV.3. $Q_{\rm test}Q_{\mathcal{A}}/r$. The charge $Q_{\mathcal{A}}$ is zero at $\mu_{1}=0$, increases to a maximum value of $2\pi n/e$ at $\mu_{1}^{2}=e^{2}v^{2}/2$, and then decreases as $2\pi nev^{2}/\mu_{1}$ for $2\mu_{1}^{2}\gg e^{2}v^{2}$. The decrease of the electric charge for large values of $\mu_{1}$ can be understood from the equations of motion (5). For large values of $\mu_{1}$ we can neglect the kinetic terms compared to the topological one. Since the kinetic terms are responsible for mediating the long-range $\rm{U(1)}_{\scriptscriptstyle A}$ force, we expect this force to be suppressed for small values of the kinetic terms (see also Fig. 7).
Before ending this section, let us also note that if the $\cal Z_{\mu}$ charge were defined from Gauss’s law, as we did for $Q_{\mathcal{A}}$, we would find zero $\cal Z_{\mu}$ charge since the $\cal Z_{\mu}$ field is screened.
IV.2 Energy of the Vortex
The energy of the vortex can be calculated starting from the Hamiltonian density of the theory defined by Eq. (4). Because the vortex is static, one can alternatively use the Euclidean version of this action. In fact, since the Chern-Simons terms do not depend on any background metric, these terms do not contribute to the energy-momentum tensor and hence to the Hamiltonian. In turn, one does not expect that both the Hamiltonian and the Euclidean action to have the same functional form. Irrespectively, we checked that both formulations give the exact same answer for the vortex energy.
The Hamiltonian density is given by222Note that our formula differs from the one given in Ref. Horvathy and Zhang (2009) by a sign of the $e^{4}Z_{0}^{2}|\varphi|^{2}$ term, which should be positive. Besides, the Chern-Simons term that they have in the Hamiltonian density should not be included as it does not contribute to the energy momentum tensor.
$$\displaystyle\begin{aligned} \displaystyle\mathcal{H}_{\rm v}&\displaystyle=%
\frac{1}{2}\left(E_{\mathcal{Z}}^{2}+B_{\mathcal{Z}}^{2}+E_{\mathcal{A}}^{2}+B%
_{\mathcal{A}}^{2}\right)+e^{4}Z_{0}^{2}|\varphi|^{2}\\
&\displaystyle\hskip 14.226378pt+|{\bf D}\varphi|^{2}+\frac{\lambda}{4}(|%
\varphi|^{2}-v^{2})^{2}\,.\end{aligned}$$
(26)
Next, we split the total Hamiltonian density into three parts: the core $\mathcal{H}_{\rm c}$, electric $\mathcal{H}_{E_{\mathcal{A}}}$ and Goldstone $\mathcal{H}_{\rm G}$. Using Eq. (IV.1),
$$\displaystyle\mathcal{H}_{\rm c}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\biggl{[}e^{2}Z_{0}^{\prime 2}+\frac{Z^{\prime 2}}{e^{%
2}r^{2}}+\frac{A^{\prime 2}}{e^{2}r^{2}}+2e^{4}v^{2}Z_{0}^{2}f^{2}$$
$$\displaystyle +2v^{2}f^{\prime 2}+\frac{\lambda v^{4}}{2}(f^{2}-1)^{2}%
\biggr{]}\,,$$
$$\displaystyle\mathcal{H}_{E_{\mathcal{A}}}$$
$$\displaystyle=$$
$$\displaystyle\frac{e^{2}}{2}A_{0}^{\prime 2}\,,$$
(27)
$$\displaystyle\mathcal{H}_{\rm G}$$
$$\displaystyle=$$
$$\displaystyle v^{2}(n-Z)^{2}\frac{f^{2}}{r^{2}}\,.$$
Integrating each term over $\mathbb{R}^{2}$, the total energy of a vortex is obtained as
$$\displaystyle\mathcal{E}_{\rm v}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{E}_{\rm c}+\mathcal{E}_{E_{\mathcal{A}}}+\mathcal{E}_{%
\rm G}\,.$$
(28)
The core contribution to the energy, $\mathcal{E}_{\rm c}$ can be found numerically (see Fig. 5),
while in the IR only the electric $\mathcal{E}_{E_{\mathcal{A}}}$ and Goldstone $\mathcal{E}_{\rm G}$ contributions remain, and the energy can be calculated analytically:
$$\displaystyle\mathcal{E}_{\rm\scriptscriptstyle IR}$$
$$\displaystyle\approx$$
$$\displaystyle\mathcal{E}_{E_{\mathcal{A}}}^{\rm\scriptscriptstyle IR}+\mathcal%
{E}_{\rm G}^{\rm\scriptscriptstyle IR}$$
(29)
$$\displaystyle=$$
$$\displaystyle\pi\biggl{[}\frac{4\mu_{1}^{2}}{e^{2}}Z(\infty)^{2}+2v^{2}(n-Z(%
\infty))^{2}\biggr{]}\int_{r_{\rm c}}^{R_{\rm\scriptscriptstyle IR}}d\ln r$$
$$\displaystyle=$$
$$\displaystyle\frac{4\pi n^{2}v^{2}\mu_{1}^{2}}{v^{2}e^{2}+2\mu_{1}^{2}}\ln%
\frac{{R_{\rm\scriptscriptstyle IR}}}{r_{\rm c}}\,,$$
where we imposed an IR cutoff at $R_{\rm\scriptscriptstyle IR}$. Exactly like an electric charge in 2+1 dimensions, the energy of a single vortex is logarithmically divergent. Note that there are two contributions to the energy. The first one comes form the electric field $\mathcal{E}_{E_{\mathcal{A}}}$, which is expected from the long-range $\rm{U(1)}_{\scriptscriptstyle A}$ outside the vortex core. The second contribution, the Goldstone energy, $\mathcal{E}_{\rm G}$ is more interesting. In a trivial background, $\varphi=0$, we can study the spectrum of the symmetry breaking of a global ${\rm U(1)}$ symmetry by writing $\varphi(x)=\left[v+h(x)\right]e^{i\Pi(x)}$, where $h$ is the Higgs boson and the phase $\Pi$ is the Goldstone boson. This Goldstone boson is a physical massless degree of freedom that exists in the spectrum of the theory. However, once we gauge ${\rm U(1)}$ this Goldstone will be eaten by the corresponding gauge field which in turn acquires a mass333This can be seen once we compute the square of the covariant derivative $|D_{\mu}\varphi|^{2}$ which gives $v^{2}(\partial_{\mu}\Pi-e{\cal Z}^{f}_{\mu})^{2}$, where ${\cal Z}_{\mu}^{f}$ is the fluctuation field. Using the gauge transformation ${\cal Z}_{\mu}\rightarrow{\cal Z}_{\mu}+\partial_{\mu}\Pi/e$ (unitary gauge), we immediately recognize $2e^{2}v^{2}$ as the Z-mass.. In fact, the physical spectrum does not contain any Goldstone boson since there is no symmetry to break: the gauged ${\rm U(1)}$ is a redundancy rather than a genuine symmetry. If the background is non-trivial, as in our vortex case, then one has to repeat the same argument in the given background. In this case, we write $\varphi(x)=v\left[f(r)+h(x)\right]e^{i\left[n\theta(r)+i\Pi(x)\right]}$, where $f(r)$ is the profile function of the vortex. We also write the field ${\cal Z}_{\mu}$ as ${\cal Z}_{\mu}(x)={\cal Z}_{\mu}^{b}(x)+{\cal Z}_{\mu}^{f}(x)$, where the background solution ${\cal Z}_{\mu}^{b}$ can be read directly from the profile functions $Z(r)$ and $Z_{0}(r)$ given above. Now the square of the covariant derivative gives $v^{2}\left[\partial_{\mu}(n\theta+\Pi)-e({\cal Z}_{\mu}^{b}+{\cal Z}_{\mu}^{f}%
)\right]^{2}$. Again, one can use an appropriate gauge transformation to kill the fluctuating Goldstone field $\Pi$, thus interpreting $2e^{2}v^{2}$ as the mass of ${\cal Z}_{\mu}$ in our non-trivial background. What remains is the Goldstone background contribution $v^{2}\left(n\partial_{\mu}\theta-e{\cal Z}_{\mu}^{b}\right)^{2}$ which gives $\mathcal{H}_{\rm G}$ in Eq. (27). As we showed above, this Goldstone background energy is logarithmically divergent444In fact, one can obtain the same behavior in the presence of a non-dynamical static magnetic $B_{\cal Z}$ field with non-integer flux. In this case, we also find that the Goldstone background energy is logarithmically divergent because of the mismatch between the winding of the Higgs and gauge fields. We thank T. Sulejmanpasic for emphasizing this point.. Therefore, unlike the Abrikosov-Nielsen-Olesen vortices or their single ${\rm U(1)}$ Chern-Simons cousins, our vortices are not genuine solitons. This forces us to consider an ensemble of an equal number of vortices and anti-vortices. These in turn will form vortex-antivortex confined pairs that lowers the total energy of the system. The next section is devoted to the study of the interaction energy of the vortex-antivortex pair.
IV.3 Interaction between Two Vortices
In this section, we show that the total energy of a system of a vortex and an antivortex pair is finite. Treating the 2-vortex system is rather complicated. However, it will be sufficient to approximate the system as a superposition of a pair of vortices with opposite winding numbers located at a large separation $R$. As long as the separation is large enough, the individual solutions do not receive a considerable modification by the presence of the other vortex. With these assumptions, the total scalar and gauge fields of a vortex-antivortex system at $|{\bf x}|\gg r_{\rm c}$ can be approximated as follows:
$$\displaystyle\begin{aligned} \displaystyle\varphi&\displaystyle\cong ve^{in%
\theta_{1}({\bf x}-{\bf x_{1}})-in\theta_{2}({\bf x}-{\bf x_{2}})}\,,\\
\displaystyle\mathcal{Z}_{0}&\displaystyle\cong 0\,,\\
\displaystyle\mathcal{Z}_{i}&\displaystyle\cong-\frac{Q_{\mathcal{A}}}{4\pi\mu%
_{1}}\epsilon_{ij}\left[\frac{({\bf x}-{\bf x_{1}})_{j}}{|{\bf x}-{\bf x_{1}}|%
^{2}}-\frac{({\bf x}-{\bf x_{2}})_{j}}{|{\bf x}-{\bf x_{2}}|^{2}}\right]\,,\\
\displaystyle\mathcal{A}_{0}&\displaystyle\cong\frac{Q_{\mathcal{A}}}{2\pi}{%
\rm ln}\frac{|{\bf x}-{\bf x_{1}}|}{|{\bf x}-{\bf x_{2}}|}\,,\\
\displaystyle\mathcal{A}_{i}&\displaystyle\cong-\frac{n\mathcal{C}_{2}}{e}%
\epsilon_{ij}\left[\frac{({\bf x}-{\bf x_{1}})_{j}}{|{\bf x}-{\bf x_{1}}|^{2}}%
-\frac{({\bf x}-{\bf x_{2}})_{j}}{|{\bf x}-{\bf x_{2}}|^{2}}\right]\,,\end{aligned}$$
(30)
where, $Q_{\mathcal{A}}$ is given by Eq. (25), ${\bf x_{1}}$ and ${\bf x_{2}}$ are the locations of the vortex and antivortex, respectively. Upon substituting Eqs. (30) in Eq. (26), the total Hamiltonian density in the IR can be obtained as:
$$\displaystyle\mathcal{H}_{\rm\scriptscriptstyle IR}(R,r,\alpha)=\frac{8n^{2}v^%
{2}\mu_{1}^{2}}{e^{2}v^{2}+2\mu_{1}^{2}}\frac{(R/2)^{2}}{r_{1}^{2}r_{2}^{2}}\,,$$
(31)
where we defined $r\equiv|{\bf x}|$,
$$\displaystyle\begin{aligned} \displaystyle r_{1}\equiv\sqrt{r^{2}-Rr\cos\alpha%
+(R/2)^{2}}~{}\,,\\
\displaystyle r_{2}\equiv\sqrt{r^{2}+Rr\cos\alpha+(R/2)^{2}}~{}\,,\end{aligned}$$
(32)
$R$ is the separation distance between the two vortices, and $\alpha$ is the polar angle.
Then, the total energy can be found upon integrating $\mathcal{H}_{\rm\scriptscriptstyle IR}(r,\alpha)$ over the polar angle $\alpha$ and radial variable $r$:
$$\displaystyle\begin{aligned} \displaystyle\mathcal{E}_{\rm\scriptscriptstyle IR%
}(R)&\displaystyle=\int_{\mathcal{C}}dr~{}r~{}\int_{0}^{2\pi}d\alpha~{}%
\mathcal{H}_{\rm\scriptscriptstyle IR}(R,r,\alpha)\\
&\displaystyle=\frac{8n^{2}v^{2}\mu_{1}^{2}}{e^{2}v^{2}+2\mu_{1}^{2}}~{}\frac{%
\pi}{2}\ln\frac{R^{4}+r_{\rm c}^{4}}{R^{2}r_{\rm c}^{2}-r_{\rm c}^{4}}\,.\end{aligned}$$
(33)
Since $R\gg r_{\rm c}$, the total energy of a vortex-antivortex system can be simply written as:
$$\displaystyle\mathcal{E}_{\rm\scriptscriptstyle IR}(R)\approx\frac{8\pi n^{2}v%
^{2}\mu_{1}^{2}}{e^{2}v^{2}+2\mu_{1}^{2}}~{}\ln\frac{R}{r_{\rm c}}+\mathcal{O}%
(r_{\rm c}/R)\,.$$
(34)
Here, for simplicity we took the region of integration for the radial variable to be $\mathcal{C}=\{0\leqslant r\leqslant R/2-r_{\rm c}~{}{\rm and}~{}R/2+r_{\rm c}%
\leqslant r\leqslant\infty\}$ to remove the contribution of the cores of the vortices. It would be more accurate to cut out just two discs of radii $r_{\rm c}$ centered at the cores of the vortex and antivortex. However, removing the contribution of this tiny strip of radius $2r_{\rm c}$ only leads to an error of order $r_{\rm c}/R$, which is a subleading effect that we ignore here.
The total energy takes an even simpler form in the $2\mu_{1}^{2}\ll e^{2}v^{2}$ limit, i.e., when the $\mu_{1}$ contribution to ${\cal Z}$-mass is small compared to the mass coming from the spontaneous breaking of the $\rm{U(1)}_{\scriptscriptstyle Z}$ symmetry:
$$\displaystyle\mathcal{E}_{\rm\scriptscriptstyle IR}(R)|_{2\mu_{1}^{2}\ll e^{2}%
v^{2}}\sim\frac{Q_{\mathcal{A}}^{2}}{2\pi}~{}\ln\frac{R}{r_{\rm c}}\,.$$
(35)
In other words, the interaction of a vortex and an antivortex in this particular limit is exactly like that of two point particles with opposite charges in $2+1$ D.
In the opposite limit $2\mu_{1}^{2}\gg e^{2}v^{2}$, the total energy takes the form
$$\displaystyle\mathcal{E}_{\rm\scriptscriptstyle IR}(R)|_{2\mu_{1}^{2}\gg e^{2}%
v^{2}}\sim 4\pi n^{2}v^{2}~{}\ln\frac{R}{r_{\rm c}}\,.$$
(36)
In this limit, $Q_{\mathcal{A}}\sim 0$, hence, the contribution is mostly due to the Goldstone background. Note that this is nothing but the energy of a global vortex-antivortex pair (see Fig.7).
To summarize, a system of a vortex-antivortex pair has a finite energy and is logarithmically confined. The total energy gets contributions both from the electric field and Goldstone background of the vortices.
V Summary and Discussion
In this work, we have obtained a new vortex solution in the $\rm{U(1)}_{\scriptscriptstyle Z}\times\rm{U(1)}_{\scriptscriptstyle A}$ Chern-Simons gauge theory. These vortices are classified by a topological number $n\in\mathbb{Z}$. Inside the core of the vortex the Higgs field is in its symmetric phase and both ${\rm U(1)}$s are topologically massive. Outside the core, the Higgs field gets a vacuum expectation value causing the spontaneous breaking of $\rm{U(1)}_{\scriptscriptstyle Z}$. The resulting Goldstone boson is eaten by the corresponding gauge field, namely the ${\cal Z}_{\mu}$ field, which now acquires an extra degree of freedom. Thus, the mass of the ${\cal Z}$-boson gets contributions from both the topological terms and the Higgs vacuum expectation value, as is evident from Eq. (17). In addition to the massive ${\cal Z}$-boson, the vortex mediates a long-range force outside its core, thanks to the unbroken $\rm{U(1)}_{\scriptscriptstyle A}$. This adds up correctly to the number of degrees of freedom (d.o.f.): inside the core the Higgs has 2 d.o.f. and each ${\rm U(1)}$ has a single d.o.f., and outside the core the massive ${\cal Z}$-boson has 2 d.o.f. while the ${\cal A}$-boson (photon) and Higgs each has a single d.o.f.
Our vortices are charged under the $\rm{U(1)}_{\scriptscriptstyle A}$ and hence two vortices will have a logarithmic interaction due to the massless $\rm{U(1)}_{\scriptscriptstyle A}$ field. The electric field $\mathbb{E}_{\cal A}$ is given by
$$\displaystyle\mathbb{E}_{\cal A}=\frac{Q_{\cal A}}{r}\hat{\bm{e}}_{r}\,,\quad Q%
_{\cal A}=\frac{4\pi nev^{2}\mu_{1}}{e^{2}v^{2}+2\mu_{1}^{2}}\,,$$
(37)
where $\hat{\bm{e}}_{r}$ is a unit vector in the radial direction. In addition to the long-range electric field, our vortices also interact due to a background Goldstone field. As we stressed at the end of Sec. IV.2, this Goldstone background is different from the Goldstone fluctuations which are completely absorbed by the ${\cal Z}$-boson. Let us define the Goldstone field:
$$\displaystyle G_{i}=v\partial_{i}\theta-{\cal Z}_{i}\,,$$
(38)
which can be rewritten as
$$\displaystyle\mathbb{G}=\frac{Q_{G}}{r}\hat{\bm{e}}_{\alpha}\,,\quad Q_{G}=%
\frac{4\sqrt{2}\pi nv\mu_{1}^{2}}{e^{2}v^{2}+2\mu_{1}^{2}}\,,$$
(39)
where $\hat{\bm{e}}_{\alpha}$ is a unit vector in the polar direction, and we have defined an effective Goldstone charge $Q_{G}$ (see Fig. 6).
Then, the total interaction energy between vortices with charges $(Q_{\cal A}^{1},G_{G}^{1})$ and $(Q_{\cal A}^{2},G_{G}^{2})$ is given by
$$\displaystyle\frac{1}{2\pi}\left(Q_{\mathcal{A}}^{1}Q_{\mathcal{A}}^{2}+Q_{G}^%
{1}Q_{G}^{2}\right)\ln\frac{R}{r_{c}}\,.$$
(40)
At $\mu_{1}=\mu_{2}=0$ our vortices reduce to the Abrikosov-Nielsen-Olesen vortices. Turning on a non zero value of $\mu_{2}$, but still setting $\mu_{1}=0$, we recover the single $\rm{U(1)}_{\scriptscriptstyle Z}$ Chern-Simons vortex. For both cases the charges $Q_{G}$ and $Q_{\cal A}$ are zero, and the vortices do not interact with long-range forces. As we turn on a non-zero value for $\mu_{1}$, our vortices start interacting logarithmically. For values of $\mu_{1}^{2}\ll e^{2}v^{2}/2$ the interaction is dominated by the electric force, while the Goldstone force is subleading. This picture is reversed for $\mu_{1}^{2}\gg e^{2}v^{2}/2$ as the Goldstone force dominates over the electric one. Both pictures are independent of the value of $\mu_{2}$. The only effect of $\mu_{2}$ is that it contributes to the ${\cal Z}$ mass, as is clear from Eq. (17), making it infinitely large as $\mu_{2}\rightarrow\infty$. On the other hand, taking $\mu_{1}\rightarrow\infty$, the electric charge as well as the magnetic flux $\Phi_{B_{\cal A}}$ vanish, the later is clear from the fact that all the gauge fields decouple in this limit, and the interaction is solely due to the Goldstone background. In fact, this is the limit where we recover global vortices. Of course, in this limit the ${\cal Z}$-boson becomes infinitely massive and decouples. The infinite ${\cal Z}$ mass can be thought of as a UV cutoff on the vortices. The parameter space diagram of these different limits is illustrated in Fig. 7. In Fig. 8, we show the profile function $f$, and its asymptotic behavior for these regimes.
Since the energy of a single vortex is logarithmically divergent, its long energy tail has to be trimmed by either putting the vortex in a container with a finite radius, or by considering an equal number of vorticies and antivortices. In the later case, the system will lower its total energy by forming confined vortex-antivortex pairs.
So far, we have not discussed the effect of the magnetic fluxes $\Phi_{B_{\cal A}}$ and $\Phi_{B_{\cal Z}}$ on the behavior of the vortices. It was shown in Bais et al. (1993) that the monodromy of a particle $(\Phi_{B_{\cal A}},\Phi_{B_{\cal Z}})$ and a remote particle $(\Phi_{B_{\cal A}}^{\prime},\Phi_{B_{\cal Z}}^{\prime})$ leads to an Aharonov-Bohm phase
$$\displaystyle\exp\left[i\mu_{1}\left(\Phi_{B_{\cal Z}}\Phi_{B_{\cal A}}^{%
\prime}+\Phi_{B_{\cal Z}}^{\prime}\Phi_{B_{\cal A}}\right)+i\mu_{2}\Phi_{B_{%
\cal Z}}\Phi_{B_{\cal Z}}^{\prime}\right]\,.$$
(41)
This phase gives rise to non-trivial statistics of the vortices, which behave as anyons. One can understand the origin of this phase as follows. Given a charge $q$ that couples to a vector potential ${\cal A}_{\mu}$, this particle acquires the Aharonov-Bohm phase $\exp\left[iq\oint dx^{\mu}{\cal A}_{\mu}\right]=\exp[iq\Phi_{B_{\cal A}}]$, where $\Phi_{B_{\cal A}}$ is the flux of ${\cal A}_{\mu}$, as the particle makes a non-contractible winding around the source ${\cal A}_{\mu}$. Now, the zeroth component of the equations of motion (5) (the Gauss’s laws) can be written in the form
$$\displaystyle\int_{\mathbb{R}^{2}}d^{2}x\nabla\cdot{\mathbb{E}_{\cal A}}$$
$$\displaystyle=$$
$$\displaystyle 2\mu_{1}\Phi_{B_{\cal Z}}\,,$$
(42)
$$\displaystyle\int_{\mathbb{R}^{2}}d^{2}x\left[\nabla\cdot{\mathbb{E}_{\cal Z}}%
+2e^{2}v^{2}f^{2}{\cal Z}_{0}\right]$$
$$\displaystyle=$$
$$\displaystyle 2\mu_{1}\Phi_{B_{\cal A}}+2\mu_{2}\Phi_{B_{\cal Z}}\,.$$
(43)
Thus, $\mu_{1}\Phi_{B_{\cal Z}}$ is an effective charge which couples to the flux of $\cal A_{\mu}$, while $\mu_{1}\Phi_{B_{\cal A}}+\mu_{2}\Phi_{B_{\cal Z}}$ is an effective charge that couples to the flux of $\cal Z_{\mu}$. Taking this into account, we arrive to the Aharonov-Bohm phase (41). In fact, Eqs. (42) and (43) give us a working definition for the charges of both the ${\cal A}_{\mu}$ and ${\cal Z}_{\mu}$ fields. The first equation gives $Q_{\cal A}=2\mu_{1}\Phi_{\cal Z}$, while the second gives $0=-Q_{\cal Z}+2\mu_{1}\Phi_{B_{\cal A}}+2\mu_{2}\Phi_{B_{\cal Z}}$. Both of these relations can be checked against their definitions given in Sec. IV.1.
Notice that according to this definition, ${\cal Z}_{\mu}$ does not carry a charge, which is expected since it is a short range field and its charge is screened. Now, in order for any number of vortices to have a zero net ${\cal A}_{\mu}$-charge, we must have $\sum_{i}n_{i}=0$ for the winding numbers $n_{i}$; violating this condition will mean that the system has a logarithmically divergent energy. In other words, any collection of vortices will be confined if and only if it has a zero net charge. Therefore, our vortices are the dynamical realization of the Cornalba-Propitius-Wilczek classical confinement phenomenon Cornalba and Wilczek (1997); de Wild Propitius (1997).
In this work we did not discuss the stability of our vortices as it is beyond the scope of this paper. However, we expect the ones with $n=\pm 1$ to be stable against decay. As we discussed above, the Abrikosov-Nielsen-Olesen and global vortices lie on the opposite sides of the interval $\mu_{1}\in[0,\infty]$ (Fig. 7). The stability of both kinds of vortices were studied in Goodband and Hindmarsh (1995), and it was found that the ones with $n=1$ are stable, as expected on topological grounds. Indeed, an analysis that follows the lines of Goodband and Hindmarsh (1995) should be repeated for our vortices to insure their stability. However, since $n=1$ vortices on the boundaries of the $\mu_{1}$ interval were found to be stable, it is implausible that they lose stability in between as we vary $\mu_{1}$.
Acknowledgements.We would like to thank G. Dunne, A.J. Long, K.D. Olum, E. Poppitz, T. Sulejmanpasic, T. Vachaspati and A.Vilenkin for useful conversations. This work is supported by the Swiss National Science Foundation. Y.B. is supported by the grant PZ00P2-142524.
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Bouncing cosmological model with general relativistic hydrodynamics in extended gravity
A.Y.Shaikh 111Department of Mathematics, Indira Gandhi Mahavidyalaya, Ralegaon 445402,India,E-mail:shaikh_ [email protected], Sankarsan Tarai 222Centre of High Energy and Condensed Matter Physics, Department of Physicss, Utkal University, Bhubaneswar 751004, India, E-mail:[email protected], S.K. Tripathy 333Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India, [email protected] B. Mishra 444Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India, email: [email protected]
Abstract
Abstract:
In this paper, in an extended theory of gravity, we have presented bouncing cosmological model at the backdrop of an isotropic, homogeneous space-time, in presence of general relativistic hydrodynamics (GRH). The scale factor has been chosen in such a manner that with appropriate normalization, the quintom bouncing scenario can be assessed. Accordingly, the bounce occurs at $t=0$ and the corresponding Hubble parameter vanishes at the bounce epoch. The equation of state (EoS) parameter and the energy conditions of the model have been analysed. The violation of strong energy condition further supports the behaviour of extended gravity. As the bouncing cosmology suffers with instability, this model also shows the similar behaviour.
Keywords: Extended gravity, Bouncing cosmology, General relativistic hydrodynamics, Stability analysis.
I Introduction
The fundamental issues in cosmology such as horizon, baryon asymmetry, flatness problem, initial singularity and most recent the dark energy and dark matter makes the standard cosmological model in a fix in spite of its several successes. To resolve the issue related to initial singularity, the bouncing cosmological models are being designed and analysed in recent years. According to this, the universe prevailed before the big bang and then near the non-vanishing minimum radius, it experienced the accelerated expansion phase. The change from the cosmic expansion regime to the current accelerating expansion phase is known as big bounce. The idea behind this theory is that the expansion phase begins immediately after the contraction phase resulted in a bounce. This may be helpful to provide some mechanism to resolve some of the issues faced by standard cosmological model without the inflationary scenario. But the cosmological model be well accepted if it is capable of being resolve the issues as answer by the inflationary mechanism. It is important to mention here that most inflationary scenarios can give the scale-invariant spectrum of the cosmological oscillations[1]. The standard cosmological model may find the solutions to the problems occurred during the contraction before the occurrence of the bounce. At the same time, the horizon problem, can be resolved if the separated regions of the present Universe would have been in the causal connection during previous contraction phase.
The cosmological observations suggest that our universe is expanding and the expansion is accelerating at least at the late phase of evolution [2, 3, 4]. Theoretically, these observations can be dealt with the postulation that certain exotic matter with negative pressure dominates the present epoch of the universe. So, the cosmological acceleration can be introduced via fluid with negative pressure. This leads to the unknown form of energy, known as dark energy that accounts for about $68\%$ of total mass-energy budget of the universe. Several candidates for dark energy have been proposed such as the cosmological constant[5], quintessence[6, 7, 8], k-essence[9, 10, 11], tachyon[12, 13, 14], phantom[15, 16, 17], holographic dark energy[18], extra dimensions[19] and so on. Among these the cosmological constant is the simplest form of the dark energy candidate. Theoretically, another approach can be used to address this late time acceleration issue i.e. by the geometrical extension of general relativity. The extended theories of gravity can be considered as a new paradigm to address the shortcomings encountered by General Relativity to address the late time cosmic acceleration issue [20]. For details on the the cosmological and astrophysical applications of extended theories of gravity, one can refer [21]. Some interesting theoretical dark energy models can be seen in [22], where the rip cosmology, $\Lambda$CDM, quintessence and phantom cosmology have been discussed. A systematic review on the development of modified gravity on late time acceleration, inflation and bouncing cosmology is available in [23]. One of such extension with minimal matter-geometry coupling is the $f(R,T)$ gravity, where $R$ and $T$ respectively denote the Ricci scalar end trace of energy momentum tensor. This has been successful to some extent to address this issue of accelerating universe. Several studies are made in $f(R,T)$ gravity to address the cosmological and astrophysical issues that includes the late time acceleration.
We shall discuss some of the bouncing models available in the literature. Solomons et al. [24] have shown the bouncing behaviour in an anisotropic universe. In Rastall’s gravity, Silva et al. [25] have given the bouncing solutions in a barotropic fluid. Sadatian [26] in a Chaplygin gas dark energy model studied the bouncing universe and rip singularity. Brevik et al. [27] have obtained the bouncing universe in an inhomogeneous dark fluid coupled with dark matter. Singh et al. [28] studied the bouncing universe by considering the matter field in the form of viscous fluid. To provide a description of the very early universe, Bradenberger and Peter [29] have reviewed the status of bouncing cosmologies as alternatives to cosmological inflation. Minas et al. [30] in the framework of general modified gravities investigated the matter bounce scenario. Bouncing models in modified theories of gravity are also discussed in the literature. Cai et al. [31] have studied the matter bounce scenario in $f(T)$ gravity and investigated the scalar and tensor modes of cosmological perturbations. Bamba et al.[32] derived the bouncing scenario for the modified gravities such as $f(R)$ gravity and $f(R)$ bi-gravity whereas, Bamba et al. [33] have investigated the bounce inflation model in the framework of $f(T)$ gravity. Mishra et al. [34] have obtained the bounce solution in $f(R,T)$ gravity at the backdrop of an anisotropic space-time whereas Shaikh and Mishra [35] have shown the bouncing behaviour in GRH model.
Bouncing cosmology in non-isometric theories has been discussed in [36]. Tripathy et al. [37] have studied the bouncing cosmology and have shown the instability of the model near the bounce. Agrawal et al. [38] have shown the matter bounce scenarion in $f(R,T)$ gravity and its reconstruction as dark energy model [39]. In the nonmetricity based gravity Agrawal et al. have shown the matter bounce scenario with an assumed form of the scale factor [40].
We shall study the dynamical and evolutionary behaviour of the Universe. The field equations are highly non-linear and in order to solve it, certain mathematical techniques to be used. The general relativistic hydrodynamics (GRH) and magneto-hydrodynamics equations (MHD) coupled to Einstein’s equations would be technically simplify the process to obtain the solution. Here we are motivated to investigate in extended theory of gravity, the bouncing scenario in presence of GRH with an isotropic space-time. The organization of the paper is as follows: Section deals with the basic formalism of extended gravity along with the brief introduction of the GRH and the field equations. In Section , the bouncing cosmology has been presented and the dynamical parameters are investigated in Section . The discussions on the geometrical diagnostics and stability analysis are performed in Section along with conclusion of the work done.
II Basic Formalism and Field Equations
The action for the geometrically extended $f(R,T)$ gravity can be expressed as [41],
$$S=\int\left[\frac{\sqrt{-g}}{2}[R+\beta f(T)]+\sqrt{-g}\mathcal{L}_{m}\right]d^{4}x,$$
(1)
where $f(T)$ is an arbitrary function of the trace $T$ of the energy momentum tensor, $\beta$ be the coupling constant and $\mathcal{L}_{m}$ is the matter Lagrangian. Varying the action $S$ with respect to $g_{\mu\nu}$, the field equations of $f(R,T)$ gravity can be expressed as,
$$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=T_{\mu\nu}-\beta f_{T}(T)T_{\mu\nu}+\frac{1}{2}\beta f(T)g_{\mu\nu}-\beta f_{T}(T)\Theta_{\mu\nu},$$
(2)
when $\beta=0$, eqn. (2) reduces to the field equations of general relativity. The energy momentum tensor can be expressed as.
$$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta g^{\mu\nu}}$$
(3)
and
$$\Theta_{\mu\nu}\equiv g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g^{\mu\nu}}$$
(4)
The tools of astrophysics like hydrodynamics, magneto-hydrodynamics, radiation transport [42] and nuclear astrophysics would be required by the observational data that involves the GRH phenomena. For an informative overview of relativistic hydrodynamics, one can refer [43]. Several techniques are availble in the literature to solve the field equations of GRH such as, (i)generalization of Roe’s approximate Riemann solver numerical method [44], (ii) special relativistic Riemann solvers [45], (iii) fully self-consistent relativistic hydrodynamics code [46]. Then, formulations of the equations of GRH and MHD, along with methods for their numerical solution was reviewed extensively [47]. Subsequently a comprehensive overview of numerical hydrodynamics and magneto hydrodynamics was presented in the context of general relativity[48].
We consider the stress-energy tensor in the form of perfect fluid as,
$$T_{\mu\nu}=\rho hu_{\mu}u_{\nu}-pg_{\mu\nu}.$$
(5)
The relativistic enthalpy $h$ can be expressed as, $h=(1+\epsilon)+\frac{p}{\rho_{0}}$, where $p$, $\rho_{0}$ and $\epsilon$ pressure, rest mass density and specific internal energy of the fluid. To note, the rest mass energy density $\rho_{0}$ is different from the energy density $\rho$. A constitutive relation of the form $p=p(\rho,\epsilon)$ and the ideal fluid equation of state parameter, $p=\rho_{0}\epsilon(\Gamma-1)$, $\Gamma$ be the adiabatic index has been considered. Now, we rewrite the relativistic specific enthalpy as,
$$h=1+\Gamma\epsilon.$$
(6)
We wish to mention here that $\epsilon$ is temperature-dependent, which can be confirmed on causal thermodynamics [49] or relativistic hydrodynamics [50]. However, in this paper our motivation is to study the bouncing scenario of the Universe in presence of GRH, therefore we prefer to choose a fixed value for $\epsilon$. If we consider the matter Lagrangian as, $L_{m}=-p$, eqn. (4) becomes, $\Theta_{\mu\nu}=-pg_{\mu\nu}-2T_{\mu\nu}$ and the field equations (2) reduce to,
$$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$$
$$\displaystyle=$$
$$\displaystyle T_{\mu\nu}+\beta f_{T}(T)T_{\mu\nu}+[f_{T}(T)p+\frac{1}{2}f(T)]\beta g_{\mu\nu}$$
(7)
$$\displaystyle=$$
$$\displaystyle[1+\beta f_{T}(T)](T_{\mu\nu}+T^{int}_{\mu\nu})$$
where,
$$T^{int}_{\mu\nu}=\frac{1}{1+\beta f_{T}(T)}\left[f_{T}(T)p+\frac{1}{2}f(T)\right]\beta g_{\mu\nu}$$
(8)
It is note here eqn. (8) vanishes for $\beta\rightarrow 0$. We consider the functional as, $\frac{1}{2}f(T)=T$, and subsequently obtained, $T^{int}_{\mu\nu}=\frac{1}{1+\beta}\left[2p+T\right]\beta g_{\mu\nu}$. Harko et al. [41] in their seminal work proposed three forms for the function $f(R,T)$ as (i) $f(R,T)=R+2f(T)$, (ii) $f(R,T)=f_{1}(R)+f_{2}(T)$, (iii) $f(R,T)=f_{1}(R)+f_{2}(R)f_{3}T)$. Here, we consider the functional form of $f(R,T)$ in such a manner that the filed equations of GR can be obtained under suitable substitution of the model parameters. One of the popular choice is; $f(R,T)=R+2\beta T$ [51, 52], $\beta$ being the coupling constant, such that $f(T)=2T$. The matter part of the extended theory of gravity has been discussed above and to set the filed equations of $f(R,T)$ gravity, the geometric part needs to be addressed. So here we consider an isotropic and homogeneous FLRW space-time as,
$$ds^{2}=dt^{2}-a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$$
(9)
where $a(t)$ be the scale factor. The $f(R,T)$ gravity field equations (7) with GRH in the form of perfect fluid can be obtained as,
$$\displaystyle 2\dot{H}+3H^{2}$$
$$\displaystyle=$$
$$\displaystyle-\alpha p+\beta\rho h$$
(10)
$$\displaystyle 3H^{2}$$
$$\displaystyle=$$
$$\displaystyle\alpha\rho h-\beta p,$$
(11)
where $\alpha=(1+3\beta)$ be the redefined coupling constant and an over dot denotes the derivative with respect to cosmic time. We have expressed the field equations in Hubble term and in order to obtain the expressions for the dynamical parameters, we need to assume a form for the Hubble parameter. In recent years bouncing cosmology, an alternative to the inflationary paradigm [53] has been studied extensively to resolve the singularity issue, therefore, we are motivated here to study the model in the matter bounce scenario.
III Bouncing Cosmological Model
We consider here the bouncing scale factor as $a(t)=\sqrt{1+\eta^{2}t^{2}}$ with $\eta>0$. This scale factor is the temporal analogue of the toy model of the traversable wormhole[53] and with proper renormalization of the scale factor parameter $\eta$, the phenomenological quintom bouncing model can be obtained [54]. The Hubble parameter can be expressed as
$$H=\frac{\dot{a}}{a}=\frac{\eta^{2}t}{1+\eta^{2}t^{2}}$$
(12)
We have presented the graphical behaviour of the scale factor and Hubble parameter in Fig. 1 to verify the occurrence of bounce during the evolution of the universe.
According to bouncing cosmology, the model should initially undergoes a phase of collapse, attains its minimum value at the bouncing point and then expands subsequently. In Fig. 1 (left panel), the scale factor shows a symmetrical behaviour. The slope of the scale factor depends upon the parameter $\eta$. For attaining the bounce in the FLRW model space-time, $a(t)$ is decreasing i.e. $(\dot{a}(t)<0)$ during the negative frame of cosmic time (contracting universe) and then in the expanding phase, the scale factor is increasing i.e. $(\dot{a}(t)>0)$. The bounce has been noticed at $t=0$ for positive $\eta$. At $t=0$, the scale factor curve shows the symmetric behaviour and attains a non-zero minimum value $a(t)\cong 1$ at the bounce. In a nutshell, more is the value of $\eta$, more is the curvature. In Fig. 1 (right panel), we have plotted the evolution of Hubble parameter for three different values of $\eta$. As we are keeping in our mind to construct a bouncing model, we may draw the cosmic time from negative domain to positive domain and consequently the evolution of Hubble parameter goes linearly from negative to positive time domain through $t=0$ at bounce. In the negative time zone, the Hubble parameter remains negative ($H<0$)in the interval $-2<t<0$, and in the positive time zone, it is positive ($H>0$) in the interval $0<t<2$. At the bouncing point the Hubble parameter vanishes irrespective of the value of $\eta>0$. The deceleration parameter explains the nature of expansion of the model and can be obtained with the second derivative of the scale factor. For this bouncing sale factor, the deceleration parameter can be obtained as,
$$q=-1-\frac{\dot{H}}{H^{2}}=-\frac{1}{\eta^{2}t^{2}}$$
(13)
Bolotin et al.[55] classified the cosmological models on the basis of time dependence on Hubble parameter and deceleration parameter.
•
For $H>0$, the model expands and the expansion is accelerating for $q<0$ or decelerating for $q>0$ and contracting behaviour for $H<0$.
•
For $H>0$ and $q=0$, the model expands, and zero deceleration or constant expansion and contracts for $H>0$.
•
For $H=0$ and $q=0$, the model remains static.
The deceleration parameter shows symmetry behaviour as in Fig. 2, at the bouncing point, $t=0$. From Fig. 2, it is observed that for both the contracting and expanding universes, the deceleration parameter $q$ is negative, and after some finite time it tends to a constant value $–1$. In the negative time zone (contacting universe), the deceleration parameter tends to the large negative values at the bouncing point even after evolving from $q=-1$. In the positive frame of the cosmic time (expanding universe), the deceleration parameter tends to $q=-1$ at late times.
IV Analysis of Dynamical Parameters of the Model
The viability of any cosmological model based on the appropriate behaviour of the dynamical parameters and its behaviour must be converging towards agreement. The disagreement, if any, reflect the systematic errors that comes either from the different methods of analysis or from the cosmological observations. It may also come from the representative values chosen for the model and free parameters involved in the expression. The late time cosmic acceleration phenomena requires the pressure and equation of state (EoS) parameter to be negative at least at the present and future time. Hence, we shall analyse here the dynamical parameters of the model in the context of bouncing behaviour. Substituting the expression of the Hubble parameter (12) in eqns. (10) and (11) and with an algebraic manipulation, we can obtain the mater pressure and energy density as,
$$\displaystyle p$$
$$\displaystyle=$$
$$\displaystyle\frac{\eta^{2}[-3(1+2\beta)\eta^{2}t^{2}+2(1+3\beta)(1-\eta^{2}t^{2})]}{(1+8\beta^{2}+6\beta)(1+\eta^{2}t^{2})^{2}}$$
(14)
$$\displaystyle\rho$$
$$\displaystyle=$$
$$\displaystyle\frac{\eta^{2}[3(1+2\beta)\eta^{2}t^{2}-3\beta(1-\eta^{2}t^{2})]}{(1+\Gamma\epsilon)(1+8\beta^{2}+6\beta)(1+\eta^{2}t^{2})^{2}}$$
(15)
The EoS parameter governs the gravitational properties of dark energy model and its evolution profile. In the cosmological models, either it appears as a time dependent function or a constant. Cosmological observations [56, 57] have suggested ranges for the EoS parameter as,
$$\displaystyle Planck+WP+Union2.1:~{}~{}~{}~{}~{}-0.92$$
$$\displaystyle\leq$$
$$\displaystyle\omega\leq-1.26,$$
$$\displaystyle Planck+WP+BAO:~{}~{}~{}~{}~{}~{}~{}-0.89$$
$$\displaystyle\leq$$
$$\displaystyle\omega\leq-1.38,$$
$$\displaystyle WMAP+eCMB+BAO+H0:~{}~{}~{}~{}~{}~{}~{}~{}-0.983$$
$$\displaystyle\leq$$
$$\displaystyle\omega\leq-1.162.$$
We can derive the EoS parameter $\omega=\frac{p}{\rho}$ as,
$$\omega=(1+\Gamma\epsilon)\left[\frac{-3(1+2\beta)\eta^{2}t^{2}+2(1+3\beta)(1-\eta^{2}t^{2})}{3(1+2\beta)\eta^{2}t^{2}-3\beta(1-\eta^{2}t^{2})}\right]$$
(16)
From eqns. (14)-(15), it is observed that the pressure and energy density of the GRH model depend on the value of the scale factor parameter $\eta$ and model parameter $\beta$. For the positive value of the model parameter $\beta$, the denominators of eqns. (14)-(15) are always positive. The positivity or the negativity of these quantities depends only on the respective numerators. In order to satisfy certain energy conditions, the energy density should remain positive throughout the cosmic evolution. For $\eta=0.24$, $\beta=-0.51,$ $\Gamma=2,$ $\epsilon=0.1$ the energy density remains positive throughout the cosmic evolution both in the positive and negative domains of cosmic time. The same can be observed in Fig. 3 (left panel). The evolution behaviour of EoS parameter, a time dependent function, has been shown in Fig. 3 (right panel). It is observed that for $\beta\rightarrow 0$, the corresponding GRH model reproduces the results of general relativity i.e. $\omega_{\beta\rightarrow 0}\approx(1+\Gamma\epsilon)\left(-1+\frac{2(1-\eta^{2}t^{2})}{3\eta^{2}t^{2}}\right)$ . From Fig. 3 (right panel), it is observed that, the EoS parameter shows symmetrical behaviour about the bouncing time. At the bounce $(t=0)$, the EoS parameter attains the value $-0.838$, which shows the quintessence behaviour of the dark energy fluid [6]. The constraints of EoS parameter of the derived GRH model coincide with the WMAP nine years observational data[56] and the Planck collaboration results [57], which is specified above before eqn. (16).
Another prescription for the bouncing model is the violation of null energy condition and strong energy condition at the bounce. In general relativity, energy conditions are used extensively to examine the singularity problem of the space-time. Essentially, the energy conditions are described by the behaviour of space like, time like or light like curves [58, 59, 60]. We have considered here the space like and time like curves. The energy conditions can be defined in extended theory of gravity as in general relativity with the new or effective pressure and energy density as in (14)-(15). We can calculate the energy conditions such as: NEC (null energy condition, WEC (weak energy condition), SEC (strong energy condition) and DEC (dominant energy condition) for the GRH bouncing model as,
$$\displaystyle\rho+p$$
$$\displaystyle=$$
$$\displaystyle\frac{\eta^{2}\left[-3\Gamma\epsilon(1+2\beta)\eta^{2}t^{2}+(1-\eta^{2}t^{2})[2(1+3\beta)(1+\Gamma\epsilon)-3\beta]\right]}{(1+\Gamma\epsilon)(1+8\beta^{2}+6\beta)(1+\eta^{2}t^{2})^{2}}\geq 0$$
$$\displaystyle\rho+p$$
$$\displaystyle=$$
$$\displaystyle\frac{\eta^{2}\left[-3\Gamma\epsilon(1+2\beta)\eta^{2}t^{2}+(1-\eta^{2}t^{2})[2(1+3\beta)(1+\Gamma\epsilon)-3\beta]\right]}{(1+\Gamma\epsilon)(1+8\beta^{2}+6\beta)(1+\eta^{2}t^{2})^{2}}\geq 0;\rho\geq 0$$
$$\displaystyle\rho+3p$$
$$\displaystyle=$$
$$\displaystyle\frac{\eta^{2}\left[3(2+\Gamma\epsilon)(1+2\beta)\eta^{2}t^{2}-(1-\eta^{2}t^{2})[2(1+3\beta)(1+\Gamma\epsilon)+3\beta]\right]}{(1+\Gamma\epsilon)(1+8\beta^{2}+6\beta)(1+\eta^{2}t^{2})^{2}}\geq 0$$
$$\displaystyle\rho-p$$
$$\displaystyle=$$
$$\displaystyle\frac{n^{2}\left[3(1+2\beta)\eta^{2}t^{2}(1-3(1+\Gamma\epsilon))+3(1-\eta^{2}t^{2})[2(1+3\beta)(1+\Gamma\epsilon)-3\beta]\right]}{(1+\Gamma\epsilon)(1+8\beta^{2}+6\beta)(1+\eta^{2}t^{2})^{2}}$$
(17)
$$\displaystyle\geq$$
$$\displaystyle 0$$
Graphically, we have represented the behaviour of energy conditions in Fig. 4. The SEC and the NEC both are violating at the bounce. In the flat FRW universe, at the bouncing point $\rho\geq 0,$ $\rho+p<0$. The energy density of the fluid decreases with contraction. It is constant at the bouncing point and then increases with subsequent expansion. Therefore, at the bounce, NEC is violated and the universe is accelerating after the bounce due to $\rho+3p<0$.
V Discussions and Conclusion
The simple candidate for the dark energy model is the cosmological constant. The cosmological observations are providing more accurate information on the dark energy properties. So, theoretically further validate the cosmological model, the geometrical diagnostic for dark energy in the form of state finder pair would be more appropriate[61]. It consists of two parameters the jerk parameter $j$ and snap parameter $s$ which can be obtained by performing the third and fourth order derivative respectively of scale factor. Both these quantities can be derived as,
$$\displaystyle j$$
$$\displaystyle=$$
$$\displaystyle\frac{\dddot{a}}{aH^{2}}=-\frac{3}{t(1+\eta^{2}t^{2})}$$
(18)
$$\displaystyle s$$
$$\displaystyle=$$
$$\displaystyle\frac{j-1}{3(q-\frac{1}{2})}=\frac{2\eta^{2}t(3+t+\eta^{2}t^{3})}{3(2+3\eta^{2}t^{2}+\eta^{4}t^{4}}$$
(19)
From eqn. (19), We can see that for $j=1$, the snap parameter should vanish and when the pair $(j,s)$ approaches to $(1,0)$, the model supports the $\Lambda$CDM behaviour. But from eqn. (18), it is evident that $j$ will approach to $1$ only in the negative time zone. Therefore, the model does not support the $\Lambda$CDM behaviour. The same behaviour can be observed from Fig. 5 graphically. In the left panel, the jerk parameter is vanishing only at very early and very late time. Both the jerk and snap parameters are showing the singularity behaviour at the bouncing epoch. Finally, while framing this cosmological model assumptions were made to solve the system because of the difficulty in the governing equations. The physical viability of these assumptions must be known. In this particular problem, we have studied the model with an assumed scale factor. Therefore it is essential to know the stability of the model. The stability of the model can be studied through the mechanical stability of the cosmic fluid. This can be performed by the adiabatic speed of sound through the cosmic fluid as, $C_{s}^{2}=\frac{dp}{d\rho}$ [62, 63, 64], which can be calculated for this model as,
$$C^{2}_{s}=(1+\Gamma\epsilon)\left[\frac{-3(1+2\beta)(1+\eta^{2}t^{2})^{2}-8\alpha\eta^{2}t}{3(1+2\beta)(1+\eta^{2}t^{2})^{2}+8\beta\eta^{2}t}\right]$$
(20)
The stability of the model depends on the behaviour of $C_{s}^{2}$ with respect to the cosmic time. If $C_{s}^{2}>0$ the model is stable and for $C_{s}^{2}<0$, the model remains unstable. In this model, we wish to know the stability at the bouncing point. Fig. 6 stability factor shows symmetry behaviour and at the bouncing point and in its neighbourhood , it remains negative. Hence we can conclude that the GRH bouncing model presented here is unstable.
The bouncing cosmological model constructed here in presence of GRH in an extended theory of gravity provides non-singular bounce at $t=0$. The energy density increases and remains maximum at the bounce epoch and then decreases. Higher the value of $\eta$, the energy density is more. The violation of NEC at the bounce further gurantees the violation of SEC. The EoS parameter remains at the quintessence phase at the bounce and showing the well-shaped, but while evolving out, it gradually increases both in the negative and positive time zone. From the geometrical diagnostic, the models does not support $\Lambda$CDM behaviour, which has already been observed from the EoS parameter. The behaviour of the deceleration parameter shows the super-exponential expansion. Some of the key results of the model has been listed down in TABLE-I
In conclusion, we say that the model is not stable at the bounce, which is in line with the statement that bouncing models suffer with instability. The other basic requirements of the bouncing model in an extended theory of gravity are satisfied.
Acknowledgements
ST acknowledges Rashtriya Uchachatar Shikshya Abhiyan(RUSA), Ministry of HRD, Govt. of India for the financial support. The authors are thankful to the honorable reviewers for their valuable suggestions and comments for the improvment of the manuscript.
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A Structural Dynamic Factor Model for Daily Global Stock Market Returns
Oliver B. Linton
University of Cambridge
Faculty of Economics, Austin Robinson Building,
Sidgwick Avenue, Cambridge, CB3 9DD. Email: [email protected]. Thanks
to the ERC for financial support.
Haihan Tang
Fudan University
Fanhai International School of Finance, Fudan University.
220 Handan Road, Yangpu District, Shanghai, 200433, China. Email:
[email protected].
Jianbin Wu
Nanjing University
School of Economics, Nanjing Universtiy, Nanjing, 210093,
China. Email: [email protected].
Abstract
Most stock markets are open for 6-8 hours per trading day. The
Asian, European and American stock markets are separated in time by time-zone
differences. We propose a statistical dynamic factor model for a large number
of daily returns across multiple time zones. Our model has a common global
factor as well as continental factors. Under a mild fixed-signs assumption,
our model is identified and has a structural interpretation. We propose
several estimators of the model: the maximum likelihood estimator-one day
(MLE-one day), the quasi-maximum likelihood estimator (QMLE), an improved
estimator from QMLE (QMLE-md), the QMLE-res (similar to MLE-one day), and a
Bayesian estimator (Gibbs sampling). We establish consistency, the rates of
convergence and the asymptotic distributions of the QMLE and the QMLE-md. We
next provide a heuristic procedure for conducting inference for the MLE-one
day and the QMLE-res. Monte Carlo simulations reveal that the MLE-one day, the
QMLE-res and the QMLE-md work well. We then apply our model to two real data
sets: (1) equity portfolio returns from Japan, Europe and the US; (2) MSCI
equity indices of 41 developed and emerging markets. Some new insights about
linkages among different markets are drawn.
Keywords:
Daily Global Stock Market Returns; Time-Zone Differences;
Structural Dynamic Factor Model; Quasi Maximum Likelihood; Minimum Distance;
Expectation Maximization Algorithm.
JEL classification
C55; C58; G15.
1 Introduction
The world’s stock markets are separated in time by substantial time-zone
differences, to the extent that for example the US and the main Asian markets
do not overlap at all (although the European markets overlap a little with
both other continents). Nevertheless, it is a common belief that international
stock markets are becoming more and more connected through international trade
and cross border investments. Linkages between different markets were
particularly evident during stressful times like the financial crisis in 2008
and the COVID-19 outbreak in 2020. The last three decades have witnessed a
heightening interest in measuring and modelling such linkages, whether dubbed
as the stock market integration, international return spillovers, cross-market
contagions etc. Gagnon and
Karolyi (2006) and Sharma and
Seth (2012) have
carefully reviewed the literature and categorized these studies according to
methodologies, data sets and findings. The difficulty in establishing the comovement between major stock markets that do not overlap in trading time is that the Efficient Markets Hypothesis (with a constant risk premium) would imply zero correlation between their observed daily returns, our work is designed to address this.
All the existing studies either examined a small number of entities such as a
few market indices, or ignored the time-zone differences whenever using daily
data. The use of daily closing prices while ignoring the time-zone differences
causes the so-called stale-price problem (Martens and
Poon (2001),
Connor
et al. (2010, p.42-44)). The standard approach is to
include the lead and lagged covariances to correct for the stale pricing
effect. This approach does not allow one to identify the source of variation
or its relative impacts.
When studying a large number of entities, it is common practice to aggregate
information. We provide a framework to model the correlations of daily stock
returns in different markets across multiple time zones. The machinery will be
a statistical dynamic factor model (first proposed by Forni
et al. (2000)),
which enables us to work with a large number of stocks. To make the framework
tractable, we make the following modelling assumption: All the markets belong
to one of three continents: Asia (A), Europe (E) and America (U). Within a
calendar day, the Asian markets close first, followed by the European and then
American markets. We suppose that the observed logarithmic 24-hr returns
(determined as the close-to-close returns) in each continent follow a dynamic
factor model with both global and continental factors. This model reflects a
situation in which new global information represented by the global factor
affects all three continents simultaneously, but is only revealed in the
observed returns in the three continents sequentially as their markets open in
turn and trade on the new information (Koch and
Koch (1991, p.235)). New
information represented by a continental factor accumulated since the last
closure of that continent’s markets will also have an impact on the upcoming
observed logarithmic 24-hr returns of those markets.
The approach of having global and continental factors, in some respects,
resembles that of Kose
et al. (2003) who modeled world yearly
macroeconomic aggregates (output, consumption and investment) using a
statistical dynamic factor model consisting of global, regional and
country-specific factors. The difference is that we work with daily stock
returns and have the feature of sequential revelation of the global
information in the observed returns in the three continents. Likewise, our
approach is related to the GVAR modeling approach (Pesaran
et al. (2004)),
which was developed to model world low-frequency macroeconomic series. Here we
do not have so many relevant variables beyond the prices themselves and hence
our model is in terms of unobserved factors.
Our approach is also closely related to the nowcasting framework
(Giannone
et al. (2008), Banbura et al. (2013),
Aruoba
et al. (2009) etc). In the nowcasting literature, researchers
use factor models to extract the information contained in the data at higher
frequencies than the target variable in order to forecast the target variable.
Here, if we make additional assumptions on the data generating
processes of the unobserved logarithmic 24-hr returns, we could also
obtain their corresponding nowcasts; this is the similarity. The difference is
that, as we shall point out in Section 2, model (2.3)
is identified under a mild fixed-signs assumption (Assumption
2.2) and hence has a structural interpretation, whereas in the
nowcasting literature, identification of factor models is usually not
addressed, and factor models are mere dimension-reducing tools with no
structural interpretations. In some sense, our model belongs to the class of
structural dynamic factor models (Stock and
Watson (2016)).
On the theoretical side, research about estimation of large factor models via
the likelihood approach has matured over the last decade. The likelihood
approach enjoys several advantages such as efficiency compared to the
principal components method (Banbura et al. (2013, p.204)).
Doz
et al. (2012) established an average rate of
convergence of the estimated factors using a quasi-maximum likelihood
estimator (QMLE) via the Kalman smoother. There is a rotation matrix attached
to the estimated factors as the authors did not address identification of
factor models. Also they did not derive consistency for the estimated factor
loadings, or the limiting distributions of any estimate.
In an important paper, Bai and Li (2012) took a different approach to study
large exact factor models. They treated factors as fixed parameters instead of
random vectors. One nice thing about this approach is that the theoretical
results obtained hold for any dynamic pattern of factors. Bai and Li (2012)
obtained consistency, the rates of convergence and the limiting distributions
of the maximum likelihood estimators (MLE) of the factor loadings,
idiosyncratic variances, and sample covariance matrix of factors. In fact,
Bai and Li (2012) called their estimators the QMLE instead of the MLE. We
decided to re-label them as the MLE since we shall reserve the phrase QMLE for
another purpose to be made specific shortly. Factors are then estimated via a
generalised least squares (GLS) method. Bai and Li (2016) generalised the
results of Bai and Li (2012) to large approximate factor models.
In practice, instead of maximizing a likelihood and finding the MLE, people
usually use the EM algorithm together with the Kalman smoother to estimate the
model (Watson and
Engle (1983), Quah and
Sargent (1992),
Doz
et al. (2012), Bai and Li (2012), Bai and Li (2016) etc). Since
the EM algorithm runs only for a finite number of iterations, strictly
speaking the estimate obtained by the EM algorithm is only an approximation to
the MLE. However, in a breakthrough study Barigozzi and
Luciani (2022) showed
that the estimate obtained by the EM algorithm converges to the MLE fast
enough so that it has the same asymptotic distribution as the MLE.
We propose several estimators of our model (2.3): the MLE-one day,
the QMLE-res, the QMLE, the QMLE-md, and the Bayesian. The MLE-one day
estimator is the usual MLE estimator of our model. The QMLE-res estimator is
the MLE estimator of the two-day representation of our model while maintaining
the working independence hypothesis (see Section
2.2); in this article we shall refer a
likelihood-based estimator obtained under the working independence hypothesis
as the QMLE rather than the MLE. The QMLE estimator differs from the QMLE-res
in the sense that only a specific subset of restrictions implied by our model
is imposed. Since the QMLE is inefficient, we propose an improved estimator,
the QMLE-md, which uses the QMLE in the first step and incorporates an
additional finite number of restrictions implied by our model via the minimum
distance method in the second step. Last, the Bayesian estimator uses the
Gibbs sampling to estimate the model (2.3). However, the Gibbs
sampling is computationally intensive and feasible only for a not-so-large
number of entities.
The large sample results of the aforementioned studies are not directly
applicable to our model because the proofs of these results are
identification-scheme dependent. In particular, Bai and Li (2012),
Bai and Li (2016) established their results under five popular identification
schemes, none of which is consistent with our model. In order to have an
identification scheme consistent with our model and at the same time utilise
the theories of Bai and Li (2012), we could only impose some, not all,
of the restrictions implied by our model to derive the first-order conditions
(FOC) of the log-likelihood. It took us a considerable amount of work to
derive the corresponding large sample results of our QMLE estimator. That is,
consistency, the rate of convergence and the asymptotic distribution of the
QMLE are established. Then the asymptotic distribution of the QMLE-md could be
derived as well. The large sample theories of the MLE-one day and the QMLE-res
are beyond the scope of this article, and we leave them for future research.
Nevertheless, we provide a heuristic procedure to approximate the standard
errors of the MLE-one day and the QMLE-res.
We conduct some Monte Carlo simulations to evaluate the MLE-one day, the
QMLE-res and the QMLE-md in terms of the root mean square errors, average of
the standard errors across the Monte Carlo samples, and the coverage
probability of the constructed confidence interval. These three estimators
perform well.
Last, we apply our model to two real data sets. The first data set consists of
daily equity portfolio returns from Japan, Europe and the US; that is, one
market per continent, over the period 1991-2020. Our methodology quantifies
how much the global factor loaded on the returns during a particular fraction
of a calendar day, as well as the relative importance of the global and
continental factors. We also uncover some interesting time-series patterns.
The second data set is MSCI equity indices of the 41 developed and emerging
markets. Taking the Asian-Pacific continent as an example, we find that
Mainland China and Hong Kong have particularly high loadings on the global
factor during the US trading time. Japan has high loadings on the continental
factor but small idiosyncratic variances, while other Asian-Pacific markets
have statistically insignificant loadings on the continental factor.
We contribute to methodology by providing a new modelling framework for daily
global stock market returns. Our framework could easily handle a large number
of stocks and at the same time take into account the time-zone differences.
Under a mild fixed-signs assumption, our model is identified and has a
structural interpretation. We also contribute to theory by deriving the
asymptotic results of the QMLE and the QMLE-md. The machinery is based on the
theoretical results of Bai and Li (2012), but we demonstrate how one could
obtain their results for almost any identified dynamic factor model.
This is an important contribution as many dynamic factor models, like ours,
are motivated by different economic theories and might not be compatible with
the five identification schemes of Bai and Li (2012). We last contribute to the
applied literature by proposing several practically usable estimators and
validate their performances in the Monte Carlo simulations. When applying our
model to two real data sets, we draw some new insights about linkages among
different stock markets.
The rest of the article is structured as follows. In Section 2
we explain our model and discuss identification while in Section 3
we introduce our estimators. Section 4 presents the large
sample theories of the QMLE and the QMLE-md. Section
5 provides a heuristic
procedure to approximate the standard errors of the MLE-one day and the
QMLE-res. Section 6 conducts the Monte Carlo simulations
to assess those advocated estimators, and Section 7
presents two empirical applications of our model. Section 8
concludes. Major proofs and technical details are to be found in Appendix;
the remaining proofs and technical materials are put in the online Supplementary Material (SM in what follows).
1.1 Notation
Let $\mathbb{R}^{n}$ and $\mathbb{Z}^{+}$ denote the $n$-dimensional Euclidean
space and set of non-negative integers, respectively. For $x\in\mathbb{R}^{n}$, let $\|x\|_{2}\mathrel{\mathop{:}}=\sqrt{\sum_{i=1}^{n}x_{i}^{2}}$ and $\|x\|_{\infty}\mathrel{\mathop{:}}=\max_{1\leq i\leq n}|x_{i}|$ denote the Euclidean ($\ell_{2}$) and
element-wise maximum ($\ell_{\infty}$) norms, respectively. Let $A$ be an
$m\times n$ matrix. Let $\operatorname*{vec}A$ denote the vector obtained by stacking columns
of $A$ one underneath the other. Let unvec denote the reverse operation of
$\operatorname*{vec}$.
The commutation matrix $K_{m,n}$ is an $mn\times mn$
orthogonal matrix which translates $\operatorname*{vec}A$ to $\operatorname*{vec}(A^{\intercal})$,
i.e., $\operatorname*{vec}(A^{\intercal})=K_{m,n}\operatorname*{vec}(A)$. If $A$ is a symmetric $n\times n$
matrix, its $n(n-1)/2$ supradiagonal elements are redundant in the sense that
they can be deduced from symmetry. If we eliminate these redundant elements
from $\operatorname*{vec}A$, we obtain a new $n(n+1)/2\times 1$ vector, denoted $\operatorname*{vech}A$.
They are related by the full-column-rank, $n^{2}\times n(n+1)/2$
duplication matrix $D_{n}$: $\operatorname*{vec}A=D_{n}\operatorname*{vech}A$. Conversely,
$\operatorname*{vech}A=D_{n}^{+}\operatorname*{vec}A$, where $D_{n}^{+}$ is $n(n+1)/2\times n^{2}$ and the
Moore-Penrose generalized inverse of $D_{n}$.
Given a vector $v$, $\operatorname*{diag}(v)$ creates a diagonal matrix whose diagonal
elements are elements of $v$.
We use $p(\cdot)$ to denote the (asymptotic) probability density function.
$\lfloor x\rfloor$ denotes the greatest integer strictly less than
$x\in\mathbb{R}$ and $\lceil x\rceil$ denotes the smallest integer greater
than or equal to $x\in\mathbb{R}$.
Landau (order) notation in this article, unless otherwise stated, should be
interpreted in the sense that $N,T\rightarrow\infty$ jointly, where $N$ and
$T$ are the cross-sectional and temporal dimensions, respectively. We use $C$
or $C$ with number subscripts to denote absolute positive constants
(i.e., constants independent of anything which is a function of $N$ and/or
$T$); identities of such $C$s might change from one place to another.
2 The Model
Our model is based on the closing prices of the stock markets on the three
continents, which occur at different calendar times. We suppose that the
closing times are ordered as follows:
$$\begin{array}[c]{ccccccccc}&A&E&U&A&E&U&A&\cdots\\
t=&1&2&3&4&5&6&7&\cdots\end{array}$$
Note that the unit of $t$ is not a day, but a fraction of a day. This
framework could be applied to three markets only (i.e., one market in each
continent), or to the case where some continent contains several markets.
Let $p_{i,t}^{c}$ denote the logarithmic closing price of stock $i$ in
continent $c$ at time $t$ for $c=A,E,U$. We shall assume that there are no
weekends as is the prevalent assumption in single market analysis.
Nevertheless, we allow that $p_{i,t}^{c}$ is not observed in two scenarios:
(i)
Missing because of non-synchronized trading. That is, $t$ might not
correspond to the closing times of continent $c$. For example, we do not
observe $p^{E}_{i,3}$ for any stock $i$ in the European continent.
(ii)
Missing because of some specific reasons. The reasons could be
continental (e.g., Chinese New Year, Christmas), market-specific (e.g.,
national holidays), or stock-specific (e.g., general meetings of shareholders).
We shall rule out scenario (ii) for the time being and address it to some
extent in SM LABEL:sec_missing_obs.
We next introduce our model. Define the logarithmic 24-hr return $y_{i,t}^{c}\mathrel{\mathop{:}}=p_{i,t}^{c}-p_{i,t-3}^{c}$ for $c=A,E,U$, $T_{A}\mathrel{\mathop{:}}=\{1,4,7,\ldots,T-2\}$,
$T_{E}\mathrel{\mathop{:}}=\{2,5,8,\ldots,T-1\}$ and $T_{U}\mathrel{\mathop{:}}=\{3,6,9,\ldots,T\}$, where $T$ is a
multiple of 3. We assume that the observed logarithmic 24-hr returns
follow the dynamic system: For $c=A,E,U$,
$$\displaystyle y_{i,t}^{c}$$
$$\displaystyle=z_{i}^{c}(L)f_{g,t}+\tilde{z}_{i}^{c}(L)f_{c,t}+e_{i,t}^{c},\quad i=1,\ldots,N_{c},\quad t\in T_{c}$$
(2.1)
where $f_{g,t}$ and $f_{c,t}$ are the scalar unobserved global and continental
factors, respectively, with
$$\displaystyle z_{i}^{c}(L)f_{g,t}$$
$$\displaystyle\mathrel{\mathop{:}}=\sum_{j\in\mathbb{Z}^{+}}z^{c}_{i,j}f_{g,t-j},\qquad\tilde{z}_{i}^{c}(L)f_{c,t}\mathrel{\mathop{:}}=\sum_{j\in\mathbb{Z}^{+}}\tilde{z}^{c}_{i,j}f_{c,t-3j}$$
for $c=A,E,U$, where $L$ is the lag operator (i.e., $Lx_{t}=x_{t-1}$). The
model is dynamic in the sense that
$$\displaystyle\phi_{g}(L)f_{g,t+1}$$
$$\displaystyle=\eta_{g,t},\qquad\qquad t=0,1,\ldots,T-1$$
$$\displaystyle\phi_{c}(L)f_{c,t+1}$$
$$\displaystyle=\eta_{c,t},\qquad\qquad t+1\in T_{c}$$
where
$$\phi_{g}(L)f_{g,t+1}\mathrel{\mathop{:}}=\sum_{j\in\mathbb{Z}^{+}}\phi_{g,j}f_{g,t+1-j},\qquad\qquad\phi_{c}(L)f_{c,t+1}\mathrel{\mathop{:}}=\sum_{j\in\mathbb{Z}^{+}}\phi_{c,j}f_{g,t+1-3j}$$
for $c=A,E,U$. Note that at every $t$ we only observe the logarithmic 24-hr
returns for one continent. The lag polynomials acting on $f_{c,t}$ are
autoregressive in terms of the time periods in $T_{c}$ as we cannot extract
$f_{c,t}$ for $t\notin T_{c}$ without additional assumptions. For the case of
several markets in one continent, one could include market-specific factors,
as Kose
et al. (2003) have done, without any further technicality, but
we avoid doing so to keep our model concise.
Although some of the aforementioned studies allow dynamics of factors, say,
factors following an AR(1) process, strictly speaking those factor models are
not dynamic factor models in the sense that the lagged factors are not allowed
to enter the equation relating factors to the observed series (see
Bai and
Wang (2015)). Exceptions are Forni
et al. (2000) and
Barigozzi and
Luciani (2022).
2.1 A Particular Form of (2.1) and Its State Space Form
It is natural to believe that only the new information accumulated since the
last closure of continent $c$ will have an impact on the upcoming observed
logarithmic 24-hr returns of continent $c$. Thus we shall assume $z_{i}^{c}(L)f_{g,t}=\sum_{j=0}^{2}z_{i,j}^{c}f_{g,t-j}$ and $\tilde{z}_{i}^{c}(L)f_{c,t}=\tilde{z}_{i,0}^{c}f_{c,t}$ for $c=A,E,U$ hereafter. Moreover,
the efficient market hypothesis (along with a time invariant risk premium)
predicts that $\phi_{g}(L)=\phi_{c}(L)=1$. However, for the case of several
markets in some continent, the presence of some AR structure in $f_{g,t}$
allows one to capture the effect caused by the fact that some markets in the
same continent might have different closing times. Thus we assume $\phi{}_{g}(L)f_{g,t+1}=f_{g,t+1}-\phi f_{g,t}$ with $|\phi|<1$ for simplicity. For
the rest of the article, we hence study the following specific form of
(2.1):
$$\displaystyle y_{i,t}^{c}$$
$$\displaystyle=\sum_{j=0}^{2}z_{i,j}^{c}f_{g,t-j}+\tilde{z}_{i,0}^{c}f_{c,t}+e_{i,t}^{c},\quad i=1,\ldots,N_{c},\quad t\in T_{c}$$
$$\displaystyle f_{g,t+1}$$
$$\displaystyle=\phi f_{g,t}+\eta_{g,t},\qquad|\phi|<1,\qquad t=0,1,\ldots,T-1$$
(2.2)
$$\displaystyle f_{c,t+1}$$
$$\displaystyle=\eta_{c,t}\qquad\qquad t+1\in T_{c}.$$
Stacking all the stocks in continent $c$, we have
$$\boldsymbol{y}_{t}^{c}=Z^{c}\boldsymbol{\alpha}_{t}+\boldsymbol{e}_{t}^{c},\qquad t\in T_{c},$$
where
$$\displaystyle\boldsymbol{y}_{t}^{c}\mathrel{\mathop{:}}=\left[\begin{array}[c]{c}y_{1,t}^{c}\\
\vdots\\
y_{N_{c},t}^{c}\\
\end{array}\right],\quad\boldsymbol{e}_{t}^{c}\mathrel{\mathop{:}}=\left[\begin{array}[c]{c}e_{1,t}^{c}\\
\vdots\\
e_{N_{c},t}^{c}\\
\end{array}\right]\quad\boldsymbol{\alpha}_{t}\mathrel{\mathop{:}}=\left[\begin{array}[c]{c}f_{g,t}\\
f_{g,t-1}\\
f_{g,t-2}\\
f_{C,t}\end{array}\right]$$
$$\displaystyle Z^{c}\mathrel{\mathop{:}}=\left[\begin{array}[c]{cccc}z_{1,0}^{c}&z_{1,1}^{c}&z_{1,2}^{c}&z_{1,3}^{c}\\
\vdots&\vdots&\vdots&\vdots\\
z_{N_{c},0}^{c}&z_{N_{c},1}^{c}&z_{N_{c},2}^{c}&z_{N_{c},3}^{c}\end{array}\right]=\mathrel{\mathop{:}}\left[\begin{array}[c]{cccc}\boldsymbol{z}_{0}^{c}&\boldsymbol{z}_{1}^{c}&\boldsymbol{z}_{2}^{c}&\boldsymbol{z}_{3}^{c}\end{array}\right],$$
where $f_{C,t}\mathrel{\mathop{:}}=f_{c,t}$ if $t\in T_{c}$, $z_{i,3}^{c}\mathrel{\mathop{:}}=\tilde{z}_{i,0}^{c}$,
and $f_{g,t},f_{C,t}\mathrel{\mathop{:}}=0$ for $t\leq 0$. We compress $f_{A,t},f_{E,t},f_{U,t}$
into a ”single” continental factor $f_{C,t}$ for the purpose of reducing the
number of state variables. Note that $\boldsymbol{y}_{t}^{c},\boldsymbol{e}_{t}^{c}$ are $N_{c}\times 1$ vectors, $\boldsymbol{\alpha}_{t}$ is a
$4\times 1$ vector, and $Z^{c}$ is an $N_{c}\times 4$ matrix. We make the
following assumptions:
Assumption 2.1.
(i)
The idiosyncratic components are i.i.d. across time: $\{\boldsymbol{e}_{t}^{c}\}_{t\in T_{c}}\overset{i.i.d.}{\sim}N(0,\Sigma_{c})$, where
$\Sigma_{c}\mathrel{\mathop{:}}=\operatorname*{diag}(\sigma^{2}_{c,1},\ldots,\sigma^{2}_{c,N_{c}})$ for
$c=A,E,U$. Moreover, $e_{i,t}^{A},e_{i,t}^{E},e_{i,t}^{U}$ are mutually
independent for all possible $i$ and $t$. Moreover, $\mathbb{E}[(e_{i,t}^{c})^{4}]\leq C$ for all $i$, $t$ and $c$.
(ii)
Assume that $\boldsymbol{\eta}_{t}\mathrel{\mathop{:}}=(\eta_{g,t},\eta_{C,t})^{\intercal}\overset{i.i.d.}{\sim}N(0,I_{2})$ for $t=0,1,\ldots,T-1$, where
$\eta_{C,t}\mathrel{\mathop{:}}=\eta_{c,t}$ if $t\in T_{c}$. Moreover, $\{\boldsymbol{\eta}_{t}\}_{t=1}^{T-1}$ are independent of $\{\boldsymbol{e}_{t}^{c}\}_{t\in T_{c}}$ for $c=A,E,U$.
Assumption 2.1(i) is the same as Assumption B of Bai and Li (2012).
We make the assumption of diagonality of $\Sigma_{c}$ for simplicity as our
model is already quite involved so we refrain from complicating the model
unnecessarily. Assumption 2.1(ii) is a white-noise assumption on
the innovations of the factors.
We now cast model (2.2) in the state space form
$$\displaystyle\boldsymbol{y}_{t}$$
$$\displaystyle=Z_{t}\boldsymbol{\alpha}_{t}+\boldsymbol{\varepsilon}_{t},\qquad\boldsymbol{\varepsilon}_{t}\sim N(0,\Sigma_{t}),\qquad t=1,\ldots,T,$$
(2.3)
$$\displaystyle\boldsymbol{\alpha}_{t+1}$$
$$\displaystyle=\left[\begin{array}[c]{cccc}\phi&0&0&0\\
1&0&0&0\\
0&1&0&0\\
0&0&0&0\end{array}\right]\boldsymbol{\alpha}_{t}+\left[\begin{array}[c]{cc}1&0\\
0&0\\
0&0\\
0&1\end{array}\right]\boldsymbol{\eta}_{t}=\mathrel{\mathop{:}}\mathcal{T}\boldsymbol{\alpha}_{t}+R\boldsymbol{\eta}_{t},\quad t=0,1,\ldots,T-1,$$
(2.12)
where $\boldsymbol{y}_{t}=\boldsymbol{y}_{t}^{c},Z_{t}=Z^{c},\boldsymbol{\varepsilon}_{t}=\boldsymbol{e}_{t}^{c},\Sigma_{t}=\Sigma_{c}$
if $t\in T_{c}$, for $c=A,E,U$. This is a non-standard dynamic factor model.
The non-standard features are: (1) The factor loading matrix $Z_{t}$ is
switching among three states $\{Z^{A},Z^{E},Z^{U}\}$. (2) The column
dimensions of $\boldsymbol{y}_{t},Z_{t},\boldsymbol{\varepsilon}_{t}$ are
switching among $\{N_{A},N_{E},N_{U}\}$. (3) The covariance matrix of
$\boldsymbol{\varepsilon}_{t}$ is switching among $\{\Sigma_{A},\Sigma_{E},\Sigma_{U}\}$.
In general for a static factor model, say, $\boldsymbol{y}_{t}=Z\boldsymbol{\alpha}_{t}+\boldsymbol{\varepsilon}_{t}$, further
identification restrictions are needed in order to separately identify $Z$ and
$\boldsymbol{\alpha}_{t}$ from the term $Z\boldsymbol{\alpha}_{t}$. In
particular, $Z\boldsymbol{\alpha}_{t}=\mathring{Z}\mathring{\boldsymbol{\alpha}}_{t}$ for any $4\times 4$ invertible matrix $C$ such that
$\mathring{Z}\mathrel{\mathop{:}}=ZC^{-1}$ and $\mathring{\boldsymbol{\alpha}}_{t}\mathrel{\mathop{:}}=C\boldsymbol{\alpha}_{t}$; we need $4^{2}$ identification restrictions so
that the only admissible $C$ is an identity matrix. A classical reference on
this issue would be Anderson and
Rubin (1956). These restrictions have been
ubiquitous in the literature (e.g., Bai and Li (2012), Bai and Li (2016)). One
exception is Bai and
Wang (2015); Bai and
Wang (2015) pointed out that by
relying on the dynamic equation of $f_{g,t}$, such as (2.2), one
could use far less identification restrictions to identify the model. In our
case, we shall only make the following mild fixed-signs assumption to identify
the model.
Assumption 2.2.
Estimators of $\boldsymbol{z}^{A}_{0},\boldsymbol{z}^{A}_{1},\boldsymbol{z}^{A}_{2},\boldsymbol{z}_{3}^{A},\boldsymbol{z}_{3}^{E},\boldsymbol{z}_{3}^{U}$ have the same column signs as
$\boldsymbol{z}^{A}_{0},\boldsymbol{z}^{A}_{1},\boldsymbol{z}^{A}_{2},\boldsymbol{z}_{3}^{A},\boldsymbol{z}_{3}^{E},\boldsymbol{z}_{3}^{U}$.
Bai and Li (2012) have made similar assumption as an implicit part of their
identification schemes (IC2, IC3 and IC5) (see Bai and Li (2012, p.445, p.463)).
Lemma 2.1.
The parameters of the dynamic factor model
(2.3) are identified under Assumption 2.2.
Our model (2.3) has a structural interpretation under Assumption
2.2, because one could not freely insert a rotation matrix
between $Z_{t}$ and $\boldsymbol{\alpha}_{t}$. In other words, under
Assumption 2.2 we are not estimating the rotations of $Z_{t}$ or
$\boldsymbol{\alpha}_{t}$; we are estimating the true $Z_{t}$ and
$\boldsymbol{\alpha}_{t}$ of the data generating process. This is a novel
feature of our model.
2.2 The Two-Day Representation
The representation of the model in state space form is of course not unique.
For simplicity, assume $N\mathrel{\mathop{:}}=N_{A}=N_{E}=N_{U}$ hereafter. We re-write our model
(2.3) in the following two-day representation:
$$\underbrace{\mathring{\boldsymbol{y}}_{t}}_{6N\times 1}=\underbrace{\Lambda}_{6N\times 14}\underbrace{\boldsymbol{f}_{t}}_{14\times 1}+\underbrace{\boldsymbol{e}_{t}}_{6N\times 1}$$
(2.13)
for $t=1,2,\ldots,T/6=\mathrel{\mathop{:}}T_{f}$, where we define $\ell\mathrel{\mathop{:}}=6(t-1)+1$,
$$\mathring{\boldsymbol{y}}_{t}\mathrel{\mathop{:}}=\left[\begin{array}[c]{c}\boldsymbol{y}_{\ell}^{A}\\
\boldsymbol{y}_{\ell+1}^{E}\\
\boldsymbol{y}_{\ell+2}^{U}\\
\boldsymbol{y}_{\ell+3}^{A}\\
\boldsymbol{y}_{\ell+4}^{E}\\
\boldsymbol{y}_{\ell+5}^{U}\end{array}\right]\qquad\boldsymbol{e}_{t}\mathrel{\mathop{:}}=\left[\begin{array}[c]{c}\boldsymbol{e}_{\ell}^{A}\\
\boldsymbol{e}_{\ell+1}^{E}\\
\boldsymbol{e}_{\ell+2}^{U}\\
\boldsymbol{e}_{\ell+3}^{A}\\
\boldsymbol{e}_{\ell+4}^{E}\\
\boldsymbol{e}_{\ell+5}^{U}\end{array}\right]\qquad\boldsymbol{f}_{t}\mathrel{\mathop{:}}=\left[\begin{array}[c]{c}f_{g,\ell+5}\\
f_{g,\ell+4}\\
f_{g,\ell+3}\\
f_{g,\ell+2}\\
f_{g,\ell+1}\\
f_{g,\ell}\\
f_{g,\ell-1}\\
f_{g,\ell-2}\\
f_{C,\ell+5}\\
f_{C,\ell+4}\\
f_{C,\ell+3}\\
f_{C,\ell+2}\\
f_{C,\ell+1}\\
f_{C,\ell}\end{array}\right]$$
$$\Lambda\mathrel{\mathop{:}}=\left[\begin{array}[c]{cccccccccccccc}\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{0}^{A}&\boldsymbol{z}_{1}^{A}&\boldsymbol{z}_{2}^{A}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{3}^{A}\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{0}^{E}&\boldsymbol{z}_{1}^{E}&\boldsymbol{z}_{2}^{E}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{3}^{E}&\mathbf{0}\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{0}^{U}&\boldsymbol{z}_{1}^{U}&\boldsymbol{z}_{2}^{U}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{3}^{U}&\mathbf{0}&\mathbf{0}\\
\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{0}^{A}&\boldsymbol{z}_{1}^{A}&\boldsymbol{z}_{2}^{A}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{3}^{A}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\
\mathbf{0}&\boldsymbol{z}_{0}^{E}&\boldsymbol{z}_{1}^{E}&\boldsymbol{z}_{2}^{E}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{3}^{E}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\
\boldsymbol{z}_{0}^{U}&\boldsymbol{z}_{1}^{U}&\boldsymbol{z}_{2}^{U}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\boldsymbol{z}_{3}^{U}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\end{array}\right],$$
(2.14)
with $\boldsymbol{z}_{k}^{c}$ being $N\times 1$ for $k=0,1,2,3$ and $c=A,E,U$,
while
$$\displaystyle\Sigma_{ee}\mathrel{\mathop{:}}=\mathbb{E}[\boldsymbol{e}_{t}\boldsymbol{e}_{t}^{\intercal}]$$
$$\displaystyle=\operatorname*{diag}(\sigma_{A,1}^{2},\ldots,\sigma_{A,N}^{2},\sigma_{E,1}^{2},\ldots,\sigma_{E,N}^{2},\sigma_{U,1}^{2},\ldots,\sigma_{U,N}^{2},\sigma_{A,1}^{2},\ldots,\sigma_{A,N}^{2},\sigma_{E,1}^{2},\ldots,\sigma_{E,N}^{2},\sigma_{U,1}^{2},\ldots,\sigma_{U,N}^{2})$$
$$\displaystyle=\mathrel{\mathop{:}}\operatorname*{diag}(\sigma_{1,1}^{2},\ldots,\sigma_{1,N}^{2},\sigma_{2,1}^{2},\ldots,\sigma_{2,N}^{2},\sigma_{3,1}^{2},\ldots,\sigma_{3,N}^{2},\sigma{}_{4,1}^{2},\ldots,\sigma_{4,N}^{2},\sigma_{5,1}^{2},\ldots,\sigma_{5,N}^{2},\sigma_{6,1}^{2},\ldots,\sigma_{6,N}^{2}).$$
Note that $\Lambda$ consists of six row blocks of dimension $N\times 14$. Let
$\boldsymbol{\lambda}_{k,j}^{\intercal}$ denote the $j$th row of the $k$th row
block of $\Lambda$. In other words, $\boldsymbol{\lambda}_{1,j}^{\intercal}$
refers to the factor loadings for the $j$th Asian stock in ”day one”, while
$\boldsymbol{\lambda}_{5,j}^{\intercal}$ refers to the factor loadings for the
$j$th European stock in ”day two”.
The reason for doing so is that we could rely on the information contained in
the covariance matrix $M\mathrel{\mathop{:}}=\mathbb{E}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]$ to estimate $\phi$, while treating $\{\boldsymbol{f}_{t}\}$
as i.i.d. across $t$ when setting up the likelihood. Then we are able to use
the theoretical results of Bai and Li (2012) to establish the large-sample
theories of the QMLE estimator of this representation. Treating
$\{\boldsymbol{f}_{t}\}$ as i.i.d. when setting up the likelihood, albeit
incorrectly, will not destroy consistency or the asymptotic normality of the
QMLE. This is the idea of working independence (Pan and
Connett (2002)).
3 Estimation
In this section, we shall outline several different estimation methods
applicable for our model (2.3). The estimators start from different
formulations of the state space model and impose different subsets of the
available parameter restrictions. The reason we consider these different
estimators is because of the difficulty of deriving the distribution theory in
some cases and the different numerical performance we have uncovered. The
different estimators also provide an understanding of the theoretical issues
we face.
MLE-one day.
This is the MLE estimator of the likelihood function $\{\boldsymbol{y}_{t}\}_{t=1}^{T}$, where $\boldsymbol{y}_{t}=Z_{t}\boldsymbol{\alpha}_{t}+\boldsymbol{\varepsilon}_{t}$. Estimation is done by the EM algorithm.
This estimator is explained in Section 3.1.
MLE-two day.
This is the MLE estimator of the likelihood function $\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}$, where $\mathring{\boldsymbol{y}}_{t}=\Lambda\boldsymbol{f}_{t}+\boldsymbol{e}_{t}$. All the restrictions
implied by $\Lambda$ and $M$, and implied by autocorrelation between
$\boldsymbol{f}_{t}$ and $\boldsymbol{f}_{t-1}$ will be taken into account.
This is equivalent to the MLE-one day. This estimator is not used in this
article, but will help readers understand the relationships among various
estimators. Estimation is done by the EM algorithm.
QMLE-res.
This is the QMLE estimator of the likelihood of $\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}$, where $\mathring{\boldsymbol{y}}_{t}=\Lambda\boldsymbol{f}_{t}+\boldsymbol{e}_{t}$. All the restrictions implied by
$\Lambda$ and $M$ are taken into account. However, autocorrelation between
$\boldsymbol{f}_{t}$ and $\boldsymbol{f}_{t-1}$ is ignored, and
$\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ are assumed as i.i.d over $t$ when
setting up the likelihood. Estimation is done by the EM algorithm. This
estimator is explained in Section 3.4.
QMLE.
This is the QMLE estimator of the likelihood of $\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}$, where $\mathring{\boldsymbol{y}}_{t}=\Lambda\boldsymbol{f}_{t}+\boldsymbol{e}_{t}$. A specific set of $14^{2}$
restrictions implied by $\Lambda$ and $M$ is employed. Autocorrelation between
$\boldsymbol{f}_{t}$ and $\boldsymbol{f}_{t-1}$ is ignored, and
$\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ are assumed as i.i.d over $t$ when
setting up the likelihood. Estimation is done by the EM algorithm. This
estimator is explained in Section 3.2.
Bai and Li (2012)’s QMLE.
These are the QMLE estimators of the likelihood of $\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}$, where $\mathring{\boldsymbol{y}}_{t}=\Lambda\boldsymbol{f}_{t}+\boldsymbol{e}_{t}$ with five specific sets of $14^{2}$
restrictions consistent with the five identification schemes of
Bai and Li (2012). Autocorrelation between $\boldsymbol{f}_{t}$ and
$\boldsymbol{f}_{t-1}$ is ignored, and $\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$
are assumed i.i.d over $t$ when setting up the likelihood. Unfortunately, our
model is not consistent with any one of the five identification schemes.
QMLE-md.
Use the QMLE as the first-step estimator and incorporate additional
finite number of restrictions implied by our model to obtain an
improved estimator via the minimum distance method. This estimator is
explained in Section 3.3.
Bayesian.
Use the Gibbs sampling to estimate $\boldsymbol{y}_{t}=Z_{t}\boldsymbol{\alpha}_{t}+\boldsymbol{\varepsilon}_{t}$. This estimator is explained in SM
LABEL:sec_Bayesian. This estimator is computationally intensive and feasible
only for not so large $N$.
3.1 MLE-one day
Define $\boldsymbol{\theta}\mathrel{\mathop{:}}=\{\phi,Z^{c},\Sigma_{c},c=A,E,U\}$,
$\Xi\mathrel{\mathop{:}}=(\boldsymbol{\alpha}_{1},\ldots,\boldsymbol{\alpha}_{T})^{\intercal}$
and $Y_{1\mathrel{\mathop{:}}T}\mathrel{\mathop{:}}=\{\boldsymbol{y}_{1}^{\intercal},\ldots,\boldsymbol{y}_{T}^{\intercal}\}^{\intercal}$. The log-likelihoods of $\Xi$ and $Y_{1\mathrel{\mathop{:}}T}|\Xi$ are
$$\displaystyle\ell(\Xi;\boldsymbol{\theta})$$
$$\displaystyle=-T\log(2\pi)-\frac{1}{2}\sum_{t=0}^{T-1}\mathinner{\bigl{[}(f_{g,t+1}-\phi f_{g,t})^{2}+f_{C,t+1}^{2}\bigr{]}}$$
(3.1)
$$\displaystyle\ell(Y_{1\mathrel{\mathop{:}}T}|\Xi;\boldsymbol{\theta})$$
$$\displaystyle=-\frac{TN}{2}\log(2\pi)-\frac{1}{2}\sum_{t=1}^{T}\log|\Sigma_{t}|-\frac{1}{2}\sum_{t=1}^{T}(\boldsymbol{y}_{t}-Z_{t}\boldsymbol{\alpha}_{t})^{\intercal}\Sigma_{t}^{-1}(\boldsymbol{y}_{t}-Z_{t}\boldsymbol{\alpha}_{t}).$$
The complete log-likelihood function of model (2.3), i.e.,
based on an observed state vector, is hence (omitting constant)
$$\displaystyle\ell(\Xi,Y_{1\mathrel{\mathop{:}}T};\boldsymbol{\theta})=\ell(Y_{1\mathrel{\mathop{:}}T}|\Xi;\boldsymbol{\theta})+\ell(\Xi;\boldsymbol{\theta})$$
$$\displaystyle=-\frac{1}{2}\sum_{t=1}^{T}\left(\log|\Sigma_{t}|+\boldsymbol{\varepsilon}_{t}^{\intercal}\Sigma_{t}^{-1}\boldsymbol{\varepsilon}_{t}\right)-\frac{1}{2}\sum_{t=1}^{T}\eta_{g,t-1}^{2}=\mathrel{\mathop{:}}-\frac{1}{2}\sum_{t=1}^{T}\left(\ell_{1,t}+\ell_{2,t}\right)$$
where $\ell_{1,t}\mathrel{\mathop{:}}=\log|\Sigma_{t}|+\operatorname*{tr}\mathinner{\bigl{(}\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t}^{\intercal}\Sigma_{t}^{-1}\bigr{)}}$
and $\ell_{2,t}\mathrel{\mathop{:}}=\eta_{g,t-1}^{2}$. The EM algorithm consists of an E-step and
an M-step.111Motivation of the EM algorithm is reviewed in SM
LABEL:sec_motivation_of_EM. In the E-step, we evaluate a conditional
expectation of the complete log-likelihood function given the observed data,
while in the M-step we maximize it with respect to parameters. The details of
the EM algorithm for this estimator are provided in Appendix
A.2.
The choice of the starting values is crucial for the EM algorithm as good
starting values ensure fast convergence of the EM algorithm. It is common
practice initialising the EM algorithm with the Principal Component (PC)
estimator (Doz
et al. (2012), Bai and Li (2012),
Barigozzi and
Luciani (2022) etc). We do not adopt this approach though because
the PC estimator does not satisfy the restrictions implied by our model
(2.3). Moreover, in finite samples the PC estimator cannot ensure
that $\hat{\boldsymbol{\alpha}}_{t,1}^{PC}=\hat{\boldsymbol{\alpha}}_{t+1,2}^{PC}$ for all $t$, where $\hat{\boldsymbol{\alpha}}_{t}^{PC}$ is the
PC estimator of $\boldsymbol{\alpha}_{t}$. Instead we estimate a restricted
version of model (2.3) via the MLE as our starting values. Take the
Asian continent as an example. All the elements of $\boldsymbol{z}_{0}^{A},\boldsymbol{z}_{1}^{A},\boldsymbol{z}_{2}^{A}$ are restricted to one
scalar, all the elements of $\boldsymbol{z}_{3}^{A}$ are restricted to one
scalar, and all the diagonal elements of $\Sigma_{A}$ are restricted to one
scalar. In some preliminary experiments, we found this gives reasonably good
starting values for estimation of our model. However, in some settings which
are different from ours, the PC estimator is perhaps the preferred choice for
the starting values.
3.2 QMLE
We work with the two-day representation (2.13). We define the following
quantities:
$$S_{yy}\mathrel{\mathop{:}}=\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\mathring{\boldsymbol{y}}_{t}\mathring{\boldsymbol{y}}_{t}^{\intercal},\qquad\Sigma_{yy}\mathrel{\mathop{:}}=\Lambda M\Lambda^{\intercal}+\Sigma_{ee},$$
where $M\mathrel{\mathop{:}}=\mathbb{E}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]$ and
$T_{f}\mathrel{\mathop{:}}=T/6$. It can be seen that the $14\times 14$ matrix $M$ and the
$8\times 8$ matrix $\Phi$ are as follows:222Given the assumption of
$|\phi|<1$ in (2.2), we have $M=O(1)$ and $M^{-1}=O(1)$.
$$\displaystyle M=\left[\begin{array}[c]{cc}\Phi&0\\
0&I_{6}\end{array}\right],\qquad\Phi\mathrel{\mathop{:}}=\frac{1}{1-\phi^{2}}\left[\begin{array}[c]{cccccccc}1&\phi&\phi^{2}&\phi^{3}&\phi^{4}&\phi^{5}&\phi^{6}&\phi^{7}\\
\phi&1&\phi&\phi^{2}&\phi^{3}&\phi^{4}&\phi^{5}&\phi^{6}\\
\phi^{2}&\phi&1&\phi&\phi^{2}&\phi^{3}&\phi^{4}&\phi^{5}\\
\phi^{3}&\phi^{2}&\phi&1&\phi&\phi^{2}&\phi^{3}&\phi^{4}\\
\phi^{4}&\phi^{3}&\phi^{2}&\phi&1&\phi&\phi^{2}&\phi^{3}\\
\phi^{5}&\phi^{4}&\phi^{3}&\phi^{2}&\phi&1&\phi&\phi^{2}\\
\phi^{6}&\phi^{5}&\phi^{4}&\phi^{3}&\phi^{2}&\phi&1&\phi\\
\phi^{7}&\phi^{6}&\phi^{5}&\phi^{4}&\phi^{3}&\phi^{2}&\phi&1\end{array}\right].$$
(3.12)
Treating $\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ as i.i.d over $t$, we can
write down the log-likelihood of $\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}$ (scaled by $1/(NT_{f})$):
$$\frac{1}{NT_{f}}\ell\mathinner{\bigl{(}\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}=-3\log(2\pi)-\frac{1}{2N}\log|\Sigma_{yy}|-\frac{1}{2N}\operatorname*{tr}(S_{yy}\Sigma_{yy}{}^{-1}).$$
(3.13)
We shall only utilise the information that $M$ is symmetric, positive definite
and that $\Sigma_{ee}$ is diagonal to derive the generic first-order
conditions (FOCs).333The word generic means that specific
forms of $\Lambda$ given by (2.14) and of $M$ given by (3.12) are not
utilised to derive the FOCs. In Appendix A.3, we derive such FOCs
and identify the QMLE estimators $\hat{\Lambda},\hat{M},\hat{\Sigma}_{ee}$
after imposing $14^{2}$ identification restrictions. The QMLE estimators
$\hat{\Lambda},\hat{M},\hat{\Sigma}_{ee}$ satisfy the equations
$$\displaystyle\hat{\Lambda}^{\intercal}\hat{\Sigma}_{yy}^{-1}(S_{yy}-\hat{\Sigma}_{yy})$$
$$\displaystyle=0$$
$$\displaystyle\operatorname*{diag}(\hat{\Sigma}_{yy}^{-1})$$
$$\displaystyle=\operatorname*{diag}(\hat{\Sigma}_{yy}^{-1}S_{yy}\hat{\Sigma}_{yy}^{-1}),$$
(3.14)
where $\hat{\Sigma}_{yy}\mathrel{\mathop{:}}=\hat{\Lambda}\hat{M}\hat{\Lambda}^{\intercal}+\hat{\Sigma}_{ee}$. Display (3.14) is the same as (2.7) and (2.8) of
Bai and Li (2012). Bai and Li (2012) considered five identification schemes,
none of which is consistent with $\Lambda$ and $M$ defined in (2.14) and
(3.12), respectively. Actually $\Lambda$ and $M$ imply more than $14^{2}$
restrictions, but in order to have a solution for the generic FOCs, we could
only impose $14^{2}$ restrictions on $\hat{\Lambda}$ and $\hat{M}$. We call
the resulting estimators the QMLE rather than the MLE because
$\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ are assumed i.i.d over $t$ when setting
up the likelihood. How to select these $14^{2}$ restrictions from those
implied by $\Lambda$ and $M$ are crucial because we cannot afford imposing a
restriction which is not instrumental for establishing large-sample theories
later. We painstakingly explain our procedure in the proofs of Proposition
4.1 and Theorems 4.1, 4.3. Our procedure is quite
innovative and does not exist in the proofs of Bai and Li (2012).
Estimation of the QMLE is also done by the EM algorithm (see Appendix
A.5 for details).
3.3 QMLE-md
The QMLE estimator we defined above only used $14^{2}$ restrictions and there
are additional restrictions implied by our model (2.3). In this
subsection, we propose an improved estimator (the QMLE-md) by including
some of these additional restrictions via the minimum distance
method. Recall that $\boldsymbol{\lambda}_{k,j}^{\intercal}$ denote the $j$th
row of the $k$th row block of $\Lambda$, so we can also use $\{\hat{\boldsymbol{\lambda}}_{k,j},\hat{\sigma}_{k,j},\hat{M}\mathrel{\mathop{:}}k=1,\ldots,6,j=1,\ldots,N\}$ to denote the QMLE estimator. Suppose that we take a
finite number $c_{1}$ of elements of the QMLE to form a column vector
$\hat{\boldsymbol{h}}$. Note that the QMLE $\hat{\boldsymbol{h}}$ is
estimating $h(\boldsymbol{\theta}_{m})$, where $\boldsymbol{\theta}_{m}\subset\boldsymbol{\theta}$ is of finite-dimension $c_{2}$ ($c_{2}<c_{1}$) and
$h(\cdot)\mathrel{\mathop{:}}\mathbb{R}^{c_{2}}\rightarrow\mathbb{R}^{c_{1}}$.
Example 3.1.
As an illustration, one could take
$$\displaystyle\underbrace{\hat{\boldsymbol{h}}}_{59\times 1}$$
$$\displaystyle=(\hat{\boldsymbol{\lambda}}_{1,2}^{\intercal},\hat{\boldsymbol{\lambda}}_{4,2}^{\intercal},\hat{\boldsymbol{\lambda}}_{3,5}^{\intercal},\hat{\boldsymbol{\lambda}}_{6,5}^{\intercal},\hat{M}_{1,1},\hat{M}_{2,1},\hat{\sigma}^{2}_{1,5})^{\intercal},$$
$$\displaystyle\underbrace{\boldsymbol{\theta}_{m}}_{10\times 1}$$
$$\displaystyle=(z^{A}_{2,0},z^{A}_{2,1},z^{A}_{2,2},z^{A}_{2,3},z^{U}_{5,0},z^{U}_{5,1},z^{U}_{5,2},z^{U}_{5,3},\phi,\sigma_{1,5}^{2})^{\intercal},$$
$$\displaystyle\underbrace{h(\boldsymbol{\theta}_{m})}_{59\times 1}$$
$$\displaystyle=(\boldsymbol{\lambda}_{1,2}^{\intercal},\boldsymbol{\lambda}_{4,2}^{\intercal},\boldsymbol{\lambda}_{3,5}^{\intercal},\boldsymbol{\lambda}_{6,5}^{\intercal},M_{1,1},M_{2,1},\sigma^{2}_{1,5})^{\intercal}$$
The expression of the $59\times 10$ derivative matrix $\partial h(\boldsymbol{\theta}_{m})/\partial\boldsymbol{\theta}_{m}$ is given in SM
LABEL:sec_SM_derivative.
Let $W$ denote a $c_{1}\times c_{1}$ symmetric, positive definite weighting
matrix and define the minimum distance estimator
$$\displaystyle\check{\boldsymbol{\theta}}_{m}\mathrel{\mathop{:}}=\arg\min_{\boldsymbol{b}\in\mathbb{R}^{c_{2}}}\mathinner{\bigl{[}\hat{\boldsymbol{h}}-h(\boldsymbol{b})\bigr{]}}^{\intercal}W\mathinner{\bigl{[}\hat{\boldsymbol{h}}-h(\boldsymbol{b})\bigr{]}}.$$
(3.15)
3.4 QMLE-res
In this subsection, we propose to estimate the two-day representation of our
model via the EM algorithm. We shall incorporate all the restrictions
implied by $\Lambda$ and $M$ defined in (2.14) and (3.12),
respectively, but assume the working independence hypothesis -
$\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ are assumed as i.i.d. over $t$ - when
setting up the likelihood. We call this the QMLE-res estimator.
The log-likelihoods of $\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}|\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ and $\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ are, respectively,
$$\displaystyle\ell\mathinner{\bigl{(}\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}|\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}$$
$$\displaystyle=-\frac{T_{f}(6N)}{2}\log(2\pi)-\frac{T_{f}}{2}\log|\Sigma_{ee}|-\frac{1}{2}\sum_{t=1}^{T}(\mathring{\boldsymbol{y}}_{t}-\Lambda\boldsymbol{f}_{t})^{\intercal}\Sigma_{ee}^{-1}(\mathring{\boldsymbol{y}}_{t}-\Lambda\boldsymbol{f}_{t})$$
$$\displaystyle\ell\mathinner{\bigl{(}\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}$$
$$\displaystyle=-\frac{T_{f}14}{2}\log(2\pi)-\frac{T_{f}}{2}\log|M|-\frac{1}{2}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}^{\intercal}M^{-1}\boldsymbol{f}_{t}.$$
The complete log-likelihood function of the two-day representation of our
model is hence (omitting constant)
$$\displaystyle\ell\mathinner{\bigl{(}\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}},\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}=\ell\mathinner{\bigl{(}\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}}|\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}+\ell\mathinner{\bigl{(}\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}$$
$$\displaystyle=-\frac{T_{f}}{2}\log|\Sigma_{ee}|-\frac{1}{2}\sum_{t=1}^{T}(\mathring{\boldsymbol{y}}_{t}-\Lambda\boldsymbol{f}_{t})^{\intercal}\Sigma_{ee}^{-1}(\mathring{\boldsymbol{y}}_{t}-\Lambda\boldsymbol{f}_{t})-\frac{T_{f}}{2}\log|M|-\frac{1}{2}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}^{\intercal}M^{-1}\boldsymbol{f}_{t}$$
$$\displaystyle=\mathrel{\mathop{:}}-\frac{1}{2}\mathinner{\biggl{(}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}+\sum_{t=1}^{T_{f}}\vec{\ell}_{2,t}\biggr{)}}$$
(3.16)
where
$$\displaystyle\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}$$
$$\displaystyle\mathrel{\mathop{:}}=T_{f}\log|\Sigma_{ee}|+\sum_{t=1}^{T}\operatorname*{tr}\mathinner{\Bigl{[}(\mathring{\boldsymbol{y}}_{t}-\Lambda\boldsymbol{f}_{t})(\mathring{\boldsymbol{y}}_{t}-\Lambda\boldsymbol{f}_{t})^{\intercal}\Sigma_{ee}^{-1}\Bigr{]}}$$
$$\displaystyle\sum_{t=1}^{T_{f}}\vec{\ell}_{2,t}$$
$$\displaystyle\mathrel{\mathop{:}}=T_{f}\log|M|+\sum_{t=1}^{T_{f}}\operatorname*{tr}\mathinner{\Bigl{[}\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}M^{-1}\Bigr{]}}.$$
Let $\vec{\mathbb{E}}$ denote the expectation with respect to the conditional
density
$p\mathinner{\bigl{(}\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}|\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\vec{\boldsymbol{\theta}}^{(i)}\bigr{)}}$
at $\vec{\boldsymbol{\theta}}^{(i)}$, where $\vec{\boldsymbol{\theta}}^{(i)}$
is the estimate of $\boldsymbol{\theta}$ from the $i$th iteration of the EM
algorithm. Taking such an expectation on both sides of (3.16), we hence
have
$$\vec{\mathbb{E}}\mathinner{\Bigl{[}\ell\mathinner{\bigl{(}\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}},\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}};\boldsymbol{\theta}\bigr{)}}\Bigr{]}}=-\frac{1}{2}\mathinner{\biggl{(}\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}+\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{2,t}\biggr{)}}.$$
This is the E-step. In the M-step, we find values of $Z^{c}$ and $\Sigma_{ee}$
to minimise $\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}$, and find
values of $\phi$ to minimise $\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{2,t}$. The details are provided in Appendix A.4.
4 Large Sample Theories
4.1 QMLE
We now present the large sample theories of the QMLE. The idea of the proof is
based on that of Bai and Li (2012), but is considerably more involved because
our identification scheme is non-standard. Hence, we provide a recipe for
obtaining results similar to those of Bai and Li (2012) for almost any
identified dynamic factor model. This does have a practical importance because
many dynamic factor models, like ours, are coming from different economic
theories and might not conform to the five identification schemes of
Bai and Li (2012).
Recall that we could use $\{\hat{\boldsymbol{\lambda}}_{k,j},\hat{\sigma}_{k,j},\hat{M}\mathrel{\mathop{:}}k=1,\ldots,6,j=1,\ldots,N\}$ to denote the QMLE. We make the
following assumption.
Assumption 4.1.
(i)
The factor loadings $\{\boldsymbol{\lambda}_{k,j}\}$ satisfy
$\|\boldsymbol{\lambda}_{k,j}\|_{2}\leq C$ for all $k$ and $j$.
(ii)
Assume $C^{-1}\leq\sigma_{k,j}^{2}\leq C$ for all $k$ and $j$. Also
$\hat{\sigma}_{k,j}^{2}$ is restricted to a compact set $[C^{-1},C]$ for all
$k$ and $j$.
(iii)
$\hat{M}$ is restricted to be in a set consisting of all positive
definite matrices with all the elements bounded in the interval $[C^{-1},C]$.
(iv)
Suppose that $Q\mathrel{\mathop{:}}=\lim_{N\to\infty}\frac{1}{N}\Lambda^{\intercal}\Sigma_{ee}^{-1}\Lambda$ is a positive definite matrix.
Assumption 4.1 is standard in the literature
of factor models and has been taken from the assumptions of Bai and Li (2012).
Proposition 4.1.
Suppose that Assumptions 2.1,
4.1 hold. When $N,T_{f}\to\infty$, with the
identification condition outlined in the proof of this proposition (i.e.,
$14^{2}$ particular restrictions imposed on $\hat{\Lambda}$ and $\hat{M}$),
and the requirement that $\hat{\Lambda}$ and $\Lambda$ have the same column
signs, we have
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle=o_{p}(1)$$
(4.1)
$$\displaystyle\frac{1}{6N}\sum_{k=1}^{6}\sum_{j=1}^{N}(\hat{\sigma}_{k,j}^{2}-\sigma_{k,j}^{2})^{2}$$
$$\displaystyle=o_{p}(1)$$
(4.2)
$$\displaystyle\hat{M}-M$$
$$\displaystyle=o_{p}(1).$$
(4.3)
for $k=1,\ldots,6,j=1,\ldots,N$.
Displays (4.1) and (4.3) establish consistency for the individual
loading estimator $\hat{\boldsymbol{\lambda}}_{k,j}$ and $\hat{M}$,
respectively, while display (4.2) establishes some average consistency
for $\{\hat{\sigma}_{k,j}^{2}\}$.
Theorem 4.1.
Under the assumptions of Proposition 4.1, we have
$$\displaystyle\|\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}\|_{2}^{2}$$
$$\displaystyle=O_{p}(T_{f}^{-1})$$
$$\displaystyle\frac{1}{6N}\sum_{k=1}^{6}\sum_{j=1}^{N}(\hat{\sigma}_{k,j}^{2}-\sigma_{k,j}^{2})^{2}$$
$$\displaystyle=O_{p}(T_{f}^{-1})$$
(4.4)
$$\displaystyle\hat{M}-M$$
$$\displaystyle=O_{p}(T_{f}^{-1/2}).$$
for $k=1,\ldots,6,j=1,\ldots,N$.
Theorem 4.1 resembles Theorem 5.1 of Bai and Li (2012) and establishes
the rate of convergence for the QMLE. The only difference is while
Bai and Li (2012) only established an average rate of convergence for
$\{\hat{\boldsymbol{\lambda}}_{k,j}\}$, we managed to establish a rate of
convergence for the individual loading estimator $\hat{\boldsymbol{\lambda}}_{k,j}$.
Theorem 4.2.
Under the assumptions of Proposition 4.1, we
have, for $k=1,\ldots,6,j=1,\ldots,N$,
(i)
$$\hat{\sigma}_{k,j}^{2}-\sigma_{k,j}^{2}=\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}e_{(k-1)N+j,t}^{2}-\sigma_{k,j}^{2}\bigr{)}}+o_{p}(T_{f}^{-1/2}),$$
where $e_{(k-1)N+j,t}$ is the $[(k-1)N+j]$th element of $\boldsymbol{e}_{t}$.
(ii)
As $N,T_{f}\to\infty$,
$$\displaystyle\sqrt{T_{f}}(\hat{\sigma}^{2}_{k,j}-\sigma^{2}_{k,j})\xrightarrow{d}N(0,2\sigma^{4}_{k,j}).$$
Theorem 4.2(i) gives the asymptotic representation of
$\hat{\sigma}_{k,j}^{2}$. Theorem 4.2(ii) is the same as
Theorem 5.4 of Bai and Li (2012) and establishes the asymptotic distribution of
$\hat{\sigma}^{2}_{k,j}$.
Theorem 4.3.
Suppose that the assumptions of Proposition 4.1 hold.
(i)
For $k=1,\ldots,6$, $j=1,\ldots,N$, we have
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})=(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+M^{-1}\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}+o_{p}(1),$$
where $\boldsymbol{e}_{t}^{\dagger}$ is a $24\times 1$ vector consisting of
$e_{(p-1)N+q,t}$ for $p=1,\ldots,6$ and $q=1,\ldots,4$, and $\Gamma$ is a
$196\times 336$ matrix, whose elements are known (but complicated) linear
functions of elements of (inverted) submatrices of $\Lambda$ and $M$,
satisfying
$$\displaystyle\operatorname*{vec}A$$
$$\displaystyle=\Gamma\times\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+o_{p}(T_{f}^{-1/2}),\qquad A\mathrel{\mathop{:}}=(\hat{\Lambda}-\Lambda)^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}(\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda})^{-1}.$$
(ii)
As $N,T_{f}\to\infty$, for $k=1,\ldots,6$, $j=5,\ldots,N$, we have
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})\xrightarrow{d}N\mathinner{\Bigl{(}\boldsymbol{0},(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(\boldsymbol{\lambda}_{k,j}\otimes I_{14})+M^{-1}\sigma_{k,j}^{2}\Bigr{)}},$$
and for $k=1,\ldots,6$, $j=1,\ldots,4$, we have
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})\xrightarrow{d}N\mathinner{\Bigl{(}\boldsymbol{0},(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(\boldsymbol{\lambda}_{k,j}\otimes I_{14})+M^{-1}\sigma_{k,j}^{2}+\text{cov}_{k,j}+\text{cov}_{k,j}^{\intercal}\Bigr{)}},$$
where $\Sigma_{ee}^{\dagger}$ is a $24\times 24$ diagonal matrix whose
$[4(p-1)+q]$th diagonal element is $\sigma_{p,q}^{2}$ for $p=1,\ldots,6$ and
$q=1,\ldots,4$. The $14\times 14$ matrix $\text{cov}_{k,j}$ is defined as
$$\displaystyle\text{cov}_{k,j}\mathrel{\mathop{:}}=(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma\mathinner{\bigl{[}\boldsymbol{\iota}_{k,j}\otimes I_{14}\bigr{]}}\sigma_{k,j}^{2},$$
where $\boldsymbol{\iota}_{k,j}$ is a $24\times 1$ zero vector with its
$[4(k-1)+j]$th element replaced by one.
Theorem 4.3 presents the asymptotic representation and distribution
of the QMLE of the factor loadings. The idea of the proof is inspired by that
for the fourth identification scheme (i.e., IC4) in Theorem 5.2 of
Bai and Li (2012). Note that the asymptotic variance of $\hat{\boldsymbol{\lambda}}_{k,j}$ depends on $\Gamma$ and $\Sigma_{ee}^{\dagger}$.
The matrix $\Sigma_{ee}^{\dagger}$ contains the idiosyncratic variances of the
first four assets in each continent. When computing $\Gamma$, we often need to
invert submatrices of the factor loadings of the first four assets in each
continent (see (A.111) in the proof of Theorem 4.3 for example).
Thus, ordering assets, with smaller idiosyncratic variances and less
multicollinearity of the factor loadings, as the first four assets in each
continent results in a $\hat{\boldsymbol{\lambda}}_{k,j}$ with smaller
asymptotic variances.
Theorem 4.4.
Suppose that the assumptions of Proposition 4.1 hold.
(i)
$$\displaystyle\sqrt{T_{f}}\operatorname*{vech}(\hat{M}-M)=-2D_{14}^{+}(I_{14}\otimes M)\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+o_{p}(1),$$
where $D_{14}^{+}$ is defined in Section 1.1.
(ii)
As $N,T_{f}\to\infty$,
$$\displaystyle\sqrt{T_{f}}\operatorname*{vech}(\hat{M}-M)\xrightarrow{d}N\mathinner{\bigl{(}0,\mathcal{M}\bigr{)}}$$
where $\mathcal{M}$ is $105\times 105$ and defined as
$$\displaystyle\mathcal{M}\mathrel{\mathop{:}}=4D_{14}^{+}(I_{14}\otimes M)\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(I_{14}\otimes M)D_{14}^{+\intercal}.$$
Theorem 4.4 presents the asymptotic representation and distribution
of the QMLE of $M$; it is similar to Theorem 5.3 of Bai and Li (2012).
We could estimate $\boldsymbol{f}_{t}$ by the generalized least squares (GLS),
as Bai and Li (2012) have done in their Theorem 6.1:
$$\displaystyle\hat{\boldsymbol{f}}_{t}=\mathinner{\bigl{(}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\bigr{)}}^{-1}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\mathring{\boldsymbol{y}}_{t}.$$
Theorem 4.5.
Suppose that the assumptions of Proposition 4.1 hold
and $\sqrt{N}/T_{f}\to 0$, $N/T_{f}\to\Delta\in[0,\infty)$. Then we have
(i)
$$\displaystyle\sqrt{N}(\hat{\boldsymbol{f}}_{t}-\boldsymbol{f}_{t})=-\sqrt{\Delta}(\boldsymbol{f}_{t}^{\intercal}\otimes I_{14})K_{14,14}\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+Q^{-1}\frac{1}{\sqrt{N}}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+o_{p}(1),$$
where $K_{14,14}$ is the commutation matrix.
(ii)
$$\displaystyle\sqrt{N}(\hat{\boldsymbol{f}}_{t}-\boldsymbol{f}_{t})|\boldsymbol{f}_{t}\xrightarrow{d}N\mathinner{\Bigl{(}\boldsymbol{0},\Delta(\boldsymbol{f}_{t}^{\intercal}\otimes I_{14})K_{14,14}\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}K_{14,14}(\boldsymbol{f}_{t}\otimes I_{14})+Q^{-1}\Bigr{)}},$$
where $Q$ is defined in Assumption 4.1.
Theorem 4.5 gives the asymptotic representation and conditional
distribution of the GLS $\hat{\boldsymbol{f}}_{t}$. Unlike Theorem 6.1 of
Bai and Li (2012), since we treat $\boldsymbol{f}_{t}$ as random, the
asymptotic normal distribution in Theorem 4.5(ii) is for $\sqrt{N}(\hat{\boldsymbol{f}}_{t}-\boldsymbol{f}_{t})$ conditioning on
$\boldsymbol{f}_{t}$.
4.2 QMLE-md
We then present the large sample theories of the QMLE-md. Recall that
$\hat{\boldsymbol{h}}$ is a vector of finite length of the QMLE estimator
$\{\hat{\boldsymbol{\lambda}}_{k,j},\hat{\sigma}_{k,j},\hat{M}\mathrel{\mathop{:}}k=1,\ldots,6,j=1,\ldots,N\}$. Relying on the asymptotic representations of the
QMLE (Theorems 4.2, 4.3, 4.4), one could
easily establish the asymptotic distribution of $\hat{\boldsymbol{h}}$, say,
$$\sqrt{T_{f}}(\hat{\boldsymbol{h}}-h(\boldsymbol{\theta}_{m}))\xrightarrow{d}N(\boldsymbol{0},\mathcal{H}).$$
Since the choice of $\hat{\boldsymbol{h}}$ varies, we omit the formula for
$\mathcal{H}$.
Theorem 4.6.
Suppose that the assumptions of Proposition
4.1 hold. Then we have
$$\sqrt{T_{f}}(\check{\boldsymbol{\theta}}_{m}-\boldsymbol{\theta}_{m})\xrightarrow{d}N(\boldsymbol{0},\mathcal{O}),$$
where
$$\displaystyle\mathcal{O}\mathrel{\mathop{:}}=\mathinner{\biggl{[}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}}\biggr{]}}^{-1}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\mathcal{H}W\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}}\mathinner{\biggl{[}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}}\biggr{]}}^{-1}.$$
In the preceding theorem, choosing $W=\mathcal{H}^{-1}$ gives the most
efficient minimum distance estimator. In that case, $\mathcal{O}$ is reduced
to
$\mathinner{\bigl{[}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}\mathcal{H}^{-1}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}}\bigr{]}}^{-1}$.
5 Inference Procedures for the MLE-one day and the QMLE-res
In classical factor analysis (i.e., fixed $N$ large $T_{f}$), the asymptotic
variances of the MLE-one day and the QMLE-res defined in Section 3
could be approximated using the numerical Hessian method as researchers have
shown that the MLE of a standard factor model is asymptotically normal but has
very complicated expressions for the asymptotic covariance matrices
(Anderson (2003, p.583)). The large sample theories of the MLE-one day
and the QMLE-res in the large $N$ large $T_{f}$ case remain as a formidable,
if not impossible, task to be completed in the future research. In SM
LABEL:sec_approximate_standard_errors, we provide the details of a heuristic procedure to
approximate the standard errors of the MLE-one day and the QMLE-res
in the large $N$ large $T_{f}$ case.
6 Monte Carlo Simulations
In this section, we shall conduct Monte Carlo simulations to evaluate the
performances of our proposed estimators. We specify the following values for
the parameters: $N=50,200$; $T=750$ (around one year’s trading data), 1500,
2250; $\phi=0.3$. For $c=A,E,U$, $\Sigma_{c,ii}$ are drawn from uniform$[0.2,2]$ for $i=1,\ldots,N$, and
$$z^{c}_{i,j}=0.6a_{c,i,j}+0.6d_{c,j}-0.2$$
where $\{a_{c,i,j}\}_{i=1,j=0}^{i=N,j=3}$ and $\{d_{c,j}\}_{j=0}^{3}$ are all
drawn from uniform$[0,1]$. For each continent, we put the assets with the
smallest four idiosyncratic variances as the first four assets. After the
logarithmic 24-hr returns are generated, the econometrician only observes
those logarithmic 24-hr returns whose $t$s correspond to the closing times of
their belonging continents. The econometrician is aware of the structure of
the true model (2.3), but does not know the values of those
parameters. In particular, he is aware of diagonality of $\Sigma_{c}$.
We estimate the model using the MLE-one day, the QMLE-res and the QMLE-md. To
initialize the EM algorithm for the MLE-one day and the QMLE-res, the starting
values of the parameters are estimated according to the procedure mentioned in
Section 3.1. We now briefly explain how to select
$\hat{\boldsymbol{h}}$ for the QMLE-md. Take the factor loading of Asia’s
$j$th asset (i.e., $z^{A}_{j,0},z^{A}_{j,1},z^{A}_{j,2},z^{A}_{j,3}$) as an
example: Select $\hat{\boldsymbol{h}}=(\hat{\boldsymbol{\lambda}}_{1,j}^{\intercal},\hat{\boldsymbol{\lambda}}_{4,j}^{\intercal},\hat{\boldsymbol{\lambda}}_{2,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{5,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{2,5}^{\intercal},\hat{\boldsymbol{\lambda}}_{5,5}^{\intercal},\hat{\boldsymbol{\lambda}}_{3,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{6,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{3,5}^{\intercal},\hat{\boldsymbol{\lambda}}_{6,5}^{\intercal})^{\intercal}$. For $\phi$: Select
$$\hat{\boldsymbol{h}}=(\hat{\boldsymbol{\lambda}}_{1,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{4,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{2,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{5,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{3,1}^{\intercal},\hat{\boldsymbol{\lambda}}_{6,1}^{\intercal},(\operatorname*{vech}M)^{\intercal})^{\intercal}.$$
For $\sigma^{2}_{k,j}$, set its QMLE-md to the QMLE.
The number of the Monte Carlo samples is chosen to be 200. From these 200
Monte Carlo samples, we calculate the following three quantities for evaluation:
(i)
the root mean square errors (RMSE),
(ii)
the average of the standard errors (Ave.se) across the Monte Carlo
samples.
(iii)
the coverage probability (Cove) of the confidence interval formed by the
point estimate $\pm$ $1.96\times$the standard error. The standard errors
differ across the Monte Carlo samples.
For a particular $j$ and $c$, it is impossible to present a evaluation
criterion of $z^{c}_{i,j}$ for all $i$, so we only report the average value
for the vector $\boldsymbol{z}^{c}_{j}$. Likewise, we report the average value
for the three diagonals of $\{\Sigma_{c}\mathrel{\mathop{:}}c=A,E,U\}$. Tables
1 and 2 report these results. To save space,
we only present the results for $T=750,2250$ (the results for $T=1500$ are
available upon request). We see that the MLE-one day and QMLE-res estimators
are very similar in terms of the three evaluation criteria. In terms of RMSE,
the MLE-one day and the QMLE-res are better than the QMLE-md, but the gap
quickly narrows when $N$ or $T$ increases. In terms of Ave.se, the QMLE-md has
slightly larger standard errors than those of the MLE-one day or the QMLE-res.
This is probably because the QMLE, the first-step estimator in the QMLE-md,
has large standard errors. When $N$ and $T$ increase, we obtain smaller RMSE,
smaller Ave.se, and better coverage in general for all estimators.
7 Empirical Work
In this section, we present two empirical applications of our model. Section
7.1 is about modelling equity portfolio returns from
Japan, Europe and the US. That is, one market per continent. Section
7.2 studies MSCI equity indices of the developed and
emerging markets (41 markets across three continents). We first list the
trading hours of the world’s top ten stock exchanges in terms of market
capitalisation in Table 3. This shows the
overlaps and lack of overlaps.
7.1 An Empirical Study of Three Markets
We now apply our model to equity portfolios of three continents/markets:
Japan, Europe and the US. Take Japan as an example. First, we consider six
equity portfolios constructed by intersections of 2 size groups (small (S) and
big (B)) and 3 book-to-market equity ratio (B/M) groups (growth (G), neutral
(N) and value (V)), in the spirit of Fama and
French (1993); we denote the six
portfolios SG, SN, SV, BG, BN and BV, respectively. Second, in a similar manner we consider
six equity portfolios constructed by intersections of 2 size groups (small (S)
and big (B)) and 3 momentum groups (loser (L), neutral (N) and winner (W)); we
denote these six portfolios SL, SN, SW, BL, BN and BW, respectively. We downloaded the daily
value-weighted portfolio returns (in percentage points) from Kenneth R.
French’s
website.444https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/
Note that these returns are not logarithmic returns, so strictly speaking our
model does not apply. Moreover we demeaned and standardised the daily
value-weighted portfolio returns so that the returns have sample variances of
one. In SM LABEL:sec_portfolio_returns, we show that our model could still be
applied by making some innocuous approximations. Since we do not have so many
assets in this application, we shall use the MLE-one day estimator with its
standard errors approximated by the numerical Hessian method.
We next discuss how to interpret the factor loadings of our model. Let
$\dot{y}_{i,t}^{c}$ denote the standardised return of portfolio $i$ of market
$c$ on period $t$. Recall that
$$\displaystyle\dot{y}^{c}_{i,t}$$
$$\displaystyle=\sum_{j=0}^{2}z^{c}_{i,j}f_{g,t-j}+z^{c}_{i,3}f_{C,t}+e_{i,t}^{c},$$
where $\operatorname*{var}(f_{g,t})=(1-\phi^{2})^{-1}$ and $\operatorname*{var}f_{C,t}=1$. Thus an
additional standard-deviation increase in $f_{g,t-1}$ predicts $z^{c}_{i,1}/\sqrt{1-\phi^{2}}$ standard-deviation increase in the standardised
return $\dot{y}^{c}_{i,t}$, while an additional standard-deviation increase in
$f_{C,t}$ predicts $z^{c}_{i,3}$ standard-deviation increase in the
standardised return $\dot{y}^{c}_{i,t}$.
We then give a formula for variance decomposition. Recall (2.3):
$\boldsymbol{\alpha}_{t+1}=\mathcal{T}\boldsymbol{\alpha}_{t}+R\boldsymbol{\eta}_{t}$, $\boldsymbol{\eta}_{t}\sim N(0,I_{2})$. We first
calculate the unconditional variance of $\boldsymbol{\alpha}_{t}$; it can be
shown that
$$\displaystyle\operatorname*{var}(\boldsymbol{\alpha}_{t})$$
$$\displaystyle=\operatorname*{unvec}\mathinner{\Bigl{\{}[I_{16}-\mathcal{T}\otimes\mathcal{T}]^{-1}(R\otimes R)\operatorname*{vec}(I_{2})\Bigr{\}}}=\left[\begin{array}[c]{cccc}\frac{1}{1-\phi^{2}}&\frac{\phi}{1-\phi^{2}}&\frac{\phi^{2}}{1-\phi^{2}}&0\\
\frac{\phi}{1-\phi^{2}}&\frac{1}{1-\phi^{2}}&\frac{\phi}{1-\phi^{2}}&0\\
\frac{\phi^{2}}{1-\phi^{2}}&\frac{\phi}{1-\phi^{2}}&\frac{1}{1-\phi^{2}}&0\\
0&0&0&1\end{array}\right].$$
Since $\dot{y}_{i,t}^{c}=\mathinner{\bigl{[}\begin{array}[]{cccc}z_{i,0}^{c}&z_{i,1}^{c}&z_{i,2}^{c}&z^{c}_{i,3}\end{array}\bigr{]}}\boldsymbol{\alpha}_{t}+e_{i,t}^{c}$, we have
$$\displaystyle\operatorname*{var}(\dot{y}_{i,t}^{c})$$
$$\displaystyle=\mathinner{\bigl{[}\begin{array}[]{cccc}z_{i,0}^{c}&z_{i,1}^{c}&z_{i,2}^{c}&z^{c}_{i,3}\end{array}\bigr{]}}\left[\begin{array}[c]{cccc}\frac{1}{1-\phi^{2}}&\frac{\phi}{1-\phi^{2}}&\frac{\phi^{2}}{1-\phi^{2}}&0\\
\frac{\phi}{1-\phi^{2}}&\frac{1}{1-\phi^{2}}&\frac{\phi}{1-\phi^{2}}&0\\
\frac{\phi^{2}}{1-\phi^{2}}&\frac{\phi}{1-\phi^{2}}&\frac{1}{1-\phi^{2}}&0\\
0&0&0&1\end{array}\right]\left[\begin{array}[c]{c}z_{i,0}^{c}\\
z_{i,1}^{c}\\
z_{i,2}^{c}\\
z^{c}_{i,3}\end{array}\right]+\operatorname*{var}(e_{i,t}^{c})$$
(7.10)
$$\displaystyle=\underbrace{\frac{1}{1-\phi^{2}}\mathinner{\bigl{[}z_{i,0}^{c,2}+z_{i,1}^{c,2}+z_{i,2}^{c,2}+2\phi z_{i,1}^{c}z_{i,0}^{c}+2\phi z_{i,1}^{c}z_{i,2}^{c}+2\phi^{2}z_{i,2}^{c}z_{i,0}^{c}\bigr{]}}}_{\text{variance due to the global factor}}$$
$$\displaystyle\qquad+\underbrace{z_{i,3}^{c,2}}_{\text{variance due to the continental
factor}}+\underbrace{\sigma_{c,i}^{2}}_{\text{variance due to the
idiosyncratic error}}.$$
(7.11)
For fixed $\operatorname*{var}(\dot{y}_{i,t}^{c})$ and $\sigma_{c,i}^{2}$, a small
$z_{i,3}^{c,2}$ means that the variance of the return is largely explained by
the global factor.
7.1.1 Two Periods of Five-Year Data
We estimate our model twice using two periods of data (20110103-20151231;
20160104-20201231). We take care of the missing returns due to continental
reasons using the technique outlined in Section LABEL:sec_missing_obs. The
starting values of the parameters for the EM algorithm are estimated according
to Section 3.
The MLE-one day estimates of the factor loading matrices are reported in Table
4. We first examine the six portfolios constructed by
intersections of size and book-to-market equity ratio (B/M) groups. In
2016-2020 the Japanese standardised returns were more likely to be affected by
the global factor during the US trading time ($\boldsymbol{z}_{1}^{A}$) and
less likely to be affected by the global factor during the European trading
time ($\boldsymbol{z}_{2}^{A}$) than they were in 2011-2015. Take the Japanese
SG portfolio as an example. In 2011-2015, an additional standard-deviation
increase in $f_{g,t-1}$ predicts $0.24/\sqrt{1-0.1654^{2}}=0.2424$
standard-deviation increase in the standardised return $\dot{y}^{A}_{SG,t}$,
while an additional standard-deviation increase in $f_{g,t-2}$ predicts
$0.51/\sqrt{1-0.1654^{2}}=0.5171$ standard-deviation increase in the
standardised return $\dot{y}^{A}_{SG,t}$. In 2016-2020, an additional
standard-deviation increase in $f_{g,t-1}$ predicts $0.71/\sqrt{1-0.2323^{2}}=0.73$ standard-deviation increase in the standardised return $\dot{y}^{A}_{SG,t}$, while an additional standard-deviation increase in $f_{g,t-2}$
predicts $0.26/\sqrt{1-0.2323^{2}}=0.2673$ standard-deviation increase in the
standardised return $\dot{y}^{A}_{SG,t}$. In 2011-2015, an additional
standard-deviation increase in $f_{C,t}$ predicts $0.75$ standard-deviation
increase in the standardised return, while in 2016-2020, an additional
standard-deviation increase in $f_{C,t}$ only predicts $0.63$
standard-deviation increase in the standardised return.
For the European portfolios, the standardised returns were less likely to be
affected by the global factor during the European and US trading times
($\boldsymbol{z}_{0}^{E},\boldsymbol{z}_{2}^{E}$), and more likely to be
affected by the continental factor ($\boldsymbol{z}_{3}^{E}$) than they were
in 2011-2015. Because of the much larger continental loadings in 2016-2020,
one could argue that the European portfolios became less integrated into the
global market than they were in 2011-2015. Take the European BG portfolio as
an example. In 2011-2015, an additional standard-deviation increase in
$f_{g,t}$ predicts $0.47/\sqrt{1-0.1654^{2}}=0.4766$ standard-deviation
increase in the standardised return. In 2016-2020, an additional
standard-deviation increase in $f_{g,t}$ predicts $0.18/\sqrt{1-0.2323^{2}}=0.1851$ standard-deviation increase in the standardised return. In
2011-2015, an additional standard-deviation increase in $f_{C,t}$ predicts
0.16 standard-deviation increase in the standardised return, while in
2016-2020 an additional standard-deviation increase in $f_{C,t}$ predicts 0.68
standard-deviation increase in the standardised return.
For the US portfolios, the standardised returns in 2015-2020 to a large extent
became slightly less affected by the global factor during the US trading time
($\boldsymbol{z}_{0}^{U}$), and became slightly more affected by the global
factor during the Asian trading time ($\boldsymbol{z}_{2}^{U}$) than they were
in 2011-2015. Take the US SN portfolio as an example. In 2011-2015, an
additional standard-deviation increase in $f_{g,t}$ predicts $0.47/\sqrt{1-0.1654^{2}}=0.4766$ standard-deviation increase in the standardised return,
while an additional standard-deviation increase in $f_{g,t-2}$ predicts
$0.52/\sqrt{1-0.1654^{2}}=0.5273$ standard-deviation increase in the
standardised return. In 2016-2020, an additional standard-deviation increase
in $f_{g,t}$ predicts $0.40/\sqrt{1-0.2323^{2}}=0.4113$ standard-deviation
increase in the standardised return, while an additional standard-deviation
increase in $f_{g,t-2}$ predicts $0.64/\sqrt{1-0.2323^{2}}=0.6580$
standard-deviation increase in the standardised return. For the US portfolios,
the loadings for the continental factor have decreased slightly; in particular
the continental loading of the BG portfolio has decreased from 0.15 to
something statistically insignificant.
Over the two periods of five years, a few general patterns emerge. First,
within the same B/M ratio category, the big portfolio has much smaller
loadings for the continental factor but larger loadings for the global factor
than the small portfolio. In particular, the variances of the standardised
returns of the US big portfolios could largely be explained by the global
factor in light of (7.11). Second, within the same size category,
the value portfolio is more affected by the global factor during the European
trading time than the growth portfolio across the three continents.
We next examine the six portfolios constructed by intersections of size and
momentum groups. The Japanese portfolios in general were more affected by the
global factor during the Asian trading time ($\boldsymbol{z}_{0}^{A}$) than
they were in 2011-2015. The effect of the global factor during the US trading
time on the Japanese portfolios ($\boldsymbol{z}_{1}^{A}$) has almost doubled
in 2016-2020. For example, in 2011-2015 an additional standard-deviation
increase in $f_{g,t-1}$ predicts $0.27$ standard-deviation increase in the
standardised return of the Japanese BN portfolio, while in 2016-2020, the same
increase predicts $0.5/\sqrt{1-0.3079^{2}}=0.5255$ standard-deviation increase
in the standardised return.
For the European portfolios, the standardised returns in 2015-2020 became more
affected by the global factor during the Asian trading time ($\boldsymbol{z}_{1}^{E}$), and became less affected by the global factor during the US
trading time ($\boldsymbol{z}_{2}^{E}$) than they were in 2011-2015. Take the
European BW portfolio as an example. In 2011-2015, an additional
standard-deviation increase in $f_{g,t-1}$ predicts $0.31$ standard-deviation
increase in the standardised return, while an additional standard-deviation
increase in $f_{g,t-2}$ predicts $0.75$ standard-deviation increase in the
standardised return. In 2016-2020, an additional standard-deviation increase
in $f_{g,t-1}$ predicts $0.94/\sqrt{1-0.3079^{2}}=0.9880$ standard-deviation
increase in the standardised return, while an additional standard-deviation
increase in $f_{g,t-2}$ predicts $0.45/\sqrt{1-0.3079^{2}}=0.4730$
standard-deviation increase in the standardised return.
For the US portfolios, the standardised returns in 2015-2020 became more
affected by the global factor during the US trading time ($\boldsymbol{z}{}_{0}^{U}$). Take the US BL portfolio as an example. In 2011-2015, an
additional standard-deviation increase in $f_{g,t}$ predicts $0.30$
standard-deviation increase in the standardised return, while in 2016-2020 the
same increase predicts $0.56/\sqrt{1-0.3079^{2}}=0.5886$ standard-deviation
increase in the standardised return.
For the size-momentum portfolios, one consistent pattern across the three
continents is that in 2011-2015 within the same size category, the winner (W)
portfolio was less affected by the global factor during the Asian trading time
than the loser (L) portfolio. In 2016-2020, again within the same size
category, the winner portfolio was less affected by the global factor during
the European trading time than the loser portfolio.
The $\tilde{\phi}$ in the application of size-B/M portfolios is significantly
negative in both five-year periods. The value is -$0.1654$ with a standard
error of 0.0266 in 2011-2015, and -$0.2323$ with a standard error of 0.0219 in
2016-2020. The $\tilde{\phi}$ in the application of size-momentum portfolios
is statistically insignificant with a point estimate of 0.0051 in 2011-2015,
and significantly negative in 2016-2020, with a value of -$0.3079$ and a
standard error of 0.0229.
7.1.2 Time Series Patterns
In this subsection, we estimate the model using thirty periods of one-year
data (1991-2020). For simplicity, we only consider the size-B/M portfolios.
The detailed point estimates and their standard errors are available upon
request; here we only discuss the main findings.
First, across the three continents, the big portfolios tended to have smaller
loadings for the continental factor but larger loadings for the global factor
than the small portfolios. The Japanese idiosyncratic variances are in general
larger than those of the US and Europe. This is especially so in 1999-2001 and
2019-2020. These observations are consistent with the observations based on
five-year data reported in the previous subsection.
Second, we discuss some year-specific patterns:
(i)
In 1998, the Japanese portfolios have particularly large loadings for
the global factor during the Asian trading time, but small loadings for the
global factor during the US trading time. This could be interpreted as the
effect of the Asian financial crisis.
(ii)
During the 2007-2008 financial crisis, the Japanese portfolios have
large loadings for the global factors during the European trading time. The
European portfolios have small loadings for the continental factor but large
loadings for the global factor during the US trading time. This could be
interpreted as the spread of the US subprime mortgage crisis.
(iii)
In 2017-2018, the Japanese portfolios have large loadings for the global
factor during the US trading time but small loadings for the continental
factor. The US portfolios have large loadings for the global factor during the
Asian trading time. The European portfolios have large loadings for the
continental factor but small loadings for the global factor during the Asian
and US trading times. This could be interpreted as Japan and US markets being
more integrated during this period but not so for Europe. This could be due to
the Sino-US trade war.
(iv)
In 2020, the European portfolios have quite small loadings for the
continental factor. Compared with the Japanese and European portfolios, the US
portfolios have relatively constant loadings for the global factor during the
three trading periods of a day.
Last, we re-estimate the model using fifteen periods of two-year data
(1991-2020) and compute the variance decomposition using (7.11). The
decompositions are plotted in Figure 1. The blue
solid and red dashed lines depict the variance proportions of the global and
continental factors, respectively. The magenta dotted lines represent the
realized volatilities computed using the standardised portfolio returns
(divided by two for a better layout). We find that the continental factor
accounts for a decreasing share of variance of the US big and value
standardised portfolio returns in the past 30 years. In the 1990s, the global
factor only accounted for small shares of variances of the US standardized
portfolio returns. During the turbulent years such as the 2008 financial
crisis, the global factor tended to account for larger shares of variances of
the European and US standardized portfolio returns.
7.2 An Empirical Study of Many Markets
We now apply our model to MSCI equity indices of the developed and emerging
markets (41 markets in total). The daily indices are obtained from
https://www.msci.com/end-of-day-data-search. There are 6 indices for each
market: Large-Growth, Mid-Growth, Small-Growth, Large-Value, Mid-Value, and
Small-Value, all in USD currency. According to the closing time of each
market, we categorize these markets into 3 continents: Asia-Pacific, Europe
and America. Since the closing time of the Israeli market is both far away
from Asia-Pacific and Europe, we exclude it from our sample.
We use the data of the period from January 1st 2018 to February 21st 2022.
Indices starting after January 1st 2018 are excluded. We estimate the model
using the QMLE-md, with choices of $\hat{\boldsymbol{h}}$ similar to those
mentioned in Section 6.555We also estimate the
model using the MLE-one day (results available upon request). Its point
estimates are close to those of the QMLE-md; its standard errors are slightly
smaller than those of the QMLE-md. The estimated $\phi$ is 0.338 with a
standard error 0.0262.
Table 6 reports the estimates of the factor loadings and
idiosyncratic variances for the Asian-Pacific continent. We present all the
indices for Mainland China, Hong Kong and Japan, but only Middle-Value and
Middle-Growth indices for other Asian-Pacific markets in the interest of
space. There are several findings. First, Mainland China and Hong Kong have
particularly large loadings on the global factors during the US trading time
(i.e, $\boldsymbol{z}_{1}^{A}$). Second, Japan has large loadings on the
continental factor (i.e, $\boldsymbol{z}_{3}^{A}$) but small idiosyncratic
variances. Third, the growth indices in general have larger idiosyncratic
variances than the value indices. Fourth, most other Asian-Pacific markets
have large loadings on the global factor during the US trading time (i.e,
$\boldsymbol{z}_{1}^{A}$), but small and insignificant loadings on the
continental factor (i.e, $\boldsymbol{z}_{3}^{A}$).
Table 7 reports the estimates of the factor loadings and
idiosyncratic variances for the European continent. We present all the indices
for the UK, but only Mid-Value and Mid-Growth indices for other European
markets in the interest of space. Most European markets have the largest
loadings on the global factor during the Asian trading time (i.e.,
$\boldsymbol{z}^{E}_{1}$). The developed European markets have large and
positive loadings on the continental factor (i.e., $\boldsymbol{z}^{E}_{3}$),
but the emerging European markets have small or negative loadings on the
continental factor (i.e., $\boldsymbol{z}^{E}_{3}$).
Table 8 reports the estimates of the factor loadings and
idiosyncratic variance for the American continent. The US and Canada have
statistically insignificant factor loadings on the continental factor (i.e.,
$\boldsymbol{z}^{U}_{3}$), while Brazil has large factor loadings on the
continental factor (i.e., $\boldsymbol{z}^{U}_{3}$ with point estimates
greater than 1). Moreover, some emerging American markets (e.g., Mexico) have
higher loadings on the global factor during the Asian trading time (i.e.,
$\boldsymbol{z}^{U}_{2}$) but small (and possibly insignificant) loadings on
the global factor during the American and European trading times (i.e.,
$\boldsymbol{z}^{U}_{0},\boldsymbol{z}^{U}_{1}$); this pattern does not hold
for the US market.
8 Conclusion
In this article we propose a new framework of using a statistical dynamic
factor model to model a large number of daily stock returns across different
time zones. The presence of global and continental factors describes a
situation in which all the new information represented by the global and
continental factors accumulated since the last closure of a continent will
have an impact on the upcoming observed logarithmic 24-hr returns of that
continent. Our model is identified under a mild fixed-signs assumption and
hence has a structural interpretation. Several estimators are outlined: the
MLE-one day, the QMLE-res, the QMLE, the QMLE-md and the Bayesian. The
asymptotic theories of the QMLE and the QMLE-md are carefully derived. In
addition, we propose a way to approximate the standard errors of the MLE-one
day and the QMLE-res. Monte Carlo simulations show good performance of the
MLE-one day, the QMLE-res and the QMLE-md. Last, we present two empirical
applications of our model. One future research direction is to work out the
asymptotic theory for the MLE-one day and the QMLE-res in the large $N$ large
$T_{f}$ case. Perhaps one could try to adapt the results of
Barigozzi and
Luciani (2022) to this purpose.
Appendix A Appendix
A.1 Proof of Lemma 2.1
Proof of Lemma 2.1.
This proof is inspired by that of Bai and
Wang (2015). Suppose that Assumption 2.2 hold. Fix a particular $t$. Recall (2.3):
$$\displaystyle\boldsymbol{y}_{t}$$
$$\displaystyle=Z_{t}\left(\begin{array}[]{c}f_{g,t}\\
f_{g,t-1}\\
f_{g,t-2}\\
f_{C,t}\end{array}\right)+\boldsymbol{\varepsilon}_{t}\quad f_{g,t+1}=\phi f_{g,t}+\eta_{g,t}\quad f_{C,t+1}=\eta_{C,t}.$$
Note that
$$\displaystyle f_{g,t}$$
$$\displaystyle=\phi^{3}f_{g,t-3}+\phi^{2}\eta_{g,t-3}+\phi\eta_{g,t-2}+\eta_{g,t-1}$$
$$\displaystyle f_{g,t-1}$$
$$\displaystyle=\phi^{2}f_{g,t-3}+\phi\eta_{g,t-3}+\eta_{g,t-2}$$
$$\displaystyle f_{g,t-2}$$
$$\displaystyle=\phi f_{g,t-3}+\eta_{g,t-3}.$$
Since $Z_{t}$ assumes one of $\{Z^{A},Z^{E},Z^{U}\}$, we need to consider three $4\times 4$ rotation matrices represented by:
$$\Delta_{1}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{1}&B_{1}&C_{1}&O_{1}\\
D_{1}&E_{1}&F_{1}&P_{1}\\
G_{1}&H_{1}&I_{1}&Q_{1}\\
R_{1}&S_{1}&T_{1}&W_{1}\end{array}\right],\Delta_{2}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{2}&B_{2}&C_{2}&O_{2}\\
D_{2}&E_{2}&F_{2}&P_{2}\\
G_{2}&H_{2}&I_{2}&Q_{2}\\
R_{2}&S_{2}&T_{2}&W_{2}\end{array}\right],\Delta_{3}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{3}&B_{3}&C_{3}&O_{3}\\
D_{3}&E_{3}&F_{3}&P_{3}\\
G_{3}&H_{3}&I_{3}&Q_{3}\\
R_{3}&S_{3}&T_{3}&W_{3}\end{array}\right].$$
Consider
$$\displaystyle\left[\begin{array}[]{cccc}A_{1}&B_{1}&C_{1}&O_{1}\\
D_{1}&E_{1}&F_{1}&P_{1}\\
G_{1}&H_{1}&I_{1}&Q_{1}\\
R_{1}&S_{1}&T_{1}&W_{1}\end{array}\right]\left[\begin{array}[]{c}f_{g,t}\\
f_{g,t-1}\\
f_{g,t-2}\\
f_{C,t}\end{array}\right]$$
$$\displaystyle=\left[\begin{array}[]{c}\tilde{f}_{g,t}\\
\tilde{f}_{g,t-1}\\
\tilde{f}_{g,t-2}\\
\tilde{f}_{C,t}\end{array}\right]$$
(A.13)
$$\displaystyle\left[\begin{array}[]{cccc}A_{3}&B_{3}&C_{3}&O_{3}\\
D_{3}&E_{3}&F_{3}&P_{3}\\
G_{3}&H_{3}&I_{3}&Q_{3}\\
R_{3}&S_{3}&T_{3}&W_{3}\end{array}\right]\left[\begin{array}[]{c}f_{g,t-1}\\
f_{g,t-2}\\
f_{g,t-3}\\
f_{C,t-1}\end{array}\right]$$
$$\displaystyle=\left[\begin{array}[]{c}\tilde{f}_{g,t-1}\\
\tilde{f}_{g,t-2}\\
\tilde{f}_{g,t-3}\\
\tilde{f}_{C,t-1}\end{array}\right]$$
(A.26)
$$\displaystyle\left[\begin{array}[]{cccc}A_{2}&B_{2}&C_{2}&O_{2}\\
D_{2}&E_{2}&F_{2}&P_{2}\\
G_{2}&H_{2}&I_{2}&Q_{2}\\
R_{2}&S_{2}&T_{2}&W_{2}\end{array}\right]\left[\begin{array}[]{c}f_{g,t-2}\\
f_{g,t-3}\\
f_{g,t-4}\\
f_{C,t-2}\end{array}\right]$$
$$\displaystyle=\left[\begin{array}[]{c}\tilde{f}_{g,t-2}\\
\tilde{f}_{g,t-3}\\
\tilde{f}_{g,t-4}\\
\tilde{f}_{C,t-2}\end{array}\right].$$
(A.39)
Considering (A.13) and (A.26), we have
$$\displaystyle\tilde{f}_{g,t-1}=D_{1}f_{g,t}+E_{1}f_{g,t-1}+F_{1}f_{g,t-2}+P_{1}f_{C,t}=A_{3}f_{g,t-1}+B_{3}f_{g,t-2}+C_{3}f_{g,t-3}+O_{3}f_{C,t-1}$$
whence we have
$$\displaystyle 0$$
$$\displaystyle=\mathinner{\bigl{[}D_{1}\phi^{3}+(E_{1}-A_{3})\phi^{2}+(F_{1}-B_{3})\phi-C_{3}\bigr{]}}f_{g,t-3}+\mathinner{\bigl{[}D_{1}\phi^{2}+(E_{1}-A_{3})\phi+(F_{1}-B_{3})\bigr{]}}\eta_{g,t-3}$$
$$\displaystyle\qquad+\mathinner{\bigl{[}D_{1}\phi+(E_{1}-A_{3})\bigr{]}}\eta_{g,t-2}+D_{1}\eta_{g,t-1}+P_{1}\eta_{C,t-1}-O_{3}\eta_{C,t-2}.$$
Note that each of $\eta_{g,t-1},\eta_{C,t-1},\eta_{C,t-2}$ is uncorrelated with any other term on the right hand side of the preceding display. We necessarily have $D_{1}\eta_{g,t-1}=0$, $P_{1}\eta_{C,t-1}=0$ and $O_{3}\eta_{C,t-2}=0$ because of the non-zero variance. Equivalently, we have $D_{1}=P_{1}=O_{3}=0$. Likewise, we deduce that $E_{1}=A_{3}$, $F_{1}=B_{3}$ and $C_{3}=0$.
Next, note that
$$\displaystyle\tilde{f}_{g,t-2}$$
$$\displaystyle=G_{1}f_{g,t}+H_{1}f_{g,t-1}+I_{1}f_{g,t-2}+Q_{1}f_{C,t}=D_{3}f_{g,t-1}+E_{3}f_{g,t-2}+F_{3}f_{g,t-3}+P_{3}f_{C,t-1}$$
whence we have
$$\displaystyle 0$$
$$\displaystyle=\mathinner{\bigl{[}G_{1}\phi^{3}+(H_{1}-D_{3})\phi^{2}+(I_{1}-E_{3})\phi-F_{3}\bigr{]}}f_{g,t-3}+\mathinner{\bigl{[}G_{1}\phi^{2}+(H_{1}-D_{3})\phi+(I_{1}-E_{3})\bigr{]}}\eta_{g,t-3}$$
$$\displaystyle\qquad\mathinner{\bigl{[}G_{1}\phi+(H_{1}-D_{3})\bigr{]}}\eta_{g,t-2}+G_{1}\eta_{g,t-1}+Q_{1}\eta_{C,t-1}-P_{3}\eta_{C,t-2}.$$
Using a similar trick, we deduce that
$$G_{1}=Q_{1}=P_{3}=0,\qquad H_{1}=D_{3}=0,\qquad I_{1}=E_{3},\qquad F_{3}=0.$$
The rotation matrices $\Delta_{1},\Delta_{3}$ are deduced to
$$\displaystyle\Delta_{1}$$
$$\displaystyle\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{1}&B_{1}&C_{1}&O_{1}\\
0&E_{1}&F_{1}&0\\
0&H_{1}&I_{1}&0\\
R_{1}&S_{1}&T_{1}&W_{1}\end{array}\right]$$
(A.44)
$$\displaystyle\Delta_{3}$$
$$\displaystyle\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}E_{1}&F_{1}&0&0\\
H_{1}&I_{1}&0&0\\
G_{3}&H_{3}&I_{3}&Q_{3}\\
R_{3}&S_{3}&T_{3}&W_{3}\end{array}\right].$$
(A.49)
Applying the trick to (A.26) and (A.39), we deduce
$$\displaystyle\Delta_{3}$$
$$\displaystyle\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{3}&B_{3}&C_{3}&O_{3}\\
0&E_{3}&F_{3}&0\\
0&H_{3}&I_{3}&0\\
R_{3}&S_{3}&T_{3}&W_{3}\end{array}\right]$$
(A.54)
$$\displaystyle\Delta_{2}$$
$$\displaystyle\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}E_{3}&F_{3}&0&0\\
H_{3}&I_{3}&0&0\\
G_{2}&H_{2}&I_{2}&Q_{2}\\
R_{2}&S_{2}&T_{2}&W_{2}\end{array}\right].$$
(A.59)
Applying the trick to (A.39) and (A.13), we deduce
$$\displaystyle\Delta_{2}$$
$$\displaystyle\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{2}&B_{2}&C_{2}&O_{2}\\
0&E_{2}&F_{2}&0\\
0&H_{2}&I_{2}&0\\
R_{2}&S_{2}&T_{2}&W_{2}\end{array}\right]$$
(A.64)
$$\displaystyle\Delta_{1}$$
$$\displaystyle\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}E_{2}&F_{2}&0&0\\
H_{2}&I_{2}&0&0\\
G_{1}&H_{1}&I_{1}&Q_{1}\\
R_{1}&S_{1}&T_{1}&W_{1}\end{array}\right].$$
(A.69)
Comparing (A.44) and (A.69), we have
$$\displaystyle A_{1}=E_{2},\quad B_{1}=F_{2},\quad H_{2}=0,\quad I_{2}=E_{1},\quad G_{1}=0,\quad F_{1}=0.$$
Comparing (A.49) and (A.54), we have
$$\displaystyle H_{1}=0,\quad F_{3}=0,\quad I_{3}=E_{2}=A_{1}.$$
Comparing (A.59) and (A.64), we have
$$\displaystyle A_{2}=E_{3}=I_{1},\quad B_{2}=F_{3}=0,\quad E_{2}=A_{1},\quad I_{2}=E_{1},\quad H_{3}=0,\quad F_{2}=0.$$
Thus the rotation matrices $\Delta_{1},\Delta_{2},\Delta_{3}$ are reduced to
$$\Delta_{1}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}A_{1}&0&0&0\\
0&E_{1}&0&0\\
0&0&I_{1}&0\\
R_{1}&S_{1}&T_{1}&W_{1}\end{array}\right],\Delta_{2}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}I_{1}&0&0&0\\
0&A_{1}&0&0\\
0&0&E_{1}&0\\
R_{2}&S_{2}&T_{2}&W_{2}\end{array}\right],\Delta_{3}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}E_{1}&0&0&0\\
0&I_{1}&0&0\\
0&0&A_{1}&0\\
R_{3}&S_{3}&T_{3}&W_{3}\end{array}\right].$$
Note that
$$\displaystyle R_{1}f_{g,t}+S_{1}f_{g,t-1}+T_{1}f_{g,t-2}+W_{1}f_{C,t}=\tilde{f}_{c,t},$$
whence we have
$$\displaystyle\mathinner{\bigl{(}R_{1}\phi^{3}+S_{1}\phi^{2}+T_{1}\phi\bigr{)}}f_{g,t-3}+\mathinner{\bigl{(}R_{1}\phi^{2}+S_{1}\phi+T_{1}\bigr{)}}\eta_{g,t-3}+(R_{1}\phi+S_{1})\eta_{g,t-2}+R_{1}\eta_{g,t-1}+W_{1}\eta_{C,t-1}$$
$$\displaystyle=\tilde{\eta}_{c,t-1}.$$
Since $\eta_{g,t}$ and $\eta_{C,t}$ are uncorrelated, we have $R_{1}=S_{1}=T_{1}=0$. Then $\Delta_{1}$ is reduced to
$$\displaystyle\Delta_{1}=\left[\begin{array}[]{cccc}A_{1}&0&0&0\\
0&E_{1}&0&0\\
0&0&I_{1}&0\\
0&0&0&W_{1}\end{array}\right].$$
Applying the similar trick, we have
$$\displaystyle\Delta_{2}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}I_{1}&0&0&0\\
0&A_{1}&0&0\\
0&0&E_{1}&0\\
0&0&0&W_{2}\end{array}\right],\qquad\Delta_{3}\mathrel{\mathop{:}}=\left[\begin{array}[]{cccc}E_{1}&0&0&0\\
0&I_{1}&0&0\\
0&0&A_{1}&0\\
0&0&0&W_{3}\end{array}\right].$$
We have
$$\displaystyle\tilde{f}_{g,t}$$
$$\displaystyle=A_{1}f_{g,t}=A_{1}(\phi f_{g,t-1}+\eta_{g,t-1})=A_{1}\phi f_{g,t-1}+A_{1}\eta_{g,t-1}\qquad\operatorname*{var}(A_{1}\eta_{g,t-1})=1$$
$$\displaystyle\tilde{f}_{g,t-1}$$
$$\displaystyle=E_{1}f_{g,t-1}=E_{1}(\phi f_{g,t-2}+\eta_{g,t-2})=E_{1}\phi f_{g,t-2}+E_{1}\eta_{g,t-2}\qquad\operatorname*{var}(E_{1}\eta_{g,t-2})=1$$
$$\displaystyle\tilde{f}_{g,t-2}$$
$$\displaystyle=I_{1}f_{g,t-2}=I_{1}(\phi f_{g,t-3}+\eta_{g,t-3})=I_{1}\phi f_{g,t-3}+I_{1}\eta_{g,t-3}\qquad\operatorname*{var}(I_{1}\eta_{g,t-3})=1$$
$$\displaystyle\tilde{f}_{c,t}$$
$$\displaystyle=W_{1}f_{C,t}=W_{1}\eta_{C,t-1}\qquad\operatorname*{var}(W_{1}\eta_{C,t-1})=1$$
$$\displaystyle\tilde{f}_{c,t-1}$$
$$\displaystyle=W_{3}f_{C,t-1}=W_{3}\eta_{C,t-2}\qquad\operatorname*{var}(W_{3}\eta_{C,t-2})=1$$
$$\displaystyle\tilde{f}_{c,t-2}$$
$$\displaystyle=W_{2}f_{C,t-2}=W_{2}\eta_{C,t-3}\qquad\operatorname*{var}(W_{2}\eta_{C,t-3})=1$$
We hence deduce that $A_{1}=\pm 1$, $E_{1}=\pm 1$, $I_{1}=\pm 1$ and $W_{i}=\pm 1$ for $i=1,2,3$. Requiring that estimators of $\boldsymbol{z}^{A}_{0},\boldsymbol{z}^{A}_{1},\boldsymbol{z}^{A}_{2},\boldsymbol{z}_{3}^{A},\boldsymbol{z}_{3}^{E},\boldsymbol{z}_{3}^{U}$ have the same column signs as those of $\boldsymbol{z}^{A}_{0},\boldsymbol{z}^{A}_{1},\boldsymbol{z}^{A}_{2},\boldsymbol{z}_{3}^{A},\boldsymbol{z}_{3}^{E},\boldsymbol{z}_{3}^{U}$ ensures that $A_{1}=1$, $E_{1}=1$, $I_{1}=1$ and $W_{i}=1$ for $i=1,2,3$. Thus $\Delta_{1},\Delta_{2},\Delta_{3}$ are reduced to identity matrices. Note that the proof works for both $\phi=0$ and $\phi\neq 0$.
∎
A.2 Computation of the MLE-one day
In this subsection, we provide the details of the EM algorithm for the MLE-one
day defined in Section 3.1.
A.2.1 E-Step
Let $\tilde{\mathbb{E}}$ denote the expectation with respect to the
conditional density $p(\Xi|Y_{1\mathrel{\mathop{:}}T};\tilde{\boldsymbol{\theta}}^{(i)})$ at
$\tilde{\boldsymbol{\theta}}^{(i)}$, where $\tilde{\boldsymbol{\theta}}^{(i)}$
is the estimate of $\boldsymbol{\theta}$ from the $i$th iteration of the EM
algorithm. Taking such expectation on both sides of the preceding display, we
hence have
$$\tilde{\mathbb{E}}\mathinner{\bigl{[}\ell(\Xi,Y_{1\mathrel{\mathop{:}}T};\boldsymbol{\theta})\bigr{]}}=\text{constant}-\frac{1}{2}\mathinner{\biggl{(}\tilde{\mathbb{E}}\sum_{t=1}^{T}\ell_{1,t}+\tilde{\mathbb{E}}\sum_{t=1}^{T}\ell_{2,t}\biggr{)}}.$$
This is the so-called ”E” step of the EM algorithm. $\tilde{\mathbb{E}}[\cdot]$ could be computed using the Kalman smoother (KS; see SM
LABEL:sec_Kalman for formulas of the KS).
A.2.2 M-Step
The M step involves maximising $\tilde{\mathbb{E}}\mathinner{\bigl{[}\ell(\Xi,Y_{1\mathrel{\mathop{:}}T};\boldsymbol{\theta})\bigr{]}}$ with respect to $\boldsymbol{\theta}$. This is
usually done by computing
$$\frac{\partial\tilde{\mathbb{E}}\mathinner{\bigl{[}\ell(\Xi,Y_{1\mathrel{\mathop{:}}T};\boldsymbol{\theta})\bigr{]}}}{\partial\boldsymbol{\theta}}$$
and setting the preceding display to zero to obtain the estimate
$\tilde{\boldsymbol{\theta}}^{(i+1)}$ of $\boldsymbol{\theta}$ for the
$(i+1)$th iteration of the EM algorithm.
M-Step of $Z_{t}$ and $\Sigma_{t}$
We now find values of $Z_{t}$ and $\Sigma_{t}$ to minimize $\tilde{\mathbb{E}}\sum_{t=1}^{T}\ell_{1,t}$. Recall that $Z_{t}=Z^{c},\Sigma_{t}=\Sigma_{c}$
if $t\in T_{c}$ for $c=A,E,U$. Without loss of generality, we shall use the
Asian continent to illustrate the procedure. We now find values of $Z^{A}$ and
$\Sigma_{A}$ to minimise $\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}$.
Since $\boldsymbol{\varepsilon}_{t}=\boldsymbol{y}_{t}-Z_{t}\boldsymbol{\alpha}_{t}$ and $\boldsymbol{\eta}_{t}=R^{\intercal}(\boldsymbol{\alpha}_{t+1}-\mathcal{T}\boldsymbol{\alpha}_{t})$ (see (2.3)), we have
$$\sum_{t\in T_{A}}\ell_{1,t}=\frac{T}{3}\log|\Sigma_{A}|+\sum_{t\in T_{A}}\operatorname*{tr}\mathinner{\biggl{(}\mathinner{\Bigl{[}\boldsymbol{y}_{t}\boldsymbol{y}_{t}^{\intercal}-2Z^{A}\boldsymbol{\alpha}_{t}\boldsymbol{y}_{t}^{\intercal}+Z^{A}\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}Z^{A\intercal}\Bigr{]}}\Sigma_{A}^{-1}\biggr{)}}$$
and hence
$$\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}=\frac{T}{3}\log|\Sigma_{A}|+\sum_{t\in T_{A}}\operatorname*{tr}\mathinner{\biggl{(}\mathinner{\Bigl{[}\boldsymbol{y}_{t}\boldsymbol{y}_{t}^{\intercal}-2Z^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}]\boldsymbol{y}_{t}^{\intercal}+Z^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]Z^{A\intercal}\Bigr{]}}\Sigma_{A}^{-1}\biggr{)}}$$
(A.70)
We now consider $Z^{A}$, and take differential of (A.70) with
respect to $Z^{A}$:
$$\displaystyle\text{d}\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}=\sum_{t\in T_{A}}\operatorname*{tr}\mathinner{\biggl{(}\mathinner{\Bigl{[}-2\text{d}Z^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}]\boldsymbol{y}_{t}^{\intercal}+2\text{d}Z^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]Z^{A\intercal}\Bigr{]}}\Sigma_{A}^{-1}\biggr{)}}$$
$$\displaystyle=-2\sum_{t\in T_{A}}\operatorname*{tr}\mathinner{\biggl{(}\text{d}Z^{A}\mathinner{\Bigl{[}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}]\boldsymbol{y}_{t}^{\intercal}-\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]Z^{A\intercal}\Bigr{]}}\Sigma_{A}^{-1}\biggr{)}},$$
whence we have
$$\displaystyle\tilde{Z}^{A}=\sum_{t\in T_{A}}\tilde{\mathbb{E}}[\boldsymbol{y}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]\mathinner{\biggl{(}\sum_{t\in T_{A}}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]\biggr{)}}^{-1}.$$
(A.71)
We next consider $\Sigma_{A}$. Define
$$\displaystyle C_{A}$$
$$\displaystyle\mathrel{\mathop{:}}=\sum_{t\in T_{A}}\mathinner{\Bigl{[}\boldsymbol{y}_{t}\boldsymbol{y}_{t}^{\intercal}-2Z^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}]\boldsymbol{y}_{t}^{\intercal}+Z^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]Z^{A\intercal}\Bigr{]}}$$
$$\displaystyle\tilde{C}_{A}$$
$$\displaystyle\mathrel{\mathop{:}}=\sum_{t\in T_{A}}\mathinner{\Bigl{[}\boldsymbol{y}_{t}\boldsymbol{y}_{t}^{\intercal}-2\tilde{Z}^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}]\boldsymbol{y}_{t}^{\intercal}+\tilde{Z}^{A}\tilde{\mathbb{E}}[\boldsymbol{\alpha}_{t}\boldsymbol{\alpha}_{t}^{\intercal}]\tilde{Z}^{A\intercal}\Bigr{]}}$$
Then (A.70) can be written as
$$\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}=\frac{T}{3}\log|\Sigma_{A}|+\operatorname*{tr}(C_{A}\Sigma_{A}^{-1})$$
Take the differential of $\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}$ with
respect to $\Sigma_{A}$
$$\text{d}\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}=\frac{T}{3}\operatorname*{tr}(\Sigma_{A}^{-1}\text{d}\Sigma_{A})-\operatorname*{tr}(\Sigma_{A}^{-1}C_{A}\Sigma_{A}^{-1}\text{d}\Sigma_{A})=\operatorname*{tr}\mathinner{\biggl{(}\Sigma_{A}^{-1}\mathinner{\biggl{[}\frac{T}{3}\Sigma_{A}-C_{A}\biggr{]}}\Sigma_{A}^{-1}\text{d}\Sigma_{A}\biggr{)}}$$
whence we have, recognising the diagonality of $\Sigma_{A}$,
$$\frac{\partial\tilde{\mathbb{E}}\sum_{t\in T_{A}}\ell_{1,t}}{\partial\Sigma_{A}}=\Sigma_{A}^{-1}\mathinner{\biggl{[}\frac{T}{3}\Sigma_{A}-C_{A}\biggr{]}}\Sigma_{A}^{-1}\circ I_{N_{A}}$$
where $\circ$ denotes the Hadamard product. The first-order condition of
$\Sigma_{A}$ is
$$\tilde{\Sigma}_{A}=\frac{3}{T}(\tilde{C}_{A}\circ I_{N}).$$
M-Step of $\phi$
We now find value of $\phi$ to minimize $\tilde{\mathbb{E}}\sum_{t=1}^{T}\ell_{2,t}$. We have
$$\displaystyle\tilde{\mathbb{E}}\sum_{t=1}^{T}\ell_{2,t}=\sum_{t=1}^{T}\tilde{\mathbb{E}}\eta_{g,t-1}^{2}=\sum_{t=1}^{T}\mathinner{\biggl{(}\tilde{\mathbb{E}}[f_{g,t}^{2}]-2\phi\tilde{\mathbb{E}}[f_{g,t}f_{g,t-1}]+\phi^{2}\tilde{\mathbb{E}}[f_{g,t-1}^{2}]\biggr{)}},$$
whence the first order condition of $\phi$ gives
$$\tilde{\phi}=\mathinner{\biggl{(}\sum_{t=1}^{T}\tilde{\mathbb{E}}[f_{g,t-1}^{2}]\biggr{)}}^{-1}\sum_{t=1}^{T}\tilde{\mathbb{E}}[f_{g,t}f_{g,t-1}].$$
(A.72)
Remark A.1.
As mentioned before, our model is perfectly geared for the scenario of missing
observations due to non-synchronized trading (scenario (i)). In SM
LABEL:sec_missing_obs, we discuss how to alter the EM algorithm if we include
the scenario of missing observations due to continental reasons such as
continent-wide public holidays (e.g., Chinese New Year). That is, both
scenario (i) and a specific form of scenario (ii) are present in the data. We
do not consider other forms of scenario (ii) - missing observations due to
market-specific, stock-specific reasons - in this article.
A.3 First-Order Conditions of (3.13)
In this subsection, we derive (3.14). Note that we only utilise
information that $M$ is symmetric, positive definite and that $\Sigma_{ee}$ is
diagonal to derive the first-order conditions; no specific knowledge of
$\Lambda$ or $M$ is utilised to derive the first-order conditions. Cholesky
decompose $M$: $M=LL^{\intercal}$, where $L$ is the unique lower triangular
matrix with positive diagonal entries. Thus
$$\displaystyle\Sigma_{yy}=\Lambda M\Lambda^{\intercal}+\Sigma_{ee}=\Lambda LL^{\intercal}\Lambda^{\intercal}+\Sigma_{ee}=BB^{\intercal}+\Sigma_{ee},$$
(A.73)
where $B\mathrel{\mathop{:}}=\Lambda L$. Recall the log-likelihood function (3.13) omitting
the constant:
$$\displaystyle-\frac{1}{2N}\log|\Sigma_{yy}|-\frac{1}{2N}\operatorname*{tr}(S_{yy}\Sigma_{yy}^{-1})=-\frac{1}{2N}\log|BB^{\intercal}+\Sigma_{ee}|-\frac{1}{2N}\operatorname*{tr}\mathinner{\Bigl{(}S_{yy}\mathinner{\bigl{[}BB^{\intercal}+\Sigma_{ee}\bigr{]}}^{-1}\Bigr{)}}.$$
Take the derivatives of the preceding display with respect to $B$ and
$\Sigma_{ee}$. The FOC of $\Sigma_{ee}$ is:
$$\displaystyle\operatorname*{diag}(\hat{\Sigma}_{yy}^{-1})$$
$$\displaystyle=\operatorname*{diag}(\hat{\Sigma}_{yy}^{-1}S_{yy}\hat{\Sigma}_{yy}^{-1}),$$
(A.74)
where $\hat{\Sigma}_{yy}\mathrel{\mathop{:}}=\hat{B}\hat{B}^{\intercal}+\hat{\Sigma}_{ee}$. The
FOC of $B$ is
$$\displaystyle\hat{B}^{\intercal}\hat{\Sigma}_{yy}^{-1}(S_{yy}-\hat{\Sigma}_{yy})=0.$$
(A.75)
Note that (A.74, A.75) has $6N(14+1)$ equations, while $B,\Sigma_{ee}$ has $6N(14+1)$ parameters. Thus $\hat{B},\hat{\Sigma}_{ee}$ can be
uniquely solved. Then we need identification conditions to kick in. Even
though we could uniquely determine $\hat{B}$, we cannot uniquely determine
$\hat{\Lambda},\hat{M}$. This is because
$$\displaystyle\hat{B}\hat{B}^{\intercal}=\tilde{\Lambda}\tilde{M}\tilde{\Lambda}^{\intercal}=\mathring{\Lambda}\mathring{M}\mathring{\Lambda}^{\intercal}=\tilde{\Lambda}CC^{-1}\tilde{M}(C^{-1})^{\intercal}C^{\intercal}\tilde{\Lambda}^{\intercal}$$
where $\mathring{\Lambda}\mathrel{\mathop{:}}=\tilde{\Lambda}C$ and $\mathring{M}\mathrel{\mathop{:}}=C^{-1}\tilde{M}(C^{-1})^{\intercal}$ for any $14\times 14$ invertible $C$.666Note that we could find at least one pair $(\tilde{\Lambda},\tilde{M})$ satisfying $\hat{B}\hat{B}^{\intercal}=\tilde{\Lambda}\tilde{M}\tilde{\Lambda}^{\intercal}$: $\tilde{\Lambda}=\hat{B}$ and $\tilde{M}=I_{14}$. We hence need to impose $14^{2}$ identification restrictions on
the estimates of $\Lambda$ and $M$ to rule out the rotational indeterminacy.
After imposing the $14^{2}$ restrictions, we obtain the unique estimates, say,
$\hat{\Lambda},\hat{M}$ (and hence $\hat{L}$). Substituting $\hat{B}=\hat{\Lambda}\hat{L}$ into (A.75), we have
$$\displaystyle\hat{\Lambda}^{\intercal}\hat{\Sigma}_{yy}^{-1}(S_{yy}-\hat{\Sigma}_{yy})=0.$$
A.4 Computation of the QMLE-res
In this subsection, we provide the details of the EM algorithm for the
QMLE-res defined in Section 3.4. We first find values of
$Z^{c}$ to minimise $\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}$. We
will illustrate the procedure using the first row of $Z^{A}$, denoted
$(Z^{A})_{1}$. Define the $4\times 14$ selection matrices $L_{1}$ and $L_{4}$,
so that $\boldsymbol{\lambda}_{1,1}^{\intercal}=(Z^{A})_{1}L_{1}$ and
$\boldsymbol{\lambda}_{4,1}^{\intercal}=(Z^{A})_{1}L_{4}$. One can show that
$$\displaystyle\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}\propto\sum_{t=1}^{T_{f}}\mathinner{\biggl{[}-2(Z^{A})_{1}L_{1}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\mathring{\boldsymbol{y}}_{1,1}]/\sigma_{1,1}^{2}+(Z^{A})_{1}L_{1}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]L_{1}^{\intercal}(Z^{A})_{1}^{\intercal}/\sigma_{1,1}^{2}\biggr{]}}$$
$$\displaystyle\qquad+\sum_{t=1}^{T_{f}}\mathinner{\biggl{[}-2(Z^{A})_{1}L_{4}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\mathring{\boldsymbol{y}}_{1,3N+1}]/\sigma_{1,1}^{2}+(Z^{A})_{1}L_{4}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]L_{4}^{\intercal}(Z^{A})_{1}^{\intercal}/\sigma_{1,1}^{2}\biggr{]}}.$$
Taking the differential with respect to $(Z^{A})_{1}$, recognising the
derivative and setting that to zero, we have
$$(\vec{Z}^{A})_{1}^{\intercal}=\left[\sum_{t=1}^{T_{f}}\mathinner{\biggl{(}L_{1}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]L_{1}^{\intercal}+L_{4}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]L_{4}^{\intercal}\biggr{)}}\right]^{-1}\left[\sum_{t=1}^{T_{f}}\mathinner{\biggl{(}L_{1}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\mathring{\boldsymbol{y}}_{1,1}]+L_{4}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\mathring{\boldsymbol{y}}_{1,3N+1}]\biggr{)}}\right].$$
In a similar way, we can obtain the QMLE-res for other factor loadings.
We now find values of $\Sigma_{ee}$ to minimise $\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}$. We can show that
$$\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{1,t}\propto T_{f}\log|\Sigma_{ee}|+\operatorname*{tr}\mathinner{\bigl{[}C_{e}\Sigma_{ee}^{-1}\bigr{]}}=T_{f}\sum_{k=1}^{3N}2\log\sigma_{k}^{2}+\sum_{k=1}^{3N}\frac{C_{e,k,k}+C_{e,3N+k,3N+k}}{\sigma_{k}^{2}}$$
where $C_{e}\mathrel{\mathop{:}}=\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\mathring{\boldsymbol{y}}_{t}\mathring{\boldsymbol{y}}_{t}^{\intercal}-2\Lambda\vec{\mathbb{E}}[\boldsymbol{f}_{t}\mathring{\boldsymbol{y}}_{t}^{\intercal}]+\Lambda\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]\Lambda^{\intercal}\bigr{)}}$, and the single-index $\sigma_{k}^{2}$ is defined as $\sigma_{k}^{2}\mathrel{\mathop{:}}=\sigma_{\lceil\frac{k}{N}\rceil,k-\lfloor\frac{k}{N}\rfloor N}^{2}$.777Note that $k\mapsto(\lceil\frac{k}{N}\rceil,k-\lfloor\frac{k}{N}\rfloor N)$ is a bijection from $\mathinner{\left\{1,...,6N\right\}}$ to $\mathinner{\left\{1,...,6\right\}}\times\mathinner{\left\{1,...,N\right\}}$. Taking the derivative with respect to $\sigma_{k}^{2}$ and
setting that to zero, we have
$$\vec{\sigma}_{k}^{2}=\frac{1}{T_{f}}\frac{C_{e,k,k}+C_{e,3N+k,3N+k}}{2}.$$
We now give the formulas for $\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]$ and $\vec{\mathbb{E}}[\boldsymbol{f}_{t}\mathring{\boldsymbol{y}}_{t}^{\intercal}]$. We know that
$$\displaystyle\left(\begin{array}[c]{c}\boldsymbol{f}_{t}\\
\mathring{\boldsymbol{y}}_{t}\end{array}\right)\sim N\left(\left(\begin{array}[c]{c}\boldsymbol{0}\\
\boldsymbol{0}\end{array}\right)\left(\begin{array}[c]{cc}M&M\Lambda^{\intercal}\\
\Lambda M&\Sigma_{yy}\end{array}\right)\right).$$
Recall that in Section 3.4 we treat $\{\boldsymbol{f}_{t}\}_{t=1}^{T_{f}}$ as i.i.d. Thus, according to the conditional
distribution of the multivariate normal, we have
$$\displaystyle\mathbb{E}[\boldsymbol{f}_{t}|\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\vec{\boldsymbol{\theta}}^{(i)}]$$
$$\displaystyle=M\Lambda^{\intercal}\Sigma_{yy}^{-1}\mathring{\boldsymbol{y}}_{t}$$
$$\displaystyle\operatorname*{var}[\boldsymbol{f}_{t}|\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\vec{\boldsymbol{\theta}}^{(i)}]$$
$$\displaystyle=M-M\Lambda^{\intercal}\Sigma_{yy}^{-1}\Lambda M=\mathbb{E}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}|\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\vec{\boldsymbol{\theta}}^{(i)}]-\mathbb{E}[\boldsymbol{f}_{t}|\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\vec{\boldsymbol{\theta}}^{(i)}]\mathbb{E}[\boldsymbol{f}_{t}^{\intercal}|\{\mathring{\boldsymbol{y}}_{t}\}_{t=1}^{T_{f}};\vec{\boldsymbol{\theta}}^{(i)}].$$
We then can show that
$$\displaystyle\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\vec{\mathbb{E}}[\mathring{\boldsymbol{y}}_{t}\boldsymbol{f}_{t}^{\intercal}]$$
$$\displaystyle=S_{yy}\Sigma_{yy}^{-1}\Lambda M$$
$$\displaystyle\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]$$
$$\displaystyle=M-M\Lambda^{\intercal}\Sigma_{yy}^{-1}\Lambda M+M\Lambda^{\intercal}\Sigma_{yy}^{-1}S_{yy}\Sigma_{yy}^{-1}\Lambda M.$$
(A.76)
Next, we find values of $\phi$ to minimise $\vec{\mathbb{E}}\sum_{t=1}^{T_{f}}\vec{\ell}_{2,t}$. It is difficult to derive the analytical solution for
$\vec{\phi}$ so we will obtain $\vec{\phi}$ in a numerical way.
A.5 Computation of the QMLE
In this subsection, we provide a way to compute the QMLE defined in Section
3.2. Again we will rely on the EM algorithm.
(i)
In the E step, calculate $T_{f}^{-1}\sum_{t=1}^{T_{f}}\vec{\mathbb{E}}[\mathring{\boldsymbol{y}}_{t}\boldsymbol{f}_{t}^{\intercal}]$ and
$T_{f}^{-1}\sum_{t=1}^{T_{f}}\vec{\mathbb{E}}[\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}]$ as in (A.76).
(ii)
In the M step, obtain the factor loading estimates similar to those of
the QMLE-res by imposing the factor loading restrictions within the $14^{2}$
restrictions. We only have to change the selection matrices, say, $L_{1}$ and
$L_{4}$ defined in Appendix A.4, accordingly.
(iii)
Iterate steps (i) and (ii) until the estimates $\hat{\Lambda},\hat{\Sigma}_{ee},\hat{M}$ satisfy (3.14) reasonably well.
(iv)
Rotate the converged $\hat{\Lambda}$ and $\hat{M}$ so that the rotated
$\hat{\Lambda}$ and $\hat{M}$ satisfy all the $14^{2}$ restrictions. This
could only be done numerically. In particular, we define a distance function
which measures the distance between the restricted elements in $\Lambda$ and
$M$ and the corresponding elements in the rotated $\hat{\Lambda}$ and $\hat{M}$.
A.6 Proof of Proposition 4.1
As Bai and Li (2012) did, we use a superscript ”*” to denote the true
parameters, $\Lambda^{*},\Sigma_{ee}^{*},M^{*}$ etc. The parameters without
the superscript ”*” denote the generic parameters in the likelihood function.
Note that the proof of (4.2) is exactly the same as that of
Bai and Li (2012), so we omit the details here.
Define
$$\displaystyle\hat{H}$$
$$\displaystyle\mathrel{\mathop{:}}=(\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda})^{-1}$$
$$\displaystyle A$$
$$\displaystyle\mathrel{\mathop{:}}=(\hat{\Lambda}-\Lambda^{\ast})^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}$$
(A.77)
$$\displaystyle K$$
$$\displaystyle\mathrel{\mathop{:}}=\hat{M}^{-1}(MA-A^{\intercal}MA).$$
Our assumptions satisfy those of Bai and Li (2012), so (A17) of
Bai and Li (2012) still holds (in our notation):
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}^{*}=K\boldsymbol{\lambda}_{k,j}^{*}+o_{p}(1),$$
(A.78)
for $k=1,\ldots,6$ and $j=1,\ldots,N$. As mentioned before, $\Lambda^{\ast}$
(i.e., $\{\boldsymbol{\lambda}_{k,j}^{\ast}\mathrel{\mathop{:}}k=1,\ldots,6,j=1,\ldots,N\}$)
defined (2.14) gives more than $14^{2}$ restrictions, but in order to
utilise the theories of Bai and Li (2012) we shall only impose $14^{2}$
restrictions on $\{\hat{\boldsymbol{\lambda}}_{k,j}\mathrel{\mathop{:}}k=1,\ldots,6,j=1,\ldots,N\}$. How to select these $14^{2}$ restrictions from those implied by $\{\boldsymbol{\lambda}_{k,j}^{\ast}\}$ are crucial because we cannot afford
imposing a restriction which is not instrumental for the proofs later. The
idea is that one restriction should pin down one free parameter in $K$. We
shall now explain our procedure. Write (A.78) in matrix form:
$$\displaystyle\hat{\Lambda}_{k}-\Lambda_{k}^{\ast}=K\Lambda_{k}^{\ast}+o_{p}(1),$$
(A.79)
where $\hat{\Lambda}_{k}\mathrel{\mathop{:}}=(\hat{\boldsymbol{\lambda}}_{k,1},\ldots,\hat{\boldsymbol{\lambda}}_{k,N})$ and $\Lambda_{k}^{\ast}\mathrel{\mathop{:}}=(\boldsymbol{\lambda}_{k,1}^{\ast},\ldots,\boldsymbol{\lambda}_{k,N}^{\ast})$ are $14\times N$ matrices. For a generic matrix $C$, let $C_{x,y}$ denote
the matrix obtained by intersecting the rows and columns whose indices are in
$x$ and $y$, respectively; let $C_{x,\bullet}$ denote the matrix obtained by
extracting the rows whose indices are in $x$ while $C_{\bullet,y}$ denote the
matrix obtained by extracting the columns whose indices are in $y$.
A.6.1 Step I Impose Some Zero Restrictions in $\{\Lambda^{*}_{k}\}_{k=1}^{6}$
Let $a\subset\{1,2,\ldots,14\}$ and $c\subset\{1,\ldots,N\}$ be two vectors of
indices, whose identities vary from place to place. From (A.79), we have
$$\displaystyle\mathinner{\bigl{(}\hat{\Lambda}_{k}-\Lambda_{k}^{*}\bigr{)}}_{a,c}$$
$$\displaystyle=K_{a,\bullet}\Lambda_{k,\bullet,c}^{\ast}+o_{p}(1)=K_{a,b}\Lambda_{k,b,c}^{\ast}+K_{a,-b}\Lambda_{k,-b,c}^{\ast}+o_{p}(1)=K_{a,b}\Lambda_{k,b,c}^{\ast}+o_{p}(1)$$
$$\displaystyle=K_{a,b_{1}}\Lambda_{k,b_{1},c}^{\ast}+K_{a,b_{2}}\Lambda_{k,b_{2},c}^{\ast}+o_{p}(1)$$
(A.80)
where $b\subset\{1,2,\ldots,14\}$ is chosen in such a way such that
$\Lambda_{k,-b,c}^{\ast}=0$ for each of the steps below, $-b$ denotes the
complement of $b$, and $b_{1}\cup b_{2}=b$ with the cardinality of $b_{2}$
equal to the cardinality of $c$.
In each of the sub-step of step I, we shall impose $\hat{\Lambda}_{k,a,c}=0$.
Step I is detailed in Table 9, and we shall use
step I.1 to illustrate. For step I.1, $\hat{\Lambda}_{k,a,c}=0$ means
$$\left[\begin{array}[c]{c}\hat{\boldsymbol{\lambda}}_{1,1}^{\intercal}\\
\hat{\boldsymbol{\lambda}}_{1,2}^{\intercal}\\
\hat{\boldsymbol{\lambda}}_{1,3}^{\intercal}\\
\hat{\boldsymbol{\lambda}}_{1,4}^{\intercal}\\
\end{array}\right]=\left[\begin{array}[c]{cccccccccccccc}0&0&0&0&0&-&-&-&0&0&0&0&0&-\\
0&0&0&0&0&-&-&-&0&0&0&0&0&-\\
0&0&0&0&0&-&-&-&0&0&0&0&0&-\\
0&0&0&0&0&-&-&-&0&0&0&0&0&-\end{array}\right].$$
This means (A.80) holds with LHS being
$\mathinner{\bigl{(}\hat{\Lambda}_{k}-\Lambda_{k}^{*}\bigr{)}}_{a,c}=0$, where
$$\displaystyle k=1,\quad c=\{1,2,3,4\},\quad a=\{1,2,3,4,5,9,10,11,12,13\},\quad b_{1}=\emptyset,\quad b_{2}=\{6,7,8,14\}.$$
Note that $c=\{1,2,3,4\}$ is arbitrary and could be replaced with any other
$c\subset\{1,\ldots,N\}$ with cardinality being 4. The crucial point is that
$c$ needs to be chosen such that $\Lambda_{k,b_{2},c}^{\ast}$ is invertible.
This is an innocuous requirement given large $N$, so we shall make this
assumption implicitly for the rest of the article. Solving (A.80) gives
$K_{a,b_{2}}=o_{p}(1)$.
A.6.2 Step II Impose Some Equality Restrictions in $\{\Lambda^{*}_{k}\}_{k=1}^{6}$
(II.1)
Note that (A.79) implies
$$\displaystyle\hat{\Lambda}_{6,9,c}-\Lambda_{6,9,c}^{\ast}$$
$$\displaystyle=K_{9,x}\Lambda_{6,x,c}^{\ast}+o_{p}(1)=K_{9,1}\Lambda_{6,1,c}^{\ast}+K_{9,9}\Lambda_{6,9,c}^{\ast}+o_{p}(1)\qquad x=\{1,2,3,9\}$$
$$\displaystyle\hat{\Lambda}_{3,12,c}-\Lambda_{3,12,c}^{\ast}$$
$$\displaystyle=K_{12,y}\Lambda_{3,y,c}^{\ast}+o_{p}(1)=K_{12,12}\Lambda_{3,12,c}^{\ast}+o_{p}(1)\qquad y=\{4,5,6,12\}.$$
We then impose $\hat{\Lambda}_{6,9,c}=\hat{\Lambda}_{3,12,c}$ for $c=\{1,2\}$.
The preceding display implies
$$K_{9,1}\Lambda_{6,1,c}^{\ast}+\left(K_{9,9}-K_{12,12}\right)\Lambda_{6,9,c}^{\ast}=o_{p}(1)$$
whence we have $K_{9,1}=o_{p}(1)$ and $K_{9,9}-K_{12,12}=o_{p}(1).$
(II.2)
We impose $\hat{\Lambda}_{4,11,c}=\hat{\Lambda}_{1,14,c}$ for
$c=\{1,2\}$. Repeating the procedure in step II.1, we have $K_{14,8}=o_{p}(1)$
and $K_{11,11}-K_{14,14}=o_{p}(1)$ in the same way.
(II.3)
We impose $\hat{\Lambda}_{6,1,c}=\hat{\Lambda}_{3,4,c}$ for
$c=\{1,2\}$, and have $K_{1,1}-K_{4,4}=o_{p}(1),K_{1,9}-K_{4,12}=o_{p}(1).$
(II.4)
We impose $\hat{\Lambda}_{6,2,c}=\hat{\Lambda}_{3,5,c}$ for
$c=\{1,2,3\}$, and have $K_{2,1}=o_{p}(1),K_{2,2}-K_{5,5}=o_{p}(1),K_{2,9}-K_{5,12}=o_{p}(1).$
(II.5)
We impose $\hat{\Lambda}_{6,3,c}=\hat{\Lambda}_{3,6,c}$ for
$c=\{1,2,3,4\}$, and have $K_{3,1}=o_{p}(1),K_{3,2}=o_{p}(1),K_{3,3}-K_{6,6}=o_{p}(1),K_{3,9}-K_{6,12}=o_{p}(1).$
(II.6)
We impose $\hat{\Lambda}_{1,8,c}=\hat{\Lambda}_{4,5,c}$ for
$c=\{1,2\}$, and have $K_{8,8}-K_{5,5}=o_{p}(1),K_{8,14}-K_{5,11}=o_{p}(1).$
(II.7)
We impose $\hat{\Lambda}_{1,7,c}=\hat{\Lambda}_{4,4,c}$ for
$c=\{1,2,3\}$, and have $K_{7,8}=o_{p}(1),K_{7,7}-K_{4,4}=o_{p}(1),K_{7,14}-K_{4,11}=o_{p}(1).$
(II.8)
We impose $\hat{\Lambda}_{1,6,c}=\hat{\Lambda}_{4,3,c}$ for
$c=\{1,2,3\}$, and have $K_{6,7}=o_{p}(1),K_{6,8}=o_{p}(1),K_{6,14}-K_{3,11}=o_{p}(1).$
A.6.3 Step III Impose Some Restrictions in $M^{*}$
After steps I and II, $K$ is reduced to
$$K=\left[\begin{array}[c]{cc}\overline{K}_{11}&\overline{K}_{12}\\
0&\overline{K}_{22}\end{array}\right]+o_{p}(1),$$
where
$$\displaystyle\overline{K}_{11}=\left[\begin{array}[c]{cccccccc}K_{1,1}&0&0&0&0&0&0&0\\
0&K_{2,2}&0&0&0&0&0&0\\
0&0&K_{3,3}&0&0&0&0&0\\
0&0&0&K_{1,1}&0&0&0&0\\
0&0&0&0&K_{2,2}&0&0&0\\
0&0&0&0&0&K_{3,3}&0&0\\
0&0&0&0&0&0&K_{1,1}&0\\
0&0&0&0&0&0&0&K_{2,2}\end{array}\right]$$
(A.89)
$$\overline{K}_{12}=\left[\begin{array}[c]{cccccc}K_{4,12}&0&0&0&0&0\\
K_{5,12}&K_{2,10}&0&0&0&0\\
K_{6,12}&K_{3,10}&K_{3,11}&0&0&0\\
0&K_{4,10}&K_{4,11}&K_{4,12}&0&0\\
0&0&K_{5,11}&K_{5,12}&K_{5,13}&0\\
0&0&0&K_{6,12}&K_{6,13}&K_{3,11}\\
0&0&0&0&K_{7,13}&K_{4,11}\\
0&0&0&0&0&K_{5,11}\end{array}\right],\overline{K}_{22}=\left[\begin{array}[c]{cccccc}K_{12,12}&0&0&0&0&0\\
0&K_{10,10}&0&0&0&0\\
0&0&K_{11,11}&0&0&0\\
0&0&0&K_{12,12}&0&0\\
0&0&0&0&K_{13,13}&0\\
0&0&0&0&0&K_{11,11}\end{array}\right].$$
In the paragraph above (A16) of Bai and Li (2012), they showed $A=O_{p}(1)$.
Given Assumption 4.1(iii), we have
$K=O_{p}(1)$. Next, (A16) of Bai and Li (2012) still holds and could be written
as
$$\displaystyle\hat{M}-(I_{14}-A^{\intercal})M^{\ast}(I_{14}-A)=o_{p}(1).$$
(A.90)
Since $M^{\ast}$ and $\hat{M}^{\ast}$ are of full rank (Assumption
4.1(iii)), (A.90) implies that $I_{14}-A$
is of full rank. Write (A.90) as
$$\hat{M}(K+I_{14})-(I_{14}-A^{\intercal})M^{\ast}=o_{p}(1).$$
As Bai and Li (2012) did in their (A20), we could premultiply the preceding
display by $\mathinner{\bigl{[}(I_{14}-A^{\intercal})M^{*}\bigr{]}}^{-1}$ to arrive at (after some
algebra and relying on (A.90)):
$$\displaystyle(I_{14}-A)(K+I_{14})-I_{14}=o_{p}(1).$$
(A.91)
Likewise, partition $A$ into $8\times 8$, $8\times 6$, $6\times 8$ and $6\times 6$
submatrices:
$$A=\left[\begin{array}[c]{cc}A_{11}&A_{12}\\
A_{21}&A_{22}\end{array}\right].$$
Then (A.91) could be written into
$$\displaystyle(I_{8}-A_{11})(\overline{K}_{11}+I_{8})-I_{8}$$
$$\displaystyle=o_{p}(1)$$
(A.92)
$$\displaystyle(I_{8}-A_{11})\overline{K}_{12}-A_{12}(\overline{K}_{22}+I_{6})$$
$$\displaystyle=o_{p}(1)$$
(A.93)
$$\displaystyle-A_{21}(\overline{K}_{11}+I_{8})$$
$$\displaystyle=o_{p}(1)$$
(A.94)
$$\displaystyle-A_{21}\overline{K}_{12}+(I_{6}-A_{22})(\overline{K}_{22}+I_{6})-I_{6}$$
$$\displaystyle=o_{p}(1).$$
(A.95)
Consider (A.92) first. Since $I_{8}+\overline{K}_{11}$ is diagonal, we
deduce that the diagonal elements of $I_{8}+\overline{K}_{11}$ could not
converge to 0, and $A_{11}$ converges to a diagonal matrix. Using the fact
that the diagonal elements of $I_{8}+\overline{K}_{11}$ could not converge to
0, (A.94) implies
$$A_{21}=o_{p}(1),$$
and $A_{21}\overline{K}_{12}=o_{p}(1)O_{p}(1)=o_{p}(1)$. Then (A.95) is
reduced to
$$(I_{6}-A_{22})(\overline{K}_{22}+I_{6})-I_{6}=o_{p}(1).$$
Since $\overline{K}_{22}+I_{6}$ is diagonal, we deduce that the diagonal
elements of $\overline{K}_{22}+I_{6}$ could not converge to 0, and $A_{22}$
should converge to a diagonal matrix as well. To sum up
$$I_{8}+\overline{K}_{11}\qquad I_{8}-A_{11}\qquad I_{6}+\overline{K}_{22}\qquad I_{6}-A_{22}$$
are diagonal or diagonal in the limit, and invertible in the limit (i.e., none
of the diagonal elements is zero in the limit). Moreover, (A.92) implies
$$\displaystyle(I_{8}+\overline{K}_{11})^{-1}=(I_{8}-A_{11})+o_{p}(1).$$
(A.96)
Via (A.91), (A.90) implies
$$\displaystyle(I_{14}+K^{\intercal})\hat{M}(I_{14}+K)-M^{\ast}=o_{p}(1).$$
(A.97)
Partition $\hat{M}$ into $8\times 8$, $8\times 6$, $6\times 8$ and $6\times 6$
submatrices:
$$\hat{M}=\left[\begin{array}[c]{cc}\hat{\overline{M}}_{11}&\hat{\overline{M}}_{12}\\
\hat{\overline{M}}_{21}&\hat{\overline{M}}_{22}\end{array}\right].$$
Then (A.97) could be written as
$$\displaystyle(I_{8}+\overline{K}_{11})\hat{\overline{M}}_{11}(I_{8}+\overline{K}_{11})-\Phi^{\ast}=o_{p}(1)$$
(A.98)
$$\displaystyle(I_{8}+\overline{K}_{11})\hat{\overline{M}}_{11}\overline{K}_{12}+(I_{8}+\overline{K}_{11})\hat{\overline{M}}_{12}(I_{6}+\overline{K}_{22})=o_{p}(1)$$
(A.99)
$$\displaystyle\mathinner{\bigl{[}\overline{K}_{12}^{\intercal}\hat{\overline{M}}_{11}+(I_{6}+\overline{K}_{22})\hat{\overline{M}}_{21}\bigr{]}}(\overline{K}_{11}+I_{8})=o_{p}(1)$$
(A.100)
$$\displaystyle\overline{K}_{12}^{\intercal}\mathinner{\bigl{(}\hat{\overline{M}}_{12}(I_{6}+\overline{K}_{22})+\hat{\overline{M}}_{11}\overline{K}_{12}\bigr{)}}+(I_{6}+\overline{K}_{22})\mathinner{\bigl{(}\hat{\overline{M}}_{22}(I_{6}+\overline{K}_{22})+\hat{\overline{M}}_{21}\overline{K}_{12}\bigr{)}}-I_{6}=o_{p}(1).$$
(A.101)
Step III.1 Considering (A.98), we have
$$\displaystyle(1+K_{3,3})^{2}\hat{M}_{6,6}$$
$$\displaystyle=\frac{1}{1-\phi^{*,2}}+o_{p}(1)$$
$$\displaystyle(1+K_{1,1})^{2}\hat{M}_{4,4}$$
$$\displaystyle=\frac{1}{1-\phi^{*,2}}+o_{p}(1)$$
$$\displaystyle(1+K_{2,2})^{2}\hat{M}_{5,5}$$
$$\displaystyle=\frac{1}{1-\phi^{*,2}}+o_{p}(1).$$
Imposing $\hat{M}_{4,4}=\hat{M}_{6,6}$, we have
$$\displaystyle\hat{M}_{4,4}\mathinner{\bigl{[}(1+K_{1,1})^{2}-(1+K_{3,3})^{2}\bigr{]}}=o_{p}(1).$$
Since (A.98) implies that $\hat{M}_{4,4}\neq o_{p}(1)$. The preceding
display implies
$$\displaystyle K_{3,3}=K_{1,1}+o_{p}(1),\qquad\text{or}\qquad 1+K_{3,3}=-(1+K_{1,1})+o_{p}(1).$$
Likewise, imposing $\hat{M}_{4,4}=\hat{M}_{5,5}$, we have
$$\displaystyle K_{2,2}=K_{1,1}+o_{p}(1),\qquad\text{or}\qquad 1+K_{2,2}=-(1+K_{1,1})+o_{p}(1).$$
Thus, there are four cases:
(a)
$K_{2,2}=K_{1,1}+o_{p}(1)$ and $K_{3,3}=K_{1,1}+o_{p}(1)$
(b)
$K_{2,2}=K_{1,1}+o_{p}(1)$ and $1+K_{3,3}=-(1+K_{1,1})+o_{p}(1)$
(c)
$1+K_{2,2}=-(1+K_{1,1})+o_{p}(1)$ and $K_{3,3}=K_{1,1}+o_{p}(1)$
(d)
$1+K_{2,2}=-(1+K_{1,1})+o_{p}(1)$ and $1+K_{3,3}=-(1+K_{1,1})+o_{p}(1)$.
Irrespective of case, (A.98) is reduced to $(1+K_{1,1})^{2}\hat{\overline{M}}_{11}=\Phi^{\ast}+o_{p}(1)$, whence we have
$$\displaystyle(1+K_{1,1})^{2}\hat{M}_{4,4}$$
$$\displaystyle=\frac{1}{1-\phi^{*,2}}+o_{p}(1)$$
$$\displaystyle(1+K_{1,1})^{2}\hat{M}_{6,4}$$
$$\displaystyle=\frac{\phi^{*,2}}{1-\phi^{*,2}}+o_{p}(1).$$
Imposing $\hat{M}_{4,4}-\hat{M}_{6,4}=1$, we have
$$(1+K_{1,1})^{2}=1+o_{p}(1)$$
whence we have $K_{1,1}=o_{p}(1)$ or $1+K_{1,1}=-1+o_{p}(1)$. Suppose that
$1+K_{1,1}=-1+o_{p}(1)$. Then (A.96) implies $(A_{11})_{1,1}=2+o_{p}(1)$.
Note that the identification scheme which we employ in Proposition
4.1 only identifies $\Lambda^{\ast}$ up to a column sign change.
Thus by assuming that $\hat{\Lambda}$ and $\Lambda^{\ast}$ have the same
column signs, we can rule out the case $(A_{11})_{1,1}=2+o_{p}(1)$
(Bai and Li (2012, p.445, p.463)). Thus we have $K_{1,1}=o_{p}(1)$ and hence
rule out cases (b)-(d). To sum up, we have
$$K_{1,1}=o_{p}(1),\quad\overline{K}_{11}=o_{p}(1),\quad\hat{\overline{M}}_{11}=\Phi^{\ast}+o_{p}(1),\quad A_{11}=o_{p}(1).$$
Step III.2 Now (A.99) is reduced to
$$\hat{\overline{M}}_{12}(I_{6}+\overline{K}_{22})=-\Phi^{\ast}\overline{K}_{12}+o_{p}(1)$$
Impose three more restrictions: Assume the 4th-6th elements of the fourth
column of $\hat{\overline{M}}_{12}$ are zero; that is $\hat{M}_{4,12}=\hat{M}_{5,12}=\hat{M}_{6,12}=0$. This implies that the corresponding three
elements of $\Phi_{12}^{\ast}\overline{K}$ are $o_{p}(1)$:
$$\frac{1}{1-\phi^{\ast 2}}\left[\begin{array}[c]{c}K_{4,12}+\phi^{\ast}K_{5,12}+\phi^{\ast 2}K_{6,12}\\
\phi^{\ast}K_{4,12}+K_{5,12}+\phi^{\ast}K_{6,12}\\
\phi^{\ast 2}K_{4,12}+\phi^{\ast}K_{5,12}+K_{6,12}\end{array}\right]=o_{p}(1)$$
whence we have $K_{4,12}=K_{5,12}=K_{6,12}=o_{p}(1)$. Similarly, assuming the
2nd-4th elements of the second column of $\hat{\overline{M}}_{12}$ are zero,
we could deduce that $K_{2,10}=K_{3,10}=K_{4,10}=o_{p}(1)$; assuming the
3rd-5th elements of the third column of $\hat{\overline{M}}_{12}$ are zero, we
could deduce that $K_{3,11}=K_{4,11}=K_{5,11}=o_{p}(1)$; assuming $\hat{M}_{5,13}=\hat{M}_{6,13}=\hat{M}_{7,13}=0,$we could deduce that
$K_{5,13}=K_{6,13}=K_{7,13}=o_{p}(1).$ We hence obtain
$$\overline{K}_{12}=o_{p}(1).$$
Step III.3 With $\overline{K}_{12}=o_{p}(1),$ (A.101) is reduced
to
$$(I_{6}+\overline{K}_{22})\hat{\overline{M}}_{22}(I_{6}+\overline{K}_{22})-I_{6}=o_{p}(1).$$
Since $I_{6}+\overline{K}_{22}$ is diagonal, $\hat{\overline{M}}_{22}$ is
asymptotically diagonal. Imposing that the 2nd-5th diagonal elements of
$\hat{\overline{M}}_{22}$ are 1 (i.e., $\hat{M}_{j,j}=1$ for $j=10,11,12,13$),
we have
$$\displaystyle(1+K_{10,10})^{2}-1$$
$$\displaystyle=o_{p}(1)$$
$$\displaystyle(1+K_{11,11})^{2}-1$$
$$\displaystyle=o_{p}(1)$$
$$\displaystyle(1+K_{12,12})^{2}-1$$
$$\displaystyle=o_{p}(1)$$
$$\displaystyle(1+K_{13,13})^{2}-1$$
$$\displaystyle=o_{p}(1)$$
whence we have $K_{j,j}=o_{p}(1)$ or $1+K_{j,j}=-1+o_{p}(1)$ for
$j=10,11,12,13$. Likewise, the case $1+K_{j,j}=-1+o_{p}(1)$ is ruled out.
Thus
$$\overline{K}_{22}=o_{p}(1),\quad A_{22}=o_{p}(1),\quad\hat{\overline{M}}_{22}=I_{6}+o_{p}(1).$$
Then (A.99) and (A.100) imply $\hat{\overline{M}}_{12}=o_{p}(1)$ and
$\hat{M}_{21}=o_{p}(1)$, respectively. Also (A.93) implies $A_{12}=o_{p}(1)$. To sum up, we have
$$K=o_{p}(1),\qquad A=o_{p}(1),\qquad\hat{M}=M^{\ast}+o_{p}(1).$$
Substituting $K=o_{p}(1)$ into (A.78), we have
$$\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}^{\ast}=o_{p}(1),$$
for $k=1,\ldots,6$ and $j=1,\ldots,N$.
A.7 Proof of Theorem 4.1
Given consistency, we can drop the superscript from the true parameters for
simplicity. The proof of Theorem 4.1 resembles that of Theorem 5.1 of
Bai and Li (2012). Most of the proof of Theorem 5.1 of Bai and Li (2012) is
insensitive to the identification condition; the only exception is their Lemma
B5. Thus we only need to prove the result of their Lemma B5 under our
identification condition. That is, we want to prove
$$\displaystyle MA$$
$$\displaystyle=O_{p}(T_{f}^{-1/2})+O_{p}\mathinner{\biggl{(}\mathinner{\biggl{[}\frac{1}{6N}\sum_{k=1}^{6}\sum_{j=1}^{N}(\hat{\sigma}_{k,j}^{2}-\sigma_{k,j}^{2})^{2}\biggr{]}}^{1/2}\biggr{)}}=\mathrel{\mathop{:}}O_{p}(\Diamond).$$
(A.102)
Following the approach of Bai and Li (2012) and using their Lemmas B1, B2, B3,
one could show that
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle=K\boldsymbol{\lambda}_{k,j}+O_{p}(\Diamond).$$
Using the same approach we adopted in the proof of Proposition 4.1,
we arrive at
$$\displaystyle K=\left[\begin{array}[c]{cc}\overline{K}_{11}&\overline{K}_{12}\\
0&\overline{K}_{22}\end{array}\right]+O_{p}(\Diamond),$$
where $\overline{K}_{11}$, $\overline{K}_{12}$, $\overline{K}_{22}$ are
defined in (A.89), and
$$\displaystyle\hat{M}-(I_{14}-A^{\intercal})M(I_{14}-A)$$
$$\displaystyle=O_{p}(\Diamond)$$
(A.103)
$$\displaystyle(I_{14}-A)(K+I_{14})-I_{14}$$
$$\displaystyle=O_{p}(\Diamond)$$
(A.104)
Note that $\sqrt{1+O_{p}(\Diamond)}=1+O_{p}(\Diamond)$ and $(1+O_{p}(\Diamond))^{-1}=1+O_{p}(\Diamond)$ because of the generalised Binomial
theorem and that $O_{p}(\Diamond)=o_{p}(1)$. Then one could repeat the
argument in the proof of Proposition 4.1 to arrive at
$$\displaystyle K$$
$$\displaystyle=O_{p}(\Diamond),\qquad A=O_{p}(\Diamond),\qquad\hat{M}=M+O_{p}(\Diamond)$$
(A.105)
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle=O_{p}(\Diamond),\qquad MA=O_{p}(\Diamond)$$
(A.106)
for $k=1,\ldots,6$ and $j=1,\ldots,N$.
A.8 Proof of Theorem 4.2
Equation (C4) of the supplement of Bai and Li (2012) still holds in our case
since its derivation does not involve identification conditions (page 17 of
the supplement of Bai and Li (2012)); in our notation it is
$$\hat{\sigma}_{m}^{2}-\sigma_{m}^{2}=\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}(e_{m,t}^{2}-\sigma_{m}^{2})+o_{p}(T_{f}^{-1/2})$$
for $m=1,\ldots,6N$, where the single-index $\sigma^{2}_{m}$ is defined as
$\sigma^{2}_{m}\mathrel{\mathop{:}}=\sigma^{2}_{\lceil\frac{m}{N}\rceil,m-\lfloor\frac{m}{N}\rfloor N}$; interpret $\hat{\sigma}^{2}_{m},e_{m,t}$ similarly.
Thus theorem 4.2 follows.
A.9 Proof of Theorem 4.3
Pre-multiply $\hat{M}$ to (A14) of Bai and Li (2012) and write in our
notation:
$$\displaystyle\hat{M}\mathinner{\bigl{(}\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}\bigr{)}}=M(\hat{\Lambda}-\Lambda)^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle-\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\hat{\Lambda}-\Lambda)M(\hat{\Lambda}-\Lambda)^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle-\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}\boldsymbol{e}_{t}^{\intercal}\biggr{)}}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle-\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{e}_{t}\boldsymbol{f}_{t}^{\intercal}\biggr{)}}\Lambda^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle-\hat{H}\mathinner{\biggl{(}\sum_{m=1}^{6N}\sum_{\ell=1}^{6N}\frac{1}{\hat{\sigma}_{m}^{2}\hat{\sigma}_{\ell}^{2}}\hat{\lambda}_{m}\hat{\lambda}_{\ell}^{\intercal}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\left[{e}_{m,t}e_{\ell,t}-\mathbb{E}({e}_{m,t}e_{\ell,t})\right]\biggr{)}}\hat{H}\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle+\hat{H}\mathinner{\biggl{(}\sum_{m=1}^{6N}\frac{1}{\hat{\sigma}_{m}^{4}}\hat{\lambda}_{m}\hat{\lambda}_{m}^{\intercal}(\hat{\sigma}_{m}^{2}-\sigma_{m}^{2})\biggr{)}}\hat{H}\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{e}_{t}\boldsymbol{f}_{t}^{\intercal}\biggr{)}}\boldsymbol{\lambda}_{k,j}+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}$$
$$\displaystyle+\hat{H}\mathinner{\biggl{(}\sum_{i=1}^{6N}\frac{1}{\hat{\sigma}_{m}^{2}}\hat{\lambda}_{m}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\left[{e}_{m,t}e_{(k-1)N+j,t}-\mathbb{E}({e}_{m,t}e_{(k-1)N+j,t})\right]\biggr{)}}-\hat{H}\hat{\boldsymbol{\lambda}}_{k,j}\frac{1}{\hat{\sigma}_{k,j}^{2}}(\hat{\sigma}_{k,j}^{2}-\sigma_{k,j}^{2}),$$
(A.107)
where $e_{i,t}$ denotes the $i$th element of $\boldsymbol{e}_{t}$, the
single-index $\boldsymbol{\lambda}_{m}$ is defined as $\boldsymbol{\lambda}_{m}\mathrel{\mathop{:}}=\boldsymbol{\lambda}_{\lceil\frac{m}{N}\rceil,m-\lfloor\frac{m}{N}\rfloor N}$; interpret $\hat{\boldsymbol{\lambda}}_{m},\sigma^{2}_{m},\hat{\sigma}^{2}_{m}$ similarly.
Consider the right hand side of (A.107). The third and fourth terms are
$o_{p}(T_{f}^{-1/2})$ by Lemma C1(e) of Bai and Li (2012). The fifth term is
$o_{p}(T_{f}^{-1/2})$ by Lemma C1(d) of Bai and Li (2012). The six term is
$o_{p}(T_{f}^{-1/2})$ by Lemma C1(f) of Bai and Li (2012). The seventh term is
$o_{p}(T_{f}^{-1/2})$ by Lemma C1(e) of Bai and Li (2012). The ninth term is
$o_{p}(T_{f}^{-1/2})$ by Lemma C1(c) of Bai and Li (2012). The tenth term is
$o_{p}(T_{f}^{-1/2})$ by Theorem 4.2. Thus (A.107) becomes
$$\displaystyle\hat{M}\mathinner{\bigl{(}\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}\bigr{)}}=MA\boldsymbol{\lambda}_{k,j}-A^{\intercal}MA\boldsymbol{\lambda}_{k,j}+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}+o_{p}(T_{f}^{-1/2}).$$
Substituting (4.4) into (A.102), we have $O_{p}(\Diamond)=O_{p}(T_{f}^{-1/2})$. Thus $A=O_{p}(T_{f}^{-1/2})$ via (A.106). The
preceding display hence becomes
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}=\hat{M}^{-1}MA\boldsymbol{\lambda}_{k,j}+\hat{M}^{-1}\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}+o_{p}(T_{f}^{-1/2}).$$
(A.108)
Note that
$$\displaystyle\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda=(\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda})^{-1}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda=(\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda})^{-1}\mathinner{\bigl{[}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}+\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\Lambda-\hat{\Lambda})\bigr{]}}$$
$$\displaystyle=I_{14}+(\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda})^{-1}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\Lambda-\hat{\Lambda})=I_{14}+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\Lambda-\hat{\Lambda})=I_{14}+O_{p}(T_{f}^{-1/2})$$
where the last equality is due to Lemma C1(a) of Bai and Li (2012).
Substituting the preceding display into (A.108), we have
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}$$
$$\displaystyle=\hat{M}^{-1}MA\boldsymbol{\lambda}_{k,j}+\hat{M}^{-1}\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}+\hat{M}^{-1}O_{p}(T_{f}^{-1/2})\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle=A\boldsymbol{\lambda}_{k,j}+(\hat{M}^{-1}-M^{-1})MA\boldsymbol{\lambda}_{k,j}+M^{-1}\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}$$
$$\displaystyle\qquad+(\hat{M}^{-1}-M^{-1})\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}+\hat{M}^{-1}O_{p}(T_{f}^{-1})+o_{p}(T_{f}^{-1/2}).$$
Given $M^{-1}=O(1)$ and $\hat{M}-M=O_{p}(T_{f}^{-1/2})$, we have $\hat{M}^{-1}-M^{-1}=O_{p}(T_{f}^{-1/2})$ and $\hat{M}^{-1}=O_{p}(1)$ (Lemma B4 of
Linton and
Tang (2021)). Hence the preceding display becomes
$$\displaystyle\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j}=A\boldsymbol{\lambda}_{k,j}+M^{-1}\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\biggr{)}}+o_{p}(T_{f}^{-1/2}).$$
Write in matrix form:
$$\displaystyle\hat{\Lambda}_{k}-\Lambda_{k}=A\Lambda_{k}+F_{k}+o_{p}(T_{f}^{-1/2}),$$
(A.109)
where $\hat{\Lambda}_{k}\mathrel{\mathop{:}}=(\hat{\lambda}_{k,1},\ldots,\hat{\lambda}_{k,N})$,
$\Lambda_{k}\mathrel{\mathop{:}}=(\lambda_{k,1},\ldots,\lambda_{k,N})$ and
$$\displaystyle F_{k}\mathrel{\mathop{:}}=M^{-1}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}\mathinner{\bigl{[}e_{(k-1)N+1,t},e_{(k-1)N+2,t},\ldots,e_{kN,t}\bigr{]}}$$
are $14\times N$ matrices. We are going to solve (A.109) for $A$ in terms
of known elements. The idea is exactly the same as that of the proof of
Proposition 4.1.
A.9.1 Step I Impose Some Zero Restrictions in $\{\Lambda_{k}\}_{k=1}^{6}$
Let $a\subset\{1,2,\ldots,14\}$ and $c\subset\{1,\ldots,N\}$ be two vectors of
indices, whose identities vary from place to place. From (A.109), we have
$$\displaystyle\mathinner{\bigl{(}\hat{\Lambda}_{k}-\Lambda_{k}\bigr{)}}_{a,c}=A_{a,\bullet}\Lambda_{k,\bullet,c}+F_{k,a,c}+o_{p}(T_{f}^{-1/2})=A_{a,b}\Lambda_{k,b,c}+A_{a,-b}\Lambda_{k,-b,c}+F_{k,a,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle=A_{a,b}\Lambda_{k,b,c}+F_{k,a,c}+o_{p}(T_{f}^{-1/2})=A_{a,b_{1}}\Lambda_{k,b_{1},c}+A_{a,b_{2}}\Lambda_{k,b_{2},c}+F_{k,a,c}+o_{p}(T_{f}^{-1/2})$$
(A.110)
where $b\subset\{1,2,\ldots,14\}$ is chosen in such a way such that
$\Lambda_{k,-b,c}=0$ for each of the sub-steps of Step I, $-b$ denotes the
complement of $b$, and $b_{1}\cup b_{2}=b$ with the cardinality of $b_{2}$
equal to the cardinality of $c$ and $A_{a,b_{1}}$ containing solved elements
for each of the sub-steps of Step I. Step I is again detailed in Table
9. Imposing $\mathinner{\bigl{(}\hat{\Lambda}_{k}\bigr{)}}_{a,c}=0$
and solving (A.110) gives
$$\displaystyle A_{a,b_{2}}=-\mathinner{\Bigl{(}A_{a,b_{1}}\Lambda_{k,b_{1},c}+F_{k,a,c}\Bigr{)}}\Lambda_{k,b_{2},c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(A.111)
A.9.2 Step II Impose $\hat{M}_{j,j}=1$ for $j=10,11,12,13$
Consider (A13) of Bai and Li (2012) and write in our notation:
$$\displaystyle\hat{M}-M$$
$$\displaystyle=-\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\hat{\Lambda}-\Lambda)M-M(\hat{\Lambda}-\Lambda)^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}$$
$$\displaystyle\qquad+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\hat{\Lambda}-\Lambda)M(\hat{\Lambda}-\Lambda)^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}$$
$$\displaystyle\qquad+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\Lambda\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}\boldsymbol{e}_{t}^{\intercal}\biggr{)}}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}$$
$$\displaystyle\qquad+\hat{H}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\mathinner{\biggl{(}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{e}_{t}\boldsymbol{f}_{t}^{\intercal}\biggr{)}}\Lambda^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\hat{H}$$
$$\displaystyle\qquad+\hat{H}\mathinner{\biggl{(}\sum_{m=1}^{6N}\sum_{\ell=1}^{6N}\frac{1}{\hat{\sigma}_{m}^{2}\hat{\sigma}_{\ell}^{2}}\hat{\lambda}_{m}\hat{\lambda}_{\ell}^{\intercal}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\left[{e}_{m,t}e_{\ell,t}-\mathbb{E}({e}_{m,t}e_{\ell,t})\right]\biggr{)}}\hat{H}$$
$$\displaystyle\qquad-\hat{H}\mathinner{\biggl{(}\sum_{m=1}^{6N}\frac{1}{\hat{\sigma}_{m}^{4}}\hat{\lambda}_{m}\hat{\lambda}_{m}^{\intercal}(\hat{\sigma}_{m}^{2}-\sigma_{m}^{2})\biggr{)}}\hat{H}.$$
(A.112)
In the paragraph following (A.107), we already showed that the last four
terms of the right hand side of the preceding display are $o_{p}(T_{f}^{-1/2})$. Since $A=O_{p}(T_{f}^{-1/2})$, $A^{\intercal}MA=o_{p}(T_{f}^{-1/2})$. Thus, (A.112) becomes
$$\displaystyle\hat{M}-M+A^{\intercal}M+MA=o_{p}(T_{f}^{-1/2}).$$
(A.113)
Imposing $\hat{M}_{10,10}=1$, the $(10,10)$th element of the preceding display
satisfies
$$o_{p}(T_{f}^{-1/2})=\hat{M}_{10,10}-1+M_{10,\bullet}A_{\bullet,10}+\left(A_{\bullet,10}\right)^{\intercal}M_{\bullet,10}=2A_{10,10}$$
whence $A_{10,10}=o_{p}(T_{f}^{-1/2})$. In the similar way, imposing $\hat{M}_{11,11}$, $\hat{M}_{12,12},\hat{M}_{13,13}=1,$ we could deduce that
$A_{11,11},A_{12,12},A_{13,13}=o_{p}(T_{f}^{-1/2}).$
A.9.3 Step III Impose Some Equality Restrictions in $\{\Lambda_{k}\}_{k=1}^{6}$
(III.1) Note that (A.109) implies
$$\displaystyle\hat{\Lambda}_{6,9,c}-\Lambda_{6,9,c}$$
$$\displaystyle=A_{9,x}\Lambda_{6,x,c}+F_{6,9,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{1,2,3,9\}$$
$$\displaystyle=A_{9,[1,9]}\Lambda_{6,[1,9],c}+A_{9,[2,3]}\Lambda_{6,[2,3],c}+F_{6,9,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle\hat{\Lambda}_{3,12,c}-\Lambda_{3,12,c}$$
$$\displaystyle=A_{12,y}\Lambda_{3,y,c}+F_{3,12,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{4,5,6,12\}.$$
We then impose $\hat{\Lambda}_{6,9,c}=\hat{\Lambda}_{3,12,c}$ for $c=\{1,2\}$;
that is, the loadings of the continent factor on day one and day two are the
same for the first two American assets. The preceding display implies
$$A_{9,[1,9]}\Lambda_{6,[1,9],c}=A_{12,y}\Lambda_{3,y,c}+F_{3,12,c}-A_{9,[2,3]}\Lambda_{6,[2,3],c}-F_{6,9,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$A_{9,[1,9]}=\mathinner{\bigl{(}A_{12,y}\Lambda_{3,y,c}+F_{3,12,c}-A_{9,[2,3]}\Lambda_{6,[2,3],c}-F_{6,9,c}\bigr{)}}\Lambda_{6,[1,9],c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(III.2) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{1}-\Lambda_{1})_{14,c}$$
$$\displaystyle=A_{14,x}\Lambda_{1,x,c}+F_{1,14,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{6,7,8,14\}$$
$$\displaystyle=A_{14,[8,14]}\Lambda_{1,[8,14],c}+A_{14,[6,7]}\Lambda_{1,[6,7],c}+F_{1,14,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{4}-\Lambda_{4})_{11,c}$$
$$\displaystyle=A_{11,y}\Lambda_{4,y,c}+F_{4,11,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{3,4,5,11\}.$$
We then impose $\hat{\Lambda}_{1,14,c}=\hat{\Lambda}_{4,11,c}$ for
$c=\{1,2\}$; that is, the loadings of the continent factor on day one and day
two are the same for the first two Asian assets. The preceding display
implies
$$A_{14,[8,14]}\Lambda_{1,[8,14],c}=A_{11,y}\Lambda_{4,y,c}+F_{4,11,c}-A_{14,[6,7]}\Lambda_{1,[6,7],c}-F_{1,14,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{14,[8,14]}=\mathinner{\bigl{(}A_{11,y}\Lambda_{4,y,c}+F_{4,11,c}-A_{14,[6,7]}\Lambda_{1,[6,7],c}-F_{1,14,c}\bigr{)}}\Lambda_{1,[8,14],c}^{-1}+o_{p}(T_{f}^{-1/2})$$
A.9.4 Step IV Impose Some Restrictions in $M$
Impose $\hat{M}_{4,12}=\hat{M}_{5,12}=\hat{M}_{6,12}=0$. Considering the
$(4,12)$th, $(5,12)$th and $(6,12)$th elements of the left hand side of
(A.113), we have
$$\displaystyle M_{4,\bullet}A_{\bullet,12}+A_{12,4}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
$$\displaystyle M_{5,\bullet}A_{\bullet,12}+A_{12,5}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
$$\displaystyle M_{6,\bullet}A_{\bullet,12}+A_{12,6}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
with the only unknown elements $A_{4,12},A_{5,12}$ and $A_{6,12}$. The
preceding display could be written as
$$M_{a,a}A_{a,12}+M_{a,b}A_{b,12}+(A_{12,a})^{\intercal}=o_{p}(T_{f}^{-1/2}),$$
where $a\mathrel{\mathop{:}}=\{4,5,6\},b\mathrel{\mathop{:}}=\{1,2,3,7,8,9,10,11,12,13,14\}$. Thus, we obtain
$$A_{a,12}=-(M_{a,a})^{-1}\left(M_{a,b}A_{b,12}+(A_{12,a})^{\intercal}\right)$$
Similarly, imposing $\hat{M}_{2,10}=\hat{M}_{3,10}=\hat{M}_{4,10}=0$, we could
solve $A_{[2,3,4],10}$. Imposing $\hat{M}_{3,11}=\hat{M}_{4,11}=\hat{M}_{5,11}=0,$ we could solve $A_{[3,4,5],11}$. Imposing $\hat{M}_{5,13}=\hat{M}_{6,13}=\hat{M}_{7,13}=0,$ we could solve $A_{[5,6,7],13}$.
A.9.5 Step V Impose Some Restrictions in $M$
(V.1) Consider the $(4,4)$th and $(6,6)$th elements of (A.113):
$$\displaystyle\hat{M}_{4,4}-1/(1-\phi^{2})+2M_{4,4}A_{4,4}+2M_{4,-4}A_{-4,4}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
$$\displaystyle\hat{M}_{6,6}-1/(1-\phi^{2})+2M_{6,6}A_{6,6}+2M_{6,-6}A_{-6,6}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
Imposing $\hat{M}_{4,4}=\hat{M}_{6,6},$ we could arrange the preceding display
into
$$M_{4,4}A_{4,4}+M_{4,-4}A_{-4,4}=M_{6,6}A_{6,6}+M_{6,-6}A_{-6,6}+o_{p}(T_{f}^{-1/2}).$$
(A.114)
Next, consider the $(4,4)$th and $(6,4)$th elements of (A.113):
$$\displaystyle\hat{M}_{4,4}-1/(1-\phi^{2})+2M_{4,4}A_{4,4}+2M_{4,-4}A_{-4,4}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
$$\displaystyle\hat{M}_{6,4}-\phi^{2}/(1-\phi^{2})+M_{6,\bullet}A_{\bullet,4}+(A_{\bullet,6})^{\intercal}M_{\bullet,4}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2}).$$
Imposing $\hat{M}_{4,4}-\hat{M}_{6,4}=1$, we could arrange the preceding
display into
$$2M_{4,4}A_{4,4}+2M_{4,-4}A_{-4,4}=M_{6,\bullet}A_{\bullet,4}+(A_{\bullet,6})^{\intercal}M_{\bullet,4}+o_{p}(T_{f}^{-1/2}).$$
(A.115)
Since there are only two unknown elements in (A.114) and
(A.115) (i.e., $A_{4,4}$ and $A_{6,6}$), we could thus solve them.
Write (A.114) and (A.115) in matrix
$$\displaystyle\left(\begin{tabular}[c]{ll}$M_{4,4}$&$-M_{6,6}$\\
$2M_{4,4}-M_{6,4}$&$-M_{4,6}$\end{tabular}\right)\left(\begin{tabular}[c]{l}$A_{4,4}$\\
$A_{6,6}$\end{tabular}\right)\ $$
$$\displaystyle=$$
$$\displaystyle\left(\begin{tabular}[c]{l}$M_{6,-6}A_{-6,6}-M_{4,-4}A_{-4,4}$\\
$M_{4,-6}A_{-6,6}-\left(2M_{4,-4}-M_{6,-4}\right)A_{-4,4}$\end{tabular}\right).$$
That is,
$$\left(\begin{tabular}[c]{l}$A_{4,4}$\\
$A_{6,6}$\end{tabular}\right)=\left(\begin{tabular}[c]{ll}$M_{4,4}$&$-M_{6,6}$\\
$2M_{4,4}-M_{6,4}$&$-M_{4,6}$\end{tabular}\right)^{-1}\left(\begin{tabular}[c]{l}$M_{6,-6}A_{-6,6}-M_{4,-4}A_{-4,4}$\\
$M_{4,-6}A_{-6,6}-\left(2M_{4,-4}-M_{6,-4}\right)A_{-4,4}$\end{tabular}\right).$$
(V.2) Consider the $(4,4)$th and $(5,5)$th elements of (A.113):
$$\displaystyle\hat{M}_{4,4}-1/(1-\phi^{2})+2M_{4,\bullet}A_{\bullet,4}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
$$\displaystyle\hat{M}_{5,5}-1/(1-\phi^{2})+2M_{5,5}A_{5,5}+2M_{5,-5}A_{-5,5}$$
$$\displaystyle=o_{p}(T_{f}^{-1/2})$$
Imposing $\hat{M}_{4,4}=\hat{M}_{5,5}$, we could solve the preceding display
for $A_{5,5}$:
$$A_{5,5}=\left(M_{4,\bullet}A_{\bullet,4}-M_{5,-5}A_{-5,5}\right)/M_{5,5}+o_{p}(T_{f}^{-1/2})$$
A.9.6 Step VI Impose Some Equality Restrictions in $\{\Lambda_{k}\}_{k=1}^{6}$
(VI.1) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{6}-\Lambda_{6})_{1,c}$$
$$\displaystyle=A_{1,x}\Lambda_{6,x,c}+F_{6,1,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{1,2,3,9\}$$
$$\displaystyle=A_{1,[1,9]}\Lambda_{6,[1,9],c}+A_{1,[2,3]}\Lambda_{6,[2,3],c}+F_{6,1,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{3}-\Lambda_{3})_{4,c}$$
$$\displaystyle=A_{4,y}\Lambda_{3,y,c}+F_{3,4,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{4,5,6,12\}.$$
We then impose $\hat{\Lambda}_{6,1,c}=\hat{\Lambda}_{3,4,c}$ for $c=\{1,2\}$;
that is, the loadings of the continent factor on day one and day two are the
same for the first two American assets. The preceding display implies
$$\displaystyle A_{1,[1,9]}\Lambda_{6,[1,9],c}=A_{4,y}\Lambda_{3,y,c}+F_{3,4,c}-A_{1,[2,3]}\Lambda_{6,[2,3],c}-F_{6,1,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{1,[1,9]}=\mathinner{\bigl{(}A_{4,y}\Lambda_{3,y,c}+F_{3,4,c}-A_{1,[2,3]}\Lambda_{6,[2,3],c}-F_{6,1,c}\bigr{)}}\Lambda_{6,[1,9],c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(VI.2) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{6}-\Lambda_{6})_{2,c}$$
$$\displaystyle=A_{2,x}\Lambda_{6,x,c}+F_{6,2,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{1,2,3,9\}$$
$$\displaystyle=A_{2,[1,2,9]}\Lambda_{6,[1,2,9],c}+A_{2,3}\Lambda_{6,3,c}+F_{6,2,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{3}-\Lambda_{3})_{5,c}$$
$$\displaystyle=A_{5,y}\Lambda_{3,y,c}+F_{3,5,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{4,5,6,12\}.$$
We then impose $\hat{\Lambda}_{6,2,c}=\hat{\Lambda}_{3,5,c}$ for
$c=\{1,2,3\}$; that is, the first lagged loadings of the global factor on day
one and day two are the same for the first two American assets. The preceding
display implies
$$\displaystyle A_{2,[1,2,9]}\Lambda_{6,[1,2,9],c}=A_{5,y}\Lambda_{3,y,c}+F_{3,5,c}-A_{2,3}\Lambda_{6,3,c}-F_{6,2,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{2,[1,2,9]}=\mathinner{\bigl{(}A_{5,y}\Lambda_{3,y,c}+F_{3,5,c}-A_{2,3}\Lambda_{6,3,c}-F_{6,2,c}\bigr{)}}\Lambda_{6,[1,2,9],c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(VI.3) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{6}-\Lambda_{6})_{2,c}$$
$$\displaystyle=A_{2,x}\Lambda_{6,x,c}+F_{6,2,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{1,2,3,9\}$$
$$\displaystyle=A_{2,[1,2,9]}\Lambda_{6,[1,2,9],c}+A_{2,3}\Lambda_{6,3,c}+F_{6,2,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{3}-\Lambda_{3})_{5,c}$$
$$\displaystyle=A_{5,y}\Lambda_{3,y,c}+F_{3,5,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{4,5,6,12\}.$$
We then impose $\hat{\Lambda}_{6,2,c}=\hat{\Lambda}_{3,5,c}$ for
$c=\{1,2,3\}$; that is, the first lagged loadings of the global factor on day
one and day two are the same for the first three American assets. The
preceding display implies
$$\displaystyle A_{2,[1,2,9]}\Lambda_{6,[1,2,9],c}=A_{5,y}\Lambda_{3,y,c}+F_{3,5,c}-A_{2,3}\Lambda_{6,3,c}-F_{6,2,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{2,[1,2,9]}=\mathinner{\bigl{(}A_{5,y}\Lambda_{3,y,c}+F_{3,5,c}-A_{2,3}\Lambda_{6,3,c}-F_{6,2,c}\bigr{)}}\Lambda_{6,[1,2,9],c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(VI.4) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{6}-\Lambda_{6})_{3,c}$$
$$\displaystyle=A_{3,x}\Lambda_{6,x,c}+F_{6,3,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{1,2,3,9\}$$
$$\displaystyle(\hat{\Lambda}_{3}-\Lambda_{3})_{6,c}$$
$$\displaystyle=A_{6,y}\Lambda_{3,y,c}+F_{3,6,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{4,5,6,12\}.$$
We then impose $\hat{\Lambda}_{6,3,c}=\hat{\Lambda}_{3,6,c}$ for
$c=\{1,2,3,4\}$; that is, the second lagged loadings of the global factor on
day one and day two are the same for the first four American assets. The
preceding display implies
$$\displaystyle A_{3,x}\Lambda_{6,x,c}=A_{6,y}\Lambda_{3,y,c}+F_{3,6,c}-F_{6,3,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{3,x}=\mathinner{\bigl{(}A_{6,y}\Lambda_{3,y,c}+F_{3,6,c}-F_{6,3,c}\bigr{)}}\Lambda_{6,x,c}^{-1}+o_{p}(T_{f}^{-1/2})$$
(VI.5) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{1}-\Lambda_{1})_{8,c}$$
$$\displaystyle=A_{8,x}\Lambda_{1,x,c}+F_{1,8,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{6,7,8,14\}$$
$$\displaystyle=A_{8,[8,14]}\Lambda_{1,[8,14],c}+A_{8,[6,7]}\Lambda_{1,[6,7],c}+F_{1,8,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{4}-\Lambda_{4})_{5,c}$$
$$\displaystyle=A_{5,y}\Lambda_{4,y,c}+F_{4,5,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{3,4,5,11\}.$$
We then impose $\hat{\Lambda}_{1,8,c}=\hat{\Lambda}_{4,5,c}$ for $c=\{1,2\}$;
that is, the second lagged loadings of the global factor on day one and day
two are the same for the first two Asian assets. The preceding display
implies
$$\displaystyle A_{8,[8,14]}\Lambda_{1,[8,14],c}=A_{5,y}\Lambda_{4,y,c}+F_{4,5,c}-A_{8,[6,7]}\Lambda_{1,[6,7],c}-F_{1,8,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{8,[8,14]}=\mathinner{\bigl{(}A_{5,y}\Lambda_{4,y,c}+F_{4,5,c}-A_{8,[6,7]}\Lambda_{1,[6,7],c}-F_{1,8,c}\bigr{)}}\Lambda_{1,[8,14],c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(VI.6) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{1}-\Lambda_{1})_{7,c}$$
$$\displaystyle=A_{7,x}\Lambda_{1,x,c}+F_{1,7,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{6,7,8,14\}$$
$$\displaystyle=A_{7,[7,8,14]}\Lambda_{1,[7,8,14],c}+A_{7,6}\Lambda_{1,6,c}+F_{1,7,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{4}-\Lambda_{4})_{4,c}$$
$$\displaystyle=A_{4,y}\Lambda_{4,y,c}+F_{4,4,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{3,4,5,11\}.$$
We then impose $\hat{\Lambda}_{1,7,c}=\hat{\Lambda}_{4,4,c}$ for
$c=\{1,2,3\}$; that is, the first lagged loadings of the global factor on day
one and day two are the same for the first three Asian assets. The preceding
display implies
$$\displaystyle A_{7,[7,8,14]}\Lambda_{1,[7,8,14],c}=A_{4,y}\Lambda_{4,y,c}+F_{4,4,c}-A_{7,6}\Lambda_{1,6,c}-F_{1,7,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{7,[7,8,14]}=\mathinner{\bigl{(}A_{4,y}\Lambda_{4,y,c}+F_{4,4,c}-A_{7,6}\Lambda_{1,6,c}-F_{1,7,c}\bigr{)}}\Lambda_{1,[7,8,14],c}^{-1}+o_{p}(T_{f}^{-1/2}).$$
(VI.7) Note that (A.109) implies
$$\displaystyle(\hat{\Lambda}_{1}-\Lambda_{1})_{6,c}$$
$$\displaystyle=A_{6,x}\Lambda_{1,x,c}+F_{1,6,c}+o_{p}(T_{f}^{-1/2})\qquad x=\{6,7,8,14\}$$
$$\displaystyle=A_{6,[7,8,14]}\Lambda_{1,[7,8,14],c}+A_{6,6}\Lambda_{1,6,c}+F_{1,6,c}+o_{p}(T_{f}^{-1/2})$$
$$\displaystyle(\hat{\Lambda}_{4}-\Lambda_{4})_{3,c}$$
$$\displaystyle=A_{3,y}\Lambda_{4,y,c}+F_{4,3,c}+o_{p}(T_{f}^{-1/2})\qquad y=\{3,4,5,11\}.$$
We then impose $\hat{\Lambda}_{1,6,c}=\hat{\Lambda}_{4,3,c}$ for
$c=\{1,2,3\}$; that is, the contemporaneous loadings of the global factor on
day one and day two are the same for the first three Asian assets. The
preceding display implies
$$\displaystyle A_{6,[7,8,14]}\Lambda_{1,[7,8,14],c}=A_{3,y}\Lambda_{4,y,c}+F_{4,3,c}-A_{6,6}\Lambda_{1,6,c}-F_{1,6,c}+o_{p}(T_{f}^{-1/2})$$
whence we have
$$\displaystyle A_{6,[7,8,14]}=\mathinner{\bigl{(}A_{3,y}\Lambda_{4,y,c}+F_{4,3,c}-A_{6,6}\Lambda_{1,6,c}-F_{1,6,c}\bigr{)}}\Lambda_{1,[7,8,14],c}^{-1}+o_{p}(T_{f}^{-1/2})$$
A.9.7 To Sum Up
Recall that (A.109) implies that for $k=1,\ldots,6$, $j=1,\ldots,N$,
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})=\sqrt{T_{f}}A\boldsymbol{\lambda}_{k,j}+M^{-1}\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}+o_{p}(1).$$
(A.116)
We have just shown that each element of the $14\times 14$ matrix $A$
could be expressed into some known (but complicated) linear function involving
elements of the following six matrices:
$$M^{-1}\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}\mathinner{\bigl{[}e_{(k-1)N+1,t},e_{(k-1)N+2,t},e_{(k-1)N+3,t},e_{(k-1)N+4,t}\bigr{]}}$$
for $k=1,\ldots,6$, plus $o_{p}(T_{f}^{-1/2})$. That is, there exists a $196\times 336$ matrix $\Gamma$, whose elements are known (but complicated) linear
functions of elements of (inverted) submatrices of $\Lambda$ and $M$,
satisfying
$$\operatorname*{vec}A=\Gamma\times\frac{1}{T_{f}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+o_{p}(T_{f}^{-1/2}),$$
where $\boldsymbol{e}_{t}^{\dagger}$ is a $24\times 1$ vector consisting of
$e_{(p-1)N+q,t}$ for $p=1,\ldots,6$ and $q=1,\ldots,4$. Thus (A.116)
could be written as
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})$$
$$\displaystyle=\sqrt{T_{f}}(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\operatorname*{vec}A+M^{-1}\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}+o_{p}(1)$$
$$\displaystyle=(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+M^{-1}\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\boldsymbol{f}_{t}e_{(k-1)N+j,t}+o_{p}(1).$$
(A.117)
In SM LABEL:sec_MDS_CLT, we show that
$$\displaystyle\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\left[\begin{array}[c]{c}(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t})\\
M^{-1}\boldsymbol{f}_{t}e_{(k-1)N+j,t}\end{array}\right]\xrightarrow{d}$$
(A.120)
$$\displaystyle N\left(\left[\begin{array}[c]{c}\boldsymbol{0}\\
\boldsymbol{0}\end{array}\right]\left[\begin{array}[c]{cc}(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(\boldsymbol{\lambda}_{k,j}\otimes I_{14})&\text{cov}_{k,j}\\
\text{cov}_{k,j}&M^{-1}\sigma_{k,j}^{2}\end{array}\right]\right)$$
(A.125)
where $\Sigma_{ee}^{\dagger}\mathrel{\mathop{:}}=\mathbb{E}[\boldsymbol{e}_{t}^{\dagger}\boldsymbol{e}_{t}^{\dagger\intercal}]$ and $\text{cov}_{k,j}$ is an
$14\times 14$ matrix defined as
$$\displaystyle\text{cov}_{k,j}\mathrel{\mathop{:}}=\text{cov}\mathinner{\Bigl{(}(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}),e_{(k-1)N+j,t}\boldsymbol{f}_{t}^{\intercal}M^{-1}\Bigr{)}}.$$
By Assumption 2.1, we have $\text{cov}_{k,j}=\boldsymbol{0}$ for
$j>4$, and
$$\displaystyle\text{cov}_{k,j}=(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma\mathbb{E}\mathinner{\bigl{[}\boldsymbol{e}_{t}^{\dagger}e_{(k-1)N+j,t}\otimes\boldsymbol{f}_{t}\boldsymbol{f}_{t}^{\intercal}\bigr{]}}M^{-1}=(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma\mathinner{\bigl{[}\sigma_{k,j}^{2}\boldsymbol{\iota}_{k,j}\otimes M\bigr{]}}M^{-1}$$
$$\displaystyle=(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma\mathinner{\bigl{[}\boldsymbol{\iota}_{k,j}\otimes I_{14}\bigr{]}}\sigma_{k,j}^{2},$$
for $j\leq 4$, where $\boldsymbol{\iota}_{k,j}$ is a $24\times 1$ zero vector
with its $[4(k-1)+j]$th element replaced by one. Thus for $j>4$, we have
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})\xrightarrow{d}N\mathinner{\Bigl{(}\boldsymbol{0},(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(\boldsymbol{\lambda}_{k,j}\otimes I_{14})+M^{-1}\sigma_{k,j}^{2}\Bigr{)}}.$$
For $j\leq 4$, we have
$$\displaystyle\sqrt{T_{f}}(\hat{\boldsymbol{\lambda}}_{k,j}-\boldsymbol{\lambda}_{k,j})\xrightarrow{d}N\mathinner{\Bigl{(}\boldsymbol{0},(\boldsymbol{\lambda}_{k,j}^{\intercal}\otimes I_{14})\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(\boldsymbol{\lambda}_{k,j}\otimes I_{14})+M^{-1}\sigma_{k,j}^{2}+\text{cov}_{k,j}+\text{cov}_{k,j}^{\intercal}\Bigr{)}}.$$
A.10 Proof of Theorem 4.4
From (A.113), we have
$$\displaystyle\sqrt{T_{f}}(\hat{M}-M)=-\sqrt{T_{f}}\mathinner{\bigl{(}A^{\intercal}M+MA\bigr{)}}+o_{p}(1).$$
whence we have
$$\displaystyle\sqrt{T_{f}}\operatorname*{vech}(\hat{M}-M)=-\sqrt{T_{f}}\operatorname*{vech}\mathinner{\bigl{(}A^{\intercal}M+MA\bigr{)}}+o_{p}(1)=-\sqrt{T_{f}}D_{14}^{+}\operatorname*{vec}\mathinner{\bigl{(}A^{\intercal}M+MA\bigr{)}}+o_{p}(1)$$
$$\displaystyle=-\sqrt{T_{f}}D_{14}^{+}\mathinner{\bigl{[}(M\otimes I_{14})\operatorname*{vec}(A^{\intercal})+(I_{14}\otimes M)\operatorname*{vec}A\bigr{]}}+o_{p}(1)$$
$$\displaystyle=-\sqrt{T_{f}}D_{14}^{+}\mathinner{\bigl{[}(M\otimes I_{14})K_{14,14}\operatorname*{vec}A+(I_{14}\otimes M)\operatorname*{vec}A\bigr{]}}+o_{p}(1)$$
$$\displaystyle=-\sqrt{T_{f}}D_{14}^{+}\mathinner{\bigl{[}K_{14,14}(I_{14}\otimes M)\operatorname*{vec}A+(I_{14}\otimes M)\operatorname*{vec}A\bigr{]}}+o_{p}(1)$$
$$\displaystyle=-\sqrt{T_{f}}D_{14}^{+}(K_{14,14}+I_{14^{2}})(I_{14}\otimes M)\operatorname*{vec}A+o_{p}(1)=-2\sqrt{T_{f}}D_{14}^{+}D_{14}D_{14}^{+}(I_{14}\otimes M)\operatorname*{vec}A+o_{p}(1)$$
$$\displaystyle=-2D_{14}^{+}(I_{14}\otimes M)\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+o_{p}(1)$$
where the second equality is due to symmetry of $A^{\intercal}M+MA$, and the
fifth and seventh equalities are due to properties of $K_{14,14}$. Thus we
have
$$\displaystyle\sqrt{T_{f}}\operatorname*{vech}(\hat{M}-M)\xrightarrow{d}N\mathinner{\bigl{(}0,\mathcal{M}\bigr{)}}$$
where $\mathcal{M}$ is $105\times 105$ and defined
$$\displaystyle\mathcal{M}\mathrel{\mathop{:}}=4D_{14}^{+}(I_{14}\otimes M)\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}(I_{14}\otimes M)D_{14}^{+\intercal}.$$
A.11 Proof of Theorem 4.5
$$\displaystyle\sqrt{N}(\hat{\boldsymbol{f}}_{t}-\boldsymbol{f}_{t})$$
$$\displaystyle=-\sqrt{N}\mathinner{\bigl{(}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\bigr{)}}^{-1}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}(\hat{\Lambda}-\Lambda)\boldsymbol{f}_{t}+\sqrt{N}\mathinner{\bigl{(}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\bigr{)}}^{-1}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\boldsymbol{e}_{t}$$
$$\displaystyle=-\sqrt{\frac{N}{T_{f}}}\sqrt{T_{f}}A^{\intercal}\boldsymbol{f}_{t}+\sqrt{N}\mathinner{\biggl{(}\frac{1}{N}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\biggr{)}}^{-1}\frac{1}{N}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\boldsymbol{e}_{t}.$$
(A.126)
Lemma D1 of Bai and Li (2012) still holds in our setting and it reads, in our
notation:
$$\displaystyle\frac{1}{N}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\boldsymbol{e}_{t}=\frac{1}{N}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+O_{p}\mathinner{\bigl{(}N^{-1/2}T_{f}^{-1/2}\bigr{)}}+O_{p}(T_{f}^{-1}).$$
(A.127)
$$\displaystyle\mathinner{\biggl{(}\frac{1}{N}\hat{\Lambda}^{\intercal}\hat{\Sigma}_{ee}^{-1}\hat{\Lambda}\biggr{)}}^{-1}=Q^{-1}+o_{p}(1).$$
(A.128)
Substituting (A.127) and (A.128) into (A.126), we have
$$\displaystyle\sqrt{N}(\hat{\boldsymbol{f}}_{t}-\boldsymbol{f}_{t})$$
$$\displaystyle=-\sqrt{\frac{N}{T_{f}}}\sqrt{T_{f}}A^{\intercal}\boldsymbol{f}_{t}+\sqrt{N}\mathinner{\bigl{(}Q^{-1}+o_{p}(1)\bigr{)}}\mathinner{\biggl{[}\frac{1}{N}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+O_{p}\mathinner{\bigl{(}N^{-1/2}T_{f}^{-1/2}\bigr{)}}+O_{p}(T_{f}^{-1})\biggr{]}}$$
$$\displaystyle=-\sqrt{\frac{N}{T_{f}}}\sqrt{T_{f}}A^{\intercal}\boldsymbol{f}_{t}+\mathinner{\bigl{(}Q^{-1}+o_{p}(1)\bigr{)}}\mathinner{\biggl{[}\frac{1}{\sqrt{N}}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+O_{p}\mathinner{\bigl{(}T_{f}^{-1/2}\bigr{)}}+O_{p}(\sqrt{N}T_{f}^{-1})\biggr{]}}$$
$$\displaystyle=-\sqrt{\Delta}\sqrt{T_{f}}A^{\intercal}\boldsymbol{f}_{t}+\mathinner{\bigl{(}Q^{-1}+o_{p}(1)\bigr{)}}\mathinner{\biggl{[}\frac{1}{\sqrt{N}}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+O_{p}\mathinner{\bigl{(}T_{f}^{-1/2}\bigr{)}}+o_{p}(1)\biggr{]}}$$
$$\displaystyle=-\sqrt{\Delta}\sqrt{T_{f}}A^{\intercal}\boldsymbol{f}_{t}+Q^{-1}\frac{1}{\sqrt{N}}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+o_{p}(1)$$
$$\displaystyle=-\sqrt{\Delta}(\boldsymbol{f}_{t}^{\intercal}\otimes I_{14})K_{14,14}\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}+Q^{-1}\frac{1}{\sqrt{N}}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}+o_{p}(1)$$
(A.129)
where the third equality is due to $\sqrt{N}/T_{f}\to 0$, $N/T_{f}\to\Delta$
and $\sqrt{T_{f}}A^{\intercal}\boldsymbol{f}_{t}=O_{p}(1)$, and the fourth
equality uses the fact that $Q^{-1}=O_{p}(1)$ and $N^{-1/2}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}=O_{p}(1)$ by the central limit theorem.
The first two terms on the right side of (A.129), conditioning on
$\boldsymbol{f}_{t}$, are asymptotically normal and asymptotically
independent. The former uses the central limit theorem over the time dimension
(only depends on the first 4 assets in each continent) and the latter uses the
central limit theorem over the cross-sectional dimension. In particular,
$$\displaystyle\left.\left(\begin{array}[c]{c}-\sqrt{\Delta}(\boldsymbol{f}_{t}^{\intercal}\otimes I_{14})K_{14,14}\Gamma\frac{1}{\sqrt{T_{f}}}\sum_{t=1}^{T_{f}}\mathinner{\bigl{(}\boldsymbol{e}_{t}^{\dagger}\otimes\boldsymbol{f}_{t}\bigr{)}}\\
Q^{-1}\frac{1}{\sqrt{N}}\Lambda^{\intercal}\Sigma_{ee}^{-1}\boldsymbol{e}_{t}\end{array}\right)\right|\boldsymbol{f}_{t}\xrightarrow{d}N\left(\left[\begin{array}[c]{c}\boldsymbol{0}\\
\boldsymbol{0}\end{array}\right]\left[\begin{array}[c]{cc}\maltese&\boldsymbol{0}\\
\boldsymbol{0}&Q^{-1}\end{array}\right]\right)$$
where
$$\maltese\mathrel{\mathop{:}}=\Delta(\boldsymbol{f}_{t}^{\intercal}\otimes I_{14})K_{14,14}\Gamma(\Sigma_{ee}^{\dagger}\otimes M)\Gamma^{\intercal}K_{14,14}(\boldsymbol{f}_{t}\otimes I_{14}).$$
Then the result of the theorem follows.
A.12 Proof of Theorem 4.6
Proof of Theorem 4.6.
Recall (3.15)
$$\displaystyle\check{\boldsymbol{\theta}}_{m}\mathrel{\mathop{:}}=\arg\min_{\boldsymbol{b}\in\mathbb{R}^{c_{2}}}\mathinner{\bigl{[}\hat{\boldsymbol{h}}-h(\boldsymbol{b})\bigr{]}}^{\intercal}W\mathinner{\bigl{[}\hat{\boldsymbol{h}}-h(\boldsymbol{b})\bigr{]}}.$$
The minimum distance estimator $\check{\boldsymbol{\theta}}_{m}$ satisfies the first-order condition:
$$\displaystyle\frac{\partial h(\check{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\mathinner{\bigl{[}\hat{\boldsymbol{h}}-h(\check{\boldsymbol{\theta}}_{m})\bigr{]}}=0.$$
(A.130)
Do a Taylor expansion
$$\displaystyle h(\check{\boldsymbol{\theta}}_{m})=h(\boldsymbol{\theta}_{m})+\frac{\partial h(\dot{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}}(\check{\boldsymbol{\theta}}_{m}-\boldsymbol{\theta}_{m}),$$
where $\dot{\boldsymbol{\theta}}_{m}$ is a mid-point between $\check{\boldsymbol{\theta}}_{m}$ and $\boldsymbol{\theta}_{m}$. Substituting the preceding display into (A.130), we have
$$\displaystyle\frac{\partial h(\check{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\mathinner{\biggl{[}\hat{\boldsymbol{h}}-h(\boldsymbol{\theta}_{m})-\frac{\partial h(\dot{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}}(\check{\boldsymbol{\theta}}_{m}-\boldsymbol{\theta}_{m})\biggr{]}}=0$$
whence we have
$$\displaystyle\sqrt{T_{f}}(\check{\boldsymbol{\theta}}_{m}-\boldsymbol{\theta}_{m})$$
$$\displaystyle=\mathinner{\biggl{[}\frac{\partial h(\check{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\frac{\partial h(\dot{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}}\biggr{]}}^{-1}\frac{\partial h(\check{\boldsymbol{\theta}}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\mathinner{\biggl{[}\hat{\boldsymbol{h}}-h(\boldsymbol{\theta}_{m})\biggr{]}}$$
$$\displaystyle\xrightarrow{d}\mathinner{\biggl{[}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}W\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}}\biggr{]}}^{-1}\frac{\partial h(\boldsymbol{\theta}_{m})}{\partial\boldsymbol{\theta}_{m}^{\intercal}}WN\mathinner{\bigl{(}0,\mathcal{H}\bigr{)}},$$
where the convergence in distribution follows from consistency of $\check{\boldsymbol{\theta}}_{m}$ (i.e., $\check{\boldsymbol{\theta}}_{m}\xrightarrow{p}\boldsymbol{\theta}_{m}$). The proof of consistency of $\check{\boldsymbol{\theta}}_{m}$ is given in SM LABEL:sec_SM_consistency_of_MD. Then we have
$$\displaystyle\sqrt{T_{f}}(\check{\boldsymbol{\theta}}_{m}-\boldsymbol{\theta}_{m})\xrightarrow{d}N\mathinner{\bigl{(}0,\mathcal{O}\bigr{)}}.$$
∎
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Efficient sampling from shallow Gaussian quantum-optical circuits with local interactions
Haoyu Qi
Xanadu, Toronto, ON, M5G 2C8, Canada
Diego Cifuentes
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA
Kamil Brádler
Xanadu, Toronto, ON, M5G 2C8, Canada
Robert Israel
Xanadu, Toronto, ON, M5G 2C8, Canada
Timjan Kalajdzievski
Xanadu, Toronto, ON, M5G 2C8, Canada
Nicolás Quesada
Xanadu, Toronto, ON, M5G 2C8, Canada
Abstract
We prove that a classical computer can efficiently sample from the photon-number probability distribution of a Gaussian state prepared by using an optical circuit that is shallow and local. Our work generalizes previous known results for qubits to the continuous-variable domain. The key to our proof is the observation that the adjacency matrices characterizing the Gaussian states generated by shallow and local circuits have small bandwidth. To exploit this structure, we devise fast algorithms to calculate loop hafnians of banded matrices.
Since sampling from deep optical circuits with exponential-scaling photon loss is classically simulable, our results pose a challenge to the feasibility of demonstrating quantum supremacy on photonic platforms with local interactions.
I Introduction
Gaussian Boson Sampling (GBS) is a model of quantum computation in which a multimode Gaussian state is probed using photon-number resolving (PNR) detectors Hamilton et al. (2017); Kruse et al. (2019); Rahimi-Keshari
et al. (2015). Originally introduced as an experimentally friendly proposal to show
quantum computational supremacy Lund et al. (2014); Barkhofen et al. (2017); Aaronson and
Arkhipov (2011), it also finds potential applications Bromley et al. (2020) in chemistry Huh et al. (2015); Jahangiri
et al. (2020a), optimization Arrazola and Bromley (2018); Arrazola et al. (2018); Banchi et al. (2020a), graph theory Schuld et al. (2020); Brádler et al. (2018, 2018), non-Gaussian state preparation Sabapathy et al. (2019); Su et al. (2019), and machine learning Banchi et al. (2020b). Experimentally, GBS is carried out by sending squeezed (and possibly displaced) single-mode states into an interferometer which mixes them. Typically, the interferometer is implemented by applying successive layers of beamsplitters that couple nearest-neighbour modes Reck et al. (1994); Clements et al. (2016); de Guise et al. (2018). If the number of layers is large enough, linearly depending on the number of modes to be precise, one can implement arbitrary passive unitary transformations.
It is precisely in this regime, when strong multi-partite entanglement is prevalent, that GBS is expected to be hard to simulate for a classical computer Hamilton et al. (2017); Aaronson and
Arkhipov (2011).
Noise and errors are inevitable in any experimental implementation of a near-term device. For photonic architectures, photon loss is the dominant source of error since it increases exponentially with the circuit depth. Therefore, it is not too surprising that a sampling device with a deep circuit loses its computational advantage asymptotically García-Patrón
et al. (2019); Qi et al. (2020). Furthermore, faster classical algorithms Neville et al. (2017); Clifford and Clifford (2018); Quesada et al. (2019); Clifford and Clifford (2020) and large-scale simulation results Wu et al. (2018); Gupt et al. (2020); Li et al. (2020) demand larger and larger circuit size or input photon number. As a result, demonstrating quantum computational supremacy on current known photonic models has become increasingly difficult.
Naturally, the next question to ask is: can we reduce the circuit depth to mitigate the adversarial effects of photon loss, but at the same time preserve the quantum advantage? In this paper we prove that the answer is no, if there are only local interactions. The classical simulability of qubit systems with shallow circuits and local gates was proved in Ref. Jozsa (2006) by representing these systems as matrix product states Vidal (2003). Tensor network theory is a powerful tool to characterize and study the entanglement properties of quantum states generated by local interactions Montangero (2018). However, for optical systems, tensor network simulation only leads to unsatisfying quasi-polynomial algorithms due to a non-constant physical dimension, a result of the photon-bunching effect García-Patrón
et al. (2019); Qi et al. (2020).
Furthermore, using tensor network methods means we need to truncate the Hilbert space, which necessarily destroys the Gaussian structure. We circumvent this issue by coarse-graining the probabilities above a certain threshold. The merits of this step are threefold: 1) it eventually leads to a strictly polynomial runtime algorithm thanks to the constant threshold; 2) It preserves the Gaussian picture in the sense that our algorithm only requires calculating loop hafnians of the adjacency matrices; 3) The coarse-graining process is actually simulating the effect of the finite resolution of PNRs Lita et al. (2008); Levine et al. (2012); Hadfield (2009).
In our simulation algorithm, the information of the circuit depth is captured by the bandwidth of the adjacency matrices. To exploit the banded structure, we introduce a new algorithm to calculate loop hafnians of banded matrices, extending previous results for permanents Cifuentes and Parrilo (2016). We also devise an algorithm to exploit photon collisions, which are unavoidable for shallow circuits, leading to further speedup. We believe these methods are also of interest in the study of combinatorial graph problems and data structures Barvinok (2016).
We organize our manuscript as follows. In Sec. II, we provide a brief review of Gaussian states and their representation in the Fock basis by using loop hafnians. We then modify the GBS simulation algorithm from Ref. Quesada and Arrazola (2020) in Sec. III to incorporate the effect of the finite resolution of PNRs. It is crucial for any GBS simulation algorithm to have a well-defined run time, otherwise the photon-number distribution would have support on the (countably infinite) non-negative integers. In Sec. IV, we study shallow circuit GBS with local interactions and show that the adjacency matrices arising from our sampling algorithm enjoy a banded structure.
We also point out that photon collisions induce repetitions in the adjacency matrices.
We exploit both of these special structures in Sec. V. Specifically, we prove that the time cost of calculating the loop hafnian of a banded matrix depends exponentially on its bandwidth. We also present an algorithm which calculates loop hafnians faster when there are repeated columns and rows.
Finally, in Sec. VI, we put all these results together to show that a classical computer can simulate GBS with local interactions in polynomial time, if the depth of the circuit is logarithmic in the number of modes. We discuss how our methods are also applicable to Boson Sampling and present the conclusion in Sec. VII.
II Gaussian boson sampling with finite-resolution photon number detectors
An $M$-mode Gaussian state $\rho$ is fully characterized by its $2M$-dimensional mean vector and its $2M\times 2M$-dimensional covariance matrix, with entries:
$$\displaystyle\alpha_{j}$$
$$\displaystyle=\text{Tr}\left\{\rho\hat{\alpha}_{j}\right\},$$
(1)
$$\displaystyle\sigma_{ij}$$
$$\displaystyle=\tfrac{1}{2}\text{Tr}\left\{\rho\left(\hat{\alpha}_{i}\hat{%
\alpha}_{j}+\hat{\alpha}_{j}\hat{\alpha}_{i}\right)\right\}-\alpha_{i}\alpha_{%
j}.$$
(2)
The vector $\hat{\alpha}:=(\hat{a}_{1},\ldots,\hat{a}_{M},\hat{a}^{\dagger}_{1},\ldots,%
\hat{a}_{M}^{\dagger})$ consists of ladder operators for each mode arranged in the shown order. We refer the readers to Refs. Weedbrook et al. (2012); Serafini (2017) for a comprehensive review on Gaussian quantum information.
Due to the ease of preparing Gaussian quantum states, GBS was proposed as a candidate to demonstrate quantum computational supremacy Hamilton et al. (2017); Kruse et al. (2019). Given a Gaussian state with vector of means $\alpha$ and covariance matrix $\sigma$, it can be shown that the probability of detecting photon pattern $s=(s_{1},\ldots,s_{M})$ is given by Quesada et al. (2019)
$$\displaystyle p(s)$$
$$\displaystyle=\frac{\exp\left(-\frac{1}{2}\alpha^{\dagger}Q^{-1}\alpha\right)}%
{\sqrt{\det(Q)}}\frac{\operatorname{lhaf}(\tilde{A}_{s})}{s_{1}!\ldots s_{M}!}%
~{},$$
(3)
$$\displaystyle Q$$
$$\displaystyle:=\sigma+\mathbb{I}_{2M}/2~{},$$
(4)
$$\displaystyle A$$
$$\displaystyle:=\begin{pmatrix}0&\mathbb{I}_{M}\\
\mathbb{I}_{M}&0\end{pmatrix}\left(\mathbb{I}_{2M}-Q^{-1}\right)~{},$$
(5)
$$\displaystyle\tilde{A}_{s}$$
$$\displaystyle:=\text{fdiag}(A_{s},\gamma_{s})~{},$$
(6)
$$\displaystyle\gamma^{T}$$
$$\displaystyle:=\alpha^{\dagger}Q^{-1}.$$
(7)
Here $A=A^{T}$ is
the adjacency matrix of the Gaussian state $\rho$. The displacement vector modifies the usual expression for the GBS probability Hamilton et al. (2017) as follows: 1) it adds an extra Gaussian prefactor which can be efficiently calculated, 2) it fills up the diagonal elements of the $A_{s}$ matrix: this is precisely what the function $\text{fdiag}(\cdot,\cdot)$ does, it replaces the diagonal entries of its first argument with the entries of its second argument.
Note that the output probability in Eq. (3) depends on the extended adjacency matrices $\tilde{A}_{s}$, which is obtained by repeating the rows and columns of $A$ to obtain $A_{s}$ and then filling its diagonal with $\gamma_{s}$. Specifically, if $s_{j}$ photons are detected at the $j$-th mode, then the $j$-th and $(j+M)$-th row and column of $A$ are repeated $s_{j}$ times to obtain $A_{s}$. If $s_{j}=0$, we simply remove the $j$-th and $(j+M)$-th rows and columns. Similarly, $\gamma_{s}$ is obtained from $\gamma$ by repeating its $i$ and $i+M$ entries a total of $s_{i}$ times.
The loop hafnian of a $d\times d$ symmetric matrix $A$ is defined as the number of perfect matchings of a weighted graph with loops that has $A$ as its adjancency matrix:
$$\displaystyle\operatorname{lhaf}(A):=\sum_{\pi\in\text{PM}(k)}\prod_{ij\in\pi}%
A_{ij}~{},$$
(8)
where PM($k$) is the set of perfect matchings of a complete graph with loops. For a complete graph with $k$ vertices that has loops the number of perfect matchings $|\text{PM}(k)|$ is given by the $k^{\text{th}}$ telephone number Björklund et al. (2019). See Fig. 2 for a graphical description of the set PM($k=4$).
It might be worthwhile to point out that the loop hafnian is closely related to the multivariate (signless) matching polynomial of the corresponding graph, a well-known quantity in graph theory Barvinok (2016). The loop hafnian is equal to the matching
polynomial evaluated at a certain point Schuld et al. (2020); Brádler et al. (2019). The matching polynomials could be a powerful tool since they enjoy a number of recursive relations Shi et al. (2016) and, in general, contain more
information about the graph than the loop hafnian.
The GBS task is to output samples according to the distribution $\left\{p(s)\right\}$. Note that the support of this probability distribution is $\mathbb{N}^{M}$, i.e., the cartesian product of the non-negative integers $M$ times. A sampling problem over an infinite sample space can be ill-defined if we are aiming for a worst-case run-time analysis, since there is non-zero probability of detecting an arbitrary high number of photons.
One might attempt to redefine the computational task as sampling over a post-selected distribution with fixed total photon number. Although such modification renders the sampling task well-defined, devising a classical algorithm to simulate such post-selected distribution might be challenging for general GBS scenarios Wu et al. (2020). This is because a system with fixed number of photons is described naturally by the particle representation, while Gaussian systems, with indefinite total photon number, are formulated under the mode representation.
However, such an issue should not be relevant for any experimental implementation, since a realistic device with finite energy cannot probe the system with infinite precision, nor can it detect an infinite amount of photons. Any realistic photon-number detector only has finite resolution power, i.e., it can only distinguish incoming photons up to a certain finite number Lita et al. (2008); Levine et al. (2012); Hadfield (2009). We say that a detector is overloaded if the number of incoming photons is actually beyond the resolution of the detector. The special case where the detectors are overloaded by one or more photons has been studied in detail in Ref. Quesada et al. (2018).
Therefore, to circumvent the divergence problem, we propose to modify the sampling task to incorporate the finite resolution of the detectors. Specifically, we modify the distribution by coarse-graining all photon patterns that overload at least one of the detectors. A two-mode example is shown in Fig. 3.
Formally, we define Gaussian Boson Threshold Sampling (GBTS), as follows:
Definition 1 (GBTS).
Consider an $M$-mode Gaussian state probed with PNR detectors with resolution $c$. Denote $\Sigma_{c}$ as the set of photon patterns which overload at least one PNR:
$$\displaystyle\Sigma_{c}$$
$$\displaystyle=\bigcup_{j=c}^{\infty}\Sigma_{j}~{},$$
(9)
$$\displaystyle\Sigma_{j}$$
$$\displaystyle=\left\{{s}:\exists j~{}s_{j}>c\right\}~{}.$$
(10)
The GBTS computational task is to output a sample from the following distribution:
$$\displaystyle\tilde{p}(x)=\begin{cases}p(s),\quad x={s}\notin\Sigma_{c}~{},\\
\sum_{s\in\Sigma_{c}}p(s),\quad x=\#~{}.\end{cases}$$
(11)
Here $p(s)$ is the output probability of the corresponding ideal GBS model given in Eq. (3). We use the symbol $\#$ to indicate that at least one of the detectors is overloaded.
Note that one can obtain bounds for the probability of occurrence of the $\#$ event in polynomial time by using marginal probabilities as shown in Appendix B of Ref. Quesada and Arrazola (2020).
III Simulating Gaussian boson sampling by calculating a polynomial number of probabilities
Naively, one can simulate any quantum device by calculating exponentially many probabilities. However, since the hardness of sampling originates from the hardness of calculating one probability Aaronson and
Arkhipov (2011); Dalzell et al. (2020), such a brute-force algorithm is expected to be far from optimal. Indeed, several fast algorithms to simulate Boson Sampling have been proposed, which output a sample by only calculating a polynomial number of probabilities Neville et al. (2017); Clifford and Clifford (2018, 2020). For GBS, the authors of Ref. Quesada et al. (2019) devised an algorithm which fully exploits the Gaussian nature of the system, in particular the fact that the reduced Gaussian states can be easily obtained. With a small modification, that algorithm can be used to sample exactly from the distribution of our GBTS problem.
The essence of the algorithm is to break down the original sampling problem into a chain of smaller GBS problems. At the $k$-th step, we sample the number of photons detected at the $k$-th mode, conditioned on the number of photons already sampled at previous steps. Specifically, at the $k$-th step, we sample from the following distribution:
$$\displaystyle q^{(k)}(x)=\begin{cases}p(x|s_{1}\ldots s_{k-1})~{},\text{ for}~%
{}~{}x=0,1,\ldots,c~{}.\\
1-\sum_{i\leq c}q^{(k)}(i)~{},\text{ for}~{}~{}x=\text{`}>\text{'}~{}.\end{cases}$$
(12)
Here the outcome labelled by ‘$>$’ represents all events that overload the $k$-th photon detector. If such outcome is produced, we simply output ‘$\#$’ and exit the algorithm. This procedure is outlined in Algorithm 1. The correctness of our algorithm, i.e., that it indeed samples according to the distribution given in Eq. (11), is shown in Appendix A.
We calculate the conditional probabilities by writing them as ratios of marginal probabilities:
$$\displaystyle p(x|s_{1},\ldots,s_{k-1})=\frac{p(s_{1},\ldots,s_{k})}{p(s_{1},%
\ldots,s_{k-1})}~{},x\leq c~{}.$$
(13)
The denominator comes for free since it must be already calculated at the previous step. The numerator is nothing but the output probability of the reduced Gaussian state on the first $k$ modes:
$$\displaystyle p(s_{1},\ldots,s_{k})=$$
$$\displaystyle\frac{\exp\left(-\frac{1}{2}\alpha^{(k)\dagger}\{Q^{(k)}\}^{-1}%
\alpha^{(k)}\right)}{\sqrt{\det(Q^{(k)})}}$$
(14)
$$\displaystyle\times\frac{\operatorname{lhaf}\left[\left(\tilde{A}^{(k)}\right)%
_{s_{1},\ldots,s_{k}}\right]}{s_{1}!\ldots s_{k}!}~{}.$$
We use superscript $(k)$ to denote quantities associated with the reduced state of the first $k$-modes. As mentioned before, calculating the mean vector (covariance matrix) of the reduced state is particularly simple for a Gaussian system: to trace off the $j$-th mode, we simply remove the $j$-th and $j+M$-th rows (and columns) of the mean vector $\alpha$ and the covariance matrix $\sigma$ (or $Q$).
This concludes the presentation of our sampling algorithm. We illustrate the basic idea behind the algorithm in Fig. 4.
It is not difficult to see that our algorithm only needs to calculate $Mc$ probabilities to output one sample. This is an exponential speed-up compared to the algorithms calculating all $c^{M}$ probabilities.
To explicitly write down the run time of this algorithm, we need to know how fast can we calculate each probability (loop hafnian). To answer this question, we need to first study the structures of the (extended) adjacency matrices appearing at each step of our algorithm.
IV Adjacency matrices of Shallow circuits with local gates
Formally, an $M$-mode optical circuit is called shallow if the number of layers of optical elements scales as $D=\log(M)$. Here, we count a group of commuting gates as one layer. We say that a two-mode gate or an optical element is local, if it acts on two adjacent modes.
Intuitively, a shallow circuit generates less quantum entanglement compared to a deep one: we expect that such system is easy to simulate. It turns out this is true, and the key to prove it lies in the observation that the relevant adjacency matrices involved in sampling from such a shallow circuit have banded structures which we can exploit to speed up the simulation.
Definition 2.
A matrix $A$ is banded with bandwidth $w$ if
$A_{i,j}=0$ for all $\left|i-j\right|>w$.
We would like to show that, for GBS with a shallow circuit of local gates, the extended adjacency matrices of each reduced Gaussian state are banded matrices. We first prove that the unitary matrix is banded.
Lemma 3.
Consider an $M$-mode optical circuit with $D$ layers of local gates, the associated unitary transformation is a banded matrix with bandwidth $D$.
Proof.
Observe that one layer of local (two-mode and single-mode) gates is represented by a unitary matrix with bandwidth equal to $1$. This is because each gate at most entangles two adjacent modes. Since the bandwidth is additive under matrix multiplication, repeatedly applying this property, we end up with a banded unitary matrix with bandwidth $D$.
∎
Lemma 4.
Consider an $M$-mode GBS circuit with D layers of local gates and uniform loss.
The adjacency matrix of the first $k$, $1\leq k\leq M$, modes has the form
$$\displaystyle A^{(k)}$$
$$\displaystyle=\begin{pmatrix}B^{(k)}&C^{(k)}\\
(C^{(k)})^{T}&(B^{(k)})^{*}\end{pmatrix},$$
(15)
where $B^{(k)}$ and $C^{(k)}$ have bandwidth $w\leq 4D$.
Proof.
We will show that the inverse of $Q^{(k)}$, which is the covariance matrix of the first $k$ modes, is block banded. If this is the case, it trivially follows that $A^{(k)}$ is block banded as it is given by
$$\displaystyle A^{(k)}$$
$$\displaystyle:=\begin{pmatrix}0&\mathbb{I}_{k}\\
\mathbb{I}_{k}&0\end{pmatrix}\left[\mathbb{I}_{2k}-\left(Q^{(k)}\right)^{-1}%
\right].$$
(16)
Before starting the proof, we set up some basic notation. We will consider a set of $M$ mixed single-mode states which are characterized by the following moments
$$\displaystyle n_{i}=\langle\delta a^{\dagger}_{i}\delta a_{i}\rangle\geq 0,%
\quad m_{i}=\langle\delta a_{i}^{2}\rangle=\langle\delta a_{i}^{\dagger 2}%
\rangle^{*},$$
(17)
where we defined $\delta a_{i}=a_{i}-\langle a_{i}\rangle$.
Pure squeezed states with squeezing parameters $r_{i}$ going through a circuit with uniform losses by overall energy transmission $\eta$ can be accommodated in the parametrization introduced above by setting
$$\displaystyle n_{i}=\eta\sinh^{2}r_{i},\quad m_{i}=\tfrac{1}{2}\eta\sinh 2r_{i},$$
(18)
where the lossless case is recovered by setting $\eta=1$.
Since we are interested in local circuits with depth $D$, the matrix $U$, describing the interferometer mixing the $M$ single-mode states, is banded with bandwidth $D$ as proved in Lemma 3.
With this notation we write
$$\displaystyle Q=$$
$$\displaystyle\;\frac{\mathbb{I}_{2M}}{2}+\sigma,\quad\sigma=VTV^{\dagger},$$
(19)
$$\displaystyle V=$$
$$\displaystyle\,\begin{pmatrix}U^{*}&0\\
0&U\end{pmatrix},$$
(20)
$$\displaystyle T=$$
$$\displaystyle\,\begin{pmatrix}\bigoplus_{i=1}^{M}\ (n_{i}+\tfrac{1}{2})&%
\bigoplus_{i=1}^{M}\ m_{i}\\
\bigoplus_{i=1}^{M}\ m_{i}^{*}&\bigoplus_{i=1}^{M}\ (n_{i}+\tfrac{1}{2})\end{%
pmatrix}.$$
(21)
We can write the adjacency matrix of the $M$ modes as Jahangiri
et al. (2020b); Rahimi-Keshari
et al. (2015)
$$\displaystyle A=$$
$$\displaystyle\;X(\mathbb{I}_{2M}-Q^{-1})=\begin{pmatrix}B&C\\
C^{T}&B^{*}\end{pmatrix},$$
(22)
$$\displaystyle B=$$
$$\displaystyle\;U\left(\bigoplus_{i=1}^{M}\lambda_{i}\right)U^{T}=B^{T},$$
(23)
$$\displaystyle C=$$
$$\displaystyle\;U\left(\bigoplus_{i=1}^{M}\mu_{i}\right)U^{\dagger}=C^{\dagger},$$
(24)
$$\displaystyle\lambda_{i}=$$
$$\displaystyle\;\frac{m_{i}}{(1+n_{i})^{2}-|m_{i}|^{2}},$$
(25)
$$\displaystyle\mu_{i}=$$
$$\displaystyle\;1-\frac{1+n_{i}}{(1+n_{i})^{2}-|m_{i}|^{2}}.$$
(26)
For pure states, one finds that $\mu_{i}=0$ for all $i$. More generally, it holds that $B$ and $C$ are banded with bandwidth $2D$.
To proceed, we first note that
$$\displaystyle Q^{(k)}=$$
$$\displaystyle\frac{\mathbb{I}_{2k}}{2}+W_{k}\sigma W_{k}^{\dagger},$$
(27)
$$\displaystyle W_{k}:=$$
$$\displaystyle\begin{pmatrix}E_{k}&0\\
0&E_{k}\end{pmatrix},$$
(28)
$$\displaystyle E_{k}:=$$
$$\displaystyle\begin{pmatrix}\mathbb{I}_{k}&0\end{pmatrix}.$$
(29)
In the last equation $E_{k}$ is a $k\times M$ matrix,
and thus $W_{k}$ has size $2k\times 2M$.
Note that Eq. (27) has precisely the form of the left hand side of the Sherman-Morrison-Woodbury identity Hager (1989)
$$\displaystyle\left(A+VCU\right)^{-1}=A^{-1}-A^{-1}V\left(C^{-1}+UA^{-1}V\right%
)^{-1}UA^{-1},$$
(30)
and thus we write
$$\displaystyle\left[Q^{(k)}\right]^{-1}=$$
$$\displaystyle\;2\mathbb{I}_{2k}-4W_{k}S^{-1}W_{k}^{\dagger},$$
(31)
$$\displaystyle S\,:=$$
$$\displaystyle\;\sigma^{-1}+2W_{k}^{\dagger}W_{k}.$$
(32)
We claim that the matrix $S^{-1}$ is block banded with bandwidth at most $4D$.
Proving this claim will conclude the proof,
as it then follows that $[Q^{(k)}]^{-1}$ is also block banded, with the same bandwidth.
Now we examine the terms in the matrix $S$.
Note that
$$\displaystyle W_{k}^{\dagger}W_{k}$$
$$\displaystyle=\begin{pmatrix}E_{k}^{\dagger}E_{k}&0\\
0&E_{k}^{\dagger}E_{k}\end{pmatrix},$$
(33)
$$\displaystyle E_{k}^{\dagger}E_{k}$$
$$\displaystyle=\begin{pmatrix}\mathbb{I}_{k}&0\\
0&0\end{pmatrix}.$$
(34)
Next we look at
$$\displaystyle\sigma^{-1}$$
$$\displaystyle=VT^{-1}V^{\dagger},$$
(35)
$$\displaystyle T^{-1}$$
$$\displaystyle=\left(\begin{array}[]{cc}\bigoplus_{i=1}^{M}\frac{n_{i}+\tfrac{1%
}{2}}{\left(n_{i}+\tfrac{1}{2}\right)^{2}-|m_{i}|^{2}}&\bigoplus_{i=1}^{M}%
\frac{-m_{i}}{\left(n_{i}+\tfrac{1}{2}\right)^{2}-|m_{i}|^{2}}\\
\bigoplus_{i=1}^{M}\frac{-m_{i}^{*}}{\left(n_{i}+\tfrac{1}{2}\right)^{2}-|m_{i%
}|^{2}}&\bigoplus_{i=1}^{M}\frac{n_{i}+\tfrac{1}{2}}{\left(n_{i}+\tfrac{1}{2}%
\right)^{2}-|m_{i}|^{2}}\\
\end{array}\right).$$
(36)
We are now ready to write
$$\displaystyle S^{-1}=$$
$$\displaystyle\;V\tilde{S}^{-1}V^{\dagger},$$
(37)
$$\displaystyle\tilde{S}\,:=$$
$$\displaystyle\;T^{-1}+2V^{\dagger}\,(W_{k}^{\dagger}W_{k})\,V.$$
(38)
Recall that the matrix $U$ describing the interferometer
is the product of precisely $D$ unitary matrices that are block diagonal,
where the blocks have either size one or two.
Similarly, the matrix $V=U^{*}\oplus U$ is also a product of block diagonal unitary matrices.
Given the special block structure of $W_{k}^{\dagger}W_{k}$,
we conclude that
$$\displaystyle V^{\dagger}\,(W_{k}^{\dagger}W_{k})\,V=\begin{pmatrix}Y&0\\
0&Y\end{pmatrix}.$$
(39)
where $Y=\mathbb{I}_{k}\oplus K\oplus 0_{M-2D-k}$
and $K$ is a $2D\times 2D$ positive semi-definite matrix.
Since the blocks of $T^{-1}$ are diagonal,
it follows that the inverse of $\tilde{S}$ has the form
$$\displaystyle\tilde{S}^{-1}=\begin{pmatrix}G&F\\
F^{T}&G^{*}\end{pmatrix},$$
(40)
where $G$ and $F$ are diagonal except for a block of size $2D\times 2D$.
After multiplying by $V$ on the left and by $V^{\dagger}$ on the right,
we conclude that $S^{-1}$ is block banded with bandwidth at most $4D$, as claimed.
∎
We can improve our upper bound on the bandwidth to $w\leq 2D$. This can be understood by noting that each layer of local gates actually has ‘half’ depth, so we have depth one for two layers of gates. Nonetheless, the upper bound of $4D$ is sufficient for our purposes.
We then observe the following:
Lemma 5.
There exists a permutation of rows and columns which transforms a block banded matrix $A^{(k)}$, where each block has bandwidth $w$, into a banded matrix with bandwidth $2w$.
Proof.
The permutation is given by
$$\displaystyle(1,\ldots,k,k+1,\ldots,2k)\rightarrow(1,k+1,\ldots,k,2k)~{}.$$
(41)
∎
Since diagonal elements do not affect the bandwidth, our results apply to $\tilde{A}^{(k)}$ as well. Combining Lemma 4 and Lemma 5, it follows that, for GBTS with threshold $c$, at each step of Algorithm 1 we need to calculate hafnians of a banded matrix with bandwidth at most $8Dc$.
V Fast computation of loop hafnians for banded matrices
The hafnian and loop hafnian are generalizations of the permanent,
which is $\sharp$P-complete to compute in the general setting Valiant (1979).
The best known algorithm for computing permanents of arbitrary matrices was introduced by Ryser Ryser (1963) and has complexity $O(n\,2^{n})$.
As for hafnians,
Björklund Björklund (2012) and Cygan and Pilipczuk Cygan and Pilipczuk (2015) derived algorithms with complexity
$O(\mathrm{poly}(n)\,2^{n/2})$ over arbitrary rings.
Subsequent work by Björklund et al. Björklund et al. (2019)
computed loop hafnians of complex matrices in time $O(n^{3}\,2^{n/2})$.
The goal of this section is to derive an algorithm for efficiently computing loop hafnians of banded matrices,
in time ${O}(n\,w\,4^{w})$ where $w$ is the bandwidth.
Previous work by Cifuentes and Parrilo Cifuentes and Parrilo (2016) proved that the computation of permanents of a banded matrix can be done in time ${O}(n\,w^{2}\,4^{w})$.
Similarly, Schwartz Schwartz (2009) gave an $O(8^{w}\log n)$
algorithm for computing hafnians of matrices which are both banded and Toeplitz. Finally, Temme and Wocjan Temme and Wocjan (2012) provided efficient algorithms for computing permanents of matrices that are block factorizable.
Theorem 6.
Let $A$ be a symmetric $n\times n$ matrix with bandwidth $w$ over an arbitrary ring.
Then we can compute its loop hafnian using ${O}(n\,w\,4^{w})$ arithmetic operations.
We first introduce some notation.
Let $G=(V,E)$ be the underlying graph structure of a banded matrix.
The vertex set is $V=[n]=\{1,2,3,\ldots,n\}$,
and the edge set $E$ consists of all pairs $(i,j)$ with $i\leq j\leq i{+}w$.
Given a list of edges $\pi\subset E$, we denote
$$\displaystyle A(\pi):=\prod_{ij\in\pi}A_{ij}.$$
(42)
With the above notation, we have that
$$\displaystyle\operatorname{lhaf}(A)=\sum_{\pi\in\operatorname{PM}(G)}A(\pi),$$
(43)
where $\operatorname{PM}(G)$ is the set of perfect matchings of $G$.
Throughout this section we assume that the matrix $A$ is fixed.
Given a subset of indices $D\subset[n]$
we denote $\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D)$ the set of perfect matchings of the restricted graph $G|_{D}$.
We consider the subhafnian
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)$$
$$\displaystyle:=\sum_{\pi\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D)}A(\pi).$$
(44)
Equivalently $\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)$ is the loop hafnian of the principal submatrix of $A$ indexed by $D$.
Our algorithm to compute $\operatorname{lhaf}(A)$ relies on dynamic programming,
and is presented as Algorithm 2.
For each $t\in[n]$ we compute a table $H_{t}$.
The loop hafnian of $A$ is one of the entries in the last table $H_{n}$. See Appendix B for an explicit example where we present how to apply our algorithm to calculate the loop hafnian of a $5\times 5$ matrix.
For each $t\in[n]$, let
$$\displaystyle a_{t}$$
$$\displaystyle:=\max\{t{-}2w,\,1\},$$
(45a)
$$\displaystyle X_{t}$$
$$\displaystyle:=\{a_{t},a_{t}{+}1,\dots,t\},$$
(45b)
$$\displaystyle\Delta_{t}$$
$$\displaystyle:=\{1,2,\dots,a_{t}{-}1\}.$$
(45c)
Note that $|X_{t}|=t$ for $t\leq 2w$,
and $|X_{t}|=2w{+}1$ for $t>2w$.
The table $H_{t}$ is indexed by subsets $\bar{D}\in X_{t}$.
In particular, the table has at most $2^{2w+1}$ entries.
As we explain next, each entry $H_{t}(\bar{D})$ is a subhafnian.
Consider the collection
$$\displaystyle\mathcal{S}=\{D\subset[n]:\Delta_{t}\subset D\subset[t]\}.$$
(46)
Note that a set $D\!\in\!\mathcal{S}$ is completely determined by its intersection with $X_{t}$.
So if we let $\bar{D}:=D\cap X_{t}$, there is a one to one correspondence between $\mathcal{S}$ and the subsets of $X_{t}$.
The subhafnians that we are interested in are
$$\displaystyle H_{t}(\bar{D})\,:=\,\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)\,=\,%
\mathrm{lhaf}_{\!A\mskip-1.0mu }(\bar{D}\cup\Delta_{t}),\quad\text{ for }\bar{%
D}\subset X_{t}.$$
(47)
In particular, the loop hafnian of $A$ is the entry $H_{n}(X_{n})=\mathrm{lhaf}_{\!A\mskip-1.0mu }([n])$ of table $H_{n}$.
The recursion used in Algorithm 2 relies on the next lemma.
Lemma 7.
Let $2\!\leq t\!\leq\!n$ and let $D$ be a subset of $[t]$
that contains $\Delta_{t}\!\cup\!\{t\}.$
Then
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)$$
$$\displaystyle=\sum_{i\in X_{t}}A_{it}\;\mathrm{lhaf}_{\!A\mskip-1.0mu }(D%
\setminus\{i,t\}).$$
(48)
Before proving the lemma,
let us see that the lemma implies the correctness of the algorithm.
Denoting $\bar{D}:=D\cap X_{t}$, the above equation can be rewritten as
$$\displaystyle H_{t}(\bar{D})$$
$$\displaystyle=\sum_{i\in X_{t}}A_{it}\;H_{t-1}(\bar{D}\setminus\{i,t\})$$
(49)
Each iteration of Algorithm 2 uses the above recursion formula.
It follows that the subhafnians $H_{t}(\bar{D})=\operatorname{lhaf}(\bar{D}\cup\Delta_{t})$ are computed correctly.
Proof of Lemma 7.
Given a matching $\pi^{\prime}\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D{\setminus}\{i,t\})$,
we can obtain a matching in $\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D)$ by adding the edge $(i,t)$.
This gives a function:
$$\displaystyle\bigcup_{i\in X_{t}}\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D{%
\setminus}\{i,t\})$$
$$\displaystyle\;\to\;\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D),\quad\pi^{%
\prime}\;\mapsto\;\pi^{\prime}\cup\{(i,t)\}.$$
Conversely, in any matching $\pi\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D)$
we have that $t$ is connected to a unique $i$,
and removing the edge $(i,t)$ gives a matching in $\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D{\setminus}\{i,t\})$.
Therefore, the function defined above is a bijection.
Hence,
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)$$
$$\displaystyle=\,\sum_{\pi\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(D)}A(\rho)$$
(50)
$$\displaystyle=\,\sum_{i\in X_{t}}\;\sum_{\pi^{\prime}\in\mathrm{PM}_{\mskip-1.%
0mu G\mskip-1.0mu }(D\setminus\{i,t\})}\!\!\!A_{it}\cdot A(\pi^{\prime})$$
(51)
$$\displaystyle=\,\sum_{i\in X_{t}}A_{ij}\cdot\mathrm{lhaf}_{\!A\mskip-1.0mu }(D%
\setminus\{i,t\}).$$
(52)
∎
Proof of Theorem 6.
We already showed correctness,
so it remains to estimate the complexity.
In each iteration we need to compute
$H_{t}(\bar{D})$ for each $\bar{D}\subset X_{t}$.
Since $|X_{t}|\leq 2w{+}1$, we need to consider at most $2\cdot 4^{w}$ subsets.
The recursion formula (49) requires $O(|X_{t}|)\!=\!O(w)$ ring operations.
Hence, each iteration of the algorithm takes $O(w\,4^{w})$,
and the overall cost is $O(n\,w\,4^{w})$.
∎
Recall that the probability of detecting a photon pattern with collisions is given by the loop hafnian of an extended adjacency matrix.
For a GBTS with threshold $c$, we have at most $c$ repetitions for each column and row.
By simply applying Theorem 6, calculating each probability in our Algorithm 1 is upper bounded by $O^{*}(4^{wc})$.
However, such scaling, with an exponential dependence on the number of repetitions, is an overestimate.
On one hand, repetitions increase the matrix size, indeed increasing the cost of calculating its hafnian.
On the other hand, the repetition structure does not carry much new information so that we also expect to see certain cost reduction.
Below we devise a faster algorithm which requires $O^{*}(n(2c{+}2)^{2w+1})$ steps to calculate the loop hafnian of a banded matrix with at most $c$ repetitions. Since this result is not necessary to prove our main result, we omit the detailed proof here. Interested readers can find the full proof in Appendix C.
Theorem 8.
Let $A$ be a complex-symmetric $n\times n$ matrix with bandwidth $w$.
Let $s=(s_{1},\dots,s_{n})$ be a vector of positive integers,
and let $A_{s}$ be the symmetric matrix
obtained from $A$ by repeating the $i$-th row and column $s_{i}$ times.
Then we can compute $\operatorname{lhaf}(A_{s})$ in time
${O}^{*}(n\,(2c{+}2)^{2w+1})$,
where $c:=\!\max\{s_{1},\dots,s_{n}\}$.
Proof.
See Appendix C.
∎
VI Gaussian threshold boson sampling with shallow circuits of local gates
We are now ready to put together everything we have prepared so far and to prove our main result.
Theorem 9.
Consider an $M$-mode, uniformly lossy GBTS problem with threshold $c>0$. If
the unitary transformation consists of, $D$ layers of commuting local gates, the sampling task can be simulated in running time $T=O^{*}(M^{2}\,\mathrm{poly}(c)\,(2c+2)^{16D})$. Consequently, when the linear-optical circuit is shallow, i.e., when $D=O(\log(M))$, the sampling can be simulated efficiently on a classical computer.
Proof.
Recall that at the $k$-th step of Algorithm 1, we need to calculate $c$ hafnians (the one correspond to zero photons is trivial to calculate): $\operatorname{lhaf}(\tilde{A}^{(k)}_{s_{1},\ldots,s_{k}})$ for $1\leq s_{k}\leq c$ . The dependence of the adjacency matrices at step $k$ on the samples from the previous step seems to be complicated to analyze. However, an upper bound on the run time of our classical algorithm, by considering the most costly outcome $s=(c,\ldots,c)$, is straightforward.
From Lemma 4, we know that the adjacency matrix of the reduced $k$-mode Gaussian state $A^{(k)}$ is a block banded matrix where the blocks have width at most $4D$.
Since the loop hafnian is invariant under permutations, invoking Lemma 5 and Theorem 8, we know the cost of finishing step $k$ when Algorithm 1 is executed is upper bounded by
$$\displaystyle T_{k}=\sum_{i=0}^{c}\left[2(k{-}1)c+2i\right]O^{*}((2c{+}2)^{16D%
+1})~{}.$$
(53)
Summing over $k=1,\ldots,M$ we have the following total run time
$$\displaystyle T=O^{*}(M^{2}\,\mathrm{poly}(c)\,(2c{+}2)^{16D})~{}.$$
(54)
For a shallow circuit, we have $D=O(\log(M))$, which gives $T=\mathrm{poly}(M)$ as $c$ is constant. This concludes our proof.
∎
VII Conclusion
We have proved that sampling from Gaussian states prepared by shallow quantum-optical circuits with local interactions can be simulated efficiently. We have introduced Gaussian Boson Threshold Sampling (GBTS), by which we obtain not only a well-defined model ready for mathematical analysis, but also a faithful representation of real experiments where photon detectors can be overloaded. We have investigated the structure of the adjacency matrices describing Gaussian states prepared by shallow circuits: they are, up to permutations, banded matrices with bandwidth that is proportional to circuit depth. We have also introduced a dynamic-programming algorithm to exploit the structure of banded matrices with repeated rows and columns, a subroutine to be called in our sampling algorithm. Putting together these steps, we obtain an efficient sampling algorithm for shallow circuits with local gates if the depth grows as a logarithm of the number of modes.
Similar results should be obtainable for regular Boson Sampling by using the faster algorithm for calculating permanents of banded matrices Cifuentes and Parrilo (2016). Actually, we expect the proof for BS can be much simpler than ours since 1) BS is defined with finite photon number and 2) the relevant matrices are simply submatrices of the unitary matrix.
However, a successful proof also requires an algorithm that calculates a polynomial number of probabilities to output one sample. A good candidate is provided by Clifford and Clifford Clifford and Clifford (2018), though their algorithm requires careful examination to make it compatible with the results from Ref. Cifuentes and Parrilo (2016). Such examination seems to be neglected in previous papers Muraleedharan et al. (2019); Lundow and Markström (2019) in which the authors attempted to leverage the idea of banded matrices.
The complexity of our algorithm agrees with the intuition that weakly-entangled multipartite states are not hard for a classical computer to simulate. Our results rule out optical systems with a shallow-depth circuit and local gates as a candidate for the demonstration of quantum supremacy.
Moreover, if the photon loss compounds exponentially with the circuit depth, which is the case for most of today’s photonic platforms, it was shown that the sampling becomes asymptotically simulable García-Patrón
et al. (2019); Qi et al. (2020). This poses a great challenge to the demonstration of quantum supremacy using optical circuits with local interactions. Therefore, our work calls for further study of the computational complexity of sampling photonic states generated by shallow-circuits with non-local gates Lubasch et al. (2018).
Acknowledgements
H.Q. thanks Raúl García-Patrón Sánchez for suggesting the initial idea of this project and Daniel Brod and Alexander M. Dalzell for helpful discussion.
Appendix A Correctness of Algorithm 1
Here we show that Algorithm 1 indeed samples from the distribution defined in Eq. 11. When the algorithm does not halt, the sampled pattern does not overload any PNR, that is $\bm{n}\notin\Sigma_{c}$ and its probability is given by
$$\displaystyle\tilde{p}(\bm{n})=\prod_{k=1}^{M}p(n_{k}|n_{1}\ldots n_{k-1})=p(%
\bm{n})~{}.$$
(55)
When the algorithm does halt, the probability that it halts at step $k$ is given by
$$\displaystyle p(\text{halt at kth step})=\sum_{n_{1},\ldots,n_{k-1}<c,n_{k}%
\geq c}\prod_{l=1}^{k}p(n_{k}|n_{1}\ldots n_{k-1})~{}.$$
(56)
Then the total probability of the algorithm halted, output $\#$, is given by
$$\displaystyle p(\#)=$$
$$\displaystyle\sum_{k=1}^{M}p(\text{halt at }k\text{th step})$$
$$\displaystyle=$$
$$\displaystyle\left(\sum_{n_{1}\geq c,n_{2},\ldots,n_{M}}+\sum_{n_{1}\leq c,n_{%
2}\geq c,n_{3}\ldots,n_{M}}+\ldots\right)p(\bm{n})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\bm{n}\in\Sigma_{c}}p(\bm{n})~{}.$$
(57)
Therefore, indeed our Algorithm 1 simulates GBTS.
Appendix B Banded hafnian example calculation
Here we illustrate our loop hafnian algorithm for banded matrices by considering an adjacency matrix with five vertices and having bandwidth equal to one ($w=1$), namely
$$\displaystyle\operatorname{lhaf}\left(\left[\begin{array}[]{ccccc}0&a&0&0&0\\
a&0&b&0&0\\
0&b&c&d&0\\
0&0&d&0&e\\
0&0&0&e&f\\
\end{array}\right]\right)\;=\;ace+adf.$$
(58)
We denote by $\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)$ the loop hafnian of the submatrix of $A$ given by indices in $D$.
Trivially, $\operatorname{lhaf}(A)=\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2,\ldots,n\})$.
The algorithm calculates many such subhafnians,
storing their values in the dynamic programming table.
By reusing previous subhafnians
as well as omitting several subsets $D$ based on the bandwidth,
the algorithm is able to calculate the hafnian of the overall matrix in ${O}(n\,w\,4^{w})$ arithmetic operations.
The algorithm proceeds in $n$ steps.
In the $t$-th step it calculates the subhafinans given by the subsets $D$
such that $\{1,2,\ldots,t{-}2w{-}1\}\subseteq D\subseteq\{1,2,\ldots,t\}$.
These subhafnians are computed using the formula:
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(D)=\sum_{i=t-w}^{t}A_{it}\,%
\mathrm{lhaf}_{\!A\mskip-1.0mu }(D\setminus\{i,t\}).$$
Step 1
$D\subseteq\{1\}$
Compute $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\emptyset)\!=\!1$ and $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1\})\!=\!0$.
Step 2
$D\subseteq\{1,2\}$
Besides the subhafnians from the previous step,
compute $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{2\})\!=\!0$ and $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2\})\!=\!a$.
Step 3
$D\subseteq\{1,2,3\}$
Four new subhafnians:
$\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{3\})\!=\!c$, $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,3\})\!=\!0$, $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{2,3\})\!=\!b$, and $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2,3\})\!=\!ac$.
Step 4
$\{1\}\subseteq D\subseteq\{1,2,3,4\}$
Four new hafnians
($D\!=\!\{1,4\}$, $\{1,2,4\}$, $\{1,3,4\}$, $\{1,2,3,4\}$).
In particular, $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2,3,4\})$ is obtained with the formula:
$$\displaystyle A_{3,4}\cdot\mathrm{lhaf}_{\!A\mskip-1.0mu }$$
$$\displaystyle(\{1,2\})+A_{4,4}\cdot\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2,3\})$$
$$\displaystyle=\,d\cdot a+0\cdot ac\,=\,ad,$$
where the needed subhafnians were already computed.
Step 5
$\{1,2\}\subseteq D\subseteq\{1,2,3,4,5\}$
Four new hafnians.
In particular,
$\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2,3,4,5\})$ is obtained with the formula:
$$\displaystyle A_{4,5}\cdot\mathrm{lhaf}_{\!A\mskip-1.0mu }$$
$$\displaystyle(\{1,2,3\})+A_{5,5}\cdot\mathrm{lhaf}_{\!A\mskip-1.0mu }(\{1,2,3,%
4\})$$
$$\displaystyle=\,e\cdot ac+f\cdot ad\,=\,ace+adf,$$
where the needed subhafnians were known.
The above value is the loop hafnian of the original matrix.
Appendix C Loop hafnian algorithm for matrices with repetitions
In this section we derive an efficient algorithm for computing loop hafnians of banded matrices with repeated entries.
Let $A$ be a symmetric $n\times n$ matrix with bandwidth $w$
and let $s=(s_{1},\dots,s_{n})\in\mathbb{N}^{n}$ be a vector of positive integers.
We will compute the loop hafnian of the matrix $A_{s}$,
obtained by repeating the $i$-th row and column $s_{i}$ times.
The special case $n=1$ is quite important for the analysis.
In this case the matrix $A$ and the vector $s$ are scalars,
so $A_{s}$ is a constant matrix.
Let $T_{k}(a)$ denote the loop hafnian of an $k\times k$ constant matrix
with all entries equal to $a$.
The sequence $\{T_{k}(a)\}_{k\in\mathbb{N}}$ satisfies the following recursion:
$$\displaystyle T_{0}(a)=1,\qquad T_{1}(a)=a,$$
$$\displaystyle T_{k}(a)=a\,\bigl{(}T_{k-1}(a)+(k{-}1)T_{k-2}(a)\bigr{)}.$$
In particular, $T_{k}(1)$ is the $k$-th telephone number Björklund et al. (2019).
Consider now an arbitrary $n$.
Let $G=(V,E)$ be the underlying graph structure of $A$,
with vertex set $V=[n]$.
We will define some generalized perfect matchings of $G$ that allow repeated edges.
We represent a list of repeated edges as a vector $\tau\!\in\!\mathbb{N}^{E}$,
i.e., $\tau$ is a vector indexed by $E$,
and for each $ij\!\in\!E$ the entry $\tau_{ij}\!\in\!\mathbb{N}$ indicates the number of times that edge $ij$ appears.
The degree vector of $\tau$
is the vector $\deg(\tau)\in\mathbb{N}^{n}$
with coordinates $\deg(\tau)_{i}:=\sum_{j\in[n]}\tau_{ij}$.
For a weight vector $s\in\mathbb{N}^{n}$,
we define an $s$-matching of $G$ as a vector $\tau\in\mathbb{N}^{E}$
such that $\deg(\tau)=s$.
Let $\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(s)$ be the set of all $s$-matchings of $G$.
The following lemma expresses $\operatorname{lhaf}(A_{s})$ as sum of some simple quantities associated to each matching $\tau\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(s)$.
The contribution of $\tau$ to $\operatorname{lhaf}(A_{s})$ is a multiple of
$$\displaystyle A(\tau)\,:=\,\left(\prod_{ii\in E_{\ell}}T_{\tau_{ii}}(A_{ii})%
\right)\,\left(\prod_{ij\in E_{0}}(A_{ij})^{\tau_{ij}}\right),$$
(59)
where $E_{\ell}\subset E$ consists of the loops in the graph,
and $E_{0}:=E\setminus E_{\ell}$ consists of the remaining edges.
Lemma 10.
Let $A$ be a symmetric $n\times n$ matrix,
and let $s\in\mathbb{N}^{n}$.
Then
$$\displaystyle\operatorname{lhaf}(A_{s})\,=\,s!\sum_{\tau\in\mathrm{PM}_{\mskip%
-1.0mu G\mskip-1.0mu }(s)}\frac{1}{\tau!}\cdot A(\tau),$$
(60)
where $s!:=\prod_{i\in[n]}s_{i}!$
and $\tau!\,:=\,\prod_{ij\in E}\tau_{ij}!\,.$
Proof.
Assume first that the graph has no loops ($E_{\ell}\!=\!\emptyset$).
Let $G_{s}$ be the graph associated to $A_{s}$.
We may view its vertices as pairs $(i,\ell_{i})$ where $i\in[n]$ and $\ell_{i}\in[s_{i}]$,
and its edges do not depend on $\ell_{i}$.
Given a perfect matching $\pi\in\operatorname{PM}(G_{s})$
there is a natural way to obtain an $s$-matching $\tau\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(s)$.
Namely, for each edge $(i,\ell_{i}),(j,\ell_{j})$ in $\pi$ we ignore the second coordinate and obtain the edge $(i,j)$.
This gives a function $f:\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(s)$.
It is clear that $A_{s}(\pi)=A(f(\pi))$.
Given $\tau\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(s)$,
a simple combinatorial argument shows that the fiber $f^{-1}(\tau)$ consists of exactly $s!/\tau!$ elements.
Then
$$\displaystyle\operatorname{lhaf}(A_{s})=\sum_{\pi\in\operatorname{PM}(G_{s})}A%
_{s}(\pi)=\sum_{\tau\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(s)}\frac{s!}{%
\tau!}\cdot A(\tau).$$
Consider now the general case, where loops are allowed.
In such a case the equation $A_{s}(\pi)=A(f(\pi))$ is no longer valid,
but we still have that
$$\displaystyle\sum_{\pi\in f^{-1}(\tau)}A_{s}(\pi)=\frac{s!}{\tau!}\cdot A(\tau).$$
Hence, the argument from above still applies.
∎
In order to compute $\operatorname{lhaf}(A_{s})$,
we will use a variant of the dynamic program in Algorithm 2.
One of the main differences is that the dynamic programming table $H_{t}$
is now indexed by vectors $\bar{d}\in\mathbb{N}^{n}$ instead of subsets $\bar{D}\subset[n]$.
We need additional notation to present the algorithm.
From now on we assume that the matrix $A$
and the weight vector $s$ (entrywise positive) are fixed.
Given another weight vector $d\in\mathbb{N}^{n}$, the associated scaled subhafnian is
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(d)\,:=\,\frac{1}{d!}\cdot%
\operatorname{lhaf}(A_{d})\,=\,\sum_{\tau\in\mathrm{PM}_{\mskip-1.0mu G\mskip-%
1.0mu }(d)}\frac{1}{\tau!}\cdot A(\tau).$$
(61)
The support of $d$ is $\operatorname{supp}(d):=\{i\!\in\![n]:d_{i}\!\neq\!0\}$.
Given $X\!\subset\![n]$, the restriction $d|_{X}\!\in\!\mathbb{N}^{n}$ is obtained by setting the entries outside of $X$ to zero, i.e.,
$$\displaystyle(d|_{X})_{i}=\begin{cases}d_{i},&\text{ if }i\in X\\
0,&\text{ if }i\in[n]\setminus X\end{cases}$$
(62)
We only consider weight vectors $d$ such that $d\leq s$ entrywise.
The saturated indices of $d$ are
$\operatorname{sat}(d):=\{i\in[n]:d_{i}\!=\!s_{i}\}$.
Note that $\operatorname{sat}(d)\subset\operatorname{supp}(d)$.
Finally, we denote $\mathbb{N}_{\leq d}^{X}$ the set of weight vectors supported by $X$ and upper bounded by $d$:
$$\displaystyle\mathbb{N}_{\leq d}^{X}:=\{e\in\mathbb{N}^{n}\,:\,\operatorname{%
supp}(e)\subset X,\;\;e\leq d\}.$$
(63)
Our dynamic program to compute $\operatorname{lhaf}(A_{s})$ is given in Algorithm 3.
For each $t\in[n]$ we compute a table $H_{t}$.
The loop hafnian of $A_{s}$ can be obtained from the last table $H_{n}$.
For each $t\in[n]$, let $a_{t},X_{t},\Delta_{t}$ be as in (45).
The table $H_{t}$ is indexed by vectors $\bar{d}\in\mathbb{N}_{\leq s}^{X_{t}}$.
Note that the number of such vectors is
$\prod_{i\in X_{t}}(s_{i}{+}1)\leq(c{+}1)^{2w+1}$,
where $c:=\!\max\{s_{1},\dots,s_{n}\}$.
The entries $H_{t}(\bar{d})$ of the table are scaled subhafnians.
Consider the collection
$$\displaystyle\mathcal{S}=\{d\in\mathbb{N}^{n}:\Delta_{t}\subset\operatorname{%
sat}(d)\subset\operatorname{supp}(d)\subset[t]\}.$$
(64)
Observe that $d\!\in\!\mathcal{S}$ is completely determined by the restriction to $X_{t}$.
So if we let $\bar{d}:=d|_{X_{t}}$, there is a one to one correspondence between $\mathcal{S}$ and $\mathbb{N}_{\leq s}^{X_{t}}$.
The scaled subhafnians that we are interested in are
$$\displaystyle H_{t}(\bar{d}):=\mathrm{lhaf}_{\!A\mskip-1.0mu }(d)=\mathrm{lhaf%
}_{\!A\mskip-1.0mu }(\bar{d}+s|_{\Delta_{t}}),\quad\text{ for }\bar{d}\in%
\mathbb{N}_{\leq s}^{X_{t}}.$$
In particular, $H_{T}(s|_{X_{n}})=\mathrm{lhaf}_{\!A\mskip-1.0mu }(s)=\frac{1}{s!}%
\operatorname{lhaf}(A_{s})$.
The recursion used in Algorithm 3 relies on the next lemma.
Lemma 11.
Let $t>1$ and let $d\in\mathbb{N}^{n}$ be such that
$$\displaystyle\Delta_{t}\subset\operatorname{sat}(d)\subset\operatorname{supp}(%
d)\subset[t],\qquad d\leq s.$$
Denoting
$Y_{t}:=X_{t}\!\setminus\!\{t\}$
and $\sigma_{t}(d^{\prime}):=d^{\prime}_{t}-\sum_{i\in Y_{t}}d^{\prime}_{i}$,
let
$$\displaystyle g(d^{\prime}):=\textstyle\frac{1}{\sigma_{t}(d^{\prime})!}\,T_{%
\sigma_{t}(d^{\prime})}(A_{tt})\cdot\prod\nolimits_{i\in Y_{t}}\frac{1}{d^{%
\prime}_{i}!}\,(A_{it})^{d_{i}^{\prime}},$$
$$\displaystyle\mathcal{D}:=\{d^{\prime}\!\in\!\mathbb{N}_{\leq d}^{X_{t}}:d^{%
\prime}_{t}\!=\!d_{t},\sigma_{t}(d^{\prime})\!\geq\!0\}.$$
Then
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(d)=\sum_{d^{\prime}\in\mathcal{D%
}}g(d^{\prime})\;\mathrm{lhaf}_{\!A\mskip-1.0mu }(d{-}d^{\prime}),$$
$$\displaystyle\Delta_{t-1}\subset\operatorname{sat}(d{-}d^{\prime})\subset%
\operatorname{supp}(d{-}d^{\prime})\subset[t{-}1]\;\;\;\forall d^{\prime}\!\in%
\!\mathcal{D}.$$
Proof of Lemma 11.
We start with the second equation.
Let $d^{\prime}\!\in\!\mathcal{D}$ and $d^{\prime\prime}:=d{-}d^{\prime}$.
Since $d^{\prime}_{t}\!=\!d_{t}$ then $d^{\prime\prime}_{t}\!=\!0$,
and hence $\operatorname{supp}(d^{\prime\prime})\!\subset\![t{-}1]$.
If $j\!\in\!\Delta_{t-1}\!\subset\!\Delta_{t}$ then $j\!\notin\!X_{t}$,
and hence $d^{\prime}_{j}\!=\!0$, $d^{\prime\prime}_{j}\!=\!d_{j}\!\neq\!0$.
Therefore, $\Delta_{t-1}\subset\operatorname{sat}(d^{\prime\prime})$.
We proceed to the first equation.
The proof is quite similar to that of Lemma 7.
Given $d^{\prime}\!\in\!\mathcal{D}$,
consider the matching
$\tau(d^{\prime})\in\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(d^{\prime})$
defined as follows:
$$\displaystyle\tau(d^{\prime})_{ij}:=\begin{cases}\sigma_{t}(d^{\prime}),&\text%
{if }i\!=\!j\!=\!t,\\
d^{\prime}_{i},&\text{if }i\!\in\!Y_{t},\,j\!=\!t,\\
0,&\text{otherwise.}\end{cases}$$
Note that
$g(d^{\prime})=A(\tau(d^{\prime}))/(\tau(d^{\prime})!)$.
Consider the function
$$\displaystyle\bigcup_{d^{\prime}\in\mathcal{D}}\mathrm{PM}_{\mskip-1.0mu G%
\mskip-1.0mu }(d{-}d^{\prime})$$
$$\displaystyle\;\to\;\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(d),\qquad\tau^{%
\prime\prime}\;\mapsto\;\tau(d^{\prime})+\tau^{\prime\prime}.$$
It can be shown that this function is a bijection.
Then
$$\displaystyle\mathrm{lhaf}_{\!A\mskip-1.0mu }(d)\,=\,\sum_{\tau\in\mathrm{PM}_%
{\mskip-1.0mu G\mskip-1.0mu }(d)}\frac{1}{\tau!}\cdot A(\tau).$$
$$\displaystyle=\,\sum_{d^{\prime}\in\mathcal{D}}\;\sum_{\tau^{\prime\prime}\in%
\mathrm{PM}_{\mskip-1.0mu G\mskip-1.0mu }(d-d^{\prime})}\frac{1}{\tau(d^{%
\prime})!\cdot\tau^{\prime\prime}!}\cdot A(\tau(d^{\prime}))\cdot A(\tau^{%
\prime\prime}).$$
$$\displaystyle=\,\sum_{d^{\prime}\in\mathcal{D}}g(d^{\prime})\cdot\mathrm{lhaf}%
_{\!A\mskip-1.0mu }(d\!-\!d^{\prime}).\qed$$
Proof of Theorem 8.
We first prove correctness.
The recursion formula in Lemma 11 is a convolution
over the vectors ${d}\!\in\!\mathbb{N}^{n}$ supported on $X_{t}$.
Each iteration of Algorithm 3 performs such a convolution.
It follows by induction on $t$ that the values $H_{t}(\bar{d})$ computed by Algorithm 3 are indeed given by $\mathrm{lhaf}_{\!A\mskip-1.0mu }(\bar{d}+s|_{\Delta_{t}})$,
so the algorithm is correct.
We proceed to estimate the complexity.
We only analyze the cost of the convolution,
since this is the dominant term.
Recall that the fast Fourier transform allows us to compute circular convolutions.
We can avoid the circular effect by appending some zeros.
It follows that the (non-circular) convolution can be computed in
${O}\left(w\,(2c{+}2)^{2w{+}1}\log c\right)$,
where $c:=\!\max\{s_{1},\dots,s_{n}\}$.
The running time of the whole algorithm is
${O}\left(n\,w\,(2c{+}2)^{2w{+}1}\log c\right)$.
∎
The complexity in Theorem 8 depends on the largest entry of $s$.
This is not convenient if, for instance, there is a single large entry.
Denoting $S_{t}:=\prod_{i=t}^{t+2w}(2s_{i}{+}2)$
and $C:=\!\max\{S_{1},\dots,S_{n}\}$,
our bound can be refined to
${O}^{*}(n\,C)$. An analogous results exists for generic (as opposed to banded) matrices derived by Kan Kan (2008).
Although Theorem 8 is stated only for complex matrices,
Algorithm 3 can be applied in more general rings.
The complexity remains the same as long as the ring admits a fast Fourier transform.
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Virtual Forward Dynamics Models for Cartesian Robot Control
Stefan Scherzinger
Stefan Scherzinger
Arne Roennau
Rüdiger Dillmann FZI Forschungszentrum Informatik
Haid-und-Neu-Str 10-14
76131 Karlsruhe, Germany
Tel.: +0049-721-9654-226
22email: {scherzinger, roennau, dillmann}@fzi.de
Arne Roennau
Stefan Scherzinger
Arne Roennau
Rüdiger Dillmann FZI Forschungszentrum Informatik
Haid-und-Neu-Str 10-14
76131 Karlsruhe, Germany
Tel.: +0049-721-9654-226
22email: {scherzinger, roennau, dillmann}@fzi.de
Rüdiger Dillmann
Stefan Scherzinger
Arne Roennau
Rüdiger Dillmann FZI Forschungszentrum Informatik
Haid-und-Neu-Str 10-14
76131 Karlsruhe, Germany
Tel.: +0049-721-9654-226
22email: {scherzinger, roennau, dillmann}@fzi.de
(Received: date / Accepted: date)
Abstract
In industrial context, admittance control represents an important scheme in
programming robots for interaction tasks with their environments.
Those robots usually implement high-gain disturbance rejection on joint-level
and hide direct access to the actuators behind velocity or position
controlled interfaces.
Using wrist force-torque sensors to add compliance to these systems, force-resolved control laws
must map the control signals from Cartesian space
to joint motion. Although forward dynamics algorithms would perfectly fit to
that task description, their application to Cartesian robot control is not well
researched. This paper proposes a general concept of virtual forward dynamics models for Cartesian
robot control and investigates how the forward mapping behaves in comparison to
well-established alternatives.
Through decreasing the virtual system’s link masses in comparison to the end effector, the
virtual system becomes linear in the operational space dynamics.
Experiments focus on stability and manipulability,
particularly in singular configurations.
Our results show that through this trick, forward
dynamics can combine both benefits of the Jacobian inverse and the Jacobian
transpose and, in this regard, outperforms the Damped Least Squares method.
Keywords:Forward dynamics robot control kinematics
††journal: Journal for Intelligent & Robotic Systems
∎
1 Introduction
In robotics, task space control is important for many applications, since it
provides a natural way for programmers to specify goals and constraints.
The according control laws can be formulated in operational space of the end-effector.
Since the robots are articulated mechanisms and are powered in their joints, these controllers
need to map the Cartesian control signals to the robots’
configurations space, i.e. the motor actuators.
We will refer to matrices that accomplish this as mapping matrices.
Two frequently-used variants of mapping matrices are the transpose of the
manipulator’s Jacobian and its inverse.
The Jacobian transpose is an important part in many classes of control schemes
for torque-actuated robots, such as in
hybrid force/motion control Whitney1977, Raibert1981, Khatib1987,
parallel force/motion control Chiaverini1993,
and impedance control Hogan1985, Villani2008.
In principal, the approaches use the Jacobian transpose as static relationship between
end-effector wrenches and joint torques for controlling robots in contact with their environments.
Although not strictly required, control performance is generally improved
through decoupling robot dynamics in operational space Khatib1987, prior
to mapping the signals to joint space.
In addition, there are also algorithmic solutions using the principle of
inverse dynamics to compute suitable joint torques from motion control signals, e.g.
with the Recursive Newton-Euler Algorithm (RNEA) Featherstone2008.
An important subset of robots, however, does not provide joint interfaces on torque level.
These systems are often found in industrial context, and are the primary focus
of this paper.
In Villani2008, those systems are referred to as
simplified systems because they hide internal dynamics decoupling behind a velocity interface.
In this paper, we will refer to them as velocity-actuated systems to underline the type of interface exposed by the robot vendors.
On these systems, velocity-resolved control variants, such as admittance control Villani2008,
usually leverage Jacobian inverse-related methods, such as the Damped Least Squares (DLS) Wampler1986 as the mapping to joint space.
Unlike inverse dynamics for torque-actuated systems,
literature on velocity-actuated robots mostly neglects forward
dynamics as an algorithmic option for control.
This is surprising, because it represents a straightforward mapping from
Cartesian wrench space to joint accelerations.
While we used this method to control robots in previous work
Scherzinger2017, Scherzinger2019Inverse, the new contribution of this paper is an in-depth analysis of
particular features of this mapping and an evaluation against other well-established methods.
The goal and novelty is a drop-in-replacement for the Jacobian inverse and DLS
in controllers for velocity-actuated robots.
Through using a dynamics-conditioned, virtual forward model, we match
the linear, decoupled behavior of the Jacobian inverse while simultaneously
keeping the inherent robustness in singularities of the Jacobian transpose
method.
The paper is structured as follows:
In 2 we briefly recapitulate the inverse kinematics mapping problem
along with established methods to make it easy for the reader to follow comparisons in the experiments.
Section 3 then presents forward dynamics-based mappings for Cartesian control.
In the experiments section 4, we investigate ill conditioning, stability and manipulability in singular configurations and evaluate our approach against suitable subsets of the Jacobian Inverse, the Jacobian transpose and the DLS method.
We finally discuss remaining points and suggestions in
5 and conclude with directions for further research in 6.
2 Problem statement and related work
The goal of this paper is to evaluate forward dynamics-based mappings against
well-established methods.
We use Singular Value Decomposition (SVD) as a tool to investigate
the characteristics of the mapping matrices.
SVD factorizes a matrix $\bm{M}$ according to
$$\bm{M}=\bm{U}\bm{\Sigma}\bm{V}^{T},\quad\text{with}~{}\sigma_{i}=\Sigma_{ii}.$$
(1)
$\bm{U}$ and $\bm{V}^{T}$ are orthogonal matrices. The entries $\sigma_{i}\geq 0$
of the diagonal matrix $\bm{\Sigma}$ are known as the singular values and
determine the scale of the mapping.
For our experiments, $\sigma_{\text{min}}$ and
$\sigma_{\text{max}}$ as the minimal and maximal singular values are
of particular interest in analyzing stability and manipulability.
To recapture some basic concepts, let the forward kinematics mapping be given with
$$\bm{x}=g\left(\bm{q}\right),$$
(2)
which computes an end-effector pose, denoted here with $\bm{x}$ from the joint state vector $\bm{q}$.
The velocity vector of generalized coordinates $\dot{\bm{q}}$ maps with
$$\dot{\bm{x}}=\bm{J}\dot{\bm{q}}$$
(3)
to end effector velocity vector $\dot{\bm{x}}$, using the manipulator Jacobian
$\bm{J}=\bm{J}(\bm{q})$. We generally omit the joint vector dependency in
further notation for brevity reasons.
For non-redundant manipulators, the inverse mapping is given by
$$\dot{\bm{q}}=\bm{J}^{-1}\dot{\bm{x}}~{}.$$
(4)
Near singular configurations, $\bm{J}$ looses rank, such that its inverse becomes numerically unstable.
The respective mapping for end-effector forces and torques to joint space with
$$\bm{\tau}=\bm{J}^{T}\bm{f}$$
(5)
does not suffer from these instabilities.
However, $\bm{J}$ becoming rank-deficient means that some components of
$\bm{f}$ will lie in the nullspace of the Jacobian transpose, i.e. they will be
balanced by the mechanism’s mechanics and will hence not be able to actuate the joints.
This effect is a severe limitation for controller implementations.
Applied to motion control, a formal investigation of the Jacobian transpose method and
a numerical solution to the Inverse Kinematics problem was presented
in Wolovich1984. The authors’ solution derives from a simple, 2nd order dynamical system
$$\ddot{\bm{q}}=\bm{K}\bm{J}^{T}\left(\bm{x}^{d}-g(\bm{q})\right)~{},$$
(6)
computing joint accelerations from the difference of a desired pose $\bm{x}^{d}$ and the current pose as determined by the forward kinematics $g(\bm{q})$.
They show with a Lyapunov stability analysis that the system is asymptotically stable for
an arbitrary positive definite matrix $\bm{K}$.
Using $\bm{J}^{T}$ will serve as a lower bound and
baseline in stability considerations of our contribution.
The DLS method is an applications of the
Levenberg-Marquardt stabilization to manipulator control Nakamura1986,
Wampler1986 and tries to remove instabilities of $\bm{J}^{-1}$ near singular configurations.
Note that the original version as proposed in Wampler1986 uses partial velocity matrices, adding the benefit of allowing different reference frames for each element.
Since we don’t make use of this feature, we replace it with the common
manipulator Jacobian $\bm{J}$ instead.
Similar to pseudo inverse methods for redundant systems, which minimize $\lVert\bm{J}\dot{\bm{q}}-\dot{\bm{x}}\rVert^{2}$, the idea is to add a damping term $\alpha$ against excessive joint velocities
that will trade-off accuracy for stability near singular configurations with
$$\lVert\bm{J}\dot{\bm{q}}-\dot{\bm{x}}\rVert^{2}+\alpha^{2}\lVert\dot{\bm{q}}%
\rVert^{2}.$$
(7)
The solution that minimizes this quantity is given by
$$\dot{\bm{q}}=(\bm{J}^{T}\bm{J}+\alpha^{2}\bm{I})^{-1}\bm{J}^{T}\dot{\bm{x}},$$
(8)
see e.g. Buss2004 for a derivation.
The matrix $(\bm{J}^{T}\bm{J}+\alpha^{2}\bm{I})$ is non-singular, which can be
shown with singular value decomposition Buss2004 and hence is guaranteed to be invertible.
This method is well established for practical control implementations and can serve as a drop-in replacement for $\bm{J}^{-1}$ in control loops.
We use this method as a baseline to compare our new forward dynamics-based
method in terms of manipulability.
A popular enhancement to DLS, using this method, is Selectively Damped Least Squares (SDLS)
Buss2005.
The method converges faster and circumvents to choose
a suitable $\alpha$ by introducing singular vector-specific damping terms of the singular
value decomposition of $\bm{J}$ at the expense of a higher runtime cost.
Other methods include the more recent Exponentially Damped Least Squares
(EDLS) Carmichael2017, which is a solution with the focus on physical
Human-Robot interaction (pHRI). Although effectively avoiding elbow-lock and
wrist-lock among other common singular phenomena, it requires explicit, albeit
easy parameterization by the user.
Both the Jacobian inverse and the Jacobian transpose have strengths and shortcomings and
present mappings for physically different control spaces.
When used on velocity-actuated systems, the Jacobian inverse does not need
dynamic decoupling, but suffers from instability, which DLS effectively
mitigates at the expense of loosing accuracy with increased damping. The Jacobian transpose needs
dynamics decoupling in the controller, but offers inherent stability
near singular configurations.
A general incentive is to obtain the best of these corner cases.
This paper proposes and empirically evaluates forward dynamics as a suitable approach to
achieve this combination at the core of closed-loop control schemes.
3 Cartesian control with forward dynamics
3.1 Forward dynamics simulations
To motivate the usage of forward dynamics in control applications let’s
illustrate its behavior with a use case:
Fig. 1 depicts an arbitrary manipulator with joints and links.
Let’s assume that the joints are pure articulations without motors and are
back-drivable, i.e. they can freely be moved.
An external force $\bm{f}$ is pulling the mechanism into singularity, in which the mechanism yields the external forces as good as it can, limited by kinematic constraints.
In the fully stretched case, increasing $\bm{f}$ will not create further motion.
The robot’s mechanics compensate the external load until a possible breakage of the links.
This behavior is in fact inverse to how $\bm{J}^{-1}$ would compute joint
motion due to an external error vector, where theoretically infinite joint
velocity would occur.
To make use of forward dynamics simulations, robotic manipulators can be modeled as a system of articulated, rigid bodies.
The equations of motion describe the relationship
between generalized loads $\bm{\tau}$ in the joints, external loads $\bm{f}$, acting on the end-effector and motion in generalized coordinates $\bm{q}$
with the following ordinary differential equations in symbolic matrix notation
$$\bm{\tau}+\bm{J}^{T}\bm{f}=\bm{H}({\bm{q}})\ddot{\bm{q}}+\bm{C}(\bm{q},\dot{%
\bm{q}})+\bm{G}(\bm{q})~{}.$$
(9)
$\bm{H}$ denotes the mechanism’s positive definite inertia matrix,
$\bm{C}$ comprises the Coriolis and
centrifugal terms and $\bm{G}$ is the vector of gravitational components.
Forward dynamics computation has the goal of solving Eq. (9) for $\bm{q}(t)$, i.e.
simulating the mechanism’s reaction motion through time, given external loads.
Literature has proposed several methods for forward dynamics computations, which can be categorized Featherstone2008 as mainly
belonging to inertia matrix methods with implementations of the
composite rigid body algorithm, e.g. in McMillan1998,
Featherstone2005, or the propagation methods with the
articulated body algorithm Featherstone1983 being an important
representative.
We refer the interested reader to Featherstone2008 for an broad coverage
of the field and recent implementation of various algorithms.
Forward dynamics is a substantial component in multi body simulations.
The fact that it is, however, mainly neglected for closed-loop control on
velocity-actuated systems may stem from the fact that computing good
approximations for $\bm{H}$ and $\bm{C}$ of the robots is
extremely difficult. The required crucial data, such as link masses and inertia
tensors is hardly available in data sheets.
On the other hand, a second reason for not being used might be that even if those data were
available, the benefit of forward simulating highly realistic motion would get
lost when executed on
velocity-actuated systems. Their internal joint servos with high-gain
disturbance rejection could not make use of the accuracy of dynamics that was
used to generate that motion.
The reference trajectory to follow would appear as any arbitrary trajectory.
This thought points to an interesting opportunity:
We could reduce Eq. (9) to a rough simplification and
investigate, if it’s possible to condition $\bm{H}$ to beneficially tweak the
behavior of this mapping when using it as a forward model in closed-loop
control.
3.2 A general closed-loop control
To motivate simplifications to Eq. (9), we investigate how a
controller would perceive the system in a possible closed loop control.
A general scheme is shown in Fig. 2.
A suitable control law computes a Cartesian control signal $\bm{f}^{c}$, using a
user specified reference input and a controlled variable as feedback from the
robot.
Note the role of the virtual system as a forward model in the controller:
We simulate how our proxy system behaves and send that as a reference to the real system.
Since we obtain joint accelerations as a response from our forward model, we
integrate those signals before sending them as reference to the real robot.
The advantage is, that this virtual model will react kinematically and
mechanically plausible to external loads $\bm{f}^{c}$, as was illustrated in
Fig. 1.
From the control law’s point of view, a linear, virtual system would be beneficial
for using constant control gains for wide regions of the robot’s joint configuration space.
By dropping the gravity term ($\bm{G}(\bm{q})=\bm{0}$) from Eq. (9), we assure that the control law does not need to constantly compensate this virtual load.
If we further consider instantaneous motion for each control cycle, i.e.
accelerate from rest with $\dot{\bm{q}}=\bm{0}$, we can drop the
non-linearities $\bm{C}(\bm{q},\dot{\bm{q}})$ and obtain
$$\ddot{\bm{q}}=\bm{H}^{-1}(\bm{q})\bm{J}^{T}\bm{f}^{c}~{}$$
(10)
as an unbiased forward mapping.
We also set $\bm{\tau}\equiv\bm{0}$ to emphasize that $\bm{f}^{c}$ shall be
the only virtual load guiding the virtual system.
While dropping these terms reduces computational complexity in our controller,
including them can offer additional configuration.
This is briefly discussed in section 5.
Note that $\bm{H}^{-1}(\bm{q})$ needs to be computed in each control cycle due
to its dependency of the current joint state. In the experiments section, we
evaluate computational cost in comparison to other mapping matrices.
Since Eq. (10) is effectively a Jacobian transpose-based
method, the next step is to decouple our virtual $\bm{H}^{-1}$.
3.3 Decoupling virtual dynamics
We start with the time derivative of Eq. (4)
$$\ddot{\bm{x}}=\dot{\bm{J}}\dot{\bm{q}}+\bm{J}\ddot{\bm{q}}~{}~{}$$
(11)
and consider instantaneous accelerations in each cycle while the virtual system is still at
rest, so that $\dot{\bm{J}}\dot{\bm{q}}=\bm{0}$.
With Eq. (10) we obtain
$$\ddot{\bm{x}}=\bm{J}\bm{H}^{-1}\bm{J}^{T}\bm{f}^{c}=\bm{\Lambda}^{-1}\bm{f}^{c%
}~{}~{},$$
(12)
which describes the Cartesian instantaneous acceleration of the virtual system
due to the Cartesian control input $\bm{f}^{c}$.
The quantity $\bm{\Lambda}$ is known as the mass matrix in operational space,
see e.g. Khatib1987, Villani2008, with
$\bm{\Lambda}=\bm{J}^{-T}\bm{H}\bm{J}^{-1}$.
The intention of our dynamic decoupling is to make $\bm{\Lambda}^{-1}$ a
time-invariant, diagonal matrix across joint configurations $\bm{q}$. This
ideal mapping is illustrated in Fig. 3.
In order to preserve consistency of our virtual system and the real robot, we
use identical kinematics for both systems. This ensures that the reference
signals for the real robot to follow agree with possible limits.
We are, however, free in changing the dynamics of the virtual $\bm{H}$ to
obtain the desired effect, in particular its mass distribution.
The Cartesian control signal $\bm{f}^{c}$ acts directly on
the virtual mechanism’s end effector.
If that end-effector link is dominant with respect to the overall dynamics, determined by the other links,
we could obtain a behavior that comes close to an idealized unit mass.
Fig. 4 illustrates this phenomenon.
As a consequence, the overall systems’ center of mass roughly stays with the end-effector.
Likewise does the operational space inertia $\bm{\Lambda}$ depend less on joint configurations,
and $\bm{f}^{c}$ experiences the same rotational inertia for both configurations.
Having a realistic link mass distribution would instead mean higher inertia
with greater distance to the rotary axis.
To measure the effect of end-effector mass dominance, we define
$$\gamma=\frac{m_{e}}{m_{l}}=\frac{ip_{e}}{ip_{l}}$$
(13)
to be the ratio of end-effector mass $m_{e}$ and link mass $m_{l}$.
The quantities $ip_{e}$ and $ip_{l}$ denote the polar momentums of inertia of the
end-effector and the other links, respectively.
In the experiments section, we empirically show that increasing $\gamma$ in deed leads to
the desired behavior and provides a decoupled virtual system for Cartesian closed-loop control.
3.4 Closed-loop stability
Comparison of Eq. (6) and Eq. (10)
shows a strong resemblance of our forward dynamics-based approach with the
dynamical system from Wolovich1984, if $\bm{f}^{c}=\bm{x}^{d}-g(\bm{q})$.
In Wolovich1984, the authors prove
with a Lyapunov stability analysis
that a closed loop system, built from this mapping, is asymptotically stable for
an arbitrary positive definite matrix $\bm{K}$.
This formal proof also includes our proposed $\bm{H}^{-1}$, which is, due to
being grounded in the manipulator’s kinetic energy $T=\dot{\bm{q}}^{T}\bm{H}\dot{\bm{q}}$, also positive definite.
4 Experiments and results
We evaluated our forward dynamics-based approach against the DLS and against the two corner cases
$\bm{J}^{-1}$ and $\bm{J}^{T}$ in various experiments.
We chose the Universal Robot UR10’s kinematics for our experiments.
Our perception is that this robot is well-known and used both in industry and academia and therefore presents a suitable platform.
Depending on the phenomena investigated, a subset of different mapping matrices was used.
An overview of these matrices and their composition is given in
Fig. 5 along with the abbreviations used in the
plots.
We implemented each mapping matrix literally, i.e. as a multiplication of the
respective symbols in C++, using the robot’s kinematics from a popular
ROS Quigley2009
package111https://github.com/ros-industrial/universal_robot and the
algorithms for computing $\bm{J}$ and $\bm{H}$ from a well-established robotics
library222https://github.com/orocos/orocos_kinematics_dynamics.
For all experiments, the following values were chosen for the forward dynamics mappings:
$$m_{e}=$1\text{\,}\mathrm{k}\mathrm{g}$,\quad m_{l}=\frac{m_{e}}{\gamma},\quad
ip%
_{e}=$1\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{2}$,\quad ip_{l}=%
\frac{ip_{e}}{\gamma}$$
(14)
The ratio $\gamma$ was then varied according to the investigation of each experiment.
4.1 Decoupling
In this experiment, we evaluated the effectiveness of our virtual model dynamics decoupling and compared the mapping to both the Jacobian inverse and the Jacobian transpose as reference.
The mapping matrices were of type (b) from Fig. 5.
Fig. 6 shows the results of the analysis.
All mean matrices are diagonal, which is to expect for sampling a vast amount of arbitrary joint configurations.
The standard deviations, however, show a strong configuration dependency for the
Jacobian transpose.
This mapping would be suboptimal if used in a closed-loop control scheme.
Instead, the Jacobian inverse behaves ideal and converges to the identity matrix.
Note that using forward dynamics with an even mass distribution ($\gamma=1$)
already improves upon the Jacobian transpose.
The experiment further shows, that with a significant end-effector mass
dominance ($\gamma=10e^{3}$) the forward dynamics mapping converges to the
Jacobian inverse and makes this mapping particularly suitable for closed-loop
control in terms of linearity.
4.2 Ill-conditioned configurations
In this experiment, we continued the evaluation of decoupling and compared FD and DLS with regard to ill conditioning
of the mapping matrices from Fig. 5(b).
Higher numbers of ill conditioning degrade control performance
Featherstone2004, but heavily depend on the manipulators configuration.
This experiment investigates how FD and DLS influence ill conditioning
by varying $\gamma$ and $\alpha$, respectively.
Based on Featherstone2004, we used $\kappa=\sigma_{\text{max}}/\sigma_{\text{min}}$ as the measure for ill conditioning.
Fig. 7 shows the results.
For each discrete point in the plots, we evaluated $1000$ random joint states
with the limits from Fig. 6.
We used quartiles on our data to effectively exclude outliers ($\sigma_{\text{min}}\rightarrow 0$), such that
the plots show the median of the ill conditioning.
It can be seen that FD converges much faster to beneficial condition numbers over its own parameter space than DLS.
In fact, most of the decoupling effect from experiment Fig. 6
is already available for low values of $\gamma$.
4.3 Behavior in singularities
Before reporting on this experiment, we briefly recapitulate desired and expected behavior in singularities.
In singular configurations, the manipulator Jacobian $\bm{J}$ becomes rank-deficient.
This is an unfortunate joint configurations for all considered approaches.
As a consequence, the manipulator is not able to achieve instantaneous motion in all directions Murray1994.
Two issues arise from this constellation:
First, Jacobian transpose-based methods tend to loose manipulability.
We measure this effect with $\sigma_{\text{min}}$ of the mapping matrix, which
is one of various established measures Murray1994.
Second, for Jacobian inverse-based methods, infinite joint velocity occurs.
We measure this effect with $\sigma_{\text{max}}$ as an indicator of how much
the mapping matrix scales $\bm{f}^{c}$ in sensitive dimensions to joint space.
The goal of the experiment is to investigate how well
each approach behaves in singular configurations concerning both measures.
As a reference, Fig. 8 shows both the Jacobian inverse and
the Jacobian transpose for a pass through four singular configurations, the first two being close together.
The curves show the expected and well-known effect: The Jacobian inverse maintains high
manipulability at the cost of an exploding $\sigma_{\text{max}}$, while the
Jacobian transpose stays stable throughout the pass but cannot avoid
$\sigma_{\text{min}}$ dropping to zero in singularities.
Fig. 9 and
Fig. 10 finally show the behavior of
forward dynamics with a set of different $\gamma$.
The curves show how FD approaches the Jacobian inverse while maintaining
$\sigma_{\text{max}}$ in stable ranges.
We added the DLS method, albeit with only one $\alpha$, for comparison.
Both FD and DLS have very similar characteristics and show a good trade-off
between both corner cases (JI, and JT).
Note how the curves for FD become more pronounced towards the Jacobian inverse for increasing $\gamma$.
4.4 Empirical analysis of stability and manipulability
In this experiment, we wanted to analyze FD in comparison to DLS on a broader scale.
The goal is an empirical analysis of varying $\alpha$ (DLS) and $\gamma$ (FD) over bigger ranges,
and evaluate how they perform in the corner cases in comparison to the Jacobian inverse and transpose.
Instead of focusing a few trajectories, we sampled a massive amount of singular constellations.
Note that in contrast to the workspace sampling for experiment
4.1, which contained singular
configurations by chance, in this investigation we exclusively used singular
configurations.
Through exclusively focusing regions of low performance (singularities) throughout the
whole workspace, the results become a feasible measure of global performance for each method.
To find a big amount of singular configurations,
we first sampled equally distributed, random joint states as start configurations.
We then used Particle Swarm Optimization (PSO) Miranda2018PySwarms,
which implements an adapted algorithm from the original work of
Kennedy1995 as a black-box optimizing strategy to converge to singular
configurations from these start states.
We used Yoshikawa’s manipulability measure $\sqrt{\text{det}(\bm{J}\bm{J}^{T})}$, which simplifies for non-redundant mechanisms to $\lvert\text{det}(\bm{J})\rvert$ Yoshikawa1985Manipulability as function to minimize,
which was faster than using SVD with $\sigma_{\text{min}}$ directly.
Alternatively, a more type-based approach of finding singularities is discussed
in Zlatanov1998, Bohigas2012Numerical.
Having a set of $1000$ singular configurations, we then computed average values
for $\sigma_{\text{min}}$ and $\sigma_{\text{max}}$ from the mapping matrices
of FD and DLS according to Fig. 5(a) for
discrete values of $\alpha$ and $\gamma$ for each of the $1000$ singular
configurations.
Fig. 11 shows the results for manipulability.
Both DLS and FD approach the Jacobian inverses behavior for decreasing $\alpha$ and increasing $\gamma$, respectively.
Note how FD approaches qualitatively faster in its own parameter space.
Fig. 12 shows the results for stability.
DLS comes closer to the Jacobian transposes stability than FD throughout most
of the observed parameter space.
However, towards reaching the Jacobian inverses high manipulability, DLS
looses stability and asymptotically approaches infinity, while FD in contrast
stays bounded.
For applications in which DLS would require very low values of $\alpha$ for
control performance, FD can be used as a safe alternative, combining
and keeping the benefits of both $\bm{J}^{-1}$ and $\bm{J}^{T}$.
4.5 Computational efficiency
Finally, we measured average execution times of the mapping approaches.
We implemented the SDLS method according to Buss2005 and included the
measurements as additional reference.
To obtain the comparison, we computed $\ddot{\bm{q}}$ as given in
Fig. 5(a) with a fictitious, constant
$\bm{f}^{c}=\bm{1}$ for $10e5$ times.
The joint state $\bm{q}$ was randomly sampled, while being identical across one single
evaluation of each method.
Fig. 13 shows the boxplots of each method’s execution
time with their quartiles. The median is plotted as vertical, orange line.
The whiskers for minimal an maximal execution times indicate a high degree of irregularity.
We expect narrower ranges for experiments on a hard real-time operating system.
The results show that the forward dynamics method is a little more
computationally intense than the DLS method, but approximately half the
execution time of the SDLS.
Being still in the low $\mu$s range makes forward dynamics in the version from this
paper suitable for real-time closed-loop control.
5 Discussion
5.1 Virtual forward models
When deriving our principal mapping for forward dynamics in Eq. (10),
we dropped gravity and non-linear terms to support dynamics decoupling of our virtual system.
For redundant manipulators, including those terms offers additional interfaces
to adjust behavior in the nullspace of the Jacobian transpose.
For those cases, switching from the composite rigid body algorithm to
propagation methods for solving the forward dynamics might be beneficial.
The articulated body algorithm Featherstone1983, e.g. allows an
intuitive integration of external loads to each link of the robot separately,
which might be used to implement collision avoidance or other optimizations
concerning the robots’ posture.
5.2 Control applications
The natural mapping of forward dynamics from Cartesian wrench space to joint
accelerations makes it particularly suitable for the implementation of
admittance-related controllers on velocity-actuated systems.
For those use cases, force-resolved control laws for disturbance rejection can
replace the velocity-resolved control laws using DLS.
The benefit of using the FD method is its ability to operate extremely
close to the ideal $\bm{J}^{-1}$ behavior without significantly sacrificing
stability.
Successful implementations of forward dynamics-based control on industrial
robots can be found e.g. in Scherzinger2019Contact for pure force control and in
Scherzinger2017, Heppner2020 for compliance control.
An application to motion control with a particular focus on sparsely sampled
targets is presented in Scherzinger2019Inverse.
6 Conclusion
This paper proposed virtual forward dynamics models for Cartesian robot control.
The core of the control loop is a simplified, virtual
model that maps Cartesian control signals to joint accelerations.
Through increasing the end effector’s mass in comparison to the other links, the
virtual system becomes linear in the operational space dynamics and matches
the exactness of the Jacobian inverse.
Further experiments showed, that this forward model’s decoupling
leads to less ill conditioning compared to the DLS method for an empirical
investigation of the joint space.
When passing through singularities, forward dynamics behaves in general similar
to DLS in terms of manipulability and stability.
Yet, when operating in singular configurations
forward dynamics models substantially differ from DLS in that they produce
bounded control signals, even when forced to approach the Jacobian inverse in
terms of manipulability.
These virtual forward models are particularly suitable for implementing
admittance controllers in industrial settings on velocity-actuated robots.
Their computation time in the low $\mu$s range makes them suitable for real-time control. |
Stateful Security Protocol Verification
Li Li1,
Jun Pang2,
Yang Liu3,
Jun Sun4,
Jin Song Dong1
1School of Computing, National University of Singapore, Singapore
2FSTC and SnT, University of Luxembourg, Luxembourg
3School of Computer Engineering, Nanyang Technological University, Singapore
4Information System Technology and Design, Singapore University of Technology and Design, Singapore
Abstract
A long-standing research problem in security protocol design is how to
efficiently verify security protocols with tamper-resistant global states.
In this paper, we address this problem by first proposing a protocol specification framework,
which explicitly represents protocol execution states and state transformations.
Secondly, we develop an algorithm for verifying security properties
by utilizing the key ingredients of the first-order reasoning for reachability analysis,
while tracking state transformation and checking the validity of newly generated states.
Our verification algorithm is proven to be (partially) correct, if it terminates.
We have implemented the proposed framework and verification algorithms in a tool named SSPA,
and evaluate it using a number of stateful security protocols.
The experimental results show that our approach is not only feasible but also practically efficient.
In particular, we have found a security flaw on the digital envelope protocol,
which could not be detected by existing security protocol verifiers.
I Introduction
Many widely used security protocols, e.g., [1, 2, 3, 4],
keep track of the protocol execution states.
These protocols maintain a global state among several sessions
and can behave differently according to the values stored in the global state.
More importantly, the protocol’s global state is tamper-resistant,
i.e., it cannot be simply cloned, faked, or reverted.
As the result, we cannot treat the global state as an input from the environment
so that the protocol becomes stateless.
In practice, such global states are usually extracted from trusted parties in protocols like
central trustworthy databases, trusted platform modules (TPMs), etc.
The global state poses new challenges for the existing verification techniques as discussed below.
First, most existing verification tools, e.g., ProVerif [5] and Scyther [6],
are designed for stateless protocols.
When they are used to verify stateful protocols,
false alarms may be introduced in the verification results.
For instance, when the protocol state is ignored in these tools,
a value generated in a later global state can be used in a former global state.
However, the execution trace is actually impractical.
Second, stateful protocols usually have sub-processes that can be executed for infinitely many times.
However, the state-of-the-art tools, e.g., [5, 6, 7], cannot handle loops.
As a consequence, only a finite number of protocol execution steps can be modeled and checked.
Therefore, valid attacks could be missed in the verification.
Even though some tools like Tamarin [8] can specify loops,
the verification cannot terminate for most stateful protocols
as they do not consider the states as tamper-resistant in the multiset rewriting rules [9].
Third, some of the abstractions made in the existing works tend to either
make the verification non-terminating for stateful protocols or introduce false alarms.
For instance, fresh nonces generated in ProVerif [5] are treated
as functions to the preceded behaviors in a session
so that the nonces with the same name could be merged under the same execution trace.
On one hand, if a stateful protocol receives some data before generating any nonce in its session,
the nonce generated in one session can be received
before the same nonce is generated in a different session.
According to the abstraction method, the nonce becomes a function applied to itself,
which could lead to infinite function applications.
Thus the verification cannot terminate.
On the other hand, if a nonce is generated without performing any session-specific behavior,
then the nonce will be the same for multiple sessions.
The query of asking whether the nonce for a particular session can be deduced may give false alarms,
because the nonce that can be deduced is actually coming from another session.
As these nonces are merged, they cannot be differentiated in the verification process.
To address the above identified challenges for verifying stateful security protocols,
we first propose a protocol specification framework (see Section IV)
that explicitly models the protocol execution state as tamper-resistant.
We specify how states are used in the protocol as well as how states are transferred.
As a result, stateful protocols can be modeled in our framework in an intuitive way.
The protocol specification is introduced with a motivating example
of the digital envelope protocol [4].
Second, a solving algorithm is developed to verify stateful protocols.
During solving, we apply a pre-order to the states and converge the states into a valid state trace.
The secrecy property checked in this work is then formulated into a reachability problem.
The partial correctness of our method is formally
defined in Section IV-E and proved in Section V.
However, as the security protocol verification problem is undecidable in general [10],
our algorithm does not guarantee the termination.
The experiments show that our method can terminate for
many stateful security protocols used in the real world.
Third, we develop a tool named SSPA (Stateful Security Protocol Analyzer) based on our approach.
Several stateful protocols including the digital envelop protocol
and the Bitlocker protocol [11] have been analyzed using SSPA.
The experiment results show that our method can both find security flaws and give proofs efficiently.
Particularly, we have found a security flaw in the digital envelope protocol which has not been identified before.
Structure of the paper.
Related works are discussed in Section II and
a motivating example is given in Section III.
In Section IV, we present our protocol specification framework
and describe how to specify cryptographic primitives, protocols and queries.
In Section V, we show how the verification algorithm works
and prove its partial correctness.
We show the implementation details and the experiment results in Section VI.
Finally, we conclude the paper with some discussions in Section VII.
II Related Works
Mödersheim developed a verification framework that works with global states [12].
His framework extends the IF language with sets
and abstracts the names based on its Set-Membership.
According to [12], this method works well for several protocols.
However, its applicability in general is unclear
since sets should be explicitly identified for the protocols
and no general solution for identifying the set is given in the paper.
Guttman extended the strand space with mutable states
to deal with stateful protocols [13, 14], but there is no tool support for his approach.
Our approach presented in this paper is different from theirs, as the protocol specification
does not need to be changed in our framework and we provide automatic tool support.
StatVerif is introduced by Arapinis et al. [7] to verify protocol with explicit states.
It extends the process calculus of ProVerif [5] with stateful operational semantics
and translates the resulting model into Horn clauses.
ProVerif is then used as an engine to perform verification. Comparing with their method that can only work with a finite number of (global) states,
our approach is more general and works for protocols with infinite states.
In [15], Delaune et al. modeled TPMs with Horn clauses and have verified three protocols using ProVerif.
However, the specifications need to be adapted according to the different protocols under study.
For instance, an additional parameter is added into the global state
when it is used for the digital envelope protocol (DEP) [4] to prevent false attacks.
More importantly, they also modified the specification of the DEP
in a way that false negatives can happen (attacks are missing)
comparing with the original DEP proposed in [4].
This is because their method does not work for infinite steps of the stateful protocols.
Specifically, they have constrained the protocol so that its second phase is not repeatable.
More discussions on the DEP can be found in Section III.
Notice that all of the previous methods can only work with protocols with finite steps.
while this is not the case with our approach.
III Motivating Example
We introduce the digital envelope protocol (DEP) [4] in this section as a motivating example.
Before going into the details of the protocol,
we give a brief introduction on the trusted platform module (TPM) [16] used in the protocol first.
TPM is an embedded cryptographic device proposed to give higher level security guarantees
than those can be offered by software alone.
Every TPM has several tamper-resistant platform configuration registers (PCRs)
that maintain the current state of the TPM.
The values stored in the PCRs can only be extended.
One possible implementation of extending a PCR $p$ with a value $n$ could be
$\mathit{extend}(n)\{p=\mathit{h}(p,n)\}$,
where $\mathit{h}$ is a one-way hash function applied to the concatenation of $p$ and $n$.
Hence, the extending actions are irreversible
unless the TPM reboot is allowed (the PCRs are reset to the default value $b$)
and the previous extending actions are replayed in an identical order.
TPM provides several APIs to help the key management,
including key generation, key usage, etc., under PCR measurement.
TPMs use several types of keys, including
the attestation identity keys (AIKs) and the warp keys.
The AIK represents the identity of the TPM in the protocol and can be used for signing.
In order to differentiate the TPMs, we assume every TPM has a unique AIK.
However, this assumption does not prevent the adversary from using multiple AIK values
as he could initiate multiple TPMs.
The warp keys form a tree structure rooted under the permanent loaded storage root key (SRK).
We usually use two kinds of warp keys in the TMP, i.e., the binding keys and the storage keys.
Data can be encrypted with the binding public key remotely,
or can be sealed with the loaded storage key in the TPM.
Typically, the TPM supports the following operations.
•
Extend. Extend the PCR value $p$ by any value $n$ to a new PCR value $h(p,n)$.
•
Read. Read the current PCR value from the TPM.
•
Quote. Certify the current PCR value.
•
CreateWrapKey. Generate a warp key under a loaded parent key
and bind it to a specific PCR value.
The new key is not yet loaded into the TPM but stored in a key blob,
which is a storage place for holding the key.
•
LoadKey2. Load the key into TPM by providing the key blob and its parent key.
•
CertifyKey. Certify a loaded key.
•
UnBind. Decrypt the data with a loaded binding key.
The PCR value for the key should be matched.
•
Seal. Encrypt the data with a storage key.
The PCR value for the key should be matched
and the encrypted data can be sealed to a particular PCR value.
•
UnSeal. Decrypt the data with the loaded storage key.
The PCR value of the seal key, the PCR value of the sealed storage
and the current PCR value are required to be the same.
As the storage key and seal/unseal operation are not used in the DEP,
we omit their specification in the following discussions.
By using TPMs, the DEP allows an agent Alice
to provide a digital secret $s$ to another agent Bob
in a way that Bob can either access $s$ without any further help from Alice,
or revoke his right to access the secret $s$ so that he can prove his revocation.
This protocol consists of two phases as shown in Figure 1.
In the first phase, Alice generates a secret nonce $n$
and uses it to extend a given PCR in Bob’s TPM with an encrypted session.
The transport session is then closed.
Since the nonce $n$ is secret, Bob cannot re-enter the current state of the TPM
if he makes any changes to the given PCR.
In the second phase, Alice and Bob read the value of the given PCR as $p$
and Bob creates a binding key pair $\langle sk,pk\rangle$ locked to the PCR value $\mathit{h(p,open)}$
and sends the key certification to Alice,
where $\mathit{open}$ is an agreed constant in the protocol.
This means the generated binding key can be used
only if the value $\mathit{open}$ is first extended to the PCR of value $p$.
After checking the correctness of the certification,
Alice encrypts the data $s$ with her public key $pk$ and sends it back to Bob.
Later, Bob can either open the digital envelope by extending the PCR with $\mathit{open}$
or revoke his right to open the envelope by extending another pre-agreed constant $\mathit{revoke}$.
If Bob revokes his right, the quote of PCR value $\mathit{h(p,revoke)}$
can be used to prove Bob’s revoke action.
The protocol is illustrated in Figure 1.
In fact, through our approach and the implemented tool,
we have found a cold-boot attack for this DEP
when the TPM reboot is allowed.
According to the DEP proposed in [4],
the authors only mentioned that Bob may lose his ability to open the envelope
or to prove his revoke action if the TPM reboot is allowed.
To the best of our knowledge, this attack has not been described before.
We present the attack scenario in Figure 2.
When the TPM reboot is allowed, Bob can reboot his TPM
immediately after the first phase.
As a consequence, the secret nonce $n$ extended to the given PCR is lost.
When Alice checks the PCR value in the beginning of the second phase,
she actually reads a PCR value that is unrelated to her previous extend action.
Hence, Bob can re-enter the current TPM state by simply performing TPM reboot again.
This attack is caused by the fact that
the PCR value in the second phase can be unrelated to the PCR value in the first phase.
On the other hand, if the TPM reboot is not allowed,
the secret nonce $n$ could never get lost.
So Alice can conduct the second protocol phase for multiple times
and the claimed properties of the DEP are always preserved.
In this way, if the TPM is maintained by a trusted server and remotely controlled by
both Alice and Bob without the right to reboot TPM, this protocol is secure.
This protocol was previously verified in [15].
However, the modifications made in [15] to the original DEP prevent the authors from detecting the attack.
In the modified version [15], Bob always does the TPM reboot before the first phase
and Alice assumes that the PCR is $h(b,n)$
without actually reading the value in the beginning of the second phase.
As a result, TPM reboot can never happen before the second phase.
The reason why they need to make such modifications is because
ProVerif, which is used in their verification, can only model finite protocol steps.
Unfortunately, this makes it impossible to find the attack as described in Figure 2.
On the contrary, in this work, we provide a framework
where protocols like this can be modeled faithfully and verified automatically.
IV Protocol Specification
In this section, we describe our specification framework
for modeling (stateful) protocols, crypto primitives and queries
as a set of first order logic rules with the protocol execution states explicitly maintained.
There are two categories of rules that can be specified in our approach,
i.e., state consistent rules and state transferring rules.
The state consistent rules specify the knowledge deductions,
while the state transferring rules describe the state transitions.
Since the protocol global state is tamper-resistant,
we assume that it can only be changed by the state transition rules.
The adversary model we consider in this work is the standard active attacker,
who can intercept all communications, compute new messages and send any messages that he can obtain or compute.
For instance, he can use all the public available functions including encryptions, decryptions and etc.
He can also ask the legitimate protocol participants to take part in the protocol.
That is, every rule specified in the framework describes a logic capability of the adversary.
Our goal is to check whether he can deduce a target fact or not.
IV-A Framework Overview
In our framework, every entity and device in the protocol
is treated as an object when it is tamper-resistant.
Every object have an object global state with a unique identity.
The protocol global state then consists of several object global states.
For simplicity, we name object global state after state for short,
and call protocol global state as protocol state in the remaining of this paper.
For instance, every TPM has a state $\mathit{tpm(aik,p)}$
which records the AIK value $\mathit{aik}$ and the PCR value $p$.
The AIK value uniquely identifies the TPM.
Initially, the protocol state of DEP is $\{\mathit{tpm(bob,p)}\}$,
where $\mathit{bob}$ stands for the AIK constant for Bob’s TPM.
After the first phase of the DEP, Alice enters a state $\mathit{alice(n)}$
where $n$ is the secret value that she extends to Bob’s TPM.
When Alice obtains the state, she can initiate the second phase of the protocol.
Because Alice could start several sessions to Bob’s TPM,
we treat $n$ as the identity of Alice’s state.
When Alice extends the nonce $n$ to Bob’s TPM,
the protocol state becomes $\{\mathit{tpm(bob,h(p,n))},\mathit{alice(n)}\}$.
A protocol state can contain several TPM states with different AIK values.
The states of the same object should be ordered in a timeline of protocol execution, forming a state trace.
For instance, the following sequence of four states is a legitimate TPM state trace in the DEP.
1.
$\mathit{tpm(bob,i)}$
2.
$\mathit{tpm(bob,h(i,n))}$
3.
$\mathit{tpm(bob,h(h(i,n),x))}$
4.
$\mathit{tpm(bob,h(h(h(i,n),x),revoke))}$
The first state is the initial state.
Then, in the first phase of the DEP,
Alice extends a secret nonce $n$ into Bob’s TPM (the second state).
Later, Bob extends a value $x$ into his TPM for other purposes (the third state)
and the second phase of the DEP begins.
At the beginning, Alice and Bob record the PCR value as $h(h(i,n),x)$.
When Bob receives Alice’s sealed secret,
Bob extends the pre-agreed constant $\mathit{revoke}$
to revoke his right of opening the envelope (the fourth state).
In most protocols, one state can be used for multiple times.
For instance, in the above example, Bob needs to use the third TPM state for several times
to generate key, load key, generate certifications and etc.
As these states are actually the same,
we need to identify them as one state when they are used in different places.
On the other hand, the first state used in the protocol is precedent to the third state.
Thus, we should also identify how states are updated,
namely the transformation an old state to a new state.
The protocol rules specified in our framework are of the form $H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}V$.
$H$ is a set of premises such as
the terms that the adversary should know and the events that the protocol should engage.
$S$ is a set of states.
Both of $H$ and $S$ must be satisfied so that the rule is applicable.
For example, when the adversary wants to load a key into the TPM,
the adversary should know its parent key and obtain the TPM state with matched PCR value.
$V$ is the conclusion of the rule with two types of values.
One type of conclusion is a fact.
Take the TPM loading key as an example, its conclusion is a fact that
the adversary can get the loaded key in the TPM.
The other type of conclusion represents
how the states are transferred from old ones to new ones.
As the states in our framework are attached to the objects,
the conclusion consists of pairs of old state and new state for the same object,
denoting that state is converted from one to another.
In TPM extending operation, the conclusion is one pair of states
$\langle tpm(aik,p),tpm(aik,h(p,n))\rangle$
in which the PCR value in the second state is extended.
$M$ and $O$ help us to organize the correspondences between facts and states.
$M$ maps the facts to the states indicating that the facts should be known at which states.
$O$ is the orderings of the states generated from the knowledge deduction.
For instance, when a fact $f$ required by a rule $R$ can be provided as the conclusion of another rule $R^{\prime}$,
we can compose these two rules together to remove the requirement of $f$.
Since the $f$ is provided by $R^{\prime}$ and used in $R$,
the states mapped by $f$ in $R$ are required later than requirement of the states in $R^{\prime}$.
The orderings are specified in the verification process
to make sure that the state trace is practical for the protocol.
We name the rule as state consistent rule when $V$ is a fact
and call the rule as state transferring rule when $V$ is a set of state conversions.
In addition, we use events and states to distinguish the protocol sessions.
The events are engaged in the rule predicates to indicate the generation of fresh nonces.
Since fresh nonces are random numbers, we assume their values can uniquely identify the events.
Whenever the nonces generated in different events have the same value,
these events should be merged.
On the other hand, the states are used to describe the objects or entities presented in the protocol.
Basically, we use states to differentiate the different phases of the objects.
As we do not bound the number of events and states,
the verification is conducted for an infinite number of sessions.
IV-B Term Syntax
We adopt the syntax in Table I to model the protocols.
Before using an event or a state in the rules,
we need to declare it with a unique identity.
For the nonce generation event, the pair of the event name and the fresh nonce is
the key111Note that it is different from a cryptographic key. and
we can merge two events if they have the same key.
While for states, the pair of the object name and the object identity is the key and
states with the same key should be ordered, describing certain phases of the same object.
Rules are used to specify the protocol execution and adversary capabilities.
They have the hierarchy structure as follows.
Terms could be defined as functions, names,
nonces, configurations or variables.
Functions can be applied to a sequence of terms;
names are globally shared constants;
nonces are freshly generated values in the sessions;
configurations are values pre-existed in the states;
and variables are memory locations for holding the terms.
States describe object stages in the protocol by maintaining a set of terms.
If two states $s$ and $s^{\prime}$ have the same key,
they are describing the same object, denoted as $s\sim s^{\prime}$.
The operator $\sim$ is an equivalence relation that can partition a state set into several disjoint subsets.
When a mutable value is encoded in the state, we name it as configuration.
It is different from variables because its value is decided by the environment,
while the value of a variable is decided by the assignment to the variable.
In other words, configuration is pre-existed while variable is post-assigned.
A fact can be the engagement of an event,
or it means that a term $t$ is known to the adversary denoted as $k(t)$.
We define mapping as a pair of fact $f$ and state $s$
denoted by $\langle f,s\rangle$, representing that $f$ is true at state $s$.
Additionally, we define state ordering by applying the binary operator $\leq$ over state pairs:
$s\leq s^{\prime}$, i.e., $s$ should be a state used no later than $s^{\prime}$.
The state set is a preorder set over $\leq$,
and each $\sim$ partition is a partially ordered set over $\leq$.
The derivation of mappings and orderings are discussed in Section V.
A conversion $c$ is a pair of states $\langle s,s^{\prime}\rangle$
which stands for the transformation from an old state $s$ to a new state $s^{\prime}$.
We call $s$ as the pre-state of $c$ denoted as $\mathit{pre}(c)$
and name $s^{\prime}$ as the post-state of $c$ denoted as $\mathit{post}(c)$.
For a set of conversions $C$, we have $\mathit{pre}(C)=\{\mathit{pre}(c)|c\in C\}$
and $\mathit{post}(C)=\{\mathit{post}(c)|c\in C\}$.
There are two kinds of rules that can be specified
in our framework as shown in the Table I.
The state consistent rule means if $f_{1},f_{2},\ldots,f_{n}$ are true
under the protocol state $s_{1},s_{2},\ldots,s_{m}$ satisfying the mappings $M$ and the orderings $O$,
$f$ is also true under the same state.
For the state transferring rule, it means if $f_{1},f_{2},\ldots,f_{n}$ are true
under the protocol state $\{s_{1},s_{2},\ldots,s_{m}\}\cup pre(C)$ satisfying the mappings $M$ and the orderings $O$,
the protocol state can be transferred into $\{s_{1},s_{2},\ldots,s_{m}\}\cup post(C)$ where $C=c_{1},c_{2},\ldots,c_{k}$.
Assume $H$ is a fact set, $S$ and $S^{\prime}$ are two state sets,
we define $H\times S=\{\langle f,s\rangle|f\in H,s\in S\}$
and $S\times S^{\prime}=\{s\leq s^{\prime}|s\in S,s^{\prime}\in S^{\prime}\}$.
Given a rule $H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}V$ directly specified from the protocol,
the predicates $H$ should be given at the exact states and all the states should be presented at the same time.
So the default value of $M$ is $H\times S$,
and the default value of $O$ is $S\times S$.
In the remaining of the paper, we omit them in the protocol specification.
IV-C Rule Modeling
In the following, we illustrate how to specify stateful protocols in our approach
by using the DEP described in Section III as a running example.
In the following protocol specification, we assume that
both of the first phase and the second phase could be conducted for infinitely many times.
We assume that all of the values extended to Bob’s TPM in the first phase
and the secrets bound to the public key in the second phase are freshly generated nonces.
So we can differentiate the sessions and values used in the sessions during the verification.
In order to clearly illustrate the modeling strategy employed in our approach,
we describe the basic functionalities of the TPM along with the rules.
Notice that our approach is not limited to the applications of TPM,
but potentially other stateful security protocols.
IV-C1 Declarations
Before specifying the protocol,
we need to declare the events and the states that are used in the rules and queries.
There are three nonce generation events in the DEP.
The $\mathit{genkey(*sk,aik,p,pcr)}$ event models that a new binding key $sk$ is generated in the TPM.
In addition to the fresh key $sk$,
the $\mathit{genkey}$ event also specifies the AIK value $aik$
and the PCR value $p$ of the TPM when the key is generated.
Moreover, the $\mathit{pcr}$ in the $\mathit{genkey}$ event models the PCR value that $sk$ is bound to.
The $\mathit{init(*n,p)}$ event is emitted
when Alice extends the nonce $n$ to Bob’s TPM of the PCR value $p$.
The $\mathit{gensrt(*s,p,pkey)}$ event is engaged
when Alice creates the secret $s$ for a new session of the second phase
after receiving a key certification of $pkey$ issued from Bob’s TPM with the PCR value $p$.
In terms of the protocol states,
Alice enters the state $\mathit{alice(*n)}$
after she extends the secret nonce $n$ to Bob’s TPM.
Alice also maintains the state $\mathit{secret(*s,p,pkey)}$
when she decides to share the secret value $s$ over Bob’s TPM with the PCR value $p$.
The $pkey$ is a public key generated from Bob’s TPM, locked to PCR $h(p,open[])$.
Beside, every TPM has a state of $\mathit{tpm(*aik,p)}$
in which the TPM is identified by the AIK value $aik$ and it has the PCR value $p$.
IV-C2 State Consistent Rules
The rules in the first category preserves the protocol execution state.
However, they can be applied only if the protocol is in some specific states.
Most of the rules related to the TPM fall into this category.
Stateless Rules.
Some stateless operations are allowed in stateful protocols
such as encryption, decryption, concatenation and etc.
For instance, public key generation and the binding operation of the TPM can be modeled as
$$\displaystyle\mathit{k(skey)}\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt%
\rightarrow}\mathit{k(pk(skey))}$$
(1)
$$\displaystyle\mathit{k(mess)},\mathit{k(pkey)}\mathrel{-\kern-2.15pt[}~{}%
\mathrel{]\kern-4.3pt\rightarrow}\mathit{k(aenc(mess,pkey))}$$
(2)
where the state set is empty in these rules.
Rule (1) means that if the adversary knows a term $\mathit{skey}$,
he could treat it as a private key and compute its corresponding public key $pk(\mathit{skey})$.
Rule (2) models the binding operation happened outside of the TPM,
which means if the adversary knows a message $\mathit{mess}$ and a binding public key $\mathit{pkey}$,
he could encrypt $\mathit{mess}$ by $\mathit{pkey}$
and get the asymmetric encryption $\mathit{aenc}(\mathit{mess},\mathit{pkey})$.
As stateless protocols can be considered a special case of stateful protocols,
our verification framework also works for stateless protocols.
Other two stateless rules in the DEP model the fact that
the agreed constant values $\mathit{open}$ and $\mathit{revoke}$ are known publicly.
$$\displaystyle\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt\rightarrow}%
\mathit{k(revoke[])}$$
(3)
$$\displaystyle\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt\rightarrow}%
\mathit{k(open[])}$$
(4)
Data Fetch Rules.
Another category of the state consistent rules contains the data fetch rules.
They model the fact that some data used in the protocol can be fetched
directly from the protocol state without other information.
In the DEP, the adversary has control over the TPM.
First of all, he can use the the storage root key (SRK) to encrypt any messages.
In addition, he can ask the TPM for its PCR value and its PCR quote without providing any information.
To specify a general case of the TPM,
the AIK value is not fixed to Bob’s TPM.
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]\kern-4.%
3pt\rightarrow}k(\mathit{srk(|aik|)})$$
(5)
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]\kern-4.%
3pt\rightarrow}k(\mathit{|p|})$$
(6)
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]\kern-4.%
3pt\rightarrow}k(\mathit{pcrcert(|aik|,|p|)})$$
(7)
As $\mathit{srk(|aik|)}$ represents the SRK itself rather than its value,
rule (5) means that the adversary has access to the SRK.
Rule (6) and (7) stand for
getting the PCR value and the PCR quote, respectively.
PCR quote is a certification issued from the TPM
that can be used to prove its PCR value.
Data Processing Rules.
The third category of the state consistent rules contains data processing rules,
which process data based on the presented information and the protocol state.
As we have illustrated in Section III,
the keys used in the TPM are well protected and strictly controlled.
In the TPM, keys can only be generated under a parent key,
and the generated key can be bound to a specific PCR value so that
it can be used only if the given PCR is of that value.
In the DEP, for the sake of simplicity,
we assume all the new keys are generated from the SRK.
Additionally, all the new keys are bound to a specific PCR value
as it is the case for the protocol.
Notice that our technique does not restrict us from
specifying the complete TPM.
$$\displaystyle k(\mathit{pcr}),k(\mathit{srk(|aik|)}),\mathit{genkey([sk],|aik|%
,|p|,pcr)}$$
$$\displaystyle~{}~{}\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]%
\kern-4.3pt\rightarrow}$$
$$\displaystyle~{}~{}k(\mathit{\langle pk([sk]),blob(|aik|,[sk],srk(|aik|),pcr)%
\rangle})$$
(8)
$$\displaystyle k(\mathit{blob(|aik|,sk,pakey,pcr)}),k(\mathit{pakey})$$
$$\displaystyle~{}~{}\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]%
\kern-4.3pt\rightarrow}k(\mathit{pcrkey(|aik|,sk,pcr)})$$
(9)
$$\displaystyle k(\mathit{pcrkey(|aik|,sk,pcr)})$$
$$\displaystyle~{}~{}\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]%
\kern-4.3pt\rightarrow}k(\mathit{keycert(|aik|,pk(sk),pcr)})$$
(10)
$$\displaystyle k(\mathit{aenc(data,pk(sk))}),k(\mathit{pcrkey(|aik|,sk,|p|)})$$
$$\displaystyle~{}~{}\mathrel{-\kern-2.15pt[}\mathit{tpm(|aik|,|p|)}\mathrel{]%
\kern-4.3pt\rightarrow}k(\mathit{data})$$
(11)
Rule (8) specifies that a new session key $sk$ can be generated
in the TPM identified by $aik$ with PCR value $p$.
In addition, the new key is bound to the PCR value $\mathit{pcr}$
so that it can only be used when the PCR is of that value.
As can be seen from rule (8),
we need to specify the target PCR value for the key and provide the SRK as well.
In addition, all of the related information should be encoded into the key generation event
so that it can be used to identify the key generation behavior.
Initially, the generated key is not loaded into the TPM but stored in a key blob.
So rule (9) models the key loading operation by providing the key blob and its parent key.
When the key is loaded, the TPM can issue key certification as illustrated in rule (10).
Rule (11) describes the bound data can be decrypted with the corresponding loaded key.
More importantly, the PCR value specified in the key should be matched with the current PCR.
When Alice receives key certification from Bob and she has already finished the first phase,
she generates a secret $[s]$, encrypts it with the public key $pkey$ and sends it to Bob.
$$\displaystyle\mathit{gensrt([s],p,pkey)},$$
$$\displaystyle\mathit{k(keycert(bob[],pkey,h(p,open[])))}$$
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{alice(|n|)}$$
$$\displaystyle\mathrel{]\kern-4.3pt\rightarrow}k(\mathit{aenc([s],|pkey|)})$$
(12)
IV-C3 State Transferring Rules
The state transferring rules change the protocol’s global state.
The PCR value extending action is modeled as follows.
$$\displaystyle k(n)\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt\rightarrow}%
\langle\mathit{tpm(|aik|,|p|)},\mathit{tpm(|aik|,h(|p|,n))}\rangle$$
(13)
Rule (13) means that if the adversary knows a value $n$,
he could extend the given PCR in the TPM by $n$.
The second state transition rule models the first phase for Alice.
$$\displaystyle\mathit{init([n],|p|)}$$
$$\displaystyle\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt\rightarrow}%
\langle,\mathit{alice([n])}\rangle,$$
$$\displaystyle\langle\mathit{tpm(bob[],|p|)},\mathit{tpm(bob[],h(|p|,[n]))}\rangle$$
(14)
The constant $bob[]$ is the AIK value of Bob’s TPM.
Alice enters a state called alice
after Alice confirms that the nonce $n$ is extended to Bob’s TPM.
Meanwhile, the nonce $n$ is extended to Bob’s TPM as described in the protocol.
After the alice state is presented,
Alice could repeatedly conduct the second phase of the protocol for infinitely many times.
The optional rule (15) below specifies the reboot behavior of the TPM.
$$\displaystyle\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt\rightarrow}%
\langle\mathit{tpm(|aik|,|p|)},\mathit{tpm(|aik|,boot[])}\rangle$$
(15)
In this work, we prove that the digital envelope protocol is secure
when the TPM reboot is disallowed.
We also show that this protocol is subject to attack otherwise.
IV-D Accessibility
Besides the rules, we also need to specify the object accessibilities for the adversary.
The accessibility describes the objects the adversary have access to.
So given a state in a rule,
we can decide whether the states can be accessed by the adversary or not.
For instance, in the DEP, the adversary can access Bob’s TPM,
and he can use additional TPMs to process messages if necessary.
$$\displaystyle\mathit{access}$$
$$\displaystyle\mathit{tpm(bob[],|p|)}$$
$$\displaystyle\mathit{access}$$
$$\displaystyle\mathit{tpm(|aik|,|p|)}$$
We match the state patterns by substituting the terms in the states.
We discuss more details about accessibility and pattern matching in Section V-B.
IV-E Query
In this paper, we focus on reachability properties such as secrecy.
For instance, we want to ensure that Bob cannot open the secret $s$
as well as obtain the proof for his revoke action
$\mathit{certpcr(bob[],h(p,\mathit{revoke}[]))}$ at the same time
for any iteration $\mathit{secret(s,p,pkey)}$ in the DEP.
If he can, it means that Bob can cheat in the protocol.
We add supplementary rules to represent
whether the adversary has the ability to obtain certain terms as events,
such that we could simply check if those events are reachable or not.
We need to first add another state transferring rule when we want to check reachability.
This rule models that Alice has indeed accepted the certification of the key.
$$\displaystyle\mathit{gensrt([s],p,pkey)},$$
$$\displaystyle\mathit{k(keycert(bob[],pkey,h(p,open[])))}$$
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{alice(|n|)}$$
$$\displaystyle\mathrel{]\kern-4.3pt\rightarrow}\mathit{\langle,secret([s],p,%
pkey)\rangle}$$
(16)
The queries are generally state consistent rules,
but they have event conclusions.
In the DEP, we are interested in the reachability properties as follows.
$$\displaystyle\mathit{gensrt([s],|p|,|pkey|)},\mathit{k([s])}$$
$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\mathrel{-\kern-2.15pt[}\mathit{secret([s%
],|p|,|pkey|)}\mathrel{]\kern-4.3pt\rightarrow}\mathit{opened()}$$
(17)
$$\displaystyle\mathit{gensrt([s],|p|,|pkey|)},\mathit{k(pcrcert(bob[],h(|p|,%
\mathit{revoke}[])))}$$
$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\mathrel{-\kern-2.15pt[}\mathit{secret([s%
],|p|,|pkey|)}\mathrel{]\kern-4.3pt\rightarrow}\mathit{revoked()}$$
(18)
$$\displaystyle\mathit{gensrt([s],|p|,|pkey|)},\mathit{k(pcrcert(bob[],h(|p|,%
\mathit{revoke}[])))}$$
$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{},\mathit{k([s])}\mathrel{-\kern-2.15pt[}%
\mathit{secret([s],|p|,|pkey|)}\mathrel{]\kern-4.3pt\rightarrow}\mathit{attack%
()}$$
(19)
The first query (rule 17) means that Bob can open the envelope and extract the nonce $[s]$.
Similarly, the second query (rule 18) means that the PCR quote can be issued from the TPM
if Bob chooses to revoke the right of opening the envelope.
The third query (rule 19), the most interesting one, checks
whether Bob can get the value of the nonce $[s]$
as well as the proof for his revoke action from his TPM at the same time.
As can be seen, we can name the events differently
and check several queries at the same time.
Because verification for security protocol is generally undecidable,
our algorithm cannot guarantee termination.
Hence we define correctness under the condition of termination (partial correctness) as follows.
In Section V, we present our verification algorithm
on reachability checking, together with its partial correctness proofs.
Definition 1 (Partial Correctness).
A verification algorithm is partially sound if and only if
the target event is reachable when the algorithm can terminate and claim that the event is reachable.
It is partially complete if and only if
the target event is unreachable when the algorithm can terminate and claim that the event is unreachable.
V Verification Algorithm
After a protocol is correctly specified (as illustrated in Section IV),
we present how to verify the protocol in details in this section.
During the verification, we divide our algorithm into two phases.
The first phase is targeted at constructing a knowledge searching base
by knowledge forward composition and state backward transformation.
Based on the knowledge base, we could then perform query searching
to find valid attacks in the second phase.
In order to verify security protocols,
the verification algorithm needs to consider all possible behaviors of the adversary.
Because the adversary adopted in this work can generate new names dynamically at runtime,
the verification process cannot be conducted in a straightforward manner.
To guide the attack searching procedure so that it can terminate,
we adopt a similar strategy as proposed in [5] that applies to the Horn theory.
Our algorithm can be briefly described as follows.
Recall that a rule of the form $H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}V$ says that
the $V$ is true when all the predicates in $H$ are satisfied and all the states $S$ are presented
under the restrictions of state mappings $M$ and orderings $O$.
On one hand, if a predicate in a rule is not yet satisfied,
we try to use a state consistent rule’s conclusion to fulfill it by rule composition.
However, if the predicate is a singleton, that is a fact of the form $k(v)$ where $v$ is a variable,
and the value of $v$ is not related to other facts in the rule,
the singleton could be automatically fulfilled as the adversary assumed in our paper can generate new names.
Additionally, events are not unifiable in our framework
as the events in the predicates and the conclusions are different.
Thus we reserve a set of facts $\mathcal{N}$ from unifying with other facts.
In this work, $\mathcal{N}$ consists of events and singletons.
On the other hand,
if several states are presented in a rule,
some of the states should be the latest ones that are presented when the conclusion is given,
while others are the outdated states.
Thus, we identify the latest states
and deduce them to their previous states with the help of rule transformation.
By performing the rule composition and rule transformation iteratively,
once the fixed-point can be reached for the knowledge base,
the query can then be answered directly from the rules in the knowledge base.
V-A Knowledge Base Construction
In this section, we compose existing rules to generate new rules
until the fixed point of the searching knowledge base is reached.
Basically, when we compose two rules together,
the term encoded in the conclusion of the first rule should be unifiable with
the term in a predicate of the second rule.
We use the most general unifier to unify the terms.
Definition 2 (Most General Unifier).
If $\sigma$ is a substitution for both terms $t_{1}$ and $t_{2}$ so that $\sigma t_{1}=\sigma t_{2}$,
we say $t_{1}$ and $t_{2}$ are unifiable and $\sigma$ is a unifier for $t_{1}$ and $t_{2}$.
If $t_{1}$ and $t_{2}$ are unifiable, the most general unifier for $t_{1}$ and $t_{2}$ is a unifier $\sigma$,
where for all unifiers $\sigma^{\prime}$ of $t_{1}$ and $t_{2}$
there exists a substitution $\sigma^{\prime\prime}$ such that $\sigma^{\prime}=\sigma^{\prime\prime}\sigma$.
The unification of the facts is defined if and only if
their predicate names are matched and the corresponding terms in the facts can be unified.
According to Section IV,
we have two kinds of rules in our framework,
i.e., state consistent rules and state transferring rules.
State consistent rules have a fact as conclusion,
so given an unsatisfied predicate in a rule,
we can compose the state consistent rule to it to provide the predicate.
The rule composition is formally defined as follows.
Definition 3 (Rule Composition).
Let $R=H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}f$ be a state consistent rule
and $R^{\prime}=H^{\prime}:M^{\prime}\mathrel{-\kern-2.15pt[}S^{\prime}:O^{\prime}%
\mathrel{]\kern-4.3pt\rightarrow}V$ be either a state consistent rule or a state transferring rule.
Assume there exists $f_{0}\in H^{\prime}$ such that $f$ and $f_{0}$ are unifiable with the most general unifier $\sigma$.
Given $S_{0}=\{s_{0}|\langle f_{0},s_{0}\rangle\in M^{\prime}\}$,
the rule composition of $R$ with $R^{\prime}$ on the fact $f_{0}$ is defined as
$$\displaystyle R\circ_{f_{0}}R^{\prime}$$
$$\displaystyle=\sigma(H\cup(H^{\prime}-\{f_{0}\})):\sigma(M\cup M^{\prime})$$
$$\displaystyle\mathrel{-\kern-2.15pt[}\sigma(S\cup S^{\prime}:O\oplus O^{\prime%
}\oplus S\times S_{0})\mathrel{]\kern-4.3pt\rightarrow}\sigma V.$$
Example 1.
For instance, given two simplified rules as follows.
We omit the mappings and orderings when they are trivial
and use special characters (e.g., $\spadesuit$, $\blacklozenge$)
to indicate the facts and states in the mappings and orderings.
$$\displaystyle\mathit{gensrt([s],|p|,pkey)}$$
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{tpm(bob[],h(|p|,open[]))}^{%
\spadesuit}\mathrel{]\kern-4.3pt\rightarrow}\mathit{k([s])}$$
$$\displaystyle\mathit{gensrt([s],|p|,pkey)}$$
$$\displaystyle,\mathit{[s]}^{\blacklozenge}:\{\langle\blacklozenge,\clubsuit\rangle\}$$
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{tpm}$$
$$\displaystyle\mathit{(bob[],h(|p|,revoke[]))}^{\clubsuit}\mathrel{]\kern-4.3pt%
\rightarrow}\mathit{attack()}$$
The first rule means that the secret $s$ can be revealed
when Bob’s TPM has the PCR value $\mathit{h(p,open[])}$.
The second rule means if Bob’s TPM has the PCR value $\mathit{h(p,revoke[])}$
and the secret $s$ is revealed (the envelope is opened),
we have found an attack.
Their rule composition on the fact $f_{0}=\mathit{k([s])}$ is
$$\displaystyle\mathit{gensrt}($$
$$\displaystyle[s],|p|,\mathit{pkey})\mathrel{-\kern-2.15pt[}\mathit{tpm(bob[],h%
(|p|,open[]))}^{\spadesuit},$$
$$\displaystyle\mathit{tpm}$$
$$\displaystyle\mathit{(bob[],h(|p|,revoke[]))}^{\clubsuit}:\spadesuit\leq%
\clubsuit\mathrel{]\kern-4.3pt\rightarrow}\mathit{attack()}$$
(20)
which means that $\mathit{open[]}$ should be extended to Bob’s TPM
before $\mathit{revoke[]}$ is extended.
This is apparent because the last state of Bob’s TPM,
according to the rules, should have $\mathit{revoke[]}$ extended.
Given a state consistent rule with a conclusion $f$,
it specifies that we can obtain $f$ if its predicates are provided and the states form a valid state trace.
Furthermore, some of the states are the latest states when the conclusion is given.
Among the latest states, the latest state transformation is taken on some of them.
If we can identify those latest states for the latest state transformation,
we then can deduce their precedent states using the corresponding state transferring rule.
We define $S_{0}$ as the cover set of $S$
if $s_{0}\in S_{0},s\in S,s_{0}\leq s$ then $s\in S_{0}$.
Assume $c$ is a conversion and $\mathit{post}(c)$ is unifiable with a state $s$ under $\sigma$,
we define the join operator $c\bowtie_{\sigma}s=\sigma\mathit{pre}(c)$.
Besides, we define $[s]^{S}$ as the $\sim$ partition of $s$ in the state set $S$.
The state transformation is then defined as follows.
Definition 4 (State Transformation).
Let $R=H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}C$ be a state transferring rule
and $R^{\prime}=H^{\prime}:M^{\prime}\mathrel{-\kern-2.15pt[}S^{\prime}:O^{\prime}%
\mathrel{]\kern-4.3pt\rightarrow}f$ be a state consistent rule.
Assume there exists a unifier $\sigma^{\prime}$ and an injective function $m:C\rightarrow\mathds{P}(S^{\prime})$
such that $\cup_{c\in C}\sigma^{\prime}m(c)$ is a cover set of
$\cup_{c\in C}[\sigma^{\prime}\mathit{post}(c)]^{\sigma^{\prime}S}$ and
$\forall c\in C,\forall s\in m(c),c\bowtie_{\sigma^{\prime}}s$ is defined.
Let $\sigma$ be the most general unifier of $\sigma^{\prime}$ and
$S_{n}=\sigma S^{\prime}-\mathit{post}(\sigma C)$,
the state transformation of applying $R$ to $R^{\prime}$ on $m$ is defined as
$$\displaystyle R\bowtie_{m}R^{\prime}=\sigma(H\cup H^{\prime}):\sigma(M\cup M^{%
\prime})\mathrel{-\kern-2.15pt[}\sigma S\cup S_{n}\cup\mathit{pre}(\sigma C)$$
$$\displaystyle:\sigma O\oplus\sigma O^{\prime}\oplus\mathit{pre}(\sigma C)%
\times\mathit{pre}(\sigma C)\oplus(\oplus_{c\in C}(([\sigma pre(c)]^{\sigma S}$$
$$\displaystyle~{}~{}~{}~{}-\sigma m(c))\times\sigma pre(C)\oplus\sigma pre(C)%
\times\sigma m(c)))\mathrel{]\kern-4.3pt\rightarrow}\sigma f.$$
Example 2.
For instance, if the PCR value extending rule (13)
is used for transferring the states in rule (2),
we first enumerate the state cover set of rule (2)
as $\{\clubsuit\},\{\spadesuit,\clubsuit\}$.
Because the states of $\{\spadesuit,\clubsuit\}$ cannot be unified,
we have only one valid rule after the state transformation.
$$\displaystyle\mathit{gensrt}([s],|p|,\mathit{pkey}),\mathit{k(revoke[])}%
\mathrel{-\kern-2.15pt[}$$
$$\displaystyle\mathit{tpm}\mathit{(bob[],|p|)}^{\heartsuit},$$
$$\displaystyle\mathit{tpm(bob[],h(|p|,open[]))}^{\spadesuit}$$
$$\displaystyle:\spadesuit\leq\heartsuit\mathrel{]\kern-4.3pt\rightarrow}\mathit%
{attack()}$$
Since the new generated rule has an unsatisfied predicate that is not in $\mathcal{N}$,
the verification algorithm continues.
However, when TPM reboot is disallowed,
these two states remained in the rule can never be unified to one state,
so the attack event cannot be reached.
The detailed discussions are available in the reachability analysis.
The adversary can generate new names.
If a singleton predicate is not related to other facts in a rule,
the adversary could generate a random fact and use it as the singleton predicate
so that it can be removed from the predicates.
In addition, given two events with the same key in the predicates,
they should be unified and merged.
Furthermore, for any two states $s\sim s^{\prime}$ and $s\leq s^{\prime}\land s^{\prime}\leq s\in O$,
they should be merged because clearly they are the same state.
Meanwhile, any mappings and orderings
related to the non-existing facts and states should be removed as well.
Definition 5 (Rule Validation).
Let $R=H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}V$ be a rule.
We define a rule as valid if and only if there exists a unifier $\sigma^{\prime}$ such that
any event in $H$ under the same key is unifiable with $\sigma^{\prime}$.
Let $\sigma$ be the most general unifier of $\sigma^{\prime}$,
The rule validation of $R$ is defined as
$$\displaystyle R\Downarrow=$$
$$\displaystyle\mathit{clear}(\mathit{merge}(\sigma H:\mathit{rm}(\sigma M)))$$
$$\displaystyle\mathrel{-\kern-2.15pt[}\mathit{elim}(\sigma S:\mathit{rm}(\sigma
O%
))\mathrel{]\kern-4.3pt\rightarrow}\sigma V$$
The function $\mathit{merge}$ merges duplicated expressions;
the function $\mathit{clear}$ removes any singleton
in which the variable does not appear in other facts in the rule;
the function $\mathit{elim}$ eliminates any isolated states and those related orderings;
and the function $\mathit{rm}$ removes the mappings and orderings related to no longer existed facts and states.
When a new rule is composed from existing ones,
we need to make sure it is not redundant.
Suppose two rules $R$ and $R^{\prime}$ can make the same conclusion,
while (1) $R$ requires less predicates, mappings and orderings than $R^{\prime}$ and (2) $R$ is no less general than $R^{\prime}$,
$R^{\prime}$ should be implicated by $R$.
The joint operator ‘$\cdot$’ between mapping $M$ and ordering $O$ is defined as
$$M\cdot O=\{\langle f,s\rangle|\langle f,s^{\prime}\rangle\in M\land s^{\prime}%
\leq s\in O\}.$$
We then define rule implication as follows.
Definition 6 (Rule Implication).
Let $R=H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}V$ and $R^{\prime}=H^{\prime}:M^{\prime}\mathrel{-\kern-2.15pt[}S^{\prime}:O^{\prime}%
\mathrel{]\kern-4.3pt\rightarrow}V^{\prime}$ be two rules.
We define $R$ implies $R^{\prime}$ denoted as $R\Rightarrow R^{\prime}$ if and only if
$\exists\sigma,\sigma V=V^{\prime}\land\sigma H\subseteq H^{\prime}\land\sigma(%
M\cdot O)\subseteq(M^{\prime}\cdot O^{\prime})\land\sigma S\subseteq S^{\prime%
}\land\sigma O\subseteq O^{\prime}$.
The knowledge base construction algorithm is shown in Algorithm 1,
where we use $\mathcal{B}_{init}$ to denote the initial set of rules as specified
and use $\mathcal{B}$ to denote the knowledge base constructed by the algorithm.
In the following discussions, we will use $\mathcal{B}$ and $\mathcal{B}_{init}$
directly assuming they are clear from the context.
In the $\mathit{add}$ procedure (Line 1 to Line 6), we use rule implication to ensure that
redundancies will not be introduced into the knowledge base.
The main procedure, starting at Line 7, first adds all the initial rules into the knowledge base (Line 8 to Line 11),
then it composes and transforms the rules until a fixed point is reached.
We discuss the rule composition and the state transformation separately as follows.
For the rule composition (Line 13 to Line 20), when rules can be composed in an unlimited method,
infinitely many composite rules can be generated,
which we shall prevent.
For instance, we can compose the rule (1) to itself
by treating the public key as a valid private key
and the composite rule becomes $\mathit{k(skey)}\mathrel{-\kern-2.15pt[}~{}\mathrel{]\kern-4.3pt\rightarrow}%
\mathit{k(pk(pk(skey)))}$,
which could then be composed to the rule (1) again.
Furthermore, as mentioned previously,
singleton predicates that are not related to other facts in the rule can be removed,
thus it is unnecessary to compose two rules on a singleton fact.
As the rules cannot compose on events, when two rules are composed in our algorithm,
we need to ensure that they can be composed on a fact $f_{0}$ such that $f_{0}\not\in\mathcal{N}$.
Moreover, when two rules are composed in the form of $R\circ_{f_{0}}R^{\prime}$
and $R$ has predicates which are not contained in $\mathcal{N}$,
we should fulfill those predicates first.
Thus we ensure that $R$’s predicates are all in $\mathcal{N}$.
For the state transformation (Line 21 to Line 28), as we deduce the states in a backward manner,
we should make sure that the states we transferred in the rule are latest,
and the target event is presented in the rule conclusion.
In addition, its predicates should be all contained in $\mathcal{N}$,
resulting from the same reason mentioned previously.
Finally, we select a subset of the rules.
Their predicates should only be singletons and events
as rules with unfulfilled predicates cannot be used to conduct attacks directly.
Their conclusion should be an event
because these rules are the only interesting rules to us.
$\mathcal{B}_{v}$ is introduced in Line 1 to help the explanation of the proof for Theorem 1.
Previously, we have reformulated our verification problem as
reachability analysis of events (see Section IV-E).
Whenever an event is derivable from the initial rules $\mathcal{B}_{init}$,
there must exist a derivation tree for that event defined as follows.
Definition 7 (Derivation Tree).
Let $\mathcal{B}$ be a set of closed rules and $e$ be an event,
where the closed rule is a rule with its conclusion initiated by its predicates and states.
$e$ can be derived from $\mathcal{B}$ if and only if
there exists a finite derivation tree defined as follows.
1.
Every edge in the tree is labeled by a fact $f$, a state set $S$ and an index $i$,
and $\forall s,s^{\prime}\in S$ we have $s\not\sim s^{\prime}$.
2.
Every node is labeled by a rule in $\mathcal{B}$.
3.
Suppose the node is labeled by a state consistent rule
as shown in Figure 2(a),
then we have $R\Rightarrow H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}f$ in which
$H=f_{1},\ldots,f_{n}$, $M=H\times S$, $O=S\times S$
and the indexes labeled on the outgoing edge and incoming edges are the same.
4.
On the other hand, if the node is labeled by a state transferring rule
as shown in Figure 2(b),
there exists $C$ such that $R\Rightarrow H:M\mathrel{-\kern-2.15pt[}S_{0}:O\mathrel{]\kern-4.3pt%
\rightarrow}C$
in which $H=f_{1},\ldots,f_{n}$, $S_{0}=S-pre(C)=S^{\prime}-post(C)$,
$M=H\times S_{0}$, $O=S_{0}\times S_{0}$
and the indexes labeled on the incoming edges equal to the index labeled on the outgoing edge plus $1$.
5.
The outgoing edge of the root is labeled by the event $e$ and the index $1$.
6.
The incoming edges of the leaves are only labeled by facts in $\mathcal{N}$ with the same index.
7.
The edges with the same index have the same state.
In the tree, every node is labeled by a rule in $\mathcal{B}_{init}$
to represent how the knowledge is deduced.
Additionally,
we label the edges with states to indicate when the knowledge deduction rule is applied
and how the state transferring rule affects the states.
Furthermore, we also label every edge with an index
to group the knowledge under the same state together as well as
to denote the valid trace of state transferring,
which eases the proof of Theorem 1.
The Lemma 1 demonstrates how to replace two directly connected nodes in the derivation tree
with one node labeled by a composite rule with the same state and the same index.
Lemma 1.
If $R_{o}\circ_{f}R^{\prime}_{o}$ is defined, $R_{t}\Rightarrow R_{o}$ and $R^{\prime}_{t}\Rightarrow R^{\prime}_{o}$,
then either there exists $f^{\prime}$ such that $R_{t}\circ_{f^{\prime}}R^{\prime}_{t}$ is defined
and $R_{t}\circ_{f^{\prime}}R_{t}^{\prime}\Rightarrow R_{o}\circ_{f}R^{\prime}_{o}$,
or $R_{t}^{\prime}\Rightarrow R_{o}\circ_{f}R^{\prime}_{o}$.
Proof.
Let $R_{o}=H_{o}:M_{o}\mathrel{-\kern-2.15pt[}S_{o}:O_{o}\mathrel{]\kern-4.3pt%
\rightarrow}f_{o}$,
$R^{\prime}_{o}=H^{\prime}_{o}:M^{\prime}_{o}\mathrel{-\kern-2.15pt[}S^{\prime}%
_{o}:O^{\prime}_{o}\mathrel{]\kern-4.3pt\rightarrow}V_{o}$,
$R_{t}=H_{t}:M_{t}\mathrel{-\kern-2.15pt[}S_{t}:O_{t}\mathrel{]\kern-4.3pt%
\rightarrow}f_{t}$,
$R^{\prime}_{t}=H^{\prime}_{t}:M^{\prime}_{t}\mathrel{-\kern-2.15pt[}S^{\prime}%
_{t}:O^{\prime}_{t}\mathrel{]\kern-4.3pt\rightarrow}V_{t}$.
There should exist a substitution $\sigma$
such that $\sigma f_{t}=f_{o}$, $\sigma H_{t}\subseteq H_{o}$, $\sigma M_{t}\subseteq M_{o}$,
$\sigma S_{t}\subseteq S_{o}$, $\sigma O_{t}\subseteq O_{o}$,
$\sigma f^{\prime}_{t}=f^{\prime}_{o}$, $\sigma H^{\prime}_{t}\subseteq H^{\prime}_{o}$, $\sigma M^{\prime}_{t}\subseteq M^{\prime}_{o}$,
and $\sigma S^{\prime}_{t}\subseteq S^{\prime}_{o}$, $\sigma O^{\prime}_{t}\subseteq O^{\prime}_{o}$.
Assume
$S_{o}\circ_{f}S^{\prime}_{o}=\sigma^{\prime}(H_{o}\cup(H^{\prime}_{o}-\{f\})):%
\sigma^{\prime}(M_{o}\cup M^{\prime}_{o})\mathrel{-\kern-2.15pt[}\sigma^{%
\prime}(S_{o}\cup S^{\prime}_{o}):\sigma^{\prime}(O_{o}\cup O^{\prime}_{o}\cup
S%
_{o}\times S)\mathrel{]\kern-4.3pt\rightarrow}\sigma^{\prime}V_{o}$
where $S=\{s|\langle f,s\rangle\in M^{\prime}_{o}\}$.
We discuss the two cases as follows.
First case. Suppose $\exists f^{\prime}\in H^{\prime}_{t}$ such that $\sigma f^{\prime}=f$.
Since $R_{o}\circ_{f}R^{\prime}_{o}$ is defined and $\sigma^{\prime}f=\sigma^{\prime}f_{o}$,
we thus have $\sigma^{\prime}\sigma f^{\prime}=\sigma^{\prime}\sigma f_{t}$.
As $f^{\prime}$ and $f_{t}$ are unifiable, $S_{t}\circ_{f^{\prime}}S^{\prime}_{t}$ is defined.
Let $\sigma_{t}$ be the most general unifier,
then $\exists\sigma^{\prime}_{t}$ such that $\sigma^{\prime}\sigma=\sigma^{\prime}_{t}\sigma_{t}$.
Suppose we have
$S_{t}\circ_{f^{\prime}}S^{\prime}_{t}=\sigma_{t}(H_{t}\cup(H^{\prime}_{t}-\{f^%
{\prime}\})):\sigma_{t}(M_{t}\cup M^{\prime}_{t})\mathrel{-\kern-2.15pt[}%
\sigma_{t}(S_{t}\cup S^{\prime}_{t}):\sigma_{t}(O_{t}\cup O^{\prime}_{t}\cup S%
_{t}\times S^{\prime})\mathrel{]\kern-4.3pt\rightarrow}\sigma_{t}V_{t}$
where $S^{\prime}=\{s|\langle f^{\prime},s\rangle\in M^{\prime}_{t}\}$.
First we prove $\sigma S^{\prime}=\{s|\langle\sigma f^{\prime},s\rangle\in\sigma M^{\prime}_{t%
}\}=\{s|\langle f,s\rangle\in\sigma M^{\prime}_{t}\}\subseteq\{s|\langle f,s%
\rangle\in M^{\prime}_{o}\}=S$.
Since $\sigma^{\prime}_{t}\sigma_{t}(H_{t}\cap(H^{\prime}_{t}-\{f^{\prime}\}))=\sigma%
^{\prime}\sigma(H_{t}\cup(H^{\prime}_{t}-f^{\prime}))\subseteq\sigma^{\prime}(%
H_{o}\cup(H^{\prime}_{o}-\{f\}))$,
$\sigma^{\prime}_{t}\sigma_{t}(S_{t}\cup S^{\prime}_{t})=\sigma^{\prime}(\sigma
S%
_{t}\cup\sigma S^{\prime}_{t})\subseteq\sigma^{\prime}(S_{o}\cup S^{\prime}_{o})$,
$\sigma^{\prime}_{t}\sigma_{t}(O_{t}\cup O^{\prime}_{t}\cup S_{t}\times S^{%
\prime})=\sigma^{\prime}(\sigma O_{t}\cup\sigma O^{\prime}_{t}\cup\sigma S_{t}%
\times\sigma S^{\prime})\subseteq\sigma^{\prime}(O_{o}\cup O^{\prime}_{o}\cup S%
_{o}\times S)$,
$\sigma^{\prime}_{t}\sigma_{t}((M_{t}\cup M^{\prime}_{t})\cdot(O_{t}\cup O^{%
\prime}_{t}\cup S_{t}\times S^{\prime}))\subseteq\sigma^{\prime}((M_{o}\cdot M%
^{\prime}_{o})\cup(O_{o}\cup O^{\prime}_{o}\cup S_{o}\times S))$,
and $\sigma^{\prime}_{t}\sigma_{t}V_{t}=\sigma^{\prime}\sigma V_{t}=\sigma^{\prime}%
V_{o}$,
we have $R_{t}\circ_{f^{\prime}}R_{t}^{\prime}\Rightarrow R_{o}\circ_{f}R^{\prime}_{o}$.
Second case. $\sigma H^{\prime}_{t}\subseteq H^{\prime}_{o}-\{f\}$,
then $\sigma^{\prime}\sigma H^{\prime}_{t}\subseteq\sigma^{\prime}(H_{o}\cup(H^{%
\prime}_{o}-\{f\}))$,
$\sigma^{\prime}\sigma(M^{\prime}_{t}\cdot O^{\prime}_{t})\subseteq\sigma^{%
\prime}(M^{\prime}_{o}\cdot O^{\prime}_{o})\subseteq\sigma^{\prime}(M_{o}\cdot
O%
_{o}\cup M^{\prime}_{o}\cdot O^{\prime}_{o})$,
$\sigma^{\prime}\sigma S^{\prime}_{t}\subseteq\sigma^{\prime}S^{\prime}_{o}%
\subseteq\sigma^{\prime}(S_{o}\cup S^{\prime}_{o})$,
$\sigma^{\prime}\sigma O^{\prime}_{t}\subseteq\sigma^{\prime}O^{\prime}_{o}%
\subseteq\sigma^{\prime}(O_{o}\cup O^{\prime}_{o}\cup S_{o}\times S)$,
and $\sigma^{\prime}\sigma V_{t}=\sigma^{\prime}V_{o}$.
Therefore $R_{t}^{\prime}\Rightarrow R_{o}\circ_{f}R^{\prime}_{o}$.
∎
Theorem 1.
Any event $e$ that is derivable from the initial rules $\mathcal{B}_{init}$
if and only if it is derivable from the knowledge base $\mathcal{B}$ constructed in Algorithm 1.
Proof.
Only if.
Assume the event $e$ is derivable from $\mathcal{B}_{init}$,
then there should exist a derivation tree $T_{i}$ for $e$
and every node in the tree is labeled by a rule in $\mathcal{B}_{init}$.
According to the $\mathit{add}$ function in Algorithm 1,
a rule is removed only if it is implied by another rule,
so we have $\forall R\in\mathcal{B}_{init},\exists R^{\prime}\in\mathcal{B}_{v},R^{\prime}\Rightarrow
R$,
where $\mathcal{B}_{v}$ appears at the line 1 in Algorithm 1.
Hence, we can replace all the rules labeled on tree with the rules in $\mathcal{B}_{v}$
and get a new derivation tree $T_{v}$.
As can be seen from Algorithm 1,
some rules are filtered out from $\mathcal{B}_{v}$ to $\mathcal{B}$,
so we need to further prove that the nodes in $T_{v}$ can be composed and transformed
until a derivation tree $T$ is formed such that all the rules labeled on $T$ are rules in $\mathcal{B}$.
To continue the proof, we consider $T_{v}$ purely as a tree structure,
and each tree consists of a root and several connected sub-trees.
Next, we prove that each sub-tree is implied by a state consistent rule in $\mathcal{B}_{v}$.
Since the leaves of $T_{v}$ are implied by the state consistent rules,
the sub-trees of the leaves are directly implied by rules in $\mathcal{B}_{v}$.
Given two nodes $n$ and $n^{\prime}$, $n$’s outgoing edge $f$ is one of incoming edges of $n^{\prime}$.
Assume the subtree $n$ is implied by a state consistent rule $R$ in $\mathcal{B}_{v}$,
the node $n^{\prime}$ is labeled by a rule $R^{\prime}$ and $n^{\prime}$ has a outgoing edge of $f^{\prime}$.
•
If $f\neq f^{\prime}$,
we have $R\Rightarrow H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}f$,
$R^{\prime}\Rightarrow H^{\prime}:M^{\prime}\mathrel{-\kern-2.15pt[}S^{\prime}:%
O^{\prime}\mathrel{]\kern-4.3pt\rightarrow}V$ and $f\in H^{\prime}$.
Since $R_{f}=(H:M\mathrel{-\kern-2.15pt[}S:O\mathrel{]\kern-4.3pt\rightarrow}f)\circ_%
{f}(H^{\prime}:M^{\prime}\mathrel{-\kern-2.15pt[}S^{\prime}:O^{\prime}\mathrel%
{]\kern-4.3pt\rightarrow}V)$ is defined,
according to Lemma 1,
the sub-tree $n^{\prime}$ is also implied by a rule in $\mathcal{B}_{v}$ in two cases.
In the first case, there exists $f^{\prime\prime}$ in the predicates of $R^{\prime}$, $R\circ_{f^{\prime\prime}}R^{\prime}\Rightarrow R_{f}$.
If $f^{\prime\prime}$ is not a singleton,
because $\mathcal{B}_{v}$ is the fixed-point of Algorithm 1,
there should exist $R^{\prime\prime}\in\mathcal{B}_{v}$ such that $R^{\prime\prime}\Rightarrow R_{f}$.
So we can merge these two nodes in the tree and the proof continues.
Otherwise, i.e., $f^{\prime\prime}$ is a singleton,
we can detach the sub-tree of $n$ from tree $T_{v}$ temporarily.
With the composition and transformation processing,
$f^{\prime\prime}$ may be unified to a non-singleton fact, so the composition could continue.
If the other part of the tree has been processed and $f^{\prime\prime}$ is still a singleton,
we will prove later that $n$ can be removed from the tree and the derivation tree is still valid.
In the second case, we can remove the node $n$ and link its incoming links directly to $n^{\prime}$,
so that the node $n^{\prime}$ with more incoming edges is still implied by $R^{\prime}$ and the proof continues.
•
If $f=f^{\prime}$, apparently we have that $R$ implies the subtree of $n^{\prime}$.
We can continue the rule composition until we reach the root
so that each subtree in $T_{v}$ is implied by a state consistent rule in $B_{v}$.
Notice that the states are not properly transferred in the rule that is labeled to the tree $T_{v}$,
so we also need to re-organize the states in the rule to form a valid state trace.
Since all the state duplications appear in the sub-tree are kept in the resulting rule,
we will merge them according to the state transformation.
Consider the root is labeled by a rule $R_{r}$,
all the states appeared in the tree $T_{v}$ should be presented in $R_{r}$.
According to the derivation tree, some of the edges are labeled by the same index.
So we prove in the following iterations, the resulting rule is still in $\mathcal{B}_{v}$.
The index starts with $1$, which is same index of the root,
and it is increased by $1$ after every iteration.
If currently the index is $i$, since the states in the rule are corresponding to the states in the edge,
so we can merge the states in the edges labeled by $i$ together.
According to the definition of the derivation tree,
from the edges labeled by $i+1$ to the edges labeled by $i$,
there exists a conversion set $C$ that converts some old states to the new states.
Hence, we could construct the mapping function $m$ defined in the state transformation,
and map each $c\in C$ to a set of states that should be merged
(the latest states for the latest state transferring rule).
After the state transformation, the largest states in the rule now are labeled by index $i+1$.
According to Algorithm 1 case 2,
the new rule should be also in $\mathcal{B}_{v}$.
Notice that we have mentioned previously that some rules cannot be composed because
the incoming edge of the rule is labeled by a singleton.
Along with the state transformations, some singleton may be unified to a non-singleton fact,
so the rule composition could continue.
In this way, the rule composition and the state transformation can be conducted
until all states left are all labeled by the largest index.
If some inner edges are still labeled by singletons,
because the adversary can generate new names,
he can actively create a new value and label it to that edge,
so that he can drop the remaining sub-tree connected by that edge
and the remaining derivation tree is still valid.
Since the facts in leaves are the events and singletons,
including those failed with unification,
the resulting rule is in the output knowledge base $\mathcal{B}$.
If.
Whenever a rule is added into $\mathcal{B}_{v}$, it should be composed or transferred from existing rules.
Thus all the rules in $\mathcal{B}_{v}$ should be derivable from $\mathcal{B}_{init}$.
Meanwhile $\mathcal{B}$ does not introduce extra rules
besides existing rules in $\mathcal{B}_{v}$,
so $\forall R^{\prime}\in\mathcal{B}$, $R^{\prime}$ is derivable from $\mathcal{B}$.
∎
V-B Reachability Analysis
When the knowledge base is constructed,
we need to check if the target event is reachable or not.
Given a rule in the base,
if the predicates are only events and singletons,
the adversary can fulfill them by asking the protocol to engage those events and generate new names.
For the remaining states in the rule,
we then need to check if the adversary has the access to the corresponding object patterns.
Assume the accessibility is modeled as a set of state patterns $P$ according to Section IV-D.
We define a state $s$ as accessible to the adversary if
$\exists p\in P$ such that $\exists\sigma$, $\sigma s=p$.
For instance, if the attack needs a TPM $tpm(cary[],p)$ from another participant Cary,
while Bob only have the access to the TPM from himself and not preciously owned TPMs.
Since there does not exist such a substitution $\sigma$ such that
$\sigma tpm(cary[],p)=tpm(bob[],p^{\prime})$ or $\sigma tpm(cary[],p)=tpm(aik,p^{\prime})$,
the attack found is impractical.
Thus, a query can be answered using a simple algorithm as shown in Algorithm 2.
It checks the target event against all the remaining rules in the knowledge base $\mathcal{B}$,
and tries to find a rule whose predicates can be fulfilled and states can be accessed by the adversary.
If there exists such a rule, the algorithm returns true;
otherwise it returns false.
We prove the partial correctness of our algorithm as follows.
Theorem 2.
An event $e$ is derivable from the initial rules $\mathcal{B}_{init}$ if and only if
there exists a rule in $\mathcal{B}$ such that its conclusion is $e$
and its states are all accessible to the adversary.
Proof.
(If - Partial Soundness)
If there is a rule in $\mathcal{B}$ that outputs $e$.
As the rules’ predicates are events and singletons,
the adversary can ask the protocol to engage those events and generate new names to fulfill the singletons.
When its states in the same partition are unifiable and all unified states are accessible to the adversary,
the adversary can have the objects to meet the requirements of those states.
Hence, $e$ is derivable by the rule.
According to Theorem 1’s if condition,
$e$ is also derivable from $\mathcal{B}_{init}$.
(Only if - Partial Completeness)
If the event $e$ is derivable from $\mathcal{B}_{init}$,
according to Theorem 1’s only if condition, $e$ is also derivable from $\mathcal{B}$.
As the derivation tree is valid,
the initial states should be accessible states for the adversary.
∎
VI Experiments
Our engineering efforts has realized the proposed approach in a tool named SSPA (Stateful Security Protocol Analyzer).
Our tool, all protocol models and evaluation results are available online at [17].
SSPA is implemented in C++ with around 11K LOC.
The experiments presented in this section are evaluated
with Mac OS X 10.9.1, 2.3 GHz Intel Core i5 and 16G 1333MHz DDR3.
We have tested our tool with three versions of the DEP [4, 15],
the Bitlocker protocol [11] and two versions of the Needham-Schroeder Public Key Protocol (NSPK) [18, 19].
All of the protocols are correctly analyzed within 30 minutes.
The results are summarized in Table II.
For the DEP example, when the TPM reboot is disallowed, the verification result shows that
Bob cannot obtain both of the secret and the proof for his revoke action at the same time.
In the meanwhile, we also found several valid traces for Bob to finish the protocol
by either opening the envelope or revoking his right.
However, when the TPM reboot is allowed,
the claimed security property of the DEP is not preserved.
In addition to the attack trace described in Section III,
SSPA also found several other traces (attacking at different states),
which are similar variants to the attack described in Section III.
The modified version of the DEP presented in [15] is also proven to be secure in our framework.
The Bitlocker [11] designed by Microsoft also uses TPM to protect its execution state.
In the machine equipped with Bitlocker,
the hard drive is assumed to be encrypted under a volume encryption key (VEK).
The VEK is in turn encrypted by a volume master key (VMK).
When the machine is booted,
an immutable pre-BIOS will load the BIOS and extend the hash value of the BIOS into the TPM.
The pre-BIOS then passes the control to the BIOS.
Later, the BIOS can load other components by first extending the hash value of that component into the TPM.
The components then could in turn load other components by doing this repeatedly, resulting in a trust chain.
Initially, the VMK is sealed by the TPM to a certain PCR value
corresponding to a correct boot state of the machine.
When the correct state is reached, the VMK can be unsealed to decrypt the hard drive and access its data.
Even though the attacker could replace the BIOS and other components in the machine,
their hash values will not be the same as the original ones.
So the correct state cannot be reached and the VMK remains secure.
We model the protocol by assuming that the attacker can read the VMK
by either replacing a fake BIOS or a fake loader (a component) in the machine.
Otherwise, the attacker cannot access the unsealed data from the machine even if it is unsealed
as it is controlled by a trusted component.
The verification result shows that Bitlocker protects the VMK from the attacker
even when the BIOS and the loader can be replaced.
Lastly, we modeled the Needham-Schroeder Public Key (NSPK) Protocol [18]
and its fixed version by Gavin Lowe [19].
We use these two examples to show that our approach also works for stateless protocols.
In order to model the nonces exchanged by the participants in NSPK as random numbers,
we add two states for the participants when their first message is sent
and they are waiting for the second message by treating them as trusted parties.
VII Discussions
In this paper, we have presented a new approach
for the stateful security protocol verification.
Different from existing tools in the literature,
our approach allows for specifying stateful protocols directly (without modifications to the protocols)
and it can deal with infinite protocol states.
Moreover, our verification procedure is sound and complete if the solving algorithm terminates.
We have implemented a tool for our new approach
and validated it on a number of protocols.
So far, the initial results are encouraging.
When rules are newly composed in the knowledge base,
the redundancy checking consumes a large amount of time.
This is mainly because of the complexity of pairing states and predicates from different rules
and finding all possible substitutions according to Definition 6.
For the future work, accelerating the redundancy checking would be very helpful
to accelerate the verification process dramatically.
In addition, analyzing more stateful protocols would be very interesting.
Moreover, adapting our approach to verify stateful protocols
with physical properties involved, e.g., time, space, etc. would be promising as well.
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1995. |
Frobenius $n$-exangulated categories
Yu Liu and Panyue Zhou111Corresponding author. Yu Liu is supported by the Fundamental Research Funds for the Central Universities (Grants No. 2682018ZT25) and the National Natural Science Foundation of China (Grants No. 11901479). Panyue Zhou is supported by the National Natural Science Foundation of China (Grants No. 11901190 and 11671221) and the Hunan Provincial Natural Science Foundation of China (Grants No. 2018JJ3205).
Abstract
Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated
categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu.
The class of $n$-exangulated categories contains
$n$-exact categories and $(n+2)$-angulated categories as examples.
In this article, we introduce the
notion of Frobenius $n$-exangulated categories
which are a generalization of Frobenius $n$-exact categories. We
show that the stable category of a Frobenius $n$-exangulated category is an
$(n+2)$-angulated category. As an application, this result
generalizes the work by Jasso. In addition, starting from $n$-exangulated categories, we obatin a new $n$-exangulated category. This construction gives $n$-exangulated categories which
are neither $n$-exact categories nor $(n+2)$-angulated categories. Finally, we discuss an application of main results and give some examples illustrating our main results.
Key words: $n$-exangulated categories; $(n+2)$-angulated categories; $n$-exact categories.
2010 Mathematics Subject Classification: 18E30; 18E10; 18G05.
1 Introduction
Higher homological algebra was introduced by Iyama [I], and it deals with $n$-cluster tilting subcategories of abelian categories (resp. exact categories). All short exact sequences in such a subcategory are split, but it has nice exact sequences with $n+2$ objects. This was recently formalized by Jasso [J] in the theory of $n$-abelian categories (resp. $n$-exact categories).
There exists also a derived version of the theory focusing on $n$-cluster tilting subcategories of triangulated categories as introduced by Geiß, Keller, and Oppermann in the theory of $(n+2)$-angulated categories in [GKO]. The properties of $(n+2)$-angulated categories have been investigated by Bergh-Thaule in [BT]. Setting $n=1$ recovers the notions of abelian, exact and
triangulated categories.
Extriangulated categories were recently introduced by Nakaoka and Palu [NP] by
extracting those properties of $\mbox{Ext}^{1}$ on exact categories and on triangulated categories
that seem relevant from the point of view of cotorsion pairs. In particular, triangulated
categories and exact categories are extriangulated categories. There are a lot of examples of extriangulated categories which are neither triangulated categories nor exact categories.
The data of such a category is a triplet $(\mathscr{C},\mathbb{E},\mathfrak{s})$, where $\mathscr{C}$ is an additive category, $\mathbb{E}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathsf{Ab}$ is an additive bifunctor and $\mathfrak{s}$ assigns to each $\delta\in\mathbb{E}(C,A)$ a class of $3$-term sequences with end terms $A$ and $C$ such that certain axioms hold. Herschend-Liu-Nakaoka [HLN] introduced an $n$-analogue of this notion called $n$-exangulated categories. Such a category is a similar triplet $(\mathscr{C},\mathbb{E},\mathfrak{s})$, with the main distinction being that the $3$-term sequences mentioned above are replaced by $(n+2)$-term sequences. We note that the case $n=1$ corresponds to extriangulated categories.
As typical examples we have that $n$-exact and $(n+2)$-angulated categories are $n$-exangulated, see [HLN, Proposition 4.34 and Proposition 4.5].
In an extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projectives and
injectives, Liu-Nakaoka [LN] introduced higher extension
groups $\mathbb{E}^{i}$ using dimension shift, and defined the notion
of an $n$-cluster tilting subcategory $\mathscr{X}$. Under suitable conditions,
Herschend-Liu-Nakaoka [HLN] showed that $\mathscr{X}$ is $n$-exangulated categories, see [HLN, Theorem 5.41].
However, there are some other examples of $n$-exangulated categories which are neither $n$-exact nor $(n+2)$-angulated, see [HLN, Section 6].
Motivated by the definitions of Frobenius $n$-exact categories and Frobenius extriangulated categories, we define Frobenius $n$-exangulated categories.
These are $n$-exangulated categories with enough
projectives and enough injectives, and such that these two classes of objects coincide. Frobenius $n$-exangulated categories are related to $(n+2)$-angulated categories. We show the first main result in this article.
Theorem 1.1.
(see Theorem LABEL:main1 for details)
Let $\mathscr{C}$ be a Frobenius $n$-exangulated category.
Then the stable category $\overline{\mathscr{C}}$ is an $(n+2)$-angulated category.
This generalizes a result by Jasso [J, Theorem 5.11] for Frobenius $n$-exact categories.
In order to give the second main result in this article, we need the following some notions.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category and $\mathscr{X}$ be a subcategory of $\mathscr{C}$.
For each pair of objects $A$ and $C$ in $\mathscr{C}$, define
$$\mathbb{F}^{\mathscr{X}}(C,A)=\{A_{0}\xrightarrow{f}A_{1}\xrightarrow{}A_{2}%
\xrightarrow{}\cdots\xrightarrow{}A_{n-1}\xrightarrow{}A_{n}\xrightarrow{}A_{n%
+1}\overset{\delta}{\dashrightarrow}\,\mid\,f\ \textrm{is an}\ \mathscr{X}%
\textrm{-monic}\}.$$
Dually, we define for each pair of objects $A$ and $C$ in $\mathscr{C}$
$$\mathbb{F}_{\mathscr{X}}(C,A)=\{A_{0}\xrightarrow{}A_{1}\xrightarrow{}A_{2}%
\xrightarrow{}\cdots\xrightarrow{}A_{n-1}\xrightarrow{}A_{n}\xrightarrow{g}A_{%
n+1}\overset{\delta}{\dashrightarrow}\,\mid\,g\ \textrm{is an}\ \mathscr{X}%
\textrm{-epic}\}.$$
Theorem 1.2.
(see Theorem LABEL:main2 for details)
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category.
If $\mathscr{X}$ is strongly functorially finite subcategory of $\mathscr{C}$, then $(\mathscr{C},\mathbb{F},\mathfrak{s}_{\mathbb{F}})$ is a Frobenius $n$-exangulated category whose projective-injective objects are precisely $\mathscr{X}$, where $\mathbb{F}:=\mathbb{F}^{\mathscr{X}}\cap~{}\mathbb{F}_{\mathscr{X}}$.
This construction gives $n$-angulated categories which are not
$n$-exact nor $(n+2)$-angulated in general.
This article is organized as follows. In Section 2, we review some elementary definitions
and facts of $n$-exangulated categories. In Section 3, we prove our first main
result. In Section 4, we prove our second main result. In section 5, we discuss an application of main results and give some examples illustrating our main results.
2 Preliminaries
Let us briefly recall some definitions and basic properties of $n$-exangulated categories from [HLN].
We omit some details here, but the reader can find
them in [HLN]. Throughout this article, let $\mathscr{C}$ be an additive category and $n$ be any positive integer.
Definition 2.1.
[HLN, Definition 2.1]
Suppose that $\mathscr{C}$ is equipped with an additive bifunctor $\mathbb{E}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathsf{Ab}$. For any pair of objects $A,C\in\mathscr{C}$, an element $\delta\in\mathbb{E}(C,A)$ is called an $\mathbb{E}$-extension or simply an extension. We also write such $\delta$ as ${}_{A}\delta_{C}$ when we indicate $A$ and $C$.
Let ${}_{A}\delta_{C}$ be any extension. Since $\mathbb{E}$ is a bifunctor, for any $a\in\mathscr{C}(A,A^{\prime})$ and $c\in\mathscr{C}(C^{\prime},C)$, we have extensions
$$\mathbb{E}(C,a)(\delta)\in\mathbb{E}(C,A^{\prime})\ \ \text{and}\ \ \mathbb{E}%
(c,A)(\delta)\in\mathbb{E}(C^{\prime},A).$$
We abbreviately denote them by $a_{\ast}\delta$ and $c^{\ast}\delta$.
In this terminology, we have
$$\mathbb{E}(c,a)(\delta)=c^{\ast}a_{\ast}\delta=a_{\ast}c^{\ast}\delta\in%
\mathbb{E}(C^{\prime},A^{\prime}).$$
For any $A,C\in\mathscr{C}$, the zero element ${}_{A}0_{C}=0\in\mathbb{E}(C,A)$ is called the split $\mathbb{E}$-extension.
Definition 2.2.
[HLN, Definition 2.3]
Let ${}_{A}\delta_{C},{}_{A^{\prime}}\delta^{\prime}_{C^{\prime}}$ be any pair of $\mathbb{E}$-extensions. A morphism $(a,c)\colon\delta\to\delta^{\prime}$ of extensions is a pair of morphisms $a\in\mathscr{C}(A,B)$ and $c\in\mathscr{C}(A^{\prime},C^{\prime})$ in $\mathscr{C}$, satisfying the equality
$$a_{\ast}\delta=c^{\ast}\delta^{\prime}.$$
Definition 2.3.
[HLN, Definition 2.7]
Let $\bf{C}_{\mathscr{C}}$ be the category of complexes in $\mathscr{C}$. As its full subcategory, define ${\bf{C}}^{n+2}_{\mathscr{C}}$ to be the category of complexes in $\mathscr{C}$ whose components are zero in the degrees outside of $\{0,1,\ldots,n+1\}$. Namely, an object in ${\bf{C}}^{n+2}_{\mathscr{C}}$ is a complex $X^{\hbox{\boldmath$\cdot$}}=\{X_{i},d^{X}_{i}\}$ of the form
$$X_{0}\overset{d^{X}_{0}}{\longrightarrow}X_{1}\overset{d^{X}_{1}}{%
\longrightarrow}\cdots\overset{d^{X}_{n-1}}{\longrightarrow}X_{n}\overset{d^{X%
}_{n}}{\longrightarrow}X_{n+1}.$$
We write a morphism $f^{\hbox{\boldmath$\cdot$}}\colon X^{\hbox{\boldmath$\cdot$}}\to Y^{\hbox{%
\boldmath$\cdot$}}$ simply $f^{\hbox{\boldmath$\cdot$}}=(f^{0},f^{1},\ldots,f^{n+1})$, only indicating the terms of degrees $0,\ldots,n+1$.
Definition 2.4.
[HLN, Definition 2.11]
By Yoneda lemma, any extension $\delta\in\mathbb{E}(C,A)$ induces natural transformations
$$\delta_{\sharp}\colon\mathscr{C}(-,C)\Rightarrow\mathbb{E}(-,A)\ \ \text{and}%
\ \ \delta^{\sharp}\colon\mathscr{C}(A,-)\Rightarrow\mathbb{E}(C,-).$$
For any $X\in\mathscr{C}$, these $(\delta_{\sharp})_{X}$ and $\delta^{\sharp}_{X}$ are given as follows.
(1)
$(\delta_{\sharp})_{X}\colon\mathscr{C}(X,C)\to\mathbb{E}(X,A)\ ;\ f\mapsto f^{%
\ast}\delta$.
(2)
$\delta^{\sharp}_{X}\colon\mathscr{C}(A,X)\to\mathbb{E}(C,X)\ ;\ g\mapsto g_{%
\ast}\delta$.
We abbreviately denote $(\delta_{\sharp})_{X}(f)$ and $\delta^{\sharp}_{X}(g)$ by $\delta_{\sharp}(f)$ and $\delta^{\sharp}(g)$.
Definition 2.5.
[HLN, Definition 2.13]
An $n$-exangle is a pair $\langle X^{\hbox{\boldmath$\cdot$}},\delta\rangle$ of $X^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}$ and $\delta\in\mathbb{E}(X^{n+1},X^{0})$ which satisfies the following conditions.
(1)
The following sequence of functors $\mathscr{C}^{\mathrm{op}}\to\mathsf{Ab}$ is exact.
$$\mathscr{C}(-,X_{0})\xLongrightarrow{\mathscr{C}(-,\ d^{X}_{0})}\cdots%
\xLongrightarrow{\mathscr{C}(-,\ d^{X}_{n})}\mathscr{C}(-,X_{n+1})%
\xLongrightarrow{~{}\delta_{\sharp}~{}}\mathbb{E}(-,X_{0})$$
(2)
The following sequence of functors $\mathscr{C}\to\mathsf{Ab}$ is exact.
$$\mathscr{C}(X_{n+1},-)\xLongrightarrow{\mathscr{C}(d^{X}_{n},\ -)}\cdots%
\xLongrightarrow{\mathscr{C}(d^{X}_{0},\ -)}\mathscr{C}(X_{0},-)%
\xLongrightarrow{~{}\delta^{\sharp}~{}}\mathbb{E}(X_{n+1},-)$$
In particular any $n$-exangle is an object in $\AE$.
A morphism of $n$-exangles simply means a morphism in $\AE$. Thus $n$-exangles form a full subcategory of $\AE$.
Definition 2.6.
[HLN, Definition 2.22]
Let $\mathfrak{s}$ be a correspondence which associates a homotopic equivalence class $\mathfrak{s}(\delta)=[{}_{A}X^{\hbox{\boldmath$\cdot$}}_{C}]$ to each extension $\delta={}_{A}\delta_{C}$. Such $\mathfrak{s}$ is called a realization of $\mathbb{E}$ if it satisfies the following condition for any $\mathfrak{s}(\delta)=[X^{\hbox{\boldmath$\cdot$}}]$ and any $\mathfrak{s}(\rho)=[Y^{\hbox{\boldmath$\cdot$}}]$.
(R0)
For any morphism of extensions $(a,c)\colon\delta\to\rho$, there exists a morphism $f^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}(X^{\hbox{\boldmath$%
\cdot$}},Y^{\hbox{\boldmath$\cdot$}})$ of the form $f^{\hbox{\boldmath$\cdot$}}=(a,f_{1},\ldots,f_{n},c)$. Such $f^{\hbox{\boldmath$\cdot$}}$ is called a lift of $(a,c)$.
In such a case, we abbreviately say that “$X^{\hbox{\boldmath$\cdot$}}$ realizes $\delta$” whenever they satisfy $\mathfrak{s}(\delta)=[X^{\hbox{\boldmath$\cdot$}}]$.
Moreover, a realization $\mathfrak{s}$ of $\mathbb{E}$ is said to be exact if it satisfies the following conditions.
(R1)
For any $\mathfrak{s}(\delta)=[X^{\hbox{\boldmath$\cdot$}}]$, the pair $\langle X^{\hbox{\boldmath$\cdot$}},\delta\rangle$ is an $n$-exangle.
(R2)
For any $A\in\mathscr{C}$, the zero element ${}_{A}0_{0}=0\in\mathbb{E}(0,A)$ satisfies
$$\mathfrak{s}({}_{A}0_{0})=[A\overset{\mathrm{id}_{A}}{\longrightarrow}A\to 0%
\to\cdots\to 0\to 0].$$
Dually, $\mathfrak{s}({}_{0}0_{A})=[0\to 0\to\cdots\to 0\to A\overset{\mathrm{id}_{A}}{%
\longrightarrow}A]$ holds for any $A\in\mathscr{C}$.
Note that the above condition (R1) does not depend on representatives of the class $[X^{\hbox{\boldmath$\cdot$}}]$.
Definition 2.7.
[HLN, Definition 2.23]
Let $\mathfrak{s}$ be an exact realization of $\mathbb{E}$.
(1)
An $n$-exangle $\langle X^{\hbox{\boldmath$\cdot$}},\delta\rangle$ is called a $\mathfrak{s}$-distinguished $n$-exangle if it satisfies $\mathfrak{s}(\delta)=[X^{\hbox{\boldmath$\cdot$}}]$. We often simply say distinguished $n$-exangle when $\mathfrak{s}$ is clear from the context.
(2)
An object $X^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}$ is called an $\mathfrak{s}$-conflation or simply a conflation if it realizes some extension $\delta\in\mathbb{E}(X_{n+1},X_{0})$.
(3)
A morphism $f$ in $\mathscr{C}$ is called an $\mathfrak{s}$-inflation or simply an inflation if it admits some conflation $X^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}$ satisfying $d_{X}^{0}=f$.
(4)
A morphism $g$ in $\mathscr{C}$ is called an $\mathfrak{s}$-deflation or simply a deflation if it admits some conflation $X^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}$ satisfying $d_{X}^{n}=g$.
Definition 2.8.
[HLN, Definition 2.27]
For a morphism $f^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}(X^{\hbox{\boldmath$%
\cdot$}},Y^{\hbox{\boldmath$\cdot$}})$ satisfying $f^{0}=\mathrm{id}_{A}$ for some $A=X_{0}=Y_{0}$, its mapping cone $M_{f}^{\hbox{\boldmath$\cdot$}}\in{\bf{C}}^{n+2}_{\mathscr{C}}$ is defined to be the complex
$$X_{1}\xrightarrow{d^{M_{f}}_{0}}X_{2}\oplus Y_{1}\xrightarrow{d^{M_{f}}_{1}}X_%
{3}\oplus Y_{2}\xrightarrow{d^{M_{f}}_{2}}\cdots\xrightarrow{d^{M_{f}}_{n-1}}X%
_{n+1}\oplus Y_{n}\xrightarrow{d^{M_{f}}_{n}}Y_{n+1}$$
where $d^{M_{f}}_{0}=\begin{bmatrix}-d^{X}_{1}\\
f_{1}\end{bmatrix},$
$d^{M_{f}}_{i}=\begin{bmatrix}-d^{X}_{i+1}&0\\
f_{i+1}&d^{Y}_{i}\end{bmatrix}\ (1\leq i\leq n-1),$
$d^{M_{f}}_{n}=\begin{bmatrix}f_{n+1}&d^{Y}_{n}\end{bmatrix}$.
Mapping cocone is defined dually, for morphisms $h^{\hbox{\boldmath$\cdot$}}$ in ${\bf{C}}^{n+2}_{\mathscr{C}}$ satisfying $h_{n+1}=\mathrm{id}$.
Definition 2.9.
[HLN, Definition 2.32]
An $n$-exangulated category is a triplet $(\mathscr{C},\mathbb{E},\mathfrak{s})$ of additive category $\mathscr{C}$, additive bifunctor $\mathbb{E}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathsf{Ab}$, and its exact realization $\mathfrak{s}$, satisfying the following conditions.
(EA1)
Let $A\overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C$ be any sequence of morphisms in $\mathscr{C}$. If both $f$ and $g$ are inflations, then so is $g\circ f$. Dually, if $f$ and $g$ are deflations then so is $g\circ f$.
(EA2)
For $\rho\in\mathbb{E}(D,A)$ and $c\in\mathscr{C}(C,D)$, let ${}_{A}\langle X^{\hbox{\boldmath$\cdot$}},c^{\ast}\rho\rangle_{C}$ and ${}_{A}\langle Y^{\hbox{\boldmath$\cdot$}},\rho\rangle_{D}$ be distinguished $n$-exangles. Then $(\mathrm{id}_{A},c)$ has a good lift $f^{\hbox{\boldmath$\cdot$}}$, in the sense that its mapping cone gives a distinguished $n$-exangle $\langle M^{\hbox{\boldmath$\cdot$}}_{f},(d^{X}_{0})_{\ast}\rho\rangle$.
(EA2${}^{\mathrm{op}}$)
Dual of (EA2).
Note that the case $n=1$, a triplet $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is a $1$-
exangulated category if and only if it is an extriangulated category, see [HLN, Proposition 4.3].
Example 2.10.
$n$-exact categories and $(n+2)$-angulated categories are $n$-exangulated categories.
There are some other examples of $n$-exangulated categories
which are neither $n$-exact nor $(n+2)$-angulated.
See [HLN, Section 6] for more details.
Lemma 2.11.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category, and
$$A_{0}\xrightarrow{\alpha_{0}}A_{1}\xrightarrow{\alpha_{1}}A_{2}\xrightarrow{%
\alpha_{2}}\cdots\xrightarrow{\alpha_{n-2}}A_{n-1}\xrightarrow{\alpha_{n-1}}A_%
{n}\xrightarrow{\alpha_{n}}A_{n+1}\overset{\delta}{\dashrightarrow}$$
a distinguished $n$-exangle. Then we have the following long exact sequences:
$$\mathscr{C}(-,A_{0})\xrightarrow{}\mathscr{C}(-,A_{1})\xrightarrow{}\cdots%
\xrightarrow{}\mathscr{C}(-,A_{n+1})\xrightarrow{}\mathbb{E}(-,A_{0})%
\xrightarrow{}\mathbb{E}(-,A_{1})\xrightarrow{}\mathbb{E}(-,A_{2});$$
$$\mathscr{C}(A_{n+1},-)\xrightarrow{}\mathscr{C}(A_{n},-)\xrightarrow{}\cdots%
\xrightarrow{}\mathscr{C}(A_{0},-)\xrightarrow{}\mathbb{E}(A_{n+1},-)%
\xrightarrow{}\mathbb{E}(A_{n},-)\xrightarrow{}\mathbb{E}(A_{n-1},-).$$
Proof.
This follows from Definition 2.13 and Corollary 3.11 in [HLN]. ∎
Definition 2.12.
[HLN, Definition 3.6 and Definition 3.7]
Let $\mathscr{C}$ be an additive category and $\mathbb{E}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathsf{Ab}$ be a additive
bifunctor.
(1)
A functor $\mathbb{F}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathsf{Sets}%
\hskip 0.7227pt$ is called a subfunctor of $\mathbb{E}$ if it satisfies the following conditions.
•
$\mathbb{F}(C,A)$ is a subset of $\mathbb{E}(C,A)$, for any $A,C\in\mathscr{C}$.
•
$\mathbb{F}(c,a)=\mathbb{E}(c,a)|_{\mathbb{F}(C,A)}$ holds, for any $a\in\mathscr{C}(A,A^{\prime})$ and $c\in\mathscr{C}(C^{\prime},C)$.
In this case, we write as $\mathbb{F}\subseteq\mathbb{E}$.
(2)
A subfunctor $\mathbb{F}\subseteq\mathbb{E}$ is said to be an additive subfunctor
if $\mathbb{F}(C,A)\subseteq\mathbb{E}(C,A)$ is an abelian subgroup for any $A,C\in\mathscr{C}$.
In this case, $\mathbb{F}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathsf{Ab}$
itself becomes a additive bifunctor.
(3)
Let $\mathbb{F}\subseteq\mathbb{E}$ be an additive subfunctor. For a realization s of $\mathbb{E}$,
define $\mathfrak{s}|_{\mathbb{F}}$ to be the restriction of s onto $\mathbb{F}$. Namely, it is defined by $\mathfrak{s}|_{\mathbb{F}}(\delta)=\mathfrak{s}(\delta)$ for any $\mathbb{F}$-extension $\delta$.
Lemma 2.13.
[HLN, Proposition 3.14]
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category. For any additive subfunctor $\mathbb{F}\subseteq\mathbb{E}$, the following statements are equivalent.
(1)
$(\mathscr{C},\mathbb{F},\mathfrak{s}_{\mathbb{F}})$ is $n$-exangulated.
(2)
$\mathfrak{s}_{\mathbb{F}}$-inflations are closed under composition.
(3)
$\mathfrak{s}_{\mathbb{F}}$-deflations are closed under composition.
3 Frobenius $n$-exangulated categories
In this section, we introduce the
notion of Frobenius $n$-exangulated categories
which are a generalization of Frobenius $n$-exact categories. Moreover, we
prove that the stable category of a Frobenius $n$-exangulated category is an
$(n+2)$-angulated category.
Let $\mathscr{C}$ be an additive category,
and $\mathscr{X}$ be a subcategory of $\mathscr{C}$.
Recall that we say a morphism $f\colon A\to B$ in $\mathscr{C}$ is an $\mathscr{X}$-monic if
$$\mbox{Hom}_{\mathscr{C}}(f,X)\colon\mbox{Hom}_{\mathscr{C}}(B,X)\to\mbox{Hom}_%
{\mathscr{C}}(A,X)$$
is an epimorphism for all $X\in\mathscr{X}$. We say that $f$ is an $\mathscr{X}$-epic if
$$\mbox{Hom}_{\mathscr{C}}(X,f)\colon\mbox{Hom}_{\mathscr{C}}(X,A)\to\mbox{Hom}_%
{\mathscr{C}}(X,B)$$
is an epimorphism for all $X\in\mathscr{X}$.
Similarly,
we say that $f$ is a left $\mathscr{X}$-approximation of $B$ if $f$ is an $\mathscr{X}$-monoic and $A\in\mathscr{X}$.
We say that $f$ is a right $\mathscr{X}$-approximation of $A$ if $f$ is an $\mathscr{X}$-epic and $B\in\mathscr{X}$.
A subcategory $\mathscr{X}$ is called contravariantly finite if any object in $\mathscr{C}$ admits a right
$\mathscr{X}$-approximation. Dually we can define covariantly finite subcategory.
Definition 3.1.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category. A subcategory $\mathscr{X}$ of $\mathscr{C}$ is called
strongly contravariantly finite, if for any object $C\in\mathscr{C}$, there exists a distinguished $n$-exangle
$$B\xrightarrow{}X_{1}\xrightarrow{}X_{2}\xrightarrow{}\cdots\xrightarrow{}X_{n-%
1}\xrightarrow{}X_{n}\xrightarrow{~{}g~{}}C\overset{}{\dashrightarrow}$$
where $g$ is a right $\mathscr{X}$-approximation of $C$ and $X_{i}\in\mathscr{X}$.
Dually we can define (strongly covariantly finite subcategory.
A strongly contravariantly finite and strongly covariantly finite subcategory is called strongly functorially finite.
Definition 3.2.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category.
(1)
An object $P\in\mathscr{C}$ is called projective if, for any distinguished $n$-exangle
$$A_{0}\xrightarrow{\alpha_{0}}A_{1}\xrightarrow{\alpha_{1}}A_{2}\xrightarrow{%
\alpha_{2}}\cdots\xrightarrow{\alpha_{n-2}}A_{n-1}\xrightarrow{\alpha_{n-1}}A_%
{n}\xrightarrow{\alpha_{n}}A_{n+1}\overset{\delta}{\dashrightarrow}$$
and any morphism $c$ in $\mathscr{C}(P,A_{n+1})$, there exists a morphism $b\in\mathscr{C}(P,A_{n})$ satisfying $\alpha_{n}\circ b=c$.
We denote the full subcategory of projective objects in $\mathscr{C}$ by $\mathcal{P}$.
Dually, the full subcategory of injective objects in $\mathscr{C}$ is denoted by $\mathcal{I}$.
(2)
We say that $\mathscr{C}$ has enough projective if
for any object $C\in\mathscr{C}$, there exists a distinguished $n$-exangle
$$B\xrightarrow{\alpha_{0}}P_{1}\xrightarrow{\alpha_{1}}P_{2}\xrightarrow{\alpha%
_{2}}\cdots\xrightarrow{\alpha_{n-2}}P_{n-1}\xrightarrow{\alpha_{n-1}}P_{n}%
\xrightarrow{\alpha_{n}}C\overset{\delta}{\dashrightarrow}$$
satisfying $P_{1},P_{2},\cdots,P_{n}\in\mathcal{P}$. We can define the notion of having enough injectives dually.
(3)
$\mathscr{C}$ is said to be Frobenius if $\mathscr{C}$ has enough projectives and enough injectives
and if moreover the projectives coincide with the injectives.
Remark 3.3.
(1) When $n=1$, these agree with
the usual definitions [NP, Definition 3.23, Definition 3.25 and Definition 7.1].
(2) If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an $n$-exact category, then these agree with
the Jasso definitions [J, Definition 3.11], [J, Definition 5.3] and [J, Definition 5.5].
(3) If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an $(n+2)$-angulated category, then
$\mathcal{P}=\mathcal{I}$ consists of zero objects. Moreover it always has enough projectives and enough injectives.
Lemma 3.4.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category. Then
following statements are equivalent for an object $P\in\mathscr{C}$.
(1)
$\mathbb{E}(P,A)=0$ for any $A\in\mathscr{C}$;
(2)
$P$ is projective;
(3)
Any distinguished $n$-exangle $A_{0}\xrightarrow{\alpha_{0}}A_{1}\xrightarrow{\alpha_{1}}A_{2}\xrightarrow{%
\alpha_{2}}\cdots\xrightarrow{\alpha_{n-2}}A_{n-1}\xrightarrow{\alpha_{n-1}}A_%
{n}\xrightarrow{\alpha_{n}}P\overset{\delta}{\dashrightarrow}$ splits.
Proof.
(1) $\Rightarrow$ (2).
For any distinguished $n$-exangle
$$A_{0}\xrightarrow{\alpha_{0}}A_{1}\xrightarrow{\alpha_{1}}A_{2}\xrightarrow{%
\alpha_{2}}\cdots\xrightarrow{\alpha_{n-2}}A_{n-1}\xrightarrow{\alpha_{n-1}}A_%
{n}\xrightarrow{\alpha_{n}}A_{n+1}\overset{\delta}{\dashrightarrow}$$
and any morphism $c$ in $\mathscr{C}(P,A_{n+1})$, by Lemma 2.11, we have the following exact sequence:
$$\mathscr{C}(P,A_{n+1})\xrightarrow{\mathscr{C}(P,~{}\alpha_{n})}\mathscr{C}(P,%
A_{n+1})\xrightarrow{}\mathbb{E}(P,A_{0})=0.$$
So there exists a morphism $b\in\mathscr{C}(P,A_{n})$ such that $\alpha_{n}\circ b=c$.
This shows that $P$ is projective.
(2) $\Rightarrow$ (3). Since $P$ is projective, there exist a morphism $u\colon P\to A_{n}$
such that $\alpha_{n}u=1_{P}$. By [HLN, Claim 2.15], we have $\delta=0$.
Hence $\delta$ splits.
(3) $\Rightarrow$ (1). It follows from [HLN, Claim 2.15]. ∎
Let $\mathscr{C}$ be an additive category.
For two objects $A,B$ in $\mathscr{X}$ denote by $\mathscr{X}(A,B)$ the subgroup of $\mbox{Hom}_{\mathscr{C}}(A,B)$ consisting of those morphisms which factor through an object in $\mathscr{X}$. Denote by $\mathscr{C}/\mathscr{X}$ the quotient category of $\mathscr{C}$ modulo $\mathscr{X}$: the objects are the same as the ones in $\mathscr{C}$, for two objects $A$ and $B$ the Hom space is given by the quotient group $\mbox{Hom}_{\mathscr{C}}(A,B)/\mathscr{X}(A,B)$.
Note that the quotient category $\mathscr{C}/\mathscr{X}$ is an additive category. We denote $\overline{f}$ the image of $f\colon A\to B$ of $\mathscr{C}$ in $\mathscr{C}/\mathscr{X}$.
From now onwards, we assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is a Frobenius $n$-exangulated category.
We refer to this category $\mathscr{C}/\mathcal{I}$ as the stable category of $\mathscr{C}$.
In keeping the convention of the classical theory, if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is a Frobenius $n$-exangulated category, then we denote its stable category by $\overline{\mathscr{C}}$.
Any object $X\in\mathscr{C}$ admits a distinguished $n$-exangle |
Chasing obscuration in type-I AGN: discovery of an eclipsing clumpy wind at the outer broad-line region of NGC 3783
M. Mehdipour
1
SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands
[email protected]
J.S. Kaastra
1
SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands
[email protected]
2Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, the Netherlands
2
G.A. Kriss
3Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
3
N. Arav
4Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA
4
E. Behar
5Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
5
S. Bianchi
6Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy
6
G. Branduardi-Raymont
7Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
7
M. Cappi
8INAF-IASF Bologna, Via Gobetti 101, I-40129 Bologna, Italy
8
E. Costantini
1
SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands
[email protected]
J. Ebrero
9European Space Astronomy Centre, P.O. Box 78, E-28691 Villanueva de la Cañada, Madrid, Spain
9
L. Di Gesu
10Department of Astronomy, University of Geneva, 16 Ch. d’Ecogia, 1290 Versoix, Switzerland
10
S. Kaspi
5Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
5
J. Mao
1
SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands
[email protected]
2Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, the Netherlands
2
B. De Marco
11Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warsaw, Poland
11
G. Matt
6Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy
6
S. Paltani
10Department of Astronomy, University of Geneva, 16 Ch. d’Ecogia, 1290 Versoix, Switzerland
10
U. Peretz
5Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
5
B.M. Peterson
3Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
312Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA
1213Center for Cosmology & AstroParticle Physics, The Ohio State University, 191 West Woodruff Ave., Columbus, OH 43210, USA
13
P.-O. Petrucci
14Univ. Grenoble Alpes, IPAG, F-38000 Grenoble, France
1415CNRS, IPAG, F-38000 Grenoble, France
15
C. Pinto
16Institute of Astronomy, Madingley Road, CB3 0HA Cambridge, UK
16
G. Ponti
17Max Planck Institute fur Extraterrestriche Physik, 85748, Garching, Germany
17
F. Ursini
8INAF-IASF Bologna, Via Gobetti 101, I-40129 Bologna, Italy
8
C.P. de Vries
1
SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, the Netherlands
[email protected]
D.J. Walton
16Institute of Astronomy, Madingley Road, CB3 0HA Cambridge, UK
16
(Received 15 May 2017 / Accepted 12 July 2017)
Key Words.:
X-rays: galaxies – galaxies: active – galaxies: Seyfert – galaxies: individual: NGC 3783 – techniques: spectroscopic
In 2016 we carried out a Swift monitoring program to track the X-ray hardness variability of eight type-I AGN over a year. The purpose of this monitoring was to find intense obscuration events in AGN, and thereby study them by triggering joint XMM-Newton, NuSTAR, and HST observations. We successfully accomplished this for NGC 3783 in December 2016. We found heavy X-ray absorption produced by an obscuring outflow in this AGN. As a result of this obscuration, interesting absorption features appear in the UV and X-ray spectra, which are not present in the previous epochs. Namely, the obscuration produces broad and blue-shifted UV absorption lines of Ly$\alpha$, C iv, and N v, together with a new high-ionisation component producing Fe xxv and Fe xxvi absorption lines. In soft X-rays, only narrow emission lines stand out above the diminished continuum as they are not absorbed by the obscurer. Our analysis shows that the obscurer partially covers the central source with a column density of few $10^{23}$ cm${}^{-2}$, outflowing with a velocity of few thousand km s${}^{-1}$. The obscuration in NGC 3783 is variable and lasts for about a month. Unlike the commonly-seen warm-absorber winds at pc-scale distances from the black hole, the eclipsing wind in NGC 3783 is located at about 10 light days. Our results suggest the obscuration is produced by an inhomogeneous and clumpy medium, consistent with clouds in the base of a radiatively-driven disk wind at the outer broad-line region of the AGN.
1 Introduction
Accretion onto supermassive black holes (SMBHs) in active galactic nuclei (AGN) is believed to be accompanied by outflows of gas, which couple the SMBHs to their environment. The observed associations between SMBHs and their host galaxies, such as the M-$\sigma$ relation (Ferrarese & Merritt 2000), point to their co-evolution through a feedback mechanism. The AGN outflows may play an important role in this feedback as they can impact star formation, chemical enrichment of the intergalactic medium, and cooling flows in galaxy clusters (e.g. review by Fabian 2012). There are however significant gaps in our understanding of the outflow phenomenon in AGN.
Winds of photoionised gas (warm absorbers - WA) are commonly observed in bright AGN through high-resolution UV and X-ray spectroscopy (e.g. Crenshaw et al. 1999; Blustin et al. 2005). They often consist of multiple ionisation components, outflowing with velocities of typically few hundred $\mathrm{km\ s^{-1}}$. From an observational point of view, other kinds of winds with different properties from WAs have been found in the X-ray band: high-ionisation ultra-fast outflows (e.g. PDS~456, Reeves et al. 2009) and obscuring outflows (e.g. NGC~5548, Kaastra et al. 2014). Compared to the common WAs at pc-scale distances from the black hole (e.g. Kaastra et al. 2012), the obscuring outflow found in NGC 5548 is a faster and more massive wind closer to the accretion disk. It produces strong absorption of the X-ray continuum, in addition to appearance of blue-shifted and broad UV absorption lines. X-ray obscuration with associated UV line absorption has been seen also in Mrk~335 (Longinotti et al. 2013) and NGC~985 (Ebrero et al. 2016). Variable X-ray absorption is commonly found in type-I AGN; e.g. NGC~1365 (Rivers et al. 2015); PDS~456 (Matzeu et al. 2016); NGC~4151 (Beuchert et al. 2017); IRAS~13224-3809 (Parker et al. 2017). However, the association to the UV broad-line absorbing outflows is unclear. Moreover, the physical connection between different kinds of AGN outflow, and their origin and driving mechanism, are still poorly understood. In this study we aim to address the nature and origin of an X-ray obscuration/eclipse through UV/X-ray spectroscopy of the absorption during an eclipsing event.
An efficient way to drive winds in quasars is via radiative acceleration of the gas through UV line absorption (e.g. Proga & Kallman 2004). However, intense X-ray radiation from the central source can over-ionise the gas, leaving insufficient line opacity to drive the wind. Shielding the UV-absorbing gas from the X-rays by an obscuring medium near the X-ray source (like that seen in NGC 5548) can prevent this. Thus, obscuration may play an important role in driving AGN outflows. A statistical study of X-ray variability by Markowitz et al. (2014) identifies obscuration events in AGN using RXTE observations. They find 12 X-ray eclipses in 8 AGN, and compute the probability of finding a type-I AGN undergoing obscuration is $\sim$1%. However, the origin, location, and physical properties of such eclipses are poorly understood. It is also uncertain whether such eclipses are a manifestation of disk winds in general. In order to broaden our understanding of this phenomenon, we have conducted a Swift monitoring program on a sample of type-I AGN to catch an obscuration event and perform a ToO multiwavelength spectroscopic study of it using XMM-Newton, NuSTAR, and HST COS.
2 Swift monitoring program and triggering of XMM-Newton, NuSTAR, and HST observations
The X-ray spectral hardness variability is a useful indicator of obscuration. We define the hardness ratio (HR) as ${(H-S)/(H+S)}$, where $H$ and $S$ are the Swift XRT count rates in the hard (1.5–10 keV) and soft (0.3–1.5 keV) bands, respectively. X-ray absorption by obscuring/eclipsing gas increases HR. During Swift Cycle 12 (April 2016–March 2017), we monitored eight suitable type I AGN: Ark~564, MR~2251-178, Mrk 335, Mrk~509, Mrk~841, NGC 3783, NGC~4593, and NGC~7469. These AGN were observed weekly by Swift during the corresponding visibility windows of the four observatories. While most of the AGN displayed stable HR throughout the year, only NGC 3783 (triggered by us) and Mrk 335 (triggered earlier by another team) showed significant X-ray spectral hardening.
Figure 1 shows the Swift lightcurve of NGC 3783 from May 2016 to January 2017. In December 2016, we found an intense X-ray spectral hardening event lasting for about 32 days. During this period we successfully executed the triggering of our XMM-Newton, NuSTAR, and HST observations (see Table 2 in Appendix A). Figure 2 (upper panel) shows the 2016 XMM-Newton EPIC-pn and NuSTAR spectra, as well as the time-averaged EPIC-pn spectra from 2000 and 2001. Strong X-ray absorption is evident in the new data (see also the RGS data in Fig. 2, bottom panel), with the 0.3–2.0 keV flux dropping from ${1.60\times 10^{-11}}$ $\rm{erg\ cm}^{-2}\ \rm{s}^{-1}$ in 2000–2001 by a factor of 8.0 (11 Dec 2016) and 4.5 (21 Dec 2016). This X-ray absorption coincides with an increase in the UV flux (Fig. 1). Strong line absorption affects the blue side of the C iv line profile in the 2016 HST/COS spectrum (Fig. 3), extending from the line center to ${\sim-3200}$ $\mathrm{km\ s^{-1}}$, with additional shallow absorption features present down to ${\sim-6200}$ $\mathrm{km\ s^{-1}}$. Blue-shifted broad UV line absorption is also detected in Ly$\alpha$ and N v in the new COS spectra (Kriss et al. in prep).
For a description of our data reduction, we refer to Appendix A in Mehdipour et al. (2015), which applies to the NGC 3783 data used here, with more details provided in our follow-up papers. The wavelength/energy bands used in our simultaneous X-ray spectral modelling of the data are 7–37 $\AA$ for RGS, 1.5–10 keV for EPIC-pn, and 10–80 keV for NuSTAR. The spectral modelling is done using the SPEX package v3.03.01 (Kaastra et al. 1996). We use C-statistics for spectral fitting with X-ray spectra optimally binned according to Kaastra & Bleeker (2016). Errors are reported at $1\sigma$ confidence level.
3 Modelling of the obscuring wind in NGC 3783
For photoionisation modelling of the WA and the new obscurer, we determined the spectral energy distribution (SED) of the central ionising source in NGC 3783. We applied a template SED model that we reported in Mehdipour et al. (2015) for NGC 5548 to fit the NGC 3783 data and determine its SED. These Seyfert-1 AGN have a SED composition consisting of an optical/UV thin disk component, an X-ray power-law continuum, a neutral X-ray reflection component, and a warm Comptonisation component for the soft X-ray excess. The exponential cut-off energy of the power-law was set to 340 keV (De Rosa et al. 2002), which is also consistent with the NuSTAR spectra. The Galactic X-ray absorption is modelled using the hot model in SPEX, with $N_{\mathrm{H}}=9.59\times 10^{20}$ cm${}^{-2}$ (Murphy et al. 1996). The redshift of NGC 3783 is set to 0.009730 (Theureau et al. 1998), and all abundances are fixed to the proto-solar values of Lodders et al. (2009). To correct for Galactic reddening, we used the ebv model, with ${E(B-V)=0.107}$ (Schlafly & Finkbeiner 2011). To take into account the host galaxy optical/UV stellar emission, we used the galactic bulge model of Kinney et al. (1996), and normalised it to the NGC 3783 host galaxy flux measured from HST (Bentz et al. 2013). In the $12\arcsec$ diameter circular aperture of OM, this is ${7.04\times 10^{-15}}$ $\rm{erg\ cm}^{-2}\ \rm{s}^{-1}\ {\AA}^{-1}$ (Bentz, priv. comm.).
Before modelling the new strong absorption by the obscurer in the 2016 data, we first derived a model for the WA from archival observations, where the WA absorption features are clearly detectable in X-rays. Previous studies have found a WA in NGC 3783 (Kaspi et al. 2002; Blustin et al. 2002; Behar et al. 2003; Scott et al. 2014). We utilised all archival data from XMM-Newton (2000 and 2001) and Chandra HETGS (2000, 2001, and 2013) to produce a set of time-averaged spectra. The HETGS spectra were obtained from TGCat (Huenemoerder et al. 2011). For photoionisation and spectral modelling of the optically-thin WA, we used the new pion model in SPEX (Mehdipour et al. 2016b). From modelling the NGC 3783 archival spectrum, we find that the WA spans a wide range of ionisation, similar to the distribution reported by Holczer et al. (2007) and Goosmann et al. (2016). We fit the absorption by the WA with multiple pion components, with outflow velocities ranging from 450 to 1200 $\mathrm{km\ s^{-1}}$. The narrow X-ray emission lines are also fitted with the pion model at zero net velocity. The total $N_{\mathrm{H}}$ of the WA is derived to be about ${4.0\times 10^{22}}$ $\rm{cm}^{-2}$. More details about this WA model will be reported by Mao et al. (in prep).
The 2016 data suggest that the photoionised emission from the X-ray narrow line region is not absorbed by the obscurer. This is evident from the clear presence of narrow emission lines and radiative recombination edges in the RGS spectrum, such as the O viii Ly$\alpha$ at 19 $\AA$ and O vii triplet lines at 22 $\AA$ (Fig. 2, bottom panel). We make a reasonable assumption that the obscurer in NGC 3783 is likely located interior to both the WA and the X-ray narrow line region. Previous studies find the WA in NGC 3783 to be at pc-scale distances from the black hole (e.g. Behar et al. 2003; Gabel et al. 2005). From our photoionisation modelling, we find that the ionisation state and turbulent velocity of both the WA and the X-ray narrow line region match each other. These indicate that they are likely at similar distances from the black hole, albeit there are modelling uncertainties associated with this interpretation. In our line of sight, the WA is effectively shielded from receiving some of the ionising radiation, thus it becomes less ionised. This lower ionisation is directly evidenced by the increased absorption in the narrow UV outflow components in the 0 to $-1500$ $\mathrm{km\ s^{-1}}$ range of the COS spectra in Fig. 3. For the WA of NGC 3783 with an electron density of ${3\times 10^{4}}$ cm${}^{-3}$ (Gabel et al. 2005), we find the recombination timescale for relevant ions is ${\lesssim 1}$ day. This means during the month-long obscuration event, the WA would be able to respond to the change in the ionising SED caused by the obscuration. Thus, we take into account the enhanced absorption by this de-ionised WA. The WA model obtained from the unobscured data is incorporated in our modelling of the new obscured data, with only the ionisation parameter $\xi$ (Krolik et al. 1981) of the WA components self-consistently lowered by the obscuration.
Continuum absorption by the obscurer is too strong to leave detectable absorption lines in soft X-rays. Therefore, to set the velocity and $\xi$ of the obscurer in our modelling, we use the broad UV absorption lines of the obscurer seen in the COS spectrum. The transmission model for the broad C iv line (Kriss et al. in prep) is shown in Fig. 3, top panel. We use the weighted average velocity of the broad C iV absorption profile in our X-ray spectral modelling (${v_{\rm out}=-1900}$ $\mathrm{km\ s^{-1}}$ and ${\sigma_{v}=1100}$ $\mathrm{km\ s^{-1}}$). The ionisation balance of the obscurer is derived using the Cloudy v13.04 photoionisation code (Ferland et al. 2013) for an optically-thick medium, to match the UV lines in the COS spectrum, with its X-ray absorption fitted using the xabs model in SPEX. This is in order to produce the observed UV lines without a massive neutral hydrogen front as there are no significant detections of C ii and Si ii in the COS spectrum of NGC 3783. We obtain a solution at ${\log\xi=1.84}$ for the obscurer.
To fit the 2016 obscured X-ray spectra we require two xabs absorption components to reproduce the observed curvature of the spectrum from soft to hard X-rays. The curvature seen in 2016 is not present in the archival spectra. The addition of the first and second xabs components improve the fit significantly with $\Delta{\rm C}$ of about 4000 and 1000, respectively. Interestingly, we find evidence for a strong high-ionisation component (HC) in the 2016 data (see Fig. 4). The Fe xxvi Ly$\alpha$ line (${E_{0}=6.966}$ keV), blue-shifted by about $-2300$ $\mathrm{km\ s^{-1}}$, overlaps with the Fe K$\beta$ emission line (${E_{0}=7.020}$ keV). This Fe xxvi absorption of the continuum causes the Fe K$\beta$ emission line to vanish in 2016, while it was present in the archival data (Fig. 4). The addition of this HC component further improves the fit with $\Delta{\rm C}$ of about 200. The X-ray transmission of all the absorption components in our line of sight to NGC 3783 are shown in Fig. 5. The final model fits the data well with C-stat / d.o.f. = 2288 / 1539 (Obs. 1) and 2285 / 1542 (Obs. 2). We note that the remaining fit residuals primarily belong to soft X-ray emission lines from the X-ray narrow line region. The modelling of these lines is independent of the obscurer. The obscurer itself is effectively featureless in X-rays (Fig. 5), detected only through continuum absorption and the characteristic curvature in the broadband X-ray continuum. The continuum is fitted well as there are no curvature residuals in our best-fit model (Fig 6, bottom panel). The intrinsic photon index $\Gamma$ of the underlying X-ray power-law continuum is found to be about 1.71 (Obs. 1) and 1.75 (Obs. 2). Our best-fit model to the data, and the corresponding SEDs, are displayed in Fig. 6. The best-fit parameters of the obscurer and the HC are given in Table 1.
4 Discussion and conclusions
The strongly diminished soft X-rays and the appearance of new spectral features in NGC 3783 can be explained by an obscuring wind at the core of this AGN. The obscurer is found to consist of two partially-covering absorption components, suggesting the obscuring medium is inhomogeneous and clumpy. A similar property was also found for the obscuring wind in NGC 5548 (Kaastra et al. 2014). However, in the case of NGC 5548, the obscuration has been lasting for several years (Mehdipour et al. 2016a), while the one in NGC 3783 was a short-lived eclipsing event, lasting for about a month (Fig. 1). Moreover, the obscurer in NGC 3783 has a significantly higher ionisation parameter $\xi$ and lower covering fraction $C_{f}$ than the one in NGC 5548. These differences can be explained if the obscurer in NGC 5548 is a spatially extended stream of cool gas in our line of sight (Kaastra et al. 2014), while the one in NGC 3783 is a hotter transient cloud, which is localised closer to the central source.
NGC 3783 has previously displayed X-ray spectral hardening in the archival Swift and RXTE data. There is a single obscured Swift observation in 2009, and from RXTE Markowitz et al. (2014) identify four eclipsing events. However, in 2016 we got the first opportunity to carry out a spectroscopic study of the obscuration with XMM-Newton, HST, and NuSTAR. Since the obscurer is found to partially cover the central X-ray source, we may assume it has a transverse size ($d$) comparable to that of the X-ray source. The transverse velocity ($v_{\rm t}$) required by the obscurer to eclipse the X-ray source is $2R_{\rm X}/t$, where $R_{\rm X}$ is the radius of the X-ray corona and $t$ is the duration of the obscuration event (i.e. 32 days). We adopt a fiducial radius of ${10\,R_{\rm g}}$ for the X-ray corona, where the gravitational radius ${R_{\rm g}=GM_{\bullet}/c^{2}}$, with $G$ the gravitational constant, $c$ the speed of light, and the black hole mass ${M_{\bullet}=2.98\times 10^{7}}$ $M_{\odot}$ (Vestergaard & Peterson 2006). These yield ${d=8.8\times 10^{13}}$ cm and ${v_{\rm t}=320}$ $\mathrm{km\ s^{-1}}$ for the obscurer. The obscurer density ${n_{\rm H}\sim N_{\mathrm{H}}/D}$, where $N_{\mathrm{H}}=2.3\times 10^{23}$ $\rm{cm}^{-2}$ (Table 1) and $D$ is the length of the obscurer in our line of sight. The length $D$ is equal to the above transverse size $d$ assuming an obscuring cloud with a spherical geometry. This gives ${n_{\rm H}}$ of about ${2.6\times 10^{9}}$ cm${}^{-3}$, which is a typical broad-line region (BLR) density (e.g. Baldwin et al. 1995). Finally, from the definition of the ionisation parameter $\xi$, we have ${r=\sqrt{L/\xi\,n_{\rm H}}}$, where the ionising luminosity $L$ over 1–1000 Ryd is about ${1.1\times 10^{44}}$ erg s${}^{-1}$ and ${\xi=10^{1.84}}$ erg cm s${}^{-1}$ from our modelling. This yields a distance ${r\sim 10}$ light days from the black hole. For comparison, the radius of the BLR in NGC 3783 from reverberation mapping (Peterson et al. 2004) ranges from about 1.4 (He ii) to 10.2 (H$\beta$) light days. The radius of the IR torus is 250–357 light days (Beckert et al. 2008). Therefore, the obscurer is likely located in the outer BLR.
Previous studies of NGC 3783 have found an outflowing HC through the detection of a narrow Fe xxv He$\alpha$ absorption line (e.g. Yaqoob et al. 2005). From our joint analysis of the stacked archival HETGS and EPIC-pn data, we find the component responsible for this narrow Fe xxv absorption line has ${\log\xi=3.0\pm 0.1}$ and ${N_{\mathrm{H}}=1.4\pm 0.1\times 10^{22}}$ $\rm{cm}^{-2}$, with ${v=-450\pm 50}$ $\mathrm{km\ s^{-1}}$ and ${\sigma_{v}=100}$ $\mathrm{km\ s^{-1}}$. However, in the 2016 obscured data a more massive ($2.3\times 10^{23}$ $\rm{cm}^{-2}$) and more ionised HC is present, which is also faster and broader than the archival HC (Table 1). Absorption by the Fe xxvi Ly$\alpha$ in the HC causes the disappearance of the Fe K$\beta$ emission line (Fig 4, middle panel). In both archival and obscured observations, the line energy and flux of the Fe K$\alpha$ line remain unchanged within errors (about $\pm 20$ eV in line energy and $\pm 10$% in flux). Therefore, according to the theoretical Fe K line calculations (Palmeri et al. 2003; Kallman et al. 2004), similar Fe K$\beta$ emission line would be expected in 2016, hence its disappearance cannot be due to a change in the ionisation state of the line-emitting region. The appearance of the new HC in 2016 data of NGC 3783 is likely associated with the obscurer. The increase in $\xi$ between the two 2016 observations (Table 1) matches the observed change in the ionising luminosity $L$ of the source between the two observations, varying from 1.0 to ${{1.2\times 10^{44}}}$ erg s${}^{-1}$. This enables us to put limits on distance $r$ of the HC from its recombination timescale $t_{\rm rec}$. From our photoionisation modelling, $n_{\rm H}\times t_{\rm rec}$ for Fe xxvi is derived. Since $t_{\rm rec}$ has to be shorter than the spacing between the two observations (9.7 days), this can be used to put constraints on $n_{\rm H}$ and hence $r$. We find the HC has ${n_{\rm H}>2.3\times 10^{5}}$ cm${}^{-3}$ and ${r<120}$ light days. Thus, it may coexist spatially with the obscurer at the outer BLR. Both the obscurer and the HC have comparable velocities (Table 1).
The eclipsing obscurer in NGC 3783 is outflowing because it produces blue-shifted and broad absorption lines. Since the obscurer is found to be inhomogeneous and clumpy, and its location matches the BLR, it is consistent with being in the base of a radiatively-driven wind at the BLR (Murray et al. 1995; Proga & Kallman 2004; Proga et al. 2014). Similar X-ray eclipses found in NGC 1365 and Mrk~766 have also been attributed to passage of BLR clouds in our line of sight to the X-ray source (Risaliti et al. 2007, 2011). Similar to NGC 3783, an association between obscuration and a HC is also found for the luminous quasar PDS 456 (Reeves et al. 2009; Nardini et al. 2015), where a partially-covering Compton-thick absorber appears together with a highly-ionised relativistic disk wind. Interestingly, the obscuring wind and the HC in this quasar are a more massive and faster version of the wind in the less luminous Seyfert-1 NGC 3783. We note that although Compton-thick obscuration is associated to outflows in PDS 456, not all Compton-thick obscurations in AGN may necessarily be related to outflows in general. Determining the origin of X-ray obscuration in nearby type-I galaxies provides key observational evidence for understanding the launching mechanisms of outflows in more powerful quasars at higher redshifts, which due to their faint signal cannot be studied with current X-ray observatories. Such winds, with significantly high outflow velocities and mass outflow rates, can play an important role in AGN feedback. The ToO multiwavelength spectroscopy of X-ray eclipses, like performed here on NGC 3783 using XMM-Newton, NuSTAR, and HST COS, is an effective way to determine the physical link between the accretion disk, BLR, and outflows in AGN.
Acknowledgements.
This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This research has made use of data obtained with the NuSTAR mission, a project led by the California Institute of Technology (Caltech), managed by the Jet Propulsion Laboratory (JPL) and funded by NASA. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We thank the Swift team for monitoring our AGN sample, and the XMM-Newton, NuSTAR, and HST teams for scheduling our ToO triggered observations. SRON is supported financially by NWO, the Netherlands Organization for Scientific Research. This work was supported by NASA through a grant for HST program number 14481 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. The research at the Technion is supported by the I-CORE program of the Planning and Budgeting Committee (grant number 1937/12). EB acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 655324. SB acknowledges financial support from the Italian Space Agency under grant ASI-INAF I/037/12/0, and from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 31278. EC is partially supported by the NWO-Vidi grant number 633.042.525. LDG acknowledges support from the Swiss National Science Foundation. BDM acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778 via the Polish National Science Center grant Polonez UMO-2016/21/P/ST9/04025. CP acknowledges support from ERC Advanced Grant Feedback 340442. GP acknowledges support from the Bundesministerium für Wirtschaft und Technologie/Deutsches Zentrum für Luft- und Raumfahrt (BMWI/DLR, FKZ 50 OR 1604) and the Max Planck Society. We thank M. Bentz for providing us the host galaxy flux in NGC 3783, and M. Giustini for useful discussions. We thank the anonymous referee for the useful comments.
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Appendix A Observation logs |
A large population of ’Lyman-break’ galaxies in a protocluster at redshift $z\approx 4.1$
George K. Miley${}^{1}$, Roderik A. Overzier${}^{1}$, Zlatan I. Tsvetanov${}^{2}$, Rychard J. Bouwens${}^{3}$, Narciso Benítez${}^{2}$, John P. Blakeslee${}^{2}$, Holland C. Ford${}^{2}$, Garth D. Illingworth${}^{3}$, Marc Postman${}^{4}$, Piero Rosati${}^{5}$, Mark Clampin${}^{4}$, George F. Hartig${}^{4}$, Andrew W. Zirm${}^{1}$, Huub J. A. Röttgering${}^{1}$, Bram P. Venemans${}^{1}$, David R. Ardila${}^{2}$, Frank Bartko${}^{6}$, Tom J. Broadhurst${}^{7}$, Robert A. Brown${}^{2}$, Chris J. Burrows${}^{2}$, E. S. Cheng${}^{8}$, Nicholas J. G. Cross${}^{2}$, Carlos De Breuck${}^{5}$, Paul D. Feldman${}^{2}$, Marijn Franx${}^{1}$, David A. Golimowski${}^{2}$, Caryl Gronwall${}^{2}$, Leopoldo Infante${}^{9}$, André R. Martel${}^{2}$, Felipe Menanteau${}^{2}$, Gerhardt R. Meurer${}^{2}$, Marco Sirianni${}^{2}$, Randy A. Kimble${}^{8}$, John E. Krist${}^{6}$, William B. Sparks${}^{4}$, Hien D. Tran${}^{2}$, Richard L. White${}^{4}$ & Wei Zheng${}^{2}$
${}^{1}$ Leiden Observatory, University of Leiden, PO Box 9513, Leiden, 2300 RA, The Netherlands
${}^{2}$ Department of Physics & Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
${}^{3}$ Lick Observatory, University of California, Santa Cruz, California 95064, USA
${}^{4}$ Space Telescope Science Institute, Baltimore, Maryland 21218, USA
${}^{5}$ European Southern Observatory, Garching, D-85748, Germany
${}^{6}$ Bartko Science & Technology, Mead, Colorado 80542-0670, USA
${}^{7}$ The Racah Institute of Physics, Hebrew University, Jerusalem, 91904, Israel
${}^{8}$ NASA-Goddard Space Flight Centre, Greenbelt, Maryland 20771, USA
${}^{9}$ Departmento de Astronomia y Astrofisica, Pontificia Universidad Catolica de Chile, Casilla 306, Santiago 22, Chile
The most massive galaxies and the richest clusters are believed to have emerged from regions with the largest enhancements of mass density${}^{1-4}$ relative to the surrounding space. Distant radio galaxies may pinpoint the locations of the ancestors of rich clusters, because they are massive systems associated with overdensities of galaxies that are bright in the Lyman-$\alpha$ line of hydrogen${}^{5-7}$. A powerful technique for detecting high-redshift galaxies is to search for the characteristic ‘Lyman break’ feature in the galaxy colour, at wavelengths just shortwards of Ly$\alpha$, due to absorption of radiation from the galaxy by the intervening galactic medium. Here we report multicolour imaging of the most distant candidate${}^{7-9}$ protocluster, TN J1338-1942 at a redshift $z\approx 4.1$. We find a large number of objects with the characteristic colours of galaxies at that redshift, and we show that this excess is concentrated around the targeted dominant radio galaxy. Our data therefore indicate that TN J1338-1942 is indeed the most distant cluster progenitor of a rich local cluster, and that galaxy clusters began forming when the Universe was only 10 per cent of its present age.
There is increasing evidence that structures of galaxies existed in the early Universe, but the detection of protoclusters at redshifts $z>1$ using conventional optical and X-ray techniques is difficult${}^{10-12}$. Some of us have developed an efficient method for pinpointing distant protoclusters. The technique is based on the hypothesis that the most powerful known high-redshift radio galaxies are frequently associated with massive forming galaxies${}^{13-16}$ in protoclusters${}^{5}$. As a first step towards testing this hypothesis, we recently conducted a large programme with the Very Large Telescope (VLT) of the European Southern Observatory to search for galaxy overdensities associated with protoclusters around luminous high-redshift radio galaxies. Deep narrow- and broad-band imaging was used to locate candidate galaxies having bright Ly$\alpha$ emission, and follow-up spectra have confirmed that most of these candidates have similar redshifts to the high-redshift radio galaxies. All five targets studied with the VLT to sufficient depth have $>20$ spectroscopically confirmed Ly$\alpha$ and/or H$\alpha$ companion galaxies, associated with galaxy overdensities${}^{6,7}$. Their formal velocity dispersions are a few hundred km s${}^{-1}$, but there was not enough time since the Big Bang for them to have become virialized. The scale sizes of the structures inferred from their spatial boundaries are $\sim 3-5$ Mpc. Assuming that the overdensities are due to a single structure, the masses derived from the observed structure sizes and overdensities are comparable to those of clusters of galaxies in the local Universe${}^{7}$. These observations led us to hypothesize that the overdensities of Ly$\alpha$ galaxies around radio sources are due to the fact that they are in protoclusters.
Galaxies that emit strong Ly$\alpha$ comprise only a small fraction of distant galaxies, and are biased towards non-dusty objects and galaxies that are undergoing the most vigorous star formation. Only about 25% of $z\approx 3$ galaxies have Ly$\alpha$ equivalent widths detectable by our VLT narrow-band imaging searches${}^{17,18}$. If the overdensities of Ly$\alpha$ galaxies are located in protoclusters, additional galaxy populations should be present and detectable on the basis of characteristic continuum features in the galaxy spectra. The most important of these features is the sharp ‘Lyman break’ blueward of Ly$\alpha$, caused by the absorption of the galaxy continuum radiation by neutral hydrogen clouds along the line of sight. Searching for Lyman-break galaxies is a powerful technique for finding highredshift galaxies${}^{11,19,20}$.
Because of its high spatial resolution, large field of view and excellent sensitivity, the Advanced Camera for Surveys${}^{21}$ (ACS) on the Hubble Space Telescope is uniquely suited for studying the morphologies of galaxies in the protoclusters and for finding additional galaxies on the basis of the Lyman-break features in their spectra. We therefore used the ACS to observe the most distant of our VLT protoclusters, TN J1338-1942 (ref. 7) at $z=4.1$. This is a structure with 21 spectroscopically confirmed Ly$\alpha$ emitters and a rest-frame velocity dispersion of 325 km s${}^{-1}$. Images were taken through three ‘Sloan’ filters–g band centred at 4,750 Å, r band centred near 6,250 Å and i band centred near 7,750 Å. These filters were chosen so that their wavelength responses bracketed redshifted Ly$\alpha$ at 6,214 Å and were sensitive to the Lyman-break feature blueward of Ly$\alpha$.
A 3.4${}^{\prime}$$\times$3.4${}^{\prime}$ field was observed, with the radio galaxy located $\sim 1\hbox{${}^{\prime}$}$ from the image centre. Besides the radio galaxy, this field covered 12 of the 21 known Ly$\alpha$ emitting galaxies in the candidate protocluster. All 12 objects were detected in both r band and i band, with i-band magnitudes ranging from 25 to 28, compared with $23.3\pm 0.03$ for the radio galaxy. As illustrated in Fig. 1, these objects were either absent or substantially attenuated in the g band, and their $g-r$ colours are generally consistent with predicted values of Lyman breaks${}^{22}$. Half of the objects are extended in $i_{775}$, and three of these are resolved into two distinct knots of continuum emission, suggestive of merging.
We next used the Lyman-break technique to search for a population of Lyman-break galaxies in the protocluster that do not emit strong Ly$\alpha$ and would therefore have been undetectable in our VLT observations. Evidence for the existence of such a population was sought by analysing the number and spatial distribution of ‘g-band dropout’ objects–that is, objects whose colours are consistent with Lyman breaks in their spectra at the redshift of the protocluster. To investigate whether there is a statistically significant excess of such g-band dropout objects, we estimated the surface density and cosmic variance of g-band dropouts in a typical ACS field observed with the same filters and to the same depth as TN J1338-1942. We did this by cloning${}^{23}$ $B_{435}$-band dropouts in 15 different pointings from the southern field of the Great Observatories Origins Deep Survey (GOODS)${}^{24}$. Results indicate that the number of g-band dropouts in our field is a factor of 2.5 times higher than the average number found in a random GOODS field. Taking account of the typical cosmic variance${}^{25}$ in the distribution of $z\approx 4$ Lyman-break galaxies, this is a $3\sigma$ excess on the assumption that the distribution function is gaussian. Further evidence that a substantial fraction of these g-dropout objects are Lyman-break galaxies associated with the protocluster is provided by the strong concentration of the g-band dropouts in a cluster-sized region around the radio galaxy. This is illustrated in Fig. 2. More than half of the g-band dropouts are located in a region of $\sim 1\hbox{${}^{\prime}$}$ in radius (corresponding to a diameter of $\sim 1$ Mpc at $z=4.1$). The number of g-band dropouts in this region is a factor of 5 times the average number encountered in similarly sized regions that are randomly drawn from the GOODS survey. This is a $5\sigma$ excess, indicating that the number of g-band dropouts in our field is anomalously high at greater than the 99% confidence level. The spatial non-uniformity of g-band dropout objects in our field becomes even more pronounced when fainter objects down to a magnitude of $=27$ are included (Fig. 2).
Are there alternative explanations for the observed excess of g-band dropout objects other than a population of Lyman-break galaxies at $z\approx 4.1$? An object with a Balmer break at $z\approx 0.5$ could also be observed as a g-band dropout object. However, a population of such $z\approx 0.5$ objects would also be present in the GOODS comparison sample. Although the existence of an intervening structure of Balmer-break galaxies at $z\approx 0.5$ cannot be completely ruled out, its coincidence in location with the $z\approx 4.1$ structure of Ly$\alpha$ galaxies and the faintness and small sizes of the observed objects make this possibility highly unlikely.
The spatial coincidence of the excess in g-band dropout objects with the previously detected overdensity of Ly$\alpha$ emitters around a forming massive galaxy is strong evidence that we are observing a new population of Lyman-break galaxies in a protocluster. This would mean that TN J1338-1942, at $z\approx 4.1$, is indeed the most distant known protocluster, and that distant luminous radio galaxies pinpoint the progenitors of nearby rich clusters. Such protoclusters provide an opportunity to study the development of galaxies and clusters in the early Universe. They provide samples of different galaxy populations at the same distance, whose morphologies and spectral energy distributions could be used to disentangle the evolution and star formation history of different types of galaxies. The topological information that could be derived by mapping the shapes and sizes of such protoclusters over larger areas could answer the question of whether the first protoclusters in the early Universe formed in sheets or filaments.
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Correspondence and requests for materials should be addressed to G.K.M. ([email protected]).
Figure 1: Deep images of Ly$\alpha$-emitting protocluster galaxies. Images show galaxy morphologies observed through three filters: g band (left), r band (middle) and i band (right). Each $2.5\hbox{${}^{\prime\prime}$}\times 2.5\hbox{${}^{\prime\prime}$}$ image has been smoothed by a gaussian function with a fullwidth at half-maximum
of 1.5 pixels ($0.074\hbox{${}^{\prime\prime}$}$). The observations were carried out between 8 and 12 July 2002 with the Wide Field Channel of the ACS${}^{21}$. The total observing time of 13 orbits was split over the broad-band filters F475W (g band, four orbits), F625W (r band, four orbits) and F775W (i band, five orbits), thereby bracketing redshifted Ly$\alpha$ at 6,214 Å. During each orbit, two 1,200-s exposures were made, to facilitate the removal of cosmic rays. The observations were processed through the ACS GTO pipeline${}^{26}$ to produce registered, cosmic-ray-rejected images. The limiting $2\sigma$ magnitudes in a 0.2-arcsec${}^{2}$ aperture were 28.71 (F475W), 28.44 (F625W) and 28.26 (F775W). Object detection and photometry were then obtained using SExtractor${}^{27}$. a, The clumpy radio galaxy TN J1338-1942 at $z=4.1$. This is the brightest galaxy in the protocluster and inferred to be the dominant cluster galaxy in the process of formation. Because the equivalent width of Ly$\alpha$ is large ($\sim 500$ Å), the r band is dominated by Ly$\alpha$. Arrows indicate the positions${}^{8}$ of the radio core (C) and the northern hotspot (H). The Ly$\alpha$ emission is elongated in the direction of the radio emission and the large-scale Ly$\alpha$ halo${{}^{7}}$ with a projected linear size of $\sim 15$ kpc (assuming $H_{0}=65$ km s${}^{-1}$ Mpc${}^{-1}$, $\Omega_{M}=0.3$, $\Omega_{M}=0.7$). b, Images of five spectroscopically confirmed Ly$\alpha$ emitters in the protocluster. Listed below each galaxy are its spectroscopic redshift${}^{7}$, the magnitude of the observed Lyman break, and the i-band magnitude. Two of the Ly$\alpha$ emitters are clumpy, as expected from young galaxies.
Figure 2: The spatial distribution of g-band dropout objects. Superimposed on the combined $3.4\hbox{${}^{\prime}$}\times 3.4\hbox{${}^{\prime}$}$ ACS greyscale image are the locations of g-band dropout objects (blue circles), selected to have colours and magnitudes of $(g-r)\geq 1.5$, $(g-r)\geq(r-i)+1.1$, $(r-i)\leq 1$ and $i<27$. In addition, objects were required to have a SExtractor${}^{27}$ stellarity parameter of less than 0.85 to ensure that the sample was not contaminated by stars. These criteria filter galaxies having Lyman breaks at $z\approx 4$, thereby providing a sample of protocluster Lyman-break galaxy candidates. We detected 30 g-dropout objects in the field around TN J1338-1942 with $i_{775}<26$, and 56 with $i_{775}<27$. The number of g-band dropout objects is anomalously large, and their distribution is concentrated within a circular region of $\sim 1\hbox{${}^{\prime}$}$ in radius that includes the radio galaxy TN J1338-1942 (large green circle). Also shown are the positions of the spectroscopically confirmed Ly$\alpha$ emitters (red squares). Because the selection criteria were optimized to detect Lyman-break galaxies, some of the Ly$\alpha$ emitters did not fall into the formal sample of Lyman-break galaxies. The measured excess and its spatial clustering are evidence that a substantial fraction of the g-band dropout objects are Lyman-break galaxies associated with the protocluster. (See Fig. 1 legend for further details about the observations and the subsequent analysis.) Scale bar, 1${}^{\prime}$. |
THRESHOLD NOISE AS A SOURCE OF VOLATILITY IN
RANDOM SYNCHRONOUS ASYMMETRIC NEURAL NETWORKS
Henrik Bohr,
Patrick McGuire,
Chris Pershing
and Johann Rafelski
$\ $
Department of Physics, University of Arizona, Tucson AZ 85721
Center for Biological Sequence Analysis,
The Technical University of Denmark, Building 206, DK-2800 Lyngby, Denmark,
EMAIL: [email protected] for Astronomical Adaptive Optics,
Steward Observatory, University of Arizona, Tucson AZ 85721,
EMAIL: [email protected]: [email protected]: [email protected]
(November 24, 1997)
Abstract
We study the diversity of complex spatio-temporal patterns of
random synchronous asymmetric neural networks (RSANNs). Specifically, we
investigate the impact of noisy thresholds on network performance and find
that there is a narrow and interesting region of noise parameters where
RSANNs display specific features of behavior desired for rapidly ‘thinking’
systems: accessibility to a large set of
distinct, complex patterns.
1 Introduction
Random Synchronous Asymmetric Neural Networks (RSANNs) with fixed synaptic coupling strengths and fixed
neuronal thresholds have been found to have access to a very limited set of
different limit-cycles (Clark, Kürten & Rafelski, 1988; Littlewort,
Clark & Rafelski, 1988; Hasan, 1989; Rand, Cohen & Holmes, 1988).
We will show here, however, that
when we add a small amount of temporal random noise by choosing the neural
thresholds not to be fixed but to vary within a narrow gaussian
distribution at each time step, we can cause transitions between different
quasi-limit-cycle attractors.
Shifting the value of neuronal thresholds randomly
from a gaussian distribution which varies on a time scale much slower than
the fast synaptic dynamics, we control the timing of the transition
between limit-cycle attractors. Hence we can gain controllable and as
we believe biologically
motivated access to a wide variety of limit-cycles, each displaying
dynamical participation by many neurons.
Perfect limit-cycles in which the network returns to some earlier state do
not exist in real biological systems, due to many complicating factors:
membrane potential noise (Little, 1974; Clark, 1990; Faure, 1997), the complexity of
biological neurons, the continuously-valued signal transmission times
between neurons, and the lack of a clock to synchronously update all neurons,
to mention a few.
Therefore the case of approximate limit-cycles is more
biologically reasonable.
Indeed, the appearance of limit-cycle behavior in
central pattern generators is evidence for such temporal behavior in biological
systems (Hasan, 1989; Marder & Hooper, 1985). We also believe that the biologically significant limit-cycles
are those in which a great fraction of the neurons actively
participate in the dynamics.
If these limit-cycles represent brain activity states which we could call ‘thoughts’,
and if a sequence of spontaneous
or externally controlled transitions from one limit-cycle to another
limit-cycle represents a ‘reasoning process’, then neural network systems which
exhibit a diversity of accessible limit-cycles (thoughts) with capability to
move rapidly between them could also exhibit a large
variety of different ‘trains of thought’. A system which can access many limit-cycles should always be able to access a novel mode; hence the system
would have the potential to be a creative system. Our current work thus
demonstrates conditions sufficient to allow access to creative dynamical behavior.
In Section 2 we introduce RSANNs along with the concept of threshold noise,
as well as the details of an RSANN’s implementation. In Section 3, we provide
the algorithm that contrasts limit-cycles, which is of prime
importance in our work.
In our quantitative investigations we need to introduce with more precision
concepts which intuitively are easy to grasp, but which mathematically are
somewhat difficult to quantify. We define eligibility in Section 4.1 as an entropy-like measure of the fraction of neurons which actively
participate in the dynamics of a limit-cycle. In order to quantify the
RSANN’s accessibility to multiple limit-cycle attractors, we define
diversity in Section 4.2 as another entropy-like measure,
calculated from the probabilities that the RSANN converges to each
of the different limit-cycles. As a measure of the creative
potential of a system, we introduce the concept of volatility as the
ability to switch from one particular highly-eligible cyclic mode to many
other highly-eligible cyclic modes. We mathematically define volatility in
Section 4.3 to be an entropy-like measure of the number of ${\em different}$ limit-cycle patterns easily available to the net, weighted by
the eligibility of each limit-cycle attractor. We find that in terms of
these variables, as the neuronal threshold noise, $\epsilon$, increases,
our RSANN exhibits a phase transformation at $\epsilon=\epsilon_{1}$ from a small
number to a large number of different ${\em accessible}$ limit-cycle
attractors (Section 4.2), and another phase transformation at
$\epsilon=\epsilon_{2}>\epsilon_{1}$ from high eligibility to low eligibility
(Section 4.1). Our main result is that the
volatility is high only in presence of threshold noise of suitable fine-tuned
strength chosen between $\epsilon_{1}\leq\epsilon\leq\epsilon_{2}$, so
allowing access to a diversity of eligible limit-cycle attractors (Section 4.3).
2 Random asymmetric neural networks with threshold noise
Random asymmetric neural networks (RSANNs)
(Bressloff & Taylor, 1989; Clark, Rafelski & Winston, 1985; Clark, 1991), with $w_{ij}\neq w_{ji}$, differ from
symmetric neural networks (SNNs) (Hopfield, 1982),
and offer considerably more biological realism, since real neural
signals are unidirectional. Due to the lack of synaptic symmetry,
RSANNs have a non-simple behavior with different limit-cycle attractors with
possible period lengths111$\Delta t$
is the time necessary for a neural signal to propagate from one
neuron to the next and is assumed to be identical for all synapses (McCullough & Pitts, 1943);
hereafter, we will take as the unit of time $\Delta t=1$, so that a limit-cycle of period $L$
will be $L$ time steps long $L\Delta t\gg\Delta t$,
even with extremely long periods (greater
than $10,000$ time steps) when the neuronal thresholds have been finely tuned
(Clark, Kürten & Rafelski, 1988; McGuire, Littlewort & Rafelski, 1991;
McGuire et al., 1992).
Consider a net of $N$ neurons with firing states which take binary values
(i.e. $a_{i}\in{0,1}$). Each neuron is connected to M other pre-synaptic
neurons by unidirectional synaptic weights. A firing pre-synaptic neuron $j$
will enhance the post-synaptic-potential (PSP) of neuron $i$ by an amount
equal to the connection strength, $w_{ij}$. Inhibitory neurons (chosen with
probability $I$) have negative connection strengths.
If the $i$th neuron’s PSP, $c_{i}$, is greater than its threshold, $V_{i}$, then
it fires an action potential: $a_{i}\rightarrow 1$. We parametrize the
thresholds, $V_{i}$, in terms of ‘normal’ thresholds, $V_{i}^{0}$, a mean bias
level, $\mu,$ and a multiplicative threshold-noise parameter, $\beta_{i}$:
$$\begin{array}[]{ccc}V_{i}(\beta_{i})&=&(\mu+\beta_{i})V_{i}^{0}\\
V_{i}^{0}&=&\frac{1}{2}\sum\limits_{k=1}^{M}w_{ik}\end{array},$$
(1)
If $\mu=>1\,$and $\beta_{i}=>0$, we recover normal thresholds ($V_{i}=>V_{i}^{0}$).
In the presence of even a small amplitude of neural-threshold noise
effective on the same time-scale as the transmission-time of neural
impulses from one neuron to the next neuron, the neural net will never
stabilize into a single limit-cycle attractor. Rather, if the noise
amplitude is not too high, the net will continually make transitions
from one almost-limit-cycle to another almost-limit-cycle222the prefix
‘almost’ here, permits occasional misfirings. This ‘non-stationary’
characteristic is common to any volatile system, but makes computer
simulation and characterization difficult because one never knows
a priori when the neural network will ‘jump’ to
a new attractor basin. In order to maintain stability and
make simulation easier, but also for reasonjs of biological
reality we will give the neural threshold noise a much
longer time scale, $t_{s}$, than the shorter signal-transmission/neural-update
time scale, $\Delta t$. Consequently, for each discrete slow-time threshold-update
step, $t_{s}$, there will be several hundred to several thousand fast-time
neural-update steps, $\Delta t$, during which the thresholds $V_{i}(\beta_{i}(t_{s}))$
are ‘quenched’ (or do not change) and the neural states $a_{i}(t)$ are allowed
to change via Eq. 2. Only at the next slow-time step are the
thresholds allowed to change,
but then kept quenched again during the many fast-time neural-update steps, see Figure 1. We see here the evolution of a threshold value and
neural activityi on slow-time, $t_{s}$, while in the insert the fine-structure of limit-cycle dynamics as a function of fast-time is displayed.
For a given limit-cycle search, labelled by the slow-time-scale parameter $t_{s}$,
the neurons initially possess random firing states;
where the fraction of initially firing neurons
is also chosen randomly to be between $0$ and $1$.
At a given slow-time step, $t_{s},$ we initialize the neuronal
threshold parameters $\beta_{i}$ with zero-mean gaussian noise of width,
$\epsilon$,
where each $\beta_{i}$ is chosen independently for all $i$.
This ‘slow threshold noise’ makes biological sense because
the concentrations of different chemicals in the brain changes
on a time-scale of seconds or even minutes, thereby changing the
effective thresholds of individual neurons on a time
scale much longer than the update time, $\Delta t$ (milliseconds).
Some biologists therefore believe that neural
thresholds are ‘constant’ and noiseless (Fitzhugh-Nagumo Model
(Pei, Bachmann & Moss, 1995));
others unequivocably feel that neurons live in a very noisy environment,
both chemically and electrically (Little, 1974); a model which
includes slow threshold noise varying randomly by $O(10^{-3}$ seems
consistent with both points of view – this will be the model developed
here.
Thereafter the firing state $a_{i}(t,t_{s})$ of the $i$th neuron as a function
of the rapidly-updating time parameter $t$ is given by the following equations:
$$\begin{array}[]{ccccc}a_{i}(t,t_{s})&=&\theta(c_{i}(t,t_{s})-V_{i}(t_{s}))&&%
\mbox{(a)}\\
c_{i}(t,t_{s})&=&\sum\limits_{j=1}^{M}w_{ij}a_{j}(t-\Delta t,t_{s})&&\mbox{(b)%
}\\
V_{i}(t_{s})&=&(\mu+\beta_{i}(t_{s}))V_{i}^{0}\,\,\,\,\,\,,&&\mbox{(c)}\end{array}$$
(2)
where $\theta$ is the step-function.
2.1 Implementation
The prescription given above of an RSANN is implemented in a computer
program on a fast workstation.
We update all neuron firing states in parallel, or ‘synchronously’, as
opposed to serial, or ‘asynchronous’, updating in which only one neuron
or a small group of neurons is
updated at a given time step.
The connection strengths $w_{ij}$ are integers chosen
randomly from a uniform distribution, $\left|W_{ij}\right|\in(0,50000]$. We have performed a cross-check simulation in which the
connection strengths are double-precision real numbers, and have found no
difference between the integer-valued connection-strength simulations and
the real-valued connection-strength simulations. The
normal thresholds are double-precision numbers. We chose $K$ incoming
connections per neuron, where $K=10$. The fraction of inhibitory connections
is randomly chosen, here with probability, $I=0.3$.
We have studied networks with
$N$ neurons, where $N\in\{10,20,30,40,50,100\}.$ The network size is
primarily constrained by the extremely long limit-cycles
or transients which get longer for larger networks, especially when
the thresholds are near-‘normal’, as given by $V_{i}^{0}$ in Eq. 1.
2.2 Updating and Limit-Cycle Search
We simulate slow threshold noise by reinitializing the thresholds via
equation 1, at each slow-time step
$t_{s}$. Initially, during this slow-time step $t_{s}$, the neural firing states
are updated by the dynamical equations 2 for $W_{n}=W_{0}=128$
fast-time steps, $t.$ After the $W_{n}$ fast-time updates, an iterative
computer routine carefully scans the record of the spatially-averaged firing
rate, $\alpha(t,t_{s})$, for $t\in[0,W_{n}],$ to check for exact periodicity:
$$\begin{array}[]{ccccc}\alpha(t,t_{s})&\equiv&\frac{1}{N}\sum\limits_{i=1}^{N}a%
_{i}(t,t_{s})&&\mbox{(a)}\\
\alpha(t+L,t_{s})&=&\alpha(t,t_{s}),&\forall t\in[0,W_{n}],&\mbox{(b)}\end{array}$$
(3)
where the limit-cycle period is is identified to be $L.$ The iterative routine
carefully scans each candidate limit-cycle to ensure perfect periodicity of
$\alpha(t,t_{s})$ for at least $P$ periods (typically $2$ or $4$ periods)
of the limit-cycle. If the scanning routine finds that the
spatially-averaged firing rate has converged to a limit-cycle (with
$L<W_{n}/P$, see Figure 2), then we halt the fast-time updating, ‘record’ the limit-cycle,
increment the slow-time ($t_{s}=t_{s}+1$), give the thresholds new initial
conditions by choosing new $\beta_{i}$ in Eq. 1 and the
neuronal firing states new initial conditions,
and begin a new limit-cycle search. On the other hand,
if the scanning routine finds that the
spatially-averaged firing rate has not yet converged to a limit-cycle, we then
let $W_{n+1}=2W_{n}$, and update the network for another $W_{n+1}$
fast-time steps. After the updating, the iterative limit-cycle scanning
routine searches again for at least $P$ periods of exact periodicity of
$\alpha(t,t_{s})$, but only within the new time-window, $t\in[W_{n},W_{n+1}]$.
We repeat this limit-cycle search algorithm until we find a limit-cycle,
or until $W_{n}=W_{\max}$ (where $W_{\max}$ is typically $4096$ or $8192$ time
steps), whichever comes first.
If the limit-cycle period is much shorter than the width of the time-window,
$$L<<\Delta W\equiv W_{n+1}-W_{n},$$
(4)
we will observe many more repetitions of the limit-cycle than the requisite
$P$. If no limit-cycle is observed during the limit-cycle search, then the
RSANN either has a very long transient or a very long period limit-cycle,
see Clark (1991) or Littlewort, Clark & Rafelski (1988) for a discussion of the correlation
between transient-length and limit-cycle period.
3 Limit-cycle comparison
From the fast-time-averaged single neural firing rates and from
estimates of each
neuron’s neural fast-time variance during a limit-cycle, we can compare two
limit-cycles found at two different slow-time steps by a chi-square measure
of the difference between two simulated distributions.
We will estimate the likelihood that two different limit-cycles (labelled by
the slow-time indices $t_{s}$ and $t_{s}^{\prime}$) belong to different basins
of attraction, by comparing the time-averaged single-neuron firing rate
spatial vectors, $A_{i}(t_{s})$ and $A_{i}(t_{s}^{\prime})$, where the subscript $i$
is the spatial index of the neuron. Often, limit-cycles
with different periods $L(t_{s})$ and $L(t_{s}^{\prime})$ will be remarkably
similar, with only an occasional slight difference between the fast-time
recordings of the spatially-averaged firing rates $\alpha(t,t_{s})$ and $\alpha(t,t_{s}^{\prime})$ (see eq. 3), caused by an occasional neuron misfiring:
$$\left|\Delta\alpha(t)\right|\equiv\left|\alpha(t,t_{s})-\alpha(t,t_{s}^{\prime%
})\right|\sim O(\frac{1}{N}).$$
(5)
Conversely, limit-cycles with the same period, $L(t_{s})=L(t_{s}^{\prime})$,
will often have grossly different $A_{i}(t_{s})$ and $A_{i}(t_{s}^{\prime})$,
especially for short-period limit-cycles.
3.1 Estimate of neuronal time-variance
We ‘record’ each limit-cycle’s period, $L(t_{s})$, and its time-averaged
single-neuron firing rates, $A_{i}(t_{s})$:
$$A_{i}(t_{s})\equiv\langle a_{i}\rangle_{t}\equiv\frac{1}{\Delta W}\sum\limits_%
{t=1+W_{n}}^{W_{n}+\Delta W}a_{i}(t,t_{s}),$$
(6)
for all neurons $i$ (see Figure 2). Since $a_{i}^{2}=a_{i}$ for $a_{i}\in\{0,1\}$,
we have $\langle a_{i}^{2}\rangle=\langle a_{i}\rangle$, and we
find that the variance (of a single measurement from the mean), $b_{i}(t_{s})$, will be:
$$b_{i}(t_{s})\equiv\langle(a_{i}-\langle a_{i}\rangle_{t})^{2}\rangle_{t}=%
\langle a_{i}^{2}(t,t_{s})\rangle_{t}-\langle a_{i}(t,t_{s})\rangle_{t}^{2}=A_%
{i}-A_{i}^{2}\,.$$
(7)
For $A_{i}\sim 0.5$, we find that the standard deviation of a single measurement
of the firing state from the mean firing rate is $\sqrt{b_{i}}\sim 0.5$. However,
for different limit-cycle attractors, the difference between firing rates of
a particular neuron $i$ is $|A_{i}(t_{s})-A_{i}(t_{s}^{\prime})|\sim 0.05\ll 0.5$.
Fortunately, we measure the firing state of each neuron many times ($\Delta W>>1$) to
determine the mean firing rate, so we can also estimate the variance of the mean, $B_{i}$, as:
$$B_{i}=b_{i}/\Delta W.$$
(8)
For $\Delta W\sim 1000$ and $A_{i}\sim 0.5$, this
gives a standard deviation of the mean of $\sqrt{B_{i}}\sim 0.01$, which
is a little smaller than the variation seen between different limit-cycle
attractors (see Figure 2). Therefore, we choose to use the variance of the mean, $B_{i}$,
rather than the variance of a single measurement from the mean for our
limit-cycle comparison tests. If we choose a longer observation time, $\Delta W$,
then we can better discriminate between different limit-cycles by using $\chi^{2}$
test discussed below.
Since the variance $B_{i}$ is $0$ for $A_{i}=0$ or $A_{i}=1$ (likely when the RSANN
has low eligibility), a $\chi^{2}$ comparison of $A_{i}(t_{s})$ and
$A_{i}(t_{s}^{\prime})$ will be plagued by division-by-zero problems. In this
case, a maximum likelihood comparison using Poisson statistics (Smith,
Hersman & Zondervan, 1993), or
maybe the Kolmogorov-Smirnov test
would be more appropriate. However, in this work, we simply chose to
‘cut-off’ the estimate (Eq. 7) of the single-measurement variance before it reaches
zero at a value of $b_{i,\mbox{min}}=0.04$.
3.2 Distinguishing the ‘fingerprints’ of different limit-cycles
Since volatility (which we define below) requires an abundance of very different limit-cycles,
we need a highly-contrasting measure of whether a given limit-cycle is
different than or similar to another limit-cycle, or in essence
a method of distinguishing between limit-cycles’ ‘fingerprints’. For each limit-cycle
(labelled by the slow time $t_{s}$), we measure each neuron’s
fast-time-averaged firing rate, $A_{i}$$(t_{s})$ and its (cutoff)
fast-time variance (of the mean), $B_{i}$$(t_{s})$ (Eq. 8),
and apply the weighted chi-square ($\chi^{2}$) method
to decide whether two limit-cycles found at
different slow-times, $t_{s}$ and $t_{s}^{\prime}$, are similar or different:
$$\begin{array}[]{ccc}\chi^{2}(N,t_{s},t_{s}^{\prime})&=&\sum\limits_{i=1}^{N}%
\frac{(A_{i}(t_{s})-A_{i}(t_{s}^{\prime}))^{2}}{B_{i}(t_{s})+B_{i}(t_{s}^{%
\prime})}\end{array}\mbox{,}$$
(9)
where N is the number of neurons in the network as well as the
number of degrees of freedom for the $\chi^{2}$-test. Each term in the $\chi^{2}$ sum should approximate the square of a gaussian-distributed variable
with unit variance. The variance of the difference of two
gaussian-distributed variables is the sum of the individual variances (not
the average), as seen in the denominator of the $\chi^{2}$ expression.
If the variance-estimate, $B_{i}(t_{s})$,
is appropriate and accurate, and if a class of similar
limit-cycles have gaussian-distributed time-averaged firing rates $A_{i}(t_{s})$
, then we can estimate a priori the maximum value of $\chi^{2}(N,t_{s},t_{s}^{\prime})$ for similar limit-cycles. For an $N$-neuron
network, two similar limit-cycles should have (for $N>>1$) a chi-squared
given by: $\chi^{2}(N,t_{s},t_{s}^{\prime})\sim N\pm\sqrt{2N}$.
Accordingly, in order to ensure that most of the
similar limit-cycles are determined to be similar by the chi-square test, for
‘similarity’ we demand that $\chi^{2}(N,t_{s},t_{s}^{\prime})<N+3\sqrt{N}$. For
large $N$, chi-square tables indicate that this is approximately a $95\%$
confidence level experiment. This similarity test corresponds to
a tolerance of about 4 misfires per time step (for $\Delta W=1000$)
From the $\chi^{2}$-test, each pair of
limit-cycles is given a difference label $d(t_{s},t_{s}^{\prime})$:
$$d(t_{s},t_{s}^{\prime})=\left\{\begin{array}[]{cc}0\mbox{,}&\mbox{if }\chi^{2}%
(N,t_{s},t_{s}^{\prime})\leq N+3\sqrt{N}\\
1\mbox{,}&\mbox{if }\chi^{2}(N,t_{s},t_{s}^{\prime})>N+3\sqrt{N}\end{array}%
\right.\mbox{.}$$
(10)
If $d(t_{s},t_{s}^{\prime})=1$, $\forall t_{s}^{\prime}<t_{s}$, then the
limit-cycle found at slow-time step, $t_{s}$, is truly a novel
limit-cycle, never having been observed before. If the limit-cycle is novel
then it is given a ‘novelty’ label of $n(t_{s})=1$, otherwise $n(t_{s})=0$ (by
default, the first observed limit-cycle will always be novel: $n(1)\equiv 1$).
Simply by counting the number of slow-time steps in which a novel
limit-cycle is found, we can estimate the number of different limit-cycle
attractors, $N_{d}$, available to the RSANN:
$$\begin{array}[]{ccc}N_{d}&\equiv&\sum\limits_{t_{s}=1}^{t_{s}^{\max}}n(t_{s})%
\,.\\
\end{array}$$
(11)
4 The performance of the random asymmetric neural network
Random asymmetric neural networks will often develop into a fixed point
where the neural firing vector, $a_{i}(t),$ does not change in time, or in
other words, the
limit-cycle has a period of one time step. For zero slow-threshold noise
($\epsilon=0)$, if the mean threshold value is much greater(less) than
normal, then the RSANN will tend to have a fixed point with very few(many)
neurons firing each time step, which is called network death(epilepsy).
Likewise, for zero noise ($\epsilon=0)$ and for normal thresholds $(\mu=1)$, limit-cycles with very long periods are possible (Figs.3,4).
When the mean threshold value is normal ($\mu=1)$, but the spatiotemporal
threshold fluctuations from normality are large($\epsilon>\epsilon_{2}$),
then there also exist many different mixed death/epilepsy fixed points
in which a fraction of the neurons are firing at each time step and the
remaining neurons never fire. Conversely, when the mean threshold value is
normal and the spatiotemporal threshold fluctuations are small ($\epsilon<\epsilon_{1}$), limit-cycles with very long periods are possible, but the
number of different limit-cycle attractors is limited. For normal mean
threshold values and intermediate-valued spatiotemporal threshold
fluctuations ($\epsilon_{1}<\epsilon<\epsilon_{2}),$ many intermediate-period
limit-cycles exist (Figs.5,6).
4.1 Eligibility as a function of noise-strength
A network is said to have a high degree of eligibility if many neurons
participate in the dynamical collective activity of the network. For an
attractor found at slow-time step $t_{s}$, the time-averaged spatial firing
vector, $A_{i}(t_{s})=$$<a_{i}(t,t_{s})>_{t}\in[0,1]$ (eq. 3), will be
maximally eligible if $A_{i}(t_{s})=0.5$ for all neurons $i,$ and minimally
eligible if $A_{i}(t_{s})$$\in\{0,1\}$ for all neurons $i.$ The Shannon
information (or entropy) has these properties, so we will adopt the form of an entropy
function as our measure of the eligibility of each limit-cycle attractor,
$e(t_{s})$:
$$e(t_{s})\equiv-\frac{1}{N}\sum_{i=1}^{N}A_{i}(t_{s})\ln A_{i}(t_{s})\,\,\,,$$
(12)
where the average eligibility, ${\cal E}$, per limit-cycle attractor is:
$${\cal E}=\frac{1}{t_{s}^{\max}}\sum_{t_{s}=1}^{t_{s}^{\max}}e(t_{s})<\frac{1}{%
2}\ln 2=0.3466\equiv{\cal E}_{\max}\,\,\,.$$
(13)
Despite its utility, we do not have a detailed dynamical motivation for
using an entropy measure for our eligibility measure.
In Figure 7, we find that
eligibility goes through a phase transformation to zero at $\epsilon_{2}\sim 0.3$, as the non-fixed-point limit-cycles gradually become fixed
points as $\epsilon$ increases. In other words, when the thresholds become
grossly ‘out-of-tune’ with the mean membrane potential, the RSANN attractors
become more trivial, with each neuron tending towards its own independent
fixed point $a_{i}=1$ or $a_{i}=0$.
4.2 Diversity
We measure the accessibility of a given attractor by estimating the
probability, $P(t_{s})$, that a given attractor (first seen at slow-time step
$t_{s}$, with the novelty label $n(t_{s}^{\prime})=1$ when $t_{s}^{\prime}=t_{s}$) is found during the slow-time observation period $t_{s}^{\prime}\in[1,t_{s}^{\max}]$:
$$\begin{array}[]{ccc}P(t_{s})&\equiv&\frac{N(t_{s})}{t_{s}^{\max}}\\
\end{array}\,,$$
(14)
where $N(t_{s})$ is the total number of times the limit-cycle $t_{s}$ (first
observed at slow-time step $t_{s}$) is observed, and $n(t_{s})$
is defined in Section 3.2. Note that if
a given attractor is only observed once (at time step $t_{s}$), then $P(t_{s})=1/t_{s}^{\max}$; also if the same attractor is observed at each
slow-time step, then $P(t_{s}=1)=1$.
If each different limit-cycle attractor can be accessed by the network with
a measured probability, $P(t_{s})$, we can define the diversity, ${\cal D}$,
as the inter-attractor occupation entropy:
$${\cal D}(t_{s}^{\max})=-\sum_{t_{s}=1}^{t_{s}^{\max}}P(t_{s})\ln P(t_{s}){\rm,%
\thinspace\thinspace\thinspace\thinspace\thinspace where}\sum_{t_{s}=1}^{t_{s}%
^{\max}}P(t_{s})=1{\rm.}$$
(15)
It is easily seen that a large ${\cal D}$ corresponds to the ability to
occupy many different cyclic modes with equal probability; the diversity
will reach a maximum value of ${\cal D}={\cal D}_{\max}=\ln t_{s}^{\max},$
when $P(t_{s}^{\max})=1/t_{s}^{\max}$ for all slow-time steps $t_{s}^{\max}.$ A
small value for ${\cal D}$ corresponds to a genuine stability of the system
– very few different cyclic modes are available. We observe a
phase transformation from low to high diversity by increasing the level of
randomness, $\epsilon$, on the threshold for each neuron past $\epsilon_{1}\sim 5\times 10^{-4}$ (Fig. 8). We have chosen not to explicitly divide
the diversity by $t_{s}^{\max}$ or $\ln t_{s}^{\max}$, since diversity should
be an extensive quantity, scaling with the observation time $t_{s}^{\max}$.
However, a diversity-production rate for the RSANN can easily be inferred, by
dividing by the observation time. We have found very little dependence of
the critical-point $\epsilon_{1}\sim 5\times 10^{-4}$ on the size of the network, $N.$
4.3 Volatility
Volatility is defined as the ability to access a large number of
highly-eligible limit-cycles, or a mixture of high eligibility and high
diversity. We have defined eligibility and diversity above each in terms of
entropy-type measure. Therefore we shall define volatility also as
an entropy-weighted entropy:
$${\cal V}=+\sum_{t_{s}=1}^{t_{s}^{\max}}e(t_{s})P(t_{s})\ln P(t_{s}){\rm%
\thinspace\thinspace.}$$
(16)
For ($\epsilon_{1}<\epsilon<\epsilon_{2})$ (Fig. 9), volatility is
high, corresponding to two distinct transformations to volatility. At $\epsilon=\epsilon_{1}\sim 5\times 10^{-4}$, the amplitude of slow threshold noise
causes a transformation to a diversity of different limit-cycles; at $\epsilon=\epsilon_{2}\sim 5\times 10^{-1}$, the slow threshold noise is so large that all
limit-cycles become fixed points. We accordingly label three different
regimes for the RSANN with slow threshold noise:
$\bullet$ Stable Regime: $\epsilon<5\times 10^{-4}$
$\bullet$ Volatile Regime: $5\times 10^{-4}\leq\epsilon<5\times 10^{-1}$
$\bullet$ Trivially Random Regime: $\epsilon\geq 5\times 10^{-1}$.
4.4 Net Size Dependence of Limit-Cycle Period and Number of
Different Attractors
From Table 1, for several different network sizes ($N\leq 50$)
with $M=10$ without slow threshold noise ($\epsilon=0$), we have found only
a handful of different limit-cycles ($N_{{\rm d}}\leq 6$) for each network,
without evidence for a systematic dependence on network size. The
limit-cycle periods for these networks at zero noise are dominated by one or
two different periods (typ($L$)), and also have no systematic dependence on
network size. We have found however that the transients (typ($W_{n-1}$))
prior to convergence to a limit-cycle tends grow very rapidly with network
size $N$. When $\epsilon=0.01$, which is in the volatile region, the
distribution of cycle lengths is broadly distributed (see Fig. 11), and from Table 1, the mean value $L_{{\rm ave}}$ of the
cycle length grows nearly exponentially with $N$ (Fig. 10). Since the
distribution of limit-cycle periods is highly non-gaussian (Fig.11), caution should be used when interpreting the properties of the average cycle-length (as in Fig. 10). Potentially the maximum or
median observed cycle length should be used rather than the average.
Also, since the cycle length distribution does not exhibit peaks at regularly spaced intervals, the possibility of an errant limit-cycle comparison algorithm is unlikely.
4.5 Observation Period-Length Dependence of Number of Different
Attractors
We observe in the volatile-regime, that the number of different attractors
observed, $N_{d}$, is nearly equal to the observation time-period $t_{s}^{\max}$,
when $\epsilon>5\times 10^{-3}$.
Conversely, in the stable-regime ($\epsilon<5\times 10^{-5}$), $N_{d}$ is largely independent of $t_{s}^{\max}.$ These two results are complementary: the former implying a
nearly inexhaustible source of different highly-eligible limit-cycle
attractors, the latter implying that we can access a small group of
different limit-cycle attractors with a high degree of predictability.
$N_{d}$ is often greater than $1$ (though small) for the
stable-regime, which means that the stable phase cannot be used to access a
particular attractor upon demand, but we can demand access to one of a small
number of different attractors. One might interpret
this result as showing that the RSANN can think the same thought in several
different ways, dependent on the initial conditions for the firing vector, $a_{i}(t=0,t_{s})$.
5 Conclusions and Summary
5.1 Comparison with other work
Kürten (1988a) has shown the existence of a dynamical phase
transition for zero-noise RSANNs with $K=3$ incoming connections per neuron
from a chaotic phase to a frozen phase. For $K<3$, RSANNs will always remain
in the frozen phase. For $K=3$, the chaotic-to-frozen transition can be
triggered either by randomly diluting the density of connections, or by
increasing an additive threshold parameter. In the chaotic phase, the RSANN
has sensitive dependence on initial conditions, and very long limit-cycle
trajectories (whose period scales as $L\propto 2^{\beta N}$). In the frozen
phase, the RSANN has insensitivity to initial conditions and very short
limit-cycle trajectories (whose period scales as $L\propto N^{\alpha}$). The
parameters $\alpha$ and $\beta$ depend on the connectivity $K$, the
additive threshold parameter, and the distribution of connection strengths.
Therefore, at zero threshold noise ($\epsilon=0$), our results are
inconclusive; however when we choose our thresholds from a sufficently wide
distribution ($\epsilon=0.01$), the limit-cycle period scales approximately
as an exponential function of $N$ (with an exponent of $\beta\sim 0.2$, see
section 4.4), so we might (cautiously) surmise that the volatile
RSANN is in the chaotic phase.
Kauffman (1993, pp. 191-235) and Kürten (1988b)
shows that random boolean automata with $K=2$ (frozen phase) typically have $N_{d}\propto N^{1/2}$ different limit-cycle attractors (for each network
realization), whose period also scales as $L\propto N^{1/2}$. This small
number of different attractors is qualitatively consistent with our
$\epsilon=0$ results. We need to perform an ensemble average of
many simulations to indeed confirm this result. We are currently unaware of any
definitive prior results on the number of different attractors for networks
in the chaotic phase (except for the observed preponderance of a very few
(long) limit-cycles, perhaps explained by ‘canalization’
(Kauffman, 1993). However, the arguments of Derrida, Gardner, and Zippelius(1987) regarding the evolution of the overlap between a configuration
and a stored configuration might apply, which would imply a strict
theoretical upper limit on the number of storable/recallable limit-cycle
attractors at $N_{d}^{\max}=2N.$ However, we do not keep our thresholds
fixed, so this upper limit is not applicable, as seen in our results.
For our implementation of noisy RSANNs, we speculate that the number of
attractors in the chaotic phase scales exponentially with $N$. This
speculation is based on our observation that with our volatile RSANN, the
number of different attractors available at $\epsilon=0.01$ is nearly
inexhaustible since in the volatile phase we are always able to observe new
limit-cycle attractors just by giving the thresholds new initial conditions
from the narrow gaussian distribution.
5.2 Volatility, Chaos, Stochastic Resonance, and Qualitative Non-Determinism
Our original interpretation of $\epsilon>0$ as providing a slowly-varying
noise on the thresholds can be reinterpreted in a slightly different
perspective. The volatile RSANNs are probably in the chaotic phase even when $\epsilon=0$, but in order to observe the expected exponential dependence of
the cycle-length on $N$, we would need to perform an ensemble average (or
observe many different realizations of the RSANN from the same class with
different connection matrices). When $\epsilon>0$, we are actually sampling
a significant fraction of the entire ensemble. Noise can give a dynamical
system access to the whole ensemble of different behaviors at different
times during the lifetime of the dynamical system. Slowly-varying threshold
noise can act as a ‘scanner’ for thoughts novel or long-lost.
Volatility can also be considered as a form of stochastic resonance,
in which an optimal noise amplitude enhances the
signal-to-noise ratio of a signal filter. Zero-noise or high-amplitude noise
tends to reduce the information-processing capability of a stochastic
resonant system. The observation of stochastic resonance without
external driving force (Gang et al., 1993) has interesting parallels to
our observation of high volatility at intermediate noise amplitudes.
By taking advantage of the chaotic threshold parameters (the connection
strength parameters are probably also chaotic), we can access a large number
of different RSANN attractors. Hence with a feedback algorithm, one might be
able to construct a system to control this chaos (Ott, Grebogi & Yorke, 1990)
and access a given attractor upon demand.
But this stability needs to be augmented by the ability to always
be able to access a novel attractor. This approach to controlled creativity
has been developed into the adaptive resonance formalism
(Carpenter & Grossberg, 1987).
There is a theoretical proof (Amit, 1989, p. 66) that thresholds with
fast-noise (Little, 1974) (chosen from a zero-mean gaussian) and sharp
step-function non-linearities is equivalent to a system with zero threshold
noise but with a rounded ‘S’-curve non-linearity.
This proof does not apply to slowly varying threshold noise. Perhaps by choosing a discontinuous non-linearity, with additive noise in the argument of the
non-linearity, the volatility or creativity that we observe is due to non-determinism (Hübler, 1992), in which prediction of final limit-cycle
attractors is nearly impossible.
The notion of ‘qualitatively’ uncertain dynamics (Heagy, Carroll & Pecora, 1994)
describes non-determinism between different attractor basins.
Qualitative non-determinism differs from the effective quantitative
non-determinism observed in ordinary deterministic chaos, in that ordinary
chaos consists of a single ‘strange’ attractor, while qualitative
non-determinism consists of multiple strange (or simple) attractors
‘riddled’ with holes for transitions to other attractors. Volatility is
precisely the same concept as qualitative non-determinism.
5.3 Summary
The main objective of this study has been to construct a volatile neural
network that could exhibit a large set of easily-accessible highly-eligible
limit-cycle attractors. Without noise, we demonstrate that random asymmetric
neural networks (RSANNs) can exhibit only a small number of different
limit-cycle attractors. With neuronal threshold noise within a rather wide
range ($\epsilon_{1}<\epsilon<\epsilon_{2}$), we show that RSANNs can access a
diversity of highly-eligible limit-cycle attractors. RSANNs exhibit a
diversity phase transformation from a small number of distinct
limit-cycle attractors to a large number at a noise amplitude of $\epsilon=\epsilon_{1}\sim 10^{-4}$. Likewise, RSANNs exhibit a
eligibility phase transformation at a threshold noise amplitude of $\epsilon=\epsilon_{2}\sim 0.5$.
Acknowledgements
H. Bohr would like to thank P. Carruthers (now deceased), J. Rafelski,
and the U. Arizona Department of
Physics for hospitality during several visits when much of this work was
completed; P. McGuire thanks the Santa Fe Institute and A. Hübler and the
University of Illinois Center for Complex Systems Research for hospitality
and atmospheres for very fertile discussions. P. McGuire was partially
supported by an NSF/U.S. Dept. of Education/State of Arizona pre-doctoral
fellowship. We all thank many individuals who have provided different
perspectives to our work, including the following:
G. Sonnenberg, D. Harley, B. Skaggs,
Z. Hasan, G. Littlewort, and J. Clark.
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On Schrödinger systems with cubic
dissipative nonlinearities of derivative type
Chunhua Li
Department of Mathematics, College of Science,
Yanbian University.
977 Gongyuan Road, Yanji, Jilin Province, 133002, China.
(E-mail: [email protected])
Hideaki Sunagawa
Department of Mathematics, Graduate School of Science,
Osaka University.
1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan.
(E-mail: [email protected])
(December 8, 2020)
Abstract:
Consider the initial value problem for systems of cubic derivative
nonlinear Schrödinger equations in one space dimension with the masses
satisfying a suitable resonance relation. We give
structural conditions on the nonlinearity under which the small data
solution gains an additional logarithmic decay as $t\to+\infty$
compared with the corresponding free evolution.
Key Words:
Derivative nonlinear Schrödinger systems; Nonlinear dissipation;
Logarithmic time-decay.
2010 Mathematics Subject Classification:
35Q55, 35B40
1 Introduction
Consider the initial value problem for the system of nonlinear Schrödinger equations of the following type:
$$\displaystyle\left\{\begin{array}[]{cl}\mathcal{L}_{m_{j}}u_{j}=F_{j}(u,%
\partial_{x}u),&t>0,\ x\in\mathbb{R},\ j=1,\ldots,N,\\
u_{j}(0,x)=\varphi_{j}(x),&x\in\mathbb{R},\ j=1,\ldots,N,\end{array}\right.$$
(1.1)
where $\mathcal{L}_{m_{j}}=i\partial_{t}+\frac{1}{2m_{j}}\partial_{x}^{2}$,
$i=\sqrt{-1}$, $m_{j}\in\mathbb{R}\backslash\{0\}$, and
$u=(u_{j}(t,x))_{1\leq j\leq N}$ is a $\mathbb{C}^{N}$-valued unknown function.
The nonlinear term $F=(F_{j})_{1\leq j\leq N}$ is always assumed to be
a cubic homogeneous polynomial in
$(u,\partial_{x}u,\overline{u},\overline{\partial_{x}u})$.
Our main interest is how the combinations of $(m_{j})_{1\leq j\leq N}$ and
the structures of $(F_{j})_{1\leq j\leq N}$ affect large-time behavior of
the solution $u$ to (1.1).
Before going into details, let us first recall some known results briefly
and clarify our motivation.
One of the most typical nonlinear Schrödinger equations appearing in
various physical settings is
$$\displaystyle i\partial_{t}u+\frac{1}{2}\partial_{x}^{2}u=\lambda|u|^{2}u,%
\qquad t>0,\ x\in\mathbb{R}$$
(1.2)
with $\lambda\in\mathbb{R}$. What is interesting in (1.2) is
that the large-time behavior of the solution is actually affected by the
nonlinearity even if the initial data is sufficiently small, smooth and
decaying fast as $|x|\to\infty$. To be more precise, it is shown in [1]
that the solution to (1.2) with small initial data behaves like
$$u(t,x)=\frac{1}{\sqrt{it}}\alpha(x/t)e^{i\{\frac{x^{2}}{2t}-\lambda|\alpha(x/t%
)|^{2}\log t\}}+o(t^{-1/2})\quad\mbox{as}\ \ t\to\infty$$
with a suitable $\mathbb{C}$-valued function $\alpha(y)$.
An important consequence of this asymptotic expression is that the solution
decays like $O(t^{-1/2})$ in $L^{\infty}(\mathbb{R}_{x})$,
while it does not behave like the free solution unless $\lambda=0$.
In other words, the additional logarithmic factor in the phase
reflects the long-range character of the cubic nonlinear Schrödinger
equations in one space dimension. If $\lambda\in\mathbb{C}$, another kind of
long-range effect can be observed. Indeed, it is verified in [15]
that the small data solution to (1.2) decays like
$O(t^{-1/2}(\log t)^{-1/2})$ in $L^{\infty}(\mathbb{R}_{x})$ as $t\to\infty$
if $\operatorname{\rm Im}\lambda<0$ (see also [17]).
This gain of additional logarithmic time decay should be interpreted
as another kind of long-range effect.
Among several extensions of this result
(see e.g., [3], [9], [11], [12],
[13] etc. and the references cited therein),
let us focus on the following two cases:
(i) the case where the nonlinearity depends also on $\partial_{x}u$,
and (ii) the case of systems.
(i)
Let us consider the single nonlinear Schrödinger equation
$$\displaystyle i\partial_{t}u+\frac{1}{2}\partial_{x}^{2}u=G(u,\partial_{x}u),%
\qquad t>0,\ x\in\mathbb{R},$$
(1.3)
where $G$ is a cubic homogeneous polynomial in
$(u,\partial_{x}u,\overline{u},\overline{\partial_{x}u})$ with complex
coefficients, and satisfies the gauge invariance
$$\displaystyle G(e^{i\theta}v,e^{i\theta}w)=e^{i\theta}G(v,w),\qquad\theta\in%
\mathbb{R},\ (v,w)\in\mathbb{C}\times\mathbb{C}.$$
(1.4)
According to [3], the solution to (1.3) decays like
$O(t^{-1/2}(\log t)^{-1/2})$ in $L^{\infty}(\mathbb{R}_{x})$ as $t\to\infty$ if
$$\displaystyle\sup_{\xi\in\mathbb{R}}\operatorname{\rm Im}G(1,i\xi)<0.$$
(1.5)
However, the approach of [3] does not work well in the case of systems,
because this additional logarithmic decay result is a consequence of the
explicit asymptotic profile of the solution $u(t,x)$,
which becomes no longer simple in the coupled case.
(ii)
For nonlinear Schrödinger systems, an additional logarithmic decay result
is first obtained by [5].
Strictly saying, two-dimensional quadratic nonlinear Schrödinger systems
are treated in [5], but we can adopt the method of [5]
directly to one-dimensional cubic nonlinear Schrödinger systems,
as pointed in [9]. When we restrict ourselves to a two-component model
$$\displaystyle\left\{\begin{array}[]{l}\mathcal{L}_{m_{1}}u_{1}=\lambda_{1}|u_{%
1}|^{2}u_{1}+\nu_{1}\overline{u_{1}}^{2}u_{2},\\
\mathcal{L}_{m_{2}}u_{2}=\lambda_{2}|u_{2}|^{2}u_{2}+\nu_{2}u_{1}^{3},\end{%
array}\right.\qquad t>0,\ x\in\mathbb{R}$$
(1.6)
with $\lambda_{1}$, $\lambda_{2}$, $\nu_{1}$, $\nu_{2}\in\mathbb{C}$
and $m_{1},m_{2}\in\mathbb{R}\backslash\{0\}$,
then the result of [5] can be read as follows:
the solution to (1.6) decays like $O(t^{-1/2}(\log t)^{-1/2})$ in
$L^{\infty}(\mathbb{R}_{x})$ as $t\to\infty$ if
$$\displaystyle m_{2}=3m_{1},$$
(1.7)
$$\displaystyle\operatorname{\rm Im}\lambda_{j}<0,\qquad j=1,2,$$
(1.8)
and
$$\displaystyle\kappa_{1}\nu_{1}=\kappa_{2}\overline{\nu_{2}}\quad\mbox{with %
some $\kappa_{1}$, $\kappa_{2}>0$}$$
(1.9)
(see Example 2.1 in [9] for the detail).
The advantage of the method of [5] is that it does not
rely on the explicit asymptotic profile at all. However, it is
not straightforward to apply this approach in the derivative
nonlinear case, because we need suitable pointwise a priori estimates
not only for the solution itself but also for its derivatives
without breaking good structure in order to apply the
method of [5].
The purpose of this paper is to unify (i) and (ii).
More precisely, we will introduce structural conditions on
$(F_{j})_{1\leq j\leq N}$ and $(m_{j})_{1\leq j\leq N}$
under which the small data solution to the derivative nonlinear
Schrödinger system (1.1) gains an additional logarithmic
decay as $t\to+\infty$ compared with the corresponding free evolution.
2 Main Results
In the subsequent sections, we will use the following notations:
We set $I_{N}=\{1,\ldots,N\}$ and
${I}^{\sharp}_{N}=\{1,\ldots,N,N+1,\ldots,2N\}$.
For $z=(z_{j})_{j\in I_{N}}\in\mathbb{C}^{N}$, we write
$${z}^{\sharp}=({z}^{\sharp}_{k})_{k\in{I}^{\sharp}_{N}}:=(z_{1},\ldots,z_{N},%
\overline{z_{1}},\ldots,\overline{z_{N}})\in\mathbb{C}^{2N}.$$
Then general cubic nonlinear term $F=(F_{j})_{j\in I_{N}}$ can be written as
$$\displaystyle F_{j}(u,\partial_{x}u)=\sum_{l_{1},l_{2},l_{3}=0}^{1}\sum_{k_{1}%
,k_{2},k_{3}\in{I}^{\sharp}_{N}}C_{j,k_{1},k_{2},k_{3}}^{l_{1},l_{2},l_{3}}(%
\partial_{x}^{l_{1}}{u}^{\sharp}_{k_{1}})(\partial_{x}^{l_{2}}{u}^{\sharp}_{k_%
{2}})(\partial_{x}^{l_{3}}{u}^{\sharp}_{k_{3}})$$
with suitable $C_{j,k_{1},k_{2},k_{3}}^{l_{1},l_{2},l_{3}}\in\mathbb{C}$.
With this expression of $F$, we define
$p=(p_{j}(\xi;Y))_{j\in I_{N}}:\mathbb{R}\times\mathbb{C}^{N}\to\mathbb{C}^{N}$ by
$$\displaystyle p_{j}(\xi;Y):=\sum_{l_{1},l_{2},l_{3}=0}^{1}\sum_{k_{1},k_{2},k_%
{3}\in{I}^{\sharp}_{N}}C_{j,k_{1},k_{2},k_{3}}^{l_{1},l_{2},l_{3}}(i\tilde{m}_%
{k_{1}}\xi)^{l_{1}}(i\tilde{m}_{k_{2}}\xi)^{l_{2}}(i\tilde{m}_{k_{3}}\xi)^{l_{%
3}}{Y}^{\sharp}_{k_{1}}{Y}^{\sharp}_{k_{2}}{Y}^{\sharp}_{k_{3}}$$
for $\xi\in\mathbb{R}$ and $Y=(Y_{j})_{j\in I_{N}}\in\mathbb{C}^{N}$, where
$$\tilde{m}_{k}=\left\{\begin{array}[]{cl}m_{k}&(k=1,\ldots,N),\\
-m_{(k-N)}&(k=N+1,\ldots,2N).\end{array}\right.$$
In what follows, we denote by $\langle\cdot,\cdot\rangle_{\mathbb{C}^{N}}$ the standard scalar
product in $\mathbb{C}^{N}$, i.e.,
$$\langle z,w\rangle_{\mathbb{C}^{N}}=\sum_{j=1}^{N}z_{j}\overline{w_{j}}$$
for $z=(z_{j})_{j\in I_{N}}$ and $w=(w_{j})_{j\in I_{N}}\in\mathbb{C}^{N}$.
Now let us introduce the following conditions:
(a)
For all $j\in I_{N}$ and
$k_{1},k_{2},k_{3}\in{I}^{\sharp}_{N}$,
$$m_{j}\neq\tilde{m}_{k_{1}}+\tilde{m}_{k_{2}}+\tilde{m}_{k_{3}}\ \ \mbox{%
implies}\ \ C_{j,k_{1},k_{2},k_{3}}^{l_{1},l_{2},l_{3}}=0,\ \ l_{1},l_{2},l_{3%
}\in\{0,1\}.$$
(b${}_{0}$)
There exists an $N\times N$ positive Hermitian matrix $A$ such that
$$\operatorname{\rm Im}\langle p(\xi;Y),AY\rangle_{\mathbb{C}^{N}}\leq 0$$
for all $(\xi,Y)\in\mathbb{R}\times\mathbb{C}^{N}$.
(b${}_{1}$)
There exist an $N\times N$ positive Hermitian matrix $A$ and
a positive constant $C_{*}$ such that
$$\operatorname{\rm Im}\langle p(\xi;Y),AY\rangle_{\mathbb{C}^{N}}\leq-C_{*}|Y|^%
{4}$$
for all $(\xi,Y)\in\mathbb{R}\times\mathbb{C}^{N}$.
(b${}_{2}$)
There exist an $N\times N$ positive Hermitian matrix $A$ and
a positive constant $C_{**}$ such that
$$\operatorname{\rm Im}\langle p(\xi;Y),AY\rangle_{\mathbb{C}^{N}}\leq-C_{**}%
\langle\xi\rangle^{2}|Y|^{4}$$
for all $(\xi,Y)\in\mathbb{R}\times\mathbb{C}^{N}$, where $\langle\xi\rangle=\sqrt{1+\xi^{2}}$.
(b${}_{3}$)
$p(\xi;Y)=0$ for all $(\xi,Y)\in\mathbb{R}\times\mathbb{C}^{N}$.
To state the main results, we introduce some function spaces. For
$s,\sigma\in\mathbb{Z}_{\geq 0}$,
we denote by $H^{s}$ the $L^{2}$-based Sobolev space of order $s$,
and the weighted Sobolev space $H^{s,\sigma}$ is defined by
$\{\phi\in L^{2}\,|\,\langle x\rangle^{\sigma}\phi\in H^{s}\}$, equipped with
the norm $\|\phi\|_{H^{s,\sigma}}=\|\langle x\rangle^{\sigma}\phi\|_{H^{s}}$.
The main results are as follows:
Theorem 2.1.
Assume the conditions (a) and (b${}_{0}$) are satisfied. Let
$\varphi=(\varphi_{j})_{j\in I_{N}}\in H^{3}\cap H^{2,1}$,
and assume $\varepsilon:=\|\varphi\|_{H^{3}}+\|\varphi\|_{H^{2,1}}$ is sufficiently
small. Then (1.1) admits a unique global solution
$u=(u_{j})_{j\in I_{N}}\in C([0,\infty);H^{3}\cap H^{2,1})$.
Moreover we have
$$\|u(t)\|_{L^{\infty}}\leq\frac{C\varepsilon}{\sqrt{1+t}},\qquad\|u(t)\|_{L^{2}%
}\leq C\varepsilon$$
for $t\geq 0$, where $C$ is a positive constant not depending on $\varepsilon$.
Theorem 2.2.
Assume the conditions (a) and (b${}_{1}$) are satisfied. Let $u$
be the global solution to (1.1), whose existence is guaranteed
by Theorem 2.1. Then we have
$$\|u(t)\|_{L^{\infty}}\leq\frac{C\varepsilon}{\sqrt{(1+t)\{1+\varepsilon^{2}%
\log(2+t)\}}}$$
for $t\geq 0$, where $C$ is a positive constant not depending on $\varepsilon$.
We also have
$$\lim_{t\to+\infty}\|u(t)\|_{L^{2}}=0.$$
Theorem 2.3.
Assume the conditions (a) and (b${}_{2}$) are satisfied. Let $u$ be as above.
Then we have
$$\|u(t)\|_{L^{2}}\leq\frac{C\varepsilon}{\sqrt{1+\varepsilon^{2}\log(2+t)}}$$
for $t\geq 0$, where $C$ is a positive constant not depending on $\varepsilon$.
Theorem 2.4.
Assume the conditions (a) and (b${}_{3}$) are satisfied. Let $u$ be as above.
For each $j\in I_{N}$, there exists $\varphi_{j}^{+}\in L^{2}(\mathbb{R}_{x})$ with
$\hat{\varphi}_{j}^{+}\in L^{\infty}(\mathbb{R}_{\xi})$ such that
$$u_{j}(t)=e^{i\frac{t}{2m_{j}}\partial_{x}^{2}}\varphi_{j}^{+}+O(t^{-1/4+\delta%
})\quad\mbox{in}\ L^{2}(\mathbb{R}_{x})$$
and
$$u_{j}(t,x)=\sqrt{\frac{m_{j}}{it}}\,\hat{\varphi}^{+}_{j}\left(\frac{m_{j}x}{t%
}\right)e^{i\frac{m_{j}x^{2}}{2t}}+O(t^{-3/4+\delta})\quad\mbox{in}\ L^{\infty%
}(\mathbb{R}_{x})$$
as $t\to+\infty$, where $\delta>0$ can be taken arbitrarily small,
and $\hat{\phi}$ denotes the Fourier transform of $\phi$, i.e.,
$$\hat{\phi}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-iy\xi}\phi(y)dy.$$
Remark 2.1.
In view of the proof of Theorem 2.4 below,
we can see that $\varphi^{+}=(\varphi_{j}^{+})_{j\in I_{N}}$ does not
identically vanish if the initial data $\varphi$
is suitably small and does not identically vanish (see Remark 6.1
for the detail).
Therefore the solution does not gain an additional logarithmic decay
under the conditions (a) and (b${}_{3}$).
Now let us give several examples which satisfy the above
mentioned conditions:
Example 2.1.
In the single case (i.e., $N=1$), we may assume $m_{1}=1$ without
loss of generality. Then we can check that the condition (a) is euqivalent to
the gauge invariance (1.4), and that the condition (1.5)
is equivalent to the condition (b${}_{1}$). Therefore our results above can be
viewed as an extension of [3] except the explicit asymptotic profile of
the solution. We can also see that our results cover the system
(1.6) under the assumptions (1.7), (1.8),
(1.9). Indeed, (1.7) plays the role of (a),
and (1.8), (1.9) correspond to (b${}_{1}$) with
$A=\begin{pmatrix}\kappa_{1}&0\\
0&\kappa_{2}\end{pmatrix}$.
Example 2.2.
Next let us consider the following two-component system
$$\left\{\begin{array}[]{l}\mathcal{L}_{m}u_{1}=\lambda_{1}|u_{1}|^{2}u_{1}+%
\lambda_{2}\overline{u_{1}}(\partial_{x}u_{1})^{2}+iu_{2}\partial_{x}(%
\overline{u_{1}}^{2}),\\
\mathcal{L}_{3m}u_{2}=\lambda_{3}|u_{2}|^{2}\partial_{x}u_{2}-i(|u_{2}|^{2}+|%
\partial_{x}u_{2}|^{2})u_{2}-iu_{1}^{2}\partial_{x}u_{1}\end{array}\right.$$
with $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}\in\mathbb{C}$ and
$m\in\mathbb{R}\backslash\{0\}$, which is a bit more complicated than
(1.6).
It is easy to check that the condition (a) is satisfied by this
system. Also it follows from simple calculations that
$$\left\{\begin{array}[]{l}p_{1}(\xi;Y)=(\lambda_{1}-\lambda_{2}m^{2}\xi^{2})|Y_%
{1}|^{2}Y_{1}+2m\xi\overline{Y_{1}}^{2}Y_{2},\\
p_{2}(\xi;Y)=i(3\lambda_{3}m\xi-1-9m^{2}\xi^{2})|Y_{2}|^{2}Y_{2}+3m\xi Y_{1}^{%
3}.\end{array}\right.$$
With $A=\begin{pmatrix}3&0\\
0&2\end{pmatrix}$, we have
$$\langle p(\xi;Y),AY\rangle_{\mathbb{C}^{2}}=3(\lambda_{1}-\lambda_{2}m^{2}\xi^%
{2})|Y_{1}|^{4}-2i(1-3\lambda_{3}m\xi+9m^{2}\xi^{2})|Y_{2}|^{4}+12m\xi%
\operatorname{\rm Re}(\overline{Y_{1}}^{3}Y_{2}),$$
whence
$$\operatorname{\rm Im}\langle p(\xi;Y),AY\rangle_{\mathbb{C}^{2}}=3\bigl{(}%
\operatorname{\rm Im}\lambda_{1}-\operatorname{\rm Im}\lambda_{2}m^{2}\xi^{2}%
\bigr{)}|Y_{1}|^{4}-\left\{2-\frac{(\operatorname{\rm Re}\lambda_{3})^{2}}{2}+%
2\biggl{(}3m\xi-\frac{\operatorname{\rm Re}\lambda_{3}}{2}\biggr{)}^{2}\right%
\}|Y_{2}|^{4}.$$
Therefore we see that
•
(b${}_{0}$) is satisfied if $\operatorname{\rm Im}\lambda_{1}\leq 0$,
$\operatorname{\rm Im}\lambda_{2}\geq 0$ and $|\operatorname{\rm Re}\lambda_{3}|\leq 2$.
•
(b${}_{1}$) is satisfied if $\operatorname{\rm Im}\lambda_{1}<0$, $\operatorname{\rm Im}\lambda_{2}\geq 0$
and $|\operatorname{\rm Re}\lambda_{3}|<2$.
•
(b${}_{2}$) is satisfied if $\operatorname{\rm Im}\lambda_{1}<0$, $\operatorname{\rm Im}\lambda_{2}>0$
and $|\operatorname{\rm Re}\lambda_{3}|<2$.
Example 2.3.
Finally we focus on the three-component system
$$\left\{\begin{array}[]{l}\mathcal{L}_{m}u_{1}=u_{2}\partial_{x}\bigl{(}%
\overline{u_{1}}u_{2}\bigr{)},\\
\mathcal{L}_{m}u_{2}=\overline{u_{1}}\overline{u_{2}}\partial_{x}u_{3}+3%
\overline{u_{1}}u_{3}\partial_{x}\overline{u_{2}},\\
\mathcal{L}_{3m}u_{3}=2u_{1}^{2}\partial_{x}u_{2}-u_{2}\partial_{x}(u_{1}^{2})%
.\end{array}\right.$$
We can immediately check that this system satisfies (a) and (b${}_{3}$).
Note that this example should be compared with [4],
where the null structure in quadratic derivative nonlinear Schrödinger
systems in $\mathbb{R}^{2}$ is considered in details
(see also [7], [8], [16]).
The rest part of this paper is organized as follows:
The next section is devoted to preliminaries on basic properties of
the operator $J_{m}$. In Scetion 4, we recall the smoothing
property of the linear Schrödinger eqautions. In Section 5,
we will get an a priori estimate. After that, The main theorems will be
proved in Section 6.
The appendix is devoted to the proof of technical lemmas.
In what follows, we will denote several positive constants by the same letter
$C$, which is possibly different from line to line.
3 Preliminaries
In this section, we collect several identities and inequalities
which are useful for our purpose. We set
$J_{m}=x+i\frac{t}{m}\partial_{x}$ for non-zero real constant $m$.
Then we can check that $[\partial_{x},J_{m}]=1$ and $[\mathcal{L}_{m},J_{m}]=0$,
where $[\cdot,\cdot]$ denotes the commutator of two linear operators.
We also note that
$$\displaystyle J_{m}\phi=\frac{it}{m}e^{im\frac{x^{2}}{2t}}\partial_{x}\bigl{(}%
e^{-im\frac{x^{2}}{2t}}\phi\bigr{)},$$
(3.1)
which yields the following useful lemmas.
Lemma 3.1.
Let $m$, $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ be non-zero real constants
satisfying $m=\mu_{1}+\mu_{2}+\mu_{3}$. We have
$$J_{m}(f_{1}f_{2}f_{3})=\frac{\mu_{1}}{m}(J_{\mu_{1}}f_{1})f_{2}f_{3}+\frac{\mu%
_{2}}{m}f_{1}(J_{\mu_{2}}f_{2})f_{3}+\frac{\mu_{3}}{m}f_{1}f_{2}(J_{\mu_{3}}f_%
{3}),$$
$$J_{m}(f_{1}f_{2}\overline{f_{3}})=\frac{\mu_{1}}{m}(J_{\mu_{1}}f_{1})f_{2}%
\overline{f_{3}}+\frac{\mu_{2}}{m}f_{1}(J_{\mu_{2}}f_{2})\overline{f_{3}}+%
\frac{\mu_{3}}{m}f_{1}f_{2}(\overline{J_{-\mu_{3}}f_{3}}),$$
$$J_{m}(f_{1}\overline{f_{2}}\overline{f_{3}})=\frac{\mu_{1}}{m}(J_{\mu_{1}}f_{1%
})\overline{f_{2}}\overline{f_{3}}+\frac{\mu_{2}}{m}f_{1}(\overline{J_{-\mu_{2%
}}f_{2}})\overline{f_{3}}+\frac{\mu_{3}}{m}f_{1}\overline{f_{2}}(\overline{J_{%
-\mu_{3}}f_{3}}),$$
$$J_{m}(\overline{f_{1}}\overline{f_{2}}\overline{f_{3}})=\frac{\mu_{1}}{m}(%
\overline{J_{-\mu_{1}}f_{1}})\overline{f_{2}}\overline{f_{3}}+\frac{\mu_{2}}{m%
}\overline{f_{1}}(\overline{J_{-\mu_{2}}f_{2}})\overline{f_{3}}+\frac{\mu_{3}}%
{m}\overline{f_{1}}\overline{f_{2}}(\overline{J_{-\mu_{3}}f_{3}})$$
for smooth $\mathbb{C}$-valued functions $f_{1}$, $f_{2}$ and $f_{3}$.
Proof.
We set $\theta=x^{2}/(2t)$. It follows from (3.1) that
$$\displaystyle mJ_{m}(f_{1}f_{2}\overline{f_{3}})$$
$$\displaystyle=ite^{i(\mu_{1}+\mu_{2}+\mu_{3})\theta}\partial_{x}\Bigl{\{}(e^{-%
i\mu_{1}\theta}f_{1})(e^{-i\mu_{2}\theta}f_{2})(\overline{e^{i\mu_{3}\theta}f_%
{3}})\Bigr{\}}$$
$$\displaystyle=\Bigl{(}ite^{i\mu_{1}\theta}\partial_{x}(e^{-i\mu_{1}\theta}f_{1%
})\Bigr{)}f_{2}\overline{f_{3}}+f_{1}\Bigl{(}ite^{i\mu_{2}\theta}\partial_{x}(%
e^{-i\mu_{2}\theta}f_{2})\Bigr{)}\overline{f_{3}}-f_{1}f_{2}\Bigl{(}\overline{%
ite^{-i\mu_{3}\theta}\partial_{x}(e^{i\mu_{3}\theta}f_{3})}\Bigr{)}$$
$$\displaystyle=(\mu_{1}J_{\mu_{1}}f_{1})f_{2}\overline{f_{3}}+f_{1}(\mu_{2}J_{%
\mu_{2}}f_{2})\overline{f_{3}}+f_{1}f_{2}(\overline{\mu_{3}J_{-\mu_{3}}f_{3}}),$$
which gives the second identity.
The other three identities can be shown in the same way.
∎
Remark 3.1.
If we do not assume $m=\mu_{1}+\mu_{2}+\mu_{3}$, we have
$$\displaystyle J_{m}(f_{1}f_{2}f_{3})=$$
$$\displaystyle\frac{\mu_{1}}{\mu_{1}+\mu_{2}+\mu_{3}}(J_{\mu_{1}}f_{1})f_{2}f_{%
3}+\frac{\mu_{2}}{\mu_{1}+\mu_{2}+\mu_{3}}f_{1}(J_{\mu_{2}}f_{2})f_{3}+\frac{%
\mu_{3}}{\mu_{1}+\mu_{2}+\mu_{3}}f_{1}f_{2}(J_{\mu_{3}}f_{3})$$
$$\displaystyle+it\left(\frac{1}{m}-\frac{1}{\mu_{1}+\mu_{2}+\mu_{3}}\right)%
\partial_{x}(f_{1}f_{2}f_{3}),$$
and so on. The last term implies a loss of time-decay in general.
(The situation is worse if $\mu_{1}+\mu_{2}+\mu_{3}=0$.)
Lemma 3.2.
Let $m$, $\mu_{1}$, $\mu_{2}$ be non-zero real constants. We have
$$\displaystyle\partial_{x}(f_{1}f_{2}f_{3})=\frac{m}{\mu_{1}}(\partial_{x}f_{1}%
)f_{2}f_{3}+\frac{R_{1}}{t}$$
(3.2)
and
$$\displaystyle\partial_{x}^{2}(f_{1}f_{2}f_{3})=\frac{m^{2}}{\mu_{1}\mu_{2}}(%
\partial_{x}f_{1})(\partial_{x}f_{2})f_{3}+\frac{R_{2}}{t},$$
(3.3)
where
$R_{1}=-imJ_{m}(f_{1}f_{2}f_{3})+im(J_{\mu_{1}}f_{1})f_{2}f_{3}$ and
$$R_{2}=-\frac{im^{2}}{\mu_{1}}J_{m}\bigl{[}(\partial_{x}f_{1})f_{2}f_{3}\bigr{]%
}+\frac{im^{2}}{\mu_{1}}(\partial_{x}f_{1})(J_{\mu_{2}}f_{2})f_{3}+\partial_{x%
}R_{1}.$$
Remark 3.2.
We do not assume any relations among $\mu_{1}$, $\mu_{2}$ and $m$
in Lemma 3.2.
Proof.
From the relation $\frac{1}{m}\partial_{x}-\frac{1}{it}J_{m}=i\frac{x}{t}$,
we see that
$$\displaystyle\frac{1}{m}\partial_{x}(f_{1}f_{2}f_{3})-\frac{1}{it}J_{m}(f_{1}f%
_{2}f_{3})=i\frac{x}{t}f_{1}f_{2}f_{3}=\left(\frac{1}{\mu_{1}}\partial_{x}f_{1%
}-\frac{1}{it}J_{\mu_{1}}f_{1}\right)f_{2}f_{3},$$
which yields (3.2).
We also have (3.3) by using (3.2) twice.
∎
Next we set
$$\bigl{(}\mathcal{U}_{m}(t)\phi\bigr{)}(x):=e^{i\frac{t}{2m}\partial_{x}^{2}}%
\phi(x)=\sqrt{\frac{|m|}{2\pi t}}e^{-i\frac{\pi}{4}\mathrm{sgn}(m)}\int_{%
\mathbb{R}}e^{im\frac{(x-y)^{2}}{2t}}\phi(y)dy$$
for $m\in\mathbb{R}\backslash\{0\}$ and $t>0$.
We also introduce the scaled Fourier transform $\mathcal{F}_{m}$ by
$$\bigl{(}\mathcal{F}_{m}\phi\bigr{)}(\xi):=|m|^{1/2}e^{-i\frac{\pi}{4}\mathrm{%
sgn}(m)}\,\hat{\phi}(m\xi)=\sqrt{\frac{|m|}{2\pi}}e^{-i\frac{\pi}{4}\mathrm{%
sgn}(m)}\int_{\mathbb{R}}e^{-imy\xi}\phi(y)dy,$$
as well as auxiliary operators
$$\bigl{(}\mathcal{M}_{m}(t)\phi\bigr{)}(x):=e^{im\frac{x^{2}}{2t}}\phi(x),\quad%
\bigl{(}\mathcal{D}(t)\phi\bigr{)}(x):=\frac{1}{\sqrt{t}}\phi\left(\frac{x}{t}%
\right),\quad\mathcal{W}_{m}(t)\phi:=\mathcal{F}_{m}\mathcal{M}_{m}(t)\mathcal%
{F}_{m}^{-1}\phi,$$
so that $\mathcal{U}_{m}$ can be decomposed into
$\mathcal{U}_{m}=\mathcal{M}_{m}\mathcal{D}\mathcal{F}_{m}\mathcal{M}_{m}=%
\mathcal{M}_{m}\mathcal{D}\mathcal{W}_{m}\mathcal{F}_{m}$.
The following lemma is well known (see e.g., [1], [14]).
Lemma 3.3.
Let $m$ be a non-zero real constant. We have
$$\displaystyle\|\phi-\mathcal{M}_{m}\mathcal{D}\mathcal{F}_{m}\mathcal{U}_{m}^{%
-1}\phi\|_{L^{\infty}}\leq Ct^{-3/4}\bigl{(}\|\phi\|_{L^{2}}+\|\mathcal{J}_{m}%
\phi\|_{L^{2}}\bigr{)}$$
and
$$\|\phi\|_{L^{\infty}}\leq t^{-1/2}\|\mathcal{F}_{m}\mathcal{U}_{m}^{-1}\phi\|_%
{L^{\infty}}+Ct^{-3/4}(\|\phi\|_{L^{2}}+\|\mathcal{J}_{m}\phi\|_{L^{2}})$$
for $t\geq 1$.
Proof.
For the convenience of the readers, we give the proof.
By the relation $J_{m}=\mathcal{U}_{m}x\mathcal{U}_{m}^{-1}$, we see that
$$\|\mathcal{F}_{m}\mathcal{U}_{m}^{-1}\phi\|_{H^{1}}\leq C\|\mathcal{U}_{m}^{-1%
}\phi\|_{H^{0,1}}\leq C(\|\phi\|_{L^{2}}+\|\mathcal{J}_{m}\phi\|_{L^{2}}).$$
Also it follows from the inequalities
$\|\phi\|_{L^{\infty}}\leq\sqrt{2}\|\phi\|_{L^{2}}^{1/2}\|\partial_{x}\phi\|_{L%
^{2}}^{1/2}$
and
$|e^{i\theta}-1|\leq C|\theta|^{1/2}$ that
$$\displaystyle\|(\mathcal{W}_{m}^{\pm 1}-1)\phi\|_{L^{\infty}}$$
$$\displaystyle\leq C\|(\mathcal{M}_{m}^{\pm 1}-1)\mathcal{F}^{-1}_{m}\phi\|_{L^%
{2}}^{1/2}\,\|\partial_{x}(\mathcal{W}_{m}^{\pm 1}-1)\phi\|_{L^{2}}^{1/2}$$
$$\displaystyle\leq C(t^{-1/2}\|\mathcal{F}^{-1}_{m}\phi\|_{H^{0,1}})^{1/2}\,\|%
\partial_{x}\phi\|_{L^{2}}^{1/2}$$
$$\displaystyle\leq Ct^{-1/4}\|\phi\|_{H^{1}}.$$
(3.4)
Combining with the inequalities obtained above, we have
$$\displaystyle\|\phi-\mathcal{M}_{m}\mathcal{D}\mathcal{F}_{m}\mathcal{U}_{m}^{%
-1}\phi\|_{L^{\infty}}$$
$$\displaystyle=\|\mathcal{M}_{m}\mathcal{D}(\mathcal{W}_{m}-1)\mathcal{F}_{m}%
\mathcal{U}_{m}^{-1}\phi\|_{L^{\infty}}$$
$$\displaystyle\leq t^{-1/2}\|(\mathcal{W}_{m}-1)\mathcal{F}_{m}\mathcal{U}_{m}^%
{-1}\phi\|_{L^{\infty}}$$
$$\displaystyle\leq Ct^{-3/4}\|\mathcal{F}_{m}\mathcal{U}_{m}^{-1}\phi\|_{H^{1}}$$
$$\displaystyle\leq Ct^{-3/4}\bigl{(}\|\phi\|_{L^{2}}+\|\mathcal{J}_{m}\phi\|_{L%
^{2}}\bigr{)}.$$
Using the result derived above, we also have
$$\displaystyle\|\phi\|_{L^{\infty}}$$
$$\displaystyle\leq\|\mathcal{M}_{m}\mathcal{D}\mathcal{F}_{m}\mathcal{U}_{m}^{-%
1}\phi\|_{L^{\infty}}+\|\phi-\mathcal{M}_{m}\mathcal{D}\mathcal{F}_{m}\mathcal%
{U}_{m}^{-1}\phi\|_{L^{\infty}}$$
$$\displaystyle\leq t^{-1/2}\|\mathcal{F}_{m}\mathcal{U}_{m}^{-1}\phi\|_{L^{%
\infty}}+Ct^{-3/4}\bigl{(}\|\phi\|_{L^{2}}+\|\mathcal{J}_{m}\phi\|_{L^{2}}%
\bigr{)}.$$
∎
Lemma 3.4.
Let $m$ be a non-zero real constant. We have
$$\|\mathcal{F}_{m}\mathcal{U}_{m}^{-1}(f_{1}f_{2}f_{3})\|_{L^{\infty}}\leq C\|f%
_{1}\|_{L^{2}}\|f_{2}\|_{L^{2}}\|f_{3}\|_{L^{\infty}}.$$
Proof.
By the relation
$\mathcal{F}_{m}\mathcal{U}_{m}^{-1}=\mathcal{W}_{m}^{-1}\mathcal{D}^{-1}%
\mathcal{M}_{m}^{-1}$
and the estimate
$\|\mathcal{W}_{m}^{-1}\phi\|_{L^{\infty}}\leq Ct^{1/2}\|\phi\|_{L^{1}}$,
we have
$$\displaystyle\|\mathcal{F}_{m}\mathcal{U}_{m}^{-1}(f_{1}f_{2}f_{3})\|_{L^{%
\infty}}$$
$$\displaystyle\leq Ct^{1/2}\|\mathcal{D}^{-1}\mathcal{M}_{m}^{-1}(f_{1}f_{2}f_{%
3})\|_{L^{1}}$$
$$\displaystyle\leq Ct^{1/2}\cdot t^{-1}\|(\mathcal{D}^{-1}f_{1})(\mathcal{D}^{-%
1}f_{2})(\mathcal{D}^{-1}\mathcal{M}_{m}^{-1}f_{3})\|_{L^{1}}$$
$$\displaystyle\leq Ct^{-1/2}\|\mathcal{D}^{-1}f_{1}\|_{L^{2}}\|\mathcal{D}^{-1}%
f_{2}\|_{L^{2}}\|\mathcal{D}^{-1}\mathcal{M}_{m}^{-1}f_{3}\|_{L^{\infty}}$$
$$\displaystyle=Ct^{-1/2}\|f_{1}\|_{L^{2}}\|f_{2}\|_{L^{2}}\cdot t^{1/2}\|f_{3}%
\|_{L^{\infty}}$$
$$\displaystyle=C\|f_{1}\|_{L^{2}}\|f_{2}\|_{L^{2}}\|f_{3}\|_{L^{\infty}}.$$
∎
We deduce the following proposition from
Lemmas 3.1–3.4,
which will play the key role in Section 5.2.
Proposition 3.1.
Suppose that the condition (a) is satisfied. For
a $\mathbb{C}^{N}$-valued function $u=(u_{j}(t,x))_{j\in I_{N}}$, we set
$\alpha_{j}(t,\xi)=\mathcal{F}_{m_{j}}[\mathcal{U}_{m_{j}}(t)^{-1}u_{j}(t,\cdot%
)](\xi)$ and $\alpha=(\alpha_{j}(t,\xi))_{j\in I_{N}}$.
Then we have
$$\left\|\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\Bigl{[}\partial_{x}^{l}F_{j%
}(u,\partial_{x}u)\Bigr{]}-\frac{(im_{j}\xi)^{l}}{t}p_{j}(\xi;\alpha)\right\|_%
{L^{\infty}_{\xi}}\leq\frac{C}{t^{5/4}}\sum_{k=1}^{N}\bigl{(}\|u_{k}(t)\|_{H^{%
3}}+\|J_{m_{k}}u_{k}(t)\|_{H^{2}}\bigr{)}^{3}$$
for $j\in I_{N}$, $l\in\{0,1,2\}$ and $t\geq 1$.
Proof.
For simplicity of exposition, we treat only the case where
$F_{j}=(\partial_{x}u_{1})(\overline{\partial_{x}u_{2}})(\partial_{x}u_{3})$ with
$m_{j}=m_{1}-m_{2}+m_{3}$.
The general case can be shown in the same way.
We set
$\alpha_{k}^{(s)}=(im_{k}\xi)^{s}\alpha_{k}$ for $s\in\mathbb{Z}_{\geq 0}$, so that
$$\partial_{x}^{s}u_{k}=\mathcal{U}_{m_{k}}\mathcal{F}_{m_{k}}^{-1}\alpha_{k}^{(%
s)}=\mathcal{M}_{m_{k}}\mathcal{D}\mathcal{W}_{m_{k}}\alpha_{k}^{(s)},\qquad%
\partial_{x}^{s}\overline{u_{k}}=\mathcal{U}_{-m_{k}}\mathcal{F}_{-m_{k}}^{-1}%
\overline{\alpha_{k}^{(s)}}.$$
Remark that
$$p_{j}(\xi;\alpha)=(im_{1}\xi)(-im_{2}\xi)(im_{3}\xi)\alpha_{1}\overline{\alpha%
_{2}}\alpha_{3}=\alpha_{1}^{(1)}\overline{\alpha_{2}^{(1)}}\alpha_{3}^{(1)}.$$
Now we consider the simplest case $l=0$. By the factorization
of $\mathcal{U}_{m_{j}}$ and the condition $m_{j}=m_{1}-m_{2}+m_{3}$,
we have
$$\displaystyle\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}F_{j}$$
$$\displaystyle=\mathcal{W}_{m_{j}}^{-1}\mathcal{D}^{-1}\mathcal{M}_{m_{j}}^{-1}%
\Bigl{[}(\mathcal{M}_{m_{1}}\mathcal{D}\mathcal{W}_{m_{1}}\alpha_{1}^{(1)})(%
\mathcal{M}_{-m_{2}}\mathcal{D}\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}}%
)(\mathcal{M}_{m_{3}}\mathcal{D}\mathcal{W}_{m_{3}}\alpha_{3}^{(1)})\Bigr{]}$$
$$\displaystyle=\frac{1}{t}\mathcal{W}_{m_{j}}^{-1}\Bigl{[}(\mathcal{W}_{m_{1}}%
\alpha_{1}^{(1)})(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}})(\mathcal{W}%
_{m_{3}}\alpha_{3}^{(1)})\Bigr{]}$$
$$\displaystyle=\frac{1}{t}p_{j}(\xi;\alpha)+\frac{1}{t}r_{0},$$
where
$$r_{0}=\mathcal{W}_{m_{j}}^{-1}\Bigl{[}(\mathcal{W}_{m_{1}}\alpha_{1}^{(1)})(%
\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}})(\mathcal{W}_{m_{3}}\alpha_{3}%
^{(1)})\Bigr{]}-\alpha_{1}^{(1)}\overline{\alpha_{2}^{(1)}}\alpha_{3}^{(1)}.$$
Since we can rewrite it as
$$\displaystyle r_{0}=$$
$$\displaystyle(\mathcal{W}_{m_{j}}^{-1}-1)\Bigl{[}(\mathcal{W}_{m_{1}}\alpha_{1%
}^{(1)})(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}})(\mathcal{W}_{m_{3}}%
\alpha_{3}^{(1)})\Bigr{]}+\bigl{\{}(\mathcal{W}_{m_{1}}-1)\alpha_{1}^{(1)}%
\bigr{\}}(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}})(\mathcal{W}_{m_{3}}%
\alpha_{3}^{(1)})$$
$$\displaystyle+\alpha_{1}^{(1)}\bigl{\{}(\mathcal{W}_{-m_{2}}-1)\overline{%
\alpha_{2}^{(1)}}\bigr{\}}(\mathcal{W}_{m_{3}}\alpha_{3}^{(1)})+\alpha_{1}^{(1%
)}\overline{\alpha_{2}^{(1)}}\bigl{\{}(\mathcal{W}_{m_{3}}-1)\alpha_{3}^{(1)}%
\bigr{\}},$$
we can apply (3.4) and the Sobolev imbedding
$H^{1}(\mathbb{R}^{1})\hookrightarrow L^{\infty}(\mathbb{R}^{1})$ to obtain
$$\|r_{0}\|_{L^{\infty}}\leq Ct^{-1/4}\|u_{1}\|_{H^{2}}\|u_{2}\|_{H^{2}}\|u_{3}%
\|_{H^{2}}.$$
Next we consider the case of $l=1$. By (3.2) with $m=m_{j}$, $\mu=m_{1}$,
$f_{1}=\partial_{x}u_{1}$, $f_{2}=\overline{\partial_{x}u_{2}}$, $f_{3}=\partial_{x}u_{3}$, we have
$$\displaystyle\partial_{x}F_{j}=\frac{m_{j}}{m_{1}}(\partial_{x}^{2}u_{1})(%
\overline{\partial_{x}u_{2}})(\partial_{x}u_{3})+\frac{R_{1}}{t},$$
(3.5)
where
$$R_{1}=-im_{j}J_{m_{j}}\Bigl{[}(\partial_{x}u_{1})(\overline{\partial_{x}u_{2}}%
)(\partial_{x}u_{3})\Bigr{]}+im_{j}(J_{m_{1}}\partial_{x}u_{1})(\overline{%
\partial_{x}u_{2}})(\partial_{x}u_{3}).$$
By applying Lemma 3.1 to the first term and using
Lemma 3.4, we see that
$$\displaystyle\|\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}R_{1}\|_{L^{\infty}}\leq$$
$$\displaystyle C\|J_{m_{1}}\partial_{x}u_{1}\|_{L^{2}}\|{\partial_{x}u_{2}}\|_{%
L^{2}}\|\partial_{x}u_{3}\|_{L^{\infty}}$$
$$\displaystyle+C\|\partial_{x}u_{1}\|_{L^{2}}\|{J_{m_{2}}\partial_{x}u_{2}}\|_{%
L^{2}}\|\partial_{x}u_{3}\|_{L^{\infty}}$$
$$\displaystyle+C\|\partial_{x}u_{1}\|_{L^{2}}\|{\partial_{x}u_{2}}\|_{L^{\infty%
}}\|J_{m_{3}}\partial_{x}u_{3}\|_{L^{2}}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{C}{t^{1/2}}\sum_{k=1}^{3}\bigl{(}\|u_{k}\|_{H^{1}}+\|J_{m_{%
k}}u_{k}\|_{H^{1}}\bigr{)}^{3},$$
(3.6)
where we have used the inequality
$\|\phi\|_{L^{\infty}}\leq Ct^{-1/2}\|\phi\|_{L^{2}}^{1/2}\|J_{m}\phi\|_{L^{2}}%
^{1/2}$ and
the commutation relation $[\partial_{x},J_{m}]=1$ in the
last line. As for the first term of (3.5), similar computations
as in the previous case lead to
$$\displaystyle\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\Bigl{[}(\partial_{x}^%
{2}u_{1})(\overline{\partial_{x}u_{2}})(\partial_{x}u_{3})\Bigr{]}$$
$$\displaystyle=\frac{1}{t}\mathcal{W}_{m_{j}}^{-1}\Bigl{[}(\mathcal{W}_{m_{1}}%
\alpha_{1}^{(2)})(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}})(\mathcal{W}%
_{m_{3}}\alpha_{3}^{(1)})\Bigr{]}$$
$$\displaystyle=\frac{1}{t}\alpha_{1}^{(2)}\overline{\alpha_{2}^{(1)}}\alpha_{3}%
^{(1)}+\frac{r_{1}}{t},$$
where
$$\displaystyle r_{1}=\mathcal{W}_{m_{j}}^{-1}\Bigl{[}(\mathcal{W}_{m_{1}}\alpha%
_{1}^{(2)})(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(1)}})(\mathcal{W}_{m_{3%
}}\alpha_{3}^{(1)})\Bigr{]}-\alpha_{1}^{(2)}\overline{\alpha_{2}^{(1)}}\alpha_%
{3}^{(1)}.$$
This can be estimated as follows:
$$\displaystyle\|r_{1}\|_{L^{\infty}}\leq Ct^{-1/4}\|\partial_{x}u_{1}\|_{H^{2}}%
\|u_{2}\|_{H^{2}}\|u_{3}\|_{H^{2}}.$$
Moreover, we observe that
$$\frac{im_{j}\xi}{t}p_{j}(\xi;\alpha)=\frac{m_{j}}{m_{1}}\frac{im_{1}\xi}{t}%
\alpha_{1}^{(1)}\overline{\alpha_{2}^{(1)}}\alpha_{3}^{(1)}=\frac{m_{j}}{m_{1}%
}\cdot\frac{1}{t}\alpha_{1}^{(2)}\overline{\alpha_{2}^{(1)}}\alpha_{3}^{(1)}.$$
Piecing them together, we arrive at
$$\displaystyle\left\|\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\partial_{x}F_{%
j}-\frac{im_{j}\xi}{t}p_{j}(\xi;\alpha)\right\|_{L^{\infty}_{\xi}}$$
$$\displaystyle=\frac{1}{t}\left\|\frac{m_{j}}{m_{1}}r_{1}+\mathcal{F}_{m_{j}}%
\mathcal{U}_{m_{j}}^{-1}R_{1}\right\|_{L^{\infty}}$$
$$\displaystyle\leq\frac{C}{t}\bigl{(}\|r_{1}\|_{L^{\infty}}+\|\mathcal{F}_{m_{j%
}}\mathcal{U}_{m_{j}}^{-1}R_{1}\|_{L^{\infty}}\bigr{)}$$
$$\displaystyle\leq\frac{C}{t^{5/4}}\sum_{k=1}^{3}\bigl{(}\|u_{k}(t,\cdot)\|_{H^%
{3}}+\|J_{m_{k}}u_{k}(t,\cdot)\|_{H^{1}}\bigr{)}^{3},$$
as desired.
Finally we consider the case of $l=2$.
By (3.3) with $m=m_{j}$, $\mu_{1}=m_{1}$ and $\mu_{2}=-m_{2}$, we have
$$\partial_{x}^{2}F_{j}=\frac{m_{j}^{2}}{-m_{1}m_{2}}(\partial_{x}^{2}u_{1})(%
\overline{\partial_{x}^{2}u_{2}})(\partial_{x}u_{3})+\frac{R_{2}}{t},$$
where
$$\displaystyle R_{2}=$$
$$\displaystyle-\frac{im_{j}^{2}}{m_{1}}J_{m_{j}}\Bigl{[}(\partial_{x}^{2}u_{1})%
(\overline{\partial_{x}u_{2}})(\partial_{x}u_{3})\Bigr{]}+\frac{im_{j}^{2}}{m_%
{1}}(\partial_{x}^{2}u_{1})(\overline{J_{m_{2}}\partial_{x}u_{2}})(\partial_{x%
}u_{3})$$
$$\displaystyle-im_{j}\partial_{x}J_{m_{j}}\bigl{[}(\partial_{x}u_{1})(\overline%
{\partial_{x}u_{2}})(\partial_{x}u_{3})\bigr{]}+im_{j}\partial_{x}\left[(J_{m_%
{1}}\partial_{x}u_{1})(\overline{\partial_{x}u_{2}})(\partial_{x}u_{3})\right].$$
As in the derivation of (3.6), we see that
$$\|\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}R_{2}\|_{L^{\infty}}\leq\frac{C}{%
t^{1/2}}\sum_{k=1}^{3}(\|u_{k}\|_{H^{2}}+\|J_{m_{k}}u_{k}\|_{H^{2}})^{3}.$$
Similarly to the previous cases, we can also show that
$$\displaystyle\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\Bigl{[}(\partial_{x}^%
{2}u_{1})(\overline{\partial_{x}^{2}u_{2}})(\partial_{x}u_{3})\Bigr{]}$$
$$\displaystyle=\frac{1}{t}\mathcal{W}_{m_{j}}^{-1}\Bigl{[}(\mathcal{W}_{m_{1}}%
\alpha_{1}^{(2)})(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(2)}})(\mathcal{W}%
_{m_{3}}\alpha_{3}^{(1)})\Bigr{]}$$
$$\displaystyle=\frac{-m_{1}m_{2}}{m_{j}^{2}}\frac{(im_{j}\xi)^{2}}{t}p_{j}(\xi;%
\alpha)+\frac{r_{2}}{t},$$
where
$$\displaystyle r_{2}=\mathcal{W}_{m_{j}}^{-1}\Bigl{[}(\mathcal{W}_{m_{1}}\alpha%
_{1}^{(2)})(\mathcal{W}_{-m_{2}}\overline{\alpha_{2}^{(2)}})(\mathcal{W}_{m_{3%
}}\alpha_{3}^{(1)})\Bigr{]}-\alpha_{1}^{(2)}\overline{\alpha_{2}^{(2)}}\alpha_%
{3}^{(1)}.$$
Note that
$$\|r_{2}\|_{L^{\infty}}\leq Ct^{-1/4}\|\partial_{x}u_{1}\|_{H^{2}}\|\partial_{x%
}u_{2}\|_{H^{2}}\|u_{3}\|_{H^{2}}.$$
Therefore we have
$$\displaystyle\left\|\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\partial_{x}^{2%
}F_{j}-\frac{(im_{j}\xi)^{2}}{t}p_{j}(\xi,\alpha)\right\|_{L^{\infty}_{\xi}}$$
$$\displaystyle\leq\frac{C}{t}\bigl{(}\|r_{2}\|_{L^{\infty}}+\|\mathcal{F}_{m_{j%
}}\mathcal{U}_{m_{j}}^{-1}R_{2}\|_{L^{\infty}}\bigr{)}$$
$$\displaystyle\leq\frac{C}{t^{5/4}}\sum_{k=1}^{3}\bigl{(}\|u_{k}(t,\cdot)\|_{H^%
{3}}+\|J_{m_{k}}u_{k}(t,\cdot)\|_{H^{2}}\bigr{)}^{3},$$
which completes the proof.
∎
4 Smoothing effect
In this section, we recall smoothing properties of the linear Schrödinger
equations. As is well known, the standard energy method causes a derivative
loss when the nonlinear term involves derivatives of the unknown functions.
Smoothing effect is a useful tool to overcome this obstacle.
Among various kinds of such techniques, we will follow the approach of
[2]. Let $\mathcal{H}$ be the Hilbert transform, that is,
$$\mathcal{H}\psi(x):=\frac{1}{\pi}\,\mathrm{p.v.}\int_{\mathbb{R}}\frac{\psi(y)%
}{x-y}dy.$$
With a non-negative weight function $\Phi(x)$ and a non-zero real constant
$m$, let us also define the operator $S_{\Phi,m}$ by
$$S_{\Phi,m}\psi(x):=\left\{\cosh\biggl{(}\int_{-\infty}^{x}\Phi(y)dy\biggr{)}%
\right\}\psi(x)-i\,\mathrm{sgn}(m)\left\{\sinh\biggl{(}\int_{-\infty}^{x}\Phi(%
y)dy\biggr{)}\right\}\mathcal{H}\psi(x).$$
Note that $S_{\Phi,m}$ is $L^{2}$-automorphism and that both
$\|S_{\Phi,m}\|_{L^{2}\to L^{2}}$, $\|S_{\Phi,m}^{-1}\|_{L^{2}\to L^{2}}$
are dominated by $C\exp(\|\Phi\|_{L^{1}})$. This operator enables us to
gain the half-derivative $|\partial_{x}|^{1/2}$.
More precisely, we have the following:
Lemma 4.1.
Let $m$, $\mu_{1},\ldots,\mu_{N}$ be non-zero real constants.
Let $v$ be a $\mathbb{C}$-valued smooth function of $(t,x)$, and
let $w=(w_{j})_{j\in I_{N}}$ be a $\mathbb{C}^{N}$-valued smooth function of $(t,x)$.
We set $\Phi=\eta(|w|^{2}+|\partial_{x}w|^{2})$ with $\eta\geq 1$, and
$S=S_{\Phi(t,\cdot),m}$.
Then we have
$$\displaystyle\frac{d}{dt}\|Sv(t)\|_{L^{2}}^{2}+$$
$$\displaystyle\frac{1}{|m|}\int_{\mathbb{R}}\Phi(t,x)\Bigl{|}S|\partial_{x}|^{1%
/2}v(t,x)\Bigr{|}^{2}dx$$
$$\displaystyle\leq 2\Bigl{|}\bigl{\langle}Sv(t),S\mathcal{L}_{m}v(t)\bigr{%
\rangle}_{L^{2}}\Bigr{|}+CB(t)\|v(t)\|_{L^{2}}^{2},$$
where
$$\displaystyle B(t)=e^{C\eta\|w\|_{H^{1}}^{2}}\left\{\eta\|w(t)\|_{W^{2,\infty}%
}^{2}+\eta^{3}\|w(t)\|_{W^{1,\infty}}^{6}+\eta\sum_{k\in I_{N}}\|w_{k}(t)\|_{H%
^{1}}\|\mathcal{L}_{\mu_{k}}w_{k}(t)\|_{H^{1}}\right\}$$
and the constant $C$ is independent of $\eta$.
We denote by $W^{s,\infty}$ the $L^{\infty}$-based Sobolev space of
order $s\in\mathbb{Z}_{\geq 0}$.
This lemma is essentially the same as Lemma 2.1 in [2], although
we need slight modifications to fit for our purpose. For the convenience of
the readers, we will give the proof of this lemma in the appendix.
By using Lemma 4.1 combined with the following
auxiliary lemma,
we can get rid of the derivative loss coming from the nonlinear terms.
Lemma 4.2.
Let $m_{1},\ldots,m_{N}$ be non-zero real constants.
Let $v=(v_{j})_{j\in I_{N}}$, $w=(w_{j})_{j\in I_{N}}$ be
$\mathbb{C}^{N}$-valued smooth functions of $x\in\mathbb{R}$.
Suppose that $q_{1,jk}$ and $q_{2,jk}$ are quadratic homogeneous
polynomials in $(w,\partial_{x}w,\overline{w},\overline{\partial_{x}w})$.
We set $\Phi=\eta(|w|^{2}+|\partial_{x}w|^{2})$ with $\eta\geq 1$, and
$S=S_{\Phi(t,\cdot),m}$ with $\eta\geq 1$, and
$S_{j}=S_{\Phi,m_{j}}$ for $j\in I_{N}$. Then we have
$$\displaystyle\sum_{j,k\in I_{N}}$$
$$\displaystyle\biggl{(}\left|\bigl{\langle}S_{j}v_{j},S_{j}\bigl{(}q_{1,jk}%
\partial_{x}v_{k}\bigr{)}\bigr{\rangle}_{L^{2}}\right|+\left|\bigl{\langle}S_{%
j}v_{j},S_{j}\bigl{(}q_{2,jk}\overline{\partial_{x}v_{k}}\bigr{)}\bigr{\rangle%
}_{L^{2}}\right|\biggr{)}$$
$$\displaystyle\leq\frac{C}{\eta}e^{C\eta\|w\|_{H^{1}}^{2}}\sum_{k\in I_{N}}\int%
_{\mathbb{R}}\Phi(x)\Bigl{|}S_{k}|\partial_{x}|^{1/2}v_{k}(x)\Bigr{|}^{2}dx$$
$$\displaystyle +Ce^{C\eta\|w\|_{H^{1}}^{2}}\bigl{(}1+\eta^{2}\|w\|_{H^{1}}^{%
4}+\eta^{2}\|w\|_{W^{1,\infty}}^{4}\bigr{)}\|w\|_{W^{2,\infty}}^{2}\|v\|_{L^{2%
}}^{2},$$
where the constant $C$ is independent of $\eta$.
We skip the proof of Lemma 4.2 because this is nothing more
than a paraphrase of Lemma 2.3 in [2].
5 A priori estimate
Let $T\in(0,+\infty]$, and
let $u=(u_{j})_{1\leq j\leq N}\in C([0,T);H^{3}\cap H^{2,1})$ be
a solution to (1.1) for $t\in[0,T)$.
As in Section 3, we set
$\alpha_{j}(t,\xi)=\mathcal{F}_{m_{j}}\Bigl{[}\mathcal{U}_{m_{j}}^{-1}u_{j}(t,%
\cdot)\Bigr{]}(\xi)$,
$\alpha(t,\xi)=(\alpha_{j}(t,\xi))_{j\in I_{N}}$, and define
$$\displaystyle E(T)=\sup_{0\leq t<T}\sum_{j\in I_{N}}\biggl{[}(1+t)^{-\frac{%
\gamma}{3}}\Bigl{(}\|u_{j}(t)\|_{H^{3}}+\|J_{m_{j}}u_{j}(t)\|_{H^{2}}\Bigr{)}+%
\sup_{\xi\in\mathbb{R}}\Bigl{(}\langle\xi\rangle^{2}|\alpha_{j}(t,\xi)|\Bigr{)%
}\biggr{]}$$
with $\gamma>0$. The goal of this section is to show the following:
Lemma 5.1.
Assume the conditions (a) and (b${}_{0}$) are satisfied.
Let $\gamma\in(0,1/4)$.
There exist positive constants
$\varepsilon_{1}$ and $K$ such that
$$\displaystyle E(T)\leq\varepsilon^{2/3}$$
(5.1)
implies
$$\displaystyle E(T)\leq K\varepsilon,$$
provided that $\varepsilon=\|\varphi\|_{H^{3}\cap H^{2,1}}\leq\varepsilon_{1}$.
The proof of this lemma will be divided into two parts.
5.1 $L^{2}$-estimates
In the first part, we consider the bounds for $\|u_{j}(t)\|_{H^{3}}$ and
$\|J_{m_{j}}u_{j}(t)\|_{H^{2}}$. It is enough to show
$$\displaystyle\sum_{j\in I_{N}}\sum_{l=0}^{1}\|J_{m_{j}}^{l}u_{j}(t)\|_{L^{2}}%
\leq C\varepsilon+C\varepsilon^{2}(1+t)^{\gamma/3}$$
(5.2)
and
$$\displaystyle\sum_{j\in I_{N}}\sum_{l=0}^{1}\|\partial_{x}^{3-l}J_{m_{j}}^{l}u%
_{j}(t)\|_{L^{2}}^{2}\leq C\varepsilon^{2}(1+t)^{2\gamma/3}$$
(5.3)
for $t\in[0,T)$ under the assumption (5.1).
First we remark that (5.1) implies a rough
$H^{1}$-bound
$$\displaystyle\|u_{j}(t)\|_{H^{1}}\leq C\|\alpha_{j}(t)\|_{H^{0,1}}\leq C\left(%
\int_{\mathbb{R}}\frac{d\xi}{\langle\xi\rangle^{2}}\right)^{1/2}\sup_{\xi\in%
\mathbb{R}}\left(\langle\xi\rangle^{2}|\alpha_{j}(t,\xi)|\right)\leq C%
\varepsilon^{2/3}$$
(5.4)
for $t\in[0,T)$.
We also deduce from (5.1) that
$$\displaystyle\|u_{j}(t)\|_{W^{2,\infty}}\leq\frac{C\varepsilon^{2/3}}{(1+t)^{1%
/2}}$$
for $t\in[0,T)$. Indeed, it follows from Lemma 3.3
and the relation $[\partial_{x},J_{m_{j}}]=1$ that
$$\displaystyle\|u_{j}(t)\|_{W^{2,\infty}}\leq\frac{C}{t^{1/2}}\sup_{\xi\in%
\mathbb{R}}|\langle\xi\rangle^{2}\alpha_{j}(t,\xi)|+\frac{C}{t^{3/4}}\bigl{(}%
\|u_{j}(t)\|_{H^{2}}+\|J_{m_{j}}u_{j}(t)\|_{H^{2}}\bigr{)}\leq\frac{C%
\varepsilon^{2/3}}{t^{1/2}}$$
for $t\geq 1$, and $H^{1}(\mathbb{R}^{1})\hookrightarrow L^{\infty}(\mathbb{R}^{1})$ yields
$\|u_{j}(t)\|_{W^{2,\infty}}\leq C\|u_{j}(t)\|_{H^{3}}\leq C\varepsilon^{2/3}$
for $t\leq 1$.
Now we consider the easier estimate (5.2). It follows from the
standard energy method that
$$\displaystyle\frac{d}{dt}\|u_{j}(t)\|_{L^{2}}$$
$$\displaystyle\leq\|F_{j}(u(t),\partial_{x}u(t))\|_{L^{2}}$$
$$\displaystyle\leq C\|u(t)\|_{W^{1,\infty}}^{2}\|u(t)\|_{H^{1}}$$
$$\displaystyle\leq C\left(\frac{\varepsilon^{2/3}}{(1+t)^{1/2}}\right)^{2}\cdot
C%
\varepsilon^{2/3}$$
$$\displaystyle\leq\frac{C\varepsilon^{2}}{1+t}.$$
Also we see from Lemma 3.1 that
$$\mathcal{L}_{m_{j}}J_{m_{j}}u_{j}=\sum_{k\in I_{N}}\Bigl{(}q_{1,jk}J_{m_{k}}%
\partial_{x}u_{k}+q_{2,jk}\overline{J_{m_{k}}\partial_{x}u_{k}}+q_{3,jk}J_{m_{%
k}}u_{k}+q_{4,jk}\overline{J_{m_{k}}u_{k}}\Bigr{)},$$
where $q_{1,jk},\ldots,q_{4,jk}$ are quadratic homogeneous
polynomials in $(u,\partial_{x}u,\overline{u},\overline{\partial_{x}u})$.
Then the standard energy method again implies
$$\frac{d}{dt}\|J_{m_{j}}u_{j}(t)\|_{L^{2}}\leq C\|u\|_{W^{1,\infty}}^{2}\sum_{k%
\in I_{N}}(\|u_{k}\|_{H^{1}}+\|J_{m_{k}}u_{k}\|_{H^{1}})\leq\frac{C\varepsilon%
^{2}}{(1+t)^{1-\gamma/3}}.$$
These lead to (5.2).
Next we consider (5.3). We set
$v_{jl}=\partial_{x}^{3-l}J_{m_{j}}^{l}u_{j}$ for $l\in\{0,1\}$ and $j\in I_{N}$.
We apply Lemma 4.1 with
$m=m_{j}$, $\mu_{k}=m_{k}$, $v=v_{jl}$, $w=u$, $\eta=\varepsilon^{-2/3}$.
Then we obtain
$$\displaystyle\frac{d}{dt}\|S_{j}v_{jl}(t)\|_{L^{2}}^{2}+\frac{1}{|m_{j}|}\int_%
{\mathbb{R}}\Phi(t,x)\Bigl{|}S_{j}|\partial_{x}|^{1/2}v_{jl}(t)\Bigr{|}^{2}dx$$
$$\displaystyle\leq 2\left|\bigl{\langle}S_{j}v_{jl},S_{j}\partial_{x}^{3-l}J_{m%
_{j}}^{l}F_{j}(u,\partial_{x}u)\bigr{\rangle}_{L^{2}}\right|+CB(t)\|v_{jl}(t)%
\|_{L^{2}}^{2},$$
(5.5)
where
$$\displaystyle B(t)$$
$$\displaystyle=e^{\frac{C}{\varepsilon^{2/3}}\|u\|_{H^{1}}^{2}}\left(%
\varepsilon^{-2/3}\|u\|_{W^{2,\infty}}^{2}+\varepsilon^{-2}\|u\|_{W^{1,\infty}%
}^{6}+\varepsilon^{-2/3}\sum_{k\in I_{N}}\|u_{k}\|_{H^{1}}\|F_{k}(u,\partial_{%
x}u)\|_{H^{1}}\right)$$
$$\displaystyle\leq\frac{C\varepsilon^{2/3}}{1+t}.$$
To estimate the first term of the right-hand side of (5.5),
we use Lemma 3.1 and the usual Leibniz rule to split
$\partial_{x}^{3-l}J_{m_{j}}^{l}F_{j}(u,\partial_{x}u)$ into the following form:
$$\sum_{k\in I_{N}}\Bigl{(}g_{1,jkl}\partial_{x}v_{kl}+g_{2,jkl}\overline{%
\partial_{x}v_{kl}}\Bigr{)}+h_{jl},$$
where $g_{1,jkl}$ and $g_{2,jkl}$ are quadratic homogeneous
polynomials in $(u,\partial_{x}u,\overline{u},\overline{\partial_{x}u})$, and $h_{jl}$
is a cubic term satisfying
$$\displaystyle\|h_{jl}\|_{L^{2}}\leq C\|u(t)\|_{W^{2,\infty}}^{2}\sum_{k\in I_{%
N}}(\|u_{k}(t)\|_{H^{3}}+\|J_{m_{k}}u_{k}(t)\|_{H^{2}})\leq\frac{C\varepsilon^%
{2}}{(1+t)^{1-\gamma/3}}.$$
Then Lemma 4.2 and the $L^{2}$-automorphism
of $S_{j}$ lead to
$$\displaystyle\sum_{j\in I_{N}}\left|\bigl{\langle}S_{j}v_{jl},S_{j}\partial_{x%
}^{3-l}J_{m_{j}}^{l}F_{j}(u,\partial_{x}u)\bigr{\rangle}_{L^{2}}\right|$$
$$\displaystyle\leq\sum_{j,k\in I_{N}}\biggl{(}\left|\bigl{\langle}S_{j}v_{jl},S%
_{j}\bigl{(}g_{1,jkl}\partial_{x}v_{kl}\bigr{)}\bigr{\rangle}_{L^{2}}\right|+%
\left|\bigl{\langle}S_{j}v_{jl},S_{j}\bigl{(}g_{2,jkl}\overline{\partial_{x}v_%
{kl}}\bigr{)}\bigr{\rangle}_{L^{2}}\right|\biggr{)}+\sum_{j\in I_{N}}\|S_{j}v_%
{jl}\|_{L^{2}}\|S_{j}h_{jl}\|_{L^{2}}$$
$$\displaystyle\leq C\varepsilon^{2/3}e^{\frac{C}{\varepsilon^{2/3}}\|u\|_{H^{1}%
}^{2}}\sum_{k\in I_{N}}\int_{\mathbb{R}}\Phi(t,x)\Bigl{|}S_{k}|\partial_{x}|^{%
1/2}v_{kl}(t,x)\Bigr{|}^{2}dx$$
$$\displaystyle +Ce^{\frac{C}{\varepsilon^{2/3}}\|u\|_{H^{1}}^{2}}\bigl{(}1%
+\varepsilon^{-4/3}\|u\|_{H^{1}}^{4}+\varepsilon^{-4/3}\|u\|_{W^{1,\infty}}^{4%
}\bigr{)}\|u\|_{W^{2,\infty}}^{2}\sum_{k\in I_{N}}\|v_{kl}\|_{L^{2}}^{2}$$
$$\displaystyle +Ce^{\frac{C}{\varepsilon^{2/3}}\|u\|_{H^{1}}^{2}}\sum_{j%
\in I_{N}}\|v_{jl}\|_{L^{2}}\|h_{jl}\|_{L^{2}}$$
$$\displaystyle\leq C_{0}\varepsilon^{2/3}\sum_{k\in I_{N}}\int_{\mathbb{R}}\Phi%
(t,x)\left|S_{k}|\partial_{x}|^{1/2}v_{kl}(t,x)\right|^{2}dx+\frac{C%
\varepsilon^{8/3}}{(1+t)^{1-2\gamma/3}}$$
with some positive constant $C_{0}$ not depending on $\varepsilon$.
Summing up, we obtain
$$\displaystyle\frac{d}{dt}\sum_{j\in I_{N}}\|S_{j}v_{jl}(t)\|_{L^{2}}^{2}$$
$$\displaystyle\leq\sum_{k\in I_{N}}\left(2C_{0}\varepsilon^{2/3}-\frac{1}{|m_{k%
}|}\right)\int_{\mathbb{R}}\Phi(t,x)\Bigl{|}S_{k}|\partial_{x}|^{1/2}v_{kl}(t,%
x)\Bigr{|}^{2}dx$$
$$\displaystyle +\frac{C\varepsilon^{8/3}}{(1+t)^{1-2\gamma/3}}+\frac{C%
\varepsilon^{2/3}}{1+t}\cdot\bigl{(}C\varepsilon^{2/3}(1+t)^{\gamma/3}\bigr{)}%
^{2}$$
$$\displaystyle\leq\frac{C\varepsilon^{2}}{(1+t)^{1-2\gamma/3}},$$
provided that
$$2C_{0}\varepsilon^{2/3}\leq\frac{1}{\displaystyle{\min_{1\leq k\leq N}|m_{k}|}}.$$
Integrating with respect to $t$, we have
$$\sum_{j\in I_{N}}\|S_{j}v_{jl}(t)\|_{L^{2}}^{2}\leq C\varepsilon^{2}+C%
\varepsilon^{2}(1+t)^{2\gamma/3}\leq C\varepsilon^{2}(1+t)^{2\gamma/3},$$
whence
$$\sum_{j\in I_{N}}\sum_{l=0}^{1}\|\partial_{x}^{3-l}J_{m_{j}}^{l}u_{j}(t)\|_{L^%
{2}}^{2}\leq e^{C\varepsilon^{-2/3}\|u(t)\|_{H^{1}}^{2}}\sum_{j\in I_{N}}\sum_%
{l=0}^{1}\|S_{j}v_{jl}(t)\|_{L^{2}}^{2}\leq C\varepsilon^{2}(1+t)^{2\gamma/3},$$
as required.
∎
5.2 Estimates for $\alpha_{j}$
In the second part, we are going to show
$\langle\xi\rangle^{2}|\alpha(t,\xi)|\leq C\varepsilon$
for $(t,\xi)\in[0,T)\times\mathbb{R}$
under the assumption (5.1).
If $t\in[0,1]$, the Sobolev imbedding yields
this estimate immediately.
Hence we have only to consider the case of $t\in[1,T)$.
We set
$$\rho_{j}(t,\xi)=\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\big{[}F_{j}(u,%
\partial_{x}u)\bigr{]}-\frac{1}{t}p_{j}(\xi;\alpha(t,\xi))$$
and $\rho=(\rho_{j})_{j\in I_{N}}$, so that
$$\displaystyle i\partial_{t}\alpha_{j}(t,\xi)$$
$$\displaystyle=\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\bigl{[}\mathcal{L}_{%
m_{j}}u_{j}\bigr{]}=\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}\bigl{[}F_{j}(u%
,\partial_{x}u)\bigr{]}$$
$$\displaystyle=\frac{1}{t}p_{j}(\xi;\alpha(t,\xi))+\rho_{j}(t,\xi).$$
(5.6)
By Proposition 3.1, we have
$$\displaystyle|\rho_{j}(t,\xi)|$$
$$\displaystyle\leq\frac{C}{\langle\xi\rangle^{2}}\sum_{l=0}^{2}\bigl{|}(im_{j}%
\xi)^{l}\rho_{j}(t,\xi)\bigr{|}$$
$$\displaystyle=\frac{C}{\langle\xi\rangle^{2}}\sum_{l=0}^{2}\left|\mathcal{F}_{%
m_{j}}\mathcal{U}_{m_{j}}^{-1}\big{[}\partial_{x}^{l}F_{j}(u,\partial_{x}u)%
\bigr{]}-\frac{(im_{j}\xi)^{l}}{t}p_{j}(\xi;\alpha(t,\xi))\right|$$
$$\displaystyle\leq\frac{C}{\langle\xi\rangle^{2}}\cdot\frac{C}{t^{5/4}}\left(E(%
T)t^{\frac{\gamma}{3}}\right)^{3}$$
$$\displaystyle\leq\frac{C\varepsilon^{2}}{\langle\xi\rangle^{2}t^{5/4-\gamma}}$$
for $t\geq 1$ and $\xi\in\mathbb{R}$,
which shows that $\rho_{j}(t,\xi)$ has enough decay rates both in
$t$ and $\xi$.
Now we put $\nu(t,\xi)=\sqrt{\langle\alpha(t,\xi),A\alpha(t,\xi)\rangle_{\mathbb{C}^{N}}}$,
where $A$ is the positive Hermitian matrix appearing in the condition (b${}_{0}$).
Remark that
$$\sqrt{\kappa_{*}}|\alpha(t,\xi)|\leq\nu(t,\xi)\leq\sqrt{\kappa^{*}}|\alpha(t,%
\xi)|,$$
where $\kappa_{*}$ and $\kappa^{*}$ are the smallest and largest
eigenvalues of $A$, respectively.
It follows from (b${}_{0}$) that
$$\displaystyle\partial_{t}\nu(t,\xi)^{2}$$
$$\displaystyle=2\operatorname{\rm Im}\langle i\partial_{t}\alpha(t,\xi),A\alpha%
(t,\xi)\rangle_{\mathbb{C}^{N}}$$
$$\displaystyle=\frac{2}{t}\operatorname{\rm Im}\langle p(\xi;\alpha(t,\xi)),A%
\alpha(t,\xi)\rangle_{\mathbb{C}^{N}}+2\operatorname{\rm Im}\langle\rho(t,\xi)%
,A\alpha(t,\xi)\rangle_{\mathbb{C}^{N}}$$
$$\displaystyle\leq 0+C|\rho(t,\xi)|\nu(t,\xi),$$
which leads to
$$\displaystyle\nu(t,\xi)\leq\nu(1,\xi)+{C}\int_{1}^{t}|\rho(\tau,\xi)|d\tau\leq%
\frac{C\varepsilon}{\langle\xi\rangle^{2}}+\frac{C\varepsilon^{2}}{\langle\xi%
\rangle^{2}}\int_{1}^{\infty}\frac{d\tau}{\tau^{5/4-\gamma}}\leq\frac{C%
\varepsilon}{\langle\xi\rangle^{2}},$$
Therefore we have
$$\langle\xi\rangle^{2}|\alpha_{j}(t,\xi)|\leq C\langle\xi\rangle^{2}\nu(t,\xi)%
\leq C\varepsilon,$$
as required.∎
6 Proof of the main theorems
Now we are in a position to prove Theorems 2.1 –
2.4.
6.1 Proof of Theorem 2.1
First let us recall the local existence theorem.
For fixed $t_{0}\geq 0$, let us consider the initial value
problem
$$\displaystyle\left\{\begin{array}[]{cl}\mathcal{L}_{m_{j}}u_{j}=F_{j}(u,%
\partial_{x}u),&t>t_{0},\ x\in\mathbb{R},\ j\in I_{N},\\
u_{j}(t_{0},x)=\psi_{j}(x),&x\in\mathbb{R},\ j\in I_{N}.\end{array}\right.$$
(6.1)
Lemma 6.1.
Let $\psi=(\psi_{j})_{j\in I_{N}}\in H^{3}\cap H^{2,1}$.
There exists a positive constant $\varepsilon_{0}$, which is independent of $t_{0}$,
such that the following holds:
for any $\underline{\varepsilon}\in(0,\varepsilon_{0})$ and $M\in(0,\infty)$,
one can choose a positive constant $\tau^{*}=\tau^{*}(\underline{\varepsilon},M)$,
which is independent of $t_{0}$,
such that (6.1) admits a unique solution
$u=(u_{j})_{j\in I_{N}}\in C([t_{0},t_{0}+\tau^{*}];H^{3}\cap H^{2,1})$,
provided that
$$\|\psi\|_{H^{1}}\leq\underline{\varepsilon}\quad\mbox{and}\quad\sum_{l=0}^{1}%
\sum_{j\in I_{N}}\Bigl{\|}\bigl{(}x+i\frac{t_{0}}{m_{j}}\partial_{x}\bigr{)}^{%
l}\psi_{j}\Bigr{\|}_{H^{3-l}}\leq M.$$
We omit the proof of this lemma because it is standard
(see e.g., Appendix of [4] for the proof of similar lemma
in the quadratic nonlinear case).
Now we are going to prove the global existence by the so-called
bootstrap argument.
Let $T^{*}$ be the supremum of all $T\in(0,\infty]$ such that
the problem (1.1) admits a unique solution
$u\in C([0,T);H^{3}\cap H^{2,1})$.
By Lemma 6.1 with $t_{0}=0$, we have $T^{*}>0$ if
$\|\varphi\|_{H^{1}}\leq\varepsilon<\varepsilon_{0}$.
We also set
$$T_{*}=\sup\bigl{\{}\tau\in[0,T^{*})\,|\,E(\tau)\leq\varepsilon^{2/3}\bigr{\}}.$$
Note that $T_{*}>0$ because of the continuity of
$[0,T^{*})\ni\tau\mapsto E(\tau)$ and
$\|\varphi\|_{H^{3}\cap H^{2,1}}=\varepsilon\leq\frac{1}{2}\varepsilon^{2/3}$
if $\varepsilon\leq 1/8$.
We claim that $T_{*}=T^{*}$ if $\varepsilon$ is small enough. Indeed, if
$T_{*}<T^{*}$, Lemma 5.1 with $T=T_{*}$ yields
$$E(T_{*})\leq K\varepsilon\leq\frac{1}{2}\varepsilon^{2/3}$$
for $\varepsilon\leq\varepsilon_{2}:=\min\{\varepsilon_{1},1/(2K)^{3}\}$,
where $K$ and $\varepsilon_{1}$ are mentioned in Lemma 5.1.
By the continuity of $[0,T^{*})\ni\tau\mapsto E(\tau)$, we can take
$T^{\flat}\in(T_{*},T^{*})$ such that
$E(T^{\flat})\leq\varepsilon^{2/3}$, which contradicts the
definition of $T_{*}$. Therefore we must have $T_{*}=T^{*}$.
By using Lemma 5.1 with $T=T^{*}$ again, we see that
$$\sum_{l=0}^{1}\sum_{j\in I_{N}}\|J_{m_{j}}^{l}u_{j}(t,\cdot)\|_{H^{3-l}}\leq K%
\varepsilon(1+t)^{\frac{\gamma}{3}},\qquad\sum_{j\in I_{N}}\sup_{\xi\in\mathbb%
{R}}\Bigl{(}\langle\xi\rangle^{2}|\alpha_{j}(t,\xi)|\Bigr{)}\leq K\varepsilon$$
for $t\in[0,T^{*})$. In particular we have
$$\sup_{t\in[0,T^{*})}\|u(t)\|_{H^{1}}\leq C\sup_{(t,\xi)\in[0,T^{*})\times%
\mathbb{R}}\Bigl{(}\langle\xi\rangle^{2}|\alpha(t,\xi)|\Bigr{)}\leq C^{\flat}\varepsilon$$
with some $C^{\flat}>0$.
Next we assume $T^{*}<\infty$. Then, by setting
$\varepsilon_{3}=\min\{\varepsilon_{2},\varepsilon_{0}/2C^{\flat}\}$
and $M=K\varepsilon_{3}(1+T^{*})^{\gamma/3}$, we have
$$\sup_{t\in[0,T^{*})}\sum_{l=0}^{1}\sum_{j\in I_{N}}\|J_{m_{j}}^{l}u_{j}(t,%
\cdot)\|_{H^{3-l}}\leq M$$
as well as
$$\sup_{t\in[0,T^{*})}\|u(t)\|_{H^{1}}\leq\varepsilon_{0}/2<\varepsilon_{0}$$
for $\varepsilon\leq\varepsilon_{3}$. By Lemma 6.1,
there exists $\tau^{*}>0$ such that (1.1) admits the solution
$u\in C([0,T^{*}+\tau^{*});H^{3}\cap H^{2,1})$. This contradicts the definition
of $T^{*}$, which means $T^{*}=+\infty$ for $\varepsilon\in(0,\varepsilon_{3}]$.
Moreover, we have
$$\|u(t)\|_{L^{2}}\leq C\sup_{\xi\in\mathbb{R}}|\langle\xi\rangle\alpha(t,\xi)|%
\leq C\varepsilon.$$
By using Lemma 3.3 and the inequality obtained above,
we also have
$$|u_{j}(t,x)|\leq\frac{C}{t^{1/2}}|\alpha_{j}(t,\xi)|+\frac{C}{t^{3/4}}\bigl{(}%
\|u_{j}(t)\|_{L^{2}}+\|J_{m_{j}}u_{j}(t)\|_{L^{2}}\bigr{)}\leq\frac{C%
\varepsilon}{t^{1/2}}$$
for $t\geq 1$ and $j\in I_{N}$.
This completes the proof of Theorem 2.1.
∎
6.2 Proof of Theorems 2.2 and 2.3
The proof of Theorems 2.2 and 2.3 heavily
relies on the following lemma due to [6].
Note that special cases of this lemma have been used previously in
[5] and [10] less explicitly.
Lemma 6.2 ([6]).
Let $C_{0}>0$, $C_{1}\geq 0$, $p>1$ and $q>1$. Suppose that
$\Psi(t)$ satisfies
$$\frac{d\Psi}{dt}(t)\leq\frac{-C_{0}}{t}|\Psi(t)|^{p}+\frac{C_{1}}{t^{q}}$$
for $t\geq 2$. Then we have
$$\Psi(t)\leq\frac{C_{2}}{(\log t)^{p^{*}-1}}$$
for $t\geq 2$, where $p^{*}$ is the Hölder conjugate of $p$
(i.e., $1/p+1/p^{*}=1$), and
$$C_{2}=\left(\frac{p^{*}}{C_{0}p}\right)^{p^{*}-1}+(\log 2)^{p^{*}-1}\Psi(2)+%
\frac{C_{1}}{\log 2}\int_{2}^{\infty}\frac{(\log\tau)^{p^{*}}}{\tau^{q}}d\tau.$$
With $\xi\in\mathbb{R}$ fixed, we set
$\Psi(t)=\langle\alpha(t,\xi),A\alpha(t,\xi)\rangle_{\mathbb{C}^{N}}$,
where $A$ is the positive Hermitian matrix appearing in the condition (b${}_{1}$).
Then we deduce from (5.6) that $\Psi$ satisfies
$$\frac{d\Psi}{dt}(t)\leq\frac{-2C_{*}}{t}|\alpha(t)|^{4}+C|\rho(t,\xi)||\alpha(%
t,\xi)|\leq\frac{-2C_{*}/\kappa_{*}^{2}}{t}|\Psi(t)|^{2}+\frac{C\varepsilon^{3%
}}{\langle\xi\rangle^{4}t^{5/4-\gamma}}$$
for $t\geq 2$, where $C_{*}$ is the positive constant appearing in
the condition (b${}_{1}$) and $\kappa_{*}$ is the smallest eigenvalue of $A$.
We also have $\Psi(2)\leq C|\alpha(2,\xi)|^{2}\leq C\varepsilon^{2}\langle\xi\rangle^{-4}$.
So we can apply Lemma 6.2 with $p=2$, $q=5/4-\gamma$ to obtain
$$|\alpha(t,\xi)|^{2}\leq C\Psi(t)\leq\frac{1}{(\log t)^{2-1}}\left(\frac{\kappa%
_{*}^{2}}{2C_{*}}+\frac{C\varepsilon^{2}}{\langle\xi\rangle^{4}}\right)\leq%
\frac{C}{\log t}.$$
From Lemma 3.3 it follows that
$$\displaystyle|u_{j}(t,x)|$$
$$\displaystyle\leq\frac{C}{t^{1/2}}\sup_{\xi\in\mathbb{R}}|\alpha_{j}(t,\xi)|+%
\frac{C}{t^{3/4}}\bigl{(}\|u_{j}(t)\|_{L^{2}}+\|J_{m_{j}}u_{j}(t)\|_{L^{2}}%
\bigr{)}$$
$$\displaystyle\leq\frac{C}{(t\log t)^{1/2}}+\frac{C\varepsilon}{t^{3/4-\gamma/3}}$$
$$\displaystyle\leq\frac{C}{(t\log t)^{1/2}},$$
for $t\geq 2$, $x\in\mathbb{R}$ and $j\in I_{N}$.
On the other hand, we already know that
$|u(t,x)|\leq C\varepsilon(1+t)^{-1/2}$ for $t\geq 0$. Hence we arrive at
$$(1+t)(1+\varepsilon^{2}\log(t+2))|u(t,x)|^{2}\leq C\varepsilon^{2}$$
for $t\geq 0$,
which implies the desired pointwise decay estimate.
By the Fatou lemma we also have
$$\limsup_{t\to+\infty}\|\alpha_{j}(t)\|_{L^{2}}^{2}\leq\int_{\mathbb{R}}\limsup%
_{t\to+\infty}|\alpha_{j}(t,\xi)|^{2}d\xi=0,$$
which leads to decay of $\|u_{j}(t)\|_{L^{2}}$ as $t\to+\infty$,
as stated in Theorem 2.2.
Under the stronger condition (b${}_{2}$), we have
$$\frac{d\Psi}{dt}(t)\leq\frac{-2C_{**}\langle\xi\rangle^{2}/\kappa_{*}^{2}}{t}|%
\Psi(t)|^{2}+\frac{C\varepsilon^{3}}{\langle\xi\rangle^{4}t^{5/4-\gamma}}$$
for $t\geq 2$. Therefore Lemma 6.2 again yields
$$|\alpha(t,\xi)|^{2}\leq\frac{1}{\log t}\left(\frac{\kappa_{*}^{2}}{2C_{**}%
\langle\xi\rangle^{2}}+\frac{C\varepsilon^{2}}{\langle\xi\rangle^{4}}\right)%
\leq\frac{C}{\langle\xi\rangle^{2}\log t},$$
whence
$$\|u(t)\|_{L^{2}}=\|\alpha(t)\|_{L^{2}}\leq C\sup_{\xi\in\mathbb{R}}\bigl{(}%
\langle\xi\rangle|\alpha(t,\xi)|\bigr{)}\leq\frac{C}{\sqrt{\log t}}$$
for $t\geq 2$. This yields Theorem 2.3.
∎
6.3 Proof of Theorem 2.4
For given $\delta>0$, we set $\gamma=\min\{\delta,1/5\}\in(0,1/4)$.
Remember that we have already shown that
$$|\alpha_{j}(t,\xi)|\leq\frac{C\varepsilon}{\langle\xi\rangle^{2}},\qquad|\rho_%
{j}(t,\xi)|\leq\frac{C\varepsilon^{2}}{\langle\xi\rangle^{2}t^{5/4-\gamma}}$$
for $t\geq 1$, $\xi\in\mathbb{R}$ and $j\in I_{N}$.
These estimates allow us to define
$\alpha^{+}=(\alpha_{j}^{+})_{j\in I_{N}}\in L^{2}\cap L^{\infty}$ by
$$\alpha_{j}^{+}(\xi):=\alpha_{j}(1,\xi)-i\int_{1}^{\infty}\rho_{j}(t^{\prime},%
\xi)dt^{\prime}.$$
On the other hand, the condition (b${}_{3}$) and (5.6) lead to
$$\alpha_{j}(t,\xi)=\alpha_{j}(1,\xi)-i\int_{1}^{t}\rho_{j}(t^{\prime},\xi)dt^{%
\prime},$$
whence
$$\|\alpha_{j}(t)-\alpha_{j}^{+}\|_{L^{2}\cap L^{\infty}}\leq\int_{t}^{\infty}\|%
\rho_{j}(t^{\prime},\cdot)\|_{L^{2}\cap L^{\infty}}dt^{\prime}\leq C%
\varepsilon^{2}t^{-1/4+{\gamma}}.$$
Now we set
$\varphi_{j}^{+}:=\mathcal{F}_{m_{j}}^{-1}\alpha_{j}^{+}$.
Then we have
$$\displaystyle\|u_{j}(t)-\mathcal{U}_{m_{j}}\varphi_{j}^{+}\|_{L^{2}}$$
$$\displaystyle=\|\mathcal{F}_{m_{j}}\mathcal{U}_{m_{j}}^{-1}u_{j}(t)-\mathcal{F%
}_{m_{j}}\varphi_{j}^{+}\|_{L^{2}}$$
$$\displaystyle=\|\alpha_{j}(t)-\alpha_{j}^{+}\|_{L^{2}}$$
$$\displaystyle\leq C\varepsilon^{2}t^{-1/4+\gamma}.$$
By Lemma 3.3 and the inequality obtained above, we also have
$$\displaystyle\|u_{j}(t)-\mathcal{M}_{m_{j}}\mathcal{D}\mathcal{F}_{m_{j}}%
\varphi_{j}^{+}\|_{L^{\infty}}$$
$$\displaystyle\leq\|u_{j}(t)-\mathcal{M}_{m_{j}}\mathcal{D}\mathcal{F}_{m_{j}}%
\mathcal{U}_{m_{j}}^{-1}u_{j}(t)\|_{L^{\infty}}+\|\mathcal{M}_{m_{j}}\mathcal{%
D}(\alpha_{j}(t)-\alpha^{+})\|_{L^{\infty}}$$
$$\displaystyle\leq Ct^{-3/4}(\|u_{j}(t)\|_{L^{2}}+\|J_{m_{j}}u_{j}(t)\|_{L^{2}}%
)+Ct^{-1/2}\|\alpha_{j}(t)-\alpha_{j}^{+}\|_{L^{\infty}}$$
$$\displaystyle\leq C\varepsilon t^{-3/4+\gamma/3}+C\varepsilon^{2}t^{-1/2-1/4+\gamma}$$
$$\displaystyle\leq C\varepsilon t^{-3/4+\delta}$$
for $t\geq 1$.
∎
Remark 6.1.
We put $\varphi_{j}=\varepsilon^{\prime}\psi_{j}$ with $\psi_{j}\not\equiv 0$ and
$\varepsilon^{\prime}\in(0,\varepsilon^{*}]$,
where $\varepsilon^{*}>0$ is chosen suitably small so that
Theorem 2.4 is valid. Then we can check that
the corresponding $\varphi_{j}^{+}$ satisfies
$$\|\varphi_{j}^{+}\|_{L^{2}}=\|\alpha_{j}^{+}\|_{L^{2}}\geq\varepsilon^{\prime}%
\|\psi_{j}\|_{L^{2}}-C^{*}(\varepsilon^{\prime})^{3}$$
with some $C^{*}>0$. Therefore $\varphi_{j}^{+}$ does not identically vanish
if $\varepsilon^{\prime}<\min\{\varepsilon^{*},\sqrt{\|\psi_{j}\|_{L^{2}}/C^{*}}\}$.
Appendix A Proof of Lemma 4.1
In this appendix, we shall give the proof of Lemma 4.1
in the similar way as Section 2 of [2] with slight modifications.
We first state the following useful lemma without proof, which is a
special case of Lemma 2.1 of [2].
Lemma A.1.
We have
$$\left\|\Bigl{[}|\partial_{x}|^{1/2},g\Bigr{]}f\right\|_{L^{2}}+\left\|\Bigl{[}%
|\partial_{x}|^{1/2}\mathcal{H},g\Bigr{]}f\right\|_{L^{2}}\leq C\|g\|_{W^{1,%
\infty}}\|f\|_{L^{2}}.$$
Proof of Lemma 4.1.
As in the standard energy method, we compute
$$\frac{1}{2}\frac{d}{dt}\|Sv\|_{L^{2}}^{2}=\operatorname{\rm Im}\langle\mathcal%
{L}_{m}Sv,Sv\rangle_{L^{2}}=\operatorname{\rm Im}\langle S\mathcal{L}_{m}v,Sv%
\rangle_{L^{2}}+\operatorname{\rm Im}\Bigl{\langle}[\mathcal{L}_{m},S]v,Sv%
\Bigr{\rangle}_{L^{2}}.$$
We also note that
$$[\mathcal{L}_{m},S]v=-\frac{i}{|m|}\Phi S|\partial_{x}|v+Q,$$
where
$$Q=\frac{1}{2m}\Phi^{2}Sv-\frac{i}{2|m|}(\partial_{x}\Phi)S\mathcal{H}v+\mathrm%
{sgn}(m)\left(\int_{-\infty}^{x}\partial_{t}\Phi(t,y)dy\right)S\mathcal{H}v.$$
Remark that $|\partial_{x}|=\mathcal{H}\partial_{x}=\partial_{x}\mathcal{H}$,
$\mathcal{H}^{2}=-1$, and that $\mathcal{H}$ is $L^{2}$-bounded.
Now we set $w_{k}^{(l)}=\partial_{x}^{l}w_{k}$ for $l\in\mathbb{Z}_{\geq 0}$.
Then, since
$$\displaystyle\partial_{t}\Phi$$
$$\displaystyle=2\eta\sum_{l=0}^{1}\sum_{k\in I_{N}}\operatorname{\rm Im}\Bigl{%
\{}(i\partial_{t}w_{k}^{(l)})\overline{w_{k}^{(l)}}\Bigr{\}}$$
$$\displaystyle=2\eta\sum_{l=0}^{1}\sum_{k\in I_{N}}\operatorname{\rm Im}\left\{%
\biggl{(}-\frac{1}{2\mu_{k}}\partial_{x}^{2}w_{k}^{(l)}+\partial_{x}^{l}%
\mathcal{L}_{\mu_{k}}w_{k}\biggr{)}\overline{w_{k}^{(l)}}\right\}$$
$$\displaystyle=2\eta\sum_{l=0}^{1}\sum_{k\in I_{N}}\operatorname{\rm Im}\left\{%
\partial_{x}\biggl{(}-\frac{1}{2\mu_{k}}(\partial_{x}w_{k}^{(l)})\overline{w_{%
k}^{(l)}}\biggr{)}+\frac{1}{2\mu_{k}}\bigl{|}\partial_{x}w_{k}^{(l)}\bigr{|}^{%
2}+(\partial_{x}^{l}\mathcal{L}_{\mu_{k}}w_{k})\overline{w_{k}^{(l)}}\right\}$$
$$\displaystyle=2\eta\sum_{l=0}^{1}\sum_{k\in I_{N}}\operatorname{\rm Im}\left\{%
\partial_{x}\biggl{(}-\frac{1}{2\mu_{k}}(\partial_{x}w_{k}^{(l)})\overline{w_{%
k}^{(l)}}\biggr{)}+(\partial_{x}^{l}\mathcal{L}_{\mu_{k}}w_{k})\overline{w_{k}%
^{(l)}}\right\},$$
we see that
$$\displaystyle\left|\int_{-\infty}^{x}\partial_{t}\Phi(t,y)dy\right|$$
$$\displaystyle=2\eta\left|\sum_{l=0}^{1}\sum_{k\in I_{N}}\operatorname{\rm Im}%
\Bigl{\{}-\frac{1}{2\mu_{k}}(\partial_{x}w_{k}^{(l)})\overline{w_{k}^{(l)}}+%
\int_{-\infty}^{x}\bigl{(}\partial_{x}^{l}\mathcal{L}_{\mu_{k}}w_{k}\bigr{)}%
\overline{w_{k}^{(l)}}dy\Bigr{\}}\right|$$
$$\displaystyle\leq C\eta\Bigl{(}\|w\|_{W^{2,\infty}}^{2}+\sum_{k\in I_{N}}\|%
\mathcal{L}_{\mu_{k}}w_{k}\|_{H^{1}}\|w_{k}\|_{H^{1}}\Bigr{)}.$$
Therefore we obtain
$$\displaystyle\frac{d}{dt}\|Sv\|_{L^{2}}^{2}+\frac{2}{|m|}\operatorname{\rm Re}%
\langle\Phi S|\partial_{x}|v,Sv\rangle_{L^{2}}\leq 2\bigl{|}\langle S\mathcal{%
L}_{m}v,Sv\rangle_{L^{2}}\bigr{|}+CB_{1}(t)\|Sv\|_{L^{2}}^{2},$$
(A.1)
where
$$\displaystyle B_{1}(t)$$
$$\displaystyle=e^{C\|\Phi\|_{L^{1}}}\Bigl{(}\|\Phi\|_{L^{\infty}}^{2}+\|%
\partial_{x}\Phi\|_{L^{\infty}}+\eta\|w\|_{W^{2,\infty}}^{2}+\eta\sum_{k\in I_%
{N}}\|\mathcal{L}_{\mu_{k}}w_{k}\|_{H^{1}}\|w_{k}\|_{H^{1}}\Bigr{)}.$$
Next we observe that
$$\displaystyle w_{k}^{(l)}S|\partial_{x}|v$$
$$\displaystyle=w_{k}^{(l)}S\partial_{x}\mathcal{H}v$$
$$\displaystyle=\partial_{x}(w_{k}^{(l)}S\mathcal{H}v)+[w_{k}^{(l)}S,\partial_{x%
}]\mathcal{H}v$$
$$\displaystyle=-|\partial_{x}|^{1/2}|\partial_{x}|^{1/2}\mathcal{H}w_{k}^{(l)}S%
\mathcal{H}v+[w_{k}^{(l)}S,\partial_{x}]\mathcal{H}v$$
$$\displaystyle=|\partial_{x}|^{1/2}\bigl{(}w_{k}^{(l)}S|\partial_{x}|^{1/2}v%
\bigr{)}+[w_{k}^{(l)}S,\partial_{x}]\mathcal{H}v-|\partial_{x}|^{1/2}\Bigl{[}|%
\partial_{x}|^{1/2}\mathcal{H},w_{k}^{(l)}S\Bigr{]}\mathcal{H}v,$$
which leads to
$$\displaystyle\bigl{\langle}w_{k}^{(l)}S|\partial_{x}|v,w_{k}^{(l)}Sv\bigr{%
\rangle}_{L^{2}}=$$
$$\displaystyle\bigl{\langle}w_{k}^{(l)}S|\partial_{x}|^{1/2}v,|\partial_{x}|^{1%
/2}\bigl{(}w_{k}^{(l)}Sv\bigr{)}\bigr{\rangle}_{L^{2}}+\bigl{\langle}[w_{k}^{(%
l)}S,\partial_{x}]\mathcal{H}v,w_{k}^{(l)}Sv\bigr{\rangle}_{L^{2}}$$
$$\displaystyle-\Bigl{\langle}\bigl{[}|\partial_{x}|^{1/2}\mathcal{H},w_{k}^{(l)%
}S\bigr{]}\mathcal{H}v,|\partial_{x}|^{1/2}(w_{k}^{(l)}Sv)\Bigr{\rangle}_{L^{2}}$$
$$\displaystyle=$$
$$\displaystyle\Bigl{\|}w_{k}^{(l)}S|\partial_{x}|^{1/2}v\Bigr{\|}_{L^{2}}^{2}+X%
_{kl},$$
where
$$\displaystyle X_{kl}=$$
$$\displaystyle\Bigl{\langle}w_{k}^{(l)}S|\partial_{x}|^{1/2}v,\bigl{[}|\partial%
_{x}|^{1/2},w_{k}^{(l)}S\bigr{]}v\Bigr{\rangle}_{L^{2}}+\Bigl{\langle}\bigl{[}%
w_{k}^{(l)}S,\partial_{x}\bigr{]}\mathcal{H}v,w_{k}^{(l)}Sv\Bigr{\rangle}_{L^{%
2}}$$
$$\displaystyle-\Bigl{\langle}\bigl{[}|\partial_{x}|^{1/2}\mathcal{H},w_{k}^{(l)%
}S\bigr{]}\mathcal{H}v,w_{k}^{(l)}S|\partial_{x}|^{1/2}v\Bigr{\rangle}_{L^{2}}%
-\Bigl{\langle}\bigl{[}|\partial_{x}|^{1/2}\mathcal{H},w_{k}^{(l)}S\bigr{]}%
\mathcal{H}v,\bigl{[}|\partial_{x}|^{1/2},w_{k}^{(l)}S\bigr{]}v\Bigr{\rangle}_%
{L^{2}}.$$
By using Lemma A.1, we can see that all the commutators
appearing in $X_{kl}$ are $L^{2}$-bounded and their operator norms
are dominated by
$$\displaystyle B_{2}(t)$$
$$\displaystyle=Ce^{C\|\Phi\|_{L^{1}}}\bigl{(}\|w\|_{W^{2,\infty}}+\|w\|_{W^{1,%
\infty}}\|\Phi\|_{L^{\infty}}\bigr{)}.$$
Hence we obtain
$$\displaystyle\Bigl{\|}\sqrt{\Phi}S|\partial_{x}|^{1/2}v\Bigr{\|}_{L^{2}}^{2}$$
$$\displaystyle-\operatorname{\rm Re}\bigl{\langle}\Phi S|\partial_{x}|v,Sv\bigr%
{\rangle}_{L^{2}}$$
$$\displaystyle=\sum_{l=0}^{1}\sum_{k\in I_{N}}\eta\operatorname{\rm Re}\biggl{(%
}\Bigl{\|}w_{k}^{(l)}S|\partial_{x}|^{1/2}v\Bigr{\|}_{L^{2}}^{2}-\bigl{\langle%
}w_{k}^{(l)}S|\partial_{x}|v,w_{k}^{(l)}Sv\bigr{\rangle}_{L^{2}}\biggr{)}$$
$$\displaystyle\leq\sum_{l=0}^{1}\sum_{k\in I_{N}}\eta|X_{kl}|$$
$$\displaystyle\leq C\eta B_{2}(t)\sum_{l=0}^{1}\sum_{k\in I_{N}}\Bigl{\|}w_{k}^%
{(l)}S|\partial_{x}|^{1/2}v\Bigr{\|}_{L^{2}}\bigl{\|}v\bigr{\|}_{L^{2}}+C\eta B%
_{2}(t)^{2}\|v\|_{L^{2}}^{2}$$
$$\displaystyle\leq\frac{1}{2}\Bigl{\|}\sqrt{\Phi}S|\partial_{x}|^{1/2}v\Bigr{\|%
}_{L^{2}}^{2}+C\eta B_{2}(t)^{2}\|v\|_{L^{2}}^{2},$$
where we have used the Young inequality in the last line.
Therefore,
$$\displaystyle\frac{2}{|m|}\operatorname{\rm Re}\langle\Phi S|\partial_{x}|v,Sv%
\rangle_{L^{2}}\geq\frac{1}{|m|}\Bigl{\|}\sqrt{\Phi}S|\partial_{x}|^{1/2}v%
\Bigr{\|}_{L^{2}}^{2}-C\eta B_{2}(t)^{2}\|v\|_{L^{2}}^{2}.$$
(A.2)
From (A.1) and (A.2)
it follows that
$$\frac{d}{dt}\|Sv\|_{L^{2}}^{2}+\frac{1}{|m|}\Bigl{\|}\sqrt{\Phi}S|\partial_{x}%
|^{1/2}v\Bigr{\|}_{L^{2}}^{2}\leq 2\bigl{|}\langle S\mathcal{L}_{m}v,Sv\rangle%
_{L^{2}}\bigr{|}+C\bigl{(}B_{1}(t)+\eta B_{2}(t)^{2}\bigr{)}\|Sv\|_{L^{2}}^{2}.$$
Finally, by using
$\|\Phi\|_{L^{1}}\leq C\eta\|w\|_{H^{1}}^{2}$,
$\|\Phi\|_{L^{\infty}}\leq C\eta\|w\|_{W^{1,\infty}}^{2}$
and $\|\partial_{x}\Phi\|_{L^{\infty}}\leq C\eta\|w\|_{W^{2,\infty}}^{2}$,
we have
$$\displaystyle B_{1}(t)+\eta B_{2}(t)^{2}\leq$$
$$\displaystyle Ce^{C\eta\|w\|_{H^{1}}^{2}}\Bigl{(}\eta^{2}\|w\|_{W^{1,\infty}}^%
{4}+\eta\|w\|_{W^{2,\infty}}^{2}+\eta\sum_{k\in I_{N}}\|\mathcal{L}_{\mu_{k}}w%
_{k}\|_{H^{1}}\|w_{k}\|_{H^{1}}\Bigr{)}$$
$$\displaystyle+C\eta e^{C\eta\|w\|_{H^{1}}^{2}}\bigl{(}\|w\|_{W^{2,\infty}}^{2}%
+C\eta^{2}\|w\|_{W^{1,\infty}}^{6}\bigr{)}$$
$$\displaystyle\leq$$
$$\displaystyle Ce^{C\eta\|w\|_{H^{1}}^{2}}\Bigl{(}\eta\|w\|_{W^{2,\infty}}^{2}+%
\eta^{3}\|w\|_{W^{1,\infty}}^{6}+\eta\sum_{k\in I_{N}}\|\mathcal{L}_{\mu_{k}}w%
_{k}\|_{H^{1}}\|w_{k}\|_{H^{1}}\Bigr{)},$$
which yields the desired conclusion.
∎
Acknowledgments
One of the authors (H.S.) would like to express his gratitude for warm
hospitailty of Department of Mathematics, Yanbian University. Main parts of
this work were done during his visit there.
The authors thank Professor Soichiro Katayama for his useful
conversations on this subject.
The work of C. L. is supported by NNSFC under Grant No. 11461074.
The work of H. S. is supported by Grant-in-Aid for Scientific Research (C)
(No. 25400161), JSPS.
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Study of the Spectral and Temporal Characteristics of X-Ray Emission of the Gamma-Ray Binary LS 5039 with
Suzaku
Tadayuki Takahashi11affiliation: Institute of Space and Astronautical Science/JAXA, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan 22affiliation: Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, 113-0033, Japan ,
Tetsuichi Kishishita11affiliation: Institute of Space and Astronautical Science/JAXA, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan 22affiliation: Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, 113-0033, Japan ,
Yasunobu Uchiyama33affiliation: Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, 2575 Sand Hill Road M/S 29, Menlo Park, CA 94025 ,
Takaaki Tanaka33affiliation: Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, 2575 Sand Hill Road M/S 29, Menlo Park, CA 94025 ,
Kazutaka Yamaoka44affiliation: Graduate School of Science and Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara, Kanagawa 229-8558, Japan ,
Dmitry Khangulyan55affiliation: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, Heidelberg 69117, Germany
Felix A. Aharonian55affiliation: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, Heidelberg 69117, Germany 66affiliation: Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland ,
Valenti Bosch-Ramon55affiliation: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, Heidelberg 69117, Germany ,
Jim A. Hinton77affiliation: School of Physics & Astronomy, University of Leeds, LS2 9JT, Leeds, UK
[email protected]
Abstract
We report on the results from Suzaku broadband X-ray observations of the galactic binary source LS 5039.
The Suzaku data, which have continuous coverage of more than one orbital period, show strong modulation of the
X-ray emission at the orbital period of this TeV gamma-ray emitting system.
The X-ray emission shows a minimum at orbital phase $\sim 0.1$,
close to the so-called superior conjunction of the compact object,
and a maximum at phase $\sim 0.7$, very close to the inferior conjunction of the
compact object.
The X-ray spectral data up to 70 keV are described by a hard power-law with a phase-dependent photon
index which varies within $\Gamma\simeq 1.45$– 1.61. The amplitude of the flux variation is a factor of 2.5,
but is significantly less than that of the factor $\sim$8 variation in the TeV flux. Otherwise the two light curves are similar, but not identical.
Although periodic X-ray emission has been found from many
galactic binary systems, the Suzaku result implies a phenomenon different from the
“standard” origin of X-rays related to the emission of the hot accretion plasma formed
around the compact companion object.
The X-ray radiation of LS 5039 is likely to be linked to very-high-energy electrons which are also responsible for the TeV gamma-ray emission.
While the gamma-rays are the result of inverse Compton scattering by electrons on optical stellar photons,
X-rays are produced via synchrotron radiation. Yet, while the modulation of the TeV gamma-ray signal can be naturally explained by the
photon-photon pair production and anisotropic inverse Compton scattering, the observed modulation of
synchrotron X-rays requires an additional process, the most natural one being
adiabatic expansion in the radiation production region.
Subject headings:acceleration of particles — X-rays: individual (LS 5039) — X-rays: binaries
1. Introduction
LS 5039 is a high-mass X-ray binary (Motch et al., 1997) with
extended radio emission (Paredes et al., 2000, 2002).
This system is formed by a main sequence O type star and a compact object of
disputed nature that has been claimed to be both
a black hole (e.g., Casares et al., 2005) and a neutron star/pulsar (e.g., Martocchia et al., 2005; Dubus, 2006).
The compact object is moving around the companion star in a moderately elliptic orbit (eccentricity $e=0.35$)
with an orbital period of $P_{\rm orb}=3.9060$ days (Casares et al., 2005).
As summarized in Bosch-Ramon et al. (2007), LS 5039 has been observed
several times in the X-ray energy band for limited phases in the orbital period. Flux variations on time
scale of days and sometimes on much shorter timescales have been reported.
The spectrum was always well represented by a power-law model with a
photon index ranging $\Gamma=1.4$–1.6 up to $\sim 10$ keV, with fluxes
changing
moderately around $\sim 1\times 10^{-11}~{}{\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$.
Softer spectra and larger fluxes had been also
inferred from RXTE observations, although background contamination
was probably behind these differences (see
Bosch-Ramon et al. 2005). Also, Chandra data taken in 2002 and 2005
showed spectra significantly harder than 1.5 (Bosch-Ramon et al., 2007), but
such a hard spectrum was probably an artifact produced by photon pile-up.
Recently, Hoffmann et al. (2009) reported the
results of INTEGRAL observations in hard X-rays. The source was detected
at energies between 25 and 60 keV. The source was detected at energies between 25 and 60 keV. The flux was estimated
to be $(3.54\pm 2.30)\times 10^{-11}~{}{\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$
(90 % confidence level)
around the inferior conjunction (INFC) of the compact object, and a flux upper limit of
$1.45\times 10^{-11}~{}{\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ (90 % confidence level) was
derived at the superior conjunction of the compact object (SUPC).
LS 5039 has also been detected in very-high-energy (VHE: $E\geq 0.1$ TeV) gamma-rays (Aharonian et al., 2005a), exhibiting a periodic
signal modulated with the orbital period (Aharonian et al., 2006). There are two other binary systems with robust detections in the
VHE range: PSR B1259$-$63 (Aharonian et al., 2005b) and LS I $+$61$\arcdeg$303 (Albert et al., 2006; Acciari et al., 2008). Evidence for TeV emission has been found also in
Cygnus X-1 (Albert et al., 2007). PSR B1259$-$63 is a clear case of a high-mass binary system containing a non-accreting pulsar
(Johnston et al., 1992), whereas Cygnus X-1 is a well known accreting black hole system (Bolton, 1972). The nature of the
compact object in LS 5039 is not yet established, and the origin of the VHE emitting electrons is unclear. They may be
related to a pulsar wind or to a black hole with a (sub)relativistic jet. In the standard pulsar-wind scenario,
the severe photon-photon absorption makes the explanation of the detection of the VHE radiation problematic, at least at the position
corresponding to the orbital phase $\phi\sim 0.0$ (see Figs. 16 and 4 from
Sierpowska-Bartosik & Torres 2008 and Dubus et al. 2008 and compare with Fig. 5 in Aharonian et al. 2006), in which the
emitter is expected to be located between the compact object and the star. However, particles may be accelerated in a
relativistic outflow formed at the interaction of the pulsar and the stellar winds (Bogovalov et al., 2008) and radiate far from
the compact object, making the pulsar wind scenario a viable option. In the microquasar scenario, the lack of
accretion features in the X-ray spectrum may be a problem unless the bulk of the accretion power is released in the form of
kinetic energy of the outflow, rather than thermal emission during accretion, as in the case of SS 433 (see
e.g., Marshall et al., 2002). At this stage, we cannot give a preference to any of these scenarios, but new data, in particular
those obtained with the Suzaku satellite, allow us to make an important step towards the understanding of the nature
of the non-thermal processes of acceleration and radiation in this mysterious object.
2. Observation
The temporal and spectral characteristics of the X-ray emission from LS 5039 along the orbit
should provide important clues for understanding the acceleration/radiation
processes in this source.
The fact that all previous X-ray observations of this object have incomplete coverage of the orbital period,
or suffered from background contamination, is therefore rather unsatisfactory.
This motivated our long, $\sim 200$ ks observation with
the Suzaku X-ray observatory (Mitsuda et al., 2007),
which gives us unprecedented coverage
of more than one orbital period, continuously
from 2007 September 9 to 15 (see Table 1).
Suzaku has four sets of X-ray telescopes (Serlemitsos et al., 2007)
each with a focal-plane X-ray CCD camera (X-ray Imaging Spectrometer(XIS); Koyama et al., 2007)
that are sensitive in the energy range of 0.3–12 keV.
Three of the XIS detectors (XIS0, 2 and 3) have front-illuminated (FI) CCDs,
whereas XIS1 utilizes a back-illuminated (BI) CCD.
The merit of the BI CCD is its improved sensitivity in the soft X-ray energy band below 1 keV.
Suzaku contains also a non-imaging collimated Hard X-ray Detector (HXD; Takahashi et al., 2007; Kokubun et al., 2007),
which covers the 10–600 keV energy band with Si PIN photodiodes (10–70 keV) and GSO scintillation detectors
(40–600 keV).
Suzaku has two default pointing positions, the XIS nominal position and the HXD nominal position.
In this observation, we used the HXD nominal position, in which the effective area of HXD
is maximized, whereas that of the XIS is reduced to on average $\sim$ 88%.
Results from XIS2 are not reported here since it has not been in operation since
an anomaly in November 2006.
In addition, we do not describe in detail the analysis of HXD-GSO data,
since the HXD-GSO detected no significant signal from the source.
3. Data Reduction
We used data sets processed using the software of the Suzaku data processing pipeline (version 2.1.6.16).
Reduction and analysis of the data were performed following the standard procedure using the HEADAS v6.4 software
package, and spectral fitting was performed with XSPEC v.11.3.2.
For the XIS data analysis, we accumulated cleaned events over good time intervals that were selected by removing
spacecraft passages through the South Atlantic Anomaly (SAA). Further, we screened the data with the following criteria —
(1) cut-off rigidity is larger than 6 GV and (2) elevation angle from the Earth’s rim is larger than 5${}^{\circ}$.
The source photons were accumulated from a circular region with a radius of $3^{\prime}$.
The background region was chosen in the same field of view with the same radius and an offset of
$9^{\prime}$ from the source region.
We have co-added the data from the two FI-CCDs (XIS0 and XIS3) to increase statistics.
The response (RMF) files and the auxiliary response (ARF) files used in this paper
were produced using xisrmfgen and xissimarfgen, respectively.
For the HXD data,“uncleaned event files” were screened with the standard event screening criteria:
the cut-off rigidity is larger than 6 GV, the elapsed time after the passage of the SAA
is more than 500 s and the time to the next SAA passage is more than 180 s, high voltages from all eight HV units are within
the normal range and the elevation angle from the Earth’s rim is more than $5^{\circ}$. We also discarded telemetry-saturated
time intervals.
In the spectral analysis in §3, we used the response file for a point-like source at the HXD-nominal
position (ae_hxd_pinhxnome4_20070914.rsp), which is released as a part of CALDB (Suzaku calibration data base).
The HXD-PIN spectrum is dominated by the time-variable instrumental background
( non-X-ray background (NXB) ) induced by cosmic rays and trapped charged particles in the satellite orbit.
To estimate the instrumental background component, we used the the time-dependent NXB
event files released by the HXD instrument team, whose reproducibility is
reported by Fukazawa et al. (2009).
In order to estimate the systematic uncertainty of the NXB model during our observation,
we compare the NXB model spectrum during Earth occultation
with the observed spectrum of the same time interval.
The event selection criteria for this study are the same as those of the cleaned event
except for the criterion on Earth elevation angle, which was chosen to
to be less than $-5^{\circ}$. The estimated uncertainty obtained is $\sim 3$%,
which is consistent with the values reported by Fukazawa et al. (2009).
Another component of the HXD-PIN background is the cosmic X-ray background (CXB).
In our analysis, we assumed the CXB spectrum reported by (Gruber et al., 1999):
$$I(\epsilon)=7.9\;\epsilon^{-1.29}_{\rm keV}\exp{\left(-\frac{\epsilon_{{\rm keV%
}}}{\epsilon_{\rm p}}\right)}~{}{\rm ph}~{}{\rm s}^{-1}~{}{\rm keV}^{-1}~{}{%
\rm cm}^{-2}~{}{\rm str}^{-1},$$
(1)
where $\epsilon_{\rm keV}=\epsilon/{\rm keV}$ and $\epsilon_{\rm p}=41.1$.
The CXB spectrum observed with HXD-PIN was simulated by using a PIN response file
for isotropic diffuse emission (ae_hxd_pinflate4_20070914.rsp) and added to
the NXB spectrum.
Based on this approach, the contribution from the CXB flux is $\sim 5$% of the NXB.
Since LS 5039 is located close to the Galactic plane, the contribution from the
Galactic ridge X-ray emission (GRXE) must be examined, especially for
HXD-PIN spectra. In order to model the shape of the GRXE, data from
Suzaku observations of the Galactic ridge region (ObsID: 500009010 and 500009020)
were analyzed.
The Suzaku spectrum from 3 keV to 50 keV can be well fitted with the Raymond-Smith plasma
with a temperature of $kT=2.2\pm 0.8$ keV and a power-law function with
$\Gamma=1.92^{+0.28}_{-0.4}$.
Although we also tried a power law with exponential cutoff, following the results from the
INTEGRAL IBIS (Krivonos et al., 2007), it turned out that the assumption on the spectral shape
of the GRXE has negligible effect on the spectral parameters of LS 5039.
The normalization of the GRXE spectrum component in the HXD-PIN spectrum of LS 5039
is determined from the XIS spectrum of the LS 5039 observation by excluding an encircled region with a radius of
$4.5^{\prime}$ centered on the LS 5039 location.
The flux of the GRXE is estimated to be $\sim 40\%$ of the contribution from the CXB.
In Figure 1, we show the time averaged HXD-PIN spectrum plotted
together with models for the NXB, CXB and GRXE.
4. Analysis and Results
4.1. Temporal analysis
The light curve obtained from the XIS detector is shown in
the top panel of Figure 2.
The continuous coverage in X-rays, longer than the orbital period of the LS 5039 system,
reveals a smooth variation of a factor 2 in the 1–10 keV count rate.
The light curve is drawn over two orbital periods.
The orbital phase is calculated with the period of
3.90603 days, and $\phi=0$
with reference epoch $T_{0}$ (${\rm HJD}-2400000.5=51942.59$)
taken from Casares et al. (2005).
The light curve from phase $\phi=1.0$ to 1.5, which was obtained in the last part of
the observation, smoothly overlaps with the one obtained at the beginning of the
observation ($\phi=0.0$–0.5).
In the middle panel of Figure 2, we present the light curve
obtained with the HXD-PIN for the energy range 15–40 keV.
Although the statistical errors are larger, the modulation behavior
is similar to that of the XIS. The amplitude of the modulation is
roughly the same between the XIS and HXD-PIN, indicating
small changes of spectral shape depending on orbital phase.
The spectral parameters obtained for each orbital phase
are reported in the following section.
The light curves obtained with Suzaku show that
the X-ray flux minimum appears around phase 0.0–0.3 and it reaches maximum around phase 0.5–0.8.
In order to quantify the amplitude of the flux variations, we fitted the XIS light curve
with a simple sinusoidal function. Due to structures in the light curve,
the fit converges with large chi-square ($\chi_{\nu}^{2}(\nu)$ = 4.92 (121)). However,
the general trend is well represented by a sinusoidal function:
$$I(\phi)=0.24\sin(\phi-0.13)+0.64\;{\rm counts}\;{\rm s}^{\rm-1}$$
(2)
where $I(\phi)$ is the count rate as a function of phase $\phi$.
The ratios between the minimum and maximum count rates are 2.21 ${}^{+0.02}_{-0.03}$ counts s${}^{-1}$ for XIS and 2.02 ${}^{+0.25}_{-0.19}$ counts s${}^{-1}$ for HXD-PIN.
Structures of the X-ray and hard X-ray light curves are similar to that discovered in the phase diagram of integral fluxes at energies $>1$ TeV
obtained on a run-by-run basis from HESS data (2004 to 2005) (Aharonian et al., 2006). The temporal X-ray behavior
was already suggested by RXTE data, as well as by a compilation of all the previous X-ray data obtained with imaging
instruments (Bosch-Ramon et al., 2005; Zabalza et al., 2008).
In addition to the continuously changing component with respect
to the orbital phase, short timescale structures are found around $\phi=0.48$ and $\phi=0.7$.
The unabsorbed flux changes about a 30% in $\Delta\phi\sim 0.05~{}(=4.7~{}{\rm hour})$.
A significant dip can be seen around $\phi=1.35$ in the top panel of Figure 2.
In comparison with the data at around $\phi=0.35$ obtained in the first half of the observation,
the flux decreased $\sim$50% only in this phase.
The time duration of this dip corresponds to $\Delta\phi\sim 0.03$.
These structures may reflect features of the (possibly changing) environment of the X-ray emitting region, it is therefore
of importance to test with further observations if these are persistent features.
4.2. Spectral Analysis
Firstly we study time-resolved (phase-resolved) X-ray spectra.
The data are divided into data segments with respect to the assigned phase,
and model fitting is performed for XIS spectra
for each segment with $\Delta\phi=0.1$.
A single power-law function with photoelectric absorption,
provides a good fit for all the segments.
In order to study the possible changes of the
amount of photoelectric absorption, we here fit the data
with $N_{\rm H}$ free.
The best-fit parameters are presented in Table 2.
The derived values of the photon index and absorption column density are consistent with previous observations
(Martocchia et al., 2005; Bosch-Ramon et al., 2007). When we fix the $N_{\rm H}$ to the value obtained from the time
averaged spectrum, resultant photon indices stay same within statistical error.
The photon index ($\Gamma$) values are plotted as a function of orbital phase
in the top panel of Figure 3.
The spectral shape varied such that the spectrum is steep around SUPC
($\Gamma\simeq 1.61$) and becomes hard ($\Gamma\simeq 1.45$)
around apastron.
The modulation behavior of $\Gamma$ is somewhat different from
that observed using HESS in the VHE range.
The amplitude of the variation is $\pm 0.1$, which is much smaller
than the change of $\pm$0.6 in the VHE region (Aharonian et al., 2006).
The 1–10 keV flux changes from $(5.18\pm 0.03)\times 10^{-12}~{}{\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$
($\phi=0.1\mbox{--}0.2$)
to $(12.05\pm 0.02)\times 10^{-12}\ \rm erg\ cm^{-2}\ s^{-1}$. ($\phi=0.6\mbox{--}0.7$).
In all the data segments, the source is significantly detected with the HXD-PIN,
indicating that hard X-ray emission extends at least up to 70 keV.
Note also that although the XIS and HXD-PIN spectra do not overlap,
they seem to be smoothly connected in the gap between 10 and 15 keV.
To study the shape of the spectrum above 10 keV, the XIS spectra and the PIN spectrum in the range 15–70 keV
after subtraction of background (NXB $+$ CXB $+$ GRXE) are jointly fitted (Figure 4 Top).
The time-averaged spectra are well represented by an absorbed power-law model with $\Gamma$= 1.51 $\pm$0.02 with reduced $\chi^{2}_{\nu}=0.99$
(235 degrees of freedom). We find no cutoff structure in the energy range of the HXD-PIN.
The spectra within the phase intervals [$0.616<\phi<0.816$] and [$\phi<0.158$ & $0.958<\phi$],
which correspond to the INFC and SUPC, respectively, are also shown
in Figure 4. The best-fit parameters are presented in Table 3.
Althought earlier observations by RXTE suggested the presence of an
iron emission line at 6.7 keV (Ribo et al. 1999), later observations by
Chandra and XMM could not find evidence of it (e.g.
Bosch-Ramon et al. 2005). A careful study of new and longer RXTE
observations, using slew data to account for background emission, revealed
that the earlierly reported 6.7 keV emission line is likely a background
feature (Bosch-Ramon et al., 2005) . The Suzaku data confirm this result.
In an attempt to find the possible signature of iron emission lines from LS 5039,
we analyzed the phase-averaged spectrum.
The upper limits on iron line structures are determined by fitting a Gaussian
at various energies and line widths at which Fe emission might be expected.
The power-law continuum model parameters are fixed with the best fit
values and a Gaussian component is added to the power-law function.
The central energy of the Gaussian line is swept from 6.0 keV to 7.1 keV in steps of 0.1 keV. Line widths are changed
from 0.01 keV to 0.09 keV in steps of 0.01 keV, together with lines with larger widths of 0.15 and 0.20 keV.
An equivalent width is determined at each grid point. The resulting upper-limit on the equivalent width is 40 eV
with 90% confidence level.
5. Discussion
The X-ray emission observed with Suzaku is characterized by
(1) a hard power law with $\Gamma\simeq 1.5$ extending from soft X-rays to $\sim 70$ keV,
(2) clear orbital modulation in flux and photon index,
(3) a moderate X-ray luminosity of
$L_{X}\sim 10^{33}(\frac{D}{1\rm kpc})^{2}\ \rm erg\ s^{-1}$,
(4) a small and constant absorbing column density, and
(5) a lack of detectable emission lines.
Although variable X-ray emission has been found from more than two hundred galactic binary systems, the Suzaku data
hardly can be explained within the “standard accretion” scenario where X-rays are produced by a hot thermal (comptonized) accretion
plasma around the compact object. The lack of X-ray emission lines (at the level of sensitivity of Suzaku)
as well as the hard $E^{-1.5}$ type energy spectrum of the X-ray continuum, extending from soft X-rays up to 70 keV,
favors a non-thermal origin of the X-rays.
This conclusion is supported by the general similarities between the properties of the observed X-rays and TeV gamma-rays.
Namely, both radiation components require a rather hard energy distribution of parent electrons with a power-law index of
$\approx 2$. This directly follows from the photon index of the synchrotron radiation $\Gamma\approx 1.5$, and agrees
quite well with the currently most favored interpretation of the TeV gamma-rays, in which they would be produced by IC scattering
off the anisotropic photon field of the massive companion star.
Assuming that the TeV gamma-ray production region is located at a
distance from the companion star of $d\sim 2\times 10^{12}$ cm (i.e. the binary system size), and taking into account that
gamma-rays are produced in the deep Klein-Nishina (KN) regime with significantly suppressed cross-section, for the well
known luminosity of the optical star $L\simeq 7\times 10^{38}\,{\rm erg}\,{\rm s}^{-1}$, one can estimate quite
robustly the strength of the magnetic field in the emission region. The numerical calculations show that the field should
be around a few Gauss (see e.g. Fig. 5). For such a magnetic field strength, the energy intervals of electrons responsible
for the two emission components overlap substantially, as shown in Figure 5. Therefore, we are most likely
dealing with the same population of parent electrons, which should be located at large distances from the compact object,
in the system periphery, to prevent the severe absorption of the TeV radiation and the subsequent intense emission from the
pair-created secondaries (Khangulyan et al., 2008a; Bosch-Ramon et al., 2008a).
It should be noted that the observed X-ray emission is very difficult to explain as synchrotron emission produced by secondary
(pair-produced) electron and positrons. Since the pair production cross-section has strong energy dependence with a distinct
maximum, for the target photons of typical energy of $\sim 10$ eV, the major fraction of the absorbed energy will be
released in the form of $\sim 100$ GeV electrons. Thus secondary pair synchrotron emission must show a spectral break in the
Suzaku energy band unless one assumes unreasonably high magnetic fields, $B\geq 1$ kG, in the surroundings of the gamma-ray
emission region (Khangulyan et al., 2008a; Bosch-Ramon et al., 2008a).
Figure 5 shows the synchrotron and IC cooling times of electrons, as a function of electron energy,
calculated for the stellar photon density at $d=2\times 10^{12}\ \rm cm$ and for a magnetic field $B=3$ G. It can be seen
that synchrotron losses dominate over IC losses at $E_{\rm e}\geq 1$ TeV. Note that the TeV gamma-ray production takes
place in the deep KN regime. This implies that the cooling time, $t_{\rm cool}=\gamma/\dot{\gamma}$, of electrons
generating GeV gamma-rays via IC scattering (Thomson regime) is shorter than the cooling time of TeV electrons responsible
for producing very high-energy gamma-rays (KN). The same applies for synchrotron cooling time of multi-TeV electrons that
produce low-energy (MeV) gamma-rays by synchrotron radiation. One should therefore expect significantly higher MeV
(synchrotron) and GeV (IC) fluxes than at keV and TeV energies, provided that the acceleration spectrum of electrons
extends from low energies to very high energies. However, in the case of existence of low-energy and very-high-energy
cutoffs in the acceleration spectrum, the gamma-ray fluxes $>10$ MeV and at GeV energies would be significantly suppressed.
To better understand the energy ranges of the electrons responsible
for X-ray and gamma-ray production, we show in Figure 5 the energy zones of
electrons relevant to the Suzaku, Fermi,
and HESS radiation domains. Note that the reconstruction
of the average energy of electrons responsible for the IC gamma-rays
depends only on the well known temperature of the companion star $T=3.8\times 10^{4}$ K.
The light green zone in Figure 5 marked as “Suzaku synchrotron”,
corresponds to electrons responsible for the synchrotron photons produced in the
energy interval
$1~{}{\rm keV}\leq\epsilon_{\rm syn}\leq 40~{}{\rm keV}$.
For a reasonable range of magnetic field values,
the energy interval of electrons relevant for Suzaku
data overlap on one hand with the HESS energy interval, and can overlap with the Fermi one. This
should allow us, in the case of detection of MeV/GeV gamma-rays by Fermi, to
considerably reduce the parameter space,
in particular, to better localize the X- and
gamma-ray production regions from electromagnetic cascade constraints, and derive the broadband energy
spectrum of electrons and the strength of the magnetic field, both as a function of the orbital phase.
Formally, when X-rays and TeV gamma-rays are produced by
the same population of very-high-energy electrons, one should expect a general
correlation between the light curves obtained by Suzaku and HESS.
In this regard, the similarity between the Suzaku and HESS light curves
seems to be natural. However, such an interpretation is not
straightforward in
the sense that two major mechanisms that might cause modulation of the TeV
gamma-ray signal are
related to interactions of electrons and gamma-rays with the photons of the companion
star, i.e.
anisotropic IC scattering and photon-photon pair production (Khangulyan et al., 2008a; Dubus et al., 2008),
and thus cannot contribute to the
X-ray modulation. The X-ray modulation requires periodic changes of the
strength of the ambient magnetic field or the number of relativistic electrons. Note,
however,
that the change of magnetic field would not have a strong impact as long as the
radiation
proceeds in the saturation regime and
synchrotron losses dominate in the relevant energy interval. One would also expect
modulation of the synchrotron X-ray flux if the energy losses of electrons are
dominated by IC scattering, although in such a case we should observe
significantly lower X-ray fluxes.
A more natural reason for the modulation of the synchrotron fluxes would come from dominantly adiabatic losses. The adiabatic cooling
of electrons in binary systems can be realized through complex (magneto)hydrodynamical processes, e.g. due to interactions
between a black hole jet or a pulsar wind with the dense stellar wind of a massive companion star (see e.g.
Bogovalov et al. 2008, Perucho & Bosch-Ramon 2008). The orbital motion could naturally produce the modulation of adiabatic cooling of
electrons around the orbit (see e.g. Khangulyan et al. 2008b). Note that because of the relatively small variation of the
X-ray flux over the orbit, a factor of only two, the requirements for this scenario are quite modest.
We note that dominant adiabatic losses have been invoked by
Khangulyan et al. (2007) to explain the variations of the X-ray and TeV gamma-ray fluxes from the binary pulsar PSR B1259$-$63.
The detected power-law spectrum of X-rays with photon index around $\Gamma=1.5$ implies that the established energy
spectrum of electrons is also a power-law with index $\alpha_{\rm e}\simeq 2$.
This agrees well with the hypothesis of
dominance of adiabatic losses, because the adiabatic losses do not change the initial spectrum of electrons. Thus the
required power-law index $\alpha_{\rm e}\simeq 2$ implies a reasonable acceleration spectrum $Q(E_{\rm e})\propto E_{\rm e}^{-2}$.
Otherwise, in an environment dominated by synchrotron losses, the acceleration spectrum should be very hard, with a power-law index $\leq 1$, or should have an unreasonably large low-energy cutoff at $E\geq 1$ TeV to explain the
observed X-ray spectra.
Obviously, adiabatic losses modulate the IC gamma-ray flux in a similar manner. However, unlike X-rays, the TeV gamma-rays suffer significant distortion due to photon-photon absorption (see
e.g. Böttcher 2007) and anisotropic IC scattering with its strong hardening of the gamma-ray spectrum (Khangulyan & Aharonian, 2005). All this leads to additional orbital modulation of the gamma-ray signal, and it is likely that these
two additional processes are responsible for the strong change of gamma-ray flux, much more pronounced than that seen
in X-rays (see Fig. 2). The Suzaku data presented in this paper implies a key additional
assumption, namely that the accelerated electrons must loose their energy adiabatically before they cool
radiatively.
In order to demonstrate that the suggested scenario can satisfactorily explain the combined Suzaku X-ray and
HESS gamma-ray data, we performed calculations of the broad-band spectral energy distributions (SEDs) of the synchrotron
and IC emission, assuming a simple model in which the same population of electrons is responsible for both X-rays and TeV
gamma-rays. We also assumed that the emission region has homogeneous physical conditions. This is a reasonable assumption
given that we deal with very short cooling timescales ($\ll 100$ s), thus electrons cannot travel significant distances
while emitting.
In the regime dominated by adiabatic energy losses, the synchrotron X-ray flux is proportional to $t_{\rm ad}$. The
X-ray modulation seen by Suzaku is then described by the modulation of the adiabatic loss rate.
In Fig. 6, we show $t_{\rm ad}(\phi)$ that is inferred from
the X-ray data.
The required adiabatic cooling timescales are $\sim 1$ s.
Any consistent calculation of the
adiabatic cooling requires the solution of the corresponding hydrodynamical problem, and one needs to know in detail
the nature of the source. At the present stage, we consider the
simple example of adiabatic cooling in a relativistically expanding source. In such a case, the adiabatic loss rate can be written as: $\dot{\gamma}_{\rm ad}(\phi,\gamma)=\gamma/t_{\rm ad}$ with $t_{\rm ad}\sim R/c\simeq 3R_{11}$ sec, where $R_{11}\equiv R/(10^{11}\ \rm cm)$ is the characteristic size of the source. The required variation of the adiabatic cooling is thus
reduced to the modulation of the size of the radiation region ($R_{11}\sim 0.3\mbox{--}1$). The size in turn depends on
the external pressure exerted by, e.g., the stellar wind from the massive star. The expected weaker external pressure around
apastron implicitly assumed in our model would be broadly consistent with the radial dependence of the wind pressure.
In Fig. 7 we show the SEDs
averaged over the INFC ($\phi=0.45$–$0.9$) and SUPC
($\phi\leq 0.45$, $\phi\geq 0.9$) phase intervals.
The corresponding gamma-ray data
have been previously reported by HESS (Aharonian et al., 2006).
Since both the absolute flux and the energy spectrum of
TeV gamma-rays vary rapidly with phase, in order to compare the
theoretical predictions with observations, we should use
smaller phase bins, ideally
speaking with the time intervals $\Delta t\leq 100$ s corresponding to
the characteristic cooling timescales of electrons.
Because of the lack of the relevant gamma-ray data available to us,
we here
use the X-ray and gamma-ray data integrated over $\Delta\phi=0.45$.
While this compromise does not allow us to
perform quantitative studies, it can be used to make a
qualitative comparison of the model calculations with observational data.
The theoretical calculations of the SEDs
are in a reasonable agreement with the observed spectra,
though they do not perfectly match the gamma-ray fluxes.
One can improve the fits by
introducing slight phase-dependent changes in the spectra of accelerated
electrons, but it is beyond the scope of this paper given the
caveat mentioned above.
We should also note that the calculations of low energy gamma-rays
(in the Fermi domain)
are performed assuming that the injection spectrum of electrons continues down to $10$ GeV. If this is not the case,
the gamma-ray fluxes in the Fermi energy domain
may be significantly suppressed. On the other hand, the detection of
gamma-rays by Fermi would allow us to recover the spectrum of electrons in
a very broad energy interval, and thus distinguish between different
acceleration models. Another important feature in this scenario is that a hard
synchrotron spectrum extending up to a few MeV,
is required by the robust detection of $\sim 10$ TeV gamma-rays from the system.
A future detection of the emission at MeV energies
may bring important information on the presence of highest energy particles in the system.
The reproduction, at least qualitatively, of the observed spectral and temporal features of the nonthermal radiation with
the simple toy model supports the production of X-ray and TeV gamma-rays by the same population of parent
particles and allows us to derive several principal conclusions.
The electron energy distribution should be a power-law with an almost constant index of $\alpha_{\rm e}=2\Gamma_{\rm X}-1\simeq 2$ to explain the X-ray spectra. Note that in an isotropic photon gas when the Compton scattering takes
place in the deep KN regime such an electron distribution results in a quite steep TeV spectrum with photon index
$\Gamma_{\gamma}\simeq 1+\alpha_{\rm e}\simeq 3$. This does not agree with HESS observations. However, the anisotropic IC
provides a remarkable hardening of the gamma-ray spectrum (Khangulyan & Aharonian, 2005), in particular, $\Gamma_{\gamma}\sim 2$ would
be expected for the INFC, and it has indeed been observed using HESS (Aharonian et al., 2006). We also note that, to explain the VHE
spectrum at SUPC, we have to assume that the emission region is located at a distance $\approx 2\times 10^{12}$ cm from the
compact object, in the direction perpendicular to the orbital plane. In the standard pulsar scenario, the production region
cannot be located far away from the compact object, and even invoking electromagnetic cascading (see e.g. Fig. 16 in
Torres & Sierpowska-Bartosik 2008), one cannot reproduce the reported fluxes around orbital phase 0.0
(Khangulyan et al., 2008a; Bosch-Ramon et al., 2008a).
We have found that the magnetic field strength cannot deviate much from a few G. We can also derive a constraint on the size of
the emission region, imposing a maximum expansion speed of $\sim$$c$ (the speed of light), and the Hillas criterion
(Hillas, 1984), in which the minimum size of a source capable of accelerating particles to a given energy $E_{\rm e}$ is
$R=R_{\rm L}$ (where $R_{\rm L}=E_{\rm eV}/300B_{\rm G}$ cm), the Larmor radius. This estimate yields a size of
$10^{10}$–$10^{11}$ cm, which agrees quite well with the estimate based on the required timescale of adiabatic cooling. Note
that the requirement of fast adiabatic losses imposes a strong constraint on the acceleration rate of electrons. Indeed,
the acceleration timescale can be expressed as: $t_{\rm acc}=\eta R_{\rm L}/{c}\sim 0.1\eta({E_{\rm e}}/{\rm 1TeV})({B}/{\rm 1%
\ G})^{-1}\ \rm s$, where $\eta\geq 1$ parametrizes the acceleration efficiency. In extreme
accelerators with the maximum possible rate allowed by classical electrodynamics $\eta=1$. The HESS spectrum provides
evidence of electron acceleration well above 10 TeV. Therefore, $t_{\rm acc}<t_{\rm ad}\sim 1$ s is required at $E_{\rm e}=10$ TeV, which translates into $\eta<3$ for $B=3$ G. Thus we arrive at the conclusion that an extremely
efficient acceleration with $\eta<3$ should operate in a compact region of $R\sim 10^{11}\ \rm cm$.
Finally, we would like to emphasize that in the scenario described here different radiation energy intervals are
characterized by fundamentally different light curves. While the synchrotron X-ray modulation is caused by adiabatic losses,
the light curve in gamma-rays depends critically, in addition, on effects related to interactions with the
optical photons of the companion star. Two of these effects, photon-photon pair production and anisotropic Compton
scattering are equally important for the formation of the light curve of TeV gamma-rays. On the other hand, only the effect of
anisotropic Compton scattering has an impact on the formation of the light curve of GeV gamma-rays. The difference of light
curves in the X-ray and GeV and TeV gamma-ray intervals in this scenario is shown in Figure 6.
6. Summary
The Suzaku X-ray satellite has observed LS 5039 for the first time with imaging capabilities over one and a half orbits.
The Suzaku data show strong modulation of the
X-ray emission at the orbital period of the system and the X-ray spectral data are described by a hard power-law up to 70 keV.
We found the close correlation of the X-ray and
TeV gamma-ray light curves, which can be interpreted as evidence of production of these two radiation components by the same
electron population via synchrotron radiation and IC scattering, respectively.
Whereas
there are at least two reasons for the formation of a periodic TeV gamma-ray light curve, both related to the interaction
with photons from the companion star (photon-photon absorption of VHE gamma-rays and IC scattering in an anisotropic
photon field), the modulated X-ray signal requires an additional effect. A simple and natural reason for the modulation in
X-rays seems to be adiabatic losses which should dominate over the radiative (synchrotron and IC) losses of electrons. We
demonstrate that this assumption allows us to explain, at least qualitatively, the
spectral and temporal characteristics of the
combined Suzaku and HESS data. In particular, the introduction of adiabatic losses
not only provides a natural explanation for the rather stable photon index of the X-ray spectrum $\Gamma_{X}\simeq 1.5$, but also allows one to approximately reproduce the TeV gamma-ray spectra.
The gamma-ray data require a location of the production region at the periphery of the binary system at $d\sim 10^{12}$ cm.
This constraint allows a quite robust estimate of the magnetic field of a few Gauss to be derived directly from the X/TeV flux
ratio, and an adiabatic loss time of a few seconds to provide the dominance of adiabatic losses. In the case of a
relativistically expanding source, the size of the production region should not exceed $10^{11}$ cm. The adiabatic
cooling cannot be shorter than several seconds, and correspondingly, the size of the production region cannot be much
smaller than $10^{11}$ cm, since otherwise the electrons could not be accelerated up to energies beyond 10 TeV, even
assuming an extreme acceleration rate close to the fundamental limit determined by quantum electrodynamics.
There is little doubt that future simultaneous observations of
LS 5039 with the Suzaku, Fermi, and HESS telescopes will provide key information for understanding the nature of this mysterious non-thermal source.
T. Kishishita and T. Tanaka are supported by research fellowships of the
Japan Society for the Promotion of Science for Young Scientists.
The authors acknowledge support by the Spanish DGI of MEC under grant
AYA2007-6803407171-C03-01, as well as partial support by the European
Regional Development Fund (ERDF/FEDER). V.B-R. acknowledges support by the
Spanish DGI of MEC under grant AYA2007-6803407171-C03-01, as well as
partial support by the European Regional Development Fund (ERDF/FEDER).
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On Zariski’s multiplicity problem at infinity
J. Edson Sampaio
J. Edson Sampaio -
Departamento de Matemática, Universidade Federal do Ceará,
Rua Campus do Pici, s/n, Bloco 914, Pici, 60440-900,
Fortaleza-CE, Brazil
[email protected]
Abstract.
We address a metric version of Zariski’s multiplicity conjecture at infinity that says that two complex algebraic affine sets which are bi-Lipschitz homeomorphic at infinity must have the same degree. More specifically, we prove that the degree is a bi-Lipschitz invariant at infinity when the homeomorphism bi-Lipschitz has Lipschitz constants close to 1. In particular, we have that a family of complex algebraic sets bi-Lipschitz equisingular at infinity has constant degree. Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In particular, we receive that if two polynomials are rugose equivalent at infinity or bi-Lipschitz contact equivalent at infinity or bi-Lipschitz right-left equivalent at infinity, then they have the same degree.
Key words and phrases:Bi-Lipschitz contact at infinity, Degree, Zariski’s Conjecture
2010 Mathematics Subject Classification: 14B05, 32S50, 58K30 (Primary) 58K20 (Secondary)
1. Introduction
Let $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ be the germ of a reduced holomorphic function at the origin and let $(V(f),0)$ be the germ of the zero set of $f$ at origin. In 1971 (see [16]), O. Zariski proposed the following problem:
Question A If $V(f)$ is topologically equivalent to $V(g)$ as germs at the origin $0\in\mathbb{C}^{n}$, i.e. there exists a homeomorphism $\varphi\colon(\mathbb{C}^{n},V(f),0)\to(\mathbb{C}^{n},V(g),0)$, then is it true that $m(V(f),0)=m(V(g),0)$?
Although many authors have presented several partial results to the question A, it remains open. We recall the multiplicity of $V(f)$ at the origin, denoted by $m(V(f),0)$, is defined as following: we write
$$f=f_{m}+f_{m+1}+\cdots+f_{k}+\cdots$$
where each $f_{k}$ is a homogeneous polynomial of degree $k$ and $f_{m}\neq 0$. Then, $m(V(f),0):=m.$ In order to know more about Zariski’s multiplicity question see, for example, [5].
By looking from a metric point of view, and in a more general setting, we have the following metric local version of the Zariski’s multiplicity question (see Chapter 2 in [2], for a definition of multiplicity of complex analytic sets):
Question Ã1($d$) Let $X\subset\mathbb{C}^{n}$ and $Y\subset\mathbb{C}^{m}$ be two complex analytic sets with $\dim X=\dim Y=d$. If their germs at $0\in\mathbb{C}^{n}$ and $0\in\mathbb{C}^{m}$, respectively, are bi-Lipschitz homeomorphic, i.e. there exists a bi-Lipschitz homeomorphism $\varphi\colon(X,0)\to(Y,0)$, then is it true that their multiplicities $m(X,0)$ and $m(Y,0)$ must be equal?
Let us remark that Question Ã1($d$) was approached by some authors and, as far as I know, it remains still open. For instance, G. Comte, in the paper [3], proved that the multiplicity of complex analytic germs in $\mathbb{C}^{n}$ is invariant under bi-Lipschitz homeomorphism with Lipschitz constant close enough to 1. Recently, the author in [14] (see also [1]) showed that multiplicity 1 is invariant by bi-Lipschitz homeomorphism and the author jointly with A. Fernandes showed in [6] that the multiplicity of a complex analytic surface singularities in $\mathbb{C}^{3}$ is a bi-Lipschitz (embedded) invariant. It was showed also in [6] that it is enough to address such a question by considering $X$ and $Y$ homogeneous complex algebraic sets. Actually, this result is stated in [6] for complex analytic hypersurfaces in $\mathbb{C}^{n}$, however, the proof works for higher codimension complex analytic subsets. Other versions of the Question Ã1($d$) were approached by some authors, for example, J.-J. Risler and D. Trotman proved in [13] that if two complex analytic functions are rugose equivalent or bi-Lipschitz right-left equivalent, then they have the same order and G. Comte, P. Milman and D. Trotman showed in [4] that two complex analytic functions $f,g:(C^{n},0)\to(\mathbb{C},0)$ have the same order, whenever there are positive constants $C$ and $D$ and a homeomorphism $\varphi:(\mathbb{C}^{n},0)\to(\mathbb{C}^{n},0)$ satisfying
(1)
$\frac{1}{C}\|z\|\leq\|\varphi(z)\|\leq C\|z\|$, for all $z$ near 0, and
(2)
$\frac{1}{D}\|f(z)\|\leq\|g\circ\varphi(z)\|\leq D\|f(z)\|$, for all $z$ near 0.
At this point, we finish this overview on local metric version of the Zariski’s multiplicity question and we start to consider the Lipschitz geometry at infinity of complex algebraic sets.
Let $f\colon\mathbb{C}^{n}\to\mathbb{C}$ be a reduced polynomial and $X=V(f)$. The degree of the polynomial $f$ is an important integer number associated to $X$; it is called the degree of $X$. According to the next example, it is hopeless that degree of $X=V(f)$ comes as a $C^{\infty}$ right invariant. In fact, the degree is not even $C^{\infty}$ right invariant in families. In particular, the degree is not a topological invariant of the embedded subset $X\subset\mathbb{C}^{n}$.
Example 1.1.
For each $t\in\mathbb{C}$, let $f_{t}:\mathbb{C}^{2}\to\mathbb{C}$ be the polynomial given by $f(x,y)=y-tx^{2}$. Let $\varphi_{t}:\mathbb{C}^{3}\to\mathbb{C}^{2}$ be the polynomial mapping given by $\varphi(x,y,t)=(x,y-tx^{2})$. Then, $\varphi_{t}:=\varphi(\cdot,t):\mathbb{C}^{2}\to\mathbb{C}^{2}$ is a polynomial automorphism (in particular it is a smooth diffeomorphism) such that $f_{t}=f_{0}\circ\varphi_{t}$, for all $t\in\mathbb{C}$. However, ${\rm deg}(V(f_{0}))=1$ and ${\rm deg}(V(f_{t}))=2$, for all $t\not=0$.
In this paper, we deal with the following metric question:
Question A1($d$) Let $X\subset\mathbb{C}^{n}$ and $Y\subset\mathbb{C}^{m}$ be two complex algebraic sets with $\dim X=\dim Y=d$. If $X$ and $Y$ are bi-Lipschitz homeomorphic at infinity, in the sense that there exist compact subsets $K_{1}\subset X$, $K_{2}\subset Y$ and a bi-Lipschitz homeomorphism $\varphi\colon X\setminus K_{1}\rightarrow Y\setminus K_{2}$, then is it true that ${\rm deg}(X)={\rm deg}(Y)$?
The author jointly with A. Fernandes showed in [7] that degree 1 comes as a bi-Lipschitz invariant at infinity of complex algebraic subsets (see Section 2, for a definition of degree for higher codimension algebraic sets in $\mathbb{C}^{n}$) and in [8] they showed that the Question A1($d$) has a positive answer for $d=1$ and, for each $d\in\mathbb{N}$, A1($d$) and Ã1($d$) are equivalent questions. Still in [8], it was showed that the degree of a complex algebraic surface in $\mathbb{C}^{3}$ is a bi-Lipschitz (embedded) invariant at infinity.
Let us describe how this paper is organized. Section 2 is dedicated to present the notions of degree of complex algebraic subsets in $\mathbb{C}^{n}$, tangent cones at infinity and, also, bi-Lipschitz homeomorphisms at infinity of such subsets. Section 3 is dedicated to prove the main results of the paper, we prove that the degree of a complex algebraic set is invariant under bi-Lipschitz homeomorphism with Lipschitz constant close enough to 1. In particular, in contrast with the example 1.1, we receive that the degree is constant in a bi-Lipschitz equisingular at infinity family.
Moreover, we prove that if two polynomials are weakly rugose equivalent at infinity, then they have the same degree. In particular, we receive that two polynomials have the same degree, if they are rugose equivalent at infinity or bi-Lipschitz contact equivalent at infinity or bi-Lipschitz right-left equivalent at infinity.
2. Preliminaries
2.1. Degree
Let us begin this subsection by recalling some basic facts about degree of complex algebraic sets, for more details see [2]. Let $\iota\colon\mathbb{C}^{n}\hookrightarrow\mathbb{P}^{n}$ be the embedding given by $\iota(x_{1},\cdots,x_{n})=[1:x_{1}:\cdots:x_{n}]$ and let $p\colon\mathbb{C}^{n+1}\setminus\{0\}\to\mathbb{P}^{n}$ be the projection mapping given by $p(x_{0},x_{1},\dots,x_{n})=[x_{0}:x_{1}:\cdots:x_{n}]$.
Remark 2.1.
Let $A$ be an algebraic set in $\mathbb{P}^{n}$ and $X$ be an algebraic set in $\mathbb{C}^{n}$. Then $\widetilde{A}=p^{-1}(A)\cup\{0\}$ is a homogeneous complex algebraic set in $\mathbb{C}^{n+1}$ and the closure $\overline{\iota(X)}$ of $\iota(X)$ in $\mathbb{P}^{n}$ is an algebraic set in $\mathbb{P}^{n}$.
Definition 2.2.
Let $A$ be an algebraic set in $\mathbb{P}^{n}$. We define the degree of $A$ by ${\rm deg}(A)=m(\widetilde{A},0)$, where $m(\widetilde{A},0)$ is the multiplicity of $\widetilde{A}$ at $0\in\mathbb{C}^{n+1}$.
Definition 2.3.
Let $X$ be a complex algebraic set in $\mathbb{C}^{n}$. We define the degree of $X$ by ${\rm deg}(X)={\rm deg}(\overline{\iota(X)})$.
Remark 2.4.
Let $f\colon\mathbb{C}^{n}\to\mathbb{C}$ is a reduced polynomial and $X=V(f)$. Then, ${\rm deg}(X)={\rm deg}(f)$.
2.2. Tangent cones
In this subsection, we set the exact notion of tangent cone that we will use along the paper and we list some of its properties.
Definition 2.5.
Let $A\subset\mathbb{R}^{n}$ be an unbounded subset. We say that $v\in\mathbb{R}^{n}$ is a tangent vector of $A$ at infinity if there is a sequence of points $\{x_{i}\}_{i\in\mathbb{N}}\subset A$ such that $\lim\limits_{i\to\infty}\|x_{i}\|=+\infty$ and there is a sequence of positive numbers $\{t_{i}\}_{i\in\mathbb{N}}\subset\mathbb{R}^{+}$ such that
$$\lim\limits_{i\to\infty}\frac{1}{t_{i}}x_{i}=v.$$
Let $C_{\infty}(A)$ denote the set of all tangent vectors of $A$ at infinity. This subset $C_{\infty}(A)\subset\mathbb{R}^{n}$ is called the tangent cone of $A$ at infinity.
Proposition 2.6 (Proposition 4.4 in [7]).
Let $Z\subset\mathbb{R}^{n}$ be an unbounded semialgebraic set. A vector $v\in\mathbb{R}^{n}$ belongs to $C_{\infty}(Z)$ if, and only if, there exists a continuous semialgebraic curve $\gamma\colon(\varepsilon,+\infty)\to Z$ such that $\lim\limits_{t\to+\infty}|\gamma(t)|=+\infty$ and $\gamma(t)=tv+o_{\infty}(t),$ where $g(t)=o_{\infty}(t)$ means $\lim\limits_{t\to+\infty}\frac{g(t)}{t}=0$.
Let $X\subset\mathbb{C}^{n}$ be a complex algebraic subset. Let $\mathcal{I}(X)$ be the ideal of $\mathbb{C}[x_{1},\cdots,x_{n}]$ given by the polynomials which vanishes on $X$. For each $f\in\mathbb{C}[x_{1},\cdots,x_{n}]$, let us denote by $f^{*}$ the homogeneous polynomial composed of the monomials in $f$ of maximum degree.
Proposition 2.7 (Theorem 1.1 in [11]).
Let $X\subset\mathbb{C}^{n}$ be a complex algebraic subset. Then, $C_{\infty}(X)$ is the affine algebraic set $V(\langle f^{*};\,f\in\mathcal{I}(X)\rangle)$.
Among other things, this result above says that tangent cones at infinity of complex algebraic sets in $\mathbb{C}^{n}$ are complex algebraic subsets as well.
2.3. Bi-Lipschitz homeomorphism at infinity
Definition 2.8.
Let $X\subset\mathbb{R}^{n}$ and $Y\subset\mathbb{R}^{m}$ be two subsets. We say that $X$ and $Y$ are bi-Lipschitz homeomorphic at infinity, if there exist compact subsets $K\subset\mathbb{R}^{n}$ and $\widetilde{K}\subset\mathbb{R}^{m}$ and a bi-Lipschitz homeomorphism $\phi\colon X\setminus K\rightarrow Y\setminus\widetilde{K}$.
We finish this Section reminding the invariance of the relative multiplicities at infinity under bi-Lipschitz homeomorphisms at infinity.
Proposition 2.9 (Theorem 3.1 in [8]).
Let $X\subset\mathbb{C}^{n}$ and $Y\subset\mathbb{C}^{m}$ be complex algebraic subsets, with pure dimension $p=\dim X=\dim Y$, and let $X_{1},\dots,X_{r}$ and $Y_{1},\dots,Y_{s}$ be the irreducible components of the tangent cones at infinity $C_{\infty}(X)$ and $C_{\infty}(Y)$ respectively. If $X$ and $Y$ are bi-Lipschitz homeomorphic at infinity, then $r=s$ and, up to a re-ordering of indices, $k_{X}^{\infty}(X_{j})=k_{Y}^{\infty}(Y_{j})$, $\forall\ j$.
3. Degree as a bi-Lipschitz Invariant at Infinity
3.1. Degree of complex algebraic sets
Theorem 3.1.
Let $X\subset\mathbb{C}^{n}$ and $Y\subset\mathbb{C}^{m}$ be two complex algebraic sets with $\dim X=\dim Y=d$ and $M=\max\{{\rm deg}(X),{\rm deg}(Y)\}$. If there are compact subsets $K\subset\mathbb{C}^{n}$ and $\widetilde{K}\subset\mathbb{C}^{m}$, constants $C_{1},C_{2}>0$ and a bi-Lipschitz homeomorphism $\varphi:X\setminus K\to Y\setminus\widetilde{K}$ such that
$$\frac{1}{C_{1}}\|x-y\|\leq\|\varphi(x)-\varphi(y)\|\leq C_{2}\|x-y\|,\quad%
\forall x,y\in X\setminus K$$
and $(C_{1}C_{2})^{2d}\leq 1+\frac{1}{M}$, then ${\rm deg}(X)={\rm deg}(Y).$
Proof.
Let $X_{1},\dots,X_{r}$ and $Y_{1},\dots,Y_{s}$ be the irreducible components of the tangent cones at infinity $C_{\infty}(X)$ and $C_{\infty}(Y)$ respectively. Looking $X$ and $Y$, respectively, as the sets $X\times\{0\}$ and $\{0\}\times Y$ in $\mathbb{C}^{n+m}=\mathbb{C}^{n}\times\mathbb{C}^{m}$, we have by proof of the Lemma 3.1 in [14], that there are $C>0$ and a bi-Lipschitz homeomorphism $\Phi:\mathbb{C}^{n+m}\to\mathbb{C}^{n+m}$ such that $\Phi|_{X\setminus K}=\varphi$ and
$$\frac{1}{C}\|x-y\|\leq\|\Phi(x)-\Phi(y)\|\leq C\|x-y\|,\quad\forall x,y\in X%
\setminus K.$$
Thus, by proof of the Theorem 4.5 in [7], there is a bi-Lipschitz homeomorphism $d\varphi:\mathbb{C}^{n+m}\to\mathbb{C}^{n+m}$ such that $d\varphi(0)=0$, $d\varphi(C_{\infty}(X))=C_{\infty}(Y)$ and
$$\frac{1}{C}\|v-w\|\leq\|d\varphi(v)-d\varphi(w)\|\leq C\|v-w\|,\quad\forall v,%
w\in C_{\infty}(X).$$
Moreover, there is a sequence $\{t_{j}\}\subset\mathbb{N}$ such that $\varphi_{n_{j}}\rightarrow d\varphi$ uniformly on compact subsets of $\mathbb{C}^{n+m}$, where each mapping $\varphi_{k}:\mathbb{C}^{n+m}\to\mathbb{C}^{n+m}$ is given by $\varphi_{k}(v)=\frac{1}{k}\Phi(kv)$ for all $v\in\mathbb{C}^{n+m}$.
Claim. $\frac{1}{C_{1}}\|v-w\|\leq\|d\varphi(v)-d\varphi(w)\|\leq C_{2}\|v-w\|,\quad%
\forall v,w\in C_{\infty}(X).$
Let $v\in C_{\infty}(X)$. By Proposition 2.6, there is a proper curve $\gamma\colon(\varepsilon,+\infty)\to X$ such that $\lim\limits_{t\to+\infty}|\gamma(t)|=+\infty$ and $\gamma(t)=tv+o_{\infty}(t)$. Then, we obtain
$$\|\frac{\Phi(t_{j}v)}{t_{j}}-\frac{\Phi(\gamma(t_{j}))}{t_{j}}\|=\frac{o_{%
\infty}(t_{j})}{t_{j}}\to 0\mbox{ as }j\to+\infty.$$
As $\Phi|_{X\setminus K}=\varphi$, we have $\frac{\Phi(\gamma(t_{j}))}{t_{j}}=\frac{\varphi(\gamma(t_{j}))}{t_{j}}\to d%
\varphi(v)$ as $j\to+\infty$. Therefore, if $v,w\in C_{\infty}(X)$, there are sequences $\{x_{j}\},\{y_{j}\}\subset X$ such that $\lim\frac{1}{t_{j}}x_{j}=v$ and $\lim\frac{1}{t_{j}}y_{j}=w$. Thus,
$$\frac{1}{C_{1}}\|\frac{x_{j}}{t_{j}}-\frac{y_{j}}{t_{j}}\|\leq\|\frac{\varphi(%
x_{j})}{t_{j}}-\frac{\varphi(y_{j})}{t_{j}}\|\leq C_{2}\|\frac{x_{j}}{t_{j}}-%
\frac{y_{j}}{t_{j}}\|.$$
Passing the limit $j\to+\infty$, we receive
$$\frac{1}{C_{1}}\|v-w\|\leq\|d\varphi(v)-d\varphi(w)\|\leq C_{2}\|v-w\|.$$
By Proposition 2.9, $r=s$ and, up to a re-ordering of indices, $k_{X}^{\infty}(X_{j})=k_{Y}^{\infty}(Y_{j})$ and $Y_{j}=d\varphi(X_{j})$, $\forall\ j$. Moreover, we know that (see Subsection 1.4 in [8])
$${\rm deg}(X)=\sum\limits_{j=0}^{r}k_{X}^{\infty}(X_{j})\cdot{\rm deg}(X_{j})$$
and
$${\rm deg}(Y)=\sum\limits_{j=0}^{r}k_{Y}^{\infty}(Y_{j})\cdot{\rm deg}(Y_{j}).$$
In particular, for each $j$, $M_{j}=\max\{{\rm deg}(X_{j}),{\rm deg}(Y_{j})\}\leq M$. Since $X_{j}$ and $Y_{j}$ are homogeneous algebraic sets, we have ${\rm deg}(X_{j})=m(X_{j},0)$ and ${\rm deg}(Y_{j})=m(Y_{j},0)$. By Theorem 1 in [3], ${\rm deg}(X_{j})={\rm deg}(Y_{j})$, for all $j$. Therefore,
$${\rm deg}(X)={\rm deg}(Y).$$
∎
Notation. Let $A\subset\mathbb{R}^{m}$, $B\subset\mathbb{R}^{k}$ and $f:A\to B$ be a Lipschitz function. We define the Lipschitz constant of $f$ by
$$Lip(f):=\sup\left\{\frac{\|f(x)-f(y)\|}{\|x-y\|};x,y\in A\mbox{ and }x\not=y%
\right\}.$$
Definition 3.2.
The family of complex algebraic sets $\{X_{t}\}_{t\in[0,1]}$ in $\mathbb{C}^{n}$ is said to be bi-Lipschitz equisingular at infinity, if there are a compact subset $K\subset\mathbb{C}^{n}$ and a mapping $\varphi:(X_{0}\setminus K)\times[0,1]\to\mathbb{C}^{n}$ such that
(i)
for each $t\in[0,1]$, $\varphi((X_{0}\setminus K)\times\{t\})=X_{t}\setminus K_{t}$ for some compact $K_{t}\subset\mathbb{C}^{n}$ and $\varphi_{t}:=\varphi(\cdot,t):X_{0}\setminus K\to X_{t}\setminus K_{t}$ is a bi-Lipschitz homeomorphism with $\varphi_{0}={\rm id}$ and
(ii)
$\lim\limits_{t\to 0^{+}}Lip(\varphi_{t})=\lim\limits_{t\to 0^{+}}Lip(\varphi_{%
t}^{-1})=1$.
In this case, we say that $\varphi$ is a bi-Lipschitz deformation of $X_{0}$ at infinity.
Theorem 3.3.
Let $\{X_{t}\}_{t\in[0,1]}$ be a family of complex algebraic sets. If $\{X_{t}\}_{t\in[0,1]}$ is bi-Lipschitz equisingular at infinity, then there is $\delta\in(0,1]$ such that ${\rm deg}(X_{t})={\rm deg}(X_{0})$, for all $t\in[0,\delta]$.
Proof.
Let $\varphi:(X_{0}\setminus K)\times[0,1]\to\mathbb{C}^{n}$ be a bi-Lipschitz deformation of $X_{0}$ at infinity. Thus, $\varphi_{t}:=\varphi(\cdot,t):(X_{0}\setminus K)\to X_{t}\setminus K_{t}$ is a bi-Lipschitz homeomorphism and $\lim\limits_{t\to 0}C_{t}=\lim\limits_{t\to 0}C_{t}^{\prime}=1$, where $C_{t}$ and $C_{t}^{\prime}$ are, respectively, the Lipschitz constants of the mappings $\varphi_{t}$ and $\varphi_{t}^{-1}$. As was made in the proof of the Theorem 3.1, for each $t\in[0,1]$ there is $\psi:C_{\infty}(X_{0})\to C_{\infty}(X_{t})$ such that
$$\frac{1}{C_{t}^{\prime}}\|v-w\|\leq\|\psi(v)-\psi(w)\|\leq C_{t}\|v-w\|,\quad%
\forall v,w\in C_{\infty}(X_{t}).$$
Thus, if $Y_{0,1},...,Y_{0,r}$ are the irreducible components of $C_{\infty}(X_{0})$, then by Lemma A.8 in [9], for each $i=1,...,r$, there is an irreducible component $Y_{t,i}$ of $C_{\infty}(X_{t})$ such that $\varphi_{t}(Y_{0,i})=Y_{t,i}$, since $\varphi_{t}$ is, in particular, a homeomorphism. By Theorem 2 in [3], there is $t_{i}\in(0,1]$ such that ${\rm deg}(Y_{t,i},0)={\rm deg}(Y_{0,i},0)$ for all $t\in[0,t_{i}]$, since $Y_{0,i}$ and $Y_{t,i}$ are homogeneous complex algebraic sets. Using that the relative multiplicities at infinity are bi-Lipschitz invariant at infinity, we obtain ${\rm deg}(X_{t},0)={\rm deg}(X_{0},0)$ for all $t\in[0,\delta]$, where $\delta=\min\{t_{1},...,t_{r}\}$.
∎
Remark 3.4.
The Corollary 3.3 above is still hold true even if the family $\{\varphi_{t}\}$ of bi-Lipschitz homeomorphisms does not satisfies $\varphi_{0}={\rm id}$.
3.2. Degree of polynomials
Definition 3.5.
We say that two polynomials $f,g:\mathbb{C}^{n}\to\mathbb{C}^{m}$ are rugose equivalent at infinity, if there are compact subsets $K,\widetilde{K}\subset\mathbb{C}^{n}$, constants $C_{1},C_{2}>0$ and a bijection $\varphi:\mathbb{C}^{n}\setminus K\to\mathbb{C}^{n}\setminus\widetilde{K}$ such that
(1)
$\frac{1}{C_{1}}\|x-y\|\leq\|\varphi(x)-\varphi(y)\|\leq C_{1}\|x-y\|$, for all $x\in\mathbb{C}^{n}\setminus K$ and $y\in f^{-1}(0)\setminus K$;
(2)
$\frac{1}{C_{2}}\|f(x)\|\leq\|g\circ\varphi(x)\|\leq C_{2}\|f(x)\|,\quad\forall
x%
\in\mathbb{C}^{n}\setminus K.$
Definition 3.6.
Let $F=(f_{1},\cdots,f_{m}):\mathbb{C}^{n}\to\mathbb{C}^{m}$ be a polynomial mapping. We define the degree of $F$ by
$${\rm deg}(F)=\max\{{\rm deg}(f_{1}),\cdots,{\rm deg}(f_{m})\}.$$
The next result was proved in ([8], Theorem 3.7). However, here we present a direct proof without to use the global Łojasiewicz inequality proved in [10].
Theorem 3.7.
Let $f,g:\mathbb{C}^{n}\to\mathbb{C}$ be two polynomials. If $f$ and $g$ are rugose equivalent at infinity, then ${\rm deg}(f)={\rm deg}(g)$.
Proof.
Let us denote $X=\{x\in\mathbb{C}^{n};\,f(x)=0\}$ and $Y=\{x\in\mathbb{C}^{n};\,g(x)=0\}$. We have that $X$ and $Y$ are bi-Lipschitz homeomorphic at infinity. By Theorem 4.5 in [7] and Proposition 2.7, $C_{\infty}(X)$ and $C_{\infty}(Y)$ are closed and bi-Lipschitz homeomorphic sets. By hypotheses, there are compact subsets $K,\widetilde{K}\subset\mathbb{C}^{n}$, positive constants $C_{1}$ and $C_{2}$ and a bijection $\varphi:\mathbb{C}^{n}\setminus K\to\mathbb{C}^{n}\setminus\widetilde{K}$ such that
$$\frac{1}{C_{1}}\|x-y\|\leq\|\varphi(x)-\varphi(y)\|\leq C_{1}\|x-y\|,\,\forall
x%
\in\mathbb{C}^{n}\setminus K\mbox{ and }y\in F^{-1}(0)\setminus K$$
and
$$\frac{1}{C_{2}}\|f(x)\|\leq\|g\circ\varphi(x)\|\leq C_{2}\|f(x)\|,\quad\forall
x%
\in\mathbb{C}^{n}\setminus K$$
Let us suppose that ${\rm deg}(f)<{\rm deg}(g)=k$. Let $S=\{n_{j}\}_{j\in\mathbb{N}}\subset\mathbb{N}$ be a sequence such that
$$n_{j}\to+\infty\quad\mbox{and}\quad\frac{\varphi(n_{j}v)}{n_{j}}\to d\varphi(v),$$
like in the Theorem 4.5 in [7]. Moreover, $d\varphi:\mathbb{C}^{n}\to\mathbb{C}^{n}$ is a bi-Lipschitz homeomorphism. Then, there is $v\in\mathbb{C}^{n}$ such that $d\varphi(v)\in\mathbb{C}^{n}\setminus\{x\in\mathbb{C}^{n};\,g^{*}(x)=0\}$, where $g^{*}$ is the homogeneous polynomial composed of the monomials in $g$ of maximum degree. Therefore,
$$\frac{\|g\circ\varphi(n_{j}v)\|}{n_{j}^{k}}\leq C_{2}\frac{\|f(n_{j}v)\|}{n_{j%
}^{k}},\quad\forall n_{j}\in S.$$
By taking $j\to+\infty$, we obtain $\|g^{*}(d\varphi(v))\|\leq 0$, which is a contraction. Then, ${\rm deg}(f)\geq{\rm deg}(g)=k$ and by using $\varphi^{-1}$ instead of $\varphi$, we obtain the other inequality. Therefore, ${\rm deg}(g)={\rm deg}(f)$.
∎
Definition 3.8.
We say that two polynomial mappings $F,G:\mathbb{C}^{n}\to\mathbb{C}^{m}$ are weakly rugose equivalent at infinity, if there are compact subsets $K,\widetilde{K}\subset\mathbb{C}^{n}$, constants $C_{1},C_{2}>0$ and a bijection $\varphi:\mathbb{C}^{n}\setminus K\to\mathbb{C}^{n}\setminus\widetilde{K}$ such that
(1)
there exist $y_{0}\in\mathbb{C}^{n}\setminus K$ and $w_{0}\in\mathbb{C}^{n}\setminus\widetilde{K}$ such that $\|\varphi(x)-\varphi(y_{0})\|\leq C_{1}\|x-y_{0}\|$, for all $x\in\mathbb{C}^{n}\setminus K$ and $\|\varphi^{-1}(z)-\varphi^{-1}(w_{0})\|\leq C_{1}\|z-w_{0}\|$, for all $z\in\mathbb{C}^{n}\setminus\widetilde{K}$;
(2)
$\frac{1}{C_{2}}\|F(x)\|\leq\|G\circ\varphi(x)\|\leq C_{2}\|F(x)\|,\quad\forall
x%
\in\mathbb{C}^{n}\setminus K.$
Let $f:\mathbb{C}^{n}\to\mathbb{C}$ be a polynomial. Then, for each $r>0$, we define
$$\delta_{r,\infty}(f)=\inf\{\delta;\,\frac{|f(z)|}{\|z\|^{\delta}}\mbox{ is %
bounded on }\mathbb{C}^{n}\setminus B_{r}(0)\}.$$
Remark that $\delta_{r,\infty}(f)$ does not depend of $r>0$. Thus, we define this common number by $\delta_{\infty}(f)$.
Proposition 3.9.
Let $f:\mathbb{C}^{n}\to\mathbb{C}$ be a polynomial. Then, ${\rm deg}(f)=\delta_{\infty}(f)$.
Proof.
If $\delta<d={\rm deg}(f)$ and writing $f=f_{0}+f_{1}+...+f_{d}$, then we choose $v\not\in V(f_{d})$. Thus, $\lim\limits_{t\to+\infty}\frac{|f(tv)|}{t^{\delta}}=+\infty$. Then, $\delta_{\infty}(f)\geq{\rm deg}(f)$.
If $\delta>{\rm deg}(f)$, then $\lim\limits_{\|z\|\to+\infty}\frac{|f(z)|}{\|z\|^{\delta}}=0.$ Thus, there exists $r>0$ such that $\frac{|f(z)|}{\|z\|^{\delta}}\leq 1$, for all $z\not\in\mathbb{C}^{n}\setminus B_{r}(0)$. This implies $\delta_{\infty}(f)\leq{\rm deg}(f)$. Therefore, $\delta_{\infty}(f)={\rm deg}(f)$.
∎
Theorem 3.10.
Let $f,g:\mathbb{C}^{n}\to\mathbb{C}$ be two polynomials. If $f$ and $g$ are weakly rugose equivalent at infinity, then ${\rm deg}(f)={\rm deg}(g)$.
Proof.
Let $r>0$ be a positive number satisfying $\widetilde{r}=C_{1}(\|\varphi(z_{0})\|+r)-\|z_{0}\|>0$ and $K\subset B_{\widetilde{r}}(0)$. Then,
$$\displaystyle\frac{|f(z)|}{\|z\|^{\delta}}$$
$$\displaystyle=$$
$$\displaystyle\frac{|f(z)|}{\|\varphi(z)\|^{\delta}}\frac{\|\varphi(z)\|^{%
\delta}}{\|z\|^{\delta}}$$
$$\displaystyle\leq$$
$$\displaystyle C_{2}\frac{|g(\varphi(z))|}{\|\varphi(z)\|^{\delta}}\left(\frac{%
\|\varphi(z)-\varphi(z_{0})\|+\|\varphi(z_{0})\|}{\|z\|}\right)^{\delta}$$
$$\displaystyle\leq$$
$$\displaystyle C_{2}\frac{|g(\varphi(z))|}{\|\varphi(z)\|^{\delta}}\left(C_{1}%
\frac{\|z-z_{0}\|}{\|z\|}+\frac{\|\varphi(z_{0})\|}{\|z\|}\right)^{\delta}$$
$$\displaystyle\leq$$
$$\displaystyle C_{2}\frac{|g(\varphi(z))|}{\|\varphi(z)\|^{\delta}}\left(C_{1}+%
C_{1}\frac{\|z_{0}\|}{r}+\frac{\|\varphi(z_{0})\|}{r}\right)^{\delta}$$
$$\displaystyle=$$
$$\displaystyle C\frac{|g(\varphi(z))|}{\|\varphi(z)\|^{\delta}},$$
for all $z\in\mathbb{C}^{n}\setminus B_{\widetilde{r}}(0)$. Thus, if $\frac{|g(w)|}{\|w\|^{\delta}}$ is bounded on $\mathbb{C}^{n}\setminus B_{r}(0)$, then $\frac{|f(z)|}{\|z\|^{\delta}}$ is bounded on $\mathbb{C}^{n}\setminus B_{\widetilde{r}}(0)$. Therefore, by Proposition 3.9, ${\rm deg}(f)\leq{\rm deg}(g)$. Similarly, we obtain ${\rm deg}(g)\leq{\rm deg}(f)$. Thus, we have the equality ${\rm deg}(g)={\rm deg}(f)$.
∎
Definition 3.11.
We say that two polynomials $f,g:\mathbb{C}^{n}\to\mathbb{C}$ are bi-Lipschitz contact equivalent at infinity, if there are compact subsets $K,\widetilde{K}\subset\mathbb{C}^{n}$, a constant $C>0$ and a bi-Lipschitz homeomorphism $\varphi:\mathbb{C}^{n}\setminus K\to\mathbb{C}^{n}\setminus\widetilde{K}$ such that
$$\frac{1}{C}\|f(x)\|\leq\|g\circ\varphi(x)\|\leq C\|f(x)\|,\quad\forall x\in%
\mathbb{C}^{n}\setminus K.$$
Definition 3.12.
We say that two polynomials $f,g:\mathbb{C}^{n}\to\mathbb{C}$ are bi-Lipschitz right-left equivalent at infinity, if there are compact subsets $K,\widetilde{K}\subset\mathbb{C}^{n}$, a constant $C>0$ and bi-Lipschitz homeomorphisms $\varphi:\mathbb{C}^{n}\setminus K\to\mathbb{C}^{n}\setminus\widetilde{K}$ and $\phi:\mathbb{C}\to\mathbb{C}$ such that $f(x)=\phi\circ g\circ\varphi(x),$ $\forall x\in\mathbb{C}^{n}\setminus K.$
It is direct of the definitions the following result
Proposition 3.13.
Let $f,g:\mathbb{C}^{n}\to\mathbb{C}$ be two polynomials. Let us consider the following statements:
(1)
$f$ and $g$ are bi-Lipschitz right-left equivalent at infinity;
(2)
$f$ and $g$ are bi-Lipschitz contact equivalent at infinity;
(3)
$f$ and $g$ are rugose equivalent at infinity;
(4)
$f$ and $g$ are weakly rugose equivalent at infinity.
Then, $(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)$.
We finish this paper by stating some direct consequences of Theorem 3.10 and Proposition 3.13.
Corollary 3.14.
Let $f,g:\mathbb{C}^{n}\to\mathbb{C}$ be two polynomials. If $f$ and $g$ are rugose equivalent at infinity, then ${\rm deg}(f)={\rm deg}(g)$.
Corollary 3.15.
Let $f,g:\mathbb{C}^{n}\to\mathbb{C}$ be two polynomials. If $f$ and $g$ are bi-Lipschitz contact equivalent at infinity, then ${\rm deg}(f)={\rm deg}(g)$.
Corollary 3.16.
Let $f,g:\mathbb{C}^{n}\to\mathbb{C}$ be two polynomials. If $f$ and $g$ are bi-Lipschitz right-left equivalent at infinity, then ${\rm deg}(f)={\rm deg}(g)$.
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Motion of charged particles around a rotating black hole
in a magnetic field
[5mm]
A. N. Aliev ${}^{\dagger}$ and N. Özdemir ${}^{\dagger\ddagger}$
[2mm]
${}^{\dagger}$ Feza Gürsey Institute, P.K. 6 Çengelköy,
81220 Istanbul, Turkey
${}^{\ddagger}$ ITU, Faculty of Sciences and Letters, Department of Physics,
80626 Maslak, Istanbul, Turkey
[5mm] December 2, 2020
111email address: [email protected]
ABSTRACT
We study the effects of an external magnetic field, which is assumed
to be uniform at infinity, on the marginally stable
circular motion of charged particles in the equatorial plane of
a rotating black hole. We show that the magnetic field has
its greatest effect in enlarging the region of stability
towards the event horizon of the black hole.
Using the Hamilton-Jacobi formalism
we obtain the basic equations governing the marginal stability
of the circular orbits and their associated energies and angular momenta.
As instructive examples, we review the case of the marginal
stability of the circular orbits in the Kerr metric,
as well as around a Schwarzschild black hole in a magnetic field.
For large enough values of the magnetic field around a maximally
rotating black hole we find the limiting analytical solutions
to the equations governing the radii of marginal stability.
We also show that the presence of a strong magnetic field
provides the possibility of relativistic motions in
both direct and retrograde innermost stable circular orbits
around a Kerr black hole.
Key words: gravitation, accretion discs - black hole physics -
magnetic fields.
1 Introduction
New observational evidence for black holes provides new motivations
for the investigation of the general relativistic dynamics of
particles and electromagnetic fields in the vicinity of the black holes.
We shall start with a brief description of the situation.
The results of astronomical observations over the last decade
continue to point insistently to the existence of
stellar-mass and supermassive black holes
in some X-ray binary systems and in galactic centres
(see Horowitz $\&$ Teukolsky 1999; Rees 1998 for reviews).
The typical examples of the stellar-mass black holes
in X-ray binaries are Cyg X-1 discovered back in 1971, the
X-ray source LMC X-3 in the Large Magellanic Cloud, as well as
the source in A0620-00 discovered in 1975
and a number of recently discovered sources, such as V404 Cyg,
GS 2000+25, GRO J0422+32 (for full list see Charles 1999).
New observational data, such as the detection of broad iron
fluorescence lines and maser emission lines of water in the spectra
provide the strongest suggestion for the
presence of supermassive black holes in the centres of
the active galaxies MCG 6-30-15, NGC 4258 and NGC 1068
Menou, Quataert $\&$ Narayan 1999; Rees 1998; Miyoshi et.al 1995;
Watson $\&$ Wallin 1994).
The supermassive black holes are also strongly believed
to be in the centres of some
low-level active, or non-active, galaxies. For instance,
recent progress in the studies of the
distributions and velocities of stars near the centres of
the giant elliptical galaxy M87, Andromeda M31 and our
own Galaxy have revealed the strong evidence
for the existence of the supermassive black holes in these centres
(Merritt $\&$ Oh 1997; Richstone et al. 1998; Rees 1982, 1998).
On the other hand, a convincing explanation for a
huge amount of energy output from the active cores of the galaxies
associated with the supermassive black holes requires the searches
for mechanisms responsible for the high-level energy release.
One of these mechanisms is the extraction of the rotational energy
from a rotating black hole surrounded by stationary magnetic fields.
The magnetic fields threading the event horizon tap the rotational energy
of the black hole due to the interaction between charged particles
and the induced electric field (Blandford $\&$ Znajek 1977;
Thorne, Price $\&$ Macdonald 1986). The interest in this model
still continues to point out new gravito-electromagnetic phenomena
in the strong and weak field domains around a rotating black hole
(Bic̆ák $\&$ Ledvinka 2000; Chamblin, Emparan $\&$ Gibbons 1998;
Mashhoon 2001; see also Punsly 2001). Another mechanism
responsible for high-level energy release
is gas accretion by a black hole (see Shapiro $\&$ Teukolsky 1983), where
energy is released mostly at the expense of the binding energy of the
particles and the strong gravitational field of the black hole.
It is well known that the binding energy in the innermost stable orbits
of the particles determines the potential efficiency of an
accretion disc. It is about 5.7 % of the rest energy in the
Schwarzschild field, but in the case of a maximally rotating black hole
it approaches 42 % of the rest energy. The observational data
from the core of some galaxies, such as the elliptical galaxy M87 reveal
that the inner part of a gas disc
around a putative supermassive black hole emits non-thermal
radiation and radio waves. The reason for this is believed
to be synchrotron emission from ultrarelativistic electrons
moving in a strong magnetic field in the inner
part of the accretion flow onto the supermassive black hole
(Fabian & Rees 1995; Narayan & Yi 1995; Rees 1998).
This gives us a new impetus to return back once again to the
investigation of the motion of charged particles in the
model of a rotating black hole in a uniform magnetic field,
though much insight into this problem was given in 1980s
(Prasanna $\&$ Vishveshwara 1978; Prasanna 1980;
Wagh, Dhurandeur & Dadhich 1985; Aliev $\&$ Gal’tsov 1989;
see also Frolov $\&$ Novikov 1998 and references therein).
In particular, the works of Prasanna & Vishveshwara (1978) and
Prasanna (1980) have given a comprehensive numerical analysis of
the charged particle motion in a magnetic field superposed on
the Kerr metric by studying the structure
of the effective potential for radial motion and integrating
the complete set of equations of the motion for appropriate
initial conditions.
In this paper we shall study a particular class of
orbits, namely the marginally stable circular orbits
of charged particles in the equatorial plane of a
Kerr black hole immersed in a uniform magnetic field.
The numerical analysis performed
by Prasanna & Vishveshwara (1978) has revealed that the presence
of a uniform magnetic field on the Kerr background
increases the range of stable circular orbits. This result
is confirmed in our model of the marginally stable circular
motion, however we also obtain significant new results:
First of all, we derive the basic equation
determining the region of the marginal stability of the circular orbits,
that comprises only two parameters, namely the rotation parameter
of black hole and the strength of the magnetic field.
We also find the closed analytical expressions for the associated angular
momentum and energy of a charged particle moving in a marginally
stable circular orbit. Further, for a sufficiently large values of
the magnetic field, as well as for a maximum value of the black hole
rotation parameter we find the limiting analytical solutions
for the radii of stability of both direct and
retrograde innermost circular orbits. This allows us not only
to emphasize that the presence of a magnetic field enlarges
the region of stability towards the event horizon, but also
to find the limiting values for this enlargement both for direct
and retrograde motions along with their associated energies
and angular momenta.
Our analytical and numerical calculations show that the combined
effects of a sufficiently strong magnetic field and the rotation of a
black hole give rise to the possibility of relativistic motion of
the charged particles in the innermost stable direct
and retrograde orbits. The existence of relativistic motion
in the innermost stable direct orbits is especially
important new feature worked out in our analysis.
These orbits lie very close to the event horizon
of a rotating black hole and they may provide a mechanism
for synchrotron emission from relativistic charged particles
with the signatures of the strong-gravity domain.
The paper is organized as follows. We shall first review
the solution of the Maxwell equations describing
a uniform magnetic field in the Kerr metric
with a small electric charge (see Section 2). In Section 3 we consider
the separation of variables in the Hamilton-Jacobi equation and derive
the effective potential for the radial motion of charged particles
around a Kerr black hole in a uniform magnetic field. These results are
used in Section 4 to obtain the basic equations governing
the region of the marginal stability of the circular orbits and their
associated energies and angular momenta. To make the analysis more
transparent in the general case, we shall first present
the results of analytical and numerical
calculations for the marginal stability of the circular orbits
in the pure Kerr metric, as well as in a uniform magnetic field in the
Schwarzchild metric (see Sections 4.1 and 4.2). Section 4.3 is devoted to
a comprehensive analysis of the effect of a uniform magnetic field
on the radii and the assigned energy and angular momentum of the
marginally stable circular orbits for various values of the magnetic field
and the rotation parameter of the black hole.
2 Uniform magnetic field around a rotating black hole
It is well known that an electrically neutral black hole can not have
intrinsic magnetic field (Ginzburg $\&$ Ozernoy 1965). However,
a magnetic field near a black hole can arise due to external factors,
such as the presence of a nearby magnetars or neutron stars. Accretion
of a matter cloud may also form a magnetosphere with a superstrong
magnetic field of magnitude up to
$B_{M}={\textstyle\frac{1}{M}}\simeq 2\times 10^{10}(M/10^{9}M_{\odot})^{-1}G$
around a supermassive black hole (Kardashev 1995).
This value corresponds to the case when the magnetic and
gravitational pressures become equal to each other
(Bic̆ák $\&$ Janis̆ 1985).
It is clear that with this value of the magnetic field
the space-time geometry near a black hole will be significantly distorted
(Aliev $\&$ Gal’tsov 1989), however, when $B\ll B_{M}$
there is definitly a region near the black hole where the space-time
is not distorted by the external magnetic field and the latter can be
considered as a perturbation. We shall consider this case i.e.
a rotating black hole with a small electric charge $(Q\ll M)$ immersed
in an external magnetic field described by a corresponding solution
of the Maxwell equations against the background of the Kerr metric
$$\displaystyle ds^{2}$$
$$\displaystyle=$$
$$\displaystyle\left(1-\frac{2Mr}{\Sigma}\right)dt^{2}+\frac{4Mar}{\Sigma}\sin^{%
2}\theta\,dt\,d\phi-\frac{A\sin^{2}\theta}{\Sigma}\,d\phi^{2}$$
(1)
$$\displaystyle-\Sigma\,\left(\frac{dr^{2}}{\Delta}+d\theta^{2}\right),$$
where $M$ is the mass of the black hole, $a=J/M$ is its angular momentum
per unit mass, $A=(r^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}\theta$,
$\Delta=r^{2}+a^{2}-2Mr$ and $\Sigma=r^{2}+a^{2}\cos^{2}\theta$. We shall
assume a magnetic field around the Kerr black hole
to be uniform at infinity. In this case there is an elegant way
to construct the solution of the Maxwell equation (Wald 1974).
We shall now give a succinct description of this solution.
The Kerr space-time is stationary and axially symmetric that implies
the existence of two commuting Killing vector fields
$\xi^{\mu}_{(t)}=(1,0,0,0)$ and $\xi^{\mu}_{(\phi)}=(0,0,0,1)$.
These fields are used as a $4$-vector potential $A_{\mu}$
describing electromagnetic fields in the Kerr metric
that are superpositions of Coulomb electric and
asymptotically uniform magnetic fields.
Indeed, for Ricci-flat space-times $(R_{\mu\nu}=0)$,
the Maxwell equations for the $4$-vector potential in the covariant
Lorentz gauge
$A^{\mu}_{\;;\;\mu}=0$
$${A^{\mu}_{\;;\;\nu}}^{;\;\nu}=0$$
(2)
and the equation
$${\xi^{\mu}_{\;;\;\nu}}^{;\;\nu}=0$$
(3)
for a Killing vector are the same. The semicolon means covariant
differentiation. We shall take the $4$-vector potential
in the form
$$A^{\mu}=\alpha\,\xi^{\mu}_{(t)}+\beta\,\xi^{\mu}_{(\phi)}$$
(4)
where $\,\alpha\,$ and $\,\beta\,$ are arbitrary parameters.
Next, we calculate the Maxwell $2$-form using
equation (4). We find
$$\displaystyle F$$
$$\displaystyle=$$
$$\displaystyle\frac{2M}{\Sigma}\left(1-\frac{2r^{2}}{\Sigma}\right)(\alpha-%
\beta a\sin^{2}\theta)(dt\wedge dr+a\sin^{2}\theta\,dr\wedge d\phi)$$
(5)
$$\displaystyle-\,\beta\,(2r\sin^{2}\theta\,dr\wedge d\phi+\frac{A}{\Sigma}\sin 2%
\theta\,d\theta\wedge d\phi)$$
$$\displaystyle+\,\frac{2Mar}{\Sigma^{2}}\sin^{2}\theta\,[(\alpha a-\beta(r^{2}+%
a^{2}))\,dt\wedge d\theta$$
$$\displaystyle+\,(r^{2}+a^{2})(\alpha-\beta a\sin^{2}\theta)\,d\theta\wedge d%
\phi]\;,$$
which in the asymptotic region $r\gg M$ reduces to
$$F=-2\beta r\sin\theta\,(\sin\theta\,dr\wedge d\phi+r\cos\theta\,d\theta\wedge d%
\phi).$$
(6)
It follows that $\,\beta=B/2\,$, where $B$ is the strength of
the uniform magnetic field, which is parallel to the rotation axis
of the black hole. As for the remaining parameter $\,\alpha\,$ it can be
specified using the surface integrals for the mass and angular momentum
of the black hole (Bardeen, Carter $\&$ Hawking 1973)
$$M=\frac{1}{8\pi}\int{\xi^{\mu}_{(t)}}^{;\;\nu}d^{2}\Sigma_{\mu\nu}\;\;\;\;\;\;%
\;\;J=-\frac{1}{16\pi}\int{\xi^{\mu}_{(\phi)}}^{;\;\nu}d^{2}\Sigma_{\mu\nu}$$
(7)
along with the Gauss integral for its electric charge
$$Q=\frac{1}{4\pi}\int F^{\mu\nu}d^{2}\Sigma_{\mu\nu}\,.$$
(8)
Finally, we obtain
$$\alpha=aB-\frac{Q}{2M}$$
(9)
Thus, the $4$-vector potential takes the form
$$A^{\mu}=\frac{1}{2}B\,\xi^{\mu}_{(\phi)}-\frac{Q-2aMB}{2M}\,\xi^{\mu}_{(t)}\,.$$
(10)
We see that in addition to the usual magnetic and Coulomb parts
of the electromagnetic field the 4-potential (10)
also contains a contribution proportional to the rotation parameter
of the black hole. It is clear that such a field appears due to
the Faraday induction; a rotation of the Kerr metric produces an induced
electric field, just as a field would be induced by a rotating loop
in a magnetic field. The underlying geometry is such that the induced
potential difference arises between the event horizon and infinitely
remote point
$$\Delta\Phi=\Phi_{H}-\Phi_{\infty}=\frac{Q-2aMB}{2M}\;.$$
(11)
Since an astrophysical object will rapidly neutralize
its electric charge by a process of the selective accretion of charges
from surrounding plasma, the potential difference
(11) will vanish, or equivalently
$$\tilde{Q}=Q-2aMB=0$$
(12)
and the black hole will acquire an inductive electric charge
$Q=2aMB$ . On the other hand, in this very simple model
we see that the Faraday induction may be a possible mechanism
for the tapping of the rotational energy from a black hole. Namely,
this mechanism lies in the basic structure of the model of a supermassive
black hole for active galactic nuclei and quasars
(Blandford $\&$ Znajek 1977; Kardashev 1995) .
3 Motion of charged particles
In a realistic model of gas accretion onto a black hole
energy is released mostly at the expense of the binding energy of the test
particles moving in a strong gravitational field of the black hole.
This may be used to propose an alternative model for the interpretation
of the observational data from the core of a number of galaxies
(see Shapiro $\&$ Teukolsky 1983 and references therein).
Moreover, as we have mentioned above the observations of the core of some
galaxies reveal the existence of
non-thermal radiation and radio waves. The latter can be explained
by synchrotron emission from ultrarelativistic electrons in
a strong magnetic field in the innermost orbits
around a supermassive black hole (Rees 1998).
This makes it very important to investigate in detail
the motion of charged particles
around a rotating black hole in an external magnetic field.
In the following we shall do it using (10) as
a 4-vector potential of the external magnetic field.
For convenience, we shall use the gauge
in which $A^{0}=0$ at infinity. Then we have
$$\tilde{A^{0}}=\frac{\tilde{Q}r(r^{2}+a^{2})}{\Delta\Sigma}\;,\;\;\;\;\;\tilde{%
A^{\phi}}=\frac{B}{2}+\frac{\tilde{Q}ra}{\Delta\Sigma}.$$
(13)
We shall study the motion of the test particles around a rotationg
black hole with zero electric charge $(\tilde{Q}\rightarrow 0)$
using the Hamilton-Jacobi equation
$$g^{\mu\nu}\left(\frac{\partial S}{\partial x^{\mu}}+e\tilde{A}_{\mu}\right)%
\left(\frac{\partial S}{\partial x^{\nu}}+e\tilde{A}_{\nu}\right)=m^{2},$$
(14)
where $e$ and $m$ are the charge and the mass of a test particle,
respectively. Since $t$ and $\phi$ are the Killing variables we
can write the action in the form
$$S=-Et+L\phi+f(r,\theta),$$
(15)
where the conserved quantities $E$ and $L$ are the energy and the angular
momentum of a test particle at infinity. Substituting it into equation
(14) we come to the equation for unseparable part of the action
$$\displaystyle\Delta\left(\frac{\partial f}{\partial r}\right)^{2}+\left(\frac{%
\partial f}{\partial\theta}\right)^{2}-\frac{A}{\Delta}\,E^{2}+\frac{\Sigma-2%
Mr}{\Delta\sin^{2}\theta}\,L^{2}+\frac{4Mra}{\Delta}\,EL-eBL\Sigma$$
$$\displaystyle+\,\frac{1}{4}\,e^{2}B^{2}A\sin^{2}\theta+m^{2}\Sigma=0$$
(16)
It is not possible to separate variables in this equation
in general case, however
one can separate it in the equatorial plane $\theta=\pi/2$. Then we
obtain the equation for radial motion
$$r^{3}\left(\frac{dr}{ds}\right)^{2}=V(E,L,r,\epsilon)$$
(17)
where $s$ is the proper time along the trajectory of a particle and
$$\displaystyle V=(r^{3}+a^{2}r+2Ma^{2})\,E^{2}-(r-2M)\,L^{2}-4MaEL$$
$$\displaystyle-\Delta r\,(1+\frac{\epsilon\,L}{M})-\frac{\Delta}{4M^{2}}\,%
\epsilon^{2}\,(r^{3}+a^{2}r+2Ma^{2})$$
(18)
can be thought of as an effective potential of the radial motion.
Here we have changed $E\rightarrow E/m$ and $L\rightarrow L/m$.
The effective potential besides the energy,
the angular momentum and the radius of the motion
also depends on the dimensionless parameter
$$\epsilon=\frac{eBM}{m}\;,$$
(19)
which characterizes the relative influence of a uniform magnetic field
on the motion of the charged particles. We shall call it as
the influence parameter of the magnetic field.
We note that even for small values of
the magnetic field strength $(B/B_{M}\ll 1)$ the parameter $\,\epsilon\,$
for a particle with high charge-to-mass ratio
(for instance, for electron $e/m\simeq 10^{21}$) may not be small.
4 Marginally stable circular orbits
We shall now describe a particular class of orbits, namely
circular orbits that play an important role in
understanding the essential features of the dynamics of test particles
around a rotating black hole in a magnetic field. Physically,
from the symmetry of the problem it is clear that the circular orbits
are possible in the equatorial plane $\theta=\pi/2$ and
to describe them one must set ${dr\over ds}$ to be zero.
This, in turn, requires vanishing of the effective potential (18)
$$V(E,L,r,\epsilon)=0$$
(20)
along with its first derivative with respect to $r$
$$\frac{\partial V(E,L,r,\epsilon)}{\partial r}=0$$
(21)
The simultaneous solution of these equations would determine
the energy and the angular momentum of the circular motion
in terms of the orbital radius, the hole’s rotation parameter and
the influence parameter of the magnetic field $\,\epsilon\,$.
However, the underlying expressions involve high order polynomial
equations and their analytical solution is
formidable and defies analysis. Therefore in
Prasanna & Vishveshwara (1978) and Prasanna (1980) the authors have
appealed to a numerical analysis of equations
(20) -(21) for different
values of the constants of motion, orbital radii, black hole’s
rotation parameter, as well as the strength parameter of
the magnetic field.
Fortunately, the situation
is changed if one wishes to restrict oneself to considering
only the marginally stable circular orbits.
After all, the stable orbits are most of interest
astrophysically, as the binding energy of the marginally stable
circular orbits is of an energy source for the potential efficiency of an
accretion disc around a black hole. The stability of the
circular motion requires the relation
$$\frac{\partial^{2}V(E,L,r,\epsilon)}{\partial r^{2}}\leq 0$$
(22)
where the case of equality corresponds to the marginally stable
circular motion. It is clear that the simultaneous solutions of
equations (20) -(22) will determine the region of
stability, the associated energy and angular momentum of
the circular orbits. It is remarkable that in this case one can
obtain the close equation governing the stability region which
depends only on the rotation parameter of the black hole and
the influence parameter of the magnetic field. Below we shall
show that for the limiting values of the parameters this equation
admits the simple analytical solutions for the radii of stability.
From equations (21) and (22)
we find that the angular momentum and the energy of a test particle
can be given in the form
$$L=-\epsilon\left(r-{a^{2}\over{3r}}\right)\pm\sqrt{\lambda}$$
(23)
$$E=\left[\eta\mp{\epsilon\over M}\left(1-{{2M}\over{3r}}\right)\sqrt{\lambda}%
\right]^{\;1\over 2},$$
(24)
where we have used the notations
$$\displaystyle\lambda=2M\left(r-{a^{2}\over 3r}\right)+{\epsilon^{2}\over 4M^{2%
}}\left[r^{2}\left(5r^{2}-4Mr+4M^{2}\right)+\right.$$
$$\displaystyle\left.{2\over 3}\,a^{2}\left(5r^{2}-6Mr+2M^{2}\right)+a^{4}\left(%
1+{4M^{2}\over 9r^{2}}\right)\right]$$
(25)
and
$$\eta=1-\frac{2M}{3r}-{\epsilon^{2}\over 6}\left[4-5{\,r^{2}\over M^{2}}-{a^{2}%
\over M^{2}}\left(3-{2M\over r}+{4M^{2}\over 3r^{2}}\right)\right]$$
(26)
These expressions depend on the radius of stability of a circular orbit
which yet has to be determined. Substituting (23) and
(24) into equation (20) we obtain the
equation
$$\displaystyle\left(6Mr-r^{2}+3a^{2}-{4a^{2}M\over r}\right)\left(1\mp{\epsilon%
\over M}\sqrt{\lambda}\right)$$
$$\displaystyle+\,\epsilon^{2}\left[r^{2}\left(6-{4r\over M}+{9\over 2}{r^{2}%
\over M^{2}}-{r^{3}\over M^{3}}\right)\right.$$
$$\displaystyle\left.-a^{2}\left(3+{8\over 3}{r\over M}-{5r^{2}\over M^{2}}%
\right)+{3\over 2}{a^{4}\over M^{2}}\left(1-{2M\over 3r}+{8M^{2}\over 9r^{2}}%
\right)\right]$$
$$\displaystyle\mp\,6a\left[\sqrt{\lambda}\mp\epsilon\left(r-{a^{2}\over{3r}}%
\right)\right]\left[\eta\mp{\epsilon\over M}\left(1-{2\over 3}{M\over r}\right%
)\sqrt{\lambda}\right]^{1\over 2}=0.$$
(27)
The solution of this equation will determine the radii of the
marginally stable circular orbits as functions of the
rotation parameter $\,a\,$, as well as
of the influence parameter of the magnetic field $\,\epsilon\,$.
It should be noted that the circular motions will occur in
direct and retrograde orbits depending on whether the particles are
corotating $\,(L>0\,)$, or counterrotating $\,(L<0)\,$ with respect
to the rotation of a black hole. In addition, the presence of
a uniform magnetic field will produce the Larmor and anti-Larmor
motions depending on whether the Lorentz force points
toward a black hole, or it points in the opposite direction.
We recall that we are considering only the case of parallel
alignment of black hole’s rotation axis and the direction of a
uniform magnetic field. Therefore, as we shall see below,
the Larmor motion $\,(L<0)\,$ will occur in retrograde orbits,
while the anti-Larmor motion $\,(L>0)\,$ will accompany
the direct orbits. Thus, in all expressions above
the upper signs corresponds to the direct,
or the anti-Larmor orbits and the lower signs refer
to the retrograde, or the Larmor orbits. Next, before carrying out
the full analysis of equations (23), (24) and
(27), it is useful to proceed with the particular cases.
4.1 Kerr black hole
In the absence of a magnetic field the energy and the angular momentum
of a test particle in the marginally stable circular orbits around
a Kerr black hole are obtained from equations (23) and
(24) for $\,\epsilon=0\,$. We have
$$E=\sqrt{1-{2M\over 3r}},\;\;\;\;\;\;L=\pm\,\sqrt{2M\left(r-{a^{2}\over{3r}}%
\right)},$$
(28)
where, the radii $r=r_{ms}$ of the orbits satisfy the equation
$$\displaystyle 6Mr^{2}-r^{3}-4a^{2}M+3a^{2}r\,\mp$$
$$\displaystyle 2\sqrt{2}\,aM^{1/2}\left[\left(3\,r-2M\right)\left(3r^{2}-a^{2}%
\right)\right]^{\,1\over 2}=0$$
(29)
which is of a particular case of equation (27)
for $\,\epsilon=0\,$. Since the radii of stability are different for
direct and retrograde motions, their appropriate energy and angular
momentum given in equations (28) are different as well.
The solution of (29) can be written in the form
$$r_{ms}=M\left\{3+\sqrt{k_{1}+k_{2}}\mp\left[2k_{1}-k_{2}-\frac{16(3-k_{1})}{%
\sqrt{k_{1}+k_{2}}}\right]^{1\over 2}\right\}$$
(30)
where
$$\displaystyle k_{1}$$
$$\displaystyle=$$
$$\displaystyle 3+{a^{2}\over M^{2}},$$
$$\displaystyle k_{2}$$
$$\displaystyle=$$
$$\displaystyle\left(3-{a\over M}\right)\left(1-{a\over M}\right)^{1\over 3}%
\left(1+{a\over M}\right)^{2\over 3}+$$
(31)
$$\displaystyle\left(3+{a\over M}\right)\left(1+{a\over M}\right)^{1\over 3}%
\left(1-{a\over M}\right)^{2\over 3}$$
This expression agrees with that found by Bardeen et al. long
ago (Bardeen et al. 1972). For a nonrotating
black hole, $\,a=0\,$, it gives $r_{ms}=6M$, while in the
maximally rotating case, $\,a=M\,$, one finds $\,r=M\,$
for the direct orbits and $\,r=9M\,$ for the retrograde orbits.
Taking these into account in (28)
we find the limiting values for the energy of the direct and retrograde
motions
$$E_{direct}=\frac{1}{\sqrt{3}}\;,\;\;\;\;\;\;E_{retrograte}=\frac{5}{3\sqrt{3}}.$$
(32)
In order to make the above limits more trasparent and also
with the purpose of comparing them with the corresponding
quantities in the presence of an external magnetic field
Table 1 provides the full list for the values of the radii,
of their assigned energies and angular momenta
as functions of the rotation parameter of a black hole.
It is important to note that the limiting value of
the energy of a direct orbit determines the maximum
binding energy that can be assigned to a last stable circular orbits.
One can easily find that this quantity is of order of $42\%$ of
the rest energy and it determines the efficiency of
an accretion disc around a maximally rotating black hole.
4.2 Schwarzschild black hole in a magnetic field
From equations (23) and (24) it is clear that the
case $\,a=0\,$ will give us the angular momentum and
the energy of a charged particle moving in a marginally
stable circular orbit around a nonrotating black hole
immersed into a uniform magnetic field
$$L=-\epsilon\,r\pm{{r^{1/2}}\over{2M}}\;\Lambda$$
(33)
and
$$E=\left[\left(1-{{2M}\over{3r}}\right)\left(1\mp\frac{\epsilon\,r^{1/2}}{2M^{2%
}}\,\Lambda\right)+\,\frac{\epsilon^{2}}{6}\,\left({{5r^{2}}\over M^{2}}-4%
\right)\right]^{1\over 2}$$
(34)
where
$$\Lambda=\left[8M^{3}+\epsilon^{2}r\,(5r^{2}-4Mr+4M^{2})\right]^{1\over 2}$$
and the stability radius $\,r=r_{ms}\,$ is determined by the equation
$$\left(6Mr-r^{2}\right)\left(1\mp\,{\epsilon\,r^{1/2}\,\over 2M^{2}}\,\Lambda%
\right)+\,\epsilon^{2}r^{2}\left(6-{4r\over M}+{9\over 2}{r^{2}\over M^{2}}-{r%
^{3}\over M^{3}}\right)=0$$
(35)
As we have mentioned above in this case there exist two different
kind of motions depending on the directions of the Lorentz force
with respect to a black hole. In turn, it is
associated with two signs in equations (33)-(35).
Following to papers by Gal’tsov $\&$ Petukhov (1978) and
Aliev $\&$ Gal’tsov (1989) we shall distinguish the two kind of motions
as the Larmor (the lower signs in (33)-(35))
and the anti-Larmor (the upper signs in (33)-(35))
motions. In order to get more insight into these motions it is useful
to obtain the expressions for the angular momentum and energy
in the asymptotically flat region $\,r\gg M\,$.
Solving equations (20) and (21) simultaneously
for $\,r\gg M\,$ we find that for orbits with $L<0$
$$L\simeq-{{eBr^{2}}\over{2m}}\;,\;\;\;\;\;\;E\simeq\sqrt{1+\,\left(\frac{eBr}{m%
}\right)^{2}}.$$
(36)
It follows that this kind of motion is nothing but an ordinary
cyclotron rotation in a uniform magnetic field under the Lorentz force
pointing towards the centre of the orbit of typical radius
$$r=\frac{mv}{eB\sqrt{1-v^{2}}}$$
(37)
This is the Larmor motion, while in the opposite case of orbits
with $\,L>0\,$ we find that $\,E\rightarrow 1\,$,
that is the anti-Larmor motion when the Lorentz force points
outwards the centre of the motion may occur only in the presence of
a black hole. Next, we need to solve equation (35)
for $\,r\,$. It is of a high order polynomial equation and only
for large values of $\,\epsilon\,$
$\,(\epsilon\gg 1)$ one can find the limiting solutions
$$r_{L}=\frac{5+\sqrt{13}}{2}\;M\,,\;\;\;\;\;\;r_{A}\rightarrow 2M.$$
(38)
Note that the radii of the marginally stable circular orbits
are different for the Larmor $\,(r_{L}\,)$ and anti-Larmor $\,(r_{A}\,)$
motions, the effect of a uniform magnetic field shifts both of them towards
the event horizon and even right up to the event horizon
for the anti-Larmor motion when $\,\epsilon\,$ is large enough.
These conclusions are in agreement with those made in
Gal’tsov $\&$ Petukhov (1978) on the basis of numerical calculations.
Substituting the radii (38) in equations
(33)-(34) respectively, we find that for
$\,\epsilon\gg 1\,$
$$E_{L}\rightarrow 5.56\,\epsilon\,M\,,\;\;\;\;\;\;L\rightarrow-23.47\,\epsilon\,M$$
(39)
for the Larmor motion and
$$E_{A}\rightarrow 0\,,\;\;\;\;\;\;L\rightarrow 2\epsilon\,M$$
(40)
for the anti-Larmor motion.
First of all, it follows from (39)
that for $\,\epsilon\gg 1\,$ there exists a stable ultarelativistic motion
of charged particles, while in the absence of a magnetic field as is well
known, a stable motion in the field of a nonrotating black hole
is only non-relativistic. In addition, the energy of
the anti-Larmor motion in the innermost stable orbit tends to zero
and the corresponding binding energy approaches $\,\simeq 100\%\,$
of the rest energy, in contrast to $\,42\%\,$ that of in
the field of a maximally rotating black hole.
A more detailed analysis of how the above conclusions are
made is given in Table 2. It exhibits the full list for the values of the
radii and of the associated energies and angular momenta as functions
of the influence parameter of the magnetic field $\,\epsilon\,$.
4.3 General case
We now turn to the consideration of the combined effects of the black hole
rotation and a uniform magnetic field on the marginally stable
circular orbits. The radii of their stability are determined by equation
(27), which in the general case can be solved only numerically.
However, in the case of a maximally rotating black hole and
large enough values of the magnetic field strength $\,(\epsilon\gg 1)\,$
we find the limiting analytical solution for a retrograde motion
$$r=2M\left\{1+2\cos\left[{1\over 3}\,tan^{-1}\,\left({\sqrt{7}\over{3}}\right)%
\right]\right\}$$
(41)
while, for a direct motion we have the two different radii
$$r_{1}\rightarrow M\;,\;\;\;\;\;\;\;r_{2}=(1+\sqrt{2})\,M.$$
(42)
The appearance of the two stable orbits can apparently be
related to the effect of the expelling of magnetic flux lines
from the horizon of a black hole as the rotation
parameter of the hole increases (Bic̆ák $\&$ Janis̆ 1985).
In our case a large enough value of the magnetic field
shifts the innermost stable
circular orbits to the event horizon and this, in turn,
along with the increase of the hole’s rotation parameter
provides the expulsion of the magnetic field lines from the black hole.
As a result of this the above two anti-Larmor orbits appear.
In the case of the first orbits the radius of stability tends
the event horizon, but the effect of the magnetic field on
it should become less and less as it approaches the horizon.
In the second case the anti-Larmor motion is expelled from the black hole
and it occurs at the limiting radius given by $\,r_{2}\,$ in (42).
A more precise description of it, of course, requires
the detailed numerical analysis of equation (27). The results
are listed in Tables 3.1-3.2. Comparing these results
and also the radii (41)-(42)
with those given in subsection 4.1 we see that
the magnetic field always plays a stabilizing role in its effect on
the circular orbits. In the Kerr metric the region
of stability for direct orbits enlarges towards the event horizon
with the increase of hole’s rotation parameter, while for prograde orbits
it moves in the opposite direction (see Table 1). The presence of
a large-scale uniform magnetic field around a Kerr black hole shifts
the innermost stable circular orbits towards the horizon
for both direct and retrograde motions.
However, as it is seen from Table 3.2,
when $\,\epsilon\,$ is large enough the
direct motion decays into two motions
and the separation becomes more significant as
the rotation parameter tends to its maximum value.
This is due to the effect of the
expelling of the magnetic field from nearby region of the horizon
as the angular momentum of the black hole increases.
At the same time, Tables 3.1-3.2. show that
for retrograde orbits the rotation of a black hole
opposes to the stabilizing effect of the magnetic field,
leading finally to the limiting value of
the radius given in (41).
Some results of the numerical analysis of equation (27)
are also depicted in Figs.1-2. Fig.1 shows the
dependence of the radii of marginally stable
circular orbits on the rotation parameter $\,a\,$ for
values of $\,\epsilon=0,1$. In Fig.2 for various values of
the black hole rotation parameter
we illustrate the enlargement of the region of the marginal stability with
the growth of the influence parameter of the magnetic field.
In both cases we observe that even
not very large value of the magnetic field produces
an essential enlargement in the region of the marginally stable
circular orbits towards the horizon of the black hole.
Next, substituting the value of the radius
(41) into equations (23)-(24)
we calculate the limiting value for the pertaining
energy and angular momentum at a retrograde motion
$$E=7.82\,\epsilon\,,\;\;\;\;\;\;L=-42.55\,\epsilon\,M.$$
(43)
As $\,\epsilon\gg 1\,$, the retrograde motion of a charged particle
at the last stable circular orbit around a maximally rotating black hole
is of a relativistic motion as in the case of a
nonrotating black hole (see equation (39)), however,
the rotation further enhances this effect. In the same way,
using the radius $\,r_{2}\,$ in (42) we find the energy and
the angular momentum
$$E=0.41\,\epsilon\,,\;\;\;\;\;\;L=3.82\,\epsilon\,M$$
(44)
pertaining to the innermost direct, anti-Larmor, motion. It follows that
the anti-Larmor motion of a charged particle at the radius $\,r_{2}\,$
may also become relativistic for $\,\epsilon\gg 1\,$,
in contrast to that in the case of a nonrotating black hole,
where the corresponding energy tends to zero (equation (40)).
In Tables 3.1-3.2 we also
give the full list for the values of the energies and angular momenta
pertaining to the marginally stable circular orbits. Comparing these
quantities with those in Table 2 we see that in all cases the energy of
the retrograde motion monotonically increases up to its
relativistic values as the influence of the magnetic field, as well as
the rotation of the black hole increase. However,
the energy of the anti-Larmor motion around a Schwarzschild black hole
systematically decreases as a test particle approaches
the event horizon. The rotation of the black hole
drastically changes the situation. The energies of direct motions
increase monotonically as the hole’s
rotation parameter increases, excepting the nearest to the horzon region,
where the effect of the expelling of the magnetic field becomes
dominant (Bic̆ák $\&$ Janis̆ 1985). Thus, we may conclude that
for large enough values of the magnetic field strength,
$\,\epsilon\gg 1\,$,
both retrograde and direct motions of charged particles
around a rotating black hole
become relativistic along the innermost stable circular orbits. It is
especially important to note the possibility of the relativistic stable
direct orbits near the event horizon of a rotationg black hole
in the presence of a strong magnetic field, unlike the case of
a nonrotating black hole. This is due to
the intimate connection between gravitation and electrodynamics
that arises in the model considered here.
5 Conclusions
The main purpose of this paper was to study the
combined effects of the rotation of a Kerr black hole and an
external magnetic field on the marginally stable circular motion
of charged particles. We have presented general equations
governing the energy, the angular momentum and the region of
the marginal stability of circular orbits around a rotating black
hole in a uniform magnetic field. The analytical results obtained
for large enough values of the magnetic field strength and for a
maximum value of the black hole angular momentum, as well as the
numerical analysis performed in general case have shown that in
all cases of the circular orbits the magnetic field essentially
enlarges the region of their marginal stability towards the event
horizon. As for the effect of the rotation of a black hole it has
been found to be different for direct and retrograde motions:
For retrograde motion the rotation opposes
to the magnetic field in its stabilizing effect, though the latter
always remains be dominant and for extreme values of the magnetic field
and rotation parameter there exists a limiting value for the
enlargement of the region of stability towards the event horizon.
In the case of direct motion the presence of a rotation
produces an additional shift of the stable orbits
to the event horizon. However, for large enough values of
the magnetic field, when the innermost stable motion occurs
at a radius that lies very close to the event horizon,
the magnetic field lines are expelled from the black hole
as the hole’s rotation parameter increases. Accordingly,
the direct motion occurs in two different orbits; one of them
is also expelled from the black hole, while the other one
being affected less and less by the magnetic field approaches
the event horizon as the rotation becomes maximum.
We have shown that the presence of a strong magnetic field
around a rotating black holes provides the possibility of retrograde
motion in the innermost stable relativistic orbits and the rotation of
the black hole further enhances this effect. It is
very important that, unlike the Schwarzschild case,
the combined effects of a sufficiently large magnetic field
and the rotation of a black hole also result in
relativistic motions in the innermost stable direct orbits that lie
very close to the event horizon. Thus, nearby the event horizon
of a rotating black hole there may exist a source of synchrotron
emission from relativistic charged particles moving in
stable circular orbits. This, of course, may play
an important role both in the searches for black holes
and in making feasible the probes of the metric
in the strong-gravity domain.
ACKNOWLEDGMENTS
We thank E. İnönü, I. H. Duru, M Hortaçsu for their stimulating
interest in our work and C. Saçlıoğlu for reading the manuscript
and valuable comments.
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Table 1. Marginally stable circular orbits around a Kerr
black hole in the absence of a magnetic field
$$\epsilon=0$$
direct orbits
retrograde
orbits
$$a/M$$
$$r/M$$
$$E$$
$$L/M$$
$$r/M$$
$$E$$
$$L/M$$
0.0
6.0
0.94
3.46
6.0
0.94
-3.46
0.2
5.32
0.93
3.26
6.63
0.94
-3.64
0.4
4.61
0.92
3.03
7.25
0.95
-3.80
0.6
3.82
0.90
2.75
7.85
0.95
-3.96
0.8
2.90
0.87
2.37
8.43
0.95
-4.10
0.999
1.18
0.65
1.34
8.99
0.96
-4.24
Table 2. Marginally stable circular orbits around a
nonrotating black hole in a uniform magnetic field
$$\hskip-5.690551pta=0\hskip-5.690551pt$$
anti-Larmor orbits
Larmor orbits
$$\epsilon$$
$$r/M$$
$$E$$
$$L/M$$
$$r/M$$
$$E$$
$$L/M$$
0.0
6.0
0.94
3.46
6.0
0.94
-3.46
0.2
3.98
0.78
3.51
4.63
1.57
-6.32
0.4
3.35
0.71
3.92
4.41
2.50
-10.35
0.6
3.05
0.65
4.34
4.35
3.52
-14.73
0.8
2.86
0.62
4.74
4.33
4.59
-19.27
1.0
2.74
0.58
5.18
4.32
5.67
-23.86
1.4
2.58
0.53
6.03
4.31
7.86
-33.11
2.0
2.43
0.49
7.19
4.30
11.16
-46.99
10.0
2.10
0.32
23.16
4.30
55.60
-234.42
100.0
2.01
0.29
203.01
4.30
556.00
-2343.99
Table 3.1. Marginally stable circular orbits around a Kerr
black hole in a uniform magnetic field with $\,\epsilon=1\,$.
$$\epsilon=1$$
direct orbits
retrograde orbits
$$a/M$$
$$r/M$$
$$E$$
$$L/M$$
$$r/M$$
$$E$$
$$L/M$$
0.0
2.74
0.58
5.18
4.32
5.67
-23.86
0.2
2.57
0.62
4.46
4.67
6.16
-27.55
0.4
2.38
0.65
3.76
4.99
6.61
-31.17
0.6
2.17
0.69
3.10
5.31
7.06
-35.02
0.8
1.89
0.71
2.38
5.61
7.49
-38.84
0.999
1.15
0.59
1.21
5.89
7.89
-42.58
Table 3.2. The influence of a sufficiently strong magnetic field
on the marginally stable circular orbits around a Kerr black hole
$\hskip-5.690551pt\epsilon=100\hskip-5.690551pt$
direct
orbits
retrograde
orbits
$a/M$
$r_{1}/M$
$E_{1}$
$L_{1}/M$
$r_{2}/M$
$E_{2}$
$L_{2}/M$
$r/M$
$E$
$L/M$
0.0
2.01
0.29
203.01
2.01
0.29
203.01
4.30
556.00
-2343.99
0.2
1.98
9.97
195.55
2.05
9.49
219.31
4.65
605.31
-2714.25
0.4
1.93
19.74
185.63
2.13
18.46
246.97
4.98
652.19
-3090.56
0.6
1.84
29.09
168.84
2.23
26.44
288.74
5.29
696.60
-3468.64
0.8
1.68
37.34
141.35
2.35
33.32
343.57
5.59
739.89
-3857.50
0.999
1.13
36.15
73.75
2.47
39.55
405.10
5.88
782.03
-4255.20
Captions to figures;
Figure 1. The dependence of the radii of marginally
stable circular orbits around a Kerr black hole on the rotation
parameter $\,a\,$ of the black hole for given $\,\epsilon=0,\,1\,$.
The solid curves refer to the innermost direct orbits,
while the dashed curves correspond to the
innermost retrograde orbits, the position of the event horizon is shown
by the bold curve $\,r_{+}\,$.
Figure 2. The dependence of the radii of marginally stable
circular orbits around a Kerr black hole in a uniform magnetic field
on the influence parameter of the magnetic field $\,\epsilon\,$.
The solid curves indicate the innermost direct
orbits, while the dashed curves correspond to the innermost
retrograde orbits $\,(\alpha=a/M=0,\,0.5,\,1)\,$. |
On the classification of rational quantum tori
and the structure of their automorphism groups
Karl-Hermann Neeb
Abstract. An ${\scriptstyle n}$-dimensional quantum torus is a twisted group algebra of the
group ${\scriptstyle{{Z}}^{n}}$. It is called rational if all invertible commutators are roots
of unity. In the present note we describe a normal form for rational
${\scriptstyle n}$-dimensional quantum
tori over any field. Moreover, we show that for
${\scriptstyle n=2}$ the natural exact sequence
describing the automorphism group of the quantum torus splits over any
field. Keyword: Quantum torus, normal form, automorphisms of quantum tori MSC: 16S35
Introduction
Let ${{K}}$ be a field and $\Gamma$ an abelian group.
A $\Gamma$-quantum torus
is a $\Gamma$-graded ${{K}}$-algebra $A=\bigoplus_{\gamma\in\Gamma}A_{\gamma}$,
for which all grading spaces are one-dimensional
and all non-zero elements in these spaces are invertible.
For any basis $(\delta_{\gamma})_{\gamma\in\Gamma}$ of such an algebra
with $\delta_{\gamma}\in A_{\gamma}$, we have
$\delta_{\gamma}\delta_{\gamma^{\prime}}=f(\gamma,\gamma^{\prime})\delta_{%
\gamma+\gamma^{\prime}}$,
where $f\colon\Gamma\times\Gamma\to{{K}}^{\times}$ is a group cocycle. In this
sense $\Gamma$-quantum tori are the same as twisted group algebras
in the terminology of [OP95].
Quantum tori arise very naturally in non-commutative geometry
as non-commutative algebras which are still very close to commutative ones
(cf. [GVF01]) and they also show up in topology (cf. [BL04, Sect. 3]).
For $\Gamma={{Z}}^{n}$, we also speak of
$n$-dimensional quantum tori, also called skew Laurent polynomial rings
if the image of $f$ lies in a cyclic subgroup of ${{K}}^{\times}$ (cf. [dCP93]).
Important special examples arise for $n=2$ and
$f(\gamma,\gamma^{\prime})=q^{\gamma_{1}\gamma_{2}^{\prime}}$, which leads to an
algebra $A_{q}$ with generators $u_{1}=\delta_{(1,0)}$ and
$u_{2}=\delta_{(0,1)}$, satisfying
$u_{1}u_{2}=qu_{2}u_{1},$
and their inverses.
Finite-dimensional quantum tori and their Jordan analogs
also play a key role in the structure theory of infinite-dimensional
Lie algebras because they are the natural coordinate structures
of extended affine Lie algebras ([BGK96], [AABGP97]).
The first problem we address in this note is the normal form
of the finite-dimensional rational quantum tori, i.e., quantum tori
with grading group $\Gamma={{Z}}^{n}$, for which $f$ takes values in the
torsion group of ${{K}}^{\times}$.
Let $P\subseteq{{K}}^{\times}$ be a subset containing for each
finite element order arising in the multiplicative group ${{K}}^{\times}$
a single representative. We then show in Section III
that any rational $n$-dimensional quantum torus $A$ is
isomorphic to a tensor product
$$A\cong A_{q_{1}}\otimes\cdots\otimes A_{q_{s-1}}\otimes A_{q_{s}^{m}}\otimes{{%
K}}[{{Z}}^{n-2s}],$$
(1)1( 1 )
where $q_{1},\ldots,q_{s}\in P$ satisfy
$1<\mathop{\rm ord}\nolimits(q_{s})|\mathop{\rm ord}\nolimits(q_{s-1})|\cdots|%
\mathop{\rm ord}\nolimits(q_{1})$
and $\mathop{\rm ord}\nolimits(q_{s}^{m})=\mathop{\rm ord}\nolimits(q_{s})$ (with $m=1$ if $2s<n$).
The existence of such a decomposition is not new.
Under the assumption that the field ${{K}}$ is algebraically closed
of characteristic zero, (1) can be found in [ABFP05], and a version for skew Laurent
polynomial rings is stated in [dCP93] (Remark in 7.2).
Our main new point in Section III are criteria for two such rational
quantum tori as in (1) to be isomorphic.
For any ${{Z}}^{n}$-quantum torus $A$, its group of automorphisms
is an abelian extension described by a short exact sequence
$${\bf 1}\to\mathop{\rm Hom}\nolimits({{Z}}^{n},{{K}}^{\times})\to\mathop{\rm Aut%
}\nolimits(A)\to\mathop{\rm Aut}\nolimits({{Z}}^{n},\lambda)\to{\bf 1},$$
(2)2( 2 )
where $\lambda\colon{{Z}}^{n}\times{{Z}}^{n}\to{{K}}^{\times},(\gamma,\gamma^{\prime}%
)\mapsto\delta_{\gamma}\delta_{\gamma^{\prime}}\delta_{\gamma}^{-1}\delta_{%
\gamma^{\prime}}^{-1}$
is the alternating biadditive
map determined by the commutator map of the unit group $A^{\times}$,
and $\mathop{\rm Aut}\nolimits({{Z}}^{n},\lambda)\subseteq\mathop{\rm GL}\nolimits_%
{n}({{Z}})\cong\mathop{\rm Aut}\nolimits({{Z}}^{n})$ is the subgroup
preserving $\lambda$. The second main result of this note is that
for $n=2$ the sequence (2) always splits.
In this case $A\cong A_{q}$ for
some $q\in{{K}}^{\times}$, and $\mathop{\rm Aut}\nolimits({{Z}}^{2},\lambda)=\mathop{\rm GL}\nolimits_{2}({{Z}})$ if
$q^{2}=1$ and $\mathop{\rm Aut}\nolimits({{Z}}^{2},\lambda)=\mathop{\rm SL}\nolimits_{2}({{Z}})$ otherwise.
The statement of
this result (in case $q$ is not a root of unity) can also be
found in [KPS94, Th. 1.5], but without any argument for the splitting
of the exact sequence (2).
According to [OP95, p.429], the determination of the automorphism groups of
general quantum tori seems to be a hopeless problem, but we think that
our splitting result stimulates some hope that more explicit descriptions might
be possible if the range of the commutator map is sufficiently well-behaved.
We thank B. Allison and A. Pianzola for stimulating discussions on the
subject matter of this paper, A. Pianzola for pointing out the
reference [OP95], B. Allison for carefully reading earlier versions of
the manuscript, K. Goodearl for pointing out references
[dCP93], [Pa96] and [BL04] and P. Gille for pointing out
that the surjectivity of the map $\Phi$ in Proposition II.3
can be derived from [Bro82]. Last, but not least,
we thank the referee for a very detailed report
that was extremely helpful in improving the exposition of this paper.
Notation
Throughout this paper ${{K}}$ denotes an arbitrary field.
We write $A^{\times}$ for the unit group of a unital ${{K}}$-algebra $A$.
Let $\Gamma$ and $Z$ be abelian groups, both written additively.
A function
$f\colon\Gamma\times\Gamma\to Z$ is called a $2$-cocycle if
$$f(\gamma,\gamma^{\prime})+f(\gamma+\gamma^{\prime},\gamma^{\prime\prime})=f(%
\gamma,\gamma^{\prime}+\gamma^{\prime\prime})+f(\gamma^{\prime},\gamma^{\prime%
\prime})$$
holds for $\gamma,\gamma^{\prime},\gamma^{\prime\prime}\in\Gamma$.
The set of all $2$-cocycles is an additive group $Z^{2}(\Gamma,Z)$
with respect to pointwise addition. The functions of the form
$h(\gamma)-h(\gamma+\gamma^{\prime})+h(\gamma^{\prime})$
are called coboundaries. They form a subgroup $B^{2}(\Gamma,Z)\subseteq Z^{2}(\Gamma,Z)$,
and the quotient group
$H^{2}(\Gamma,Z):=Z^{2}(\Gamma,Z)/B^{2}(\Gamma,Z)$ is called the second
cohomology group of $\Gamma$ with values in $Z$. It classifies central
extensions of $\Gamma$ by $Z$ up to equivalence. Here we assign to
$f\in Z^{2}(\Gamma,Z)$ the central extension
$Z\times_{f}\Gamma$, which is the set $Z\times\Gamma$, endowed with the
group multiplication
$$(z,\gamma)(z^{\prime},\gamma^{\prime})=(z+z^{\prime}+f(\gamma,\gamma^{\prime})%
,\gamma+\gamma^{\prime})\quad z,z^{\prime}\in Z,\gamma,\gamma^{\prime}\in\Gamma.$$
(0.1)0.1( 0.1 )
We also write $\mathop{\rm Ext}\nolimits(\Gamma,Z)\cong H^{2}(\Gamma,Z)$
for the group of all central extensions of
$\Gamma$ by $Z$, and
$\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)$ for the subgroup corresponding to the
abelian extensions of the group
$\Gamma$ by $Z$, which correspond to symmetric $2$-cocycles.
We call a biadditive map $\Gamma\times\Gamma\to Z$
vanishing on the diagonal alternating and denote the set of these
maps by $\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)$.
A function $q\colon\Gamma\to Z$ is called a quadratic form if the map
$$\beta_{q}\colon\Gamma\times\Gamma\to Z,\quad(\gamma,\gamma^{\prime})\mapsto q(%
\gamma+\gamma^{\prime})-q(\gamma)-q(\gamma^{\prime})$$
is biadditive.
Note that we do not require here that $q(n\gamma)=n^{2}q(\gamma)$ holds for
$n\in{{Z}}$ and $\gamma\in\Gamma$.
For $n\in{{N}}:=\{1,2,3,\ldots\}$,
we write $Z[n]:=\{z\in Z\colon nz=0\}$ for the $n$-torsion
subgroup of $Z$. We also write ${{N}}_{0}:={{N}}\cup\{0\}$.
I. The correspondence between quantum tori and central extensions
Definition I.1.
Let $\Gamma$ be an abelian group. A unital associative ${{K}}$-algebra
$A$ is said to be a $\Gamma$-quantum torus if it is $\Gamma$-graded,
$A=\bigoplus_{\gamma\in\Gamma}A_{\gamma},$
with one-dimensional grading spaces $A_{\gamma}$, and each
non-zero element of $A_{\gamma}$ is invertible.**In [OP95], these algebras are
called twisted group algebras.
For $\Gamma\cong{{Z}}^{d}$ we call a $\Gamma$-quantum torus also a $d$-dimensional
quantum torus.
Remark I.2. In each $\Gamma$-quantum torus $A$ the set
$A^{\times}_{h}:=\bigcup_{\gamma\in\Gamma}{{K}}^{\times}\delta_{\gamma}$
of homogeneous units (called trivial units in [OP95])
is a subgroup containing ${{K}}^{\times}{\bf 1}\cong{{K}}^{\times}$ in its center.
We thus obtain a central extension
$${\bf 1}\to{{K}}^{\times}\to A^{\times}_{h}\to\Gamma\to{\bf 1}$$
of abelian groups.
It is instructive to see how this can be made more explicit in terms of cocycles,
which shows in particular that each central extension of $\Gamma$ by
${{K}}^{\times}$ arises as $A^{\times}_{h}$ for some $\Gamma$-quantum torus $A$.
Let $A$ be a $\Gamma$-quantum torus and pick non-zero
elements $\delta_{\gamma}\in A_{\gamma}$,
so that $(\delta_{\gamma})_{\gamma\in\Gamma}$ is a basis of $A$.
Then each $\delta_{\gamma}$
is an invertible element of $A$, so that we get
$$\delta_{\gamma}\delta_{\gamma^{\prime}}=f(\gamma,\gamma^{\prime})\delta_{%
\gamma+\gamma^{\prime}}\quad\hbox{ for }\quad\gamma,\gamma^{\prime}\in\Gamma,$$
(1.1)1.1( 1.1 )
where
$f\in Z^{2}(\Gamma,{{K}}^{\times})$ is a $2$-cocycle for which
$A_{h}^{\times}\cong{{K}}^{\times}\times_{f}\Gamma$ (cf. (0.1)).
Conversely, starting with a cocycle $f\in Z^{2}(\Gamma,{{K}}^{\times})$, we define a
multiplication on the vector space $A:=\bigoplus_{\gamma\in\Gamma}{{K}}\delta_{\gamma}$
with basis $(\delta_{\gamma})_{\gamma\in\Gamma}$
by
$\delta_{\gamma}\delta_{\gamma^{\prime}}:=f(\gamma,\gamma^{\prime})\delta_{%
\gamma+\gamma^{\prime}}.$
Then the cocycle property implies that we get a unital associative algebra,
and it is clear from the construction that it is a $\Gamma$-quantum torus.
Definition I.3.
There
are two natural equivalence relations between quantum tori. The finest
one is the notion of graded equivalence: Two $\Gamma$-quantum tori
$A$ and $B$ are called graded equivalent if there is an algebra isomorphism
$\varphi\colon A\to B$ with $\varphi(A_{\gamma})=B_{\gamma}$ for all $\gamma\in\Gamma$.
A slightly weaker notion is
graded isomorphy: Two $\Gamma$-quantum tori
$A$ and $B$ are called graded isomorphic if there is an isomorphism
$\varphi\colon A\to B$ and an automorphism $\varphi_{\Gamma}\in\mathop{\rm Aut}\nolimits(\Gamma)$
with $\varphi(A_{\gamma})=B_{\varphi_{\Gamma}(\gamma)}$ for all $\gamma\in\Gamma$.
The following theorem reduces the corresponding classification problems to
purely group theoretic ones.
Theorem I.4.
The graded equivalence classes of $\Gamma$-quantum
tori are in one-to-one correspondence with the central extensions of the group
$\Gamma$ by the multiplicative group ${{K}}^{\times}$, hence
parametrized by the cohomology group $H^{2}(\Gamma,{{K}}^{\times})$.
The graded isomorphy classes of $\Gamma$-quantum
tori are parametrized by the set
$$H^{2}(\Gamma,{{K}}^{\times})/\mathop{\rm Aut}\nolimits(\Gamma)$$
of orbits of the group
$\mathop{\rm Aut}\nolimits(\Gamma)$ in the cohomology group $H^{2}(\Gamma,{{K}}^{\times})$, where
the action is given on the level of cocycles
by $\psi.f:=(\psi^{-1})^{*}f=f\circ(\psi^{-1}\times\psi^{-1})$.
Proof. If $\varphi\colon A\to B$ is a graded equivalence of $\Gamma$-quantum tori,
then the restriction to the group $A^{\times}_{h}$ of homogeneous units
leads to the commutative diagram
$$\matrix{{{K}}^{\times}&\to&A^{\times}_{h}&\to&\Gamma\cr\Big{\downarrow}\hbox t%
o 0.0pt{$\vbox{\hbox{$\scriptstyle\mathop{\rm id}\nolimits_{{{K}}^{\times}}$}}%
$}&&\Big{\downarrow}\hbox to 0.0pt{$\vbox{\hbox{$\scriptstyle\varphi$}}$}&&%
\Big{\downarrow}\hbox to 0.0pt{$\vbox{\hbox{$\scriptstyle\mathop{\rm id}%
\nolimits_{\Gamma}$}}$}\cr{{K}}^{\times}&\to&B^{\times}_{h}&\to&\Gamma.\cr}$$
This means that the central extensions $A^{\times}_{h}$ and $B^{\times}_{h}$ of $\Gamma$
by ${{K}}^{\times}$ are equivalent. If, conversely, these extensions are equivalent,
then any equivalence $\varphi\colon A^{\times}_{h}\to B^{\times}_{h}$ extends linearly
to a graded equivalence $A\to B$.
Now the observation from Remark I.2 implies that
the graded equivalence classes of $\Gamma$-quantum
tori are parametrized by the cohomology group $H^{2}(\Gamma,{{K}}^{\times})\cong\mathop{\rm Ext}\nolimits(\Gamma,{{K}}^{%
\times})$.
If $\varphi\colon A\to B$ is a graded isomorphism of $\Gamma$-quantum tori,
then the diagram
$$\matrix{{{K}}^{\times}&\to&A^{\times}_{h}&\to&\Gamma\cr\Big{\downarrow}\hbox t%
o 0.0pt{$\vbox{\hbox{$\scriptstyle\mathop{\rm id}\nolimits_{{{K}}^{\times}}$}}%
$}&&\Big{\downarrow}\hbox to 0.0pt{$\vbox{\hbox{$\scriptstyle\varphi$}}$}&&%
\Big{\downarrow}\hbox to 0.0pt{$\vbox{\hbox{$\scriptstyle\varphi_{\Gamma}$}}$}%
\cr{{K}}^{\times}&\to&B^{\times}_{h}&\to&\Gamma\cr}$$
commutes, which means that the corresponding central extensions
$A^{\times}_{h}$ and $B^{\times}_{h}$ are contained in the same orbit of
$\mathop{\rm Aut}\nolimits(\Gamma)$ on $\mathop{\rm Ext}\nolimits(\Gamma,{{K}}^{\times})\cong H^{2}(\Gamma,{{K}}^{%
\times})$ (we leave the
easy verification to the reader).
Conversely, any isomorphism $\varphi\colon A^{\times}_{h}\to B^{\times}_{h}$ of central
extensions extends linearly to an isomorphism of algebras $A\to B$.
II. Central extensions of abelian groups
In this section $\Gamma$ and $Z$ are abelian groups, written
additively. We shall derive some
general facts on the set of equivalence classes $\mathop{\rm Ext}\nolimits(\Gamma,Z)\cong H^{2}(\Gamma,Z)$
of central extensions of $\Gamma$ by $Z$. In Sections III and IV below we shall apply these
to the special case $Z={{K}}^{\times}$ for a field ${{K}}$.
Remark II.1. Let
$Z\hookrightarrow\mathaccent 866{\Gamma}\smash{\mathop{\hbox to 20.0pt{%
\rightarrowfill}}\limits^{q}}\Gamma$ be a central extension of the
abelian group $\Gamma$ by the abelian group $Z$ and
$$\mathaccent 866{\lambda}\colon\mathaccent 866{\Gamma}\times\mathaccent 866{%
\Gamma}\to Z,\quad(x,y)\mapsto[x,y]:=xyx^{-1}y^{-1}$$
the commutator map of $\mathaccent 866{\Gamma}$. Its values lie in $Z$ because
$\Gamma$ is abelian. Obviously, $\mathaccent 866{\lambda}(x,x)=0$,
and $\mathaccent 866{\lambda}$ is an alternating biadditive map (cf. [OP95, p.418]).
Moreover, the commutator map is constant on the fibers of the map $q$,
hence factors through a biadditive map
$\lambda\in\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)$.
Next we write $\mathaccent 866{\Gamma}$
as $Z\times_{f}\Gamma$ with a $2$-cocycle $f\in Z^{2}(\Gamma,Z)$.
For the map $\sigma\colon\Gamma\to\mathaccent 866{\Gamma},\gamma\mapsto(0,\gamma)$
we then have
$\sigma(\gamma)\sigma(\gamma^{\prime})=\sigma(\gamma+\gamma^{\prime})f(\gamma,%
\gamma^{\prime}),$
which leads to
$$\eqalign{\lambda(\gamma,\gamma^{\prime})&=\mathaccent 866{\lambda}(\sigma(%
\gamma),\sigma(\gamma^{\prime}))=\sigma(\gamma)\sigma(\gamma^{\prime})\big{(}%
\sigma(\gamma^{\prime})\sigma(\gamma)\big{)}^{-1}\cr&=\sigma(\gamma+\gamma^{%
\prime})f(\gamma,\gamma^{\prime})\big{(}\sigma(\gamma+\gamma^{\prime})f(\gamma%
^{\prime},\gamma)\big{)}^{-1}=f(\gamma,\gamma^{\prime})f(\gamma^{\prime},%
\gamma)^{-1}=f(\gamma,\gamma^{\prime})-f(\gamma^{\prime},\gamma).\cr}$$
Therefore the map
$\lambda_{f}\in\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)$ defined by
$$\lambda_{f}(\gamma,\gamma^{\prime}):=f(\gamma,\gamma^{\prime})-f(\gamma^{%
\prime},\gamma)$$
(2.1)2.1( 2.1 )
can be identified with the commutator map of $\mathaccent 866{\Gamma}$.
Note that the commutator map $\lambda_{f}$ only depends on
the cohomology class $[f]\in H^{2}(\Gamma,Z)$. We thus obtain a group homomorphism
$$\Phi\colon H^{2}(\Gamma,Z)\to\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z),\quad[f]%
\mapsto\lambda_{f}.$$
Remark II.2. Each biadditive map $f\colon\Gamma\times\Gamma\to Z$ is a cocycle,
but it is not true that
each cohomology class in $H^{2}(\Gamma,Z)$ has a biadditive representative.
A typical example is the class corresponding to the exact sequence
${\bf 0}\to m{{Z}}\to{{Z}}\to{{Z}}/m{{Z}}\to{\bf 0}$.
Proposition II.3.
For abelian groups $\Gamma$ and $Z$ we have a
split short exact sequence
$${\bf 0}\to\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)\to\mathop{\rm Ext}%
\nolimits(\Gamma,Z)\cong H^{2}(\Gamma,Z)\smash{\mathop{\hbox to 20.0pt{%
\rightarrowfill}}\limits^{\Phi}}\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)\to{\bf
0},$$
describing the kernel of the map $\Phi$.
Proof. For the exactness in $\mathop{\rm Ext}\nolimits(\Gamma,Z)$, we only have to
observe that an extension $\mathaccent 866{\Gamma}$ of $\Gamma$ by $Z$
is an abelian group if and only if the commutator map of $\mathaccent 866{\Gamma}$ is
trivial (cf. Remark II.1).
The remaining assertions can be found as Exercise 5 in [Bro82, §V.6].
The main point of the argument is to use the short exact
Universal Coefficient Sequence
$${\bf 0}\to\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)\to H^{2}(\Gamma,Z)%
\smash{\mathop{\hbox to 20.0pt{\rightarrowfill}}\limits^{\Psi}}\mathop{\rm Hom%
}\nolimits(H_{2}(\Gamma),Z)\to{\bf 0},$$
then show that $H_{2}(\Gamma)\cong\Lambda^{2}(\Gamma)$, which leads to
an isomorphism
$\mathop{\rm Hom}\nolimits(H_{2}(\Gamma),Z)\cong\mathop{\rm Alt}\nolimits^{2}(%
\Gamma,Z)$
([Bro82, Thm. 6.4]), and then to verify that $\Phi$ corresponds to
$\Psi$ under this identification.
In [Bro82], the proof of the surjectivity of $\Phi$ is based on
the observation that each abelian group is a direct limit of its
finitely generated subgroups which in turn are products of cyclic groups.
Below we give a direct argument for the surjectivity of $\Phi$
if $\Gamma$ is a direct sum of cyclic groups (the only case relevant
in the following). We thus obtain
an explicit description of $H^{2}(\Gamma,Z)$.
For the following proposition we recall that, as a consequence of the
Well-Ordering Theorem, each set $I$ carries a total order. We also recall
the notation $Z[n]=\{z\in Z\colon nz=0\}$.
Proposition II.4.
Let $\Gamma=\bigoplus_{i\in I}\Gamma_{i}$
be a direct sum of cyclic groups
$\Gamma_{i}\cong{{Z}}/m_{i}{{Z}}$, $m_{i}\in{{N}}_{0}$. Further let $\leq$ be a total order
on $I$. Then
$$H^{2}(\Gamma,Z)\cong\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)\oplus\mathop{%
\rm Alt}\nolimits^{2}(\Gamma,Z)\cong\prod_{m_{i}\not=0}Z/m_{i}Z\oplus\prod_{i<%
j}Z[\gcd(m_{i},m_{j})],$$
(2.2)2.2( 2.2 )
where we put $\gcd(m,0):=m$ for $m\in{{N}}_{0}$.
If, in addition, $\Gamma$ is free, then $\Phi$ is an isomorphism,
$H^{2}(\Gamma,Z)\cong Z^{\{(i,j)\in I^{2}\colon i<j\}},$
and each cohomology class has a biadditive representative.
Proof. To see that $\Phi$ is surjective, let
$\eta\in\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)$. If $\gamma_{i}$ is a generator of $\Gamma_{i}$,
we have $\eta(n\gamma_{i},m\gamma_{i})=nm\eta(\gamma_{i},\gamma_{i})=0$ for
$n,m\in{{Z}}$, so that
$\eta$ vanishes on $\Gamma_{i}\times\Gamma_{i}$.
We define a biadditive map $f_{\eta}\colon\Gamma\times\Gamma\to Z$ by
$$f_{\eta}(\gamma_{i},\gamma_{j}):=\cases{\eta(\gamma_{i},\gamma_{j})&for $i>j$,%
$\gamma_{i}\in\Gamma_{i},\gamma_{j}\in\Gamma_{j}$,\cr 0&for $i\leq j$, $%
\gamma_{i}\in\Gamma_{i},\gamma_{j}\in\Gamma_{j}$.\cr}$$
Then $f_{\eta}$ is biadditive, hence a $2$-cocycle (Remark II.2), and
$\Phi(f_{\eta})=\eta$.
Clearly, the assignment $\eta\mapsto f_{\eta}$ defines an injective homomorphism
$\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)\to H^{2}(\Gamma,Z)$, splitting $\Phi$.
We know from Proposition II.3, that $\ker\Phi=\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)$.
We next observe that
$\mathop{\rm Alt}\nolimits(\Gamma,Z)\cong\prod_{i<j}\mathop{\rm Hom}\nolimits(%
\Gamma_{i}\otimes\Gamma_{j},Z),$
and $\Gamma_{i}\otimes\Gamma_{j}\cong{{Z}}/\gcd(m_{i},m_{j}){{Z}},$
which leads to
$\mathop{\rm Hom}\nolimits(\Gamma_{i}\otimes\Gamma_{j},Z)\cong Z[\gcd(m_{i},m_{%
j})].$
On the other hand,
$$\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)\cong\prod_{i\in I}\mathop{\rm Ext%
}\nolimits_{\rm ab}(\Gamma_{i},Z)\cong\prod_{m_{i}\not=0}Z/m_{i}Z$$
(cf. [Fu70, §52]), which leads to (2.2).
If, in addition, $\Gamma$ is free, then $m_{i}=0$ for each $i\in I$, and the assertion
follows from $\mathop{\rm Ext}\nolimits_{\rm ab}(\Gamma,Z)={\bf 0}$.
III. The Normal form of rational quantum tori
In this section we write $\Gamma:={{Z}}^{n}$ for the free abelian group of rank $n$.
For an abelian group $Z$ we write $\mathop{\rm Alt}\nolimits_{n}(Z)$ for the set of alternating
$(n\times n)$-matrices with entries in $Z$, i.e.,
$a_{ii}=0$ for each $i$ and $a_{ij}=-a_{ji}$ for $i\not=j$.
This is an abelian group with respect to matrix addition.
Clearly the map $\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)\to\mathop{\rm Alt}\nolimits_{n}(Z),f%
\mapsto(f(e_{i},e_{j}))_{i,j=1,\ldots,n}$
is an isomorphism of abelian groups, so that $\mathop{\rm Alt}\nolimits_{n}(Z)\cong H^{2}(\Gamma,Z)$ by
Proposition II.4. Writing $\lambda_{A}\in\mathop{\rm Alt}\nolimits^{2}(\Gamma,Z)$ for the alternating
form $\lambda_{A}(\alpha,\beta):=\beta^{\top}A\alpha$ determined by the alternating
matrix $A$, we have for $g\in\mathop{\rm GL}\nolimits_{n}({{Z}})\cong\mathop{\rm Aut}\nolimits(\Gamma)$ the relation
$\lambda_{A}(g.\alpha,g.\beta)=\beta g^{\top}Ag\alpha,$
so that the orbits of the natural action of $\mathop{\rm Aut}\nolimits(\Gamma)\cong\mathop{\rm GL}\nolimits_{n}({{Z}})$ on
the set of alternating forms correspond to the orbits of the action of $\mathop{\rm GL}\nolimits_{n}({{Z}})$
on $\mathop{\rm Alt}\nolimits_{n}(Z)$ by
$$g.A:=gAg^{\top},$$
(3.1)3.1( 3.1 )
where we multiply matrices in $M_{n}({{Z}})$ with matrices in $M_{n}(Z)$ in the
obvious fashion. We conclude that
$$H^{2}(\Gamma,Z)/\mathop{\rm Aut}\nolimits(\Gamma)\cong\mathop{\rm Alt}%
\nolimits_{n}(Z)/\mathop{\rm GL}\nolimits_{n}({{Z}}),$$
(3.2)3.2( 3.2 )
the set of $\mathop{\rm GL}\nolimits_{n}({{Z}})$-orbits in $\mathop{\rm Alt}\nolimits_{n}(Z)$.
If $n=n_{1}+\ldots+n_{r}$ is a partition of $n$ and
$A_{i}\in M_{n_{i}}(Z)$, then we write
$$A_{1}\oplus A_{2}\oplus\ldots\oplus A_{r}:=\mathop{\rm diag}\nolimits(A_{1},%
\ldots,A_{r}),$$
for the block diagonal matrix with entries $A_{1},\ldots,A_{r}$. For
$h_{1},\ldots,h_{s}\in Z$ we further write
$$N(h_{1},\ldots,h_{s}):=\pmatrix{0&h_{1}\cr-h_{1}&0\cr}\oplus\pmatrix{0&h_{2}%
\cr-h_{2}&0\cr}\oplus\ldots\oplus\pmatrix{0&h_{s}\cr-h_{s}&0\cr}\oplus{\bf 0}_%
{n-2s}\in\mathop{\rm Alt}\nolimits_{n}(Z).$$
In the following we shall assume that $Z$ is a cyclic group, hence of the
form ${{Z}}/(m)$ for some $m\in{{N}}_{0}$. If $m=0$, then $Z={{Z}}$ is a principal
ideal domain. This is not the case for $m>0$,
but $Z$ still carries a natural
ring structure given by $\overline{x}\cdot\overline{y}:=\overline{xy}$ for
$x,y\in{{Z}},\overline{x}:=x+m{{Z}}$, turning it into a principal ideal ring.
We write $Z^{\times}$ for the set of units in $Z$ and note that
if $m=p_{1}^{\ell_{1}}\cdots p_{k}^{\ell_{k}}$ is the prime factorization of $m$, then the set
$$P:=\{\overline{p}_{1}^{j_{1}}\cdots\overline{p}_{k}^{j_{k}}\colon 0\leq j_{i}%
\leq\ell_{i},i=1,\ldots,k\}\subseteq Z$$
is a multiplicatively closed set of representatives for the
multiplicative cosets of the unit group $Z^{\times}$.
We say that $a$ divides $b$ in $Z$, written
$a|b$, if $bZ\subseteq aZ$. Since each subgroup of $Z$ is cyclic and
determined by its order, we have
$$a|b\quad\Longleftrightarrow\quad\mathop{\rm ord}\nolimits(b)|\mathop{\rm ord}%
\nolimits(a)\quad\hbox{ and }\quad aZ=bZ\quad\Longleftrightarrow\quad b\in aZ^%
{\times}.$$
If $h_{1},h_{2}\in P$ are non-zero and $h_{1}|h_{2}$, then the explicit description of the
set $P$ shows that there exists a unique element
$h\in P$ with $h_{2}=h_{1}h$. We then write $h_{2}/h_{1}:=h$.
Although $Z$ is not a principal ideal domain for $m>0$, we define
for a matrix $A\in M_{n}(Z)$ the determinantal divisor
$d_{i}(A)\in P$, $i=1,\ldots,n$, as the unique element in $P$
generating the additive subgroup of $Z$ generated by all $j$-minors
of the matrix $A$. As a consequence of the Cauchy–Binet Formula
([New72, II.12]),
$$d_{i}(AB)=d_{i}(BA)=d_{i}(A)\quad\hbox{ for}\quad A\in M_{n}(Z),B\in\mathop{%
\rm GL}\nolimits_{n}(Z).$$
(3.3)3.3( 3.3 )
We thus obtain a set of $n$ $P$-valued invariants for the action of
$\mathop{\rm GL}\nolimits_{n}(Z)$ on $\mathop{\rm Alt}\nolimits_{n}(Z)$ satisfying for $N:=N(h_{1},\ldots,h_{s})$ with
$h_{i}\in P$ and $h_{1}|h_{2}|\ldots|h_{s}$:
$$d_{1}(N)=h_{1},\quad d_{2}(N)=h_{1}^{2},\quad d_{3}(N)=h_{1}^{2}h_{2},\quad%
\ldots,\quad d_{2s}(N)=h_{1}^{2}\cdots h_{s}^{2},\quad d_{j}(N)=0,\ j>2s.$$
Unfortunately, these invariants do not separate the orbits for
a finite cyclic group $Z$, but they do for $Z={{Z}}$
(cf. Theorem III.2 below and [New72, Th. II.9]).
Theorem III.1.
(Smith normal form over cyclic rings)
We consider the action of the group $\mathop{\rm GL}\nolimits_{n}(Z)\times\mathop{\rm GL}\nolimits_{n}(Z)$ on
$M_{n}(Z)$ by $(g,h).A:=gAh^{-1}$.
(1) Each $\mathop{\rm GL}\nolimits_{n}(Z)^{2}$-orbit contains a unique matrix of
the form
$$\mathop{\rm diag}\nolimits(h_{1},\ldots,h_{n})\quad\hbox{ with }\quad h_{i}\in
P%
,\ h_{1}|h_{2}|\ldots|h_{n}.$$
(2) Each $\mathop{\rm SL}\nolimits_{n}(Z)^{2}$-orbit contains a unique matrix of
the form
$$\mathop{\rm diag}\nolimits(h_{1},\ldots,zh_{n})\quad\hbox{ with }\quad h_{i}%
\in P,\ h_{1}|h_{2}|\ldots|h_{n},z\in Z^{\times}.$$
(3) For $h_{i}\in P$ with $h_{1}|h_{2}|\ldots|h_{n}$, we
consider the multiplicative subgroup
$$D_{(h_{1},\ldots,h_{s})}:=\{\mathop{\rm det}\nolimits(g)\colon g\in\mathop{\rm
GL%
}\nolimits_{n}(Z),gN(h_{1},\ldots,h_{s})g^{\top}=N(h_{1},\ldots,h_{s})\}\leq Z%
^{\times}.$$
Then
$D_{(h_{1},\ldots,h_{s})}=Z^{\times}$ for $2s<n$, and if $2s=n$, then
$$\{z\in Z^{\times}\colon zh_{s}=h_{s}\}\subseteq D_{(h_{1},\ldots,h_{s})}%
\subseteq\{z\in Z^{\times}\colon z^{2}h_{s}=h_{s}\}.$$
Proof. (1) [Br93, Th. 15.24]
(2) For $z\in Z$ we write
$\sigma(z):=\mathop{\rm diag}\nolimits(1,\ldots,1,z)$
and observe that $\sigma\colon Z^{\times}\to\mathop{\rm GL}\nolimits_{n}(Z)$ is an embedding, which
leads to a semidirect product decomposition
$\mathop{\rm GL}\nolimits_{n}(Z)=\mathop{\rm SL}\nolimits_{n}(Z)\sigma(Z)\cong%
\mathop{\rm SL}\nolimits_{n}(Z)\mathchar 9583\relax Z^{\times}.$
Existence:
For each $A\in M_{n}(Z)$, (1) implies the
existence of $g,h\in\mathop{\rm GL}\nolimits_{n}(Z)$ such that
$N:=gAh^{-1}=\mathop{\rm diag}\nolimits(h_{1},\ldots,h_{n})$ as in (1).
Writing $g=\sigma(z)g_{1}$ and $h=\sigma(w)h_{1}$ with $g_{1},h_{1}\in\mathop{\rm SL}\nolimits_{n}(Z)$,
it follows that
$$g_{1}Ah_{1}^{-1}=\sigma(z)^{-1}N\sigma(w)=\sigma(z^{-1}w)N=\mathop{\rm diag}%
\nolimits(d_{1},\ldots,d_{n-1},z^{-1}wd_{n}).$$
Uniqueness: Suppose first that $Z={{Z}}$ is infinite.
If $d_{n}=0$, then there is nothing to show. If $d_{n}\not=0$,
then the fact that the determinant function is constant on the
orbits of $\mathop{\rm SL}\nolimits_{n}(Z)^{2}$ implies the assertion.
We may therefore assume that $Z={{Z}}/(m)$ for some $m>0$.
Writing $m=p_{1}^{m_{1}}\cdots p_{k}^{m_{k}}$ for its prime factorization,
we obtain a direct product of rings
$${{Z}}/(m)\cong{{Z}}/(p_{1}^{m_{1}})\times\ldots\times{{Z}}/(p_{k}^{m_{k}}).$$
For $Z_{i}:={{Z}}/(p_{i}^{m_{i}})$, we accordingly have
$\mathop{\rm SL}\nolimits_{n}(Z)\cong\prod_{i=1}^{k}\mathop{\rm SL}\nolimits_{n%
}(Z_{i})$ and
$M_{n}(Z)\cong\prod_{i=1}^{k}M_{n}(Z_{i}),$
as direct products of groups, resp., rings.
Therefore it suffices to prove the assertion for the case
$Z={{Z}}/(p^{m})$, where $m\in{{N}}$ and $p$ is a prime.
Each element $z\in Z$ can be uniquely written as
$$z=a_{0}+a_{1}p+\ldots+a_{m-1}p^{m-1}\quad\hbox{ with }\quad 0\leq a_{i}<p.$$
It is a unit if and only if $a_{0}\not=0$, i.e.,
$p\not|z$. If $p^{k}$, $0\leq k\leq m-1$,
is the maximal power of $p$ dividing $z$, then
$$z=\sum_{j=k}^{m-1}a_{j}p^{j}=p^{k}\sum_{j=k}^{m-1}a_{j}p^{j-k},$$
where the second factor is a unit. Therefore
$\{1,p,p^{2},\ldots,p^{m-1},p^{m}=0\}$
is a system of representatives of the multiplicative cosets of $Z^{\times}$ in
$Z$.
Step 1: We have to show that if two matrices $D(z_{1})$ and $D(z_{2})$ of the form
$$D(z):=\mathop{\rm diag}\nolimits(p^{k_{1}},\ldots,zp^{k_{n}})=\sigma(z)D(1),%
\quad 0\leq k_{1}\leq\ldots\leq k_{n}<m,$$
lie in the same orbit of $\mathop{\rm SL}\nolimits_{n}(Z)^{2}$, then $D(z_{1})=D(z_{2})$.
Since the orbit of $D(z)=\sigma(z)D(1)$ under
$\mathop{\rm SL}\nolimits_{n}(Z)^{2}$ coincides with the set $\sigma(z)(\mathop{\rm SL}\nolimits_{n}(Z)^{2}.D(1))$,
it suffices to consider the case $z_{1}=1$ and $z_{2}=z\in Z^{\times}$.
Step 2: We proceed by induction on the size $n$ of the matrices.
For $n=1$ the group $\mathop{\rm SL}\nolimits_{n}(Z)$ is trivial, which immediately
implies the assertion.
Step 3: We reduce the assertion to the
special case $k_{1}=0$. So let us assume that the
assertion is correct if $k_{1}=0$ and assume that
there are $g,h\in\mathop{\rm SL}\nolimits_{n}(Z)$ with
$gD(1)h=D(z).$
Writing
$D(z)=p^{k_{1}}D^{\prime}(z)$ with
$D^{\prime}(z)=\mathop{\rm diag}\nolimits(1,p^{k_{2}-k_{1}},\ldots,zp^{k_{n}-k_%
{1}}),$
this means that
$p^{k_{1}}(gD^{\prime}(1)h-D^{\prime}(z))=0,$
i.e., that $p^{m-k_{1}}$ divides each entry of the matrix
$gD^{\prime}(1)h-D^{\prime}(z)$. Over the quotient ring $Z^{\prime}:={{Z}}/(p^{m-k_{1}})$ we
then have
$gD^{\prime}(1)h=D^{\prime}(z)$ with $k_{1}=0$. Since we assume that
the theorem holds in this situation,
we derive that
$$zp^{k_{n}-k_{1}}\equiv p^{k_{n}-k_{1}}\mathop{\rm mod}\nolimits p^{m-k_{1}}.$$
This means that
$p^{m-k_{1}}|(z-1)p^{k_{n}-k_{1}}$, and hence that
$p^{k_{1}}(z-1)p^{k_{n}-k_{1}}=(z-1)p^{k_{n}}=0$
in $Z$.
Step 4: Now we consider the special case $k_{1}=0$.
Let $n_{1}$ be maximal with $k_{n_{1}}=0$, $n_{2}:=n-n_{1}$, and write
elements of $M_{n}(Z)$ accordingly as $(2\times 2)$-block matrices.
We further put $k:=k_{n_{1}+1}>0$. Suppose that
$gD(1)h=D(z)$ for
$$g=\pmatrix{a&b\cr c&d\cr},\quad h=\pmatrix{a^{\prime}&b^{\prime}\cr c^{\prime}%
&d^{\prime}\cr}\in\mathop{\rm SL}\nolimits_{n}(Z).$$
We write
$$D(z)=\pmatrix{{\bf 1}&0\cr 0&p^{k}D^{\prime}(z)\cr},\quad\hbox{ where }\quad D%
^{\prime}(z)=\mathop{\rm diag}\nolimits(1,p^{k_{n_{1}+2}-k},\ldots,zp^{k_{n}-k%
}).$$
If $n_{1}=n$, then $D(1)={\bf 1}$ is the identity matrix,
and $z=\mathop{\rm det}\nolimits(D(z))=\mathop{\rm det}\nolimits(gD(1)h)=1$
proves the assertion in this case. We may therefore
assume that $1\leq n_{1}<n$. We now have
$$\eqalign{\pmatrix{{\bf 1}&0\cr 0&p^{k}D^{\prime}(z)\cr}&=\pmatrix{a&b\cr c&d%
\cr}\pmatrix{{\bf 1}&0\cr 0&p^{k}D^{\prime}(1)\cr}\pmatrix{a^{\prime}&b^{%
\prime}\cr c^{\prime}&d^{\prime}\cr}\cr&=\pmatrix{aa^{\prime}+p^{k}bD^{\prime}%
(1)c^{\prime}&ab^{\prime}+p^{k}bD^{\prime}(1)d^{\prime}\cr ca^{\prime}+p^{k}dD%
^{\prime}(1)c^{\prime}&cb^{\prime}+p^{k}dD^{\prime}(1)d^{\prime}\cr}.\cr}$$
From $aa^{\prime}+p^{k}bD^{\prime}(1)c^{\prime}={\bf 1}$, it follows that
$aa^{\prime}\equiv{\bf 1}\mathop{\rm mod}\nolimits\ p$, hence that $\mathop{\rm det}\nolimits(a)\in Z^{\times}$,
which means that $a\in\mathop{\rm GL}\nolimits_{n_{1}}(Z)$. Multiplication of
$g$ from the right with the matrix
$$g^{\prime}:=\pmatrix{a^{-1}&-a^{-1}b\sigma(\mathop{\rm det}\nolimits a)\cr 0&%
\sigma(\mathop{\rm det}\nolimits a)\cr}\in\mathop{\rm SL}\nolimits_{n}(Z),$$
leads to the relations
$$gg^{\prime}=\pmatrix{a&b\cr c&d\cr}\pmatrix{a^{-1}&-a^{-1}b\sigma(\mathop{\rm
det%
}\nolimits a)\cr 0&\sigma(\mathop{\rm det}\nolimits a)\cr}=\pmatrix{{\bf 1}&0%
\cr*&*\cr}$$
and
$$\eqalign{(g^{\prime})^{-1}D(1)&=\pmatrix{a&b\cr 0&\sigma(\mathop{\rm det}%
\nolimits a)^{-1}\cr}D(1)=\pmatrix{a&p^{k}bD^{\prime}(1)\cr 0&p^{k}\sigma(%
\mathop{\rm det}\nolimits a)^{-1}D^{\prime}(1)\cr}\cr&=\pmatrix{a&p^{k}bD^{%
\prime}(1)\cr 0&p^{k}D^{\prime}(1)\sigma(\mathop{\rm det}\nolimits a)^{-1}\cr}%
=D(1)\pmatrix{a&p^{k}bD^{\prime}(1)\cr 0&\sigma(\mathop{\rm det}\nolimits a)^{%
-1}\cr}.\cr}$$
With $h^{\prime}:=\pmatrix{a&p^{k}bD^{\prime}(1)\cr 0&\sigma(\mathop{\rm det}%
\nolimits a)^{-1}\cr},$
we thus arrive at
$$D(z)=gD(1)h=gg^{\prime}(g^{\prime})^{-1}D(1)h=(gg^{\prime})D(1)(h^{\prime}h).$$
We may now replace $g$ by $gg^{\prime}$ and $h$ by $h^{\prime}h$,
so that we may assume that $a={\bf 1}$ and $b=0$.
Now
$$\pmatrix{{\bf 1}&0\cr 0&p^{k}D^{\prime}(z)\cr}=\pmatrix{{\bf 1}&0\cr c&d\cr}%
\pmatrix{{\bf 1}&0\cr 0&p^{k}D^{\prime}(1)\cr}\pmatrix{a^{\prime}&b^{\prime}%
\cr c^{\prime}&d^{\prime}\cr}=\pmatrix{a^{\prime}&b^{\prime}\cr ca^{\prime}+p^%
{k}dc^{\prime}&cb^{\prime}+p^{k}dD^{\prime}(1)d^{\prime}\cr}$$
leads to $a^{\prime}={\bf 1}$ and $b^{\prime}=0$, which in turn implies
$p^{k}dD^{\prime}(1)d^{\prime}=p^{k}D^{\prime}(z).$
In view of $\mathop{\rm det}\nolimits(g)=\mathop{\rm det}\nolimits(d)=1$ and
$\mathop{\rm det}\nolimits(h)=\mathop{\rm det}\nolimits(d^{\prime})=1$, we may now use our induction hypothesis
that the theorem holds for matrices of smaller size.
Since we have
$dD^{\prime}(1)d^{\prime}=D^{\prime}(z)$ in the ring ${{Z}}/(p^{m-k})$,
we thus obtain $p^{k_{n}-k}(1-z)=0$ modulo $p^{m-k}$.
This leads to
$0=p^{k}p^{k_{n}-k}(1-z)=p^{k_{n}}(1-z)$ modulo $p^{m}$, and from that
we derive $D(z)=D(1)$.
(3) If $2s<n$ and $N(z):=N(h_{1},\ldots,zh_{s})$, then
$\sigma(z).N(1)=N(z)=N(1)$ for each $z\in Z^{\times}$, so that
$D_{(h_{1},\ldots,h_{s})}=Z^{\times}$.
Assume $2s=n$. If $zh_{s}=h_{s}$, then $\sigma(z).N(1)=N(1)$
implies that $z=\mathop{\rm det}\nolimits(\sigma(z))\in D_{(h_{1},\ldots,h_{s})}$.
If, conversely, $z\in D_{(h_{1},\ldots,h_{s})}$, then we pick
$g\in\mathop{\rm GL}\nolimits_{n}(Z)$ with $\mathop{\rm det}\nolimits(g)=z$ and
$g.N(1)=N(1)$. Then $\sigma(z)^{-1}g\in\mathop{\rm SL}\nolimits_{n}(Z)$ implies that
$\sigma(z)^{-1}g.N(1)=N(z^{-1})$
lies in the $\mathop{\rm SL}\nolimits_{n}(Z)^{2}$-orbit of
$\mathop{\rm diag}\nolimits(h_{1},h_{1},\ldots,h_{s},z^{-2}h_{s}),$
and the assertion follows from (2). For this last argument we use that
for $w\in Z^{\times}$,
$$\pmatrix{0&-w^{-1}\cr w&0\cr}\in\mathop{\rm SL}\nolimits_{2}(Z)\quad\hbox{ %
satisfies}\quad\pmatrix{0&-w^{-1}\cr w&0\cr}\pmatrix{0&wh_{s}\cr-wh_{s}&0\cr}=%
\pmatrix{h_{s}&0\cr 0&w^{2}h_{s}\cr}.$$
Conjecture III. We believe that if $2s=n$, then
$zh_{s}=h_{s}$ for each $z\in D_{(h_{1},\ldots,h_{s})}$, so that we have equality in
Theorem III.1(3), whose present version only implies that
$D_{(h_{1},\ldots,h_{s})}\cdot h_{s}$ can be identified with an elementary
abelian $2$-group, hence is of cardinality $2^{k}$ for some $k$.
The conjecture is true if all $h_{i}$ coincide. In fact,
for $h_{1}=\ldots=h_{s}$ and $N:=N(h_{1},\ldots,h_{s})$ we write
$N=h_{s}N^{\prime}$, so that the relation
$g^{\top}Ng=N$ implies that $h_{s}\cdot(g^{\top}N^{\prime}g-N^{\prime})=0$.
We conclude that $g^{\top}N^{\prime}g\equiv N^{\prime}\mathop{\rm mod}\nolimits\mathop{\rm ord}%
\nolimits(h_{s})$,
so that [New72, Th. VII.21] implies the
existence of some $\mathaccent 869{g}\in\mathop{\rm Sp}\nolimits_{2n}({{Z}})$ with
$g\equiv\mathaccent 869{g}\mathop{\rm mod}\nolimits\mathop{\rm ord}\nolimits(h_%
{s})$.
Therefore $\mathop{\rm det}\nolimits\mathaccent 869{g}=1$ implies $\mathop{\rm det}\nolimits g\equiv 1\mathop{\rm mod}\nolimits\mathop{\rm ord}%
\nolimits(h_{s})$,
i.e., $\mathop{\rm det}\nolimits(g)\cdot h_{s}=h_{s}$.
The following theorem provides a normal form for the orbits of
$\mathop{\rm GL}\nolimits_{n}({{Z}})$ in $\mathop{\rm Alt}\nolimits_{n}(Z)$ for any cyclic group $Z$.
For $Z={{Z}}$ it follows from Theorem 2.19 in [Pa96].
Theorem III.2.
For any cyclic group $Z$ the following assertions hold:
(1) Each $\mathop{\rm GL}\nolimits_{n}({{Z}})$-orbit in $\mathop{\rm Alt}\nolimits_{n}(Z)$ contains a matrix of the
form
$$N(h_{1},\ldots,h_{s}),\ 2s<n\quad\hbox{ or }\quad N(h_{1},\ldots,zh_{s}),2s=n,$$
with $z\in Z^{\times}$ and $0\not=h_{i}\in P$ satisfying
$h_{1}|h_{2}|\cdots|h_{s}$.
(2) If the matrices
$N(h_{1},\ldots,zh_{s})$ and $N(h_{1}^{\prime},\ldots,z^{\prime}h_{s}^{\prime})$ lie in the same
$\mathop{\rm GL}\nolimits_{n}({{Z}})$-orbit, then $s=s^{\prime}$ and $h_{i}^{\prime}=h_{i}$ for each $i$.
(3) If $2s<n$ or $Z\cong{{Z}}$, then any corresponding
$\mathop{\rm GL}\nolimits_{n}({{Z}})$-orbit contains
a unique matrix of the form $N(h_{1},\ldots,h_{s})$.
If $2s=n$, then two matrices $N(h_{1},\ldots,h_{s-1},zh_{s})$
and $N(h_{1},\ldots,h_{s-1},wh_{s})$ lie in the same orbit if and only if
$d:=zw^{-1}\in\pm D_{(h_{1},\ldots,h_{s})}$.
In this case, $d^{2}h_{s}=h_{s}$
Proof. (1) Let $q\colon{{Z}}\to Z$ be a surjective homomorphism and
$q_{n}\colon M_{n}({{Z}})\to M_{n}(Z)$ the induced homomorphism
which is equivariant with respect to the action (3.1) of $\mathop{\rm GL}\nolimits_{n}({{Z}})$ on both groups.
If $A\in\mathop{\rm Alt}\nolimits_{n}(Z)$, then its diagonal vanishes and
$a_{ij}=-a_{ji}$, and there exists a matrix $\mathaccent 869{A}\in\mathop{\rm Alt}\nolimits_{n}({{Z}})$ with
$q_{n}(\mathaccent 869{A})=A$.
As ${{Z}}$ is a principal ideal domain, the Theorem on the Skew Normal Form
([New72, Thms. IV.1,IV.2]) implies the existence of
$g\in\mathop{\rm GL}\nolimits_{n}({{Z}})$ with
$$g.\mathaccent 869{A}=N(\mathaccent 869{h}_{1},\ldots,\mathaccent 869{h}_{t})%
\quad\hbox{ and }\quad\mathaccent 869{h}_{1}|\mathaccent 869{h}_{2}|\cdots|%
\mathaccent 869{h}_{t}.$$
We then have
$g.A=q_{n}(g.\mathaccent 869{A})=N(z_{1}h_{1},\ldots,z_{s}h_{s}),$
where $q(\mathaccent 869{h}_{j})=z_{j}h_{j}$ with $z_{j}\in Z^{\times}$,
$h_{j}\in P$, and $s$ is maximal with $h_{s}\not=0$.
We further get $h_{1}|h_{2}|\cdots|h_{s}$.
Next we recall from [New72, Th. VII.6] that
$q_{n}(\mathop{\rm SL}\nolimits_{n}({{Z}}))=\mathop{\rm SL}\nolimits_{n}(Z)$, which implies that
$$q_{n}(\mathop{\rm GL}\nolimits_{n}({{Z}}))=\{g\in\mathop{\rm GL}\nolimits_{n}(%
Z)\colon\mathop{\rm det}\nolimits g\in\{\pm 1\}\}.$$
(3.4)3.4( 3.4 )
For $2s=n$ the matrix
$$d:=\mathop{\rm diag}\nolimits(z_{1}^{-1},1,z_{2}^{-1},1,\ldots,z_{s-1}^{-1},1,%
\ldots,1,z_{1}\cdots z_{s-1})\in\mathop{\rm SL}\nolimits_{n}(Z)$$
now satisfies
$$d.N(z_{1}h_{1},\ldots,z_{s}h_{s})=N(h_{1},\ldots,h_{s-1},z_{1}\cdots z_{s}h_{s%
}),$$
and for $2s<n$, the matrix
$$d:=\mathop{\rm diag}\nolimits(z_{1}^{-1},1,z_{2}^{-1},1,\ldots,z_{s}^{-1},1,%
\ldots,1,z_{1}\cdots z_{s})\in\mathop{\rm SL}\nolimits_{n}(Z)$$
satisfies
$$d.N(z_{1}h_{1},\ldots,z_{s}h_{s})=N(h_{1},\ldots,h_{s}).$$
Since $d\in q(\mathop{\rm GL}\nolimits_{n}({{Z}}))$, this implies (1).
(2) The Smith Normal Form of the matrix
$N(h_{1},\ldots,zh_{s})$ is
$\mathop{\rm diag}\nolimits(h_{1},h_{1},\ldots,h_{s},h_{s},0,\ldots,0)$
and for the matrix
$N(h_{1}^{\prime},\ldots,z^{\prime}h_{s}^{\prime})$ we have the normal form
$\mathop{\rm diag}\nolimits(h_{1}^{\prime},h_{1}^{\prime},\ldots,h_{s^{\prime}}%
^{\prime},h_{s^{\prime}}^{\prime},0,\ldots,0).$
Therefore Theorem III.1 implies (2).
(3) In view of (2), the number $s$ and $h_{1},\ldots,h_{s}$ are uniquely
determined by the $\mathop{\rm GL}\nolimits_{n}({{Z}})$-orbit.
If $2s<n$, then it follows already that the
corresponding orbit contains a uniqe matrix of the form $N(h_{1},\ldots,h_{s})$.
If $Z={{Z}}$, then the uniqueness assertion
follows from the uniqueness of the Skew Normal Form
([New72, Thms. IV.1,IV.2]), which follows from
the fact that the determinantal divisors of
$N=N(h_{1},\ldots,h_{s})$ satisfy
$$h_{1}=d_{1}(N)=d_{2}(N)/d_{1}(N),\ldots,h_{s}=d_{2s-1}(N)/d_{2s-2}(N)=d_{2s}(N%
)/d_{2s-1}(N)$$
and $d_{j}(N)=0$ for $j>2s$.
It remains to consider the case $2s=n$.
For $\sigma(z):=\mathop{\rm diag}\nolimits(1,\ldots,1,z)$
and $N(z):=N(h_{1},\ldots,zh_{s})$,
we get
$N(z)=\sigma(z).N(1)$,
and if there exists a $g\in\mathop{\rm GL}\nolimits_{n}({{Z}})$ with
$g.(\sigma(z).N(1))=\sigma(w).N(1),$
then
$\mathop{\rm det}\nolimits(\sigma(w)^{-1}g\sigma(z))=w^{-1}z\mathop{\rm det}%
\nolimits(g)\in D_{(h_{1},\ldots,h_{s})}.$
In view of $\mathop{\rm det}\nolimits(g)\in\{\pm 1\}$, this implies that
$w^{-1}z\in\pm D_{(h_{1},\ldots,h_{s})}$.
If, conversely,
$w^{-1}z\in\pm D_{(h_{1},\ldots,h_{s})}$,
then there exists a matrix $g\in\mathop{\rm GL}\nolimits_{n}(Z)$ fixing
$N(1)$ with $\mathop{\rm det}\nolimits(g)\in\{\pm zw^{-1}\}$.
Hence
$\mathop{\rm det}\nolimits(\sigma(w)g\sigma(z)^{-1})\in\{\pm 1\},$
and (3.4) imply the existence
of $g_{1}\in\mathop{\rm GL}\nolimits_{n}({{Z}})$ with $q_{n}(g_{1})=\sigma(w)g\sigma(z)^{-1}$. We now have
$$\eqalign{g_{1}.N(z)&=g_{1}\sigma(z).N(1)=\sigma(w)g\sigma(z)^{-1}\sigma(z).N(1%
)=\sigma(w)g.N(1)=\sigma(w)N(1)=N(w).\cr}$$
Definition III.3.
(a) We call a $\Gamma$-quantum torus rational if
the commutator group $C_{A}$ of $A^{\times}=A_{h}^{\times}$ (cf. Proposition A.1)
consists of roots of unity in ${{K}}$. We call it of cyclic type if
$C_{A}$ is a cyclic subgroup of ${{K}}^{\times}$.
(b) For each $q\in{{K}}^{\times}$ we write $A_{q}$ for the ${{Z}}^{2}$-quantum torus
corresponding to the biadditive cocycle $f\colon{{Z}}^{2}\times{{Z}}^{2}\to{{K}}^{\times}$ determined by
$$f(e_{1},e_{1})=f(e_{2},e_{2})=f(e_{2},e_{1})=1\quad\hbox{ and }\quad f(e_{1},e%
_{2})=q.$$
Then the algebra $A_{q}$ is generated by $u_{1}=\delta_{e_{1}}$, $u_{2}=\delta_{e_{2}}$
satisfying
$u_{1}u_{2}=qu_{2}u_{1},$
and their inverses. Then $C_{A_{q}}=\langle q\rangle$, so that the
quantum torus $A_{q}$ is rational if and only if $q$ is a root of unity.
Theorem III.4.
(Normal form of rational quantum tori)
Let ${{K}}$ be any field.
(a) For
any rational $n$-dimensional quantum torus $A$ over ${{K}}$, the
commutator group $C_{A}\subseteq{{K}}^{\times}$ is cyclic. Let
$q$ be a generator of $C_{A}$ and choose $P\subseteq{{Z}}/(m)$
for $m=|C_{A}|=\mathop{\rm ord}\nolimits(q)$ as above.
Then there exists an $s\in{{N}}_{0}$ with $2s\leq n$ and
$h_{2}|\ldots|h_{s}$ in $P\setminus\{0\}$
such that
$$A\cong A_{q}\otimes A_{q^{h_{2}}}\otimes\ldots\otimes A_{q^{h_{s}}}\otimes{{K}%
}[{{Z}}^{n-2s}]\quad\hbox{ and }\quad 2s<n$$
(3.5)3.5( 3.5 )
or
$$A\cong A_{q}\otimes A_{q^{h_{2}}}\otimes\ldots\otimes A_{q^{h_{s-1}}}\otimes A%
_{q^{zh_{s}}}\quad\hbox{ and }\quad 2s=n$$
(3.6)3.6( 3.6 )
for some $z\in{{N}}$ with $\mathop{\rm ord}\nolimits(q^{zh_{s}})=\mathop{\rm ord}\nolimits(q^{h_{s}})$.
(b) If two $n$-dimensional rational quantum tori $A$ and $A^{\prime}$
are (graded) isomorphic, then $C_{A}=C_{A^{\prime}}$, both can be described
by some data $(h_{2},\ldots,zh_{s})$ and
$(h_{2}^{\prime},\ldots,z^{\prime}h^{\prime}_{s^{\prime}})$ as in (a) related to the same choice
of generator $q$ of $C_{A}=C_{A^{\prime}}$.
(c) Two
$n$-dimensional rational quantum tori $A$ and $A^{\prime}$ given by such data
are (graded) isomorphic if and only if
$s=s^{\prime}$, $h_{i}=h_{i}^{\prime}$ for $i=2,\ldots,s$, and
$$z^{\prime}\in\pm z\cdot D_{(1,h_{2},\ldots,h_{s})},$$
where
$$D_{(h_{1},\ldots,h_{s})}=\{\mathop{\rm det}\nolimits(g)\colon g\in\mathop{\rm
GL%
}\nolimits_{n}(Z),gN(h_{1},\ldots,h_{s})g^{\top}=N(h_{1},\ldots,h_{s})\}\leq Z%
^{\times}$$
for the ring $Z:={{Z}}/\mathop{\rm ord}\nolimits(q)$.
In this case $z^{2}h_{s}=(z^{\prime})^{2}h_{s}$ holds in $Z$.
Proof. (a) We know from Theorem I.4 and (3.2)
that the $\Gamma$-quantum tori over ${{K}}$
are classified by the orbits of $\mathop{\rm Aut}\nolimits(\Gamma)\cong\mathop{\rm GL}\nolimits_{n}({{Z}})$ in
$H^{2}(\Gamma,{{K}}^{\times})\cong\mathop{\rm Alt}\nolimits^{2}(\Gamma,{{K}}^{%
\times})$.
In this picture, the rational quantum tori correspond to alternating
forms $f\in\mathop{\rm Alt}\nolimits^{2}(\Gamma,{{K}}^{\times})$
on $\Gamma$ whose values are roots of unity.
Since the group $C_{A}$ generated by the image of $f$ is generated by
the finite set
$f(e_{i},e_{j})$, $i,j=1,\ldots,n$, it is a finite subgroup of
${{K}}^{\times}$, hence cyclic (cf. [La93, Th. IV.1.9]).
Therefore Theorem III.2 applies,
and we see that for $s<2n$ the quantum torus
$A$ is isomorphic to one defined by a biadditive
cocycle $f\colon\Gamma\times\Gamma\to C_{A}\subseteq{{K}}^{\times}$, satisfying
$(\lambda_{f}(e_{i},e_{j}))_{i,j}=N(q,q^{h_{2}},\ldots,q^{h_{s}})$,
where $h_{2}|\ldots|h_{s}$. Here $h_{1}=1$ follows from the fact that the
commutator subgroup of $A^{\times}$ is generated by $q$.
The quantum torus $A_{f}\cong A$ defined by $f$ then satisfies (3.5).
In the other case we have $2s=n$ and (3.6) holds.
(b) That (graded) isomorphic quantum tori have the same commutator group is clear.
Therefore (b) follows from (a).
(c) The remaining assertion now follows from
Theorem I.4, combined with Theorem III.2.
Remark III.5. If $A$ is a ${{Z}}^{n}$-quantum torus of cyclic type
and the group of commutators in $A^{\times}$ is generated by $q\in{{K}}^{\times}$,
then the Skew Normal Form over ${{Z}}$ and the argument from the proof of
Theorem III.1 imply the existence of
$h_{2}|\ldots|h_{s}\in{{N}}$ and $s\in{{N}}_{0}$ such that
$$A\cong A_{q}\otimes A_{q^{h_{2}}}\otimes\ldots\otimes A_{q^{h_{s}}}\otimes{{K}%
}[{{Z}}^{n-2s}]$$
(see the Remark in 7.2 of [dCP93]).
If $\mathop{\rm ord}\nolimits(q)=\infty$, two such decompositions describe isomorphic algebras
if and only if $s=s^{\prime}$ and $h_{i}=h_{i}^{\prime}$ for all $i$ (Theorem 2.19 in [Pa96]
or Theorem III.2).
The main point of the preceding theorem is that it gives more precise
information on the isomorphism classes in the rational case.
IV. Graded automorphisms of quantum tori
In this section we briefly discuss the group of automorphisms of a general
quantum torus, but our main result only concerns the $2$-dimensional case:
For $A=A_{q}$ and the corresponding alternating form
$\lambda$ on ${{Z}}^{2}$, the group $\mathop{\rm Aut}\nolimits(A)$ it is a semi-direct product
$\mathop{\rm Hom}\nolimits({{Z}}^{2},{{K}}^{\times})\mathchar 9583\relax\mathop%
{\rm Aut}\nolimits({{Z}}^{2},\lambda).$
Definition IV.1.
Let $A$ be a $\Gamma$-quantum torus. We write
$\mathop{\rm Aut}\nolimits_{\rm gr}(A)$ for the group of graded automorphisms of $A$,
i.e., all those automorphisms $\varphi\in\mathop{\rm Aut}\nolimits(A)$ for which there exists an
automorphism $\varphi_{\Gamma}\in\mathop{\rm Aut}\nolimits(\Gamma)$ with
$\varphi(A_{\gamma})=A_{\varphi_{\Gamma}(\gamma)}$ for all $\gamma\in\Gamma$.
Note that Proposition A.1 in the appendix implies that if $\Gamma$ is torsion free,
then all units are homogeneous, which implies that each automorphism of $A$ is graded.
Remark IV.2. We fix a basis $(\delta_{\gamma})_{\gamma\in\Gamma}$ of $A$
and suppose that $f\in Z^{2}(\Gamma,Z)$ is the corresponding cocycle determined by (1.1).
Then for each graded automorphism $\varphi$ of $A$ there is an automorphism
$\varphi_{\Gamma}\in\mathop{\rm Aut}\nolimits(\Gamma)$ and a function $\chi\colon\Gamma\to{{K}}^{\times}$ such
$$\varphi(\delta_{\gamma})=\chi(\gamma)\delta_{\varphi_{\Gamma}(\gamma)},\quad%
\gamma\in\Gamma.$$
(4.1)4.1( 4.1 )
Conversely, for a pair $(\chi,\varphi_{\Gamma})$ of a function
$\chi\colon\Gamma\to{{K}}^{\times}$ and an automorphism
$\varphi_{\Gamma}\in\mathop{\rm Aut}\nolimits(\Gamma)$ the prescription
$\varphi(\delta_{\gamma}):=\chi(\gamma)\delta_{\varphi_{\Gamma}(\gamma)}$
defines an automorphism of $A$ if and only if
$${(\varphi_{\Gamma}^{*}f)(\gamma,\gamma^{\prime})\over f(\gamma,\gamma^{\prime}%
)}={\chi(\gamma+\gamma^{\prime})\over\chi(\gamma)\chi(\gamma^{\prime})}\quad%
\hbox{ for all }\quad\gamma,\gamma^{\prime}\in\Gamma.$$
(4.2)4.2( 4.2 )
Note that if $f$ is biadditive, then $\varphi_{\Gamma}^{*}f/f$ is biadditive, so that
$\chi$ is a corresponding ${{K}}^{\times}$-valued quadratic form.
If $f$ and $\varphi_{\Gamma}$ are given, then a $\chi$ satisfying (4.2) exists if
and only if $[\varphi_{\Gamma}^{*}f]=[f]$ holds in $H^{2}(\Gamma,Z)$.
Lemma IV.3. The image of the map
$Q\colon\mathop{\rm Aut}\nolimits_{\rm gr}(A)\to\mathop{\rm Aut}\nolimits(%
\Gamma),\varphi\mapsto\varphi_{\Gamma}$
is the group
$$\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}:=\{\psi\in\mathop{\rm Aut}\nolimits(%
\Gamma)\colon[\psi^{*}f]=[f]\},$$
which is contained in
$\mathop{\rm Aut}\nolimits(\Gamma,\lambda_{f}):=\{\psi\in\mathop{\rm Aut}%
\nolimits(\Gamma)\colon\psi^{*}\lambda_{f}=\lambda_{f}\},$
where $\lambda_{f}(\gamma,\gamma^{\prime})={f(\gamma,\gamma^{\prime})\over f(\gamma^{%
\prime},\gamma)}$.
If, in addition, $\Gamma$ is free, then
$\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}=\mathop{\rm Aut}\nolimits(\Gamma,%
\lambda_{f}).$
Proof. Let $\varphi_{\Gamma}\in\mathop{\rm Aut}\nolimits(\Gamma)$.
In view of Remark IV.2, the existence of $\varphi\in\mathop{\rm Aut}\nolimits_{\rm gr}(A)$
with $Q(\varphi)=\varphi_{\Gamma}$ is equivalent to the existence of
$\chi$ satisfying (4.2), which is equivalent to $[\varphi_{\Gamma}^{*}f]=[f]$ in
$H^{2}(\Gamma,{{K}}^{\times})$.
Since (4.2) implies that $\varphi_{\Gamma}^{*}f/f$ is symmetric, we have
$\varphi_{\Gamma}^{*}\lambda_{f}=\lambda_{\varphi_{\Gamma}^{*}f}=\lambda_{f}.$
If, in addition, $\Gamma$ is free, then Proposition II.4 entails that
$\varphi_{\Gamma}^{*}\lambda_{f}=\lambda_{f}$ is equivalent to
$[\varphi_{\Gamma}^{*}f]=[f]$ in $H^{2}(\Gamma,{{K}}^{\times})$
(cf. [OP95, Lemma 3.3(iii)]).
From (4.2) we derive in particular that $(\chi,{\bf 1})$ defines an automorphism of
$A$ if and only if $\chi\in\mathop{\rm Hom}\nolimits(\Gamma,{{K}}^{\times})$, so that we obtain
the exact sequence
$${\bf 1}\to\mathop{\rm Hom}\nolimits(\Gamma,{{K}}^{\times})\to\mathop{\rm Aut}%
\nolimits_{\rm gr}(A)\to\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}\to{\bf 1}$$
(4.3)4.3( 4.3 )
(cf. [OP95, Lemma 3.3(iii)]). We call the automorphisms of the form $(\chi,{\bf 1})$ scalar.
Remark IV.4. If the map $\Phi$ from Proposition II.4 is not injective,
then the groups $\mathop{\rm Aut}\nolimits(\Gamma,\lambda_{f})$ and $\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}$ need not coincide,
but with Proposition II.3 we obtain a $1$-cocycle
$$I\colon\mathop{\rm Aut}\nolimits(\Gamma,\lambda_{f})\to\mathop{\rm Ext}%
\nolimits_{\rm ab}(\Gamma,{{K}}^{\times}),\quad\psi\mapsto[\psi^{*}f-f]$$
with respect to the right action of $\mathop{\rm Aut}\nolimits(\Gamma,\lambda_{f})$ on
$\mathop{\rm Ext}\nolimits(\Gamma,{{K}}^{\times})\cong H^{2}(\Gamma,{{K}}^{%
\times})$ by
$\psi.[f]:=[\psi^{*}f]$. We then have
$\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}=I^{-1}(0).$
In the remainder of this section we restrict our attention to the case,
where $\Gamma={{Z}}^{n}$ is a free abelian group of rank $n$, which
implies that $\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}=\mathop{\rm Aut}\nolimits(\Gamma,%
\lambda_{f})$ and
that $\mathop{\rm Aut}\nolimits(A)=\mathop{\rm Aut}\nolimits_{\rm gr}(A)$ (Corollary A.2).
Remark IV.5. (a) For $n=1$, each alternating biadditive map $\lambda$ on
$\Gamma$ vanishes, so that $\mathop{\rm Aut}\nolimits(\Gamma,\lambda)=\mathop{\rm Aut}\nolimits(\Gamma)%
\cong\{\pm\mathop{\rm id}\nolimits_{\Gamma}\}$.
(b) For each alternating form $\lambda\colon\Gamma\times\Gamma\to{{K}}^{\times}$ we
have $-\mathop{\rm id}\nolimits_{\Gamma}\in\mathop{\rm Aut}\nolimits(\Gamma,\lambda)$.
(c) In [OP95], it is shown that if $n\geq 3$ and the subgroup $\langle\mathop{\rm im}\nolimits(\lambda)\rangle$
of ${{K}}^{\times}$ generated by the
image of $\lambda$ is free of rank ${n\choose 2}$, then
$\mathop{\rm Aut}\nolimits(\Gamma,\lambda_{f})=\{\pm\mathop{\rm id}\nolimits_{%
\Gamma}\}$.
Moreover, for $n=3$ and $\langle\mathop{\rm im}\nolimits(\lambda)\rangle$ free of rank $2$,
[OP95, Prop. 3.7] implies the existence of
a basis $\gamma_{1},\gamma_{2},\gamma_{3}\in\Gamma$
with $\lambda(\gamma_{1},\gamma_{2})=1$ and
$$\eqalign{\mathop{\rm Aut}\nolimits(\Gamma,\lambda)&\cong\{\sigma\in\mathop{\rm
Aut%
}\nolimits(\Gamma)\colon(\exists a,b\in{{Z}},\varepsilon\in\{\pm 1\})\ \sigma(%
\gamma_{1})=\gamma_{1}^{\varepsilon},\sigma(\gamma_{2})=\gamma_{2}^{%
\varepsilon},\sigma(\gamma_{3})=\gamma_{1}^{a}\gamma_{2}^{b}\gamma_{3}^{%
\varepsilon}\}\cr&\cong{{Z}}^{2}\mathchar 9583\relax\{\pm\mathop{\rm id}%
\nolimits_{{{Z}}^{2}}\}.\cr}$$
We now take a closer look at the case $n=2$.
Any alternating form $\lambda\in\mathop{\rm Alt}\nolimits^{2}({{Z}}^{2},{{K}}^{\times})$
is uniquely determined by $q:=\lambda(e_{1},e_{2})$, which implies
$\lambda(\gamma,\gamma^{\prime})=q^{\gamma_{1}\gamma_{2}^{\prime}-\gamma_{2}%
\gamma_{1}^{\prime}}.$
We may therefore assume that a corresponding bimultiplicative
cocycle $f$ satisfies
$f(\gamma,\gamma^{\prime})=q^{\gamma_{1}\gamma_{2}^{\prime}}$, which leads to the quantum torus
$A_{q}$ with two generators $u_{i}=\delta_{e_{i}}$ and their inverses, satisfying
$u_{1}u_{2}=qu_{2}u_{1}$, as defined in the introduction.
We start with two simple observations:
Lemma IV.6. $\mathop{\rm Aut}\nolimits({{Z}}^{2},\lambda)=\cases{\mathop{\rm SL}\nolimits_{%
2}({{Z}})&for $q^{2}\not=1$\cr\mathop{\rm GL}\nolimits_{2}({{Z}})&for $q^{2}=1%
$.\cr}$
Proof. Clearly $\mathop{\rm SL}\nolimits_{2}({{Z}})\subseteq\mathop{\rm Aut}\nolimits({{Z}}^{2%
},\lambda)\subseteq\mathop{\rm GL}\nolimits_{2}({{Z}})$.
The map $g_{0}(\gamma)=(\gamma_{2},\gamma_{1})$ satisfies
$\mathop{\rm GL}\nolimits_{2}({{Z}})\cong\mathop{\rm SL}\nolimits_{2}({{Z}})%
\mathchar 9583\relax\langle g_{0}\rangle$, and we have
$${g_{0}^{*}\lambda(e_{1},e_{2})\over\lambda(e_{1},e_{2})}={\lambda(e_{2},e_{1})%
\over\lambda(e_{1},e_{2})}=q^{-2}.$$
Example IV.7. (a) On ${{Z}}^{2}$ the map
$\chi(\gamma):=\gamma_{1}\gamma_{2}$ is a quadratic form with
$$\chi(\gamma+\gamma^{\prime})-\chi(\gamma)-\chi(\gamma^{\prime})=\gamma_{1}%
\gamma_{2}^{\prime}+\gamma_{2}\gamma_{1}^{\prime}.$$
(b) On ${{Z}}$ the map $\chi(n):={n\choose 2}$ is a quadratic form with
$$\chi(n+n^{\prime})-\chi(n)-\chi(n^{\prime})={(n+n^{\prime})(n+n^{\prime}-1)-n(%
n-1)-n^{\prime}(n^{\prime}-1)\over 2}={nn^{\prime}+n^{\prime}n\over 2}=nn^{%
\prime}.$$
From $\mathop{\rm SL}\nolimits_{2}({{Z}})\subseteq\mathop{\rm Aut}\nolimits({{Z}}^{2%
},\lambda)$, it follows in particular that
each matrix
$g=\pmatrix{a&b\cr c&d\cr}\in\mathop{\rm SL}\nolimits_{2}({{Z}})$
can be lifted to an automorphism of $A_{q}$.
To determine a corresponding quadratic form $\chi\colon{{Z}}^{2}\to{{K}}^{\times}$,
we have to solve the equation (4.2):
$${(g^{*}f)(\gamma,\gamma^{\prime})\over f(\gamma,\gamma^{\prime})}={\chi(\gamma%
+\gamma^{\prime})\over\chi(\gamma)\chi(\gamma^{\prime})}.$$
The form $g^{*}f/f$ is determined by its values on the pairs
$(e_{1},e_{1}),(e_{1},e_{2})$ and $(e_{2},e_{2})$:
$$(g^{*}f/f)(e_{1},e_{1})=f(g.e_{1},g.e_{1})=q^{ac},\quad(g^{*}f/f)(e_{1},e_{2})%
=f(g.e_{1},g.e_{2})q^{-1}=q^{ad-1}$$
and
$(g^{*}f/f)(e_{2},e_{2})=f(g.e_{2},g.e_{2})=q^{bd}.$
This means that
$$(g^{*}f/f)(\gamma,\gamma^{\prime})=q^{ac\gamma_{1}\gamma_{1}^{\prime}+(ad-1)(%
\gamma_{1}\gamma_{2}^{\prime}+\gamma_{1}^{\prime}\gamma_{2})+bd\gamma_{2}%
\gamma_{2}^{\prime}}.$$
Before we turn to lifting the full groups $\mathop{\rm Aut}\nolimits({{Z}}^{2},\lambda)$
to an automorphism group of
$A$, we discuss certain specific elements of finite order separately.
Remark IV.8. (a) For the central element $z=-{\bf 1}\in\mathop{\rm SL}\nolimits_{2}({{Z}})$, any lift
$\mathaccent 866{z}\in\mathop{\rm Aut}\nolimits(A_{q})$ is of the form
$$\mathaccent 866{z}.\delta_{\gamma}=r^{\gamma_{1}}s^{\gamma_{2}}\cdot\delta_{-%
\gamma}\quad\hbox{ for some}\quad r,s\in{{K}}^{\times},$$
and any such element satisfies
$\mathaccent 866{z}^{2}.\delta_{\gamma}=r^{\gamma_{1}}s^{\gamma_{2}}\cdot%
\mathaccent 866{z}.\delta_{-\gamma}=r^{\gamma_{1}-\gamma_{1}}s^{\gamma_{2}-%
\gamma_{2}}\cdot\delta_{\gamma}=\delta_{\gamma}.$
Hence each lift $\mathaccent 866{z}$ of $z$ is an element of order $2$.
(b) The matrices
$$g_{1}:=\pmatrix{0&1\cr-1&0\cr}\quad\hbox{ and }\quad g_{2}:=\pmatrix{1&1\cr-1&%
0\cr}$$
satisfy $g_{1}^{2}=z=g_{2}^{3}$, which leads to
$\mathop{\rm ord}\nolimits(g_{1})=4$ and $\mathop{\rm ord}\nolimits(g_{2})=6$.
From the preceding paragraph we conclude that for any lift
$\mathaccent 866{g}_{j}$ of $g_{j}$, $j=1,2$, we have
$\mathaccent 866{g}_{1}^{4}={\bf 1}=\mathaccent 866{g}_{2}^{6}.$
In view of
$$(g_{1}^{*}f/f)(\gamma,\gamma^{\prime})=q^{-(\gamma_{1}\gamma_{2}^{\prime}+%
\gamma_{1}^{\prime}\gamma_{2})},$$
a lift $\mathaccent 869{g}_{1}$ of $g_{1}$ is given by
$\mathaccent 869{g}_{1}.\delta_{\gamma}=q^{-\gamma_{1}\gamma_{2}}\delta_{g_{1}.\gamma}$
(Example IV.7(a)).
We then have
$$\mathaccent 869{g}_{1}^{2}.\delta_{\gamma}=q^{-\gamma_{1}\gamma_{2}}%
\mathaccent 869{g}_{1}.\delta_{(\gamma_{2},-\gamma_{1})}=q^{-\gamma_{1}\gamma_%
{2}}q^{\gamma_{2}\gamma_{1}}.\delta_{-\gamma}=\delta_{-\gamma}.$$
Any other lift $\mathaccent 866{g}_{1}$ of $g_{1}$ is of the form
$$\mathaccent 866{g}_{1}.\delta_{g}=r_{1}^{\gamma_{1}}s_{1}^{\gamma_{2}}q^{-%
\gamma_{1}\gamma_{2}}\delta_{g_{1}.\gamma}$$
for two elements $r_{1},s_{1}\in{{K}}^{\times}$. The square of this element is given by
$$\mathaccent 866{g}_{1}^{2}.\delta_{g}=r_{1}^{\gamma_{1}}s_{1}^{\gamma_{2}}%
\mathaccent 866{g}_{1}\mathaccent 869{g}_{1}.\delta_{\gamma}=r_{1}^{\gamma_{1}%
+\gamma_{2}}s_{1}^{\gamma_{2}-\gamma_{1}}\mathaccent 869{g}_{1}^{2}.\delta_{%
\gamma}=\Big{(}{r_{1}\over s_{1}}\Big{)}^{\gamma_{1}}(r_{1}s_{1})^{\gamma_{2}}%
\cdot\delta_{-\gamma}.$$
(4.4)4.4( 4.4 )
For the matrix $g_{2}$ we have
$$(g_{2}^{*}f/f)(\gamma,\gamma^{\prime})=q^{-\gamma_{1}\gamma_{1}^{\prime}-(%
\gamma_{1}\gamma_{2}^{\prime}+\gamma_{1}^{\prime}\gamma_{2})},$$
so that we obtain a lift $\mathaccent 869{g}_{2}$ of $g_{2}$ by
$\mathaccent 869{g}_{2}.\delta_{\gamma}=q^{-{\gamma_{1}\choose 2}-\gamma_{1}%
\gamma_{2}}\delta_{(\gamma_{1}+\gamma_{2},-\gamma_{1})}$
(Example IV.7(b)).
Hence each lift $\mathaccent 866{g}_{2}$ of $g_{2}$ is of the form
$$\mathaccent 866{g}_{2}.\delta_{g}=r_{2}^{\gamma_{1}}s_{2}^{\gamma_{2}}q^{-{%
\gamma_{1}\choose 2}-\gamma_{1}\gamma_{2}}\delta_{(\gamma_{1}+\gamma_{2},-%
\gamma_{1})},$$
for some $r_{2},s_{2}\in{{K}}^{\times}$.
In view of
$g_{2}^{2}=\pmatrix{0&1\cr-1&-1\cr},$
we get with Example IV.7(b):
$$\eqalign{\mathaccent 869{g}_{2}^{3}.\delta_{\gamma}&=q^{-{\gamma_{1}\choose 2}%
-\gamma_{1}\gamma_{2}}\mathaccent 869{g}_{2}^{2}.\delta_{\gamma_{1}+\gamma_{2}%
,-\gamma_{1}}=q^{-{\gamma_{1}\choose 2}-\gamma_{1}\gamma_{2}}q^{-{\gamma_{1}+%
\gamma_{2}\choose 2}+(\gamma_{1}+\gamma_{2})\gamma_{1}}\mathaccent 869{g}_{2}.%
\delta_{\gamma_{2},-\gamma_{1}-\gamma_{2}}\cr&=q^{-2{\gamma_{1}\choose 2}-{%
\gamma_{2}\choose 2}-\gamma_{1}\gamma_{2}+\gamma_{1}^{2}}q^{-{\gamma_{2}%
\choose 2}+(\gamma_{1}+\gamma_{2})\gamma_{2}}\delta_{-\gamma}=q^{-\gamma_{1}(%
\gamma_{1}-1)-\gamma_{2}(\gamma_{2}-1)+\gamma_{1}^{2}+\gamma_{2}^{2}}\delta_{-%
\gamma}=q^{\gamma_{1}+\gamma_{2}}\delta_{-\gamma}.\cr}$$
This further leads to
$$\leqalignno{\mathaccent 866{g}_{2}^{3}.\delta_{\gamma}&=r_{2}^{\gamma_{1}}s_{2%
}^{\gamma_{2}}\mathaccent 866{g}_{2}^{2}\mathaccent 869{g}_{2}.\delta_{\gamma}%
=r_{2}^{2\gamma_{1}+\gamma_{2}}s_{2}^{-\gamma_{1}+\gamma_{2}}\mathaccent 866{g%
}_{2}\mathaccent 869{g}_{2}^{2}.\delta_{\gamma}=r_{2}^{2\gamma_{1}+2\gamma_{2}%
}s_{2}^{-2\gamma_{1}}\mathaccent 869{g}_{2}^{3}.\delta_{\gamma}\cr&=r_{2}^{2(%
\gamma_{1}+\gamma_{2})}s_{2}^{-2\gamma_{1}}q^{\gamma_{1}+\gamma_{2}}.\delta_{-%
\gamma}=\Big{(}{r_{2}^{2}\over s_{2}^{2}}q\Big{)}^{\gamma_{1}}(r_{2}^{2}q)^{%
\gamma_{2}}\delta_{-\gamma}.&(4.5)\cr}$$
(c) If, in addition, $q^{2}=1$, then
$\mathop{\rm Aut}\nolimits(\Gamma,\lambda_{f})=\mathop{\rm Aut}\nolimits(\Gamma%
)\cong\mathop{\rm GL}\nolimits_{2}({{Z}})$
(Lemma IV.6). For the involution
$$g_{0}:=\pmatrix{0&1\cr 1&0\cr}$$
we have $\mathop{\rm GL}\nolimits_{2}({{Z}})=\mathop{\rm SL}\nolimits_{2}({{Z}})%
\mathchar 9583\relax\langle g_{0}\rangle,$
and the elements $g_{0},g_{1},g_{2}$ satisfy
$$g_{0}g_{1}g_{0}=g_{1}^{-1}=g_{1}^{3}\quad\hbox{ and }\quad g_{0}g_{2}g_{0}=g_{%
2}^{5}=g_{2}^{-1}.$$
(4.6)4.6( 4.6 )
To lift $g_{0}$ to an automorphism of $A_{q}$, we first note
that $q^{2}=1$ implies that
$$(g_{0}^{*}f/f)(\gamma,\gamma^{\prime})=q^{\gamma_{2}\gamma_{1}^{\prime}-\gamma%
_{1}\gamma_{2}^{\prime}},=q^{\gamma_{2}\gamma_{1}^{\prime}+\gamma_{1}\gamma_{2%
}^{\prime}},$$
which shows that each lift $\mathaccent 866{g}_{0}$ of $g_{0}$ is of the form
$\mathaccent 866{g}_{0}.\delta_{\gamma}=r_{0}^{\gamma_{1}}s_{0}^{\gamma_{2}}q^{%
\gamma_{1}\gamma_{2}}\delta_{(\gamma_{2},\gamma_{1})}$
for some $r_{0},s_{0}\in{{K}}^{\times}$.
In view of
$$\mathaccent 866{g}_{0}^{2}.\delta_{\gamma}=r_{0}^{\gamma_{1}}s_{0}^{\gamma_{2}%
}q^{\gamma_{1}\gamma_{2}}\mathaccent 866{g}_{0}.\delta_{(\gamma_{2},\gamma_{1}%
)}=r_{0}^{\gamma_{1}+\gamma_{2}}s_{0}^{\gamma_{2}+\gamma_{1}}q^{2\gamma_{1}%
\gamma_{2}}\delta_{\gamma}=(r_{0}s_{0})^{\gamma_{1}+\gamma_{2}}\delta_{\gamma},$$
$\mathaccent 866{g}_{0}^{2}={\bf 1}$ is equivalent to
$r_{0}s_{0}=1.$ If this condition is satisfied, then
$\mathaccent 866{g}_{0}.\delta_{\gamma}=r_{0}^{\gamma_{1}-\gamma_{2}}q^{\gamma_%
{1}\gamma_{2}}\delta_{(\gamma_{2},\gamma_{1})}.$
Before we state the following theorem, we recall that for any split extension
$${\bf 1}\to A\to\mathaccent 866{G}\smash{\mathop{\hbox to 20.0pt{%
\rightarrowfill}}\limits^{q}}G\to{\bf 1}$$
of a group $G$ by some (abelian) $G$-module $A$, the set of
all splittings is parametrized by the group
$$Z^{1}(G,A)=\{f\colon G\to A\colon(\forall x,y\in G)\ f(xy)=f(x)+x.f(y)\}$$
of $A$-valued $1$-cocycles. This parametrization is obtained by choosing a
homomorphic section $\sigma_{0}\colon G\to\mathaccent 866{G}$ and then
observing that any other homomorphic section
$\sigma\colon G\to\mathaccent 866{G}$ is of the form $\sigma=f\cdot\sigma_{0}$, where
$f\in Z^{1}(G,A)$.
Theorem IV.9.
For each element $q\in{{K}}^{\times}$ and
$\lambda(\gamma,\gamma^{\prime})=q^{\gamma_{1}\gamma_{2}^{\prime}-\gamma_{2}%
\gamma_{1}^{\prime}}$
the exact sequence
$${\bf 1}\to\mathop{\rm Hom}\nolimits({{Z}}^{2},{{K}}^{\times})\to\mathop{\rm Aut%
}\nolimits(A_{q})\to\mathop{\rm Aut}\nolimits({{Z}}^{2},\lambda)\to{\bf 1}$$
splits.
For $q^{2}=1$, the homomorphisms
$\sigma\colon\mathop{\rm GL}\nolimits_{2}({{Z}})\to\mathop{\rm Aut}\nolimits(A_%
{q})$ splitting the
sequence are parametrized by the abelian group
$$Z^{1}(\mathop{\rm GL}\nolimits_{2}({{Z}}),\mathop{\rm Hom}\nolimits({{Z}}^{2},%
{{K}}^{\times}))\cong\{(r_{0},r_{1},r_{2})\in({{K}}^{\times})^{3}\colon r_{2}^%
{4}r_{0}^{2}=r_{1}^{2}\},$$
and for $q^{2}\not=1$, the homomorphisms $\sigma\colon\mathop{\rm SL}\nolimits_{2}({{Z}})\to\mathop{\rm Aut}\nolimits(A_%
{q})$ splitting the
sequence are parametrized by
$$Z^{1}(\mathop{\rm SL}\nolimits_{2}({{Z}}),\mathop{\rm Hom}\nolimits({{Z}}^{2},%
{{K}}^{\times}))\cong({{K}}^{\times})^{2}\times\{z\in{{K}}^{\times}\colon z^{2%
}=1\}.$$
Proof. First we consider the case $q^{2}\not=1$,
where $\mathop{\rm Aut}\nolimits({{Z}}^{2},\lambda)=\mathop{\rm SL}\nolimits_{2}({{Z}})$
(Remark IV.8). We shall use the description of the lifts of
$g_{1},g_{2}$ given in Remark IV.8.
Since $\mathop{\rm SL}\nolimits_{2}({{Z}})$ is presented by the relations
$$g_{1}^{4}=g_{2}^{6}={\bf 1},\quad g_{1}^{2}=g_{2}^{3}$$
([Ha00, p.51]), Remark IV.8 implies that
a pair of elements $(\mathaccent 866{g}_{1},\mathaccent 866{g}_{2})$ lifting $(g_{1},g_{2})$ leads to a lift
$\mathop{\rm SL}\nolimits_{2}({{Z}})\to\mathop{\rm Aut}\nolimits(A_{q})$ if and only if
$\mathaccent 866{g}_{1}^{2}=\mathaccent 866{g}_{2}^{3}$.
Comparing (4.4) and (4.5), we see that $\mathaccent 866{g}_{1}^{2}=\mathaccent 866{g}_{2}^{3}$ is equivalent to
$${r_{1}\over s_{1}}={r_{2}^{2}\over s_{2}^{2}}q\quad\hbox{ and }\quad r_{1}s_{1%
}=r_{2}^{2}q,$$
which is equivalent to
$$s_{1}^{2}=s_{2}^{2}\quad\hbox{ and }\quad s_{1}={r_{2}^{2}q\over r_{1}},$$
(4.8)4.8( 4.8 )
These equations have the simple solution
$r_{1}=q,r_{2}=s_{1}=s_{2}=1,$
showing that the action of the group $\mathop{\rm SL}\nolimits_{2}({{Z}})$ on $\Gamma$ lifts to an action on $A_{q}$. Moreover, for each pair $(r_{1},r_{2})$, the set of all
solutions is determined by the choice of sign in
$s_{2}:=\pm s_{1}$, which is vacuous if $\mathop{\rm char}\nolimits({{K}})=2$.
Next we consider the case $q^{2}=1$.
We assume that the lift $\mathaccent 866{g}_{0}$ of $g_{0}$ satisfies
$\mathaccent 866{g}_{0}^{2}={\bf 1}$ (cf. Remark IV.8(c)).
Now the
relation $\mathaccent 866{g}_{0}\mathaccent 866{g}_{1}\mathaccent 866{g}_{0}=\mathaccent
8%
66{g}_{1}^{-1}$ is equivalent to
$(\mathaccent 866{g}_{0}\mathaccent 866{g}_{1})^{2}={\bf 1}$. We calculate
$$\mathaccent 866{g}_{0}\mathaccent 866{g}_{1}.\delta_{\gamma}=r_{1}^{\gamma_{1}%
}s_{1}^{\gamma_{2}}q^{-\gamma_{1}\gamma_{2}}\mathaccent 866{g}_{0}.\delta_{(%
\gamma_{2},-\gamma_{1})}=(r_{0}r_{1})^{\gamma_{1}}(r_{0}s_{1})^{\gamma_{2}}%
\delta_{(-\gamma_{1},\gamma_{2})}$$
to get
$$(\mathaccent 866{g}_{0}\mathaccent 866{g}_{1})^{2}.\delta_{\gamma}=(r_{0}r_{1}%
)^{\gamma_{1}}(r_{0}s_{1})^{\gamma_{2}}\mathaccent 866{g}_{0}\mathaccent 866{g%
}_{1}.\delta_{(-\gamma_{1},\gamma_{2})}=(r_{0}s_{1})^{2\gamma_{2}}\delta_{%
\gamma}.$$
Hence $\mathaccent 866{g}_{0}\mathaccent 866{g}_{1}\mathaccent 866{g}_{0}=\mathaccent
8%
66{g}_{1}^{-1}$ is equivalent to
$$r_{0}^{2}s_{1}^{2}=1.$$
(4.9)4.9( 4.9 )
To see when $\mathaccent 866{g}_{0}\mathaccent 866{g}_{2}\mathaccent 866{g}_{0}=\mathaccent
8%
66{g}_{2}^{-1}$ holds,
we first observe that
$$\mathaccent 866{g}_{2}^{-1}.\delta_{\gamma}=r_{2}^{\gamma_{2}}s_{2}^{-\gamma_{%
1}-\gamma_{2}}q^{{-\gamma_{2}\choose 2}-\gamma_{2}(\gamma_{1}+\gamma_{2})}%
\delta_{(-\gamma_{2},\gamma_{1}+\gamma_{2})}.$$
Further
$$\eqalign{\mathaccent 866{g}_{0}\mathaccent 866{g}_{2}.\delta_{\gamma}&=r_{2}^{%
\gamma_{1}}s_{2}^{\gamma_{2}}q^{-{\gamma_{1}\choose 2}-\gamma_{1}\gamma_{2}}%
\mathaccent 866{g}_{0}.\delta_{(\gamma_{1}+\gamma_{2},-\gamma_{1})}=(r_{0}^{2}%
r_{2})^{\gamma_{1}}(r_{0}s_{2})^{\gamma_{2}}q^{{\gamma_{1}\choose 2}+\gamma_{1%
}\gamma_{2}+(\gamma_{1}+\gamma_{2})\gamma_{1}}\cdot\delta_{(-\gamma_{1},\gamma%
_{1}+\gamma_{2})}\cr&=(r_{0}^{2}r_{2})^{\gamma_{1}}(r_{0}s_{2})^{\gamma_{2}}q^%
{{\gamma_{1}\choose 2}+\gamma_{1}^{2}}\cdot\delta_{(-\gamma_{1},\gamma_{1}+%
\gamma_{2})}=(r_{0}^{2}r_{2}q)^{\gamma_{1}}(r_{0}s_{2})^{\gamma_{2}}q^{{\gamma%
_{1}\choose 2}}\cdot\delta_{(-\gamma_{1},\gamma_{1}+\gamma_{2})}\cr}$$
because $q^{2}=1$ implies $q^{n^{2}}=q^{n}=q^{-n}$ for each $n\in{{Z}}$.
On the other hand, we have
$$\eqalign{\mathaccent 866{g}_{2}^{-1}\mathaccent 866{g}_{0}.\delta_{\gamma}&=r_%
{0}^{\gamma_{1}}r_{0}^{-\gamma_{2}}q^{\gamma_{1}\gamma_{2}}\mathaccent 866{g}_%
{2}^{-1}.\delta_{(\gamma_{2},\gamma_{1})}=r_{0}^{\gamma_{1}}r_{0}^{-\gamma_{2}%
}q^{\gamma_{1}\gamma_{2}}r_{2}^{\gamma_{1}}s_{2}^{-\gamma_{2}-\gamma_{1}}q^{{-%
\gamma_{1}\choose 2}-\gamma_{1}(\gamma_{1}+\gamma_{2})}\delta_{(-\gamma_{1},%
\gamma_{1}+\gamma_{2})}\cr&=(r_{0}r_{2}s_{2}^{-1})^{\gamma_{1}}(r_{0}s_{2})^{-%
\gamma_{2}}q^{{-\gamma_{1}\choose 2}-\gamma_{1}^{2}}\delta_{(-\gamma_{1},%
\gamma_{1}+\gamma_{2})}=(r_{0}r_{2}s_{2}^{-1})^{\gamma_{1}}(r_{0}s_{2})^{-%
\gamma_{2}}q^{-{\gamma_{1}\choose 2}}\delta_{(-\gamma_{1},\gamma_{1}+\gamma_{2%
})}\cr&=(r_{0}r_{2}s_{2}^{-1})^{\gamma_{1}}(r_{0}s_{2})^{-\gamma_{2}}q^{{%
\gamma_{1}\choose 2}}\delta_{(-\gamma_{1},\gamma_{1}+\gamma_{2})}.\cr}$$
Therefore $\mathaccent 866{g}_{0}\mathaccent 866{g}_{2}\mathaccent 866{g}_{0}=\mathaccent
8%
66{g}_{2}^{-1}$ is equivalent to
$r_{0}r_{2}s_{2}^{-1}=r_{0}^{2}r_{2}q$ and $(r_{0}s_{2})^{2}=1,$
which is equivalent to
$$r_{0}s_{2}=q,$$
(4.10)4.10( 4.10 )
because this relation implies $(r_{0}s_{2})^{2}=q^{2}=1$.
We conclude that the numbers $r_{0},r_{1},r_{2},s_{1},s_{2}$ which determine
$\mathaccent 866{g}_{0},\mathaccent 866{g}_{1},\mathaccent 866{g}_{2}$ define a lift of $\mathop{\rm GL}\nolimits_{2}({{Z}})$ to
$\mathop{\rm Aut}\nolimits(A_{q})$ if and only if the equations (4.8), (4.9) and (4.10) are satisfied:
$$s_{1}^{2}=s_{2}^{2},\quad s_{1}={r_{2}^{2}q\over r_{1}},\quad r_{0}^{2}s_{1}^{%
2}=1,\quad\hbox{ and }\quad r_{0}s_{2}=q.$$
If $r_{0},r_{1}$ and $r_{2}$ are given, we determine
$s_{1}$ and $s_{2}$ by
$s_{1}:={r_{2}^{2}q\over r_{1}}$ and $s_{2}:={q\over r_{0}}.$
Then
$${s_{1}^{2}\over s_{2}^{2}}={r^{4}_{2}r_{0}^{2}\over r_{1}^{2}}=r_{0}^{2}s_{1}^%
{2},$$
so that we obtain only the relation
$r_{2}^{4}r_{0}^{2}=r_{1}^{2}$ for $r_{0},r_{1},r_{2}$.
This completes the proof.
Remark IV.10. (a) From the
proof of the preceding theorem, we see that if $q^{2}=1$, we obtain the particularly simple solution
$$r_{0}=r_{1}=r_{2}=1,\quad s_{1}=s_{2}=q.$$
(b) For $\mathop{\rm char}\nolimits{{K}}=2$ the equation $q^{2}=1$ has the unique solution
$q=1$, so that $A_{q}\cong{{K}}[{{Z}}^{2}]$, and the action of $\mathop{\rm GL}\nolimits_{2}({{Z}})$ has a canonical
lift to an action on $A_{q}$.
Problem IV.1. Does the sequence (4.3) always split?
We have seen above, that this is true for
$\Gamma={{Z}}^{2}$.
If the answer is no, it would be of some interest to understand the
cohomology groups
$$H^{2}(\mathop{\rm Aut}\nolimits(\Gamma)_{[f]},\mathop{\rm Hom}\nolimits(\Gamma%
,{{K}}^{\times}))$$
parametrizing the possible abelian extensions of $\mathop{\rm Aut}\nolimits(\Gamma)_{[f]}$ by the module
$\mathop{\rm Hom}\nolimits(\Gamma,{{K}}^{\times})$.
Problem IV.2. Let $\lambda\in\mathop{\rm Alt}\nolimits^{2}({{Z}}^{n},Z)$, where $Z$ is a cyclic group.
Determine the structure of the group
$\mathop{\rm Aut}\nolimits({{Z}}^{n},\lambda)$. It should have a semidirect product structure,
where the normal subgroup is something like a Heisenberg group
and the quotient is the automorphism group of ${{Z}}^{n}/\mathop{\rm rad}\nolimits(\lambda)$,
endowed with the induced non-degenerate form. Can this group
be described in a conventient way by generators and relations?
Maybe the results in [Is03] can be used to deal with degenerate cocycles.
A. The group of units if $\Gamma$ is torsion free
The following result is used in [OP95, Lemma 3.1] without reference. Here we provide
a detailed proof.
Proposition A.1.
If the group $\Gamma$ is torsion free and $A$ a $\Gamma$-quantum torus,
then $A^{\times}=A^{\times}_{h}$, i.e., each unit of $A$ is graded.
Proof. Let $a\in A^{\times}$ be a unit and
write $a=\sum_{\gamma}a_{\gamma}\delta_{\gamma}$ in terms of some graded
basis. We do the same with its inverse
$a^{-1}=\sum_{\gamma}(a^{-1})_{\gamma}\delta_{\gamma}$, and observe that the set
$\mathop{\rm supp}\nolimits(a):=\{\gamma\in\Gamma\colon a_{\gamma}\not=0\}$
is finite. The same holds for $\mathop{\rm supp}\nolimits(a^{-1})$, so that both sets
generate a free subgroup $F$ of $\Gamma$.
Then $A_{F}:=\mathop{\rm span}\nolimits\{\delta_{\gamma}\colon\gamma\in F\}$
is an $F$-quantum torus with $a\in A_{F}^{\times}$. We may therefore assume
that $\Gamma={{Z}}^{d}$ for some $d\in{{N}}_{0}$.
We prove by induction on $k\in\{0,\ldots,d\}$ that the subalgebra
$A_{k}:=\mathop{\rm span}\nolimits\{\delta_{\gamma}\colon\gamma\in{{Z}}^{k}%
\times\{0\}\}$
has no zero-divisors (cf. Th. 1.2 in [Pa96])
and that all its units are homogeneous. This holds trivially for $k=0$.
Let $u_{i}:=\delta_{e_{i}}$, where $e_{1},\ldots,e_{d}$ is the canonical basis of
${{Z}}^{d}$. We write
$0\not=x\in A$ as a finite sum $\sum_{k=k_{0}}^{k_{1}}x_{k}u_{d}^{k}$ with
$x_{k}\in A_{d-1}$ and $x_{k_{0}}$ and $x_{k_{1}}$ non-zero.
Likewise we write $0\not=y\in A$ as
$\sum_{m=m_{0}}^{m_{1}}y_{m}u_{d}^{m}$ with
$y_{m}\in A_{d-1}$ and $y_{m_{0}}$ and $y_{m_{1}}$ non-zero.
Then the lowest degree term with respect to $u_{d}$ in $xy$
is
$$x_{k_{0}}u_{d}^{k_{0}}y_{m_{0}}u_{d}^{m_{0}}=x_{k_{0}}\big{(}u_{d}^{k_{0}}y_{m%
_{0}}u_{d}^{-k_{0}}\big{)}u_{d}^{k_{0}+m_{0}},$$
and the induction hypothesis implies
$x_{k_{0}}u_{d}^{k_{0}}y_{m_{0}}u_{d}^{-k_{0}}\not=0$
because conjugation with $u_{d}$ preserves the subalgebra $A_{d-1}$.
This implies that $xy\not=0$.
Now assume that $x\in A$ is a unit and $y=x^{-1}$.
Since $A_{d-1}$ has no zero-divisors,
$$x_{k_{0}}u_{d}^{k_{0}}y_{m_{0}}u_{d}^{-k_{0}}\in A_{d-1}\setminus\{0\}$$
leads to $k_{0}+m_{0}=0$.
A similar consideration for the highest order term implies
$k_{1}+m_{1}=0$, which leads to $k_{0}=k_{1}$ and $m_{0}=m_{1}$.
Now we can argue by induction.
Corollary A.2.
([OP95, Lemma 3.1]) If
the group $\Gamma$ is torsion free, then each
automorphism of $A$ is graded, i.e.,
$\mathop{\rm Aut}\nolimits(A)=\mathop{\rm Aut}\nolimits_{\rm gr}(A).$ (cf. Def. IV.1)
References
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7 (1996), 129–143.
Karl-Hermann Neeb
Technische Universität Darmstadt
Schlossgartenstrasse 7
D-64289 Darmstadt
Deutschland
[email protected] |
November 25, 2020
Simulation of Ultra High Energy Neutrino Interactions in Ice and Water
(the ACoRNE Collaboration)${}^{a}$
S. Bevan${}^{1}$, S. Danaher${}^{2}$, J. Perkin${}^{3}$, S. Ralph${}^{3\dagger}$, C. Rhodes${}^{4}$, L. Thompson${}^{3}$, T. Sloan${}^{5b}$ and
D. Waters${}^{1}$.
${}^{1}$ Physics and Astronomy Dept, University College London, UK.
${}^{2}$ School of Computing Engineering and Information Sciences, University of Northumbria, Newcastle-upon-Tyne, UK.
${}^{3}$ Dept of Physics and Astronomy, University of Sheffield, UK.
${}^{4}$ Institute for Mathematical Sciences, Imperial College London, UK.
${}^{5}$ Department of Physics, University of Lancaster,
Lancaster, UK
${}^{\dagger}$ Deceased
${}^{a}$ Work supported by the UK Particle Physics and Astronomy Research
Council and by the Ministry of Defence Joint Grants Scheme
${}^{b}$ Author for correspondence, email [email protected]
The CORSIKA program, usually used to simulate extensive cosmic ray
air showers, has been adapted to work in a water or ice medium.
The adapted CORSIKA code was used to simulate hadronic showers
produced by neutrino interactions. The simulated showers
have been used to study the spatial distribution of the
deposited energy in the showers. This allows a more precise
determination of the acoustic signals produced by ultra high energy
neutrinos than has been possible previously. The properties of
the acoustic signals generated by such showers are described.
(Submitted to Astroparticle Physics)
1 Introduction
In recent years interest has grown in the detection of very high energy
cosmic ray neutrinos [1]. Such particles could be produced
in the cosmic particle accelerators which make the charged primaries
or they could be produced by the interactions of the primaries with
the Cosmic Microwave Background, the so called GZK effect [2].
The flux of neutrinos expected from these two sources has been
calculated [3, 4]. It is found to be very low so
that large targets are needed for a measurable detection rate.
It is interesting to measure
this neutrino flux to see if it is compatible with the values
expected from these sources, incompatibility implying
new physics.
Searches for cosmic ray neutrinos are ongoing in AMANDA [5],
IceCube [6], ANTARES [7] and NESTOR [8],
detecting upward going
muons from the Cherenkov light in either ice or water.
In general, these experiments are sensitive to lower energies than
discussed here since the Earth becomes opaque to neutrinos
at very high energies. The experiments could detect
almost horizontal higher energy neutrinos but have limited target
volume due to the attenuation of the light signal in the ice.
The Pierre Auger Observatory, an extended air shower array detector,
will also search for upward and almost horizontal showers from
neutrino interactions [9].
In addition to these detectors there are ongoing experiments to detect
the neutrino interactions by either radio or acoustic emissions from
the resulting particle showers [1].
These latter techniques, with much longer attenuation lengths,
allow very large target volumes utilising either large ice fields
or dry salt domes for radio or ice fields, salt domes and the oceans
for the acoustic technique.
In order to assess the feasibility of each technique the production of
the particle shower from neutrino interactions needs to be simulated.
Since experimental data on the interactions of such high energy particles
do not exist it is necessary to use theoretical models to simulate them.
The most extensive ultra high energy simulation program which has so far
been developed is CORSIKA [10]. However, this program has been
used previously only for the simulation of cosmic ray air showers.
The program is readily available [10].
Different simulations are necessary for the radio and acoustic techniques.
Radio emission occurs due to coherent Cherenkov
radiation from the particles in the shower, the Askaryan Effect [11].
The emitted energy is sensitive to the distribution of the
electron-positron asymmetry which develops in the shower and which grows
for lower energy electromagnetic particles. Hence, to simulate radio
emission, the
electromagnetic component of the shower must be followed down to
very low kinetic energies ($\sim 100$ keV)[12]. In contrast,
an acoustic signal is generated by the sudden local heating of the
surrounding medium induced by the particle shower [13].
Thus to simulate the acoustic signal the spatial distribution of
the deposited energy is needed. Once the electromagnetic energy in
the shower reaches the MeV level (electron range $\sim 1$ cm) the
energy can be simply added to the total deposited energy
and the simulation of such particles discontinued.
Extensive simulations have been carried
out for the radio technique [14]. However, the simulations for
the acoustic technique are less advanced. Some work has been done
[15, 16]
using the Geant4 package [17]. However, this work is
restricted to energies less than $10^{5}$ GeV for hadron showers since
the range of validity of the physics models in this package does not
extend to higher energy hadrons.
In this paper the energy distributions of showers produced by
neutrino interactions in sea water at energies up to $10^{12}$ GeV
are discussed. The distributions are generated using the
air shower program CORSIKA [10] modified to work in a sea
water medium. The salt component of the sea water has a negligible
effect111The shower maximum was observed to peak at a depth
$2.4\pm 1.1\%$ less in sea water than in fresh water with the same peak
energy deposited, for protons of energy $10^{5}$ GeV.
and the results are presented in distance units of g cm${}^{-2}$,
hence they should be applicable to ice also.
The computed distributions have been parameterised and this
parameterisation is used to develop a simple program
to simulate neutrino interactions and the resulting particle showers.
The properties of the acoustic signals from the generated showers
are also presented.
2 Adaptation of the CORSIKA program to a water medium
The air shower program, CORSIKA (version 6204) [10], has
been adapted to run
in sea water i.e. a medium of constant density of 1.025 g cm${}^{-3}$
rather than the variable density needed for an air atmosphere. Sea
water was assumed to consist of a medium in which $66.2\%$ of the
atoms are hydrogen, $33.1\%$ of the atoms are oxygen and $0.7\%$ of the
atoms are made of common salt, NaCl. The salt was assumed to be
a material with atomic weight and atomic number A=29.2 and Z=14, the mean
of sodium and chlorine. The purpose of this is to maintain the structure
of the program as closely as possible to the air shower version
which had two principal atmospheric components (oxygen and nitrogen)
with a trace of argon. The presence of the salt component had an
almost undetectable effect on the behaviour of the showers.
Other changes made to the program to accommodate the water medium
include a modification of the stopping power formula to allow for
the density effect in water
222The stopping power was computed using the Bethe-Bloch
formula [20] and the density effect from the formulae of
Sternheimer et al [21].. This only affects the energy loss
for hadrons since the stopping powers for electrons are part of the
EGS [18] package which is used by CORSIKA to simulate the
propagation of the electromagnetic component of the shower.
Smaller radial binning of the shower was also required since shower
radii in water are much smaller than those in air. In addition the
initial state energy for electrons and photons
above which the LPM effect [19] was simulated
in the program was reduced to the much lower value necessary for
water 333The level was set at 1 TeV compared to the
characteristic energy for water $E_{LPM}=270$ TeV [20]..
The LPM effect suppresses pair production from high energy photons
and bremsstrahlung from high energy electrons.
Similarly, the interactions of neutral pions had to be simulated at
lower energy than in air because of the higher density water medium.
In all about 100 detailed changes needed to be made to the CORSIKA
program to accommodate the water medium.
To test the implementation of the LPM effect [19] in the program
100 showers from incident gamma ray photons at several different energies
were generated and the mean
depth of the first interaction (the mean free path) calculated.
The observed mean free path was found to be in agreement with the
expected behaviour when both the suppression of pair production
and photonuclear interactions were taken into account
(see Figure 1). This showed that
the LPM effect had been properly implemented in CORSIKA.
Considerable fluctuations between showers occurred. These are expressed
in terms of the ratio of the root mean square (RMS) deviation of a given
parameter to its mean value: the RMS peak energy deposit to
the mean peak energy deposit was observed to be $14\%$ at $10^{5}$ GeV reducing
to $4\%$ at $10^{11}$ GeV, that for the depth of the peak position varied
from $19\%$ to $7.4\%$ and for the full width at half maximum
of the shower from $63\%$ to $18\%$.
To smooth out such fluctuations averages of 100 generated showers will
be taken in the following. The statistical error on the averages is then
given by these RMS values divided by 10. The hadronic energy
contributes only about $10\%$ to the shower energy at the shower peak,
the remainder being carried by the electromagnetic part of the shower.
The simulations were all carried out in a vertical column of sea water
20 m long. The deposited energy generated by CORSIKA was binned into
20 g cm${}^{-2}$ slices longitudinally and 1.025 g cm${}^{-2}$ annular
cylinders radially for $0<r<10.25$ g cm${}^{-2}$ and 10.25 g cm${}^{-2}$ for
$10.25<r<112.75$ g cm${}^{-2}$ where $r$ is the distance from the
vertical axis. To reduce computing times, the thinning option was
used i.e. below a certain fraction of the
primary energy (in this case $10^{-4}$) only one of the particles
emerging from the interaction is followed and an appropriated weight
is given to it [22]. The simulation of particles
continued down to cut-off energies of 3 MeV for electromagnetic particles
and 0.3 GeV for hadrons. When a particle reached this cut-off, the energy
was added to the slice where this occurred. The
QGSJET [23] model was used to simulate the hadronic interactions.
3 Comparison with other simulations
3.1 Comparison with Geant4
Proton showers were generated in sea water using the program
Geant4 (version 8.0) [17] and compared with those
generated in CORSIKA.
Unfortunately, the range of validity of Geant4 physics models for hadronic
interactions does not extend beyond an energy of $10^{5}$ GeV. Hence the
comparison is restricted to energies below this.
Figure 2 shows the longitudinal distributions of proton showers
at energies of $10^{4}$ and $10^{5}$ GeV (averaged over 100 showers)
as determined from Geant4 and CORSIKA. The showers from CORSIKA tend to
be slightly broader and with a smaller peak energy than those generated
by Geant4. The difference in the peak height is $\sim 5\%$ at $10^{4}$ GeV
rising to $\sim 10\%$ at energy $10^{5}$ GeV. Figure 3 shows the
radial distributions. The differences in the longitudinal distributions
are reflected in the radial distributions. However, the shapes of the
radial distributions are very similar between Geant4 and CORSIKA, with
CORSIKA producing $\sim 10\%$ more energy near
the shower axis at depths between 450 and 850 g cm${}^{-2}$ where most
of the energy is deposited. The acoustic signal from a shower is most
sensitive to the radial distribution, particularly near the
axis ($r\sim 0$). It is relatively insensitive to the shape of the
longitudinal distribution.
3.2 Comparison with the simulation of Alvarez-Muñiz
and Zas
The CORSIKA simulation was also compared with the longitudinal shower
profile for protons computed in the simulation by Alvarez-Muñiz and
Zas (AZ) [24].
There was a reasonable agreement between the longitudinal shower shapes
from CORSIKA and those shown in Figure 2 of ref. [24]. However,
the numbers of electrons and positrons at the peak of the CORSIKA showers
was $\sim 20\%$ lower than those from ref. [24].
This number is sensitive to the energy below which these
particles are counted and this is not specified in [24].
Hence the agreement between CORSIKA and their simulation is probably
satisfactory within this uncertainty.
In conclusion, the modifications made to CORSIKA to simulate high energy
showers in a water medium give results which are compatible with the
predictions from the Geant4 simulations for energy less than $10^{5}$ GeV
and the simulation of AZ within $20\%$. This is
taken to be the accuracy of the simulation program assuming that there
are no unexpected and unknown interactions between the centre of mass
energy explored at current accelerators and those studied in
these simulations.
Studies of the sensitivity of the CORSIKA simulation to the different
models of the hadronic interactions have been reported in
reference [25]. They find that the peak number of electrons plus
positrons varies by $\sim 20\%$ for proton showers in air depending on
the choice of the hadron interaction model used. These differences are
similar in magnitude to the differences between the AZ, Geant4 and CORSIKA
simulations reported here. Hence the observed
differences between the Geant4, AZ and CORSIKA simulations in water
could be within the uncertainties of the hadronic interaction models.
4 Simulation of neutrino induced showers
Neutrinos interact with the nuclei of the detection medium by either
the exchange of a charged vector boson ($W^{+}$), i.e. charged current (CC)
interactions or the exchange of the neutral vector boson ($Z^{0}$),
i.e. neutral current (NC) deep inelastic scattering interactions
(see for example [26]). The ratio of the CC to NC interaction
cross sections is approximately 2:1. The CC interactions produce
charged secondary scattered leptons while the NC interactions produce
neutrinos. The hadron shower carries a fraction $y$ of the energy of
the incident neutrino and the scattered lepton the remaining fraction
$1-y$. We assume that the neutrino flavours are homogeneously mixed
when they arrive at the Earth by neutrino oscillations. Hence in the CC
interactions electrons, $\mu$ and $\tau$ leptons will be produced as the
scattered leptons in equal proportions. At the energies we shall consider,
these particles behave in a manner similar to minimum ionising particles
for $\mu$ and $\tau$ leptons. This is almost true also for electrons
for which the bremsstrahlung process will be suppressed by the
LPM effect. Hence the charged scattered leptons contribute little
to the energy producing an acoustic signal. In the
case of NC interactions there is no contribution to this energy from
the scattered lepton. For these reasons the contribution of the scattered
lepton to the shower profile is ignored beyond $z=20$ m in what follows.
It is interesting to note that a $\tau$ lepton can decay to hadrons or
a very high energy electron or muon can produce bremsstrahlung photons at
large distances from the interaction point. These can initiate further
distant showers, the so called “double bang” effect. The stochastic
nature of such electron showers is studied in [15, 16].
These effects are not considered in this study.
4.1 Neutrino-nucleon interaction cross sections.
A number of groups have computed the high energy neutrino-nucleon
interaction cross sections, $\sigma$, [29, 27, 28].
In the quark parton model of the nucleon for the single vector boson
exchange process, the differential cross section for
CC interactions can be expressed in terms of the measured
structure functions of the target nucleon $F_{2}$ and $xF_{3}$ as
$$\frac{d^{2}\sigma}{dQ^{2}dy}=\frac{G_{F}^{2}}{2\pi y}\bigg{(}\frac{M_{W}^{2}}{%
Q^{2}+M_{W}^{2}}\bigg{)}^{2}(F_{2}(x,Q^{2})(1-y+y^{2}/2)\pm y(1-y/2)xF_{3}(x,Q%
^{2}))$$
(1)
where $G_{F}$ is the Fermi weak coupling, $M_{W}$ is the mass of the
weak vector boson, $Q^{2}$ is the square of the four momentum transferred to
the target nucleon, $y=\nu/E$ where $\nu$ is the energy transferred to
the nucleon ($\nu=E-E^{\prime}$ with $E$ and $E^{\prime}$ the energies of the
incident and scattered leptons) and $x=Q^{2}/2M\nu$ is the fraction of the
momentum of the target nucleon carried by the struck quark (here $x$ and $y$
are defined for a stationary target nucleon).
The plus (minus) sign is for neutrino (anti-neutrino) interactions. It can
be seen that $y$ is the fraction of the neutrino’s energy which is converted
into the energy of the hadron shower. A similar expression can be
written down for the NC interaction (see for example
[26]) which has a ratio to the CC cross section
varying from 0.33 to 0.41 as the neutrino energy increases from $10^{4}$ to
$10^{13}$ GeV. The structure functions $F_{2}$ and $xF_{3}$ are the sum of
the quark distribution functions which have been parameterised by fitting
data [30, 31]. It can be shown that $Q^{2}=sxy$ where $s=2ME$ is
the square of the centre of mass energy ($M$ is the target nucleon mass).
To compute the cross sections the structure functions must be calculated at
values of $x\lesssim M_{W}^{2}/s$
i.e. at values well outside the region of the fits to the
parton distribution functions (PDFs) which have been performed
for $x\gtrsim 10^{-5}$,
the range of current measurements. The extrapolation outside the measurement
range is discussed in [27], [29] and [32, 33].
Here we adopt the procedure of extrapolating linearly on a log-log scale
from the parameterised parton distribution functions of [30]
computed at $x=10^{-4}$ and $x=10^{-5}$. By considering various theoretical
evolution procedures it is estimated in [29] that the procedure
has an accuracy of $\sim 32\%$ per decade and we use this as an
estimate of the accuracy of the calculation. However, this could be an
underestimate [34].
The expression in equation
1 for charged current interactions and the one for neutral current
interactions were integrated to obtain the total neutrino-nucleon
interaction cross section, the value of
the fraction of events per interval of $y$, $1/\sigma d\sigma/dy$, and
the mean value of $y$. The total cross section was found to be in good
agreement with the values in [27, 29] and in reasonable
agreement with [28] which is based on a model different from the
quark parton model. Figure 4 shows the mean value of
$y$ obtained from this procedure (solid curve) and the effect of
multiplying or dividing the PDFs by a factor 1.32 per decade (dashed
curves) as an indication of the possible range of uncertainties in
the extrapolation of the PDFs. Figure 5 shows the $y$ dependence
of the cross section for different neutrino energies.
4.2 A simple generator for neutrino interactions.
A simple generator for neutrino interactions in a column of water of
thickness 20 m was constructed as follows. The neutrino interacts at the
top of the water column (z=0, with the z axis along the axis of the column).
The energy fraction transferred, $y$, for the interaction was generated,
distributed according to
the curve for the energy of the neutrino shown in Figure 5.
This allows the energy of the hadron shower to be calculated for the event.
The assumption was made that these hadron showers will have approximately
the same distributions as those of a proton interaction at z=0 (see
Section 4.3 for a test of this assumption).
A series of files of 100 such proton interactions were generated at
energies in steps of half an order of magnitude between $10^{5}$ and
$10^{12}$ GeV. The hadron shower for each neutrino interaction was
selected at random from the 100 showers in the file at the proton
energy closest to the energy of the hadron shower. The deposited energy
in each bin was then multiplied by the ratio of the energy of the
hadron shower to that of the proton shower. This is made possible
because the shower shapes vary slowly with shower
energy. For example, the ratio of the peak energy deposit per 20
g cm${}^{-2}$ slice to the shower energy varies from 0.037 to 0.030
as the proton shower energy varies from $10^{5}$ to $10^{12}$ GeV.
4.3 The HERWIG neutrino generator.
The CORSIKA program has an option to simulate the interactions
of neutrinos at a fixed point [35]. The first interaction
is generated by the HERWIG package [36]. This option
was adapted to our version of CORSIKA in sea water.
Some problems were encountered with the $y$ dependence of the
resulting interactions due to the extrapolation of the PDFs
to very small $x$ at high energies. This only affects the rate
of the production of the showers at different $y$ and
the distribution of the hadrons produced in the interaction
at a given $y$ should be unaffected.
A total of 700 neutrino
interactions were generated at an incident neutrino energy of
$2\cdot 10^{11}$ GeV. These were divided into the shower energy
intervals $0.5-2\cdot 10^{10}$, $2-4\cdot 10^{10}$, $4-7.5\cdot 10^{10}$,
$0.75-1.3\cdot 10^{11}$ and $1.3-2\cdot 10^{11}$. The showers in which the
scattered lepton energy disagreed with the shower energy by more
than $20\%$ were eliminated leading to a loss of $17\%$ of the
events with shower energy greater than $0.5\cdot 10^{10}$ GeV.
This is due to radiative effects and misidentification
of the scattered lepton. Approximately 70 events remained in each
energy interval. The energy depositions from these were averaged
and compared to the averages from proton showers scaled by
the ratio of the shower energy to the proton energy. Figure 6
shows the longitudinal distributions of the hadronic shower energy
deposited for the different energy intervals (labelled $E_{W}$)
compared to the scaled proton distributions. Figure 7
shows a sample of the transverse distributions.
There is a good consistency between the proton and neutrino induced
showers. The proton showers peak, on average, 20 g cm${}^{-2}$ shallower
in depth with a peak energy $2\%$ larger
than the neutrino induced showers. This is small compared to the
overall uncertainty. The slight shift in the longitudinal distribution
is reflected as a normalisation shift in the radial distributions.
We conclude therefore that to equate a proton induced shower
starting at the neutrino interaction point to that
from a neutrino is a satisfactory approximation.
5 Parameterisation of showers
In this section a parameterisation of the energy deposited by the
showers generated by CORSIKA (averaged over 100 showers depositing
the same total energy) is described. Other available
parameterisations will then
be compared with the showers generated by CORSIKA.
The acoustic signal generated by a hadron shower depends mainly on
the energy deposited in the inner core of the shower. This is illustrated
in figure 8 which shows the contribution to the acoustic
signal from cores of different radii. This figure shows that it is
crucial to represent the deposited energy well at radius less than
2.05 g cm${}^{-2}$. The calculation of the acoustic signal
from the deposited energy is described in section 6.
5.1 Parameterisation of the CORSIKA Showers
The differential energy deposited was parameterised as follows
$$\frac{d^{2}E}{drdz}=L(z,E_{L})\cdot R(r,z,E_{L})$$
(2)
where the function $L(z,E_{L})$ represents the longitudinal distribution
of deposited energy and $R(r,z,E_{L})$ the radial distribution. Here $E_{L}$
is $\log_{10}E$ with $E$ the total shower energy.
The function $L(z,E_{L})=dE/dz$ is a modified444
The modification is to replace the shape parameter $\lambda$
in equation 3.5 of reference [37] by the quadratic expression
in $z$ in equation 3.
version of the Gaisser-Hillas function [37]. This function
represents the longitudinal distribution of the energy deposited.
$$L(z,E_{L})=P_{1L}\bigg{(}\frac{z-P_{2L}}{P_{3L}-P_{2L}}\bigg{)}^{\frac{(P_{3L}%
-P_{2L})}{P_{4L}+P_{5L}z+P_{6L}z^{2}}}\exp\bigg{(}{\frac{P_{3L}-z}{P_{4L}+P_{5%
L}z+P_{6L}z^{2}}}\bigg{)}$$
(3)
Here the parameters $P_{nL}$ were fitted to quadratic functions
of $E_{L}=\log_{10}E$ with values
$$\frac{P_{1L}}{E}=2.760\cdot 10^{-3}-1.974\cdot 10^{-4}E_{L}+7.450\cdot 10^{-6}%
E_{L}^{2}$$
(4)
$$P_{2L}=-210.9-6.968\cdot 10^{-3}E_{L}+0.1551E_{L}^{2}$$
(5)
$$P_{3L}=-41.50+113.9E_{L}-4.103E_{L}^{2}$$
(6)
$$P_{4L}=8.012+11.44E_{L}-0.5434E_{L}^{2}$$
(7)
$$P_{5L}=0.7999\cdot 10^{-5}-0.004843E_{L}+0.0002552E_{L}^{2}$$
(8)
$$P_{6L}=4.563\cdot 10^{-5}-3.504\cdot 10^{-6}E_{L}+1,315\cdot 10^{-7}E_{L}^{2}.$$
(9)
The parameter $P_{1L}$ represents the peak energy deposited and $P_{3L}$ the
depth in the $z$ coordinate at this peak while $P_{2L}$,
$P_{4L}$, $P_{5L}$ and $P_{6L}$ are related to the shower width and shape
in $z$.
The radial distribution was represented by the NKG function [37]
$$R(r,z,E_{L})=\frac{1}{I}\bigg{(}\big{(}\frac{r}{P_{1R}}\big{)}^{(P_{2R}-1)}%
\big{(}1+\frac{r}{P_{1R}}\big{)}^{(P_{2R}-4.5)}\bigg{)}$$
(10)
where the integral
$$I=\int_{0}^{\infty}\bigg{(}\big{(}\frac{r}{P_{1R}}\big{)}^{(P_{2R}-1)}\big{(}1%
+\frac{r}{P_{1R}}\big{)}^{(P_{2R}-4.5)}\bigg{)}~{}dr=P_{1R}\frac{\Gamma(4.5-2P%
_{2R})\Gamma(P_{2R})}{\Gamma(4.5-P_{2R})}.$$
The parameter $P_{1R}$ was found
to vary strongly with depth while $P_{2R}$ was only a weak function of
depth. The parameters $P_{nR}$ (with $n=1$,$2$) were each represented by the
quadratic form
$$P_{nR}=A+Bz+Cz^{2}$$
(11)
and the quantities $A,B,C$ parameterised as quadratic
functions of $E_{L}$. This gave for $P_{1R}$
$$A=0.01287E_{L}^{2}-0.2573E_{L}+0.9636$$
(12)
$$B=-0.4697\cdot 10^{-4}E_{L}^{2}+0.0008072E_{L}+0.0005404$$
(13)
$$C=0.7344\cdot 10^{-7}E_{L}^{2}-1.375\cdot 10^{-6}E_{L}+4.488\cdot 10^{-6}$$
(14)
and for the parameter $P_{2R}$
$$A=-0.8905\cdot 10^{-3}E_{L}^{2}+0.007727E_{L}+1.969$$
(15)
$$B=0.1173\cdot 10^{-4}E_{L}^{2}-0.0001782E_{L}-5.093\cdot 10^{-6}$$
(16)
$$C=-0.1058\cdot 10^{-7}E_{L}^{2}+0.1524\cdot 10^{-6}E_{L}-0.1069\cdot 10^{-8}.$$
(17)
The fit was made in a depth range where $dE/dz$ was greater than $10\%$ of
the peak value defined by equation 4. The program MINUIT [38]
was used to minimise the squared fractional deviations
$$\chi^{2}=\sum_{i}\bigg{(}\frac{F_{i}-D_{i}}{F_{i}+D_{i}}\bigg{)}^{2}$$
(18)
where $F_{i}$ and $D_{i}$ refer to the fitted value and the value observed
in the $i$th bin from
the CORSIKA showers, respectively. In order to improve the fit at small
radii the contributions to $\chi^{2}$ were arbitrarily
weighted by 10 for $r<2.05$ g cm${}^{-2}$, 4 for $2.05<r<3.075$
g cm${}^{-2}$, unity for $3.075<r<51.25$ g cm${}^{-2}$ and
0.25 for $r>51.25$ g cm${}^{-2}$.
The RMS value of the fractional deviations was $3.4\%$ for radii less
than 51.25 g cm${}^{-2}$ and for energies greater than $10^{6.5}$ GeV.
The fit becomes poorer at lower energies and greater radii than these.
Integrating the parameterisation shows that the fraction of the total
energy computed from the fit within the fit range was $91\%$
averaged over the deposited energy range $10^{7}$ to $10^{12}$ GeV.
The corresponding fraction directly from the CORSIKA distributions
was $92.5\%$, averaged over the same energy range.
When applying this parameterisation at depths with smaller energy
deposit than $10\%$ of the peak value, the energy was
assumed to be confined to an annular radius of 1.025 g cm${}^{-2}$.
There was a good agreement (within $5\%$ at the peak) between the
acoustic signal computed using this parameterisation and that taken
directly from the CORSIKA showers.
5.2 The parameterisation used by the SAUND Collaboration
The SAUND Collaboration [39] uses the following parameterisation
[40], based on the NKG formulae
(e.g. see reference [37]), for the energy deposited per unit
depth, $z$, and per unit annular thickness at radius $r$ from a shower
of energy $E$
$$\frac{d^{2}E}{drdz}=Ek(\frac{z}{z_{max}})^{t}\exp{(t-z/\lambda)}~{}2\pi r\rho(r)$$
(19)
where $z_{max}=0.9X_{0}\ln(E/E_{c})$ is the maximum shower depth,
$X_{0}=36.1$ g cm${}^{-2}$ is the radiation length and $E_{c}=0.0838$ GeV.
The constants
$t=z_{max}/\lambda$ where $\lambda=130-5\log_{10}(E/10^{4}\mathrm{GeV})$
g cm${}^{-2}$ and $k=t^{t-1}/\exp{(t)}\lambda\Gamma(t)$.
The radial density is given by
$$\rho(r)=\frac{1}{r_{M}^{2}}a^{s-2}(1+a)^{s-4.5}\frac{\Gamma(4.5-s)}{2\pi\Gamma%
(s)\Gamma(4.5-2s)}$$
(20)
where $a=r/r_{M}$ with $r_{M}=9.04$ g cm${}^{-2}$, the Molière radius in water,
and $s=1.25$.
Figure 9 shows the
radial distributions from CORSIKA compared with the absolute predictions of
this parameterisation.
There is qualitative agreement between the parameterisation and
the CORSIKA results. The difference in normalisation is explained
by the somewhat different longitudinal profiles of the CORSIKA showers
from the SAUND parameterisation. The latter are broader with a lower
peak energy deposit and a depth of the maximum which is larger than
the CORSIKA showers. CORSIKA predicts more energy at small
$r$ than the SAUND parameterisation. Quantitatively, $51\%$ of the shower
energy is contained within a cylinder of radius 4 cm for the CORSIKA
showers compared to $35\%$ from the SAUND parameterisation.
These fractions are approximately independent of energy.
Hence, in acoustic detectors a harder frequency spectrum for
the acoustic signals is predicted by CORSIKA than by the SAUND
parameterisation. Note that in the fit described in Section 5.1
the values of the parameter $P_{1R}$ (equivalent to $R_{M}$ in equation
20) were strongly depth dependent
and much lower than the Molière radius in water, assumed by the SAUND
collaboration. In addition, the value of $P_{2R}$
(equivalent to $s$ in equation 20) while relatively
constant tended to be at a higher value ($\sim 1.9$) than that
assumed by SAUND.
5.3 The parameterisation used by Niess and Bertin
Hadron showers, generated by Geant4 (version 4.06 p03), were studied up
to energies of $10^{5}$ GeV and electromagnetic showers to higher energies
by Niess and Bertin [15, 16]. The hadronic showers were
parameterised as follows.
$$\frac{d^{2}E}{drdz}=rf(z)g(r,z)$$
(21)
with
$$f(z)=\frac{E}{X_{0}}b\frac{(bz^{\prime})^{a-1}\exp{-bz^{\prime}}}{\Gamma(a)}$$
(22)
where $E$ is the energy of the hadron shower, $X_{0}$ is the radiation
length in water, $z^{\prime}=z/X_{0}$, $b=0.56$ as
determined from the fit and $a$ is chosen to satisfy
$z_{max}^{\prime}=(a-1)/b$.
Here $z_{max}^{\prime}$ is the depth in radiation lengths at which
the shower maximum occurs. This is parameterised as
$$z_{max}^{\prime}=0.65\log(\frac{E}{E_{c}})+3.93$$
(23)
with $E_{c}=0.05427$ GeV. The radial distribution function is parameterised as
$$g(r,z)=g_{0}\bigg{(}\frac{r_{i}}{r}\bigg{)}^{n}$$
(24)
where $r_{i}=3.5$ cm, $n=n_{1}=1.66-0.29(z/z_{max})$ for $r<r_{i}$ and
$n=n_{2}=2.7$ for $r>r_{i}$. The constant $g_{0}$ is chosen to be
$(2-n_{1})(n_{2}-2)/((n_{2}-n_{1})r_{i}^{2})$ so that the integral of the radial
distribution is unity.
Figure 10 shows the radial distributions from this parameterisation
compared with the predictions of CORSIKA. There is quite good agreement
between the two. There is a difference in the normalisation with depth
since Geant4, on which this parameterisation is based, produces showers
which tend to develop more slowly with depth than those from CORSIKA
(see Figure 2). Furthermore, both this and the SAUND
parameterisation (Section 5.2) assume a linear variation of
the shower peak depth with $\log E$ whereas CORSIKA gives a clear parabolic
shape (see equation 6). This is illustrated in Figure 11.
The Niess-Bertin parameterisation predicts that $56\%$ of the shower
energy is contained within a cylinder of radius 4 cm in reasonable
agreement with the value of $51\%$
from CORSIKA (these values are almost independent of energy).
6 The acoustic signals from the showers.
The pressure, $P$, from a hadron shower depositing total energy $E$ at
time $t$ resulting from the deposition of relative energy density
$\epsilon=(1/E)(1/2\pi r)d^{2}E/drdz$ at
a point distant $d$ from the volume, $dV$, follows the form [13]
$$P(d,t)=\frac{E\beta}{4\pi C_{p}}\int\frac{\epsilon}{d}\frac{d}{dt}\bigg{(}%
\delta(t-d/c)\bigg{)}dV$$
(25)
where the integral is over the total volume of the shower. Here
$\beta=2.0\cdot 10^{-4}$ is the thermal expansion
coefficient of the medium at $14^{\circ}$C, $C_{p}=3.8\cdot 10^{3}$
J kg${}^{-1}$ K${}^{-1}$ is the
specific heat capacity and $c=1500$ ms${}^{-1}$ is the velocity of
sound in the sea water.
Acoustic signals seen by an observer at distance $r$ from the shower centre
are computed from equation (25) as follows.
Points are produced randomly throughout the volume of the shower with
density proportional to the deposited energy density and the time of
flight from every produced point to the observer calculated. The flight
times to the observer are histogrammed over $2^{n}$ bins (in this case
$n=10$ is chosen) centred on the mean flight time and with a suitable
bin width, $\tau$ (chosen here to be 1$\mu$s). The counts in each bin
of the histogram are divided by $\tau$ yielding the function $E_{xyz}(t)$.
The Fourier transform of the pressure wave is then
$$P(\omega)=\frac{1}{r}\int_{-\infty}^{\infty}\frac{E\beta}{4\pi C_{p}}\frac{d}{%
dt}E_{xyz}(t)e^{-i\omega t}dt=\frac{1}{r}\frac{E\beta}{4\pi C_{p}}i\omega\int_%
{-\infty}^{\infty}E_{xyz}(t)e^{-i\omega t}dt=\frac{1}{r}\frac{E\beta}{4\pi C_{%
p}}i\omega E_{xyz}(\omega)$$
(26)
using the standard Fourier transform theorem, that taking the derivative
in the time domain is the same as multiplying by $i\omega$ in the frequency
domain. The Fourier transform $E_{xyz}(\omega)$ at
angular frequency $\omega$ is evaluated numerically by a fast Fourier
Transform (FFT) from the histogram
$E_{xyz}(t)$. A correction is applied for attenuation in the water by
a factor $A(\omega)=e^{-\alpha(\omega)r}$ where $\alpha(\omega)$ is
the frequency dependent attenuation coefficient.
The pressure as a function of time is then evaluated numerically
by an inverse FFT using frequency steps from zero to the
sampling frequency (the inverse of the bin width $\tau$ i.e. 1 MHz
in this case). This gives
$$P(t)=\frac{1}{1024}\sum_{n=-512}^{n=511}P(\omega_{n})A(\omega_{n})e^{in\Omega}$$
(27)
where $\Omega=2\pi/1024$ radians and $\omega_{n}/2\pi=n\Omega/2\pi$ MHz
is the $n$th frequency.
The attenuation coefficient $\alpha(\omega)$ is computed either
according to the formulae in [42] or using the complex attenuation
given in [15, 16].
This method of calculation was computationally much faster than the
evaluation of the space integral given in equation 18 of reference
[13] and gave identical results.
Acoustic pulses, computed with the complex attenuation described
in [15, 16],
using the parameterisations of the shower profile given
above are shown in Figure 12. It can be seen that the
parameterisation developed here gives similar results to that
described in [15, 16] despite the fact that the latter was
an extrapolation from low energy simulations. The parameterisation
used by SAUND [39, 40] gives smaller signals
concentrated at somewhat lower frequencies.
Further properties of the acoustic signals are shown in Figures
13 to 16. The pulses tend to be somewhat asymmetric
with the asymmetry defined by $|P_{max}|-|P_{min}|/|P_{max}|+|P_{min}|$.
The complex nature of the attenuation enhances this asymmetry.
This is most evident in the far field conditions e.g. at 5km
where non complex attenuation would yield a totally symmetric pulse.
Figure 13 shows the angular dependence of the peak pressure.
Here the angle is that subtended by the acoustic detector relative
to the plane, termed the median plane, through the shower maximum
at right angles to the axis of the shower. The parameterisation
derived here gives a somewhat narrower angular spread than the others.
This could be due to the slightly longer showers predicted by CORSIKA
than the others. Figure 13 also
shows the asymmetry of the pulse as a function of this angle. The
pulse initially becomes more symmetric moving out of the median plane
and then the asymmetry becomes negative at larger angles.
Figure 14 shows the decrease of the pulsed peak pressure
with distance from the shower in the median plane and the asymmetry
with distance in this plane. Figures 15 and 16
show the frequency composition of the pulses at different angles
to the median plane at 1 km from the shower
and at different distance in the median plane, respectively.
7 Conclusions
The simulation program for high energy cosmic ray air showers, CORSIKA,
has been modified to work in a water or ice medium. This allows both
hadron and neutrino showers to be generated in the medium over a wide
range of energy ($10^{5}$ to $10^{12}$ GeV). The properties of
hadronic showers in water simulated by CORSIKA agree with those from other
simulations to within $10-20\%$. A similar uncertainty has been noted
previously from the variations in CORSIKA showers in air generated by
different models of the hadron interactions. However, none of the other
available simulations for water cover the range of energies accessible
to CORSIKA. The hadronic showers produced by neutrino interactions are
shown to have similar profiles to proton showers which deposit the same
amount of energy to that from the neutrino and which start at the
interaction point of the neutrino. The properties of the neutrino
interactions are described.
A parameterisation of the shower profiles generated by CORSIKA is given.
There is reasonable agreement with the parameterisation based on the
Geant4 simulations at low energy ($<10^{5}$ GeV) developed by Niess and
Bertin. However, the agreement with the parameterisation used by the
SAUND Collaboration, which is based on the NKG formalism, is less good.
The position of the shower maximum, determined from the CORSIKA program,
is found to vary quadratically with $\log E$ rather than linearly
as assumed in the latter two parameterisations.
The acoustic signals generated by neutrino interactions using
CORSIKA and by the two other parameterisations are described
and their properties are studied. The acoustic signal is found to
be very sensitive to the energy deposited close to the shower axis.
7.1 Acknowledgments
We wish to thank Ralph Engel, Dieter Heck, Johannes Knapp and Tanguy
Pierog for their assistance in modifying the CORSIKA program. We also
thank Valentin Niess and Justin Vandenbroucke for valuable discussions.
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\renewbibmacro
in: |
A New Approach for Distributed Hypothesis Testing with Extensions to Byzantine-Resilience
Aritra Mitra, John A. Richards and Shreyas Sundaram
A. Mitra and S. Sundaram are with the School of Electrical and Computer Engineering at Purdue University. J. A. Richards is with Sandia National Laboratories. Email: {mitra14, sundara2}@purdue.edu, [email protected]. This work was supported in part by NSF CAREER award
1653648, and by a grant from Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Abstract
We study a setting where a group of agents, each receiving partially informative private observations, seek to collaboratively learn the true state (among a set of hypotheses) that explains their joint observation profiles over time. To solve this problem, we propose a distributed learning rule that differs fundamentally from existing approaches, in the sense, that it does not employ any form of “belief-averaging”. Specifically, every agent maintains a local belief (on each hypothesis) that is updated in a Bayesian manner without any network influence, and an actual belief that is updated (up to normalization) as the minimum of its own local belief and the actual beliefs of its neighbors. Under minimal requirements on the signal structures of the agents and the underlying communication graph, we establish consistency of the proposed belief update rule, i.e., we show that the actual beliefs of the agents asymptotically concentrate on the true state almost surely. As one of the key benefits of our approach, we show that our learning rule can be extended to scenarios that capture misbehavior on the part of certain agents in the network, modeled via the Byzantine adversary model. In particular, we prove that each non-adversarial agent can asymptotically learn the true state of the world almost surely, under appropriate conditions on the observation model and the network topology.
1 Introduction
Various distributed learning problems arising in social networks (such as opinion formation and spreading), and in engineering systems (such as target recognition by a group of aerial robots) can be studied under the formal framework of distributed hypothesis testing. Within this framework, a group of agents repeatedly observe certain private signals, and aim to infer the “true state of the world” that explains their joint observations. While much of the earlier work on this topic assumed the existence of a centralized fusion center for performing computational tasks [1, 2], more recent endeavors focus on a distributed setting where interactions among agents are captured by a communication graph [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Our work here falls in the latter class. A typical belief update rule in the distributed setting combines a local Bayesian update with a consensus-based opinion pooling of neighboring beliefs. Specifically, linear opinion pooling is studied in [3, 4, 5], whereas the log-linear form of belief aggregation is studied in the context of distributed hypothesis testing in [6, 7, 8, 9, 10], and distributed parameter estimation in [11, 12]. Notably, exponential convergence rates are achieved in [4, 6, 7, 8, 9], while a finite-time analysis is presented in [10]. Extensions to time-varying graphs have also been studied in [5, 6, 7].
In [7, Section III], the authors explain that the commonly studied linear and log-linear forms of belief aggregation are specific instances of a more general class of opinion pooling known as g-Quasi-Linear Opinion pools (g-QLOP), introduced in [13]. The main contribution of our paper is the development of a novel belief update rule that deviates fundamentally from the broad family of g-QLOP learning rules discussed above. Specifically, the learning algorithm that we propose in Section 3.1 does not rely on any linear consensus-based belief aggregation protocol. Instead, each agent maintains two sets of beliefs: a local belief that is updated in a Bayesian manner based on the private observations (without neighbor interactions), and an actual belief that is updated (up to normalization) as the minimum of the agent’s own local belief and the actual beliefs of its neighbors. In Section 6, we establish that under minimal requirements on the agents’ signal structures and the communication graph, the actual beliefs of the agents asymptotically concentrate on the true state almost surely. In Section 5, we argue that our approach works under graph-theoretic conditions that are milder than the standard assumption of strong-connectivity.
In addition to the above contribution to the distributed hypothesis testing problem, we also show in this paper that our approach is capable of handling agents that do not follow the prescribed learning algorithm. Indeed, despite the wealth of literature on distributed inference, there is limited understanding of the impact of misbehaving agents for the problem under consideration. Such agents may represent stubborn individuals, ideological extremists in the context of a social network, or model faults (either benign or malicious) in a networked control system. In the presence of such misbehaving entities, how should the remaining agents process their private observations and the beliefs of their neighbors to eventually learn the truth? To answer this question, we model misbehaving agents via the classical Byzantine adversary model, and develop a provably correct, resilient version of our proposed learning rule in Section 3.2. The only related work (that we are aware of) in this regard is reported in [9]. As we discuss in Section 3.2, our proposed approach is significantly less computationally intensive relative to those in [9]. We identify conditions on the observation model and the network structure that guarantee applicability of our Byzantine-resilient learning rule, and argue (in Section 5) that such conditions can be checked in polynomial time.
2 Model and Problem Formulation
Network Model: We consider a group of agents $\mathcal{V}=\{1,2,\ldots,n\}$ interacting over a time-invariant, directed communication graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. An edge $(i,j)\in\mathcal{E}$ indicates that agent $i$ can directly transmit information to agent $j$. If $(i,j)\in\mathcal{E}$, then agent $i$ will be called a neighbor of agent $j$, and agent $j$ will be called an out-neighbor of agent $i$. The set of all neighbors of agent $i$ will be denoted $\mathcal{N}_{i}$. Given two disjoint sets $\mathcal{C}_{1},\mathcal{C}_{2}\subseteq{\mathcal{V}}$, we say that $\mathcal{C}_{2}$ is reachable from $\mathcal{C}_{1}$ if for every $i\in\mathcal{C}_{2}$, there exists a directed path from some $j\in\mathcal{C}_{1}$ to agent $i$ (note that $j$ will in general be a function of $i$). We will use $|\mathcal{C}|$ to denote the cardinality of a set $\mathcal{C}$.
Observation Model: Let $\Theta=\{\theta_{1},\theta_{2},\ldots,\theta_{m}\}$ denote $m$ possible states of the world; each $\theta_{i}\in\Theta$ will be called a hypothesis. Let $\mathbb{N}$ and $\mathbb{N}_{+}$ denote the set of non-negative integers and positive integers, respectively. Then at each time-step $t\in\mathbb{N}_{+}$, every agent $i\in\mathcal{V}$ privately observes a signal $s_{i,t}\in\mathcal{S}_{i}$, where $\mathcal{S}_{i}$ denotes the signal space of agent $i$. The joint observation profile so generated across the network is denoted ${s}_{t}=(s_{1,t},s_{2,t},\ldots,s_{n,t})$, where $s_{t}\in\mathcal{S}$, and $\mathcal{S}=\mathcal{S}_{1}\times\mathcal{S}_{2}\times\ldots\mathcal{S}_{n}$.
The signal $s_{t}$ is generated based on a conditional likelihood function $l(\cdot|\theta^{\star})$, governed by the true state of the world $\theta^{\star}\in\Theta$. Let $l_{i}(\cdot|\theta^{\star}),i\in\mathcal{V}$ denote the $i$-th marginal of $l(\cdot|\theta^{\star})$.
The signal structure of each agent $i\in\mathcal{V}$ is then characterized by a family of parameterized marginals $\{l_{i}(w_{i}|\theta):\theta\in\Theta,w_{i}\in\mathcal{S}_{i}\}$.111Whereas $w_{i}\in\mathcal{S}_{i}$ will be used to refer to a generic element of the signal space of agent $i$, $s_{i,t}$ will denote the random variable (with distribution $l_{i}(\cdot|\theta^{\star})$) that corresponds to agent $i^{\prime}$s observation at time-step $t$.
We make the following standard assumptions [3, 4, 5, 6, 7, 8, 9, 10]: (i) The signal space of each agent $i$, namely $\mathcal{S}_{i}$, is finite. (ii) Each agent $i$ has knowledge of its local likelihood functions $\{l_{i}(\cdot|\theta_{p})\}_{p=1}^{m}$, and it holds that $l_{i}(w_{i}|\theta)>0,\forall w_{i}\in\mathcal{S}_{i}$, and $\forall\theta\in\Theta$. (iii) The observation sequence of each agent is described by an i.i.d. random process over time; however, at any given time-step, the observations of different agents may potentially be correlated. (iv) There exists a fixed true state of the world $\theta^{\star}\in\Theta$ (unknown to the agents) that generates the observations of all the agents.222The approach in [6, 7] applies to a more general setting where there may not exist such a true hypothesis. Finally, we define a probability triple $(\Omega,\mathcal{F},\mathbb{P}^{\theta^{\star}})$, where $\Omega\triangleq\{\omega:\omega=(s_{1},s_{2},\ldots),\forall s_{t}\in\mathcal{%
S},\forall t\in\mathbb{N}_{+}\}$, $\mathcal{F}$ is the $\sigma$-algebra generated by the observation profiles, and $\mathbb{P}^{\theta^{\star}}$ is the probability measure induced by sample paths in $\Omega$. Specifically, $\mathbb{P}^{\theta^{\star}}=\prod\limits_{t=1}^{\infty}l(\cdot|\theta^{\star})$. For the sake of brevity, we will say that an event occurs almost surely to mean that it occurs almost surely w.r.t. the probability measure $\mathbb{P}^{\theta^{\star}}$.
Given the above setup, the goal of each agent in the network is to discern the true state of the world $\theta^{\star}$. The challenge associated with such a task stems from the fact that the private signal structure of any given agent is in general only partially informative. To make this notion precise, define $\Theta^{\theta^{\star}}_{i}\triangleq\{\theta\in\Theta:l_{i}(w_{i}|\theta)=l_{%
i}(w_{i}|\theta^{\star}),\forall w_{i}\in\mathcal{S}_{i}\}.$ In words, $\Theta^{\theta^{\star}}_{i}$ represents the set of hypotheses that are observationally equivalent to the true state $\theta^{\star}$ from the perspective of agent $i$. In general, for any agent $i\in\mathcal{V}$, we may have $|\Theta^{\theta^{\star}}_{i}|>1$, necessitating collaboration among agents. While inter-agent collaboration is implicitly assumed in the distributed hypothesis testing literature, in this paper we will also allow for potential misbehavior on the part of certain agents in the network, modeled as follows.
Adversary Model: We assume that a certain fraction of the agents are adversarial, and model their behavior based on the Byzantine fault model [14]. In particular, Byzantine agents possess complete knowledge of the observation model, the network model, the algorithms being used, the information being exchanged, and the true state of the world. Leveraging such information, adversarial agents can behave arbitrarily and in a coordinated manner, and can in particular, send incorrect, potentially inconsistent information to their out-neighbors. In terms of their distribution in the network, we will consider an $f$-local adversarial model, i.e., we assume that there are at most $f$ adversaries in the neighborhood of any non-adversarial agent.333Note that the $f$-local adversarial model assumed here is more general than the $f$-total adversarial model considered in [9], where the total number of adversaries in the entire network is upper bounded by $f$. Finally, we emphasize that the non-adversarial agents are unaware of the identities of the adversaries in their neighborhood. As is fairly standard in the distributed fault-tolerant literature [15, 16, 17, 18, 19, 20], we only assume that non-adversarial agents know the upper bound $f$ on the number of adversaries in their neighborhood. The adversarial set will be denoted by $\mathcal{A}\subset\mathcal{V}$, and the remaining agents $\mathcal{R}=\mathcal{V}\setminus\mathcal{A}$ will be called the regular agents.
Our objective in this paper will be to design a distributed learning rule that allows each regular agent $i\in\mathcal{R}$ to identify the true state of the world almost surely, despite (i) the partially informative signal structures of the agents, and (ii) the actions of any $f$-local Byzantine adversarial set. To this end, we introduce the following notion of source agents.
Definition 1.
(Source agents) An agent $i$ is said to be a source agent for a pair of distinct hypotheses $\theta_{p},\theta_{q}\in\Theta$, if $D(l_{i}(\cdot|\theta_{p})||l_{i}(\cdot|\theta_{q}))>0$, where $D(l_{i}(\cdot|\theta_{p})||l_{i}(\cdot|\theta_{q}))$ represents the KL-divergence between the distributions $l_{i}(\cdot|\theta_{p})$ and $l_{i}(\cdot|\theta_{q})$, and is given by:
$$D(l_{i}(\cdot|\theta_{p})||l_{i}(\cdot|\theta_{q}))=\sum\limits_{w_{i}\in%
\mathcal{S}_{i}}l_{i}(w_{i}|\theta_{p})\log\frac{l_{i}(w_{i}|\theta_{p})}{l_{i%
}(w_{i}|\theta_{q})}.$$
(1)
The set of all source agents for the pair $\theta_{p},\theta_{q}$ is denoted by $\mathcal{S}(\theta_{p},\theta_{q})$.444Notice that $\mathcal{S}(\theta_{p},\theta_{q})=\mathcal{S}(\theta_{q},\theta_{p})$, since $D(l_{i}(\cdot|\theta_{p})||l_{i}(\cdot|\theta_{q}))>0\iff D(l_{i}(\cdot|\theta%
_{q})||l_{i}(\cdot|\theta_{p}))>0$.
In words, a source agent for a pair $\theta_{p},\theta_{q}\in\Theta$ is an agent that can distinguish between the pair of hypotheses $\theta_{p},\theta_{q}$ based on its private signal structure. In our developments, we will require the following two definitions.
Definition 2.
($r$-reachable set) [16] For a graph $\mathcal{G}=(\mathcal{V,E})$, a set $\mathcal{C}\subseteq\mathcal{V}$, and an integer $r\in\mathbb{N}_{+}$, $\mathcal{C}$ is an $r$-reachable set if there exists an $i\in\mathcal{C}$ such that $|\mathcal{N}_{i}\setminus\mathcal{C}|\geq r$.
Definition 3.
(strongly $r$-robust graph w.r.t. $\mathcal{S}(\theta_{p},\theta_{q})$) For $r\in\mathbb{N}_{+}$ and $\theta_{p},\theta_{q}\in\Theta$, a graph $\mathcal{G}=(\mathcal{V,E})$ is strongly $r$-robust w.r.t. the set of source agents $\mathcal{S}(\theta_{p},\theta_{q})$, if for every non-empty subset $\mathcal{C}\subseteq\mathcal{V}\setminus\mathcal{S}(\theta_{p},\theta_{q})$, $\mathcal{C}$ is $r$-reachable.
3 Proposed Learning Rules
3.1 A Novel Belief Update Rule
In this section, we propose a novel belief update rule and discuss the intuition behind it. To introduce the key ideas underlying our basic approach, we first consider a scenario where all agents are regular, i.e., $\mathcal{R}=\mathcal{V}$. Every agent $i$ maintains and updates (at every time-step) two separate sets of belief vectors, namely, $\boldsymbol{\pi}_{i,t}$ and $\boldsymbol{\mu}_{i,t}$. Each of these vectors are probability distributions over the hypothesis set $\Theta$. We will refer to $\boldsymbol{\pi}_{i,t}$ and $\boldsymbol{\mu}_{i,t}$ as the “local” belief vector (for reasons that will soon become obvious), and the “actual” belief vector, respectively, maintained by agent $i$. The goal of each agent $i\in\mathcal{V}$ in the network will be to use its own private signals, and the information available from its neighbors, to update $\boldsymbol{\mu}_{i,t}$ sequentially so that $\lim_{t\to\infty}\mu_{i,t}(\theta^{*})=1$ almost surely. To do so, for each $\theta\in\Theta$, and at each time-step $t+1,t\in\mathbb{N}$, agent $i$ first generates $\pi_{i,t+1}(\theta)$ via a local Bayesian update rule that incorporates the private observation $s_{i,t+1}$ using $\pi_{i,t}(\theta)$ as a prior. Having generated $\pi_{i,t+1}(\theta)$, agent $i$ updates $\mu_{i,t+1}(\theta)$ (up to normalization) by setting it to be the minimum of its locally generated belief $\pi_{i,t+1}(\theta)$, and the actual beliefs $\mu_{j,t}(\theta),j\in\mathcal{N}_{i}$ of its neighbors at the previous time-step. It then reports its actual belief $\mu_{i,t+1}(\theta)$ to each of its out-neighbors.555Note that based on our algorithm, agents only exchange their actual beliefs, and not their local beliefs. The belief vectors are initialized as $\mu_{i,0}(\theta)>0,\pi_{i,0}(\theta)>0,\forall\theta\in\Theta,\forall i\in%
\mathcal{V}$. Subsequently, these vectors are updated at each time-step $t+1$ (where $t\in\mathbb{N}$) as follows:
•
Step 1: Update of the local beliefs:
$$\pi_{i,t+1}(\theta)=\frac{l_{i}(s_{i,t+1}|\theta)\pi_{i,t}(\theta)}{\sum%
\limits_{p=1}^{m}l_{i}(s_{i,t+1}|\theta_{p})\pi_{i,t}(\theta_{p})}.$$
(2)
•
Step 2: Update of the actual beliefs:
$$\mu_{i,t+1}(\theta)=\frac{\min\{\{\mu_{j,t}(\theta)\}_{{j\in\mathcal{N}_{i}}},%
\pi_{i,t+1}(\theta)\}}{\sum\limits_{p=1}^{m}\min\{\{\mu_{j,t}(\theta_{p})\}_{{%
j\in\mathcal{N}_{i}}},\pi_{i,t+1}(\theta_{p})\}}.$$
(3)
Intuition behind the learning rule: Consider the set of source agents $\mathcal{S}(\theta^{*},\theta)$ who can differentiate between a certain false hypothesis $\theta$ and the true state $\theta^{\star}$. Suppose for now that this set is non-empty. We ask: how do the agents in the set $\mathcal{S}(\theta^{\star},\theta)$ contribute to the process of collaborative learning? To answer this question, we note that the signal structures of such agents are rich enough for them to be able to eliminate $\theta$ on their own, i.e., without the support of their neighbors. Thus, the agents in $\mathcal{S}(\theta^{\star},\theta)$ should contribute towards driving the actual beliefs of their out-neighbors (and eventually, of all the agents in the set $\mathcal{V}\setminus\mathcal{S}(\theta^{\star},\theta)$) on the hypothesis $\theta$ to zero. To achieve the above objective, we are especially interested in devising a rule that ensures that the capability of the source agents $\mathcal{S}(\theta^{\star},\theta)$ to eliminate $\theta$ is not diminished due to neighbor interactions. As we shall see later, such a property will be particularly useful when certain agents in the network are adversarial. It is precisely these considerations that motivate us to employ (i) an auxiliary belief vector $\boldsymbol{\pi}_{i,t+1}$ generated via local processing (i.e., without any network influence) of the private signals, and (ii) a min-rule of the form (3). Specifically, if $i\in\mathcal{S}(\theta^{\star},\theta)$, then the sequence of local beliefs $\pi_{i,t+1}(\theta)$ will almost surely converge to $0$ based on the update rule (2). Hence, for a source agent $i\in\mathcal{S(\theta^{\star},\theta)}$, $\pi_{i,t+1}(\theta)$ will play the key role of an external network-independent input in the min-rule (3). This in turn will trigger a process of belief reduction on the hypothesis $\theta$ originating at the source set $\mathcal{S(\theta^{\star},\theta)}$, and eventually propagating via the proposed min-rule to each agent in the network reachable from such a source set. The above discussion will be made precise in Section
6.
Remark 1.
We emphasize that the proposed min-rule (3) does not employ any form of “belief-averaging”. This feature is in stark contrast with existing approaches to distributed hypothesis testing that rely either on linear opinion pooling [3, 4, 5], or log-linear opinion pooling[11, 12, 10, 6, 7, 8, 9]. As such, the lack of linearity in our belief update rule precludes (direct or indirect) adaptation of existing analysis techniques to suit our needs. Consequently, we develop a novel sample path based proof technique in Section 6 to establish consistency of the proposed learning rule. As one of the main outcomes of this analysis, we argue that our learning rule works under graph-theoretic conditions that are in general weaker than strong-connectivity (see also Section 5).
3.2 A Byzantine-Resilient Belief Update Rule
As pointed out in the Introduction, a key benefit of our approach is that it can be extended to account for the worst-case Byzantine adversarial model described in Section 2. A standard way to analyze the impact of such adversarial agents while designing resilient distributed consensus-based protocols (for applications in consensus [16, 15], optimization [17, 18], hypothesis testing [9], and multi-agent rendezvous [21]) is to construct an equivalent matrix representation of the linear update rule that involves only the regular agents [22]. In particular, this requires expressing the iterates of a regular agent as a convex combination of the iterates of its regular neighbors, based on appropriate filtering techniques, and under certain assumptions on the network structure. While this can indeed be achieved efficiently for scalar consensus problems, for problems requiring consensus on vectors (like the belief vectors in our setting), such an approach becomes computationally prohibitive [9]. To bypass such heavy computations, and yet accommodate Byzantine attacks, we now develop a resilient version of the learning rule introduced in Section 3.1, as follows. Each agent $i\in\mathcal{R}$ acts as follows at every time-step $t+1$ (where $t\in\mathbb{N}$).
•
Step 1: Update of the local beliefs: The local belief $\pi_{i,t+1}(\theta)$ is updated as before, based on (2).
•
Step 2: Filtering extreme beliefs: If $|\mathcal{N}_{i}|\geq(2f+1)$, then agent $i$ performs a filtering operation as follows. It collects the actual beliefs $\mu_{j,t}(\theta)$ from each neighbor $j\in\mathcal{N}_{i}$ and sorts them from highest to lowest. It rejects the highest $f$ and the lowest $f$ of such beliefs (i.e., it throws away $2f$ beliefs in all). In other words, for each hypothesis, a regular agent retains only the moderate beliefs received from its neighbors.
•
Step 3: Update of the actual beliefs:
If $|\mathcal{N}_{i}|\geq(2f+1)$, then agent $i$ updates $\mu_{i,t+1}(\theta)$ as follows. Let the set of neighbors whose beliefs on $\theta$ are not rejected by agent $i$ (based on the previous filtering step) be denoted by $\mathcal{M}^{\theta}_{i,t}\subset\mathcal{N}_{i}$. The actual belief $\mu_{i,t+1}(\theta)$ is then updated as follows:
$$\mu_{i,t+1}(\theta)=\frac{\min\{\{\mu_{j,t}(\theta)\}_{j\in\mathcal{M}^{\theta%
}_{i,t}},\pi_{i,t+1}(\theta)\}}{\sum\limits_{p=1}^{m}\min\{\{\mu_{j,t}(\theta_%
{p})\}_{j\in\mathcal{M}^{\theta_{p}}_{i,t}},\pi_{i,t+1}(\theta_{p})\}}.$$
(4)
If $|\mathcal{N}_{i}|<(2f+1)$, then agent $i$ updates $\mu_{i,t+1}(\theta)$ as follows:
$$\mu_{i,t+1}(\theta)=\pi_{i,t+1}(\theta).$$
(5)
As with the learning rule presented in Section 3.1, agent $i$ transmits $\mu_{i,t+1}(\theta)$ to each of its out-neighbors on completion of the above steps. We will refer to the above sequence of actions as the Local-Filtering based Resilient Hypothesis Elimination (LFRHE) algorithm.
4 Main Results
In this section, we state our main results, and then comment on them in Section 5; detailed proofs of the results are presented in Section 6.
Our first result establishes the correctness of the learning rule proposed in Section 3.1.
Theorem 1.
Suppose $\mathcal{R}=\mathcal{V}$, and that the following are true:
(i)
For every pair of hypotheses $\theta_{p},\theta_{q}\in\Theta$, the corresponding source set $\mathcal{S}(\theta_{p},\theta_{q})$ is non-empty.
(ii)
For every pair of hypotheses $\theta_{p},\theta_{q}\in\Theta$, $\mathcal{V}\setminus\mathcal{S}(\theta_{p},\theta_{q})$ is reachable from the source set $\mathcal{S}(\theta_{p},\theta_{q})$.
(iii)
Every agent $i\in\mathcal{V}$ has a non-zero prior belief on each hypothesis, i.e., $\pi_{i,0}(\theta)>0,\mu_{i,0}(\theta)>0$ for all $i\in\mathcal{V}$, and for all $\theta\in\Theta$.
Then, the learning rule described by equations (2) and (3) leads to collaborative learning of the true state, i.e., $\mu_{i,t}(\theta^{\star})\rightarrow 1$ almost surely $\forall i\in\mathcal{V}$.
Our second result establishes the correctness of the LFRHE algorithm proposed in Section 3.2.
Theorem 2.
Suppose the following are true:
(i)
For every pair of hypotheses $\theta_{p},\theta_{q}\in\Theta$, the graph $\mathcal{G}$ is strongly $(2f+1)$-robust w.r.t. the corresponding source set $\mathcal{S}(\theta_{p},\theta_{q})$.
(ii)
Each regular agent $i\in\mathcal{R}$ has a non-zero prior belief on each hypothesis, i.e., $\pi_{i,0}(\theta)>0,\mu_{i,0}(\theta)>0$ for all $i\in\mathcal{R}$, and for all $\theta\in\Theta$.
Then, the LFRHE algorithm described by equations (2), (4) and (5) leads to collaborative learning of the true state despite the actions of any $f$-local set of Byzantine adversaries, i.e., $\mu_{i,t}(\theta^{\star})\rightarrow 1$ almost surely $\forall i\in\mathcal{R}$.
Remark 2.
For any pair $\theta_{p},\theta_{q}\in\Theta$, notice that condition (i) of Theorem 2 (together with the definition of strong-robustness in Def. 3) requires $|\mathcal{S}(\theta_{p},\theta_{q})|\geq(2f+1)$, if $\mathcal{V}\setminus\mathcal{S}(\theta_{p},\theta_{q})$ is non-empty.
5 Discussion
(Assumptions in Theorem 1):
While the first condition in Theorem 1 is a basic global identifiability condition, the second condition on the network structure is in general weaker than the standard assumption of strong-connectivity made in [3, 4, 11, 12, 10, 8]. To see why the latter statement is true, consider a scenario where $\Theta=\{\theta_{1},\theta_{2}\}$. Clearly, any agent $i\in\mathcal{S}(\theta_{1},\theta_{2})$ can discern the true state without neighbor interactions, precluding the need for incoming edges to such agents.666For the problem under consideration, the argument that the strong connectivity assumption can be relaxed applies to more general scenarios as well, where there does not necessarily exist any one agent that can identify the true state based on just its private signal structure. The underlying reason for this stems from information heterogeneity and information redundancy among agents [23], features shared by distributed estimation and detection type problems, but lacking in a standard consensus setting. Finally, the assumption of non-zero initial beliefs is fairly standard, and can be easily met by maintaining a uniform support over the hypotheses set initially.
(Assumptions in Theorem 2): The first condition in Theorem 2 blends requirements on the signal structures of the agents with those on the communication graph. To gain intuition about this condition, suppose $\Theta=\{\theta_{1},\theta_{2}\}$, and let there exist at least one agent $i\in\mathcal{V}\setminus\mathcal{S}(\theta_{1},\theta_{2})$. To enable agent $i$ to discern the truth despite potential adversaries in its neighborhood, one requires (i) redundancy in the signal structures of the agents (see Remark 2), and (ii) redundancy in the network structure to facilitate reliable information flow from $\mathcal{S}(\theta_{1},\theta_{2})$ to agent $i$. These requirements are captured by condition (i), a point made apparent in Section 6.2.
(Complexity of Checking Condition (i) in Theorem 2): Given a network of agents with associated signal structures, condition (i) in Theorem 2 can be checked in polynomial time. Specifically, for every pair $\theta_{p},\theta_{q}\in\Theta$, finding the source set $\mathcal{S}(\theta_{p},\theta_{q})$ can easily be done in polynomial time via inspection of the agents’ signal structures. For a fixed source set $\mathcal{S}(\theta_{p},\theta_{q})$, checking whether $\mathcal{G}$ is strongly $(2f+1)$-robust w.r.t. $\mathcal{S}(\theta_{p},\theta_{q})$ amounts to simulating a bootstrap percolation process on $\mathcal{G}$, with $\mathcal{S}(\theta_{p},\theta_{q})$ as the initial active set, and $(2f+1)$ as the threshold. This too can be achieved in polynomial time, as discussed in [19].
(Analogy with Distributed State Estimation):
Consider the problem of collaboratively estimating the state of an LTI process based on information exchanges among agents that receive partial measurements of the state. There are natural connections between this setting, and the problem studied in this paper. For the state estimation scenario, one can fix an unstable mode of the process, and define source agents for that mode to be agents that can detect the eigenspaces associated with that mode. Interestingly, with source agents defined for each unstable mode in the manner described above, [23, Theorem 3] and [19, Theorem 7] (in the context of distributed state estimation) can be viewed as analogues of Theorem 1 and Theorem 2, respectively.
(Convergence Rate): Consider any false hypothesis $\theta\neq\theta^{\star}$. We conjecture that based on our learning rules, the actual beliefs of all the regular agents on $\theta$ will almost surely decay exponentially fast after a transient period, with the rate of decay lower bounded by $\min_{i\in\mathcal{S}(\theta^{\star},\theta)\cap\mathcal{R}}D(l_{i}(\cdot|%
\theta^{\star})||l_{i}(\cdot|\theta))$.
6 Proofs of the Main Results
We start with the following simple lemma that characterizes the asymptotic behavior of the local belief sequences generated based on (2); we provide a proof (adapted to our notation) to keep the paper self-contained.
Lemma 1.
Consider an agent $i\in\mathcal{S}(\theta^{\star},\theta)\cap\mathcal{R}$. Suppose $\pi_{i,0}(\theta^{\star})>0$. Then, the update rule (2) ensures that (i) $\pi_{i,t}(\theta)\rightarrow 0$ almost surely, and (ii) $\pi_{i,\infty}(\theta^{\star})\triangleq\lim_{t\to\infty}\pi_{i,t}(\theta^{%
\star})$ exists almost surely, and satisfies $\pi_{i,\infty}(\theta^{\star})\geq\pi_{i,0}(\theta^{\star})$.
Proof.
Pick an agent $i\in\mathcal{S}(\theta^{\star},\theta)\cap\mathcal{R}$, and define:
$$\rho_{i,t}(\theta)\triangleq\log\frac{\pi_{i,t}(\theta)}{\pi_{i,t}(\theta^{%
\star})},\hskip 5.690551pt\lambda_{i,t}(\theta)\triangleq\log\frac{l_{i}(s_{i,%
t}|\theta)}{l_{i}(s_{i,t}|\theta^{\star})}.$$
(6)
Then, based on (2), we obtain the following recursion:
$$\rho_{i,t+1}(\theta)=\rho_{i,t}(\theta)+\lambda_{i,t+1}(\theta),\forall t\in%
\mathbb{N}.$$
(7)
Rolling out the above equation over time yields:
$$\rho_{i,t}(\theta)=\rho_{i,0}(\theta)+\sum\limits_{k=1}^{t}\lambda_{i,k}(%
\theta),\forall t\in\mathbb{N}_{+}.$$
(8)
Notice that $\{\lambda_{i,t}(\theta)\}$ is a sequence of i.i.d. random variables with finite means and variances. In particular, it is easy to verify that each random variable $\lambda_{i,t}(\theta)$ has mean777More precisely, the mean here is obtained by using the expectation operator $\mathbb{E}^{\theta^{\star}}[\cdot]$ associated with the measure $\mathbb{P}^{\theta^{\star}}$. given by $-D(l_{i}(\cdot|\theta^{\star})||l_{i}(\cdot|\theta))$. Thus, based on the strong law of large numbers, we have $\frac{1}{t}\sum\limits_{k=1}^{t}\lambda_{i,k}(\theta)\rightarrow-D(l_{i}(\cdot%
|\theta^{\star})||l_{i}(\cdot|\theta))$ almost surely. Dividing both sides of (8) by $t$, and taking the limit as $t$ goes to infinity, we then obtain:
$$\lim_{t\to\infty}\frac{1}{t}\rho_{i,t}(\theta)=-D(l_{i}(\cdot|\theta^{\star})|%
|l_{i}(\cdot|\theta))\hskip 2.845276pt\textrm{almost surely}.$$
(9)
Finally, note that based on the definition of the set $\mathcal{S}(\theta^{\star},\theta)$, $D(l_{i}(\cdot|\theta^{\star})||l_{i}(\cdot|\theta))>0$. It then follows from (9) that $\rho_{i,t}(\theta)\rightarrow-\infty$ almost surely, and hence $\pi_{i,t}(\theta)\rightarrow 0$ almost surely. For any $\theta\in\Theta^{\theta^{\star}}_{i}$, observe that $\lambda_{i,t}(\theta)=0,\forall t\in\mathbb{N}_{+}$. It then follows from (7) that for each $\theta\in\Theta^{\theta^{\star}}_{i}$, $\rho_{i,t}(\theta)=\rho_{i,0}(\theta),\forall t\in\mathbb{N}_{+}$. From the above discussion, we conclude that a limiting belief vector $\boldsymbol{\pi}_{i,\infty}$ exists almost surely, with non-zero entries corresponding to only those $\theta\in\Theta^{\theta^{\star}}_{i}$ for which $\pi_{i,0}(\theta)>0$. Part (ii) of the lemma then follows readily by noting that $\pi_{i,0}(\theta^{\star})>0$.
∎
We are now in position to prove Theorems 1 and 2.
6.1 Proof of Theorem 1
Proof.
Let $\bar{\Omega}\subseteq\Omega$ denote the set of sample paths along which for each agent $i\in\mathcal{V}$, the following hold: (i) for each $\theta\in\Theta\setminus{\Theta^{\theta^{\star}}_{i}}$, $\pi_{i,t}(\theta)\rightarrow 0$, and (ii) $\pi_{i,\infty}(\theta^{\star})\triangleq\lim_{t\to\infty}\pi_{i,t}(\theta^{%
\star})$ exists, and satisfies $\pi_{i,\infty}(\theta^{\star})\geq\pi_{i,0}(\theta^{\star})$. Recall that $\Theta^{\theta^{\star}}_{i}$ represents the set of hypotheses that are observationally equivalent to the true state $\theta^{\star}$ from the point of view of agent $i$. Hence, for each $\theta\in\Theta\setminus{\Theta^{\theta^{\star}}_{i}}$, we have $i\in\mathcal{S}(\theta^{\star},\theta)$. Based on the third condition in the statement of Theorem 1, and Lemma 1, we infer that $\bar{\Omega}$ has measure $1$. Thus, to prove the desired result, it suffices to confine our attention to the set $\bar{\Omega}$. Specifically, fix any sample path $\omega\in\bar{\Omega}$, and pick any $\epsilon>0$. Our goal will be to establish that along the sample path $\omega$, there exists $t(\omega,\epsilon)$ such that for all $t\geq t(\omega,\epsilon)$, $\mu_{i,t}(\theta)<\epsilon$ for all $i\in\mathcal{V}$, and for all $\theta\neq\theta^{\star}$ in the dynamics given by (3). This would be equivalent to establishing that the actual beliefs of all the agents on the true state can be made arbitrarily close to $1$ (since the proposed min-rule (3) generates a valid probability distribution over the hypothesis set at each time-step). We complete the proof in the following two steps.
Step 1: Lower bounding the actual beliefs on the true state: Consider the following scenario. During a transient phase, certain agents see private signals that cause them to temporarily lower their local beliefs on the true state. This in turn gets propagated via the min-rule (3) to the actual beliefs of the agents in the network. For sample paths in the set $\bar{\Omega}$, we rule out the possibility of such a transient phenomenon triggering a cascade of progressively lower beliefs on the true state. To this end, define $\gamma_{1}\triangleq\min_{i\in\mathcal{V}}\pi_{i,0}(\theta^{\star})$. Notice that $\gamma_{1}>0$ based on condition (iii) of the theorem. Given the choice of the sample path $\omega$, we notice that $\pi_{i,\infty}(\theta^{\star})$ exists for each $i\in\mathcal{V}$, and that $\pi_{i,\infty}(\theta^{\star})\geq\gamma_{1}$. Pick a small number $\delta>0$ such that $\delta<\gamma_{1}$. The following statement is then immediate. For each agent $i\in\mathcal{V}$, there exists $t_{i}(\omega,\delta)$, such that for all $t\geq t_{i}(\omega,\delta)$, $\pi_{i,t}(\theta^{\star})\geq\gamma_{1}-\delta>0$. Define $\bar{t}_{1}(\omega,\delta)\triangleq\max_{i\in\mathcal{V}}t_{i}(\omega,\delta)$. In words, $\bar{t}_{1}(\omega,\delta)$ represents the time-step beyond which the local beliefs of all the agents on the true state are lower-bounded by $\gamma_{1}-\delta$. We ask: At such a time-step, what is the lowest actual belief held by an agent on the true state? More precisely, we define $\gamma_{2}(\omega)\triangleq\min_{i\in\mathcal{V}}\{\mu_{i,\bar{t}_{1}(\omega,%
\delta)}(\theta^{\star})\}$. We claim $\gamma_{2}(\omega)>0$. To see this, observe that given the assumption of non-zero prior beliefs on the true state, and the structure of the proposed min-rule (3), $\gamma_{2}(\omega)$ can be $0$ if and only if there exists some time-step $t^{{}^{\prime}}(\omega)\leq\bar{t}_{1}(\omega,\delta)$ such that $\pi_{i,t^{{}^{\prime}}(\omega)}(\theta^{\star})=0$, for some $i\in\mathcal{V}$. However, given the structure of the local Bayesian update rule (2), we would then have $\pi_{i,t}(\theta^{\star})=0$, for all $t\geq{t}^{{}^{\prime}}(\omega)$, contradicting the fact that $\pi_{i,t}(\theta^{\star})\geq\gamma_{1}-\delta>0,\forall t\geq\bar{t}_{1}(%
\omega,\delta)\geq t^{{}^{\prime}}(\omega),\forall i\in\mathcal{V}$ (the latter fact has already been established above). Having thus established that $\gamma_{2}(\omega)>0$, define $\eta(\omega)\triangleq\min\{\gamma_{1}-\delta,\gamma_{2}(\omega)\}>0$. In other words, $\eta(\omega)$ lower-bounds the lowest belief (considering both local and actual beliefs) on the true state $\theta^{\star}$ held by an agent at time-step $\bar{t}_{1}(\omega,\delta)$. We claim the following:
$$\mu_{i,t}(\theta^{\star})\geq\eta(\omega),\forall t\geq\bar{t}_{1}(\omega,%
\delta),\forall i\in\mathcal{V}.$$
(10)
To see why (10) is true, fix an agent $i\in\mathcal{V}$, and consider the following chain of inequalities:
$$\displaystyle\mu_{i,\bar{t}_{1}(\omega,\delta)+1}(\theta^{\star})$$
$$\displaystyle\overset{(a)}{=}\frac{\min\{\{\mu_{j,\bar{t}_{1}(\omega,\delta)}(%
\theta^{\star})\}_{{j\in\mathcal{N}_{i}}},\pi_{i,\bar{t}_{1}(\omega,\delta)+1}%
(\theta^{\star})\}}{\sum\limits_{p=1}^{m}\min\{\{\mu_{j,\bar{t}_{1}(\omega,%
\delta)}(\theta_{p})\}_{{j\in\mathcal{N}_{i}}},\pi_{i,\bar{t}_{1}(\omega,%
\delta)+1}(\theta_{p})\}}$$
(11)
$$\displaystyle\overset{(b)}{\geq}\frac{\eta(\omega)}{\sum\limits_{p=1}^{m}\min%
\{\{\mu_{j,\bar{t}_{1}(\omega,\delta)}(\theta_{p})\}_{{j\in\mathcal{N}_{i}}},%
\pi_{i,\bar{t}_{1}(\omega,\delta)+1}(\theta_{p})\}}$$
$$\displaystyle\overset{}{\geq}\frac{\eta(\omega)}{\sum\limits_{p=1}^{m}\pi_{i,%
\bar{t}_{1}(\omega,\delta)+1}(\theta_{p})}$$
$$\displaystyle\overset{(c)}{=}\eta(\omega),$$
where $(a)$ is given by (3), $(b)$ follows from the way $\eta(\omega)$ is defined and by noting that $\pi_{i,t}(\theta^{\star})\geq\eta(\omega),\forall t\geq\bar{t}_{1}(\omega,%
\delta),\forall i\in\mathcal{V}$, and $(c)$ follows by noting that the local belief vectors generated via (2) (at each time-step) are valid probability distributions over the hypothesis set $\Theta$, and hence $\sum\limits_{p=1}^{m}\pi_{i,\bar{t}_{1}(\omega,\delta)+1}(\theta_{p})=1$. Since the above reasoning applies to every agent in the network, we can keep repeating it to establish (10) via induction.
Step 2: Upper bounding the actual beliefs on each false hypothesis: The key observation that guides the rest of the proof is as follows. While Step 1 of the proof ensures that the beliefs (both local and actual) of each agent on the true state $\theta^{\star}$ are lower-bounded by $\eta(\omega)>0$ after a finite period of time (given by $\bar{t}_{1}(\omega,\delta)$), Lemma 1 guarantees that the local beliefs on any false hypothesis $\theta$ will eventually become arbitrarily small (and in particular, smaller than $\eta(\omega)$) for each agent $i\in\mathcal{S}(\theta^{\star},\theta)$, on the sample path $\omega\in\bar{\Omega}$ under consideration. In what follows, we investigate how this impacts the actual beliefs of the agents in the network. To this end, given an $\epsilon>0$, pick a small $\bar{\epsilon}(\omega)>0$ such that $\bar{\epsilon}(\omega)<\min\{\eta(\omega),\epsilon\}$. Fix a hypothesis $\theta\neq\theta^{\star}$. By virtue of condition (i) of the theorem, we know that $|\mathcal{S}(\theta^{\star},\theta)|>0$. Let $q=d(\mathcal{G})+2$, where $d(\mathcal{G})$ represents the diameter of the graph $\mathcal{G}$. Then, based on Lemma 1, for each $i\in\mathcal{S}(\theta^{\star},\theta)$, there exists $t^{\theta}_{i}(\omega,\bar{\epsilon}(\omega))$ such that for all $t\geq t^{\theta}_{i}(\omega,\bar{\epsilon}(\omega))$, $\pi_{i,t}(\theta)\leq\bar{\epsilon}^{q}(\omega)$. Define
$$\bar{t}^{\theta}_{2}(\omega,\delta,\bar{\epsilon}(\omega))\triangleq\max\{\bar%
{t}_{1}(\omega,\delta),\max_{i\in\mathcal{S}(\theta^{\star},\theta)}\{t^{%
\theta}_{i}(\omega,\bar{\epsilon}(\omega))\}\}.$$
(12)
Throughout the rest of the proof, we suppress the dependence of $\bar{t}_{2}$ on $\theta,\omega,\delta$ and $\bar{\epsilon}(\omega)$ to avoid cluttering the exposition.
For any agent $i\in\mathcal{S}(\theta^{\star},\theta)$, we obtain the following chain of inequalities:
$$\displaystyle\mu_{i,\bar{t}_{2}+1}(\theta)$$
$$\displaystyle\overset{(a)}{=}\frac{\min\{\{\mu_{j,\bar{t}_{2}}(\theta)\}_{{j%
\in\mathcal{N}_{i}}},\pi_{i,\bar{t}_{2}+1}(\theta)\}}{\sum\limits_{p=1}^{m}%
\min\{\{\mu_{j,\bar{t}_{2}}(\theta_{p})\}_{{j\in\mathcal{N}_{i}}},\pi_{i,\bar{%
t}_{2}+1}(\theta_{p})\}}$$
(13)
$$\displaystyle\overset{(b)}{\leq}\frac{\bar{\epsilon}^{q}(\omega)}{\sum\limits_%
{p=1}^{m}\min\{\{\mu_{j,\bar{t}_{2}}(\theta_{p})\}_{{j\in\mathcal{N}_{i}}},\pi%
_{i,\bar{t}_{2}+1}(\theta_{p})\}}$$
$$\displaystyle\overset{}{\leq}\frac{\bar{\epsilon}^{q}(\omega)}{{\min\{\{\mu_{j%
,\bar{t}_{2}}(\theta^{\star})\}_{{j\in\mathcal{N}_{i}}},\pi_{i,\bar{t}_{2}+1}(%
\theta^{\star})\}}}$$
$$\displaystyle\overset{(c)}{\leq}\frac{\bar{\epsilon}^{q}(\omega)}{\eta(\omega)}$$
$$\displaystyle\overset{(d)}{<}\bar{\epsilon}^{(q-1)}(\omega)\leq\bar{\epsilon}(%
\omega)<\epsilon,$$
where $(a)$ is given by (3), $(b)$ follows from the fact that for each $i\in\mathcal{S}(\theta^{\star},\theta)$, we have $\pi_{i,t}(\theta)\leq\bar{\epsilon}^{q}(\omega),\forall t\geq\bar{t}_{2}$, $(c)$ follows from (10) and (12), and $(d)$ follows from the way $\bar{\epsilon}$ has been chosen. In particular, note that the above chain of reasoning used to arrive at (13) applies to subsequent time-steps as well. We thus conclude:
$$\mu_{i,t}(\theta)<\bar{\epsilon}^{(q-1)}(\omega),\forall t\geq\bar{t}_{2}+1,%
\forall i\in\mathcal{S}(\theta^{\star},\theta).$$
(14)
We now wish to investigate how the effect of (14) propagates through the rest of the network. If $\mathcal{V}\setminus\mathcal{S}(\theta^{\star},\theta)$ is empty, then we have reached the desired conclusion w.r.t. the false hypothesis $\theta$. If not, define
$$\mathcal{L}^{(\theta^{\star},\theta)}_{1}\triangleq\{i\in\{\mathcal{V}%
\setminus\mathcal{S}(\theta^{\star},\theta)\}\hskip 2.845276pt{:}\hskip 2.8452%
76pt|\mathcal{N}_{i}\cap\mathcal{S}(\theta^{\star},\theta)|>0\}$$
(15)
as the set of immediate out-neighbors of the source set $\mathcal{S}(\theta^{\star},\theta)$. By virtue of condition (ii) of the theorem, if $\mathcal{V}\setminus\mathcal{S}(\theta^{\star},\theta)$ is non-empty, then $\mathcal{L}^{(\theta^{\star},\theta)}_{1}$ as defined above is also non-empty. Consider any agent $i\in\mathcal{L}^{(\theta^{\star},\theta)}_{1}$. By definition, agent $i$ has a neighbor in $\mathcal{S}(\theta^{\star},\theta)$ satisfying (14). This observation coupled with equations (10), (12) can be used to obtain a similar chain of inequalities as the ones featuring in (13). Specifically, we obtain:
$$\mu_{i,t}(\theta)<\bar{\epsilon}^{(q-2)}(\omega),\forall t\geq\bar{t}_{2}+2,%
\forall i\in\mathcal{L}^{(\theta^{\star},\theta)}_{1}.$$
(16)
With $\mathcal{L}^{(\theta^{\star},\theta)}_{0}\triangleq\mathcal{S}(\theta^{\star},\theta)$, the above arguments can be repeated by successively defining the sets $\mathcal{L}^{(\theta^{\star},\theta)}_{r},1\leq r\leq d(\mathcal{G})$ as follows:
$$\mathcal{L}^{(\theta^{\star},\theta)}_{r}\triangleq\{i\in\mathcal{V}\setminus%
\{\bigcup_{c=0}^{r-1}\mathcal{L}^{(\theta^{\star},\theta)}_{c}\}\hskip 2.84527%
6pt{:}\hskip 2.845276pt|\mathcal{N}_{i}\cap\{\bigcup_{c=0}^{r-1}\mathcal{L}^{(%
\theta^{\star},\theta)}_{c}\}|>0\}.$$
(17)
Whenever $\mathcal{V}\setminus\{\bigcup_{c=0}^{r-1}\mathcal{L}^{(\theta^{\star},\theta)}%
_{c}\}$ is non-empty, condition (ii) of the theorem implies that $\mathcal{L}^{(\theta^{\star},\theta)}_{r}$ will also be non-empty.
One can then easily verify via induction on $r$ that:
$$\mu_{i,t}(\theta)<\bar{\epsilon}^{(q-(r+1))}(\omega),\forall t\geq\bar{t}_{2}+%
(r+1),\forall i\in\mathcal{L}^{(\theta^{\star},\theta)}_{r},$$
(18)
where $1\leq r\leq d(\mathcal{G})$. Noting that $q=d(\mathcal{G})+2$, we obtain the desired result that $\mu_{i,t}(\theta)<\bar{\epsilon}(\omega)<\epsilon$, $\forall t\geq\bar{t}_{2}+d(\mathcal{G})+1,\forall i\in\mathcal{V}$. An identical argument as the one presented above can be made for each false hypothesis $\theta\neq\theta^{\star}$. This completes the proof.
∎
6.2 Proof of Theorem 2
Proof.
Consider an $f$-local adversarial set $\mathcal{A}\subset\mathcal{V}$, and let $\mathcal{R}=\mathcal{V}\setminus\mathcal{A}$. We study two separate cases.
Case 1: Consider a regular agent $i\in\mathcal{R}$ such that $|\mathcal{N}_{i}|<(2f+1)$. Based on condition (i) of the theorem, we claim that $i\in\mathcal{S}(\theta_{p},\theta_{q})$, for every pair $\theta_{p},\theta_{q}\in\Theta$. We prove this claim via contradiction. To do so, suppose there exists a pair $\theta_{p},\theta_{q}\in\Theta$, such that $i\in\mathcal{V}\setminus\mathcal{S}(\theta_{p},\theta_{q})$. As $|\mathcal{N}_{i}|<(2f+1)$, the set $\{i\}$ is clearly not $(2f+1)$-reachable (see Def. 2). Thus, $\mathcal{G}$ is not strongly $(2f+1)$-robust w.r.t. the source set $\mathcal{S}(\theta_{p},\theta_{q})$, a fact that contradicts condition (i) of the theorem. Thus, we have established that for networks satisfying condition (i) of the theorem, regular agents with fewer than $(2f+1)$ neighbors can distinguish between every pair of hypotheses. Lemma 1 then implies that such agents can discern the true state $\theta^{\star}$ by simply running the local Bayesian estimator (2), and updating actual beliefs via (5).
Case 2: We now focus only on regular agents $i$ satisfying $|\mathcal{N}_{i}|\geq(2f+1)$. For this case, the structure of the proof mirrors that of Theorem 1; we thus only elaborate on details that are specific to tackling the aspect of adversarial agents. A key property of the proposed LFRHE algorithm that will be used throughout the proof is as follows. For any $i\in\mathcal{R}$, and any $\theta\in\Theta$, the filtering operation of the LFRHE algorithm ensures that at each time-step $t\in\mathbb{N}$, we have:
$$\mu_{j,t}(\theta)\in Conv(\Psi^{\theta}_{i,t}),\forall j\in\mathcal{M}^{\theta%
}_{i,t},$$
(19)
where
$$\Psi^{\theta}_{i,t}\triangleq\{\mu_{j,t}(\theta)\hskip 2.845276pt{:}\hskip 2.8%
45276ptj\in\mathcal{N}_{i}\cap\mathcal{R}\},$$
(20)
and $Conv(\Psi^{\theta}_{i,t})$ is used to denote the convex hull formed by the points in the set $\Psi^{\theta}_{i,t}$. In other words, any neighboring belief (on a particular hypothesis) that agent $i$ uses in the update rule (4) lies in the convex hull of the actual beliefs of its regular neighbors (on that particular hypothesis). To see why (19) is true, partition the neighbor set $\mathcal{N}_{i}$ of a regular agent into three sets $\mathcal{U}^{\theta}_{i,t},\mathcal{M}^{\theta}_{i,t}$, and $\mathcal{J}^{\theta}_{i,t}$ as follows. Sets $\mathcal{U}^{\theta}_{i,t}$ and $\mathcal{J}^{\theta}_{i,t}$ are each of cardinality $f$, and contain neighbors of agent $i$ that transmit the highest $f$ and the lowest $f$ actual beliefs respectively, on the hypothesis $\theta$, to agent $i$ at time-step $t$. The set $\mathcal{M}^{\theta}_{i,t}$ contains the remaining neighbors of agent $i$, and is non-empty at every time-step since $|\mathcal{N}_{i}|\geq(2f+1)$. If $\mathcal{M}^{\theta}_{i,t}\cap\mathcal{A}=\emptyset$, then (19) holds trivially. Thus, consider the case when there are adversaries in the set $\mathcal{M}^{\theta}_{i,t}$, i.e., $\mathcal{M}^{\theta}_{i,t}\cap\mathcal{A}\neq\emptyset$. Given the $f$-locality of the adversarial model, and the nature of the filtering operation in the LFRHE algorithm, we infer that for each $j\in\mathcal{M}^{\theta}_{i,t}\cap\mathcal{A}$, there exist regular agents $u,v\in\mathcal{N}_{i}\cap\mathcal{R}$, such that $u\in\mathcal{U}^{\theta}_{i,t}$, $v\in\mathcal{J}^{\theta}_{i,t}$, and $\mu_{v,t}(\theta)\leq\mu_{j,t}(\theta)\leq\mu_{u,t}(\theta)$. This establishes our claim regarding equation (19).
With the above property in hand, our goal will be to now establish each of the two steps in the proof of Theorem 1. To this end, let $\bar{\Omega}\subseteq\Omega$ denote the set of sample paths along which for each agent $i\in\mathcal{R}$, the following hold: (i) for each $\theta\in\Theta\setminus{\Theta^{\theta^{\star}}_{i}}$, $\pi_{i,t}(\theta)\rightarrow 0$, and (ii) $\pi_{i,\infty}(\theta^{\star})\triangleq\lim_{t\to\infty}\pi_{i,t}(\theta^{%
\star})$ exists, and satisfies $\pi_{i,\infty}(\theta^{\star})\geq\pi_{i,0}(\theta^{\star})$. Based on condition (ii) of the theorem, and Lemma 1, we infer that $\bar{\Omega}$ has measure $1$. Thus, as in Theorem 1, fix a sample path $\omega\in\bar{\Omega}$, and pick $\epsilon>0$. Define $\gamma_{1}=\min_{i\in\mathcal{R}}\pi_{i,0}(\theta^{\star})$, pick a small number $\delta>0$ satisfying $\delta<\gamma_{1}$, and observe that for each agent $i\in\mathcal{R}$, there exists $t_{i}(\omega,\delta)$, such that for all $t\geq t_{i}(\omega,\delta)$, $\pi_{i,t}(\theta^{\star})\geq\gamma_{1}-\delta>0$. Define $\bar{t}_{1}(\omega,\delta)\triangleq\max_{i\in\mathcal{R}}t_{i}(\omega,\delta)$ and $\gamma_{2}(\omega)\triangleq\min_{i\in\mathcal{R}}\{\mu_{i,\bar{t}_{1}(\omega,%
\delta)}(\theta^{\star})\}$. As before, we claim $\gamma_{2}(\omega)>0$. To establish this claim, we need to answer the following question: Can an adversarial agent cause its out-neighbors to set their actual beliefs on $\theta^{\star}$ to be $0$ by setting its own actual belief on $\theta^{\star}$ to be $0$? We argue that this is impossible under the LFRHE algorithm. By way of contradiction, suppose there exists a time-step ${t}^{\prime}(\omega)$ satisfying:
$${t}^{\prime}(\omega)=\min\{t\in\mathbb{N}\hskip 2.845276pt{:}\hskip 2.845276pt%
\exists i\in\mathcal{R}\hskip 2.845276pt\textrm{with}\hskip 2.845276pt\mu_{i,t%
}(\theta^{\star})=0\}.\ $$
(21)
In words, $t^{\prime}(\omega)$ represents the first time-step when some regular agent $i$ sets its actual belief on the true hypothesis to be zero. Clearly, $t^{\prime}(\omega)\neq 0$ based on condition (ii) of the theorem. Suppose ${t}^{\prime}(\omega)$ is some positive integer, and focus on how agent $i$ updates $\mu_{i,{t}^{\prime}(\omega)}(\theta^{\star})$ based on (4). Following similar arguments as in the proof of Theorem 1, we know that $\pi_{i,t}(\theta^{\star})>0,\forall t\in\mathbb{N},\forall i\in\mathcal{R}.$ At the same time, every belief featuring in the set $\Psi^{\theta^{\star}}_{i,{t}^{\prime}(\omega)-1}$ (as defined in equation (20)) is strictly positive based on the way ${t}^{\prime}(\omega)$ is defined. In light of the above arguments, and based on (19), (20), we infer:
$$\min\{\{\mu_{j,{t}^{\prime}(\omega)-1}(\theta^{\star})\}_{j\in\mathcal{M}^{%
\theta^{\star}}_{i,{t}^{\prime}(\omega)-1}},\pi_{i,{t}^{\prime}(\omega)}(%
\theta^{\star})\}>0.$$
(22)
Thus, based on (4), we must have $\mu_{i,{t}^{\prime}(\omega)}(\theta^{\star})>0$, yielding the desired contradiction. With $\eta(\omega)\triangleq\min\{\gamma_{1}-\delta,\gamma_{2}(\omega)\}>0$, one can easily verify the following:
$$\mu_{i,t}(\theta^{\star})\geq\eta(\omega),\forall t\geq\bar{t}_{1}(\omega,%
\delta),\forall i\in\mathcal{R}.$$
(23)
In particular, (23) follows by (i) noting that for each $i\in\mathcal{R}$, $\pi_{i,\bar{t}_{1}(\omega,\delta)+1}(\theta^{\star})\geq\eta(\omega)$, and each belief featuring in the set $\Psi^{\theta^{\star}}_{i,\bar{t}_{1}(\omega,\delta)}$ is lower bounded by $\eta(\omega)$, (ii) leveraging (19), (20), and (iii) using a similar string of arguments as those used to arrive at (11). This completes Step 1.
To proceed with Step 2 (i.e., upper-bounding the actual beliefs on each false hypothesis), given an $\epsilon>0$, pick a small $\bar{\epsilon}(\omega)>0$ such that $\bar{\epsilon}(\omega)<\min\{\eta(\omega),\epsilon\}$. Fix a hypothesis $\theta\neq\theta^{\star}$, let $q=n+1$, and note that based on Lemma 1, for each $i\in\mathcal{S}(\theta^{\star},\theta)\cap\mathcal{R}$, there exists $t^{\theta}_{i}(\omega,\bar{\epsilon}(\omega))$ such that for all $t\geq t^{\theta}_{i}(\omega,\bar{\epsilon}(\omega))$, $\pi_{i,t}(\theta)\leq\bar{\epsilon}^{q}(\omega)$. Define
$$\bar{t}_{2}\triangleq\max\{\bar{t}_{1}(\omega,\delta),\max_{i\in\mathcal{S}(%
\theta^{\star},\theta)\cap\mathcal{R}}\{t^{\theta}_{i}(\omega,\bar{\epsilon}(%
\omega))\}\},$$
(24)
where we have suppressed the dependence of $\bar{t}_{2}$ on $\theta,\omega,\delta$ and $\bar{\epsilon}(\omega)$ as in the proof of Theorem 1. For any agent $i\in\mathcal{S}(\theta^{\star},\theta)\cap\mathcal{R}$, observe that
$$\min\{\{\mu_{j,\bar{t}_{2}}(\theta^{\star})\}_{j\in\mathcal{M}^{\theta^{\star}%
}_{i,\bar{t}_{2}}},\pi_{i,\bar{t}_{2}+1}(\theta^{\star})\}\geq\eta(\omega).$$
(25)
Combining the above with a similar line of argument as used to arrive at (13), we obtain:
$$\mu_{i,t}(\theta)<\bar{\epsilon}^{(q-1)}(\omega),\forall t\geq\bar{t}_{2}+1,%
\forall i\in\mathcal{S}(\theta^{\star},\theta)\cap\mathcal{R}.$$
(26)
If $\mathcal{V}\setminus\mathcal{S}(\theta^{\star},\theta)$ is empty, then we are done. Else, define
$$\mathcal{L}^{(\theta^{\star},\theta)}_{1}\triangleq\{i\in\{\mathcal{V}%
\setminus\mathcal{S}(\theta^{\star},\theta)\}\hskip 2.845276pt{:}\hskip 2.8452%
76pt|\mathcal{N}_{i}\cap\mathcal{S}(\theta^{\star},\theta)|\geq(2f+1)\}.$$
(27)
Whenever $\mathcal{V}\setminus\mathcal{S}(\theta^{\star},\theta)$ is non-empty, we claim that $\mathcal{L}^{(\theta^{\star},\theta)}_{1}$ (as defined above) is also non-empty based on condition (i) of the theorem. To see this, note that if $\mathcal{L}^{(\theta^{\star},\theta)}_{1}$ is empty, then $\mathcal{C}=\mathcal{V}\setminus\mathcal{S}(\theta^{\star},\theta)$ is not $(2f+1)$-reachable, violating the fact that $\mathcal{G}$ is strongly $(2f+1)$-robust w.r.t. $\mathcal{S}(\theta^{\star},\theta)$. We claim
$$\min_{j\in\mathcal{M}^{\theta}_{i,\bar{t}_{2}+1}}{\mu_{j,\bar{t}_{2}+1}(\theta%
)}<\bar{\epsilon}^{(q-1)}(\omega),\forall i\in\mathcal{L}^{(\theta^{\star},%
\theta)}_{1}\cap\mathcal{R}.$$
(28)
To verify the above claim, pick any agent $i\in\mathcal{L}^{(\theta^{\star},\theta)}_{1}\cap\mathcal{R}$. When $|\mathcal{M}^{\theta}_{i,\bar{t}_{2}+1}\cap\{\mathcal{S}(\theta^{\star},\theta%
)\cap\mathcal{R}\}|>0$, the claim follows immediately based on (26). Consider the case when $|\mathcal{M}^{\theta}_{i,\bar{t}_{2}+1}\cap\{\mathcal{S}(\theta^{\star},\theta%
)\cap\mathcal{R}\}|=0$. Since $i\in\mathcal{L}^{(\theta^{\star},\theta)}_{1}$, it has at least $(2f+1)$ neighbors in $\mathcal{S}(\theta^{\star},\theta)$, out of which at least $f+1$ are regular based on the $f$-locality of the adversarial model. Since the set $\mathcal{J}^{\theta}_{i,\bar{t}_{2}+1}$ has cardinality $f$, it must then be that $|\mathcal{U}^{\theta}_{i,\bar{t}_{2}+1}\cap\{\mathcal{S}(\theta^{\star},\theta%
)\cap\mathcal{R}\}|>0$. Let $u\in\mathcal{U}^{\theta}_{i,\bar{t}_{2}+1}\cap\{\mathcal{S}(\theta^{\star},%
\theta)\cap\mathcal{R}\}$. Based on the way $\mathcal{M}^{\theta}_{i,\bar{t}_{2}+1}$ is defined, it must be that $\mu_{j,\bar{t}_{2}+1}(\theta)\leq\mu_{u,\bar{t}_{2}+1}(\theta)<\bar{\epsilon}^%
{(q-1)}(\omega),\forall j\in\mathcal{M}^{\theta}_{i,\bar{t}_{2}+1}$, where the last inequality follows from (26). This establishes our claim regarding (28). Consider the update of $\mu_{i,\bar{t}_{2}+2}(\theta)$ based on (4). In light of the above arguments (that apply identically to subsequent time-steps as well), the numerator of the fraction on the RHS of (4) is upper-bounded by $\bar{\epsilon}^{(q-1)}(\omega)$, while the denominator is lower-bounded by $\eta(\omega)$. This leads to the following conclusion:
$$\mu_{i,t}(\theta)<\bar{\epsilon}^{(q-2)}(\omega),\forall t\geq\bar{t}_{2}+2,%
\forall i\in\mathcal{L}^{(\theta^{\star},\theta)}_{1}\cap\mathcal{R}.$$
(29)
With $\mathcal{L}^{(\theta^{\star},\theta)}_{0}\triangleq\mathcal{S}(\theta^{\star},\theta)$, we recursively define the sets $\mathcal{L}^{(\theta^{\star},\theta)}_{r},1\leq r\leq(n-1)$ as follows:
$$\mathcal{L}^{(\theta^{\star},\theta)}_{r}\triangleq\{i\in\mathcal{V}\setminus%
\{\bigcup_{c=0}^{r-1}\mathcal{L}^{(\theta^{\star},\theta)}_{c}\}\hskip 2.84527%
6pt{:}\hskip 2.845276pt|\mathcal{N}_{i}\cap\{\bigcup_{c=0}^{r-1}\mathcal{L}^{(%
\theta^{\star},\theta)}_{c}\}|\geq(2f+1)\}.$$
(30)
We complete the proof by inducting on $r$. To this end, suppose the following holds for all $0\leq r\leq(n-2)$:
$$\mu_{i,t}(\theta)<\bar{\epsilon}^{(q-(r+1))}(\omega),\forall t\geq\bar{t}_{2}+%
(r+1),\forall i\in\mathcal{L}^{(\theta^{\star},\theta)}_{r}\cap\mathcal{R}.$$
(31)
The claim extends to the case when $r=(n-1)$ by noting that (i) $\mathcal{L}^{(\theta^{\star},\theta)}_{(n-1)}$ is non-empty if $\mathcal{V}\setminus\{\bigcup_{c=0}^{(n-2)}\mathcal{L}^{(\theta^{\star},\theta%
)}_{c}\}$ is non-empty (based on condition (i) of the theorem), (ii) any agent $i\in\mathcal{L}^{(\theta^{\star},\theta)}_{(n-1)}\cap\mathcal{R}$ has at least $(2f+1)$ neighbors in the set $\bigcup_{c=0}^{(n-2)}\mathcal{L}^{(\theta^{\star},\theta)}_{c}$, of which at least $f+1$ are regular (based on the $f$-locality of the adversarial model), and (iii) using the induction hypothesis and arguments similar to those used for arriving at (29). Finally, note that the sets $\mathcal{L}^{(\theta^{\star},\theta)}_{r}$ are constructed in a way such that all agents in $\mathcal{R}$ are covered. The rest of the proof is identical to that of Theorem 1.
∎
7 Conclusion
In this paper, we introduced a distributed learning rule that differs fundamentally from those existing in the literature, in the sense, that it does not rely on any consensus-based belief aggregation protocol. Using a novel sample path based analysis technique, we established its consistency under minimal requirements on the information structures of the agents and the communication graph. We then showed that a significant benefit of the proposed learning rule is that it can be easily and efficiently modified to account for the presence of misbehaving agents in the network, modeled via the Byzantine adversary model.
Ongoing work involves performing a detailed convergence rate analysis to see how such rates compare with those existing in literature. Extensions to time-varying graphs are also of interest.
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Heavy Hybrids with Constituent Gluons
Eric S. Swanson${}^{1,2}$ and Adam P. Szczepaniak${}^{3}$
${}^{1}$
Department of Physics,
North Carolina State University,
Raleigh, North Carolina 27695-8202
${}^{2}$
Jefferson Laboratory,
12000 Jefferson Avenue,
Newport News, VA 23606
${}^{3}$
Physics Department,
Indiana University,
Bloomington, Indiana 47405-4202
Abstract
Hybrid meson energies are calculated in the static quark limit with the
Dynamical Quark Model (DQM). In the DQM, transverse gluons are
represented as effective constituents with a dynamically generated mass.
Hybrid masses are
determined within the Tamm-Dancoff approximation for the resulting
relativistic Salpeter equation. Although the general features of the
adiabatic potential surfaces correspond with lattice data, the results
disagree on level orderings. Similar problems appear to exist in all
constituent glue models of hybrids. We conclude that constituent gluons
do not accurately represent soft gluonic degrees of freedom. The steps
necessary to correct this deficiency are discussed.
pacs:
Dec, 1997
I Introduction
A decade of experimental signals[1] for QCD hybrids
(in particular with $J^{PC}=1^{-+}$) has culminated in the
claimed observation of three such states
at Brookhaven[2] in the last year.
The question of the nature of QCD hybrids has thus become topical.
Furthermore, lattice gauge calculations are now at the point of accurately
determining light hybrid masses. In view of these developments, it is of
interest to compare models of strong (low energy) QCD
with lattice data to determine their viability and to explicate
and guide current experimental efforts.
It is often stated that a hybrid is a hadron consisting of valence quarks
and glue. However, one must specify what is meant by the notion of valence
glue for this statement to be useful.
There are two broad ideas in this regard: it is some sort of string or flux
tube[3, 4] or it is an effective constituent confined by a
bag[5, 6] or potential[7, 8, 9].
As an example of the importance of choosing correct degrees of freedom, we
mention the simple problem of determining the
number of components of a constituent gluon.
It has been suggested that a massive
constituent gluon should be transverse so as to maintain consistency with
Yang’s theorem[8]. However it was noted that this is inconsistent
with the requirements of Lorentz invariance.
Thus, for example, $J=1$ glueballs are
expected to exist and lattice calculations indicate that they are quite heavy
(roughly 3 GeV)[10]. Such a state may not be constructed from two
transverse constituent gluons (Yang’s theorem) and therefore may be expected to
have a mass of roughly $3m_{g}\sim 3$ GeV. However massive vector gluons have
no such constraint and one therefore expects them to have a mass of
approximately $2m_{g}\sim 2$ GeV.
The nature of the appropriate effective degrees of freedom for glue can only be
determined by a long process of calculation and comparison with experimental
and lattice data.
There are, however, a few indications that low energy glue is string-like.
Perhaps the most compelling of these are lattice calculations of energy,
action, or field densities between static color sources which are
reminiscent of flux tubes[11]. An intriguing clue is also provided
by the spin splittings of heavy quarkonia.
It is known that an effective interaction free of long range exchange
spin-orbit coupling is needed to reproduce the mass splitting of the ${}^{3}P_{J}$
heavy quarkonium multiplets. However, an analysis of QCD in the heavy quark limit
convincingly demonstrates that obtaining such an effective potential
requires that low energy glue must be string-like[12].
Alternatively, pointlike models of low energy glue have a
long history,
originating with MIT bag model calculations of Barnes[5] and
others[6].
Horn and Mandula[7] were the first to consider a potential constituent
glue model
of hybrids. Their hybrids consisted of constituent quarks and pointlike,
massless, spinless, and colorless glue in a nonrelativistic potential model.
The confining potential was taken to be linear with a string tension given
by the ratio of color Casimir operators: $b_{qg}=9/8b_{q\bar{q}}$. The
authors noted that the two body $q\bar{q}$ potential is anti-confining in the
color octet channel and has a repulsive Coulomb spike at short distances.
They argued that this is
unphysical and hence choose to neglect this term in the interaction. It is
clear that a great many simplifying assumptions have gone into the
construction of this model. It is our purpose to compare a more
sophisticated version of the model to lattice data to learn something about
these assumptions.
In the following we employ a model field theoretic Hamiltonian of QCD. The model
incorporates linear confinement at low energy and evolves into perturbative QCD
at high energy. A nontrivial vacuum is used to generate constituent quark
and gluon masses. The eigenvalue equation is derived for a $q\bar{q}g$
system where the quarks are static. The resulting adiabatic potential surfaces
are then compared to recent lattice results. We conclude that the simple picture
of glue as a pointlike constituent particle reproduces the general behaviour
of the lattice results but fails to yield the correct level orderings.
Furthermore, other models which regard the gluonic degrees of freedom as
pointlike (eg., [18], [7]) do not contain sufficient degrees of
freedom to generate all of the adiabatic surfaces. Thus constituent glue
models appear to fail to describe hybrids. This stands in contrast to
string or bag-like models which, although disagreeing on details, capture
the rough structure of the lattice data.
II A Constituent Glue Model of Static Hybrids
II.1 The Dynamical Quark Model
The starting point for our description of hybrids is the following model
Hamiltonian:
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle\int d{\bf x}\psi^{{\dagger}}({\bf x})\left[-i{\vec{\alpha}}\cdot%
{\vec{\nabla}}+\beta m\right]\psi({\bf x})+{1\over 2}\int d{\bf x}\left[|{\bf E%
}^{A}({\bf x})|^{2}+|{\bf B}^{A}(x)|^{2}\right]$$
(1)
$$\displaystyle+$$
$$\displaystyle{1\over 2}\int d{\bf x}d{\bf y}\rho^{A}({\bf x})V(|{\bf x}-{\bf y%
}|)\rho^{A}({\bf y})$$
where the color charge density is $\rho^{A}({\bf x})=\psi^{{\dagger}}({\bf x}){\rm T}^{A}\psi({\bf x})-f^{ABC}{%
\bf A}^{B}({\bf x})\cdot{\bf E}^{C}({\bf x})$ and the
potential is given by
$$V(r)={\alpha_{s}\over r}-{2{N_{c}b}\over{N_{c}^{2}-1}}r\,\left(1-{\rm e}^{-%
\Lambda_{UV}r}\right)$$
(2)
and $N_{c}=3$. The quark mass appearing in this Hamiltonian is the current
mass. To be phenomenologically successful, constituent quark masses must
be generated in some way. This may be achieved by employing a BCS vacuum
Ansatz; the gap equation which follows from minimizing the vacuum energy
density, $\langle\Omega|H|\Omega\rangle$, (where
$|\Omega\rangle$
represents the BCS trial vacuum) gives rise to a low energy constituent
quark mass
of roughly 200 MeV[13]. A similar calculation in the glue sector
yields a gluon dispersion relation which is well-approximated by
$$\omega(k)=\sqrt{k^{2}+m_{g}^{2}{\rm e}^{-k/\kappa}}$$
(3)
with $m_{g}=800$ MeV and $\kappa=6.5$ GeV. One sees that a
constituent gluon mass of approximately 800 MeV has been generated[14].
Hadrons are then constructed on top of the BCS vacuum $|\Omega\rangle$
by employing a basis truncation (typically Tamm Dancoff or Random Phase) and
solving the resulting Bethe-Salpeter equation. This approach has been used
to derive the low lying spectrum of glueballs and agrees remarkably well with
lattice data[14]. It should be noted that the dynamical gluons are
transverse so that Yang’s theorem holds and the difficulty mentioned in the
Introduction does not arise.
In the following, the parameters of the model are fixed to the $q\bar{q}$
potential derived in the lattice calculation of Juge, Kuti, and
Morningstar[15] (see Fig. 3 below).
The fit yields $\alpha_{s}=0.29$ and $b=0.24$ GeV${}^{2}$.
The final parameter, $\Lambda_{UV}$ serves as an ultraviolet cutoff on
the linear
confinement potential. Its value was determined in Ref. [14] by fitting
the gluon condensate, and will be set at 4 GeV in the following. We shall
consider the static quark limit in the remainder of this work so that quark
masses (and the quark sector of the BCS vacuum) become irrelevant.
The model presented here may be considered as a simplified version of the
Coulomb gauge QCD Hamiltonian where the effects of nonabelian gauge couplings
have been modeled by the linear confinement term. Furthermore, second order
transverse gluon exchange is suppressed by the heavy quark masses. In principle,
this approach allows the elimination of the ultraviolet scale $\Lambda_{UV}$ via
renormalization. A method for achieving this which is appropriate for
nonperturbative Hamiltonian-based calculations is described in Ref. [13].
II.2 Static Hybrids
In the Tamm Dancoff approximation, hybrids are constructed as $q\bar{q}g$
excitations of the BCS vacuum. For the heavy hybrids considered here, the
(anti)quarks serve as static
color sources (sinks) and the gluons are constituent particles as described
above. We choose to work in the “diatomic molecule” basis because this
facilitates comparison with the lattice results of Ref. [15]. Thus
the hybrid state may be written as
$$|\vec{R}\,n_{g};j_{g}\,\Lambda\,\xi\rangle=\int d\vec{k}\varphi_{n_{g}j_{g}}(k%
){\cal D}^{j_{g}}_{\mu\Lambda}(\hat{R}){\cal D}^{j_{g}*}_{\mu\lambda^{\prime}}%
(\hat{k})\sqrt{2j_{g}+1\over 4\pi}\chi_{\lambda\lambda^{\prime}}^{\xi}{\rm T^{%
A}_{ab}\over 2}b^{\dagger}_{\vec{R}/2,a}d^{\dagger}_{-\vec{R}/2,b}a^{\dagger}_%
{k,A,\lambda}|\Omega\rangle.$$
(4)
Wigner rotation functions are written as ${\cal D}$ in this expression,
$R$ is the distance between the $q\bar{q}$ pair, $\lambda$ is the gluon
helicity, and $\varphi$ is the radial hybrid wavefunction in momentum
space. The gluon polarization wavefunction, $\chi^{\xi}_{\lambda\lambda^{\prime}}$ is given by
$\chi^{1}_{\lambda\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}/\sqrt{2}$ and
$\chi^{-1}_{\lambda\lambda^{\prime}}=\lambda\delta_{\lambda\lambda^{\prime}}/%
\sqrt{2}$.
The two cases, $\xi=$1, -1, represent transverse magnetic and transverse
electric hybrids respectively. Finally, $\Lambda$ is the projection of the
gluonic angular momentum onto the $q\bar{q}$ axis, $j_{g}$ is the total
angular momentum of the gluon, and $n_{g}$ labels the radial basis state. We note that
employing
helicity basis gluon creation operators makes this expression significantly
more compact than the canonical basis.
The Salpeter equation which follows from this ansatz and the Hamiltonian in
Eqn. (1) may be obtained from the following matrix element:
$$\displaystyle\langle\vec{R}^{\prime}\,n_{g}^{\prime};j_{g}^{\prime}\,\Lambda^{%
\prime}\,\xi^{\prime}|$$
$$\displaystyle H$$
$$\displaystyle|\vec{R}\,n_{g};j_{g}\,\Lambda\,\xi\rangle=\int d\vec{k}\,\varphi%
_{n_{g}^{\prime}j_{g}^{\prime}}^{*}(k)\varphi_{n_{g}j_{g}}(k){1\over 2}[\omega%
(k)+{k^{2}\over\omega(k)}]$$
(5)
$$\displaystyle+$$
$$\displaystyle{3\over 8}\int\int d\vec{q}d\vec{k}\,\varphi_{n_{g}^{\prime}j_{g}%
^{\prime}}^{*}(k)\varphi_{n_{g}j_{g}}(k)\,V(k-q)\left[{\omega(k)^{2}+\omega(q)%
^{2}\over\omega(k)\omega(q)}(1+(\hat{q}\cdot\hat{k})^{2})\right]$$
$$\displaystyle-$$
$$\displaystyle{3\over 4}\int\int d\vec{q}d\vec{k}\,\varphi_{n_{g}^{\prime}j_{g}%
^{\prime}}^{*}(q)\varphi_{n_{g}j_{g}}(k)\,V(k-q)\left({\rm e}^{i{R\over 2}%
\cdot(k-q)}+{\rm e}^{-i{R\over 2}\cdot(k-q)}\right){\omega(k)+\omega(q)\over%
\sqrt{\omega(k)\omega(q)}}\times$$
$$\displaystyle\times{\cal D}^{j_{g}*}_{\Lambda\lambda^{\prime}}(\hat{k}){\cal D%
}^{j_{g}^{\prime}}_{\Lambda^{\prime}\sigma^{\prime}}(\hat{q})\sqrt{2j_{g}+1%
\over 4\pi}\sqrt{2j_{g}^{\prime}+1\over 4\pi}\chi_{\lambda\lambda^{\prime}}^{%
\xi}\chi_{\sigma\sigma^{\prime}}^{\xi^{\prime}}{\cal D}^{1}_{\mu\lambda}(\hat{%
k}){\cal D}^{1*}_{\mu\sigma}(\hat{q})$$
$$\displaystyle+$$
$$\displaystyle\left({1\over 6}V(R)+{4\over 3}V(0)\right){\cal N}_{n_{g}^{\prime%
}j_{g}^{\prime}n_{g}j_{g}}\delta_{j_{g}^{\prime}j_{g}}\delta_{\Lambda^{\prime}%
\Lambda}\delta_{\xi^{\prime}\xi}$$
An overall $\delta(\vec{R}^{\prime}-\vec{R})$ is understood and ${\cal N}_{n_{g}^{\prime}j_{g}^{\prime}n_{g}j_{g}}=\int dkk^{2}\varphi_{n_{g}^{%
\prime}j_{g}^{\prime}}^{*}(k)\varphi_{n_{g}j_{g}}(k)$ is the
wavefunction normalization factor. The two extra Wigner rotation matrices arise
from converting the Cartesian basis implicit in Eqn. (1) and in the expression
for the color current to the helicity basis, $a_{kiA}=\epsilon_{H}^{i}(k\lambda)a_{k\lambda A}={\cal D}^{1}_{m\lambda}(\phi,%
\theta,-\phi)\epsilon_{C}^{i}(m)a_{k\lambda A}$, where $\epsilon_{C}$ and $\epsilon_{H}$ are canonical and
helicity polarization vectors respectively.
The first term in this expression is the gluon kinetic energy, the second
is the gluon self energy, the third is the gluon potential, and the fourth
is the $q\bar{q}$ potential and self energies for static quarks in a color
octet. The presence of the gluon and quark self energies assures the
infrared finiteness of the Salpeter equation. This appears to be a general
feature of color singlet states in our approach[14].
The last task is to identify the diatomic quantum numbers used to label the
states.
In the Jacob-Wick convention the action of parity on glue is given by
$$Pa_{\vec{k},\lambda,A}P^{\dagger}=\eta^{P}_{g}{\rm e}^{-2i\lambda\phi}a_{-\vec%
{k},-\lambda,A}$$
(6)
where $\phi$ is the azimuthal angle of $\hat{k}$ and $\eta^{P}_{g}=-1$ is the intrinsic gluon parity.
Thus the hybrid states given in Eqn. (4) are eigenstates of gluonic parity with
$$P_{g}|\vec{R}n_{g};j_{g}\Lambda\xi\rangle=\xi\eta^{P}_{g}(-)^{j_{g}+1}|\vec{R}%
n_{g};j_{g}\Lambda\xi\rangle.$$
(7)
A reflection of the glue degrees of freedom through a plane containing the
$q\bar{q}$ axis leaves
the Hamiltonian invariant and when acting on the states it takes $\Lambda\rightarrow-\Lambda$.
For $|\Lambda|>0$ one thus has doubly degenerate states.
We call this operation $Y$-parity and perform it by setting
$\vec{R}\to R\hat{z}$ and taking $y\rightarrow-y$ which may be achieved
by a parity operation followed by a rotation through $\pi$ about the y-axis.
The action of $Y$ on a single transverse gluon state,
$|k\lambda A\rangle=a^{\dagger}_{k\lambda A}|0\rangle$ is
therefore given by,
$$Y|k\lambda A\rangle={\rm e}^{-i\pi J_{y}}P|k\lambda A\rangle=\eta^{P}_{g}{\rm e%
}^{2i\lambda\phi}|k^{\prime},-\lambda,A\rangle,$$
(8)
where $\vec{k}^{\prime}=(k_{x},-k_{y},k_{z})$. For the states defined in
Eq. (4) one has
$$Y|\vec{R}n_{g};j_{g}\Lambda\xi\rangle=\xi\eta^{P}_{g}(-)^{\Lambda+1}|\vec{R}n_%
{g};j_{g}\,-\Lambda\xi\rangle,$$
(9)
where the relations ${\cal D}^{j}_{\mu-\mu^{\prime}}(k^{\prime})=(-)^{2j+\mu+\mu^{\prime}}{\rm e}^{%
-2i\mu^{\prime}\phi}{\cal D}^{j}_{-\mu\mu^{\prime}}(k)$
and ${\cal D}^{j}_{\mu\mu^{\prime}}(\hat{z})=\delta_{\mu\mu^{\prime}}$ were used.
For $\Lambda\neq 0$ the Y-diagonal states may thus be written as
$$|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle={1\over\sqrt{2}}\Bigg{(}|\vec{%
R}n_{g};j_{g}|\Lambda|\xi\rangle+\eta_{Y}|\vec{R}n_{g};j_{g}-|\Lambda|\xi%
\rangle\Bigg{)},$$
(10)
where $\eta_{Y}=\pm 1$ and,
$$Y|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle=\xi\eta^{P}_{g}\eta_{Y}(-)^{%
\Lambda+1}|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle.$$
(11)
For $\Lambda=0$ we simply have,
$$Y|\vec{R}n_{g};j_{g}0\xi\rangle=-\xi\eta^{P}_{g}|\vec{R}n_{g};j_{g}0\xi\rangle.$$
(12)
To allow for easy comparison with conventions used elsewhere, we summarize the
transformation properties of $\bar{q}qg$ states under total parity,
$P=P_{q}P_{g}$, and charge conjugation $C=C_{q}C_{g}$, including the quark degrees
of freedom.
$$P|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle=\xi\eta^{P}_{g}\eta^{P}_{\bar%
{q}q}(-)^{j_{g}+1}|-\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle$$
(13)
$$C|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle=\eta^{C}_{g}\eta^{C}_{\bar{q}%
q}(-)^{S_{\bar{q}q}+1}|-\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle$$
(14)
where $\eta^{P}_{\bar{q}q}$ = $\eta^{C}_{q\bar{q}}$ = $\eta^{C}_{g}$ = -1. The states
introduced in
Eq. (10) are therefore also eigenstates of combined $PC$, with
$$PC|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle=\xi\eta^{P}_{g}\eta^{C}_{g}(%
-)^{j_{g}+S_{\bar{q}q}}|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle$$
(15)
After dividing out the quark portion of $PC$ we are left with
$$(PC)_{g}|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle=\xi\eta^{P}_{g}\eta^{C%
}_{g}(-)^{j_{g}+1}|\vec{R}n_{g};j_{g}|\Lambda|\xi;\eta_{Y}\rangle$$
(16)
II.3 Results
The Salpeter equation is solved by expanding the radial wavefunction in a
complete basis and by diagonalizing the resulting Hamiltonian matrix. The
evaluation of the matrix elements is greatly facilitated by performing the
angular integrals analytically. Furthermore, we found it expedient to do
the remaining numerical integrals with the potential in position space since
the integrand is less oscillatory for large argument in this case. The
plane waves
and potential were expanded in a double series of Wigner functions yielding
a total of ten Wigner functions. The angular integral then evaluates
to a sum over a product of six Clebsch-Gordon coefficients.
As discussed above eigenstates of the Hamiltonian may be labelled with the projection of the
angular momentum onto the $q\bar{q}$ axis, the product of gluonic parity with
charge conjugation, $(PC)_{g}$ and the Y-parity eigenvalue.
For $\Lambda\neq 0$ the two Y-parity
eigenstates are degenerate. States are therefore denoted by
$\Lambda_{(PC)_{g}}^{Y}$
where $\Lambda=0,1,2$ are denoted by $\Sigma$, $\Pi$, $\Delta$; $(PC)_{g}=\eta^{P}_{g}\eta^{C}_{g}\xi(-)^{j_{g}+1}$ =
$g$ or $u$ for even or
odd parity respectively; and $Y=\xi\eta^{P}_{g}\eta_{Y}(-)^{\Lambda+1}$ = $\pm$.
The total gluonic angular momentum, $j_{g}$ is not a good quantum number.
We have found however, that the sum over radial basis states and gluon angular
momentum saturates quickly. For example, Fig 1. shows the (approximately) exact wavefunction
for $\Sigma_{g}^{+}$ with $j_{g}=1$ and a variational single Gaussian orbital.
The next contribution to the $\Sigma_{g}^{+}$
eigenstate has $j_{g}=3$, this is shown as the dotted line in the figure.
Evidently
the eigenstate is dominated by the lowest gluonic angular
momentum component and the single Gaussian approximation is quite accurate.
This remains true for all $R$ studied here, with the $j_{g}=3$ component
rising to 13% of the wavefunction for $R=10$ GeV${}^{-1}$. These observations
support traditional approximations made for hybrids: the use of simple Gaussian
wavefunctions [19] and the truncation of the adiabatic Schrödinger
equation to lowest $j_{g}$[21].
Fig. 1. $\Sigma_{g}^{+}$ wavefunctions. Exact result (dashed line), single
Gaussian approximation (solid line), $j=3$ component (dotted line).
The results for the gluon spectrum are presented as a function of the
static quark separation in Figs. 2 and 3.
They are plotted in terms of the lattice scale, $R_{0}$ = 2.32 GeV${}^{-1}$[15] and the potentials have
all been normalized by subtracting an overall constant given by $V_{q\bar{q}}(2R_{0})$.
In Figs. 2a-c we compare the
recent lattice results from Ref. [15]
with the predictions of the flux tube model [4].
The flux tube model was motivated by the strong coupling limit of the QCD lattice Hamiltonian.
It is based upon a nonrelativistic, small oscillation approximation to motion of the colored links
in a topological sector where there are no overlapping (color representations of higher dimension)
or disconnected links. Under parity (charge) conjugation the spatial (color) orientation
of lattice links is reversed, thus the nonrelativistic “beads” of the flux tube model are assumed
to flip orientation with respect to the position of the
quarks under parity and to have positive intrinsic charge conjugation parity.
The lowest solid line in Fig. 2a is a fit to the $q\bar{q}$ potential given by
$E(R)=-4\alpha_{S}/(3R)+bR+const$ which corresponds to the ground state lattice $\Sigma^{+}_{g}$ potential.
The other solid lines in Fig. 2 show the flux tube potential as given by
$$E(R)=bR+{N\pi\over R}(1-e^{-fb^{1/2}R})$$
(17)
with $f\sim 1$ and
$N=\sum_{m=1}m(n_{m+}+n_{m-})$. The latter represents the total number of right-handed ($n_{m+}$) and
left-handed ($n_{m-}$) transverse phonon modes weighted by the phonon momentum ($m$). We note that
the authors of Ref. [4] included a Coulomb term in this expression, which is incompatible with the lattice results.
The flux tube model predicts the first excited $\Sigma^{\prime+}_{g}$ to be split by
$N=2$
from the ground states, and two degenerate $\Sigma^{+}_{u}$ and $\Sigma^{-}_{u}$ potentials at $N=3$.
In the flux tube model the lowest $\Pi$ state is predicted to be the $\Pi_{u}$. It is split
from the $q\bar{q}$ ground state by $N=1$ and is followed by the $N=2$, $\Pi_{g}$ potential.
The two lowest $\Delta$ states, $\Delta_{g}$ and $\Delta_{u}$ correspond to $N=2$ and $N=3$ respectively.
The flux tube model fits the first excited state, $\Pi_{u}$ quite well
over a wide range of the quark separation. It is, however, the only surface to
do so at small distance. Furthermore,
this may be a fluke due to the particular choice of the short distance cutoff of the
$\pi/R$ term employed in Ref. [4]. Alternatively, at large distances, the system is expected to act
like some sort of string with an excitation energy given by $\pi/R$. The splittings
do indeed appear reasonable for all the surfaces considered. It is, however, disconcerting that the
$\Pi$ surfaces diverge from the flux tube model predictions for $r\gtrsim 4R_{0}$. This must be
taken as an indication that the simple flux tube model considerations fail for more complex gluonic configurations.
In Fig.3a-c we plot the results of our calculations for $\Sigma$, $\Pi$ and
the $\Delta$ potentials with the flux tube results (solid lines) for comparison.
It is apparent that the $\Pi_{g}$ surface lies below the $\Pi_{u}$ surface while
the $\Delta_{u}$ lies below the $\Delta_{g}$. This is opposite to the lattice
results, indicating that the model fails to reproduce the expected level
orderings. The $\Sigma$ levels are also permuted with respect to the lattice
calculations at small $R$. We note, however, that the correct level
orderings may be reinstated by simply flipping the intrinsic parity of
the gluon (set $\eta^{P}_{g}=1$ in the expressions
above).
The resulting surfaces agree reasonably well with
the lattice at small distances. For example, the $\Pi_{u}$ and $\Pi_{g}$ potentials
are roughly 1 and 2 at the origin in both calculations. We obtain values of
approximately 3 and 3.5 for the $\Delta_{u}$ and $\Delta_{g}$ surfaces at the
origin, similar to the lattice values of 2 and 3 respectively.
The agreement persists to intermediate $r$, after which it is apparent that
the model results approach the linear regime much too quickly with respect to
the lattice. This is a strong
indication that more degrees of freedom become active as the quark separation
increases beyond $r\sim R_{0}\sim 1/2$ fm.
This is certainly sensible from the flux tube model point of view, where a
large number of degrees of freedom
are necessary to construct string phonons.
Our results include the contribution due to direct interaction between the
static quarks (the last term of Eqn. (5)). This term was set to zero by Horn and Mandula[7]
because it anti-confines and therefore may be expected to not produce a flux tube between the
quarks. However, we note that it is responsible for producing a hybrid potential slope equal to
that of the $q\bar{q}$ ground state potential, in keeping (roughly) with the lattice data. Thus we
have chosen to retain this term. Note however, that this implies that there is a short distance
repulsive Coulomb spike which should appear at very short quark separation. The appearance
of this spike is, however, unphysical because the hybrid may emit a gluon and convert into a
$q\bar{q}$ color singlet and a low lying glueball and this is energetically favorable for small $r$.
Equivalently, $V_{\Pi_{u}}>V_{q\bar{q}}+m_{\rm gb}$ for $r\lesssim 0.2$ fm. Thus the Coulomb spike
mentioned above is irrelevant and one should see, instead, a Coulomb core for small $r$. This effect
may be easily incorporated into the model presented here by allowing for coupling to the $q\bar{q}gg$
channel. Notice that no core is visible in the lattice data (especially in $\Pi_{u}$, which is measured down to
$r\approx 0.04$ fm). The authors of Ref. [15] are currently
examining this issue[20].
Fig. 2a. $\Sigma$ surfaces. Lattice (symbols) and flux tube (lines).
Fig. 2b. $\Pi$ surfaces. Lattice (symbols) and flux tube (lines).
Fig. 2.c. $\Delta$ surfaces. Lattice (symbols) and flux tube (lines).
Fig. 3a. $\Sigma$ surfaces. Model (symbols) and flux tube (lines).
Fig. 3b. $\Pi$ surfaces. Model (symbols) and flux tube (lines).
Fig. 3c. $\Delta$ surfaces. Model (symbols) and flux tube (lines).
III Conclusions
The model presented here agrees moderately well with lattice calculations of adiabatic
potential surfaces. However, it disagrees in detail. In particular, it is necessary to
ignore the intrinsic parity of the gluon to obtain the expected level
orderings. Although one may argue that this is in keeping with the expectations for the parity
of a lattice link operator (such as is employed in the development of the Flux Tube model), the
lack of consistency is disconcerting. Furthermore, the model predicts surfaces which become
linear too quickly with respect to the lattice. We expect this flaw to persist in all
constituent glue potential models (such as [7] and [18]). This is a
strong indication that more degrees of freedom are required to describe soft glue at large
interquark separation.
It is important to note that models which employ single spinless gluons
do not contain sufficient degrees of freedom to reproduce the potential spectrum. In particular,
“single bead” Flux Tube models ([18]) cannot make $\Pi_{g}$ or $\Sigma_{u}$ states while
three dimensional bead models ([7]) cannot make $\Sigma^{+}$ states. Thus including gluon
spin is a minimal necessity in this class of models (although the level ordering problem must
be overcome as discussed above).
We also note that it is possible for spherical bag models to reproduce the lattice calculations
at small quark separation, but that they fail at large $R$. Furthermore, Flux Tube model or
Nambu-Goto string models[22] reproduce the lattice reasonably well for intermediate to
large quark separations, but do not perform well at small $R$ or in detail at large $R$. In
particular the potential separation does not appear to be the expected $\pi/R$ and the slopes
do not appear to agree. Finally, the bag model of Ref. [6, 21] works reasonably well
over all quark separations, although problems remain to be resolved in the $\Sigma$ states and
for large $R$.
In summary, it appears that some sort of flux tube is required to explain the lattice
adiabatic hybrid potential surfaces at large quark separation. This is in keeping with
the conclusions of Ref. [12], where flux tubes were required to explain the spin
splittings in heavy quarkonia. However, any flux tube model must attempt to incorporate
the small $R$ behaviour of the potential surfaces (and the seemingly anomalous behaviour at
large $R$). This appears to be an indication that different degrees of freedom are required
at small distances (as is incorporated in, for example, the bag model of Ref. [6]).
Finally, the model presented here (with the gluon parity reversal) works moderately well in
describing the potential surfaces and could provide a useful starting point to simple
models of gluonic hadron properties (note that the early transition to a linear potential
seen in Figs. 3 should not be relevant since the hadron wavefunction is exponentially
suppressed in this region).
Thus we expect that the success of the previous glueball
spectrum calculation[14] in this model (where the gluon parity was irrelevant) was
not a fluke.
Several benefits of the model are particularly relevant to hybrids; light quarks may be easily
incorporated, the effects of coupled channels may be examined, and the effects of light quark
coupling to virtual transverse or Coulomb gluons may be included. The latter effect is of interest
because it is excluded in quenched lattice calculations and may have a substantial effect on the
hidden flavor hybrid spectrum.
Acknowledgements.The authors are grateful to C. Morningstar for discussions and for providing us the preliminary
lattice data of Ref. [15]. This work was supported by the DOE
under grants DE-FG02-96ER40944 (ES) and DE-FG02-87ER40365 (AS).
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Luther-Emery Stripes, RVB Spin Liquid Background and High $T_{c}$ Superconductivity
WenJun Zheng
Department of Physics and Center for Material Research, Stanford University, Palo Alto, California 94305
(Received 1 June 1999)
Abstract
The stripe phase in high $T_{c}$ cuprates is modeled as a single stripe coupled to
the RVB spin liquid background by the single particle hopping process. In normal state, the strong
pairing correlation inherent in RVB state is thus transfered into the Luttinger stripe and
drives it toward spin-gap formation described by Luther-Emery Model. The establishment of
global phase coherence in superconducting state contributes to a more relevant coupling to
Luther-Emery Stripe and leads to gap opening in both spin and charge sectors. Physical
consequences of the present picture are discussed, and emphasis is put on the unification of
different energy scales relevant to cuprates, and good agreement is found with
the available experimental results, especially in ARPES.
pacs: 74.20.Mn, 74.20.Mn, 74.72.Dn
The universal presence of phase separation in high $T_{c}$ cuprates has been
confirmed by extensive experiments, including elastic and inelastic neutron scatterings[1], NMR and NQR[2], and Angular Resolved Photon Emission Spectroscopy [3](ARPES) in $La_{2-x}Sr_{x}CuO_{4}$, $YBa_{2}Cu_{3}O_{7-x}$ and
$Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$ (Bi-2212) etc. The emerging picture is , upon hole doping beyond $x=0.06$, quarter-filled [4] hole rich stripes begin to form, separating the
copper oxide plane into slices of antiferromagnetic insulating regions,
with the inter-stripe distance in proportion to $1/x$, where $x$ is the density of
doped holes. Above $x=1/8$ and inside the overdoped regime, incommensurate stripe modulation persists, although the inter-stripe spacing saturates, with
the excessive holes overflowing into insulating regions , signifying the crossover to
conventional metallic phase with overall homogeneity. Besides, the stripes are dynamically fluctuating and may coexist with superconductivity. Dated back to the late 1980’s, the relevance of phase separation and dormain walls
to high $T_{c}$ cuprates was already under considerable discussions[5]. In 1993, Emery and Kivelson
suggested a scenario of mesoscopic phase separation frustrated by long-range Coulomb interactions [6], as a general consequence of doping a strongly correlated insulator, and they also pointed out the relevance of dynamical stripes to high $T_{c}$ superconductivity [7]. The origin of phase separation is still under hot debate. Another equally important issue that will be treated here is: assuming the presence of stripes coupled to an undoped background, can we improve our understanding of the interesting and even puzzling physical features revealed in
both normal state and superconducting state of cuprates? The importance of such exploration has been recently emphasized in [8][9], and some
interesting results have been reported . Most of these attempts treat the stripe as
a 1D or quasi-1D Luttinger Liquid coupled to neighbouring stripes [10] or insulating background which is either modeled as a canonical antiferromagnet [11] or as another 1D Luttinger Liquid [9]. The couplings through pair tunneling [9] or spin exchange [11] have been discussed. Here, in contrast to [9] and [11],
I emphasize the importance of coupling a stripe to a truly anomalous 2D insulating background, which has its hidden unconventional nature of RVB (Resonating Valence Bond ) spin liquid, under the classical apparel of antiferromagnetic (AFM) order. It is shown that , through single particle hopping [12] between
1D stripes and 2D RVB background, a normal state pseudo-gap $\Delta_{n}$ is ”induced” inside the stripe’s spin sector, which coincides with the mechanism of spin gap formation in a class of 1D electron systems named after Luther and Emery [13] . Further more, inside the superconducting state, the presence of global phase coherence in RVB order parameter contributes to an even more relevant pairing coupling to 1D stripe, which results in a gap of $\Delta_{sc}$ opening in both spin and charge sectors. Experimental consequences of 2 quantitatively different gaps are discussed, and good agreement is found with ARPES results [14].
The brilliant idea of RVB states was advanced by Anderson soon
after the discovery of high $T_{c}$ superconductivity [15]. The RVB state is described by a coherent superstition of different configurations of valence bonds, which was expected to
be a reasonable approximation to the ground state of insulating spin 1/2 Heisenberg Model, especially with frustration or hole doping, although the ground state of undoped cuprates clearly has a Neel order. Lately there has been renewed interest in the plausible relevance of RVB correlation to cuprate physics at relatively high energy scale, motivated both experimentally and theoretically. Recent ARPES result on $Ca_{2}CuO_{2}Cl_{2}$ by F. Ronning et al. [16] reveals the presence of a d-wave dispersion along the remnant Fermi surface and a Dirac like dispersion isotropically focused around $(\pi/2,\pi/2)$, which is exactly what was predicted for the ”$\pi$-flux phase” [21] of RVB spin liquid, where $\epsilon(k)=J\sqrt{\cos^{2}k_{x}+\cos^{2}k_{y}}$, and can not be described within the spin density wave picture although the latter can account for the low-lying spin excitations in the Neel state. Further numerical results also support the presence of a local RVB spin liquid state around a doped hole with momentum $\bf{k}=(\pi,0)$[17], accompanied by an anti-phase of spins around the hole which may be relevant to the generation of anti-phase domain walls in striped phase of cuprates. Theoretically, Kim and Lee show that Neel order can be restored in $\pi$-flux phase through dynamical mass generation of gauge fluctuations at low temperature [18] , which points toward an emerging consistent RVB picture spanning from ground state to high energy scale physics [19].
Based on the above results, I suggest that one can model the environment of a quarter-filled stripe as a RVB spin liquid, which is coupled with the stripe by a single particle hopping term that conserves the momentum along the stripe direction . To get started, one can first ignore the inter-stripe correlation in the normal state of underdoped cuprates, considering the strong incoherence revealed by experiments.
The total Hamiltonian is given by
$$\displaystyle H(c,c^{+},d,d^{+})$$
$$\displaystyle=$$
$$\displaystyle H_{1D}(d,d^{+})+H_{RVB}(c,c^{+})$$
$$\displaystyle+$$
$$\displaystyle H_{couple}(c,c^{+},d,d^{+}),$$
where $c$, $c^{+}$ and $d$, $d^{+}$ represent the annihilation and creation operators of a single particle in 2D RVB background and 1D stripe, respectively.
$H_{couple}(c,c^{+},d,d^{+})=\sum_{{\bf{k}},q,\sigma}Vc^{+}_{\bf{k},\sigma}d_{q%
,\sigma}\delta_{k_{x},q}+h.c.$,
where only horizontal stripe is considered, ${\bf{k}}=(k_{x},k_{y})$, and momentum conservation is ensured by requiring
$k_{x}=q$. $V$ gives the hopping matrix element, which is vital in deciding different energy scales relevant to cuprates , as will be discussed later [12].
A routine Hartree-Fork decoupling is applied to the RVB Hamiltonian $H_{RVB}$ [15],
$$\displaystyle H_{RVB}$$
$$\displaystyle=$$
$$\displaystyle-J\sum_{<ij>}b^{+}_{ij}b_{ij}$$
$$\displaystyle=$$
$$\displaystyle-J\sum_{<ij>}(\Delta^{*}_{ij}b_{ij}+\Delta_{ij}b^{+}_{ij}-|\Delta%
_{ij}|^{2}),$$
where $b^{+}_{ij}=\frac{1}{\sqrt{2}}[c^{+}_{i,\uparrow}c^{+}_{j,\downarrow}-c^{+}_{i,%
\downarrow}c^{+}_{j,\uparrow}]$, $\Delta_{ij}$ is
the RVB order parameter defined on each bond between 2 nearest neighbors, which is reduced to the mean field average of $b_{ij}$ operator at the saddle point level.
Then one can change into the momentum space, that is
$$\displaystyle H_{RVB}$$
$$\displaystyle=$$
$$\displaystyle\frac{-J}{\sqrt{2V}}\sum_{\bf{k},\bf{q}}[\Delta^{*}_{\bf{k},\bf{q%
}}(c_{\bf{q},\downarrow}c_{\bf{k}-\bf{q},\uparrow}-c_{\bf{q},\uparrow}c_{\bf{k%
}-\bf{q},\downarrow})+h.c.],$$
(3)
where $\Delta_{\bf{k},\bf{q}}=\sum_{\bf{\hat{\delta}}}\Delta_{\bf{k},\bf{\hat{\delta}%
}}e^{i\bf{q}\bf{\hat{\delta}}}$, and
$\Delta_{\bf{k},\bf{\hat{\delta}}}=\frac{1}{\sqrt{V}}\sum_{\bf{r_{i}}}\Delta_{i%
,i+\bf{\hat{\delta}}}e^{-i\bf{k}\bf{r_{i}}}$ ($\bf{\widehat{\delta}}=\pm\widehat{x},\pm\widehat{y}$).
There are many possible mean field states in RVB theory [22], among which the ”$\pi$-flux
phase” is selected here, because of its low energy, conservation of time-reversal symmetry and possible connection to AFM long range order[18].
In ”$\pi$-flux phase”, $\Delta_{ij}=\Delta_{0}e^{i\phi_{ij}}$ is chosen to have uniform amplitude $\Delta_{0}$, while its phase $\phi_{ij}$ is selected to ensure that staggered $+\pi$ and $-\pi$ flux is threaded through each plaquette. For convenience, one can choose $\phi_{ij}=\pm\pi/4$.
Therefore, $H_{RVB}$ is simplified to
$$\displaystyle H_{RVB}$$
$$\displaystyle=$$
$$\displaystyle-J\sum_{\bf{q}}[\Delta^{*}_{0}\gamma({\bf{q}})(c_{{\bf{q}},%
\downarrow}c_{-{\bf{q}},\uparrow}-c_{{\bf{q}},\uparrow}c_{-{\bf{q}},\downarrow%
})+h.c.]$$
$$\displaystyle+$$
$$\displaystyle iJ\sum_{\bf{q}}[\Delta^{*}_{0}\eta({\bf{q}})(c_{{\bf{q}},%
\downarrow}c_{{\hat{\pi}-\bf{q}},\uparrow}-c_{{\bf{q}},\uparrow}c_{{\hat{\pi}-%
\bf{q}},\downarrow})-h.c.],$$
where $\gamma({\bf{q}})=\cos{q_{x}}+\cos{q_{y}}$, $\eta({\bf{q}})=\cos{q_{x}}-\cos{q_{y}}$ and
${\widehat{\pi}}=(\pi,\pi)$.
Then perform Euclidean path integral over the 2D degrees of freedom and obtain the low energy effective action for 1D stripe as follows
$$\displaystyle e^{-S_{eff}}$$
$$\displaystyle=$$
$$\displaystyle\exp\{-\int_{0}^{\beta}H_{eff}(d,d^{+})d\tau+\frac{1}{\beta}\sum_%
{n}i\omega_{n}d^{+}d\}$$
$$\displaystyle=$$
$$\displaystyle\int dcdc^{+}\exp\{-\int_{0}^{\beta}[H_{1D}(d,d^{+})+H_{couple}(c%
,c^{+},d,d^{+})$$
$$\displaystyle+$$
$$\displaystyle H_{RVB}(c,c^{+})]d\tau+\frac{1}{\beta}\sum_{n}i\omega_{n}(c^{+}c%
+d^{+}d)\},$$
where $\omega_{n}=\frac{\pi n}{\beta}$ ($n$ is odd integer), $\beta=\frac{1}{k_{B}T}$.
Assuming $J|\Delta_{0}|>>V$ , one can change back to 1D coordinate system and get
$$\displaystyle H_{eff}$$
$$\displaystyle=$$
$$\displaystyle H_{1D}(d,d^{+})-\frac{V^{2}}{16J}\sum_{l}[\frac{1}{\Delta^{*}_{0%
}}(d_{l,\downarrow}d_{l+1,\uparrow}-d_{l,\uparrow}d_{l+1,\downarrow})$$
$$\displaystyle+$$
$$\displaystyle h.c.]-i\frac{V^{2}}{16J}\sum_{l}[\frac{(-1)^{l}}{\Delta^{*}_{0}}%
(d_{l,\downarrow}d_{l+1,\uparrow}-d_{l,\uparrow}d_{l+1,\downarrow})-h.c.].$$
Then go to the continuum limit, $d_{l,\sigma}\rightarrow\sqrt{a}\Psi(x=la)_{\sigma}$, with the size of unit cell $a~{}\rightarrow 0$
and retain only the slow varying part of $H_{eff}$, we finally arrive at the following correction to $H_{1D}$ due to its coupling to RVB background,
$$\displaystyle\Delta H_{eff}=g\int dx\Psi_{\downarrow}(x)\Psi_{\uparrow}(x)+h.c.,$$
(7)
where $g=-\frac{V^{2}\cos{k_{F}}}{8J\Delta^{*}_{0}}$, and $k_{F}=\pi/4$ for quarter-filled stripe.
We note that the 1D anomalous propagator is induced in stripes through hopping $V$ by the strong pairing correlation inherent to the RVB background. This mechanism is central to the pairing process among mobile carriers inside stripes, via which superconductivity becomes viable.
Based on the above result, I will discuss both normal state and superconducting state ,
respectively. Let us first come to the issue of normal state pseudo-gap , which is
deemed as very important but remains controversial. Theorists are sharply divided in
whether it is precursor pairing or otherwise has nothing to do with pairing, but caused by proximity to quantum critical point of , for example AFM phase transition. To treat normal state here, one can take the strong phase fluctuation in RVB order
parameter into account , while its amplitude is basically non-zero and much less fluctuating.
Therefore, one can integrate out the phase of $\Delta_{0}$, and
get
$$H_{eff}=H_{1D}(d,d^{+})+g_{1}\int dx\Psi^{+}_{\uparrow}\Psi_{\uparrow}\Psi^{+}%
_{\downarrow}\Psi_{\downarrow},$$
where $g_{1}\approx\frac{-g^{2}a^{2}}{2v}$, and $v$ is the bare Fermi velocity,
which can be treated with the standard bosonization technique [23]as follows
$$\displaystyle H_{eff}$$
$$\displaystyle=$$
$$\displaystyle\int dx\{[(\frac{K_{c}u_{c}}{2}-\frac{g_{1}}{2\pi})\Pi_{c}^{2}+%
\frac{u_{c}}{2K_{c}}{(\partial_{x}\Phi_{c})}^{2}]$$
$$\displaystyle+$$
$$\displaystyle[(\frac{u_{s}}{2}+\frac{g_{1}}{2\pi})\Pi_{s}^{2}+(\frac{u_{s}}{2}%
+\frac{g_{1}}{2\pi}){(\partial_{x}\Phi_{s})}^{2}]$$
$$\displaystyle+$$
$$\displaystyle g_{1}\cos(\sqrt{8\pi}\Phi_{s})\},$$
where $\Phi_{c},~{}\Pi_{c}$, and $\Phi_{s},~{}\Pi_{s}$ , are conjugated boson operators representing
density fluctuations in charge and spin sectors of 1D Luttinger Liquid [23], respectively. $u_{c}$, $u_{s}$ are the corresponding propagating velocities, and $K_{c}$ is
a parameter of interaction.
In terms of renormalization group formulation, $g_{1}\cos(\sqrt{8\pi}\Phi_{s})$ in $H_{eff}$ is marginally relevant, which results in the opening of a spectral gap in spin sector
$$\Delta_{s}\propto\sqrt{|g_{1}|}\exp{(\frac{v}{2\pi g_{1}})},$$
where $v$ is the bare Fermi
velocity. $\Delta_{s}$ is here identified as
the normal state pseudo-gap $\Delta_{n}$ that leads to spectral weight depletion in low energy spin fluctuations and single particle spectrum, while the charge excitations remain gapless, which give rise to metallic transporting along the stripe. This is of the same principle as the early results by Luther and Emery in exploration of spin gap formation as an instability of Luttinger Liquid [13]. Further more, the effect of $g_{1}$ on charge sector is also physically important, it leads to $K_{c}>1$ [24], which ensures that singlet superconducting fluctuation dominates over CDW (charge density wave) correlation, and drives the system close to the opening of charge gap and superconducting phase transition that accompanies it ( as will be clarified later).
Now let’s turn to the superconducting state. It is generally agreed that global phase coherence
is established at $T<T_{c}$, so that strong phase fluctuation in $\Delta_{ij}=\Delta_{0}e^{i\phi_{ij}}$ is quenched, and the relevant correction to $H_{1D}$ becomes Eq[7] itself with $\Delta_{0}$ replaced by its average magnitude.
Then standard bosonization gives
$$\Delta H_{eff}=2gd_{u}\int\cos(\sqrt{2\pi}\Theta_{c})\sin(\sqrt{2\pi}\Phi_{s})dx,$$
The scaling dimension of $\Delta H_{eff}$ is $\frac{1}{2}+\frac{1}{2K_{c}}$, so it is generally
relevant except for very strong repulsive interactions(i.e. $K_{c}<1/3$). Unlike the normal state case discussed before, in $\Delta H_{eff}$ both spin sector and charge sector are
coupled together by a relevant effective interaction and spin-charge separation typical of
a Luttinger Liquid is thus broken and this kind of ” spin-charge recombination ” may be relevant to
the generation of well-defined quasi-particles in superconducting state[25]. Under scaling to lower energy, $2gd_{u}$
is renormalized to divergence, so $\Theta_{c}$ and $\Phi_{s}$ oscillate around stable equilibrium positions and gaps open in both spin and charge excitations, which leads to non-magnetic ground state dominated by singlet superconducting fluctuations. For clarity,
let’s discuss the special case of $K_{c}=1$ and $u_{s}=u_{c}$. Then $H_{1D}$ can be decoupled
into 2 independent Sine-Gordon models of $\Phi_{\pm}=\frac{1}{\sqrt{2}}(\Theta_{c}\pm\Phi_{s})$,
corresponding to 2 branches of free massive fermions. In this case, both spin gap and charge
gap are equal, that is
$$\Delta_{c}=\Delta_{s}\propto 2\pi|g|d_{u}\propto\frac{V^{2}}{J\Delta_{0}}.$$
In general, the effect of small $|K_{C}-1|>0$ is only to mix the above two branches together, while the qualitative picture of gap formation remains robust. Further more,
at leading order , it is expected that $\Delta_{s,c}\propto\frac{V^{2}}{J\Delta_{0}}\propto|g_{1}|^{1/2}$ is a fairly good approximation to start with [25],
. One can associate this gap with the superconducting gap $\Delta_{sc}$,
identified as the quasiparticle gap measured for example by ARPES in superconducting state.
Provided with two quantitatively different energy scales $\Delta_{n}$ and $\Delta_{sc}$ derived above , one can
explore their experimental consequences. It is emphasized that, without considering the
inter-stripe coherent couplings, $V$ represents the strength of local hopping
between a single stripe and its insulating background (its range is limited by inter-stripe distance), through which the strong pairing interaction intrinsic
to RVB spin liquid is ”transfered” into the stripe, and leads to gap openings in both normal state and superconducting state. In going toward overdoped region, because RVB correlation is
significantly suppressed, the relevant energy scale $g$ is reduced to $J\Delta_{0}$, instead of
$V^{2}/J\Delta_{0}$ [25]. Because $\frac{\Delta_{n}}{\Delta_{sc}}\propto\exp{(\frac{-v}{\pi ag^{2}})}$, $\Delta_{n}$ is much more suppressed compared with $\Delta_{sc}$[25], which is well consistent with the ARPES results [14], and extensive experimental evidences supporting the ”absence” of normal state gap in overdoped region [26]. Besides, by combining the
present scenario with the spectral properties of Luther-Emery system [27]
, one can understand the broad ”edge” feature near ($\pi$,0) in ARPES of underdoped normal state, as due to the proximity toward charge gap formation that turns the power law singularity ($\propto\omega^{\alpha-1/2}$,$0<\alpha<<1/2$ ) into a non-singular edge in $A(k,\omega)\propto\omega^{\alpha-1/2}$ ($\alpha>1/2$) [25]. However, this singularity is restored in overdoped region where the effect of RVB background on stripe is
much weakened, therefore singular peaks with long
tails are preserved in $A(k,\omega)$ spectrum, as is consistent with what was observed in ARPES [28] .
In superconducting state, global phase coherence allows single particle hopping
between adjacent stripes through higher order process. From the calculation of corresponding matrix element $t^{\prime}\approx\frac{\hbar^{2}}{2m^{*}d^{2}}$($d$ is inter-stripe distance) , one can extract the effective mass $1/m^{*}\propto\frac{V^{2}}{J\Delta_{0}}$ (underdoped case), and thus estimate the Josephson coupling energy
$E_{J}\approx\frac{\hbar^{2}\rho_{s}}{2m^{*}d}\propto\frac{\Delta_{sc}}{d},$
where $\rho_{s}$ is the superfluid density of a single stripe [25] . In underdoped region, one can
attribute superconducting transition to the global phase ordering [29] and therefore $T_{c}\propto E_{J}\propto x\Delta_{sc}$, which agrees well with two facts: first, $T_{c}\propto x$;
second, $T_{c,max}$ scales with $\Delta_{sc}$ among the cuprates family.
In overdoped region, $T_{c}\propto\Delta_{sc}\propto J\Delta_{0}$ because $d$ is saturated and a new energy scale $J\Delta_{0}$ takes the place of $\frac{V^{2}}{J\Delta_{0}}$, this is consistent with the BCS like relation observed in overdoped cuprates .
Before end, three comments are in order. First, the present scenario opens new route toward the understanding of the subtle relation between pseudo-gap and superconducting gap, in that both $\Delta_{n}$ and $\Delta_{sc}$ have the same origin :
strong pairing interaction in RVB background, but can be quantitatively different in their
dependences on $V$ and $J\Delta_{0}$. Secondly, one can unify the important energy scales : $\Delta_{n}$, $\Delta_{sc}$, $E_{J}$, $T_{c}$, by determining
their unique dependences on a single parameter ($V^{2}/J\Delta_{0}$ in underdoped region and $J\Delta_{0}$ in overdoped region), this explains the material-independent scaling in $\Delta_{sc}:\Delta_{n}:T_{c,max}$ among cuprates family, while a single material-independent $J$ can not.
Thirdly, one can treat the ”heavy mass” issue raised recently in [32] within the present picture: in underdoped cuprates, $\frac{k_{B}T_{c}}{x}=\hbar v^{*}\propto\Delta_{sc}\propto V^{2}/J\Delta_{0}$ and is roughly doping-independent. It can be connected to the flat dispersion perpendicular to horizontal stripes ( $\Gamma$ to (0,$\pi$) direction), as suggested in [32], and can be attributed to slow hole motion transverse to stripes[25],
which limits the achievement of higher $T_{c}$.
In conclusion, I model the stripe phase in high $T_{c}$ cuprates as a single stripe coupled to
the RVB spin liquid background by the single particle hopping. In normal state, the strong
pairing interaction inherent in RVB state is therefore transfered into the Luttinger stripe and
drives it toward Luther-Emery Stripe with spin-gap formation. The establishment of
global coherence in superconducting state contributes to a more relevant coupling to
the stripe and leads to gap opening in both spin and charge sectors. Physical
consequences of the present picture are discussed, and good agreement is found with
the available experimental results in ARPES.
I thank S. A. Kivelson, Z. X. Shen , S. Doniach, D. L. Feng and J. P. Hu for discussions and comments on this work. The support from Stanford Graduate Fellowship (SGF) is acknowledged.
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Statistical Modeling of an astro-comb for high precision radial velocity observation
Fei Zhao,${}^{1}$
Gang Zhao,${}^{1}$ Yujuan Liu,${}^{1}$
Liang Wang,${}^{4,5}$
Huijuan Wang,${}^{1}$
Hongbin Li,${}^{1}$
Huiqi Ye,${}^{2,3}$
Zhibo Hao,${}^{2,3}$
Dong Xiao,${}^{2,3}$
Junbo Zhang,${}^{1}$
Hanna Kellermann${}^{4,5}$
and Frank Grupp${}^{4,5}$
${}^{1}$Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Datun Road 20A, Beijing,
100012, China
${}^{2}$National Astronomical Observatories Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing,
Jiangsu 210042, China
${}^{3}$Key Laboratory of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing, Jiangsu 210042, China
${}^{4}$Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse, D-85748 Garching, Germany
${}^{5}$Universitäts-Sternwarte München, Scheinerstr. 1, D-81679 München, Germany
(Accepted 2018 October 06. Received 2018 October 06; in original form 2018 August 31)
Abstract
The advent of the laser frequency comb as the wavelength calibration unit allows us to measure the radial velocity at $cm\ s^{-1}$ precision level with high stability in long-term, which enable the possibility of the detection of Earth-twins around solar-like stars.
Recent study shows that the laser frequency comb can also be used to measure and study the precision of the instrumental system including the variations of line profile and the systematic uncertainty and instrumental drift.
In this paper, we present the stringent analysis of a laser frequency comb(LFC) system with 25GHz repetition frequency on a R$\sim$50,000 spectrograph with the wavelength spanning from 5085Å to 7380Å.
We report a novel fitting model optimized for the comb line profile, the constrained double Gaussian. The constraint condition is set as $\left|\mu_{1,2}-\mu\right|<\sqrt{2ln2}\sigma$. We introduce Bayesian information criterion to test various models. Compared to the traditional Gaussian model, the CDG(Constrained Double Gaussians) model provides much better goodness of fit.
We apply the CDG model to the observed comb data to demonstrate the improvement of RV precision with CDG model. We find that the improvement of CDG model is about 40%$\sim$60% for wavelength calibration precision.
We also consider the application to use the LFC and CDG model as a tool to characterize the line shape variation across the detector.
The motivation of this work is to measure and understand the details of the comb lines including their asymmetry and behaviors under various conditions, which plays a significant role in the simultaneous calibration process and cross-correlation function method to determine the Doppler shift at high precision level.
keywords:
methods: data analysis – techniques: radial velocities – techniques: spectroscopic
††pubyear: 2018††pagerange: Statistical Modeling of an astro-comb for high precision radial velocity observation–Statistical Modeling of an astro-comb for high precision radial velocity observation
1 Introduction
The golden era for the detection of exoplanets is enabled by the radial velocity(RV) method with precise astronomical spectroscopy. Beginning from the discovery of substellar companions(Campbell
et al., 1988; Latham et al., 1989) and the validation of the first extra-solar planet orbiting a main-sequence star(Mayor &
Queloz, 1995), the study of Doppler meansurements has become a vibrant field in exoplanets research(Winn &
Fabrycky, 2015).
To date, more than 3800 exoplanets are confirmed in which nearly 750 were discovered by Doppler method111https://exoplanet.eu/ & https://exoplanetarchive.ipac.caltech.edu/. Among them, there are about 12% super Earth and about 17% Mini-Neptune(by mass classification scheme)(Valencia et al., 2007; Charbonneau
et al., 2009). On the road of finding terrestrial exoplanets in habitable zone or even Earth-twins, high precision Doppler measurements play a vital role.
Combined with transit observations, they provide the planetary mass for the measurement of the bulk density and characterization of the planet’s structure(Mills &
Mazeh, 2017; Cloutier
et al., 2017; Sarkis
et al., 2018).
The amplitude of the RV for a solar mass star due to an Earth mass planet at 1 $AU$ is about 9.6 $cm\ s^{-1}$ (or 174 kHz shift in frequency at 5500Å). For this planet at 0.05$AU$, this amplitude can increases to about 1 $m\ s^{-1}$.
With the goal of achieving a precision better than 1 m/s, it is necessary to identify and overcome the limitation on Doppler measurements. To understand and characterize the RV uncertainties, we briefly divide them into two major categories depending on their origins: instrumental(Pepe
et al., 2002) and stellar(Kjeldsen
et al., 2005; Hatzes, 2013; Dumusque
et al., 2014).
The instrumental uncertainty could dominate the RV precision especially in $cm\ s^{-1}$ level.
The research on RV noise caused by stellar photospheres will become more effective if we have a better understanding of the noise characteristics from instruments.
There are several main factors from instrumental aspect that induce RV uncertainties, such as wavelength calibration, the stability of the light injection for optics, detector effects etc.(Lo Curto
et al., 2012).
Among these, the wavelength calibration is a crucial step especially for an instrumental system based on simultaneous calibration method and cross-correlation function(CCF) technique to determine Doppler shifts(Baranne
et al., 1996; Lovis &
Pepe, 2007).
Originally, the detector can only record the number of photons or the light’s intensity as a function of pixel position without the information of wavelength. In order to determine the continuous wavelength distribution, we should convert the pixel space into the wavelength space by observing a calibration lamp as reference. One of these solutions is to use a Thorium-Argon hollow cathode lamp(ThAr lamp). With the knowledge of the standard wavelength of thorium emission lines measured in laboratory, we can fit a calibration curve (or wavelength solution) with high order polynomials. For each echelle order, this fitting function translates the pixel position information into the wavelength information.
A typical precision level provided by ThAr lamps is $10^{-7}$ to $10^{-9}$(or a few $m\ s^{-1}$ to several tens of $cm\ s^{-1}$ scaled by the speed of light)(Lo Curto
et al., 2012).
In the case of ThAr lamp, the wavelength calibration precision is limited by the line blending, the irregular intensity and space density of thorium emission lines, the lamps age, the contamination of strong emission lines from adjacent orders etc.
Another calibration method is the absorption cell. Iodine cell technique is widely used to calculate the Doppler shifts. With the narrow absorption $I_{2}$ lines, we can model the spectrum by taking into account the variations of instrumental profile(IP)(Butler et al., 1996). The limitation of iodine cell method mainly includes the loss of the starlight via the absorption gas cell, the contamination of the spectral lines and the narrow wavelength coverage.A typical RV precision level obtained by iodine cell is about several $m\ s^{-1}$(Endl
et al., 2000).
However, such a calibration precision level is not sufficient for detecting the population of terrestrial exoplanets. To estimate how small the shift is, we consider an R=50,000 spectrograph, its resolution unit will be expressed as 0.11Å at 5500Å. Meanwhile, the radial velocity resolution unit is $\Delta v$ = c/R = 6000 $m\ s^{-1}$. Typically, $\Delta\lambda$ should cover 2 pixels on detector based on Nyquist sampling. Thus the velocity resolution is equal to 3000 $m\ s^{-1}$ per pixel. For a 0.1 $m\ s^{-1}$ Doppler shift on a 12$\mu$m pixel size CCD, the physical shift in pixels is $4\times 10^{-10}$m which is about 7.5 times the radius of a Hydrogen atom.
For such a exquisite shift, it is a challenge for wavelength calibration methods to measure and calculate.
Beyond the traditional wavelength calibration methods like iodine cells and Th-Ar lamps, laser frequency combs(LFC)(Reichert et al., 1999; Udem et al., 2002) can offer a promising solution to the wavelength calibration with ultra high precision and long-term stability(Murphy
et al., 2007; Li
et al., 2008; Steinmetz
et al., 2008). An LFC, based on femtosecond pulse mode-locked laser, can produce thousands of uniformly spaced modes in frequency space as a function of $f_{n}=f_{0}+n\cdot f_{rep}$, where $f_{n}$ is the frequency of the nth mode. The $f_{0}$ is the carrier envelope offset frequency which provide an offset for each mode. Here n is typically a large integer approximately in the range of $10^{5}-10^{6}$. The frequency spacing between each adjacent mode is a constant(which equals to the repetition frequency $f_{rep}$). The frequencies are synchronized with the absolute ratio frequency reference such as an atomic clock or a GPS(global positioning system), which makes the astronomical laser frequency comb(astro-comb) meet the requirements of an ideal tools for high precision wavelength calibration.
In the work of Murphy et al. (2012), the authors studied the variation of Intra-pixel sensitivity(IPS) by using a moded-locked fiber-based laser frequency comb with $f_{rep}$=90MHz and 1.03Ghz after filtered by a designed Fabry-Perot(FP) cavity. With the symmetrical fitting model, they found the change of averaged IPS deviates by less than 8% level.
Dumusque
et al. (2015) reported the observations of the Sun as a star with an astro-comb as the simultaneous wavelength calibrator. This work showed several tens $cm\ s^{-1}$ RV precision during 7 days time scale.
Nowadays, the laser frequency comb becomes a mature technology applied in several astronomical spectrographs worldwide. One of the available turn-key LFC is from Menlo system(Steinmetz
et al., 2008; Probst
et al., 2014).
In recent years, the Menlo combs are used or in operation at several telescopes, such as HARPS at 3.6m telescope in La Silla, the VTT solar telescope in Tenerife, the FOCES spectrograph in Wendelstain observatory at USM and the HRS(High Resolution Spectrograph) on 2.16m telescope in Xinglong observatory in China.
In Wilken
et al. (2010), the authors showed the evidence of the CCD stitching pattern revealed by using the Menlo system’s 18GHz LFC mounted on HARPS. There are discontinuities on each 512 pixel position for the wavelength solution curve due to the variations of the intra-pixel distance and sensitivity at the borders of the stitching. By comparing the signals in both fibers(channels) simultaneously, a remarkable short-term repeatability of 2.5cm/s can be achieved on HARPS(Wilken
et al., 2012). Since February 2016, a 25GHz LFC was installed in HRS(Zhao &
Li, 2001; Fan
et al., 2016), which is a fiber-fed spectrograph with R$\sim$50,000 at 0.19mm slit width. In this paper, we demonstrate the calibration results of this HRS-comb system and analysis the line profile variations based on an optimized fitting model.
Ideally, for a well-designed astro-comb system, each individual comb mode(line) should be well resolved at the giving line spacing and should be unresolved at the scale of the FWHM(no intrinsic structures).
With respect to the HRS spectrography for R$\sim$50,000 at 5500Å, the 25GHz line spacing equals to 2.29 times of FWHM. This can be derived based on the equation below:
$$|df|=\frac{c}{\lambda^{2}}|d\lambda|$$
(1)
where c is the speed of light.
Considering that an infinitely narrow mode of LFC passes through the spectrograph, the instrumental response of the spectrograph will be the instrumental profile(IP). Note that the LFC’s mode is much narrower than the resolution of our spectrograph, the comb line represents the IP and can be used as a tool to model the line profile and study the asymmetries and shape variation as a function of wavelength, pixel position and signal intensity etc.
In general, the comb signal can be described as:
$$C(x)=\sum_{i=1}^{N}[E_{i}\int_{i-n}^{i+m}IP(x)dx]$$
(2)
Where $x$ is the pixel position in the spectral direction on each echelle order and $i$ is the line center for each comb line while $m$ and $n$ are the ends position for each individual comb line. $E$ is the envelope function. $N$ is the total number of the comb lines in each echelle order.
For a 18GHz line spacing comb on R=110,000 spectragraph, $N$ is about 275$\pm$5 while for a 25GHz comb on R=50,000, $N$ is around 345$\pm$5 depending on each order’s wavelength range and average SNR.
Thus, the LFC can be use as a powerful tool to characterize the line profile variation across the bandpass, to trace the instrumental shift at long-term and to probe the detector effects that induce errors to the RV measurements.
The significance of studying the line profile consist of:
1). Although the comb modes are filtered to eliminate side modes, the superfluous modes are not completely removed(Quinlan et al., 2010; Ycas
et al., 2012) and the mode remnants may have influences on the line profile that can provide asymmetries and potential shift to the line center. This can be reflected by studying the comb line profile.
2). The signal of LFC undergoes two non-linear process during the optical path. One is the frequency doubling, the another is the spectra broadening. They will have impacts on the remnant side modes and may induce unwanted shift to the RV measurements.
3). The errors from detector effects such as quantization uncertainty(Zhao
et al., 2014) and the influence of the CCD stitching effects can be studied in details by scanning all the pixels with the LFC lines.
This paper is structured as follows: In Section 2 we explain the instrumental setup of the LFC and describe the data reduction methods. In Section 3 we demonstrate the new model of line profile, estimate the precision derived from various models and analyze the parameters. In Section 4 we discuss our results, and we present the conclusions and summaries in Section 5.
2 Instrumental setup and Data reduction
Our astro-comb calibration system is based on a commercial mode-locked Yb-fiber Menlosystem laser frequency comb(Wilken
et al., 2012) with the mode repetition rate of 250MHz which is locked to a rubidium atomic clock. Three Fabry-Perot cavities are employed as filters to suppress the unwanted intermediate modes and increase the line spacing to 25GHz which can be well resolved by the HRS spectrograph at R$\sim$50,000. Then we use a high power amplifier to reduce the affects due to low pulse energies. After the amplification, a second harmonic generator(SHG) is adopted to double the frequency of the light and converts the center of wavelength coverage to the optical spectral region.
We then use photonic crystal fibers(PCFs) to broaden the spectrum and obtain sufficient wavelength coverage.
The LFC signal is then coupled to a multimode fiber.
A fiber scrambler(Ye
et al., 2016) is then implemented to increase the occupancy of the spatial modes of the light from the multimode fiber. By averaging a large number of modes, the scrambler can reduce the sensitivity of the light injection. Finally, the light from LFC is coupled to the HRS calibration fiber and illuminates the CCDs.
We installed the laser frequency comb in HRS spectrograph on 2.16m telescope at Xinglong observation in early 2016 with the efforts of the teams from Nanjing Institute of Astronomical Optics & Technology (NIAOT), National Astronomical Observatories of China(NAOC) and Menlosystem. Several measurement campaigns has been carried out during the past two years. We highlight some typical acquisitions obtained from the HRS-LFC system and report the data analysis. In January 2017, we measured the performance of the wavelength calibration for this system. During this campaign, we obtained more than 60 consecutive exposures in one observing night with the light of LFC in the reference fiber(diameter=2.4") by using the typical exposure time of 20s and read-out time for the CCD of 40s.
The CCD is a E2V 4k$\times$4k 12$\mu$ scientific chip working at -106${}^{\circ}$C cooling with Liquid nitrogen. The tempreature variation in short-term is $\pm$0.05${}^{\circ}$C and $\pm$0.34${}^{\circ}$C in a week(Fan
et al., 2016).
A python and iraf based pipeline is used to reduce the data from the 2D raw fits to the flat-fielded, background-subtracted and extracted 1D spectrum. The default wavelength frame is measured by a ThAr lamp. The wavelength of comb line $i$ can also be computed independently by the setup of the LFC with the equation $\lambda_{i}=c/f_{i}=c/(f_{0}+n\cdot f_{rep})$ for giving $f_{0}$=9.52GHz, $f_{rep}$=25GHz and set the range of $n$ in [16000, 25000].
In each acquisition obtained in this campaign, the wavelength coverage of LFC is about 2300 Å(from 5085Å to 7382Å). There are 35 echelle orders(25th to 60th) can be used with high enough signal to noise ratio(SNR). The averaged SNR in the whole wavelength coverage is 234.9 with the two ends lower than 50. For each echelle order $j$ we can estimate the wavelength calibration precision $\sigma_{j}$ by calculating the rms(root mean square) around the wavelength solution curve.
We find that when j<25 and j>60, we have $\sigma_{j}$>3$\sigma_{total}$ where $\sigma_{total}$ is the mean of $\sigma_{j}$ with 25<j<60.
Thus, with the limitation of both SNR and $\sigma_{j}$, we set the available comb line range between echelle order 25th and 60th, which is corresponding to [5085Å, 7382Å].
In October 2017, we observed the $\tau$-Ceti(HD10700) in simultaneous calibration mode with the starlight in the science fiber and LFC light in the reference fiber. A set of neutral density(ND) filters were employed to control the signal intensity of LFC in the aim of exposing appropriately with the star. For a 200s exposure and ND=3dB setup, Fig 1 shows a part of the reduced spectrum in the echelle order 43th for two channels simultaneously. The mean FWHM(Full width at half maximum) of this order is 0.125Å.
We fit each individual comb line with Gaussian model. The red profiles in the middle panel of Fig 1 represent the Gaussian fitting results. Their residuals are plotted in the bottom panel with blue dots.
The rms of residuals of Gaussian model in the wavelength range of [5888Å, 5896Å] is 1348.7 counts $e^{-}$ while for the double Gaussian model it is 376.9 counts $e^{-}$. We can see from Fig 1 that each comb line occupied about $\sim$14 pixels in this spectrum range which represents 0.288Å.
We noticed that the amplitude of each individual comb line varies along the spectral direction. In Fig 1, the maximum flux is 66112 counts $e^{-}$ and the minimum is 17579 counts $e^{-}$. Currently the reason of this variation is not fully understood(Milakovic
et al., 2017). It may come from the non-linear process during the optical pass or the affects from the fibers or the combination of them. One of our aim is to study whether this variation gives influence to the wavelength calibration precision which is discussed in the following Section 3. We also notice that the Gaussian fitting residuals of each pixel is changing with its intensity. It can be seen from the bottom panel of Fig 1. For a pixel near the peak of one comb line with the flux of 66112 counts $e^{-}$, the Gaussian fitting residuals of this pixel is 2989 counts $e^{-}$. For a pixel located near the gap between two adjacent lines, its residuals are close to zero. As can be seen from Fig 1, the double Gaussian model(gray dots) can improve this effect. In the following sections, we demonstrate the several models of comb line profile and discuss the correlation between the models and wavelength calibration precision.
3 Characterizing the line profile with various models
Fitting an appropriate model of comb lines can help develop a better understanding of the RV uncertainties induced by the line profile variation.
A widely used model of the the IP is the sum of several Gaussians(Valenti
et al., 1995), which parameterizes a central Gaussian plus several satellite Gaussians.
The main component is chosen as Gaussian because the instrumental profile at first order follows a Gaussian profile. The satellite Gaussians as components are used to generate asymmetries to the whole profile.
3.1 Optimized double Gaussian Model
Recalling that in a typical exposure acquisition, there are about 12$\sim$14 pixels for each individual comb line. In this case, we mainly adopt two Gaussians to fit the line profile to achieve a high enough degree of freedom(DOF).
We use the Levenberg-Marquardt algorithm for fitting to find local minima.
In particular, we set a constraint to the parameters of double Gaussians to avoid unwanted fitting results.
These situations occur mainly to the low SNR comb lines where the line wings may suffer from the remnants of side modes and the other noises.
When constraining the parameters of double Gaussians to an optimized range, they can properly represents the intrinsic components of line profile and give out a better goodness of fit.
Considering that $x$ is the pixel or wavelength unit along the spectrum’s main dispersion direction, and $G_{1}$ and $G_{2}$ represent the two Gaussians function: $G_{1}(x,A_{1},\mu_{1},\sigma^{2}_{1})=A_{1}exp[-(x-\mu_{1})^{2}/2\sigma_{1}^{2}]$ and $G_{2}(x,A_{2},\mu_{2},\sigma^{2}_{2})=A_{2}exp[-(x-\mu_{2})^{2}/2\sigma_{2}^{2}]$. We assume that the line profile reaches the maximum when x=$x_{m}$. This can be described as:
$$\frac{\partial G_{1}+\partial G_{2}}{\partial x}\bigg{|}_{x=x_{m}}=0$$
(3)
With this equation, we can simply derived that:
$$x_{m}=\frac{\frac{\mu_{1}G_{1}}{\sigma_{1}^{2}}+\frac{\mu_{2}G_{2}}{\sigma_{2}%
^{2}}}{\frac{G_{1}}{\sigma_{1}^{2}}+\frac{G_{2}}{\sigma_{2}^{2}}}=\frac{\sigma%
_{2}^{2}G_{1}\cdot\mu_{1}+\sigma_{1}^{2}G_{2}\cdot\mu_{2}}{\sigma_{2}^{2}G_{1}%
+\sigma_{1}^{2}G_{2}}$$
(4)
In this way, if we define a parameter $W$ as the weights:
$$W=\frac{\sigma_{1}^{2}G_{2}}{\sigma_{2}^{2}G_{1}+\sigma_{1}^{2}G_{2}}$$
(5)
where 0<W<1. Then Equation 4 can be written as:
$$x_{m}=\mu_{1}-W\cdot(\mu_{1}-\mu_{2})$$
(6)
When $W=0$ we can get $x_{m}=\mu_{1}$ and when $W=1$ we have $x_{m}=\mu_{2}$. Equation 6 indicates that the maximum position(mode) lies in the range of [$\mu_{1}$, $\mu_{2}$]. The distance between $\mu_{1}$ and $\mu_{2}$(marked as $\left|\mu_{1}-\mu_{2}\right|$) can be used to set constraints to the two Gaussians components. Considering a free $\left|\mu_{1}-\mu_{2}\right|$ situation, when there is a ’jump’ data point near one of the line wing due to certain noises or side mode remnants, the satellite Gaussian profile may arrange its peak around this jump area to give more weights and to find minimum chi-square for the whole profile. This could generate unwanted fitting results such as ’double-peaks’ shape which are obviously not the real intrinsic profile.
We found this phenomenon in the comb exposure acquisitions especially at low SNR level. Typically, taking the 45th echelle order for example($\sim$ 356 lines), we compared the 45th orders from the exposure acquisitions of different SNR. When the mean SNR is less than 60 the ’double-peak’ lines are 1.12%. When the mean SNR<50 the ratio grows to 1.96%.
To avoid this situation, one of the conservative estimation for the constraint is to use:
$\left|\mu_{1}-\mu_{2}\right|<FWHM$ or $\left|\mu_{1,2}-\mu\right|<\frac{1}{2}FWHM$ where $\mu_{1}$ and $\mu_{2}$ are centers of the two components Gaussians $G_{1}$ and $G_{2}$ and $\mu$ is the center of single Gaussian fit. To express the FWHM, we use the relations of $A\cdot exp[-(x-\mu)^{2}/2\sigma^{2}]=(1/2)\cdot G_{max}$ and $G_{max}=G(\mu)$ to derive that $FWHM=2\sqrt{-2\cdot ln(1/2)}\sigma\approx 2.355\sigma$. Then the constraint correlation can be written as:
$$\mu-1.177\sigma<\mu_{1,2}<\mu+1.177\sigma$$
(7)
Here $\mu$ and $\sigma$ are calculated from single Gaussian fit.
Considering a low SNR line with certain asymmetric profile, the $\mu$ and $\sigma$ from Gaussian fit may already induce errors compared to the real line center and line width. Thus, we use the first raw moment and second central moment instead of $\mu$ and $\sigma$. For the $i$th pixel with flux $x_{i}$, the first raw moment equals to center of gravity(CG here after) which is $CG=E[x_{i}]$. The second raw moment is the variance of the $x_{i}$ distribution which can be written as: $V=\sqrt{E[x_{i}-CG]^{2}}=K^{-1}\cdot\sigma$. With regards to our situation that each comb line occupied about 12$\sim$14 pixels, we computed the coefficient $K=1.107\pm 0.000874$.
In this way, the constraint conditions can be described as:
$$\left|\mu_{1,2}-CG\right|<\sqrt{2ln2}K\cdot V$$
(8)
where the first raw moment and second central moment can be calculated directly from the extracted 1D spectra without any fitting process and its errors.
For the example we mentioned above, by using this constraint to the 45th ehcelle order for double Gaussians fitting, we find no unwanted fitting situation(e.g. double-peak shape) in the whole process.
We apply this constrained double Gaussians fitting method to all the comb lines in our acquisitions(765337 lines in consecutive 63 exposures) and compared it to other fitting models.
To evaluate the goodness of the fit, one of the direct approaches is to calculate the reduced $\chi^{2}$ for each individual line. Figure 2 shows the comparison of the single Gaussian and the constrained double Gaussians(CDG hereafter) fitting. The mean reduced $\chi^{2}$ of the Gaussian model is $<\chi^{2}_{red,G}>=1.406$ with the standard deviation of 0.0611. For the CDG model, we have $\chi^{2}_{red,CDG}=1.252\pm 0.0505$. The CDG model has $\chi^{2}_{red}$ more closer to 1 in most cases. However, we should be careful about adding more parameters to increase the likelihood by considering the situation of overfitting. For a more stringent study of the model selection, we attempt to use another advanced indicator in the next subsection with the aim to analyze which model is preferred.
3.2 BIC analysis of various models
In statistics, one of the pervasively used tools in candidate model selection is the Schwarz criterion(Schwarz, 1978) or Bayesian information criterion(BIC, hereafter)(Kass &
Wasserman, 1995; Liddle, 2007) which is an asymptotic approximation applied to a form of the posterior probability in Bayesian statistics for determining candidate models.
The standard definition of BIC is as follow:
$$BIC=-2ln[L(\hat{\theta}_{k}|x)]+k\cdot ln(n)$$
(9)
where $x$ is the observed data, $n$ is the sample size, e.g. the number of data points in x, $k$ is the number of parameters in the model and $\theta_{k}$ stands for the set of all parameters. The components in the k-dimensional parametric vectors are functionally independent. The $L(\theta_{k}|x)$ represents the likelihood corresponding to the density function $p(x|\theta_{k})$. The $\hat{\theta}_{k}$ denotes the estimated parameter values obtained by maximizing the likelihood function.
For the line shape analysis, we assume that the errors of the different models are independent and identically distributed following a normal distribution.
We also suppose that the derivative of the likelihood in log scale as the boundary condition with respect to the variance is zero(Priestley, 1981). In this case, equation 9 can be derived as:
$$BIC=n\cdot ln\left[\frac{\sum_{i=1}^{n}(Resid_{i})^{2}}{n}\right]+k\cdot ln(n)$$
(10)
Here $Resid_{i}$ denotes the fitting residuals of position $i$. In this way, we can calculate the BIC value for each comb line and for different models such as n=14 and k=3 as Gaussian profile.
Comparing various models with the Bayesian information criterion can simply refer to calculate the BIC value for each model. The model with the lowest BIC value is considered the best one.
Figure 3 shows the BIC comparison for four models: Gaussian, constrained double Gaussians, Gaussian multiplied by an error function and Gaussian plus a constant. The error function is set with the aim to help generate skewness components to find the best fit especially for the comb lines at low SNR level. A constant as free parameter is added to a Gaussian for considering the contributions of the remnants of background.
For CDG model, it has the lowest BIC value’s range which is from 170.8 to 206.6. With regards to the other three models, the BIC’s range are similar which is about [187.4, 230.38]. The ’Gaussian+C’ model is relatively $\sim$7% better for the BIC’s range.
Recalling from Fig.1 bottom panel, the residuals have different distributions when the line intensity varies. We measure the standard deviation of the fitting residuals and grouped by each order as it is shown in y-axis of Fig.3.
We find that the "flux-flat" orders tends to have smaller BICs. While the orders with stronger variation of line intensity may have larger BICs. For the CDG model, this correlation follows a 2th-order polynomial:
$$STD_{Res}=1.206\times 10^{4}+143.5BIC+0.439BIC^{2}$$
(11)
When we study the target comb lines in a certain wavelength range or pixel position range, we are able to simply obtain the $STD_{Res}$ by the fitting code and then we use the equation 11 to estimate the BIC values and check the goodness of fit.
We also calculate the $\Delta BIC$ for the comparison, which is the difference of BIC values between a particular model and the target model. The $\Delta BIC$ can be used as an argument against the other models.
Generally, if the value of $\Delta BIC$ is between 2 and 5, we can say the evidence against the target model is positive.
If $\Delta BIC$ is between 5 and 10, it can be believed that the evidence for the best model compared to the weaker model is strong. If $\Delta BIC>10$, it means the evidence for the best model against the alternate models is very strong(Liddle, 2007).
We choose the Gaussian as the target model, and use $\Delta BIC$ to evaluate the other models compared to Gaussian.
Figure 4 depicts the $\Delta BIC$ changing as a function of the line intensity.
For each individual comb line $i$, assuming it has $\sim$14 data points, we calculate its BIC values of Gaussain model, Gaussian$\times$error function model(Ge, hereafter), Gaussian+constant model(Gc, hereafter) and of CDG model by computing the difference of them. Then they are grouped by each exposure acquisition which is displayed as the black dots in the top panel of Figure 4. During the measurements, we used neutral density filters (0.2dB to 4dB) to control the average light intensity of the exposures.
For the CDG model, we find that the $\Delta BIC$ increases when the averaged flux of lines’ peak is growing. For the exposure with lowest mean flux, it has the $\Delta BIC=14.61$, while for the acquisitions with mean flux of peak greater than $6\times 10^{4}$, the $\Delta BIC$ increased to $>15.7$. The whole trend in our measurement range follows a polynomial as:
$$\Delta BIC=14.15+4.91\times 10^{-5}F-3.7\times 10^{-10}F^{2}$$
(12)
Where $F$ denotes the flux as the averaged comb line intensity for each exposure acquisition.
This evidence means that the CDG model is significantly better for fitting comb lines in all the flux range and especially tends to have better fitting results for the stronger lines.
Equation 12 can be used to estimate the $\Delta BIC$ in various flux levels for given exposures in our following observations. For the Ge model, we find it works worse than Gaussian due to the negative $\Delta BIC$ in the range of (-2.38, -2.2). With regard to the Gc model as its $\Delta BIC$ in (4.2, 5), it shows a positive evidence against the Gaussian model.
On the other hand, we already notice that the strong lines has larger residuals(e.g. Fig. 1, bottom panel) for Gaussian profile fitting. The CDG can effectively reduce the fitting residuals for the comb lines in high flux level.
In the bottom panel of Fig.4, averagely speaking, the difference between the fitting residuals of Gaussian and CDG grows up with the increase of peak flux, which is roughly along a linear function as $\Delta Resi=-30.5+0.0366\times Flux_{p}$. This relationship is corresponding to equation 12 which indicates that the CDG model fit the line profile better especially for strong lines.
The relationship between the amplitude of comb line and the fitting residuals for various models is shown in Fig.5. We find that 35.5% comb lines concentrate in the line amplitudes range of [4.2, 4.4] in log scale. The mean of residuals for Gaussian model is 1279.2 counts $e^{-}$, comparing with the CDG model as $550.5\pm 251.1$ counts $e^{-}$. With regards to Gaussian$\times$error function model, we estimate its residuals as $Res_{Ge}=1265.4\pm 631.5$ counts $e^{-}$ which is almost the same with Gaussian fit. For the model of Gaussian+constant, we have $Res_{Gc}=992.1\pm 540.8$ counts $e^{-}$ that is about 22.4% better than Gaussian fit. From the comparison, it is evident that CDG model is currently the best one.
In the next step, we will discuss the results
when we apply the CDG model to the observed comb data.
3.3 Improving the RV precision by CDG model
In the subsections above, we introduce the CDG model and demonstrate that it has a better goodness of fit than other models for comb line profile. Thus, an important question may arises that how the CDG model can improve the wavelength calibration precision. In this section, we discuss the performance when applying the CDG model to the comb data.
To be clarified, there are several definitions of RV precision referring to Doppler method which basically stem from two approaches: iodine cell technique(Butler et al., 1996) and cross correlation technique(Lovis &
Pepe, 2007). For different time scales, they can be classified as single measurement precision and long-term radial velocity rms. To investigate the LFC as the wavelength calibrator, we adopt the method of Wilken
et al. (2012), in which the single measurement precision is calculated from the standard deviation around the wavelength solution curve for each echelle order.
$$\sigma_{i}=std\left[\frac{3\times 10^{8}\times(\lambda_{j}-\hat{\lambda}_{j})}%
{\lambda_{j}}\bigg{|}_{j=1,2...N}\right]$$
(13)
here for the $i$th order, $\lambda_{j}$ is the wavelength of the $j$th comb line and $\hat{\lambda}_{j}$ stands for its wavelength values from the wavelength calibration curve(5th order polynomial). The N is the total number of the comb lines in the $i$th order.
Figure 6 shows the wavelength dependence of the precision. We calculated the $\sigma_{i}$ for $i$ in [25, 60] and grouped by each order. For the Gaussian fit situation, there is a steep decrease trend from about 5100Åto 6600Å. This trend grows up near the red end of the coverage. With regards to the orders whose wavelength is less than 5500Å, their mean precision is 0.56$m\ s^{-1}$ and maximum is 0.62$m\ s^{-1}$. It drops to the minimum of 0.097$m\ s^{-1}$ at 6898Å. Then it increases to 4.74$\pm$1.13 $m\ s^{-1}$ at 7304Å.
For the CDG model, the inclination is relatively weak and smooth. Approximately, it follows the function as:
$$\sigma_{CDG}=0.676+0.000415\times\lambda+2.74\times 10^{-8}\times\lambda^{2}$$
(14)
As it is depicted in Fig.6, the precision of the red side is relatively better than the blue side. One of the reason is the varying line interval in different orders due to the comb’s characteristic. Recalling equation 1, when $\Delta f$ is fixed to $f_{rep}$, we will have a larger $\Delta\lambda$ at greater $\lambda$. In this way, a larger distance between neighbour comb lines could reduce the influence from its conjoint neighbour comb lines and enable a better determination of each line’s position from the fitting models. Thus, the results of wavelength calibration precision is better. However, on the blue side, the narrower interval distance makes the comb lines slightly suffer the signals from their neighbour lines or remnant of side modes between them. Besides, we also notice that there are several scatted points of Gaussian model near red end, which may be due to the low SNR in this area. The mean SNR in the order close to the red edge is 39.8.
In a word, we find the precision derived by CDG model is generally better than traditional Gaussian model in the wavelength coverage shorter than $\sim$6500Å. Near the blue end, the improvement of CDG model is about 56%.
In Fig.6, we also notice that there exists a outlying data point near the red edge of the comb wavelength coverage due to the low SNR level in the red end. If we only examine the comb lines without considering the red edge($\sim$7300Å), the performance of the two models are similar at the wavelength longer than $\sim$6500Å.
Figure 7 exhibits the precision grouped by each exposure acquisition. For the 63 consecutive exposures, the precision of CDG model is $\sigma_{CDG}=0.17\pm 0.034$ and for Gaussian model is $\sigma_{G}=0.32\pm 0.11$. During the whole test series, the improvement of CDG model is about 47% compared with Gaussian model. We also investigate the photon noise limit of all these exposure acquisitions, and compare it to the two fitting models. An optimum photon-limited precision is claimed in Murphy
et al. (2007). We compute the $(S/N)_{max}=244$ for a typical acquisition. Then we estimate the precision by given:
$$\sigma^{opt}=0.45\cdot\left(\frac{500}{244}\right)\left(\frac{1.5\times 10^{5}%
}{R}\right)^{\frac{3}{2}}=4.79\ [cm\ s^{-1}]$$
(15)
where R=50,000 for the current setup of HRS. We also compute the photon noise and quality factor for each exposure individually based on (Bouchy
et al., 2001). Considering a small enough RV change is detected from pixel $i$ with the intensity $I(i)$, the pixels with a larger gradient of flux tends to have more contributions to the RV precise, which can be expressed as:
$$\frac{\delta(i)}{c}=\frac{\Delta I(i)}{\lambda(i)\cdot\left[\frac{\partial I(i%
)}{\partial\lambda(i)}\right]}$$
(16)
where $\Delta I(i)$ is the change of intensity relative to a reference(epoch 0). By giving an optimum weight $W(i)=[\delta(i)/c]^{-2}$, one can derive the quality factor $Q=\sqrt{\Sigma W(i)}/\sqrt{\Sigma I_{0}(i)}$. Then the uncertainty of the RV change is given by:
$$\delta_{rv}=\frac{c}{Q\cdot\sqrt{N_{e^{-}}}}$$
(17)
Here $N_{e^{-}}$ is the whole number of the photoelectrons gathered in the entire spectral range. By applying this algorithm, we calculate the photon noise limit(red dots in Fig.7) and the corresponding quality factor $Q$ for all the available exposure acquisitions as $\sigma_{v}=0.088\pm 0.019m\ s^{-1}$ and $Q=63018\pm 2484$.
We pick out 10 successive and relative stable exposures with $\Delta\sigma_{v}=1.3\times 10^{-3}m\ s^{-1}$ and show the details in Table1. From Fig.7 and Table1, we can see the average precision of CDG is about 46% better than the Gaussian model but still above the photon noise limit.
4 Discussion
Using the asymmetric fitting models for line profile or calculating from the raw moment analysis, we find the comb line shapes are not perfectly symmetric. Approximately, the skewness of line shape can be seen as a function of line intensity and the line’s position on the detector. The origins of the asymmetric component is still not clear. With the clues given by the analysis methods in this paper, it may source from the asymmetric instrumental profile, the CTI effect(Toyozumi &
Ashley, 2005; Murphy et al., 2012), the imperfect light injection and the envelop-background problem of the current LFCs (Milakovic
et al., 2017), or a mixture of all the sources above which are estimated to generate about several tens of $cm\ s^{-1}$ uncertainties.
To understand the origins of line skewness can help push down the RV precision below 1 $m\ s^{-1}$ and keep stable for long-term scale.
In the next step, the CDG model is considered to be used as a tool to study the asymmetric properties of line profile by designing more experiments and observations.
There are several ongoing RV surveys that have achieved the RV precision from a few $m\ s^{-1}$ to the level less than 1 $m\ s^{-1}$, such as ELODIE(Baranne
et al., 1996), HARPS(Pepe
et al., 2002; Mayor
et al., 2003) and HARPS-N(Cosentino
et al., 2012). The up-coming instruments with the main goals to search for terrestrial planets at several tens of $cm\ s^{-1}$ precision includes ESPRESSO(Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations)(Pepe
et al., 2014), HARPS3(Thompson
et al., 2016), EXPRES(The EXtreme PREcision Spectrograph)(Jurgenson
et al., 2016) and CARMENES(Quirrenbach
et al., 2014).
For the RV precision level below 1 $m\ s^{-1}$, some instrumental effects and uncertainties must be carefully studied. With the application of LFC and the CDG model, it is possible to characterize the spectrometer drift, estimate the PSF across the instrument bandpass and investigate the detector imperfections.
In the current test, when we apply CDG model analysis into the pipeline for routine observations, the computational cost of time is about extra 10 to 20 seconds for each typical exposure which contains about 12,000 comb lines.
Moreover, the precise radial velocity measurements with LFC are significant for the science of the current and future space missions.
With the high precision RV follow-up observation with astro-comb, we can enhance the productivity of the transit missions such as TESS(Ricker
et al., 2014), CHEOPS(Cessa
et al., 2017) and PLATO(Rauer et al., 2016).
5 Conclusions and Summary
We reported the successful observation and measurement with a 25GHz LFC system on HRS spectrograph in Xinglong observatory. In the measurement campaigns, we obtained more than 60 consecutive exposure acquisitions during one observing night. More than 2000Åwavelength range(from 5085Åto 7382Å) is covered by comb lines in each exposure acquisition. With the statistic investigation about multitudes of comb lines as samples, we develop a novel model to fit the line profile.
The Double Gaussians model with a constraint can effectively give out a better goodness of fit even at low SNR level.
The constraint is given as $\left|\mu_{1,2}-\mu\right|<\sqrt{2ln2}\sigma$ where $\mu$ and $\sigma$ are from traditional Gaussian fit. To evaluate and compare the goodness of fit among various models, we introduce Bayesian information criterion to test the data. After comparison, it is obvious that the CDG model is significantly better than any other models. We apply the CDG model to the obtained comb data to testify the improvement of RV precision by CDG model. The results shows that the CDG model can give better wavelength calibration precision about 20 $cm\ s^{-1}$.
In the next step, we also consider to use the LFC and CDG model to characterize the line shape variation across the detector.
Acknowledgements
We would like to thank the referee for the helpful comments that have improved the paper. We are grateful to the team in Xinglong observatory for their work to provide the data.
We also thank the team of NIAOT(Nanjing Institute of Astronomical Optics and Technology) for the setup of LFC and technical support.
This work is supported by the National Science Foundation of China(NSFC) under the grant numbers 11703052, 11390371 and 11303042.
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Holographic Complexity from the Crofton’s Formula in Lorentzian ${\rm AdS}_{3}$
Xing Huang
[email protected]
Le Zhang
[email protected]
()
Abstract
We study the Crofton’s formula in the Lorentzian AdS${}_{3}$ and find that the area of a generic space-like two dimensional surface is given by the flux of space-like geodesics. The “complexity=volume” conjecture then implies a new holographic representation of complexity in terms of the number of geodesics. Finally, we explore the possible explanation of this result from the standpoint of information theory.
Contents
1 Introduction
2 Integral geometry in Euclidean space
2.1 Kinematic space and invariant measure
2.2 Crofton’s formula in the Euclidean ${\rm AdS}_{2}$
2.3 The generic Crofton’s formula
3 Integral geometry in the Lorentzian ${\rm AdS}_{3}$
3.1 Geodesics
3.2 The Crofton’s formula on the ${\rm AdS}_{3}$
4 Discussions
A The Crofton’s formula on the Euclidean plane
1 Introduction
As a standout of the holographic principle, the ${\rm AdS/CFT}$ correspondence [1] offers possibly the best playground for exploring quantum gravity as it indicates that the latter (likely to be string theory) in the Anti-de Sitter spacetime is equivalent to some well-defined conformal field theory on the boundary. It has long been conjectured that the bulk geometry shall emerge from the boundary entanglement as both should be universal i.e., insensitive to
the discrepancies between the CFTs that admit gravity duals. A strong supporting evidence comes from the Ryu-Takayanagi formula [2, 3], which associates the area of a minimal surface anchored on the boundary with the entanglement entropy of some subregion in the boundary CFT.
Motivated by the Ryu-Takayanagi formula, Balasubramanian, et al. [4] connected the length of a closed bulk curve on the time slice of ${\rm AdS}_{3}$ to the differential entropy, which thus provides entropic interpretation for more general geometric objects. This has later been incorporated by Czech, et al. [5] into the program of integral geometry, in which the length of a curve follows from the number of geodesics it has intersections with, which is known as the Crofton’s formula in the mathematical literature [6].
Complexity is another concept whose dual may help to uncover the information theoretic origin of gravitational physics. It is defined as the number of quantum gates (unitary operators) to produce a given state from a reference state. We can imagine the system to be a collection of qubits each located on a site of a spatial lattice. We would like the quantum gates to act only on a small number of neighboring qubits for our interests. Such restriction of locality is physically reasonable (as interactions are generally local particularly in the field theories we consider) and it is crucial to the definition of complexity. Accordingly the reference state is usually the one with no pre-existing long range correlation. For example, we can choose a state with all qubits set into the state $|0\rangle$. The quantum gates form a quantum circuit i.e., a unitary operator and hence it is natural to assign the complexity to the circuit itself. Such operator/circuit complexity can be defined in a geometric way pioneered by Nielsen and collaborators [7, 8, 9] (see also [10] for recent developments), as the length of a geodesic in the space of all the unitary operators on the Hilbert space, measured by the cost function. Complexity provides ordering to the states in the Hilbert space. In this sense, it is argued that complexity can be understood as an analog of entropy (see e.g. [11]). Naturally complexity can be also used as a probe of the chaotic behavior (see e.g. [12]).
It is difficult to define complexity in a field theory. We will instead work with those theories with holographic duals and focus on the holographic dual of complexity. There are two proposals on such duality that could turn out to be equivalent. Complexity may either be dual to the volume of the spatial slice (so-called complexity=volume, a.k.a. CV [13]) or the gravity action inside the Wheeler-DeWitt patch (complexity=action, a.k.a. CA [14, 15]).
On the boundary CFT side, complexity can be defined using the Liouville action [16, 17, 18, 19]. Recently the correspondence between a quantum circuit and a co-dimension one surface was proposed in [19]. More precisely, the quantum circuit is realized as a path integral on the co-dimension one bulk surface $M[{\Sigma}(0)]$ whose boundary $\partial M={\Sigma}(0)$ is the constant time surface (taken to be $t=0$) on the boundary. The evolution of the Euclidean time gives a one parameter family of quantum states, which correspond (in the sense of surface/state correspondence [20]) to the codimension-two surfaces ${\Sigma}(t)$ on $M[{\Sigma}(0)]$. This process is essentially the renormalization group flow and $t$ serves as the energy scale. Different surfaces $M[{\Sigma}(0)]$ lead to different induced metrics in the path integral, which can be written in the canonical form of $e^{2\phi}\eta_{\mu\nu}$. Due to conformal symmetry, all these path integrals are supposed to provide the same state with different normalization factors $e^{C(M[{\Sigma}(0)])}$ given by the Liouville action $C(M[{\Sigma}(0)])=C_{L}(e^{2\phi}\eta_{\mu\nu})$ (the so-called PI complexity), which can be taken as an alternate definition of complexity (see [21, 22] for the equivalence with the circuit complexity in Nielsen’s approach). 111In fact it might be possible to go one-step further to think that the bulk region enclosed by the surface is determined entirely by the path integral, which seems quite natural from the standpoint of surface/state correspondence even though the precise dictionary is unclear. The CV conjecture then implies the areas of these surfaces are equal to the values of the cost function for the circuits, with the complexity being the minimum given by the optimized one 222It should be noted that the area does not necessarily agree with PI complexity. In our opinion it is perfectly acceptable to have different definitions for complexity as long as they can be good enough approximation of each other. For example, the number of gates might be only polynomial in the circuit complexity defined geometrically [8]. However, such generalization of CV does not seem to work in Lorentzian AdS as the extremal surface used to define complexity is the maximal one.. Both the cost function and the path integral complexity can be defined for a generic circuit and we will loosely refer to them as the cost of a circuit.
It is known [6, 23] that the area of a codimension-one surface on a time slice or more generally in any Euclidean AdS-like space can be computed using the flux of geodesics. Although the same Crofton’s formula is expected to hold in Lorentzian space, the precise form suitable for practical
computation is less clear. We study this problem in the Lorentzian AdS${}_{3}$ and find the area of generic space-like two dimensional surface can be reproduced by the flux of space-like geodesics, which are associated with the entanglement entropies of intervals in the boundary CFT. In our opinions this may be the first step towards understanding the CV conjecture from the perspective of information theory. We analyze this result using some toy model and find it reasonable to interpret the geodesic number contributing to the complexity as counting the entanglement of a state constructed from the path integral with two boundaries. More importantly we realize that these geodesics are circuit-independent and hence contribute as lower bound on the cost of all the circuits.
In section 2, we review some basic facts in the integral geometry, particularly the Crofton’s formula in the Euclidean space. In section 3 we figure out the precise form of the Crofton’s formula in the Lorentzian AdS${}_{3}$ and show that the area of a generic 2d space-like surface is given by the number of space-like geodesics it intersects. Finally in section 4, we discuss the possible reason why the complexity can be expressed in terms of the geodesic number.
2 Integral geometry in Euclidean space
Integral geometry is not a new subject. In fact most of the conclusions we are going to use are probably well known to mathematicians [6]. It was not until very recently that [5] they were introduced to the community of AdS/CFT and provided some interesting new insights. Here we will briefly touch on various useful results for later convenience.
2.1 Kinematic space and invariant measure
In integral geometry, geometric objects are expressed in terms of the
integrals of some probe objects, the collection of which forms the so-called kinematic space. In the remainder of the paper, we mostly focus on the kinematic
space of the geodesics in which every point represents a single geodesic. In a symmetric space like ${\rm AdS}_{3}$, the
geometry of the kinematic space can be determined entirely on symmetry ground
(as it can be expressed in terms of the coset of symmetry groups).
To see this explicitly, we can pick two points $x_{1}$ and $x_{2}$ on the boundary to denote a geodesic
and express the metric as follows
$${\rm d}s^{2}=f_{\mu\nu}(x_{1},x_{2}){\rm d}x_{1}^{\mu}{\rm d}x_{2}^{\nu}\quad(%
\mu,\nu=0,1)\,.$$
(1)
The precise form of $f_{\mu\nu}$ can be obtained using
the following trick in [24]: Its transformation under the conformal group is the same as the two-point function of two spin-$1$ currents of conformal dimension $1$. The requirement of conformal invariance then fixes its
form to be
$$f_{\mu\nu}(x_{1},x_{2})=\frac{4I_{\mu\nu}(x_{1}-x_{2})}{|x_{1}-x_{2}|^{2}},$$
(2)
where $4$ is an normalization constant and $I_{\mu\nu}(x)=\eta_{\mu\nu}-2x_{\mu}x_{\nu}/x^{2}$.
For simplicity, we will work in the Poincare patch (i.e., $x^{0,1}=t,x$) with metric
$${\rm d}s^{2}=\frac{{\rm d}u^{2}+{\rm d}z{\rm d}\bar{z}}{u^{2}}=\frac{{\rm d}u^%
{2}+{\rm d}x^{2}-{\rm d}t^{2}}{u^{2}},$$
(3)
where we introduce the null coordinates
$$z=x+t\qquad\bar{z}=x-t\,.$$
(4)
It is not different to check using eq.(2) and the precise form
of $I_{\mu\nu}$ that the density for geodesics space becomes
$${\rm d}s^{2}=\frac{2{\rm d}z_{1}{\rm d}z_{2}}{(z_{1}-z_{2})^{2}}+\frac{2{\rm d%
}\bar{z}_{1}{\rm d}\bar{z}_{2}}{(\bar{z}_{1}-\bar{z}_{2})^{2}}.$$
(5)
The same result can be deduced from the second derivatives of the geodesic length as $f_{\mu\nu}(x_{1},x_{2})=\partial_{\mu}\partial_{\nu}L(x_{1},x_{2})$, the latter of which in the null coordinates reads (${\epsilon}$ being the cutoff)
$$L(z_{1},z_{2})=\log\left[\frac{(z_{1}-z_{2})(\bar{z}_{1}-\bar{z}_{2})}{{%
\epsilon}}\right]\,.$$
(6)
The Euclidean ${\rm AdS}_{2}$ or rather $\mathbb{H}^{2}$ can be regarded as a constant time slice of the ${\rm AdS}_{3}$ like the one specified by $z_{i}=\bar{z}_{i}$, and we get the kinematic space on the $\mathbb{H}^{2}$ with metric
$${\rm d}s^{2}=\frac{4{\rm d}z_{1}{\rm d}z_{2}}{(z_{1}-z_{2})^{2}}\,.$$
(7)
2.2 Crofton’s formula in the Euclidean ${\rm AdS}_{2}$
In two-dimensional space, the Crofton’s formula says that the length of a curve is given by the number of geodesics it meets. As explained above, this number
follows from a volume integral in the kinematic space. We leave the derivation in Euclidean plane in the appendix A. The more interesting case to us is Euclidean
AdS${}_{2}$ (E${\rm AdS}_{2}$), where the formula takes the following form
$$L_{\gamma}={\frac{1}{2}}\int_{M_{1}\cap L_{1}\neq 0}\frac{2\sigma_{0}(M_{1}%
\cap L_{1})}{(z_{1}-z_{2})^{2}}{\rm d}z_{2}\wedge{\rm d}z_{2}\,,$$
(8)
where the integration is over all geodesics $L_{1}$ with nonvanishing intersection
number $\sigma_{0}(M_{1}\cap L_{1})$ with the curve ${\gamma}$. As we can see, the denominator comes from the measure in eq.(7). In general the measure
is given by the second derivatives of the geodesic length and the Crofton’s formula goes
like
$$L_{\gamma}=\frac{1}{2}\int_{M_{1}\cap L_{1}\neq 0}\sigma_{0}(M_{1}\cap L_{1})%
\frac{\partial^{2}L(z_{1},z_{2})}{\partial z_{1}\partial z_{2}}{\rm d}z_{1}%
\wedge{\rm d}z_{2}.$$
(9)
Moreover the length of a geodesic is related to the entanglement entropy via the RT formula:
$$S(z_{1},z_{2})=\frac{L(z_{1},z_{2})}{4G}$$
(10)
where $S(z_{1},z_{2})$ is the entanglement entropy of an interval $(z_{1},z_{2})$ on the boundary. Putting eq.(10) into eq.(9), we get
$$\frac{L_{\gamma}}{4G}=\frac{1}{2}\int_{M_{1}\cap L_{1}\neq 0}\sigma_{0}(M_{1}%
\cap L_{1})\frac{\partial^{2}S(z_{1},z_{2})}{\partial z_{1}\partial z_{2}}{\rm
d%
}z_{1}\wedge{\rm d}z_{2}.$$
(11)
So the length $L_{\gamma}$ as a bulk geometric quantity is connected with the entanglement entropy $S(z_{1},z_{2})$. In fact, given the entanglement entropies of all intervals, one can reconstruct the geometry in the kinematic space and hence the geometry in the bulk. This is a perfect example of notion of “entanglement=geometry”.
2.3 The generic Crofton’s formula
The generic Crofton’s formula first proposed in [25] (see also [6, 5, 23, 26]) establishes the relationship between $q$-dimensional target object $M_{q}$ and the sets of $r$-planes (geodesically complete submanifolds) for any constant curvature space. In a $d$ dimensional
Euclidean space, it takes the following form
$$\int_{M_{q}\cap L_{r}\neq 0}\sigma_{q+r-d}(M_{q}\cap L_{r})\epsilon_{{\cal K}}%
=\frac{O_{d}...O_{d-r}O_{q+r-d}}{O_{r}...O_{1}O_{0}O_{q}}\sigma_{q}(M_{q})\,.$$
(12)
We note that the $r$-planes $L_{r}$ are unoriented 333This convention is the same as [23, 6], but different from [5]. The difference leads to a factor $2$ in the Crofton’s formula.. The volume element $\epsilon_{{\cal K}}$ of the kinematic space ${\cal K}$ measures the number density of the $r$-planes. The symbols $\sigma_{q}(M_{q})$ and $\sigma_{q+r-d}(M_{q}\cap L_{r})$ denote the volumes of $n$ and $q+r-d$ dimensional objects, the latter of
which is the cross section between $M_{q}$ and $L_{r}$. The numeric factors $O_{k}$
are the area of $k$ dimensional unit-sphere,
$$O_{k}=\frac{2\pi^{\frac{k+1}{2}}}{\Gamma(\frac{k+1}{2})}.$$
In this paper, we only consider the kinematic space of geodesics (that is $r=1$) and in this case eq.(12) reduces to
$$\int_{M_{d-1}\cap L_{1}\neq 0}N(M_{d-1}\cap L_{1})\epsilon_{{\cal K}}=\frac{O_%
{d}}{O_{1}}\sigma_{d-1}(M_{d-1})\,.$$
(13)
where $N(M_{d-1}\cap L_{1})\equiv\sigma_{0}(M_{d-1}\cap L_{1})$ is the number of intersection points $M_{d-1}\cap L_{1}$. There are two special cases, $d=2$ and $d=3$, respectively,
$$\begin{split}\displaystyle\left\{\begin{array}[]{ll}\sigma_{1}(M_{1})=\frac{1}%
{2}\int_{M_{1}\cap L_{1}\neq 0}N(M_{1}\cap L_{1})\epsilon_{{\cal K}}&\hbox{(d=%
2,r=1,q=1),}\\
\sigma_{2}(M_{2})=\frac{1}{\pi}\int_{M_{2}\cap L_{1}\neq 0}N(M_{2}\cap L_{1})%
\epsilon_{{\cal K}}&\hbox{(d=3,r=1,q=2)}\,.\end{array}\right.\end{split}$$
(14)
One merit of this choice 444Such a choice also extend the Crofton’s formula to general Riemannian surface but this is irrelevant in
the current context. is that the measure is always given by the
second derivative of the lengths of geodesics even in the absence of maximal
symmetry,
$${\epsilon}_{\cal K}=\det\left[\frac{\partial^{2}L(\vec{x}_{1},\vec{x}_{2})}{%
\partial\vec{x}_{1}\partial\vec{x}_{2}}\right]\prod_{i=1}^{d-1}{\rm d}x^{i}_{2%
}\wedge{\rm d}x^{i}_{1}\,,$$
(15)
where we still use $L$ to denote the length of a geodesic (even though it no longer has any connection with entanglement entropy) and a geodesic is parameterized
using its coordinates $(\vec{x}_{1},\vec{x}_{2})$ ($x_{1,2}^{i},\;i=1,\dots d-1$) of the end points.
3 Integral geometry in the Lorentzian ${\rm AdS}_{3}$
In this section, we will study the Crofton’s formula on Lorentzian ${\rm AdS}_{3}$. We will stick with the geodesics as the probe (i.e., $r=1$) but now they are
no longer restricted to a time slice. One subtlety about the Lorentzian space is that there are three different types of geodesics (time-like, space-like and null). Moreover, the time-like geodesics never hit the boundary twice and they usually have no known information theoretic meaning in the boundary theory (neither are the null geodesics). Fortunately, it turns out that space-like geodesics are enough to see the space-like
2-surface and we have the following Crofton’s formula similar to eq.(14)
$$\sigma_{2}(M_{2})={\kappa}\int_{M^{2}\cap LS_{1}\neq 0}\sigma_{0}(M^{2}\cap LS%
_{1})\epsilon_{{\cal K}},$$
(16)
where $LS_{1}$ denote space-like geodesics and $M_{2}$ is a space-like 2-surface
and ${\kappa}$ is a numeric factor to be determined later. In the remainder of this section, we will prove this formula.
3.1 Geodesics
To prove eq.(16) it is necessary to find all geodesics passing through a given surface $M_{2}$. In Poincare coordinate (3), the parametrization of a geodesic from one boundary point $(z_{1},\bar{z}_{1},0)$ to another $(z_{2},\bar{z}_{2},0)$ is [27]
$$z(\lambda)=\frac{z_{1}+z_{2}}{2}+\frac{z_{1}-z_{2}}{2}\tanh{\lambda},$$
(17)
$$\bar{z}(\lambda)=\frac{\bar{z}_{1}+\bar{z}_{2}}{2}+\frac{\bar{z}_{1}-\bar{z}_{%
2}}{2}\tanh{\lambda},$$
(18)
$$u(\lambda)=\frac{\sqrt{(z_{1}-z_{2})(\bar{z}_{1}-\bar{z}_{2})}}{2\cosh{\lambda%
}},$$
(19)
where $(z,\bar{z},u)$ denote a point along the geodesic and $\lambda$ is a parameter ranging from negative infinity to positive infinity.
A geodesic can provide a nonzero contribution to the integral when it hits the target object $M_{2}$, which can be parameterized by the function $\bar{z}=\bar{z}(z,u)$. A bulk point $(z,\bar{z},u)$ on $M_{2}$ and a boundary point $(z_{1},\bar{z}_{1},0)$ determine the other
$$\begin{split}&\displaystyle z_{2}=z+\frac{u^{2}}{\bar{z}-\bar{z}_{1}},\\
&\displaystyle\bar{z}_{2}=\bar{z}+\frac{u^{2}}{z-z_{1}}.\end{split}$$
(20)
So we can instead use $(z,u,z_{1},\bar{z}_{1})$ to denote the geodesics, all of
which are space-like and therefore, we have
$$\begin{split}&\displaystyle u^{2}+(z-z_{1})(\bar{z}-\bar{z_{1}})>0,\\
\displaystyle\textrm{or equivalently}&\displaystyle u^{2}+(z-x_{1}-t_{1})(\bar%
{z}-x_{1}+t_{1})>0,\end{split}$$
(21)
Now we consider the target surface. Expressing the derivative of $\bar{z}$ with respect to $z$ and $u$ as
$$A=\frac{\partial\bar{z}}{\partial z},\quad B=\frac{\partial\bar{z}}{\partial u%
}\,,$$
the space-like constraint of the surface requires that the normal dual vector $(A,-1,B)$ is time-like, that is
$$-4A+B^{2}<0.$$
3.2 The Crofton’s formula on the ${\rm AdS}_{3}$
We have already calculated the measure of kinematic space of Lorentzian ${\rm AdS}_{3}$ in the sec 2.1, which gives the following integral from eq.(16),
$$\tilde{\sigma}_{2}(M_{2})={\kappa}\int{\rm d}z_{2}\int{\rm d}\bar{z}_{2}\int{%
\rm d}z_{1}\int{\rm d}\bar{z}_{1}\;\frac{\sigma_{0}(LS_{1}\cap M_{2})}{(z_{1}-%
z_{2})^{2}(\bar{z}_{1}-\bar{z}_{2})^{2}}.$$
(22)
Here we use the notation $\tilde{\sigma}_{2}(M_{2})$ to denote the integral and eventually we will see that it is equal to the area of $M_{2}$. Under coordinate transformation (20), the right hand side becomes
$$\begin{split}\displaystyle\tilde{\sigma}_{2}(M_{2})={\kappa}\int{\rm d}z\int{%
\rm d}u\int{\rm d}z_{1}\int{\rm d}\bar{z}_{1}\,\frac{1}{2}\left|\frac{B[u^{2}-%
(z-z_{1})(\bar{z}-\bar{z}_{1})]+2u[A(z-z_{1})-(\bar{z}-\bar{z}_{1})]}{[u^{2}+(%
z-z_{1})(\bar{z}-\bar{z_{1}})]^{3}}\right|\end{split}$$
(23)
The range of the parameters $z,u$ is determined by the surface $M_{2}$. Given $z$ and $u$, $(z_{1},\bar{z}_{1},0)$ take all points satisfying eq.(21). It is noteworthy that we compute all the geodesics twice in the case, therefore the eq.(23) contains a factor $\frac{1}{2}$. The expression in eq.(23) only depends on the relative position of the bulk and boundary points, and therefore, we introduce the new coordinates $x,t$
$$\begin{split}&\displaystyle\hat{z}=z_{1}-z=x^{\prime}+t^{\prime}\\
&\displaystyle\check{z}=\bar{z}_{1}-\bar{z}=x^{\prime}-t^{\prime}\end{split}$$
(24)
which satisfies
$$u^{2}+x^{\prime 2}-t^{\prime 2}>0.$$
After a boost transformation in $x^{\prime},t^{\prime}$, one gets
$$\begin{split}&\displaystyle\tilde{\sigma}_{2}(M_{2})\\
\displaystyle=&\displaystyle{\kappa}\int{\rm d}z\int{\rm d}u\int{\rm d}x^{%
\prime}\int{\rm d}t^{\prime}\frac{|B(u^{2}-x^{\prime 2}+t^{\prime 2})-2ut^{%
\prime}\sqrt{4A}|}{(u^{2}+x^{\prime 2}-t^{\prime 2})^{3}}\\
\displaystyle=&\displaystyle{\kappa}\int{\rm d}z\int{\rm d}u\int{\rm d}x^{%
\prime}\int{\rm d}t^{\prime}\frac{\sqrt{4A-B^{2}}|\sinh\xi(u^{2}-x^{\prime 2}+%
t^{\prime 2})-2ut^{\prime}\cosh\xi|}{(u^{2}+x^{\prime 2}-t^{\prime 2})^{3}},%
\end{split}$$
(25)
where
$$\begin{split}\displaystyle\sinh\xi=\frac{B}{\sqrt{4A-B^{2}}}\,,\quad\cosh\xi=%
\frac{\sqrt{4A}}{\sqrt{4A-B^{2}}}.\end{split}$$
(26)
To take care of the absolute value, we have to go to the angular coordinates
$$\begin{split}&\displaystyle\sin\beta\sinh\alpha=\frac{2ut^{\prime}}{u^{2}+x^{%
\prime 2}-t^{\prime 2}},\\
&\displaystyle\sin\beta\cosh\alpha=\frac{u^{2}-x^{\prime 2}+t^{\prime 2}}{u^{2%
}+x^{\prime 2}-t^{\prime 2}},\\
&\displaystyle\cos\beta=\frac{2ux^{\prime}}{u^{2}+x^{\prime 2}-t^{\prime 2}}.%
\end{split}$$
(27)
Physically, we pick three unit vectors $\hat{x},\hat{t},\hat{u}$ (vielbeins) along $\partial_{x},\partial_{t},\partial_{u}$ and ${\alpha},{\beta}$ are the angles between the unit tangent vector $v$ of the geodesic and the vielbeins. More precisely, ${\beta}$ is the angle with $\hat{x}$ and ${\alpha}$ is the (hyperbolic) angle between $v-(v\cdot\hat{x})\hat{x}$ and $\hat{u}$. The Jacobian then reads
$$\left|\frac{\partial(x^{\prime},t^{\prime})}{\partial(\alpha,\beta)}\right|=%
\frac{u^{2}|\sin\beta|}{(1+\cosh\alpha\sin\beta)^{2}}\,,$$
(28)
and eq.(25) becomes
$$\begin{split}&\displaystyle\tilde{\sigma}_{2}(M_{2})\\
\displaystyle=&\displaystyle{\kappa}\int{\rm d}z\int{\rm d}u\int_{0}^{2\pi}{%
\rm d}\beta\int_{-\infty}^{+\infty}{\rm d}\alpha\frac{\sqrt{4A-B^{2}}}{4u^{2}}%
|\sin\beta||\sin\beta\sinh(\xi-\alpha)|\\
\displaystyle=&\displaystyle{\kappa}\int{\rm d}z\int{\rm d}u\frac{\sqrt{4A-B^{%
2}}}{4u^{2}}(\cosh\alpha|^{-\frac{1}{\chi}}_{0}+\cosh\alpha|^{+\frac{1}{\chi}}%
_{0})\\
\displaystyle=&\displaystyle{\kappa}\int{\rm d}z\int{\rm d}u\frac{\sqrt{4A-B^{%
2}}}{2u^{2}}\left(\cosh\frac{1}{\chi}-1\right),\end{split}$$
(29)
where $\chi$ is a cutoff for $\alpha$. We note that $|\sin{\beta}|{\rm d}{\beta}{\rm d}{\alpha}$ is the volume element of the solid angle and $\sin{\beta}\sinh(\xi-\alpha)$ is the inner product between $v$ and the normal vector $\hat{n}$ of the surface, which implies this integral should be independent of the $\hat{n}$. Practically the parameter $\xi$ drops out after a shift in ${\alpha}$ (which is equivalent to choosing new vielbeins with the normal vector as $\hat{t}$).
As a quick consistent check, we can perform the same integral in the Euclidean space. With $\hat{n}$ being one the of axes, the integral (${\theta}$ being the angle with $\hat{n}$)
$$\frac{1}{2}\int|\cos{\theta}|{\rm d}{\Omega}_{d-1}=\int_{0}^{\frac{\pi}{2}}%
\cos{\theta}\sin^{d-2}{\theta}{\rm d}{\theta}{\rm d}{\Omega}_{d-2}=\frac{O_{d}%
}{O_{1}}\,,$$
(30)
gives precisely the numeric factor on the right hand side of eq.(13).
We can now compare the final result (29) with the area of $M_{2}$. From the induced line element
$${\rm d}s^{2}=\frac{A{\rm d}z^{2}+B{\rm d}z{\rm d}u+{\rm d}u^{2}}{u^{2}},$$
(31)
one may get
$$\sigma_{2}(M_{2})=\int{\rm d}z\int{\rm d}u\frac{\sqrt{4A-B^{2}}}{2u^{2}}\,,$$
(32)
which agrees with (29) up to an infinite factor, which is canceled by ${\kappa}$
$${\kappa}^{-1}={\frac{1}{2}}\int_{0}^{2\pi}{\rm d}\beta\int_{-\infty}^{+\infty}%
{\rm d}\alpha\sin^{2}\beta|\sinh(\alpha)|=\cosh\frac{1}{\chi}-1\,.$$
(33)
As a result, with the CV assumption complexity (whether it is that of a pure state or the reduced density matrix of a subregion [28]) can be expressed in terms of the number of geodesics.
4 Discussions
We examined the precise form of the Crofton’s formula in the Lorentzian AdS${}_{3}$ and showed that the area of a space-like two dimensional surface is given by the flux of space-like geodesics. Based on the validity of the Crofton’s formula in general Euclidean AdS, we expect the same conclusion to hold for space-like codimension-one surfaces in higher dimensional Lorentzian asymptotically AdS spaces. In AdS${}_{3}$, the geodesics have entropic interpretation and hence it is tempting to think that this conclusion may provide an information theoretic explanation of the CV conjecture. We would like to share some of our observations in that regard, leaving the more complete analysis to future study. For simplicity, we only consider the complexity of a pure state.
It was proposed in [23] that one can heuristically associate every geodesic with a Bell pair located at the two end points on the boundary. By no means this naive picture captures all the physics as the entanglement structure is not entirely bipartite. It does however offer a very nice interpretation of the entanglement entropy (of a single interval) as counting the number of Bell pairs crossing the entangling surface. Moreover, the length of a convex bulk curve (i.e., differential entropy) can also be understood as the amount of long-range entanglement in this framework.
We find this picture also very illuminating in the current context and hence decide to stay with it in subsequent discussions despite its apparent flaw. To avoid the issue of Bell pairs, one can simply take the geodesic density as a type of measure of the two-point entanglement. Based on this picture, it was pointed out [23] that under renormalization group flow, the short-range entanglement is removed while the long-range entanglement is reshuffled to shorter scales. These two operations are the “geodesic” versions of disentangler and isometry in MERA. Complexity counts the total number of these two operations. From the perspective of one geodesic, it contributes one removing (disentangler) operation or one reshuffling (isometry) operation for each step of RG, with the total number proportional to the length of the geodesic (see Fig.1). Consequently, the complexity is given by counting the total number of geodesics weighted by the length of each, which is precisely the area of the codimension-one bulk surface according to the Crofton’s formula applied to the constant time slice $\mathbb{H}^{2}$ alone [29].
The Crofton’s formula in AdS${}_{3}$ can reproduce not only the complexity (area of the corresponding optimized surface) of a state but also the cost of a non-optimized circuit, which may help to understand the CV conjecture from the viewpoint of information theory. It is however very unfortunate that henceforth we have to restrict ourselves to the Euclidean AdS. It is unlikely that the area of a generic codimension-one surface gives the cost of the circuit in the Lorentzian case. Obviously that is not the case for a time-like or null-like surface. Moreover, in the Lorentzian case it is usually the maximal surface that corresponds to the optimized circuit and gives the complexity. The infinite factor (33) between the flux of geodesics and the area makes it difficult to connect the former to any information theoretic interpretation. Nevertheless, we still hope the subsequent discussions in the Euclidean case may shed some lights on the Lorentzian problem that we eventually have to tackle.
In the Euclidean case, the correspondence between circuit and co-dimension one surface, combined with the CV conjecture implies that the cost of a circuit is measured by the area of the surface, which in turn follows from the flux of geodesics via Crofton’s formula. It is not clear to us why the number density of geodesic actually accounts for the cost. The good news is that the former does follow from entanglement entropy associated with the quantum circuit, which was computed in [30]. The conformal factors $e^{2\phi}$ at the end points provide corrections $c/6\,\phi$ to the entanglement entropy of a single interval
$$S(x_{1},x_{2})=\frac{c}{3}\log|x_{1}-x_{2}|+\frac{c}{6}\left[\phi(x_{1})+\phi(%
x_{2})\right]\,,$$
(34)
where $x_{1},x_{2}$ are the coordinates of the end points. Let us consider a simple example of the entanglement entropy at $t=\mu$, i.e., that of the excited state corresponding to the bulk curve on the $t=0$ time slice specified by $u=\mu$. The change in the conformal factor from $e^{2\phi(t={\epsilon},x)}=\frac{1}{{\epsilon}^{2}}$ to $e^{2\phi(t=\mu,x)}=\frac{1}{\mu^{2}}$ implies that the entanglement entropy is given by $\frac{c}{3}\log\left(\ell/\mu\right)$ for an interval of length $\ell$, which agrees with the length of a geodesic on the new cutoff surface. We would like to remind the reader that the number density is obtained from the length on geodesics ending on new surface $M[{\Sigma}(0)]$ (see e.g. Fig.2(a)), which is guaranteed by the nontrivial fact that the measure in the kinematic space of geodesics always follows from eq.(15) even in a general space without any symmetry. The same conclusion does not necessarily hold for the probes of higher dimensions.
It is very clear that the flux of geodesics depends on the circuit/surface. Graphically, we know that the optimized surface (for vacuum state at $t=0$) receives no contribution from geodesics with both end points in the $t<0$ region. Instead, every circuit receives contribution from the geodesics connecting the $t>0$ and $t<0$ regions (see Fig.2(a)). Such a circuit independent contribution serves as the lower bound of the cost, which is saturated by the optimized circuit (corresponding to the $t=0$ slice in the bulk, henceforth $M_{0}$).
We would like to explore the physical meaning of such a contribution. In [19], it is shown that the relevant geodesics come from the entanglement entropy between subsystem $AA^{\prime}$ and $BB^{\prime}$, with $A,B$ being subsystems on a slice in the $t<0$ region while $A^{\prime},B^{\prime}$ being subsystems on a slice in the $t>0$ region. The quantum state of the total system is obtained from the mapping given by the path integral with the two slices as the boundaries. More precisely, a mapping like $|i\rangle A_{ij}\langle j|$ leads to an in general entangled state $|i\rangle|j\rangle A_{ij}$ by turning bras into kets. Such a practice is common in the study of tensor network. The identity map $|i\rangle\langle i|$ becomes Bell pairs (more precisely a maximally entangled state) after the move.
For better demonstration, let us assume the system is discrete and the state takes the form of a tensor network obtained from RG flow (realized as mappings between various Hilbert spaces with different dimensions, see e.g. [31] for a review). The PI integral then becomes the mapping between two slices $t_{2}=-t_{1}>0$, which is given by $W^{\dagger}(0,t_{1})W(0,t_{1})$ with $W(0,t_{1})$ being the mapping of RG flow. It is known that such a product should be proportional to identity $W^{\dagger}(0,t_{1})W(0,t_{1})={\alpha}I$ 555$W$ is a map between Hilbert spaces at different energy scales and therefore we need ${\alpha}$ to take into account the difference in dimensions. and hence there are Bell pairs between the subsystems at $t_{1}$ ($AB$) and $t_{2}$ ($A^{\prime}B^{\prime}$) whose total number gives the entanglement entropy between the two subsystems.
The next step is to consider states constructed from two arbitrary slices $t_{1},t_{2}$ (assuming $0<t_{2}<|t_{1}|$). The unitary $W(0,t_{1})$ can be broken up as $W(0,-t_{2})W(-t_{2},t_{1})$. The mapping from $t_{1}$ to $t_{2}$ then becomes
$$W^{\dagger}(0,-t_{2})W(0,t_{1})={\alpha}I_{-t_{2}}\,W(-t_{2},t_{1})\,.$$
The reduced density matrix for slice $t_{1}$ is obtained by tracing out the system at $-t_{2}$
$$\rho_{t_{1}}\sim W^{\dagger}(-t_{2},t_{1})W(-t_{2},t_{1})\sim I_{t_{1}}\,,$$
which is equal to the identity operator again after appropriate normalization and hence the total entanglement entropy is determined by the size of the Hilbert space at the scale of $t_{1}$. The interesting part is that the specific forms of $W(0,t_{1})$’s, which lead to new quantum circuits do not affect the total number of Bell pairs i.e., the entanglement entropy. To see that we first rewrite the new circuit $W^{\prime}(0,t_{1})$ as the product of (see Fig. 2(b))
$$W^{\prime}(0,t_{1})=U({\delta}t)U^{\dagger}({\delta}t)W(0,t_{1})U({\delta}t)\,,$$
where $U({\delta}t)$ is the time translation by the amount ${\delta}t$. The mapping to consider is
$$W^{\dagger}(0,-t_{2})W^{\prime}(0,t_{1})=W(-t_{2},t_{1})U({\delta}t)\,,$$
(or $W^{\dagger}(t_{1},-t_{2})U({\delta}t)$ if $t_{2}>|t_{1}|$) and one can then use the same argument to show the invariance of the entanglement between $t_{1}$ and $t_{2}$. The slice $t_{2}$ is on the surface $M_{0}$ while the slice $t_{1}$ is on $M$. In the heuristic picture discussed above a Bell pair turns into a geodesic in the continuous limit and the entanglement is measured by the geodesics with one point on $M_{0}$ and the other on $M_{1}$. In fact since $M_{1}$ is closed and shares the boundary with $M_{0}$, every geodesic going through $M_{0}$ must also hit $M_{1}$. What we learn from the circuit point of view is that the flux of these geodesics corresponds to the circuit-independent entanglement. For comparison, we can also take a look at the circuit-dependent contribution. As we can see from Fig. 3, the extra flux follows from the geodesics between $s_{2}$ and $s_{1}$, which is the consequence of the additional piece $s_{2}$ corresponding to the circuit $U({\delta}t)$.
Despite only a hand-waving argument, it does give a reasonable picture in which the flux of geodesics going through $M_{0}$ measures the entanglement between the different Euclidean times (or energy scales in the RG sense) and provides the circuit-independent contribution to the cost. This lower bound is saturated when other circuit-dependent contributions all drop out i.e., when surface is $M_{0}$.
Acknowledgments
XH is supported by the NWU Starting Grant No.0115/338050048 and the Double First-class University Construction Project of Northwest University.
Appendix A The Crofton’s formula on the Euclidean plane
In this section we prove the Crofton’s formula on the Euclidean plane $E_{2}$
(with
the metric ${\rm d}s^{2}={\rm d}x^{2}+{\rm d}y^{2}$), which measures a smooth convex closed curve $\gamma$ by a set of geodesics and verify that its integrand is invariant under isometry.
We start with a geodesic in $E_{2}$. Shoot a ray $OR$ from the origin which is perpendicular to the given geodesic at point $H$. Let $\theta$ be the angle between the ray $OR$ and the $x$-axis, and let $p$ be the distance of the line segment $OH$ as in figure 4. The equation of geodesic is then given by
$$x\cos\theta+y\sin\theta-p=0.$$
(35)
A geodesic specified by $\theta$ and $p$ is identified with the other parameterized by $\theta+\pi$ and $-p$, as we can see in figure 4. With
the introduction of the orientation to the geodesics, the degeneracy is lifted.
For convenience, let us consider a smooth convex closed curve $\gamma$. For a given angle $\theta$, there are two straight-lines tangent to the curve $\gamma$ and we make such a convention that $p(\theta)$ is the larger of the two. In fact, $p(\theta+\pi)$ will correspond to the other one. So $p$ is a single-value function of $\theta$ with period $2\pi$. All such geodesics forming an envelope of ${\gamma}$ and their equations (35) can be expressed in terms of a single implicit function as $F(x,y,p(\theta),\theta)=0$. According to the envelope theorem, $F=0$ and $\partial_{\theta}F=0$ determine the curve
$$\begin{split}&\displaystyle x=p\cos\theta-\frac{{\rm d}p}{{\rm d}\theta}\sin%
\theta,\\
&\displaystyle y=p\sin\theta+\frac{{\rm d}p}{{\rm d}\theta}\cos\theta.\end{split}$$
(36)
The conditions for ${\gamma}$ to be smooth, convex and closed implies that $p+\frac{{\rm d}^{2}p}{{\rm d}{\theta}^{2}}>0$ and $\frac{{\rm d}p}{{\rm d}\theta}|_{\theta}=\frac{{\rm d}p}{{\rm d}\theta}|_{%
\theta+2\pi}$. Now we can compute the length of the curve $\gamma$ as
$$L_{\gamma}=\oint{\rm d}s=\int_{0}^{2\pi}{\rm d}\theta\>|p+\frac{{\rm d}^{2}p}{%
{\rm d}{\theta}^{2}}|=\int_{0}^{2\pi}{\rm d}\theta\>p\,,$$
(37)
which can be rewritten as
$$\begin{split}\displaystyle L_{\gamma}=&\displaystyle\frac{1}{2}\int_{0}^{2\pi}%
{\rm d}\theta\>p(\theta)+\frac{1}{2}\int_{0}^{2\pi}{\rm d}\theta\>p(\theta+\pi%
)\\
\displaystyle=&\displaystyle\frac{1}{2}\int_{0}^{2\pi}{\rm d}\theta\int_{-p(%
\theta+\pi)}^{p(\theta)}{\rm d}p=\frac{1}{4}\int_{0}^{2\pi}{\rm d}\theta\int_{%
-\infty}^{\infty}{\rm d}p\;\sigma_{0}(\gamma\cap L_{1})\\
\displaystyle=&\displaystyle\frac{1}{2}\int_{0}^{\pi}{\rm d}\theta\int_{-%
\infty}^{\infty}{\rm d}p\;\sigma_{0}(\gamma\cap L_{1}),\end{split}$$
(38)
where $\sigma_{0}(\gamma\cap L_{1})$ denotes the number of intersections between
${\gamma}$ and $L_{1}$ parameterized by $p,\theta$. Introducing a differential form $\epsilon_{\cal K}={\rm d}p\wedge{\rm d}\theta$, the equation (38) becomes
$$L_{\gamma}=\frac{1}{2}\int_{\gamma\cap L_{1}\neq\varnothing}\;\sigma_{0}(%
\gamma\cap L_{1})\epsilon_{\cal K}\,,$$
(39)
which is the Crofton’s formula in $\mathbb{E}_{2}$.
Let us derive the measure of the geodesics on the Euclidean plane from symmetry considerations. The measure $f(p,\theta){\rm d}p\wedge{\rm d}\theta$ shall be
invariant under the isometry transformation
$$\left(\begin{array}[]{c}x\\
y\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi&-\sin\phi\\
\sin\phi&\cos\phi\\
\end{array}\right)\left(\begin{array}[]{c}x^{\prime}\\
y^{\prime}\\
\end{array}\right)+\left(\begin{array}[]{c}a\\
b\\
\end{array}\right),$$
(40)
where $\phi$ is the rotation angle and $a,b$ describe the translation. Plugging (40) into (35), the relation between new parameters $p^{\prime},\theta^{\prime}$ and old ones $p,\theta$ is
$$\theta^{\prime}=\theta-\phi;\qquad p^{\prime}=p-a\cos\theta-b\sin\theta.$$
(41)
Symmetry then requires the new measure $f(p^{\prime},{\theta}^{\prime}){\rm d}p^{\prime}\wedge{\rm d}{\theta}^{\prime}$ to agree with the original one. It’s easy to check ${\rm d}p\wedge{\rm d}\theta={\rm d}p^{\prime}\wedge{\rm d}\theta^{\prime}$. The invariance of measure implies for any set X
$$\int_{X}\;f(p,\theta){\rm d}p\wedge{\rm d}\theta=\int_{X}\;f(p^{\prime},\theta%
^{\prime}){\rm d}p^{\prime}\wedge{\rm d}\theta^{\prime}=\int_{X}\;f(p-a\cos%
\theta-b\sin\theta,\theta-\phi){\rm d}p\wedge{\rm d}\theta\,,$$
(42)
which forces $f(p,\theta)$ to be a constant.
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An unbalanced Optimal Transport splitting scheme for general advection-reaction-diffusion problems
T.O. Gallouët, M. Laborde, L. Monsaingeon
Résumé
In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric framework.
We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations.
We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit.
Moreover, some numerical simulations are included.
1 Introduction
Since the seminal works of Jordan-Kinderlehrer-Otto [19], it is well known that certain diffusion equations can be interpreted as gradient flows in the space of probability measures, endowed with the quadratic Wasserstein distance $\mathtt{W}$.
The well-known JKO scheme (a.k.a. minimizing movement), which is a natural implicit Euler scheme for such gradient flows, naturally leads to constructive proofs of existence for weak solutions to equations or systems with mass conservation such as, for instance, Fokker-Planck equations [19], Porous Media Equations [32], aggregation equation [9], double degenerate diffusion equations [31], general degenerate parabolic equation [1] etc.
We refer to the classical textbooks of Ambrosio, Gigli and Savaré [4] and to the books of Villani [43, 44] for a detailed account of the theory and extended bibliography. Recently, this theory has been extended to study the evolution of interacting species with mass-conservation, see for examples [15, 45, 23, 20, 8].
Nevertheless in biology, for example for diffusive prey-predator models, the conservation of mass may not hold, and the classical optimal transport theory does not apply.
An unbalanced optimal transport theory was recently introduced simultaneously in [11, 12, 21, 25, 26], and the resulting Wasserstein-Fisher-Rao ($\mathtt{WFR}$) metrics (also referred to as the Hellinger-Kantorovich distance $\mathtt{HK}$) allows to compute distances between measures with variable masses while retaining a convenient Riemannian structure.
See section 2 for the definition and a short discussions on this $\mathtt{WFR}$ metric.
We also refer to [37, 16] for earlier attempts to account for mass variations within the framework of optimal transport.
The $\mathtt{WFR}$ metrics can be seen as an inf-convolution between Wasserstein/transport and Fisher-Rao/reaction processes, and is therefore extremely convenient to control both in a unified metric setting.
This allows to deal with non-conservative models of population dynamics, see e.g. [21, 22].
In [18], the first and third authors proposed a variant of the JKO scheme for $\mathtt{WFR}$-gradient flows corresponding to some particular class of reaction-diffusion PDEs: roughly speaking, the reaction and diffusion were handled separately in two separate $\mathtt{FR},\mathtt{W}$ metrics, and then patched together using a particular uncoupling of the inf-convolution, namely $\mathtt{WFR}^{2}\approx\mathtt{W}^{2}+\mathtt{FR}^{2}$ in some sense (see [18, section 3] for a thorough discussion).
However, the analysis was restricted to very particular structures for the PDE, corresponding to pure $\mathtt{WFR}$ gradient-flows.
In this work we aim at extending this splitting scheme in order to handle more general reaction-diffusion problems, not necessarily corresponding to gradient flows.
Roughly speaking, the structure of our splitting scheme is the following: the transport/diffusion part of the PDE is treated by a single Wasserstein JKO step
$$\rho^{k}\xrightarrow[\mbox{transport}]{\mathtt{W}}\rho^{k+1/2},$$
and the next Fisher-Rao JKO step
$$\rho^{k+1/2}\xrightarrow[\mbox{reaction}]{\mathtt{FR}}\rho^{k+1}$$
handles the reaction part of the evolution.
As already mentioned, the $\mathtt{WFR}$ metric will allow to suitable control both steps in a unified metric framework.
We will first state a general convergence result for scalar reaction-diffusion equations, and then illustrate on a few particular examples how the general idea can be adapted to treat e.g. prey-predator systems or very degenerate Hele-Shaw diffusion problems.
In this work we do not focus on optimal results and do not seek full generality, but rather wish to illustrate the efficiency of the general approach.
Another advantage of the splitting scheme is that is well adapted to existing Monge/Kantorovich/Wasserstein numerical solvers, and the Fisher-Rao step turns out to be a simple pointwise convex problem which can be implemented in a very simple way.
See also [10, 13] for a more direct numerical approach by entropic regularization.
Throughout the paper we will illustrate the theoretical results with a few numerical tests.
All the numerical simulations were implemented with the augmented Lagrangian ALG2-JKO scheme from [6] for the Wasserstein step, and we used a classical Newton algorithm for the Fisher-Rao step.
The paper is organized as follows.
In section 2 we recall the basic definitions and useful properties of the Wasserstein-Fisher-Rao distance $\mathtt{WFR}$.
Section 3 contains the precise description of the splitting scheme and a detailed convergence analysis for a broad class of reaction-diffusion equations.
In section 4 we present an extension to some prey-predator multicomponent systems with nonlocal interactions.
In section 5 we extend the general result from section 3 to a very degenerate tumor growth model studied in [34], corresponding to a pure $\mathtt{WFR}$ gradient flow: we show that the splitting scheme captures fine properties of the model, particularly the $\Gamma$-convergence of discrete gradient flows as the degenerate diffusion parameter of Porous Medium type $m\to\infty$.
The last section 6 contains an extension to a tumor-growth model coupled with an evolution equation for the nutrients.
2 Preliminaries
Let us first fix some notations.
Throughout the whole paper, $\Omega$ denotes a possibly unbounded convex subset of $\operatorname*{\mathbb{R}}^{d}$, $Q_{T}$ represents the product space $[0,T]\times\Omega$, for $T>0$, and we write $\mathcal{M}^{+}=\mathcal{M}^{+}(\Omega)$ for the set of nonnegative finite Radon measures on $\Omega$.
We say that a curve of measures $t\mapsto\rho_{t}\in\mathcal{C}_{w}([0,1];\mathcal{M}^{+})$ is narrowly continuous if it is continuous with respect to the narrow convergence of measures, namely for the duality with $\mathcal{C}_{b}(\Omega)$ test-functions.
Definition 2.1.
The Fisher-Rao distance between $\rho_{0},\rho_{1}\in\mathcal{M}^{+}$ is
$$\mathtt{FR}(\rho_{0},\rho_{1}):=\min_{(\rho_{t},r_{t})\in\mathcal{A}_{\mathtt{%
FR}}[\rho_{0},\rho_{1}]}\int_{0}^{1}\int_{\Omega}|r_{t}|^{2}\,d\rho_{t}(x)dt,$$
where the admissible set $A_{\mathtt{FR}}[\rho_{0},\rho_{1}]$ consists in curves $[0,1]\ni t\mapsto(\rho_{t},r_{t})\in\mathcal{M}^{+}\times\mathcal{M}$ such that $t\mapsto\rho_{t}\in\mathcal{C}_{w}([0,1];\mathcal{M}^{+})$ is narrowly continuous with endpoints $\rho_{t}(0)=\rho_{0},\rho_{t}(1)=\rho_{1}$, and
$$\partial_{t}\rho_{t}=\rho_{t}r_{t}$$
in the sense of distributions $\mathcal{D}^{\prime}((0,1)\times\Omega)$.
The Monge-Kantorovich-Wasserstein admits several equivalent definitions and formulations, and we refer e.g. to [43, 44, 4, 41] for a complete description.
For our purpose we shall only need the dynamical Benamou-Brenier formula:
Theorem 2.2 (Benamou-Brenier formula, [5, 4]).
There holds
$$\mathtt{W}^{2}(\rho_{0},\rho_{1})=\min\limits_{(\rho,\mathbf{v})\in\mathcal{A}%
_{\mathtt{W}}[\rho_{0},\rho_{1}]}\int_{0}^{1}\int_{\Omega}|\mathbf{v}_{t}|^{2}%
\mathrm{d}\rho_{t}\mathrm{d}t,$$
(2.1)
where the admissible set $\mathcal{A}_{\mathtt{W}}[\rho_{0},\rho_{1}]$ consists in curves $(0,1)\ni t\mapsto(\rho_{t},\mathbf{v}_{t})\in\mathcal{M}^{+}\times\mathcal{M}(%
\Omega;\operatorname*{\mathbb{R}}^{d})$ such that $t\mapsto\rho_{t}$ is narrowly continuous with endpoints $\rho_{t}(0)=\rho_{0}$, $\rho_{t}(1)=\rho_{1}$ and solving the continuity equation
$$\partial_{t}\rho_{t}+\operatorname*{div}(\rho_{t}\mathbf{v}_{t})=0$$
in the sense of distributions $\mathcal{D}^{\prime}((0,1)\times\Omega)$.
According to the original definition in [11] we have
Definition 2.3.
The Wasserstein-Fisher-Rao distance between $\rho_{0},\rho_{1}\in\mathcal{M}^{+}(\Omega)$ is
$$\mathtt{WFR}^{2}(\rho_{0},\rho_{1}):=\inf_{(\rho,\mathbf{v},r)\in\mathcal{A}_{%
\mathtt{WFR}}[\rho_{0},\rho_{1}]}\int_{0}^{1}\int_{\Omega}(|\mathbf{v}_{t}(x)|%
^{2}+|r_{t}|^{2})\,d\rho_{t}(x)dt,$$
(2.2)
where the admissible set $\mathcal{A}_{\mathtt{WFR}}[\rho_{0},\rho_{1}]$ is the set of curves $t\in[0,1]\mapsto(\rho_{t},v_{t},r_{t})\in\mathcal{M}^{+}\times\mathcal{M}(%
\Omega;\operatorname*{\mathbb{R}}^{d})\times\mathcal{M}$ such that $t\mapsto\rho_{t}\in\mathcal{C}_{w}([0,1],\mathcal{M}^{+})$ is narrowly continuous with endpoints $\rho_{|t=0}=\rho_{0}$, $\rho_{|t=1}=\rho_{1}$ and solves the continuity equation with source
$$\partial_{t}\rho_{t}+\operatorname*{div}(\rho_{t}v_{t})=\rho_{t}r_{t}.$$
Comparing definition 2.3 with definition 2.1 and Theorem 2.2, this dynamical formulation à la Benamou-Brenier shows that the $\mathtt{WFR}$ distance can be viewed as an inf-convolution of the Wasserstein and Fisher-Rao distances $\mathtt{W},\mathtt{FR}$.
From [11, 12, 21, 25] the infimum in (2.2) is always a minimum, and the corresponding minimizing curves $t\mapsto\rho_{t}$ are of course constant-speed geodesics $\mathtt{WFR}(\rho_{t},\rho_{s})=|t-s|\mathtt{WFR}(\rho_{0},\rho_{1})$.
Then $(\mathcal{M}^{+},\mathtt{WFR})$ is a complete metric space, and $\mathtt{WFR}$ metrizes the narrow convergences of measures (see again [11, 12, 21, 25]).
Interestingly, there are other possible formulations of the distance in terms of static unbalanced optimal transportation, primal-dual characterizations with relaxed marginals, lifting to probability measures on a cone over $\Omega$, duality with subsolutions of Hamilton-Jacobi equations, and we refer to [11, 12, 21, 26, 25] for more details.
As a first useful interplay between the distances $\mathtt{WFR},\mathtt{W},\mathtt{FR}$ we have
Proposition 2.4 ([18]).
Let $\rho_{0},\rho_{1}\in\mathcal{M}^{+}_{2}$ such that $|\rho_{0}|=|\rho_{1}|$. Then
$$\mathtt{WFR}^{2}(\rho_{0},\rho_{1})\leqslant\mathtt{W}^{2}(\rho_{0},\rho_{1}).$$
Similarly for all $\mu_{0},\mu_{1}\in\mathcal{M}^{+}$ (with possibly different masses) there holds
$$\mathtt{WFR}^{2}(\mu_{0},\mu_{1})\leqslant\mathtt{FR}^{2}(\mu_{0},\mu_{1}).$$
Finally, for all $\nu_{0},\nu_{1}\in\mathcal{M}^{+}_{2}$ such that $|\nu_{0}|=|\nu_{1}|$ and all $\nu\in\mathcal{M}^{+}$, there holds
$$\mathtt{WFR}^{2}(\nu_{0},\nu)\leqslant 2(\mathtt{W}^{2}(\nu_{0},\nu_{1})+%
\mathtt{FR}^{2}(\nu_{1},\nu)).$$
Moreover, we have the following link between the reaction and the velocity in (2.2), which was the original definition in [21]:
Proposition 2.5 ([18]).
The definition (2.3) of the $\mathtt{WFR}$ distance can be restricted to the subclass of admissible paths $(\mathbf{v}_{t},r_{t})=(\nabla u_{t},u_{t})$ for potentials $u_{t}\in H^{1}(\mathrm{d}\rho_{t})$ and continuity equations
$$\partial_{t}\rho_{t}+\operatorname*{div}(\rho_{t}\nabla u_{t})=\rho_{t}u_{t}.$$
This shows that $(\mathcal{M}^{+},\mathtt{WFR})$ can be endowed with the formal Riemannian structure constructed as follow: any two tangent vectors $\xi^{1}=\partial_{t}\rho^{1},\xi^{2}=\partial_{t}\rho^{2}$ can be uniquely identified with potentials $u^{i}$ by solving the elliptic equations
$$\xi^{i}=-\operatorname*{div}(\rho\nabla u^{i})+\rho u^{i}.$$
Then the Riemaniann tensor is naturally constructed on the $H^{1}(\mathrm{d}\rho)$ scalar product, i-e
$$g_{\rho}(\xi^{1},\xi^{2}):=\langle u^{1},u^{2}\rangle_{H^{1}(\mathrm{d}\rho)}=%
\int_{\Omega}(\nabla u^{1}\cdot\nabla u^{2}+u^{1}u^{2})\mathrm{d}\rho.$$
This is purely formal, and we refer again to [18] for discussions.
Given a functional
$$\mathcal{F}(\rho):=\int_{\Omega}F(\rho)+\int_{\Omega}\rho V+\frac{1}{2}\int_{%
\Omega}(K\ast\rho)\rho,$$
this Riemannian structure also allows to compute $\mathtt{WFR}$ gradients as
$$\operatorname{grad}_{\mathtt{WFR}}\mathcal{F}(\rho)=-\operatorname*{div}\left(%
\rho\nabla\frac{\delta\mathcal{F}}{\delta\rho}\right)+\rho\frac{\delta\mathcal%
{F}}{\delta\rho}=\operatorname{grad}_{\mathtt{W}}\mathcal{F}(\rho)+%
\operatorname{grad}_{\mathtt{FR}}\mathcal{F}(\rho),$$
where $\frac{\delta\mathcal{F}}{\delta\rho}=F^{\prime}(\rho)+V+K\ast\rho$ denotes the Euclidean first variation of $\mathcal{F}$ with respect to $\rho$.
In other words, the Riemannian tangent vector $\operatorname{grad}_{\mathtt{WFR}}\mathcal{F}(\rho)$ is represented in the previous $H^{1}(\mathrm{d}\rho)$ duality by the scalar potential $u=\frac{\delta\mathcal{F}}{\delta\rho}$.
3 An existence result for general parabolic equations
In this section, we propose to solve scalar parabolic equations of the form
$$\left\{\begin{array}[]{l}\partial_{t}\rho=\operatorname*{div}(\rho\nabla(F^{%
\prime}_{1}(\rho)+V_{1}))-\rho(F^{\prime}_{2}(\rho)+V_{2})\\
\rho|_{t=0}=\rho_{0}\\
\left.\rho\nabla(F^{\prime}_{1}(\rho)+V_{1})\right|_{\partial\Omega}\cdot\nu=0%
\end{array}\right.$$
(3.1)
in a bounded domain $\Omega\subset\mathbb{R}^{d}$ with Neumann boundary condition and suitable initial conditions.
Our goal is to extend to the case $F_{1}\neq F_{2},V_{1}\neq V_{2}$ the method initially introduced in [18] for variational $\mathtt{WFR}$-gradient flows, i-e (3.1) with $F_{1}=F_{2}$ and $V_{1}=V_{2}$.
We assume for simplicity that $F_{1}\,:\,\operatorname*{\mathbb{R}}\rightarrow\operatorname*{\mathbb{R}}$ is given by
$$\displaystyle F_{1}(z)=\left\{\begin{array}[]{ll}z\log z-z&\mbox{(linear %
diffusion)}\\
\mbox{or}\\
\frac{1}{m_{1}-1}z^{m_{1}}&\mbox{(Porous Media diffusion)}\end{array}\right.,$$
(3.2)
and $F_{2}\,:\,\operatorname*{\mathbb{R}}\rightarrow\operatorname*{\mathbb{R}}$ is given by
$$\displaystyle F_{2}(z)=\frac{1}{m_{2}-1}z^{m_{2}},\qquad\mbox{for some }m_{2}>1.$$
(3.3)
Note that we cannot take $F_{2}(z)=z\log z-z$ because the Boltzmann entropy is not well behaved (neither regular nor convex) with respect to the Fisher-Rao metric in the reaction step, see [18, 26, 25] for discussions.
In addition, we assume that
$$V_{1}\in W^{1,\infty}(\Omega)\qquad\mbox{and}\qquad V_{2}\in L^{\infty}(\Omega).$$
We denote $\mathcal{E}_{1},\mathcal{E}_{2}\,:\,\mathcal{M}^{+}\rightarrow\operatorname*{%
\mathbb{R}}$ the energy functionals
$$\mathcal{E}_{i}(\rho):=\mathcal{F}_{i}(\rho)+\mathcal{V}_{i}(\rho),$$
where
$$\mathcal{F}_{i}(\rho):=\left\{\begin{array}[]{ll}\int_{\Omega}F_{i}(\rho)&%
\text{ if }\rho\ll\mathcal{L}_{|\Omega}\\
+\infty&\text{ otherwise, }\end{array}\right.\qquad\text{and}\qquad\mathcal{V}%
_{i}(\rho):=\int_{\Omega}V_{i}\rho.$$
Although more general statements with suitable structural assumptions could certainly be proved, we do not seek full generality here and choose to restrict from the beginning to the above simple (but nontrivial) setting for the sake of exposition.
Definition 3.1.
A weak solution of (3.1) is a curve $[0,+\infty)\ni t\mapsto\rho(t,\cdot)\in L^{1}_{+}\cap L^{\infty}(\Omega)$ such that for all $T<\infty$ the pressure $P_{1}(\rho):=\rho F_{1}^{\prime}(\rho)-F_{1}(\rho)$ satisfies $\nabla P_{1}(\rho)\in L^{2}([0,T]\times\Omega)$, and
$$\int_{0}^{+\infty}\left(\int_{\Omega}(\rho\partial_{t}\phi-\nabla V_{1}\cdot%
\nabla\phi\rho-\nabla P_{1}(\rho)\cdot\nabla\phi-\rho(F_{2}^{\prime}(\rho)+V_{%
2})\phi)\,dx\right)\,dt=-\int_{\Omega}\phi(0,x)\rho_{0}(x)\,dx$$
for every $\phi\in\mathcal{C}^{\infty}_{c}([0,+\infty)\times\mathbb{R}^{d})$.
Note that the pressure $P_{1}$ is defined so that the diffusion term $\operatorname*{div}(\rho\nabla F_{1}^{\prime}(\rho))=\Delta P_{1}(\rho)$, at least for smooth solutions.
The starting point of our analysis is that (3.1) can be written, at least formally as,
$$\partial_{t}\rho=\operatorname*{div}(\rho\nabla(F^{\prime}_{1}(\rho)+V_{1}))-%
\rho(F^{\prime}_{2}(\rho)+V_{2})\quad\leftrightarrow\quad\partial_{t}\rho=-%
\operatorname{grad}_{\mathtt{W}}\mathcal{E}_{1}(\rho)-\operatorname{grad}_{%
\mathtt{FR}}\mathcal{E}_{2}(\rho).$$
Our splitting scheme is a variant of that originally introduced in [18], and can be viewed as an operator splitting method: each part of the PDE above is discretized (in time) in its own $\mathtt{W},\mathtt{FR}$ metric, and corresponds respectively to a $\mathtt{W}$/transport/diffusion step and to a $\mathtt{FR}$/reaction step.
More precisely, let $h>0$ be a small time step.
Starting from the initial datum $\rho_{h}^{0}:=\rho_{0}$, we construct two recursive sequences $(\rho_{h}^{k})_{k}$ and $(\rho_{h}^{k+1/2})_{k}$ such that
$$\displaystyle\left\{\begin{array}[]{l}\rho_{h}^{k+1/2}\in\operatorname*{argmin%
}\limits_{\rho\in\mathcal{M}^{+},|\rho|=|\rho_{h}^{k}|}\left\{\frac{1}{2h}%
\mathtt{W}^{2}(\rho,\rho_{h}^{k})+\mathcal{E}_{1}(\rho)\right\},\\
\\
\rho_{h}^{k+1}\in\operatorname*{argmin}\limits_{\rho\in\mathcal{M}^{+}}\left\{%
\frac{1}{2h}\mathtt{FR}_{2}^{2}(\rho,\rho_{h}^{k+1/2})+\mathcal{E}_{2}(\rho)%
\right\}.\end{array}\right.$$
(3.4)
With our structural assumptions on $F_{i},V_{i}$ and arguing as in [18], the direct method shows that this scheme is well-posed, i-e that each minimizing problem in (3.4) admits a unique minimizer.
We construct next two piecewise-constant interpolating curves
$$\displaystyle\left\{\begin{array}[]{l}\rho_{h}(t)=\rho_{h}^{k+1},\\
\tilde{\rho}_{h}(t)=\rho_{h}^{k+1/2},\end{array}\right.\text{ for all }t\in(kh%
,(k+1)h].$$
(3.5)
Our main results in this section is the constructive existence of weak solutions to (3.1):
Theorem 3.2.
Assume that $\rho_{0}\in L^{1}_{+}\cap L^{\infty}(\Omega)$.
Then, up to a discrete subsequence (still denoted $h\to 0$ and not relabeled here), $\rho_{h}$ and $\tilde{\rho}_{h}$ converge strongly in $L^{1}((0,T)\times\Omega)$ to a
weak solution $\rho$ of (3.1).
Note that any uniqueness for (3.1) would imply convergence of the whole (continuous) sequence $\rho_{h},\tilde{\rho}_{h}\to\rho$ as $h\to 0$, but for the sake of simplicity we shall not address this issue here.
The main technical obstacle in the proof of Theorem 3.2 is to retrieve compactness in time.
For the classical minimizing scheme of any energy $\mathcal{E}$ on any metric space $(X,d)$, suitable time compactness is usually retrieved in the form of the total-square distance estimate
$\frac{1}{2h}\sum\limits_{k\geq 0}d^{2}(x^{k},x^{k+1})\leqslant\mathcal{E}(x_{0%
})-\inf\mathcal{E}$.
This usually works because only one functional is involved, and $\mathcal{E}(x_{0})-\inf\mathcal{E}$ is obtained as a telescopic sum of one-step energy dissipations $\mathcal{E}(x^{k+1})-\mathcal{E}(x^{k})$.
Here each of our elementary step in (3.1) involves one of the $\mathtt{W},\mathtt{FR}$ metrics, and we will use the $\mathtt{WFR}$ distance to control both simultaneously: this strongly leverages the inf-convolution structure, the $\mathtt{WFR}$ distance being precisely built on a compromise between $\mathtt{W}$/transport and $\mathtt{FR}$/reaction.
On the other hand we also have two different functionals $\mathcal{E}_{1},\mathcal{E}_{2}$, and we will have to carefully estimate the dissipation of $\mathcal{E}_{1}$ during the $\mathtt{FR}$ reaction step (driven by $\mathcal{E}_{2}$) as well as the dissipation of $\mathcal{E}_{2}$ during the $\mathtt{W}$ transport/diffusion step (driven by $\mathcal{E}_{1}$).
We start by collecting one-step estimates, exploiting the optimality conditions for each elementary minimization procedure, and postpone the proof of Theorem 3.2 to the end of the section.
3.1 Optimality conditions and pointwise $L^{\infty}$ estimates
The optimality conditions for the first Wasserstein step $\rho^{k}\to\rho^{k+1/2}$ in (3.4) are by now classical, and can be written for example
$$\displaystyle\frac{-\nabla\varphi_{h}^{k+1/2}}{h}\rho_{h}^{k+1/2}=\nabla P_{1}%
(\rho_{h}^{k+1/2})+\rho_{h}^{k+1/2}\nabla V_{1}\qquad\mbox{a.e.}$$
(3.6)
Here $\varphi_{h}^{k+1/2}$ is an optimal (backward) Kantorovich potential from $\rho_{h}^{k+1/2}$ to $\rho_{h}^{k}$.
Lemma 3.3.
For all $k\geqslant 0$,
$$\|\rho_{h}^{k+1/2}\|_{L^{1}}=\|\rho_{h}^{k}\|_{L^{1}}$$
(3.7)
and for all constant $C$ such that $V_{1}\leqslant C$,
$$\rho_{h}^{k}(x)\leqslant(F_{1}^{\prime})^{-1}(C-V_{1}(x))\,\mbox{a.e.}\qquad%
\Rightarrow\qquad\rho_{h}^{k+1/2}(x)\leqslant(F_{1}^{\prime})^{-1}(C-V_{1}(x))%
\,\mbox{a.e.}$$
(3.8)
Démonstration.
The Wasserstein step is mass conservative by construction, so the first part is obvious.
The second part is a direct consequence of a generalization [36, lemma 2] of Otto’s maximum principle [32].
∎
Remark 3.4.
Note that if $\rho_{h}^{k}\leqslant M$, we may take $C=F_{1}^{\prime}(M)+\|V_{1}\|_{L^{\infty}}$ in (3.8).
Formally, this corresponds to taking $\overline{\rho}(x):=(F_{1}^{\prime})^{-1}(C-V_{1}(x))$ as a stationary Barenblatt supersolution for $\partial_{t}\rho=\operatorname*{div}(\rho\nabla(F_{1}^{\prime}(\rho)+V_{1}))$ at the continuous level.
In addition, if $V_{1}\equiv 0$ we recover Otto’s maximum principle [32] in the form $\|\rho^{k+1/2}\|_{L^{\infty}}\leqslant\|\rho^{k}\|_{L^{\infty}}$.
For the second Fisher-Rao reaction step, the optimality condition has been obtained in [18, section 4.2] in the form
$$\displaystyle\left(\sqrt{\rho_{h}^{k+1}}-\sqrt{\rho_{h}^{k+1/2}}\right)\sqrt{%
\rho_{h}^{k+1}}=-\frac{h}{2}\rho_{h}^{k+1}\left(F_{2}^{\prime}(\rho_{h}^{k+1})%
+V_{2}\right)\qquad\mbox{a.e.}$$
(3.9)
As a consequence we have
Lemma 3.5.
There is $C\equiv C(V_{2})>0$ such that for $h\leqslant h_{0}(V_{2})$ small enough we have
$$\rho_{h}^{k+1}(x)\leqslant(1+Ch)\rho_{h}^{k+1/2}(x)\qquad\mbox{a.e.},$$
(3.10)
and for all $M>0$ there is $c\equiv c(M,V_{2})$ such that if $\|\rho_{h}^{k+1/2}\|_{\infty}\leqslant M$ then
$$(1-ch)\rho_{h}^{k+1/2}(x)\leqslant\rho_{h}^{k+1}(x)\qquad\mbox{a.e.}$$
(3.11)
Note in particular that this immediately implies
$$\mathop{\rm supp}\,\rho_{h}^{k+1}=\mathop{\rm supp}\,\rho_{h}^{k+1/2},$$
(3.12)
which was to be expected since the reaction part $\partial_{t}\rho=-\rho(F_{2}^{\prime}(\rho)+V_{2})$ of the PDE (3.1) preserves strict positivity.
Démonstration.
We start with the upper bound: inside $\mathop{\rm supp}\rho_{h}^{k+1}$, (3.9) and $F_{2}^{\prime}\geqslant 0$ give
$$\displaystyle\sqrt{\rho_{h}^{k+1}(x)}-\sqrt{\rho_{h}^{k+1/2}(x)}$$
$$\displaystyle=$$
$$\displaystyle-h\sqrt{\rho_{h}^{k+1}(x)}(F_{2}^{\prime}(\rho_{h}^{k+1}(x))+V_{2%
}(x))$$
$$\displaystyle\leqslant$$
$$\displaystyle-hV_{2}(x)\sqrt{\rho_{h}^{k+1}(x)}\leqslant h\|V_{2}\|_{\infty}%
\sqrt{\rho_{h}^{k+1}(x)}$$
whence
$$\sqrt{\rho_{h}^{k+1}(x)}\leqslant\frac{1}{1-h\|V_{2}\|_{\infty}}\sqrt{\rho_{h}%
^{k+1/2}(x)}.$$
Taking squares and using
$$\frac{1}{(1-h\|V_{2}\|_{\infty})^{2}}=1+2\|V_{2}\|_{L^{\infty}}h+\mathcal{O}(h%
^{2})\leqslant 1+3\|V_{2}\|_{L^{\infty}}h$$
for small $h$ gives the desired inequality.
For the lower bound (3.11), we first observe that since $F_{2}^{\prime\prime}\geqslant 0$ and from (3.10) we have $F_{2}^{\prime}(\rho^{k+1}_{h})\leqslant F_{2}^{\prime}((1+Ch)\rho^{k+1/2}_{h})%
\leqslant F_{2}^{\prime}(2M)$ if $h$ is small enough.
Then (3.9) gives inside $\mathop{\rm supp}\rho^{k+1}$
$$\displaystyle\sqrt{\rho_{h}^{k+1}(x)}-\sqrt{\rho_{h}^{k+1/2}(x))}$$
$$\displaystyle=$$
$$\displaystyle-h\sqrt{\rho_{h}^{k+1}(x)}(F_{2}^{\prime}(\rho_{h}^{k+1}(x))+V_{2%
}(x))$$
$$\displaystyle\geqslant$$
$$\displaystyle-h(F_{2}^{\prime}(2M)+\|V_{2}\|_{\infty})\sqrt{\rho_{h}^{k+1}(x)},$$
hence
$$\rho_{h}^{k+1}(x)\geqslant\frac{1}{(1+h(F_{2}^{\prime}(2M)+\|V_{2}\|_{\infty})%
)^{2}}\rho_{h}^{k+1/2}(x)\geqslant(1-ch)\rho_{h}^{k+1/2}(x)$$
for small $h$.
∎
Combining Lemma 3.3 and Lemma 3.5, we obtain at the continuous level
Proposition 3.6.
For all $T>0$ there exist constants $M_{T},M_{T}^{\prime}$ such that for all $t\in[0,T]$,
$$\|\rho_{h}(t)\|_{L^{1}\cap L^{\infty}},\|\tilde{\rho}_{h}(t)\|_{L^{1}\cap L^{%
\infty}}\leqslant M_{T}$$
and
$$\|\rho_{h}(t)-\tilde{\rho}_{h}(t)\|_{L^{1}}\leqslant hM_{T}^{\prime}$$
uniformly in $h\geqslant 0$.
Note from the second estimate that strong $L^{1}((0,T)\times\Omega)$ convergence of $\rho_{h}$ will immediately imply convergence of $\tilde{\rho}_{h}$ to the same limit.
Démonstration.
By induction combining (3.8) and (3.10), we obtain, for all $t\in[0,T]$,
$$\|\rho_{h}(t)\|_{L^{\infty}},\|\tilde{\rho}_{h}(t)\|_{L^{\infty}}\leqslant C_{%
T},$$
where $C_{T}$ is a constant depending on $\|V_{1}\|_{L^{\infty}}$, see [36, lemma 2].
The $L^{1}$ bound is even easier: since the Wasserstein step is mass preserving, we can integrate (3.10) in space to get
$$\|\rho^{k+1}_{h}\|_{L^{1}}\leqslant(1+Ch)\|\rho_{h}^{k+1/2}\|_{L^{1}}=(1+Ch)\|%
\rho_{h}^{k+1}\|_{L^{1}}.$$
For $t\leqslant T\Leftrightarrow k\leqslant\lfloor T/h\rfloor$ the $L^{1}$ bounds immediately follow by induction, with $(1+Ch)^{\lfloor T/h\rfloor}\lesssim e^{CT}$.
and we conclude again by induction.
In order to compare now $\rho_{h}$ and $\tilde{\rho}_{h}$, we take advantage of the above upper bound to write $\rho^{k+1/2}_{h}\leqslant M_{T}$ as long as $kh\leqslant T$.
Taking $c=c(M_{T})$ in (3.11) and combining with (3.10), we have
$$-ch\rho^{k+1/2}_{h}\leqslant\rho^{k+1/2}_{h}-\rho^{k+1}_{h}\leqslant Ch\rho^{k%
+1/2}_{h}\qquad\mbox{a.e.}$$
Integrating in $\Omega$ we conclude that
$$\|\rho_{h}(t)-\tilde{\rho}_{h}(t)\|_{1}=\|\rho^{k+1}_{h}-\rho_{h}^{k+1/2}\|_{1%
}\leqslant h\max\{c,C\}\|\rho_{h}^{k+1/2}\|_{1}\leqslant h\max\{c,C\}M_{T}=hM^%
{\prime}_{T}$$
and the proof is complete.
∎
3.2 Energy dissipation
Our goal is here to estimate the crossed dissipation along each elementary $\mathtt{W},\mathtt{FR}$ step.
Testing $\rho=\rho_{h}^{k}$ in the first Wasserstein step in (3.4), we get as usual
$$\displaystyle\frac{1}{2h}\mathtt{W}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k})%
\leqslant\mathcal{F}_{1}(\rho_{h}^{k})-\mathcal{F}_{1}(\rho_{h}^{k+1/2})+\int_%
{\Omega}V_{1}(\rho_{h}^{k}-\rho_{h}^{k+1/2}).$$
(3.13)
Since $V_{1}$ is Globally Lipschitz we can first use standard methods from [15, 23] to control $\int_{\Omega}V_{1}(\rho_{h}^{k}-\rho_{h}^{k+1/2})$ in terms of $\mathtt{W}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k})$, and suitably reabsorb in the left-hand side to obtain
$$\displaystyle\frac{1}{4h}\mathtt{W}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k})%
\leqslant\mathcal{F}_{1}(\rho_{h}^{k})-\mathcal{F}_{1}(\rho_{h}^{k+1/2})+C_{T}h.$$
(3.14)
The dissipation of $\mathcal{F}_{1}$ along the Fisher-Rao step is controlled as
Proposition 3.7.
For all $T>0$ there exists a constant $C_{T}>0$ such that, for all $k\geqslant 0$ and $k\leq\lfloor T/h\rfloor$,
$$\mathcal{F}_{1}(\rho_{h}^{k+1})\leqslant\mathcal{F}_{1}(\rho_{h}^{k+1/2})+C_{T%
}h.$$
(3.15)
Démonstration.
We first treat the case of $F_{1}(z)=\frac{1}{m_{1}-1}z^{m_{1}}$ with $m_{1}>1$.
Since $F_{1}$ is increasing, we use (3.10) to obtain
$$\displaystyle\mathcal{F}_{1}(\rho_{h}^{k+1})-\mathcal{F}_{1}(\rho_{h}^{k+1/2})$$
$$\displaystyle\leqslant$$
$$\displaystyle\frac{((1+Ch)^{m_{1}}-1)}{m_{1}-1}\int_{\Omega}(\rho_{h}^{k+1/2})%
^{m_{1}}$$
$$\displaystyle\leqslant$$
$$\displaystyle Ch\|\rho^{k+1/2}\|_{L^{\infty}}^{m_{1}-1}\,\|\rho^{k+1/2}\|_{L^{%
1}},$$
and we conclude from Proposition 3.6.
In the second case $F_{1}(z)=z\log(z)-z$, we have
$$\mathcal{F}_{1}(\rho_{h}^{k+1})=\int_{\{\rho_{h}^{k+1}\leqslant e^{-1}\}}\rho_%
{h}^{k+1}\log(\rho_{h}^{n+1})+\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}\rho_{h%
}^{k+1}\log(\rho_{h}^{k+1})-\int_{\Omega}\rho_{h}^{k+1}.$$
Note from Proposition 3.6 that the $z$ contribution in $F_{1}(z)=z\log z-z$ is immediately controlled by $|\int\rho_{h}^{k+1}-\int\rho^{k+1/2}_{h}|\leqslant\|\rho^{k+1}_{h}-\rho_{h}^{k%
+1/2}\|_{L^{1}}\leqslant hM_{T}^{\prime}$, so we only have to estimate the $z\log z$ contribution.
Since $z\mapsto z\log z$ is increasing on $\{z\geqslant e^{-1}\}$ and using (3.10), the second term in the right hand side becomes
$$\displaystyle\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}\rho_{h}^{k+1}\log(\rho_%
{h}^{k+1})$$
$$\displaystyle\leqslant$$
$$\displaystyle\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}(1+Ch)\rho_{h}^{k+1/2}%
\log((1+Ch)\rho_{h}^{k+1/2})$$
$$\displaystyle\leqslant$$
$$\displaystyle\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}\rho_{h}^{k+1/2}\log(%
\rho_{h}^{k+1/2})+Ch\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}\rho_{h}^{k+1/2}%
\log(\rho_{h}^{k+1/2})$$
$$\displaystyle +(1+Ch)\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}\rho%
_{h}^{k+1/2}\log(1+Ch)$$
$$\displaystyle\leqslant$$
$$\displaystyle\int_{\{\rho_{h}^{k+1}\geqslant e^{-1}\}}\rho_{h}^{k+1/2}\log(%
\rho_{h}^{k+1/2})+C_{T}h,$$
where we used $\|\rho_{h}^{k+1/2}\|_{L^{1}}\leqslant M_{T}$ from Proposition 3.6 as well as $\log(1+Ch)\leqslant Ch$ in the last inequality.
Using the same method with the bound from below (3.11) on $\{\rho^{k+1}_{h}\leqslant e^{-1}\}$ (where $z\mapsto z\log z$ is now decreasing), we obtain similarly
$$\int_{\{\rho_{h}^{k+1}\leqslant e^{-1}\}}\rho_{h}^{k+1}\log(\rho_{h}^{k+1})%
\leqslant\int_{\{\rho_{h}^{k+1}\leqslant e^{-1}\}}\rho_{h}^{k+1/2}\log(\rho_{h%
}^{k+1/2})+C_{T}h.$$
Combining both inequalities gives
$$\int_{\Omega}\rho_{h}^{k+1}\log(\rho_{h}^{k+1})\leqslant\int_{\Omega}\rho_{h}^%
{k+1/2}\log(\rho_{h}^{k+1/2})+C_{T}h$$
and the proof is complete.
∎
Summing (3.14) and (3.15) over $k$ we obtain
$$\frac{1}{2h}\sum_{k=0}^{N-1}\mathtt{W}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k})%
\leqslant\mathcal{F}_{1}(\rho_{0})-\mathcal{F}_{1}(\rho_{h}^{N})+C_{T},$$
(3.16)
where $N=\lfloor\frac{T}{h}\rfloor$.
In the above estimate we just controlled the dissipation of $\mathcal{F}_{1}$ along the $\mathtt{FR}$/reaction steps, and the goal is now to similarly estimate the dissipation of $\mathcal{F}_{2}$ along the Wasserstein step.
Testing $\rho=\rho_{h}^{k+1/2}$ in the second Fisher-Rao step in (3.4), we obtain
$$\displaystyle\frac{1}{2h}\mathtt{FR}_{2}(\rho_{h}^{k+1},\rho_{h}^{k+1/2})%
\leqslant\mathcal{F}_{2}(\rho_{h}^{k+1/2})-\mathcal{F}_{2}(\rho_{h}^{k+1})+%
\int_{\Omega}V_{2}(\rho_{h}^{k+1/2}-\rho_{h}^{k+1}).$$
(3.17)
Since we assumed $V_{2}\in L^{\infty}(\Omega)$ and because $\rho_{h}(t)=\rho^{k+1}_{h}$ remains close to $\tilde{\rho}_{h}(t)=\rho_{h}^{k+1/2}$ in $L^{1}$ uniformly in $t,h$ by Proposition 3.6, we immediately control the potential part as
$$\int_{\Omega}V_{2}(\rho_{h}^{k+1/2}-\rho_{h}^{k+1})\leqslant\|V_{2}\|_{\infty}%
C_{T}h.$$
(3.18)
For the internal energy we argue exactly as in the proof Proposition 3.7 (for the Porous Media part, since we chose here $F_{2}(z)=\frac{1}{m_{2}-1}z^{m_{2}}$), and obtain
$$\mathcal{F}_{2}(\rho_{h}^{k+1/2})-\mathcal{F}_{2}(\rho_{h}^{k+1})\leqslant C_{%
T}h.$$
(3.19)
Combining (3.17), (3.18) and (3.19), we immediately deduce that
$$\displaystyle\frac{1}{2h}\sum_{k=0}^{N-1}\mathtt{FR}^{2}(\rho_{h}^{k+1/2},\rho%
_{h}^{k+1})\leqslant C_{T},$$
(3.20)
where $N=\lfloor\frac{T}{h}\rfloor$ as before.
Finally, we recover an approximate compactness in time in the form
Proposition 3.8.
There exists a constant $C_{T}>0$ such that for all $h$ small enough and $k\leqslant N=\lfloor T/h\rfloor$,
$$\frac{1}{h}\sum_{k=0}^{N-1}\mathtt{WFR}^{2}(\rho_{h}^{k},\rho_{h}^{k+1})%
\leqslant 4\mathcal{F}_{1}(\rho_{0})+C_{T}.$$
(3.21)
Démonstration.
Adding (3.16) and (3.20) gives
$$\frac{1}{h}\sum_{k=0}^{N-1}\mathtt{W}^{2}(\rho_{h}^{k},\rho_{h}^{k+1/2})+%
\mathtt{FR}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k+1})\leqslant 2\left(\mathcal{F}_{%
1}(\rho_{0})-\mathcal{F}_{1}(\rho_{h}^{N})+C_{T}\right)+2C_{T}\leqslant 2%
\mathcal{F}_{1}(\rho_{0})+C_{T},$$
since in any case $F_{1}(z)=\frac{1}{m_{1}-1}z^{m_{1}}\geqslant 0$ and $F_{1}(z)=z\log z-z\geqslant-1$ is bounded from below on the bounded domain $\Omega$, hence $\mathcal{F}_{1}(\rho_{h}^{N})\geqslant-C_{\Omega}$ uniformly.
It then follows from Proposition 2.4 that $\mathtt{W}^{2}(\rho_{h}^{k},\rho_{h}^{k+1/2})+\mathtt{FR}^{2}(\rho_{h}^{k+1/2}%
,\rho_{h}^{k+1})\geqslant\frac{1}{2}\mathtt{WFR}^{2}\rho_{h}^{k},\rho_{h}^{k+1}$ in the left-hand side, and the result immediately follows.
∎
3.3 Estimates and convergences
From the total-square distance estimate (3.21) we recover as usual the approximate $\frac{1}{2}$-Hölder estimate
$$\displaystyle\mathtt{WFR}(\rho_{h}(t),\rho_{h}(s))+\mathtt{WFR}(\tilde{\rho}_{%
h}(t),\tilde{\rho}_{h}(s))\leqslant C_{T}|t-s+h|^{1/2}$$
(3.22)
for all fixed $T>0$ and $t,s\in[0,T]$.
From (3.20) and Proposition 2.4 we have moreover
$$\displaystyle\mathtt{WFR}(\rho_{h}(t),\tilde{\rho}_{h}(t))\leqslant\mathtt{FR}%
(\rho_{h}(t),\tilde{\rho}_{h}(t))\leqslant C\sqrt{h}.$$
(3.23)
Using a refined version of Ascoli-Arzelà theorem, [4, prop. 3.3.1] and arguing exactly as in [18, prop. 4.1], we see that for all $T>0$ and up to extraction of a discrete subsequence, $\rho_{h}$ and $\tilde{\rho}_{h}$ converge uniformly to the same $\mathtt{WFR}$-continuous curve $\rho\in\mathcal{C}^{1/2}([0,T],\mathcal{M}^{+}_{\mathtt{WFR}})$ as
$$\sup_{t\in[0,T]}(\mathtt{WFR}(\rho_{h}(t),\rho(t))+\mathtt{WFR}(\tilde{\rho}_{%
h}(t),\rho(t)))\rightarrow 0.$$
In order to pass to the limit in the nonlinear terms, we first strengthen this $\mathtt{WFR}$-convergence into a more tractable $L^{1}$ convergence.
The first step is to retrieve compactness in space:
Proposition 3.9.
For all $T>0$, $\rho_{h}$ and $\tilde{\rho}_{h}$ satisfies
$$\|P_{1}(\tilde{\rho}_{h})\|_{L^{2}([0,T];H^{1}(\Omega))}\leqslant C_{T}.$$
(3.24)
Démonstration.
From (3.6) and the $L^{1}\cap L^{\infty}$ bounds from Proposition 3.6 we see that
$$\displaystyle\int_{\Omega}|\nabla P_{1}(\rho_{h}^{k+1/2})|^{2}$$
$$\displaystyle\leqslant$$
$$\displaystyle\frac{1}{2h^{2}}\int_{\Omega}|\nabla\varphi_{h}^{k+1/2}|^{2}(\rho%
_{h}^{k+1/2})^{2}+\frac{1}{2}\int_{\Omega}|\nabla V_{1}|^{2}(\rho_{h}^{k+1/2})%
^{2}$$
$$\displaystyle\leqslant$$
$$\displaystyle\frac{C_{T}}{2h^{2}}\int_{\Omega}|\nabla\varphi_{h}^{k+1/2}|^{2}%
\rho_{h}^{k+1/2}+\frac{1}{2}\|\nabla V_{1}\|_{\infty}^{2}\int_{\Omega}(\rho_{h%
}^{k+1/2})^{2}$$
$$\displaystyle\leqslant$$
$$\displaystyle C_{T}\left(\frac{\mathtt{W}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k})}{%
h^{2}}+1\right)$$
since $\varphi^{k+1/2}_{h}$ is the optimal (backward) Kantorovich potential from $\rho_{h}^{k+1/2}$ to $\rho_{h}^{k}$.
Multiplying by $h>0$, summing over $k$, and exploiting (3.16) gives
$$\|P_{1}(\tilde{\rho}_{h})\|^{2}_{L^{2}([0,T];H^{1}(\Omega))}\leqslant\sum_{k=0%
}^{N-1}h\|P_{1}(\rho^{k+1/2}_{h})\|^{2}_{H^{1}}\leqslant C_{T}(\mathcal{F}_{1}%
(\rho_{0})-\mathcal{F}_{1}(\rho_{h}^{N})+1)\leqslant C_{T},$$
where we used as before $\mathcal{F}_{1}(\rho_{h}^{N})\geqslant-C_{\Omega}$ in the last inequality.
∎
We are now in position of proving our main result:
Proof of Theorem 3.2.
Exploiting (3.21) and (3.24), we can apply the extension of the Aubin-Lions lemma established by Rossi and Savaré in [39] to obtain that $\tilde{\rho}_{h}$ converges to $\rho$ strongly in $L^{1}(Q_{T})$ (see [23]).
By diagonal extraction if needed, we can assume that the convergence holds in $L^{1}(Q_{T})$ for all fixed $T>0$.
Then by Proposition 3.6 we have
$$\|\rho_{h}-\rho\|_{L^{1}(Q_{T})}\leqslant\|\rho_{h}-\tilde{\rho}_{h}\|_{L^{1}(%
Q_{T}}+\|\tilde{\rho}_{h}-\rho\|_{L^{1}(Q_{T})}\leqslant C_{T}h+\|\tilde{\rho}%
_{h}-\rho\|_{L^{1}(Q_{T})}\to 0$$
hence $\rho_{h}\to\rho$ as well.
Moreover, since $P_{1}(\tilde{\rho}_{h})$ is bounded in $L^{2}((0,T),H^{1}(\Omega))$ we can assume that $\nabla P_{1}(\tilde{\rho}_{h})\rightharpoonup\nabla P_{1}(\rho)$ in $L^{2}((0,T),H^{1}(\Omega))$ for all $T>0$.
Exploiting the Euler-Lagrange equations (3.6)(3.9) and arguing exactly as in [18, Theorem 4], it is easy to pass to the limit to conclude that
$$\int_{\Omega}\rho(t_{2})\varphi-\rho(t_{1})\varphi=-\int_{t_{1}}^{t_{2}}\int_{%
\Omega}\Big{\{}\nabla P(\rho)\cdot\nabla\varphi+\rho\nabla V_{1}\cdot\nabla%
\varphi-\rho(F^{\prime}_{2}(\rho)+V_{2})\varphi\Big{\}}$$
for all $0<t_{1}<t_{2}$ and $\varphi\in\mathcal{C}^{1}_{b}(\Omega)$.
Since $\rho\in\mathcal{C}([0,T];\mathcal{M}^{+}_{\mathtt{WFR}})$ takes the initial datum $\rho(0)=\rho_{0}$ and $\mathtt{WFR}$ metrizes the narrow convergence of measures, this is well-known to be equivalent to our weak formulation in Definition 3.1, and the proof is complete.
∎
Remark 3.10.
In the above proofs one can check that Theorem 3.2 extends in fact to all $\mathcal{C}^{1}$ nonlinearities $F_{2}$ such that $F_{2}^{\prime}\geqslant C$ for some $C\in\operatorname*{\mathbb{R}}$.
Likewise, we stated and proved our main result in bounded domains for convenience: all the above arguments immediately extend to $\Omega=\operatorname*{\mathbb{R}}^{d}$ at least for $F_{1}(z)=\frac{1}{m_{1}-1}z^{m_{1}}\geqslant 0$.
The only place where we actually used the boundedness of $\Omega$ was in the proof of Proposition 3.8, when we bounded from below $\mathcal{F}_{1}(\rho^{N}_{h})\geqslant-C_{\Omega}$ in order to retrieve the total-square distance estimate.
When $\Omega=\operatorname*{\mathbb{R}}^{d}$ and $F_{1}(z)=z\log z-z$ a lower bound $\mathcal{F}_{1}(\rho^{N}_{h})\geqslant-C_{T}$ still holds, but the proof requires a tedious control of the second moments $\mathfrak{m}_{2}(\rho)=\int_{\operatorname*{\mathbb{R}}^{d}}|x|^{2}\rho$ hence we did not address this technical issue for the sake of brevity.
4 Application to systems
In this section we shall try to illustrate that the previous scheme is very tractable and allows to solve systems of the form
$$\displaystyle\left\{\begin{array}[]{l}\partial_{t}\rho_{1}=\operatorname*{div}%
(\rho_{1}\nabla(F_{1}^{\prime}(\rho_{1})+V_{1}[\rho_{1},\rho_{2}]))-\rho_{1}(G%
_{1}^{\prime}(\rho_{1})+U_{1}[\rho_{1},\rho_{2}]),\\
\partial_{t}\rho_{2}=\operatorname*{div}(\rho_{2}\nabla(F_{2}^{\prime}(\rho_{2%
})+V_{2}[\rho_{1},\rho_{2}]))-\rho_{2}(G_{2}^{\prime}(\rho_{2})+U_{2}[\rho_{1}%
,\rho_{2}]),\\
{\rho_{1}}_{|t=0}=\rho_{1,0},\,{\rho_{2}}_{|t=0}=\rho_{2,0}.\end{array}\right.$$
(4.1)
For simplicity we assume again that $\Omega$ is a smooth, bounded subset of $\mathbb{R}^{d}$.
Then the system (4.1) is endowed with Neumann boundary conditions,
$$\rho_{1}\nabla(F_{1}^{\prime}(\rho_{1})+V_{1}[\rho_{1},\rho_{2}])\cdot\nu=0%
\text{ and }\rho_{2}\nabla(F_{2}^{\prime}(\rho_{2})+V_{2}[\rho_{1},\rho_{2}])%
\cdot\nu=0\qquad\text{ on }\mathbb{R}^{+}\times\partial\Omega,$$
where $\nu$ is the outward unit normal to $\partial\Omega$.
In system of the form (4.1), we allow interactions between densities in the potential terms $V_{i}[\rho_{1},\rho_{2}]$ and $U_{i}[\rho_{1},\rho_{2}]$. In the mass-conservative case (without reaction terms), this system has already been studied in [15, 23, 8], using a semi-implicit JKO scheme introduced by Di Francesco and Fagioli, [15]. This section combines the splitting scheme introduced in the previous section and semi-implicit schemes both for the Wasserstein JKO step and for the Fisher-Rao JKO step.
For the ease of exposition we keep the same assumptions for $F_{i}$ and $G_{i}$ as in the previous section, i.e the diffusion terms $F_{i}$ satisfy (3.2) and the reaction terms $G_{i}$ satisfy (3.3).
Moreover, since the potentials depend now on the densities $\rho_{1}$ and $\rho_{2}$, we need stronger hypotheses:
we assume that $V_{i}\,:\,L^{1}(\Omega;\operatorname*{\mathbb{R}}^{+})^{2}\rightarrow\mathcal{%
C}^{1}(\Omega)$ are continuous and verify, uniformly in $\rho_{1},\rho_{2}\in L^{1}(\Omega;\operatorname*{\mathbb{R}}^{+})$,
$$\displaystyle\begin{array}[]{c}\|V_{i}[\rho_{1},\rho_{2}]\|_{W^{1,\infty(%
\Omega)}}\leqslant K(1+\|\rho_{1}\|_{L^{1}(\Omega)}+\|\rho_{2}\|_{L^{1}(\Omega%
)}),\\
\|\nabla(V_{i}[\rho_{1},\rho_{2}])-\nabla(V_{i}[\mu_{1},\mu_{2}])\|_{L^{\infty%
}(\Omega)}\leqslant K(\|\rho_{1}-\mu_{1}\|_{L^{1}(\Omega)}+\|\rho_{2}-\mu_{2}%
\|_{L^{1}(\Omega)}).\end{array}$$
(4.2)
The interacting potentials we have in mind are of the form $V_{i}[\rho_{1},\rho_{2}]=K_{i,1}\ast\rho_{1}+K_{i,2}\ast\rho_{2}$, where $K_{i,1},K_{i,2}\in W^{1,\infty}(\Omega)$ and then $V_{i}$ satisfies (4.2).
For the reaction, we assume that the potentials $U_{i}$ are continuous from $L^{1}(\Omega)_{+}^{2}$ to $L^{1}$ with moreover
$$U_{i}[\rho_{1},\rho_{2}]\geqslant-K,\qquad\forall\,\rho_{1},\rho_{2}\in L^{1}(%
\Omega;{\operatorname*{\mathbb{R}}}^{+})$$
(4.3)
for some $K\in\operatorname*{\mathbb{R}}$, and
$$\|U_{i}[\rho_{1},\rho_{2}]\|_{L^{\infty}(\Omega)}\leqslant K_{M},\qquad\forall%
\|\rho_{1}\|_{L^{1}(\Omega)},\|\rho_{2}\|_{L^{1}(\Omega)}\leqslant M$$
(4.4)
for some nondecreasimg function $K_{M}\geqslant 0$ of $M$.
The examples we have in mind are of the form
$$U_{1}[\rho_{1},\rho_{2}]=C_{1}\frac{\rho_{2}}{1+\rho_{1}},\quad U_{2}[\rho_{1}%
,\rho_{2}]=-C_{2}\frac{\rho_{1}}{1+\rho_{1}}$$
for some constants $C_{i}\geq 0$, or nonlocal reactions
$$U_{i}[\rho_{1},\rho_{2}](x)=\int_{\Omega}K_{i,1}(x,y)\rho_{1}(y)\,dy+\int_{%
\Omega}K_{i,2}(x,y)\rho_{2}(y)\,dy$$
for some nonnegative kernels $K_{i,j}\in L^{1}\cap L^{\infty}$.
Such reaction models appear for example in biological adaptive dynamics [33].
Definition 4.1.
We say that $(\rho_{1},\rho_{2})\,:\,\operatorname*{\mathbb{R}}^{+}\rightarrow L^{1}_{+}%
\cap L^{\infty}_{+}(\Omega)$ is a weak solution of (4.1) if, for $i\in\{1,2\}$ and all $T<+\infty$, the pressure $P_{i}(\rho_{i}):=\rho_{i}F_{i}^{\prime}(\rho_{i})-F_{i}(\rho_{i})$ satisfies $\nabla P_{i}(\rho_{i})\in L^{2}([0,T]\times\Omega)$, and
$$\displaystyle\int_{0}^{+\infty}\left(\int_{\Omega}(\rho\partial_{t}\phi_{i}-%
\rho_{i}\nabla V_{i}[\rho_{1},\rho_{2}]\cdot\nabla\phi_{i}-\nabla P_{i}(\rho_{%
i})\cdot\nabla\phi_{i}-\rho_{i}(G_{i}^{\prime}(\rho_{i})+U_{i}[\rho_{1},\rho_{%
2}])\phi_{i})\,dx\right)\,dt\\
\displaystyle=-\int_{\Omega}\phi_{i}(0,x)\rho_{i,0}(x)\,dx,$$
(4.5)
for all $\phi_{i}\in\mathcal{C}^{\infty}_{c}([0,+\infty)\times\mathbb{R}^{d})$.
Then, the following result holds,
Theorem 4.2.
Assume that $\rho_{1,0},\rho_{2,0}\in L^{1}\cap L^{\infty}_{+}(\Omega)$ and that $V_{i},U_{i}$ satisfy (4.2)(4.3)(4.4).
Then (4.1) admits at least one weak solution.
Note that this result can be easily adapted to systems with an arbitrary number of species $N\geqslant 2$, coupled by nonlocal terms $V_{i}[\rho_{1},\mathellipsis,\rho_{N}]$ and $U_{i}[\rho_{1},\mathellipsis,\rho_{N}]$.
Remark 4.3.
A refined analysis shows that our approach would allow to handle systems of the form
$$\left\{\begin{array}[]{l}\partial_{t}\rho_{1}-\operatorname*{div}(\rho_{1}%
\nabla(F_{1}^{\prime}(\rho_{1})+V_{1}))=-\rho_{1}h_{1}(\rho_{1},\rho_{2}),\\
\partial_{t}\rho_{2}-\operatorname*{div}(\rho_{2}\nabla(F_{2}^{\prime}(\rho_{2%
})+V_{2}))=+\rho_{2}h_{2}(\rho_{1}),\end{array}\right.$$
where $h_{1}$ is a nonnegative continuous function and $h_{2}$ is a continuous functions.
Indeed since $h_{1}\geq 0$ the reaction term is the first equation is nonpositive, hence
$\|\rho_{1}(t)\|_{L^{\infty}(\Omega)}\leqslant C_{T}$.
Then it follows that $-h_{2}(\rho_{1})$ satisfies assumptions (4.3) and (4.4).
A classical example is $h_{2}(\rho_{1})=\rho_{1}^{\alpha}$ and $h_{1}(\rho_{1},\rho_{2})=\rho_{1}^{\alpha-1}\rho_{2}$, where $\alpha\geqslant 1$, see for example [38] for more discussions.
As already mentioned, the proof of theorem 4.2 is based on a semi-implicit splitting scheme.
More precisely, we construct four sequences $\rho_{1,h}^{k+1/2},\rho_{1,h}^{k+1},\rho_{2,h}^{k+1/2},\rho_{2,h}^{k+1}$ defined recursively as
$$\displaystyle\left\{\begin{array}[]{l}\rho_{i,h}^{k+1/2}\in\operatorname*{%
argmin}\limits_{\rho\in\mathcal{M}^{+},|\rho|=|\rho_{i,h}^{k}|}\left\{\frac{1}%
{2h}\mathtt{W}^{2}(\rho,\rho_{i,h}^{k})+\mathcal{F}_{i}(\rho)+\mathcal{V}_{i}(%
\rho|\rho_{1,h}^{k},\rho_{2,h}^{k})\right\}\\
\\
\rho_{i,h}^{k+1}\in\operatorname*{argmin}\limits_{\rho\in\mathcal{M}^{+}}\left%
\{\frac{1}{2h}\mathtt{FR}^{2}(\rho,\rho_{i,h}^{k+1/2})+\mathcal{G}_{i}(\rho)+%
\mathcal{U}_{i}(\rho|\rho_{1,h}^{k},\rho_{2,h}^{k})\right\}\end{array}\right.,$$
(4.6)
where the fully implicit terms
$$\mathcal{F}_{i}(\rho):=\left\{\begin{array}[]{ll}\int_{\Omega}F_{i}(\rho)&%
\text{ if }\rho\ll\mathcal{L}_{|\Omega}\\
+\infty&\text{ otherwise }\end{array}\right.\quad\text{and}\quad\mathcal{G}_{i%
}(\rho):=\left\{\begin{array}[]{ll}\int_{\Omega}G_{i}(\rho)&\text{ if }\rho\ll%
\mathcal{L}_{|\Omega}\\
+\infty&\text{ otherwise }\end{array}\right.,$$
and the semi-implicit terms
$$\mathcal{V}_{i}(\rho|\mu_{1},\mu_{2}):=\int_{\Omega}V_{i}[\mu_{1},\mu_{2}]\rho%
\quad\text{and}\quad\mathcal{U}_{i}(\rho|\mu_{1},\mu_{2}):=\int_{\Omega}U_{i}[%
\mu_{1},\mu_{2}]\rho.$$
In the previous section, the proof of theorem 3.2 for scalar equations strongly leveraged the uniform $L^{\infty}(\Omega)$-bounds on the discrete solutions.
Here an additional difficulty arises due to the nonlocal terms $\nabla V_{i}[\rho_{1},\rho_{2}]$ and $U_{i}[\rho_{1},\rho_{2}]$, which are a priori not uniformly bounded in $L^{\infty}(\Omega)$.
Using assumption (4.3) we will first obtain a uniform $L^{1}(\Omega)$-bound on $\rho_{1},\rho_{2}$, and then extend proposition 3.6 to the system (4.1).
This in turn will give a uniform $W^{1,\infty}$ control on $V_{i}[\rho_{1},\rho_{2}]$ and $L^{\infty}$ control on $U_{i}[\rho_{1},\rho_{2}]$ through our assumptions (4.2)-(4.3)-(4.4), which will finally allow to argue as in the previous section and give $L^{\infty}$ control on $\rho_{1},\rho_{2}$.
Numerical simulations for a diffusive prey-predator system are presented at the end of this section.
4.1 Properties of discrete solutions
Arguing as in the case of one equation, the optimality conditions for the Wasserstein step and for the Fisher-Rao step first give
Lemma 4.4.
For all $k\geqslant 0$ and $i\in\{1,2\}$, we have
$$\|\rho_{i,h}^{k+1/2}\|_{L^{1}}=\|\rho_{i,h}^{k}\|_{L^{1}}.$$
(4.7)
Moreover, there exists $C_{i}\equiv C(U_{i})>0$ (uniform in $k$) such that
$$\rho_{i,h}^{k+1}(x)\leqslant(1+C_{i}h)\rho_{i,h}^{k+1/2}(x)\qquad a.e.$$
(4.8)
Démonstration.
The first part is simply the mass conservation in the Wasserstein step, and the second part follows the lines of the proof of (3.10) in Lemma 3.5 using assumption (4.3).
∎
As a direct consequence we have uniform control on the $L^{1}$-norms:
Lemma 4.5.
For all $T>0$ there exist constants $C_{T},C_{T}^{\prime}>0$ such that, for all $t\in[0,T]$,
$$\|\rho_{i,h}(t)\|_{L^{1}},\|\tilde{\rho}_{i,h}(t)\|_{L^{1}}\leqslant C_{T}$$
and
$$\|V_{i}[\rho_{1,h}(t),\rho_{2,h}(t)]\|_{W^{1,\infty}},\|V_{i}[\tilde{\rho}_{1,%
h}(t),\tilde{\rho}_{2,h}(t)]\|_{W^{1,\infty}}\leqslant C^{\prime}_{T}.$$
(4.9)
Démonstration.
Integrating (4.8) and iterating with (4.7), we obtain for all $t\leqslant T$ and $k\leqslant\lfloor T/h\rfloor$
$$\|\rho_{i,h}^{k+1}\|_{L^{1}}\leqslant(1+C_{i}h)\|\rho_{i,h}^{k}\|_{L^{1}}%
\leqslant(1+C_{i}h)^{k}\|\rho_{i,0}\|_{L^{1}}\leqslant e^{C_{i}T}\|\rho_{i,0}%
\|_{L^{1}}.$$
Then (4.9) follows from our assumption (4.2) on the interactions.
∎
Combining (4.8) and (4.9), we deduce
Proposition 4.6.
For all $T>0$, there exists $M_{T}$ such that for all $t\in[0,T]$,
$$\|\rho_{i,h}(t)\|_{L^{\infty}},\|\tilde{\rho}_{i,h}(t)\|_{L^{\infty}}\leqslant
M%
_{T}.$$
Then, there exists $c_{i}\equiv c(M_{T},U_{i})\geq 0$, such that, for all $k\leqslant\lfloor T/h\rfloor$ and $h\leqslant h_{0}(U_{1},U_{2})$,
$$(1-c_{i}h)\rho_{i,h}^{k+1/2}\leqslant\rho_{i,h}^{k+1}.$$
In particular, there exist $M_{T}^{\prime}>0$ such that for all $t\in[0,T]$,
$$\|\rho_{i,h}(t)-\tilde{\rho}_{i,h}(t)\|_{L^{1}}\leqslant hM_{T}^{\prime}.$$
Démonstration.
The first $L^{\infty}$ estimate can be found in [36, Lemma 2], and the rest of our statement can be proved exactly as in Lemma 3.5 and Proposition 3.6.
∎
4.2 Estimates and convergences
Since we proved that $V_{1}[\rho_{1,h},\rho_{2,h}]$ and $V_{2}[\rho_{1,h},\rho_{2,h}]$ are bounded in $L^{\infty}([0,T],W^{1,\infty}(\Omega))$, we can argue exactly as in the previous section for the Wasserstein step and obtain
$$\displaystyle\frac{1}{4h}\mathtt{W}^{2}(\rho_{i,h}^{k+1/2},\rho_{i,h}^{k})%
\leqslant\mathcal{F}_{i}(\rho_{i,h}^{k})-\mathcal{F}_{i}(\rho_{i,h}^{k+1/2})+C%
_{T}h,$$
(4.10)
see (3.13)-(3.14) for details.
Since $\tilde{\rho}_{1,h}$ and $\tilde{\rho}_{2,h}$ are uniformly bounded in $L^{1}(\Omega)$ (Lemma 4.5), our assumption (4.4) ensures that $U_{1}[\rho_{1,h}^{k+1/2},\rho_{2,h}^{k+1/2}]$ and $U_{2}[\rho_{1,h}^{k+1/2},\rho_{2,h}^{k+1/2}]$ are uniformly bounded in $L^{\infty}(\Omega)$.
Proposition 4.6 then allows to argue exactly as in (3.17)-(3.18)-(3.19) for the Fisher-Rao step, and we get
$$\displaystyle\frac{1}{2h}\mathtt{FR}^{2}(\rho_{h}^{k+1},\rho_{h}^{k+1/2})%
\leqslant\mathcal{G}_{i}(\rho_{i,h}^{k+1/2})-\mathcal{G}_{i}(\rho_{i,h}^{k+1})%
+C_{T}h.$$
(4.11)
The dissipation of $\mathcal{F}_{i}$ along the Fisher-Rao step is obtained in the same way as Proposition 3.7 and we omit the details:
Proposition 4.7.
For all $T>0$ and $i\in\{1,2\}$, there exist constants $C_{T},C^{\prime}_{T}>0$ such that, for all $k\geqslant 0$ with $hk\leqslant T$,
$$\begin{array}[]{c}\mathcal{F}_{i}(\rho_{i,h}^{k+1})\leqslant\mathcal{F}_{i}(%
\rho_{i,h}^{k+1/2})+C_{T}h,\\
\mathcal{G}_{i}(\rho_{i,h}^{k+1/2})\leqslant\mathcal{G}_{i}(\rho_{i,h}^{k+1})+%
C^{\prime}_{T}h.\end{array}$$
From (4.10) and (4.11) this immediately gives a telescopic sum
$$\frac{1}{2h}\left(\mathtt{W}^{2}(\rho_{i,h}^{k},\rho_{i,h}^{k+1/2})+\mathtt{FR%
}^{2}(\rho_{h}^{k+1/2},\rho_{h}^{k})\right)\leqslant 2[\mathcal{F}_{i}(\rho_{i%
,h}^{k})-\mathcal{F}_{i}(\rho_{i,h}^{k+1})]+C_{T}h$$
which in turn yields an approximate $\frac{1}{2}$-Hölder estimate (with respect to the $\mathtt{WFR}$ distance) as in Proposition 3.8.
The rest of the proof of Theorem 4.2 is then identical to section 3 and we omit the details.
4.3 Numerical application: prey-predator systems
Our constructive scheme can be implemented numerically, by simply discretizing (4.6) in space.
We use the augmented Lagrangian method ALG-JKO from [6] to solve the Wasserstein step, and the Fisher-Rao step is just a convex pointwise minimization problem.
Indeed, it is known [18, 27] that $\mathtt{FR}^{2}(\rho,\mu)=4\|\sqrt{\rho}-\sqrt{\mu}\|^{2}_{L^{2}}$, hence the Fisher-Rao step in (4.6) is a mere convex pointwise minimization problem of the form: for all $x\in\Omega$ (and omitting all indexes $\rho_{i,h}$),
$$\rho^{k+1}(x)=\operatorname*{argmin}\limits_{\rho\geq 0}\left\{4\left|\sqrt{%
\rho}-\sqrt{\rho^{k+1/2}(x)}\right|^{2}+2hF(\rho)\right\}.$$
This is easily solved using any simple Newton procedure.
Figure (1) shows the numerical solution of the following diffusive prey-predator system
$$\left\{\begin{array}[]{l}\partial_{t}\rho_{1}-\Delta\rho_{1}-\operatorname*{%
div}(\rho_{1}\nabla V_{1}[\rho_{1},\rho_{2}])=A\rho_{1}\left(1-\rho_{1}\right)%
-B\frac{\rho_{1}\rho_{2}}{1+\rho_{1}},\\
\partial_{t}\rho_{2}-\Delta\rho_{2}-\operatorname*{div}(\rho_{2}\nabla V_{2}[%
\rho_{1},\rho_{2}])=\frac{B\rho_{1}\rho_{2}}{1+\rho_{1}}-C\rho_{2},\end{array}%
\right..$$
Here the $\rho_{1}$ species are preys and $\rho_{2}$ are predators, see for example [30], the parameters $A=10,C=5,B=70$, and the interactions are chosen as
$$V_{1}[\rho_{1},\rho_{2}]=|x|^{2}\ast\rho_{1}-|x|^{2}\ast\rho_{2},\quad V_{2}[%
\rho_{1},\rho_{2}]=|x|^{2}\ast\rho_{1}+|x|^{2}\ast\rho_{2}.$$
In (4.1) this corresponds to
$$G_{1}(\rho_{1})=A\frac{\rho_{1}^{2}}{2},\quad G_{2}(\rho_{2})=0,\quad U_{1}[%
\rho_{1},\rho_{2}]=\frac{B\rho_{2}}{1+\rho_{1}}-A,\quad U_{2}[\rho_{1},\rho_{2%
}]=-\frac{B\rho_{1}}{1+\rho_{1}}+C.$$
Of course, $U_{1}$ and $U_{2}$ satisfy assumptions (4.3) and (4.4), and then Theorem 4.2 gives a solution of the prey-predator system.
As before, we shall disregard the uniqueness issue for the sake of simplicity.
Figure (2) depicts the mass evolution of the prey and predator species: we observe the usual oscillations in time with phase opposition, a characteristic behaviour for Lotka-Volterra types of systems.
5 Application to a tumor growth model with very degenerate enery
In this section we take interest in the equation
$$\displaystyle\left\{\begin{array}[]{l}\partial_{t}\rho=\operatorname*{div}(%
\rho\nabla p)+\rho(1-p),\\
p\geqslant 0\quad\mbox{and}\quad p(1-\rho)=0\\
0\leqslant\rho\leqslant 1,\\
\rho_{|t=0}=\rho_{0}.\end{array}\right.$$
(5.1)
This equation is motivated by tumor growth models [34, 35] and exhibits a Hele-Shaw patch dynamics: if $\rho_{0}=\chi_{\Omega_{0}}$ then the solution remains an indicator $\rho(t)=\chi_{\Omega(t)}$ and the boundary moves with normal velocity $V=-\nabla p|_{\partial\Omega(t)}$, see [2] for a rigorous analysis in the framework of viscosity solutions.
At least formally, we remark that (5.1) is the Wasserstein-Fisher-Rao gradient flow of the singular functional
$$\mathcal{F}(\rho):=\mathcal{F}_{\infty}(\rho)-\int_{\Omega}\rho,$$
where
$$\mathcal{F}_{\infty}(\rho):=\left\{\begin{array}[]{ll}0&\text{ if }\rho%
\leqslant 1\,\mbox{ a.e},\\
+\infty&\text{ otherwise.}\end{array}\right.$$
Indeed, the compatibility conditions $p\geqslant 0$ and $p(1-\rho)=0$ in (5.1) really mean that the pressure $p$ belongs to the subdifferential $\partial\mathcal{F}_{\infty}(\rho)$, and (5.1) thus reads as the gradient flow
$$\partial_{t}\rho=\operatorname*{div}(\rho\nabla u)-\rho u,\qquad u=p-1\in-%
\partial\mathcal{F}(\rho).$$
However, this functional is too singular for the previous splitting scheme to correctly capture the very degenerate diffusion.
Indeed, the naive and direct approach from section 3 would lead to
$$\left\{\begin{array}[]{l}\rho_{h}^{k+1/2}\in\operatorname*{argmin}\limits_{%
\rho\leqslant 1,\,|\rho|=|\rho_{h}^{k}|}\left\{\frac{1}{2h}\mathtt{W}^{2}(\rho%
,\rho_{h}^{k})-\int_{\Omega}\rho\right\},\\
\\
\rho_{h}^{k+1}\in\operatorname*{argmin}\limits_{\rho\leqslant 1}\left\{\frac{1%
}{2h}\mathtt{FR}^{2}(\rho,\rho_{h}^{k+1/2})-\int_{\Omega}\rho\right\}.\end{%
array}\right.$$
Since the Wasserstein step is mass-conservative by definition, the $\int\rho$ term has no effect in the first step and the latter reads as “project $\rho_{h}^{k}$ on $\{\rho\leqslant 1\}$ w.r.t to the $\mathtt{W}$ distance”.
Since the output of the reaction step $\rho^{k+1}_{h}\leqslant 1$, the Wasserstein step will never actually project anything, and the diffusion is completly shut down.
As an example, it is easy to see that if the initial datum is an indicator $\rho_{0}=\chi_{\Omega_{0}}$ then the above naive scheme leads to a stationary solution $\rho^{k+1}_{h}=\rho^{k+1/2}_{h}=\rho_{0}$ for all $k\geqslant 0$, while the real solution should evolve according to the aforementioned Hele-Shaw dynamics $\rho(t)=\chi_{\Omega(t)}$ [2, 34].
One could otherwise try to write a semi-implicit scheme as follows: 1) keep the projection on $\{\rho\leqslant 1\}$ in the first Wasserstein step.
As in [29] a pressure term $p^{k+1/2}_{h}$ appears as a Lagrange multiplier in the Wasserstein projection.
2) in the $\mathtt{FR}$/reaction step, relax the constraint $\rho\leqslant 1$ and minimize instead $\rho^{k+1}\in\operatorname*{argmin}\left\{\frac{1}{2h}\mathtt{FR}^{2}(\rho)+%
\int\rho p^{k+1/2}-\int\rho\right\}$, and keep iterating.
This seems to correctly capture the diffusion at least numerically speaking, but raises technical issues in the rigorous proof of convergence and most importantly destroys the variational structure at the discrete level (due to the fact that the reaction step becomes semi-explicit).
We shall use instead an approximation procedure, which preserves the variational structure at the discrete level: it is well-known that the Porous-Medium functional
$$\mathcal{F}_{m}(\rho):=\left\{\begin{array}[]{ll}\int_{\Omega}\frac{\rho^{m}}{%
m-1}&\text{ if }\rho^{m}\in L^{1}(\Omega)\\
+\infty&\text{ otherwise}\end{array}\right.$$
$\Gamma$-converges to $\mathcal{F}_{\infty}$ as $m\to\infty$, see [7].
In the spirit of [40], one should therefore expect that the gradient flow $\rho_{m}$ of $\mathcal{F}_{m}(\rho)-\int\rho$ converges to the gradient flow $\rho_{\infty}$ of the limiting functional $\mathcal{F}(\rho)=\mathcal{F}_{\infty}(\rho)-\int\rho$.
Implementing the splitting scheme for the regular energy functional $\mathcal{F}_{m}(\rho)-\int\rho$ gives a sequence $\rho_{h,m}$, and we shall prove below that $\rho_{h,m}$ converges to a solution of the limiting gradient flow as $m\to\infty$ and $h\to 0$.
However, it is known [17] that the limit depends in general on the interplay between the time-step $h$ and the regularization parameter ($m\to\infty$ here), and for technical reasons we shall enforce the condition
$$mh\to 0\qquad\mbox{as }m\to\infty\mbox{ and }h\to 0.$$
Note that [34] already contained a similar approximation $m\to\infty$ but without exploiting the variational structure of the $m$- gradient flow, and our approach is thus different.
The above gradient-flow structure was already noticed and fully exploited in the ongoing work [10], where existence and uniqueness of weak solutions is proved and numerical simulations are performed needless of any splitting an using directly the $\mathtt{WFR}$ structure.
Here we rather emphasize the fact that the splitting does capture delicate $\Gamma$-convergence phenomena.
In order to make this rigorous, we fix a time step $h>0$ and construct two sequences $(\rho_{h,m}^{k+1/2})_{k}$ and $(\rho_{h,m}^{k})_{k}$, with $\rho_{h,m}^{0}=\rho_{0}$, defined recursively as
$$\displaystyle\left\{\begin{array}[]{l}\rho_{h}^{k+1/2}\in\operatorname*{argmin%
}\limits_{\rho\in\mathcal{M}^{+},\,|\rho|=|\rho_{h}^{k}|}\left\{\frac{1}{2h}%
\mathtt{W}^{2}(\rho,\rho_{h,m}^{k})+\mathcal{F}_{m}(\rho)-\int_{\Omega}\rho%
\right\},\\
\\
\rho_{h}^{k+1}\in\operatorname*{argmin}\limits_{\rho\in\mathcal{M}^{+}}\left\{%
\frac{1}{2h}\mathtt{FR}^{2}(\rho,\rho_{h}^{k+1/2})+\mathcal{F}_{m}(\rho)-\int_%
{\Omega}\rho\right\}.\end{array}\right.$$
(5.2)
As is common in the classical theory of Porous Media Equations [42], we define the pressure as the first variation
$$p_{m}:=F_{m}^{\prime}(\rho)=\frac{m}{m-1}\rho^{m-1}.$$
We accordingly write
$$p_{h,m}^{k+1/2}:=\frac{m}{m-1}(\rho_{h,m}^{k+1/2})^{m-1}\qquad\text{and}\qquad
p%
_{h,m}^{k+1}:=\frac{m}{m-1}(\rho_{h,m}^{k+1})^{m-1}$$
for the discrete pressures.
As in section 3 we denote by $\rho_{h,m}(t),p_{h,m}(t)$ and $\tilde{\rho}_{h,m}(t),\tilde{p}_{h,m}(t)$ the piecewise constant interpolations of $\rho_{h,m}^{k+1},p_{h,m}^{k+1}$ and $\rho_{h,m}^{k+1/2},p_{h,m}^{k+1/2}$, respectively.
Our main result is
Theorem 5.1.
Assume that $\rho_{0}\in BV(\Omega)$, $\rho_{0}\leqslant 1$, and $mh\to 0$ as $h\to 0$ and $m\to\infty$.
Then
for all $T>0$, $\rho_{h,m},\tilde{\rho}_{h,m}$ both converge to some $\rho$ strongly in $L^{1}((0,T)\times\Omega)$, the pressures $p_{h,m},\tilde{p}_{h,m}$ both converge to some $p$ weakly in $L^{2}((0,T),H^{1}(\Omega))$, and $(\rho,p)$ is the unique weak solution of (5.1).
Since we have a $\mathtt{WFR}$ gradient-flow structure, uniqueness should formally follows from the $-1$ geodesic convexity of the driving functional $\mathcal{E}_{\infty}(\rho)-\int_{\Omega}\rho$ with respect to the $\mathtt{WFR}$ distance [24, 26] and the resulting contractivity estimate $\mathtt{WFR}(\rho^{1}(t),\rho^{2}(t))\leq e^{t}\mathtt{WFR}(\rho^{1}_{0},\rho^%
{2}_{0})$.
This is proved rigorously in [10], and therefore we retrieve convergence of the whole sequence $\rho_{h,m}\to\rho$ in Theorem 5.1 (and not only for subsequences).
Given this uniqueness, it is clearly enough to prove convergence along any discrete (sub)sequence, and this is exactly what we show below.
The strategy of proof for Theorem 5.1 is exactly as in section 3, except that we need now the estimates to be uniform in both in $h\to 0$ and $m\to\infty$.
5.1 Estimates and convergences
In this section, we improve the previous estimates from section 3.
We start with an explicit $L^{\infty}$-bound:
Lemma 5.2.
Assume that $\rho_{0}\leqslant 1$, then for all $t\in\operatorname*{\mathbb{R}}^{+}$,
$$\|\rho_{h,m}(t,\cdot)\|_{\infty},\|\tilde{\rho}_{h,m}(t,\cdot)\|_{\infty}%
\leqslant 1.$$
Démonstration.
We argue by induction at the discrete level, starting from $\rho_{0}=\rho^{0}_{h,m}\leqslant 1$ by assumption.
If $\|\rho_{h,m}^{k}\|_{\infty}\leqslant 1$, Otto’s maximum principle [31] implies that $\|\rho_{h,m}^{k+1/2}\|_{\infty}\leqslant\|\rho_{h,m}^{k}\|_{\infty}\leqslant 1$ in the Wasserstein step.
Assume now by contradiction that $E:=\{\rho_{h,m}^{k+1}>1\}$ has positive Lebesgue measure.
The optimality condition (3.9) for the Fisher-Rao minimization step gives, dividing by $\sqrt{\rho_{h,m}^{k+1}}>0$ in $E$,
$$\sqrt{\rho_{h,m}^{k+1}}-\sqrt{\rho_{h,m}^{k+1/2}}=\frac{h}{2}\sqrt{\rho_{h,m}^%
{k+1}}\left(1-\frac{m}{m-1}(\rho_{h,m}^{k+1})^{m-1}\right)$$
Then $1-\frac{m}{m-1}(\rho_{h,m}^{k+1})^{m-1}\leqslant 1-\frac{m}{m-1}<0$ in the right-hand side, hence the desired contradiction $\rho_{h,m}^{k+1}<\rho_{h,m}^{k+1/2}\leqslant 1$.
∎
Noticing that the functional $\frac{1}{m-1}\int\rho^{m}-\int\rho$ corresponds to taking explicitly $F_{2}(z)=z^{m}/m-1$ and $V_{2}(x)\equiv-1$ in section 3, it is easy to reproduce the computations from the proof of Lemma 3.5 and carefully track the dependence of the constants w.r.t $m>1$ to obtain
Lemma 5.3.
There exists $c>0$ such that, for all $m>m_{0}$ large enough and all $h\leq h_{0}$ small enough,
$$(1-ch)\rho^{k+1/2}_{h,m}(x)\leqslant\rho^{k+1}_{h,m}(x)\leqslant(1+h)\rho^{k+1%
/2}_{h,m}(x)\qquad\mbox{a.e.}$$
(5.3)
Note that this holds regardless of any compatibility such as $hm\to 0$.
The key point is here that the lower bound $c$ previously depended on an upper bound $M$ on $\rho^{k+1/2}$ in Lemma 3.5, but since we just obtained in Lemma 5.2 the universal upper bound $\rho^{k+1/2}\leqslant 1$ we end up with a lower bound which is also uniform in $h,m$.
The proof is identical to that of Lemma 3.5 and we omit the details for simplicity.
Recalling that the Wasserstein step is mass-preserving, we obtain by immediate induction and for all $0\leq t\leq T$
$$\|\rho_{h,m}(t)\|_{L^{1}},\,\|\tilde{\rho}_{h,m}(t)\|_{L^{1}}\leqslant e^{T}\|%
\rho_{0}\|_{L^{1}}$$
as well as
$$\|\rho_{h,m}(t)-\tilde{\rho}_{h,m}(t)\|_{L^{1}}\leqslant C_{T}h.$$
(5.4)
Testing successively $\rho=\rho_{h,m}^{k}$ and $\rho=\rho_{h,m}^{k+1/2}$ in (5.2), we get
$$\frac{1}{2h}\left(\mathtt{W}^{2}(\rho_{h,m}^{k},\rho_{h,m}^{k+1/2})+\mathtt{FR%
}^{2}(\rho_{h,m}^{k+1/2},\rho_{h,m}^{k+1})\right)\leqslant\mathcal{F}_{m}(\rho%
_{h,m}^{k})-\mathcal{F}_{m}(\rho_{h,m}^{k+1})+\int_{\Omega}(\rho_{h,m}^{k+1/2}%
-\rho_{h,m}^{k+1}).$$
Using Proposition 2.4 to control $\mathtt{WFR}^{2}\lesssim 2(\mathtt{W}^{2}+\mathtt{FR}^{2})$ and the lower bound in (5.3) yields
$$\displaystyle\frac{1}{4h}\mathtt{WFR}^{2}(\rho^{k+1}_{h,m},\rho^{k}_{h,m})$$
$$\displaystyle\leqslant$$
$$\displaystyle\frac{1}{2h}\left(\mathtt{W}^{2}(\rho_{h,m}^{k},\rho_{h,m}^{k+1/2%
})+\mathtt{FR}^{2}(\rho_{h,m}^{k+1/2},\rho_{h,m}^{k+1})\right)$$
$$\displaystyle\leqslant$$
$$\displaystyle\mathcal{F}_{m}(\rho_{h,m}^{k})-\mathcal{F}_{m}(\rho_{h,m}^{k+1})%
+\int_{\Omega}(\rho_{h,m}^{k+1/2}-\rho_{h,m}^{k+1})$$
$$\displaystyle\leqslant$$
$$\displaystyle\mathcal{F}_{m}(\rho_{h,m}^{k})-\mathcal{F}_{m}(\rho_{h,m}^{k+1})%
+ch\int_{\Omega}\rho_{h,m}^{k+1/2}$$
$$\displaystyle\leqslant$$
$$\displaystyle\mathcal{F}_{m}(\rho_{h,m}^{k})-\mathcal{F}_{m}(\rho_{h,m}^{k+1})%
+che^{T}$$
for all $k\leqslant N:=\lfloor T/h\rfloor$.
Summing over $k$ we get
$$\displaystyle\frac{1}{4h}\sum_{k=0}^{N-1}\mathtt{WFR}^{2}(\rho_{h,m}^{k},\rho_%
{h,m}^{k+1})$$
$$\displaystyle\leqslant$$
$$\displaystyle\mathcal{F}_{m}(\rho_{0})-\mathcal{F}_{m}(\rho_{h,m}^{N})+C_{T}$$
$$\displaystyle\leqslant$$
$$\displaystyle\frac{1}{m-1}\int_{\Omega}\rho_{0}^{m}+C_{T}\leqslant\frac{1}{m-1%
}\int_{\Omega}\rho_{0}+C_{T}\leqslant C_{T},$$
where we used successively $F_{m}\geq 0$ to get rid of $\mathcal{F}_{m}(\rho^{N}_{h,m})$, and $\rho_{0}^{m}\leq\rho_{0}$ for $\rho_{0}\leq 1$ and $m>1$.
Consequently, for all fixed $T>0$ and any $t,s\in[0,T]$ we obtain the classical $\frac{1}{2}$-Hölder estimate
$$\displaystyle\left\{\begin{array}[]{l}\mathtt{WFR}(\rho_{h,m}(t),\rho_{h,m}(s)%
)\leqslant C_{T}|t-s+h|^{1/2},\\
\mathtt{WFR}(\tilde{\rho}_{h,m}(t),\tilde{\rho}_{h,m}(s))\leqslant C_{T}|t-s+h%
|^{1/2}.\end{array}\right.$$
(5.5)
Exploiting the explicit algebraic structure of $F_{m}(z)=\frac{1}{m-1}z^{m}$, compactness in space will be given here by
Lemma 5.4.
If $\rho_{0}\in BV(\Omega)$ then
$$\sup_{t\in[0,T]}\left\{\|\rho_{h,m}(t,\cdot)\|_{BV(\Omega)},\|\tilde{\rho}_{h,%
m}(t,\cdot)\|_{BV(\Omega)}\right\}\leqslant e^{T}\|\rho_{0}\|_{BV(\Omega)}.$$
Démonstration.
The argument closely follows the lines of [18, prop. 5.1].
We first note from [14, thm. 1.1] that the $BV$-norm is nonincreasing during the Wasserstein step,
$$\|\rho_{h,m}^{k+1/2}\|_{BV(\Omega)}\leqslant\|\rho_{h,m}^{k}\|_{BV(\Omega)}.$$
Using as before the implicit function theorem, we show below that $\rho_{h,m}^{k+1}=R(\rho_{h,m}^{k+1/2})$ for some suitable $(1+h)$-Lispchitz function $R$.
By standard $Lip\circ BV$ composition [3] this will prove that
$$\|\rho_{h,m}^{k+1}\|_{BV(\Omega)}\leqslant(1+h)\|\rho_{h,m}^{k+1/2}\|_{BV(%
\Omega)}$$
and will conclude the proof by immediate induction.
Indeed, we already know from (5.3) that $\rho_{h,m}^{k+1/2}$ and $\rho_{h,m}^{k+1}$ share the same support.
In this support and from (3.9) it is easy to see that $\rho=\rho_{h,m}^{k+1}(x)$ is the unique positive solution of
$f(\rho,\rho_{h,m}^{k+1/2}(x))=0$ with
$$f(\rho,\mu)=\sqrt{\rho}\left(1-\frac{h}{2}\left(1-\frac{m}{m-1}\rho^{m-1}%
\right)\right)-\sqrt{\mu}.$$
For $\mu>0$, the implicit function theorem gives the existence of a $\mathcal{C}^{1}$ map $R$ such that $f(\rho,\mu)=0\Leftrightarrow\rho=R(\mu)$, with $R(0)=0$.
An algebraic computation shows moreover that $0<\frac{dR}{d\mu}=-{\frac{\partial_{\mu}f}{\partial_{\rho}f}}_{|\rho=R(\mu)}%
\leqslant(1+h)$ uniformly in $m>1$, hence $R$ is $(1+h)$-Lipschitz as claimed and the proof is complete.
∎
Proposition 5.5.
Up to extraction of a discrete sequence $h\to 0,m\to\infty$, there holds
$$\rho_{h,m},\,\tilde{\rho}_{h,m}\to\rho\qquad\mbox{strongly in }L^{1}(Q_{T})$$
$$p_{h,m}\rightharpoonup p\quad\mbox{and}\quad\tilde{p}_{h,m}\rightharpoonup%
\tilde{p}\qquad\mbox{weakly in all }L^{q}(Q_{T})$$
for all $T>0$.
If in addition $mh\to 0$ then $p=\tilde{p}$.
Démonstration.
The first part of the statement follows exactly as in section 3, exploiting the $\frac{1}{2}$-Hölder estimates (5.5) and the space compactness from Proposition 5.4 in order to apply the Rossi-Savaré theorem [39].
The fact that $\rho_{h,m},\tilde{\rho}_{h,m}$ have the same limit comes from (5.4).
For the pressures, we simply note from $\rho_{h,m}\leqslant 1$ and $m\gg 1$ that $p_{h,m}=\frac{m}{m-1}\rho_{h,m}^{m-1}\leqslant 2\rho_{h,m}$ is bounded in $L^{1}\cap L^{\infty}(Q_{T})$ uniformly in $h,m$ in any finite time interval $[0,T]$.
Thus up to extraction of a further sequence we have $p_{h,m}\rightharpoonup p$ in all $L^{q}(Q_{T})$, and likewise for $\tilde{p}_{h,m}\rightharpoonup\tilde{p}$.
Finally, we only have to check that $p=\tilde{p}$ if $hm\to 0$.
Because $\rho_{h,m},\tilde{\rho}_{h,m}\leqslant 1$ and $z\mapsto z^{m-1}$ is $(m-1)$-Lipschitz on $[0,1]$ we have for all fixed $t\geqslant 0$ that
$$\displaystyle\int_{\Omega}|p_{m,h}(t,\cdot)-\tilde{p}_{m,h}(t,\cdot)|$$
$$\displaystyle=$$
$$\displaystyle\int_{\Omega}\frac{m}{m-1}|\rho_{h,m}^{m-1}(t,\cdot)-\tilde{\rho}%
_{h,m}^{m-1}(t,\cdot)|$$
$$\displaystyle\leqslant$$
$$\displaystyle m\int_{\Omega}|{\rho_{h,m}(t)}-{\tilde{\rho}_{h}(t)}|\leqslant C%
_{T}hm\longrightarrow 0,$$
where we used (5.4) in the last inequality.
Hence $p=\tilde{p}$ and the proof is complete.
∎
In order to pass to the limit in the diffusion term $\operatorname*{div}(\rho\nabla p)$ we first improve the convergence of $\tilde{p}_{h,m}$:
Lemma 5.6.
There exists a constant $C_{T}$, independent of $h$ and $m$, such that
$$\|\tilde{p}_{h,m}\|_{L^{2}((0,T),H^{1}(\Omega))}\leqslant C_{T}$$
for all $T>0$.
Consequently, up to a subsequence, $\tilde{p}_{h,m}$ converges weakly in $L^{2}((0,T),H^{1}(\Omega))$ to $p$.
Démonstration.
The proof is based on the flow interchange technique developed by Matthes, McCann and Savaré in [28].
Let $\eta$ be the (smooth) solution of
$$\left\{\begin{array}[]{l}\partial_{t}\eta=\Delta\eta^{m-1}+\varepsilon\Delta%
\eta,\\
\eta|_{t=0}=\rho^{k+1/2}_{h,m}.\end{array}\right.$$
It is well known [4] that $\eta$ is the Wasserstein gradient flow of
$$\mathcal{G}(\rho):=\int_{\Omega}\frac{\rho^{m-1}}{m-2}+\varepsilon\int_{\Omega%
}\rho\log(\rho).$$
Since $\mathcal{G}$ is geodesically $0$-convex, $\eta$ satisfies the Evolution Variational Inequality (EVI)
$$\left.\frac{1}{2}{\frac{d^{+}}{dt}}\right|_{t=s}\mathtt{W}^{2}(\eta(s),\rho)%
\leqslant\mathcal{G}(\rho)-\mathcal{G}(\eta(s)),$$
for all $s>0$ and for all $\rho\in\operatorname*{\mathcal{P}^{ac}}(\Omega)$, where $\frac{d^{+}}{dt}f(t):=\limsup\limits_{s\rightarrow 0^{+}}\frac{f(t+s)-f(t)}{s}$.
By optimality of $\rho^{k+1/2}_{h,m}$ in (5.2), we obtain that
$$\left.\frac{1}{2}{\frac{d^{+}}{dt}}\right|_{t=s}\mathtt{W}^{2}(\eta(s),\rho_{h%
,m}^{k})\geqslant-h\left.{\frac{d^{+}}{dt}}\right|_{t=s}\mathcal{F}_{m}(\eta(s%
)).$$
Since $\eta$ is smooth due to the regularizing $\varepsilon\Delta$ term, we can legitimately integrate by parts for all $s>0$
$$\displaystyle\frac{d}{ds}\mathcal{F}_{m}(\eta(s))$$
$$\displaystyle=$$
$$\displaystyle\int_{\Omega}\frac{m}{m-1}\eta(s)^{m-1}(\Delta\eta(s)^{m-1}+%
\varepsilon\Delta\eta(s))$$
$$\displaystyle=$$
$$\displaystyle-\int_{\Omega}\frac{m}{m-1}|\nabla\eta(s)^{m-1}|^{2}-\varepsilon%
\int_{\Omega}m\eta(s)^{m-2}|\nabla\eta(s)|^{2}$$
$$\displaystyle\leqslant$$
$$\displaystyle-\int_{\Omega}\frac{m}{m-1}|\nabla\eta(s)^{m-1}|^{2}=-\frac{m-1}{%
m}\int_{\Omega}\left|\nabla\left(\frac{m}{m-1}\eta(s)^{m-1}\right)\right|^{2}$$
Remarking that $\frac{m}{m-1}\eta(s)^{m-1}\to\frac{m}{m-2}\rho_{h,m}^{k+1/2}=p_{h,m}^{k+1/2}$ as $s\to 0$, an easy lower semi-continuity argument gives that
$$\int_{\Omega}\frac{m-1}{m}|\nabla p_{h,m}^{k+1/2}|^{2}=\int_{\Omega}\frac{m}{m%
-1}|\nabla(\rho_{h,m}^{k+1/2})^{m-1}|^{2}\leqslant\liminf_{s\searrow 0}\left.{%
\frac{d^{+}}{dt}}\right|_{t=s}\mathcal{F}_{m}(\eta(s)).$$
Then we have
$$\displaystyle h\int_{\Omega}\frac{m-1}{m}|\nabla p_{h,m}^{k+1/2}|^{2}$$
$$\displaystyle\leqslant\mathcal{F}_{m-1}(\rho_{h,m}^{k})-\mathcal{F}_{m-1}(\rho%
_{h,m}^{k+1/2})$$
$$\displaystyle+\varepsilon\left(\int_{\Omega}\rho_{h,m}^{k}\log(\rho_{h,m}^{k})%
-\int_{\Omega}\rho_{h,m}^{k+1/2}\log(\rho_{h,m}^{k+1/2})\right).$$
First arguing as in Proposition 3.7 to control
$$\mathcal{F}_{m-1}(\rho_{h,m}^{k+1})\leqslant\mathcal{F}_{m-1}(\rho_{h,m}^{k+1/%
2})+C_{T}h,$$
and then passing to the limit $\varepsilon\searrow 0$, we obtain
$$h\int_{\Omega}\frac{m-1}{m}|\nabla p_{h,m}^{k+1/2}|^{2}\leqslant\mathcal{F}_{m%
-1}(\rho_{h,m}^{k})-\mathcal{F}_{m-1}(\rho_{h,m}^{k+1})+C_{T}h.$$
Summing over $k$ gives
$$\int_{0}^{T}\int_{\Omega}|\nabla\tilde{p}_{h,m}(t,x)|^{2}\,dxdt\leqslant\frac{%
m}{m-1}(\mathcal{F}_{m-1}(\rho_{0})-\mathcal{F}_{m-1}(\rho^{N}_{h,m})+C_{T})%
\leqslant 2\mathcal{F}_{m-1}(\rho_{0})+C_{T}$$
for all $T<+\infty$.
Due to $\rho_{0}\leqslant 1$ and $m\gg 1$ we can bound $\mathcal{F}_{m-1}(\rho_{0})=\frac{1}{m-2}\int\rho_{0}^{m-1}\leqslant\frac{1}{m%
-2}\int\rho_{0}\leqslant\|\rho_{0}\|_{L^{1}(\Omega)}$ and the result finally follows.
∎
5.2 Properties of the pressure $p$ and conclusion
We start by showing that the limits $\rho,p$ satisfy the compatibility conditions in (5.1).
Lemma 5.7.
There holds
$$0\leqslant\rho,p\leqslant 1\quad\text{and}\quad p(1-\rho)=0\,\text{ a.e. in }Q%
_{T}.$$
Démonstration.
By Lemma 5.2 it is obvious that $0\leqslant\rho\leqslant 1$ and $0\leqslant p\leqslant 1$ are inherited from $0\leqslant\rho_{h,m}\leqslant 1$ and $0\leqslant p_{h,m}=\frac{m}{m-1}\rho^{m-1}_{h,m}\leqslant\frac{m}{m-1}$.
In order to prove that $p(1-\rho)=0$, we first observe that
$$p_{h,m}(1-\rho_{h,m})\to 0\qquad\mbox{a.e. in }Q_{T}.$$
Indeed, since $\rho_{h,m}\to\rho$ strongly in $L^{1}(Q_{T})$ we have $\rho_{h,m}(t,x)\to\rho(t,x)$ a.e.
If the limit $\rho(t,x)<1$ then $\rho_{h,m}(t,x)\leqslant(1-\varepsilon)$ for small $h$ and large $m$.
Hence $p_{h,m}(t,x)=\frac{m}{m-1}\rho_{h,m}^{m-1}\leqslant\frac{m}{m-1}(1-\varepsilon%
)^{m-1}\to 0$ while $1-\rho_{h,m}$ remains bounded, and therefore the product $p_{h,m}(1-\rho_{h,m})\to 0$.
Now if the limit $\rho(t,x)=1$ then the pressure $p_{h,m}=\frac{m}{m-1}\rho_{h,m}^{m-1}\leqslant\frac{m}{m-1}$ remains bounded, while $1-\rho_{h,m}(t,x)\to 0$ hence the product goes to zero in this case too.
Thanks to the uniform $L^{\infty}$ bounds $\rho_{h,m}\leqslant 1$ and $p_{h,m}\leqslant\frac{m}{m-1}\leqslant 2$ we can apply Lebesgue’s convergence theorem to deduce from this pointwise a.e. convergence that, for all fixed nonnegative $\varphi\in\mathcal{\mathcal{missing}}C^{\infty}_{c}(Q_{T})$, there holds
$$\lim\int_{Q_{T}}p_{h,m}(1-\rho_{h,m})\varphi=0.$$
On the other hand since $\rho_{h,m}\to\rho$ strongly in $L^{1}(Q_{T})$ hence a.e, and because $0\leqslant\rho_{h,m}\leqslant 1$, we see that $(1-\rho_{h,m})\varphi\to(1-\rho)\varphi$ in all $L^{q}(Q_{T})$.
From Proposition 5.5 we also had that $p_{h,m}\rightharpoonup p$ in all $L^{q}(Q_{T})$, hence by strong-weak convergence we have that
$$\int_{Q_{T}}p(1-\rho)\varphi=\lim\int_{Q_{T}}p_{h,m}(1-\rho_{h,m})\varphi=0$$
for all $\varphi\geqslant 0$.
Because $p(1-\rho)\geqslant 0$ we conclude that $p(1-\rho)=0$ a.e. in $Q_{T}$ and the proof is achieved.
∎
We end this section with
Proof of Theorem 5.1.
We only sketch the argument and refer to [18] for the details.
Fix any $0<t_{1}<t_{2}$ and $\varphi\in\mathcal{C}^{2}_{c}(\operatorname*{\mathbb{R}}^{d})$.
Exploiting the Euler-Lagrange equations (3.6)(3.9) and summing from $k=k_{1}=\lfloor t_{1}/h\rfloor$ to $k=k_{2}-1=\lfloor t_{2}/h\rfloor-1$, we first obtain
$$\int_{\operatorname*{\mathbb{R}}^{d}}\rho_{h,m}(t_{2})\varphi-\rho_{h,m}(t_{1}%
)\varphi+\int_{k_{1}h}^{k_{2}h}\int_{\operatorname*{\mathbb{R}}^{d}}\tilde{%
\rho}_{h,m}\nabla\tilde{p}_{h,m}\cdot\nabla\varphi=-\int_{k_{1}h}^{k_{2}h}\int%
_{\operatorname*{\mathbb{R}}^{d}}\rho_{h,m}(1-p_{h,m})\varphi+R(h,m),$$
where the remainder $R(h,m)\to 0$ for fixed $\varphi$.
The strong convergence $\rho_{h,m},\tilde{\rho}_{h,m}\to\rho$ and the weak convergences $\nabla\tilde{p}_{h,m}\rightharpoonup\nabla\tilde{p}=\nabla p$ and $p_{h,m}\rightharpoonup p$ are then enough pass to the limit to get the corresponding weak formulation for all $0<t_{1}<t_{2}$.
Moreover since the limit $\rho\in\mathcal{C}([0,T];\mathcal{M}^{+}_{\mathtt{WFR}})$ the initial datum $\rho(0)=\rho_{0}$ is taken at least in the sense of measures.
This gives an admissible weak formulation of (5.1), and the proof is complete.
∎
5.3 Numerical simulation
The constructive scheme (5.2) naturally leads to a fully discrete algorithm, simply discretizing the minimization problem in space for each $\mathtt{W},\mathtt{FR}$ step.
We use again the ALG2-JKO scheme [6] for the Wasserstein steps.
As already mentioned the Fisher-Rao step is a mere convex pointwise minimization problem, here explicitly given by: for all $x\in\Omega$,
$$\rho_{h,m}^{k+1}(x)=\operatorname*{argmin}_{\rho\geq 0}\left\{4\left|\sqrt{%
\rho}-\sqrt{\rho_{h,m}^{k+1/2}(x)}\right|^{2}+2h\left(\frac{\rho^{m}}{m-1}-1%
\right)\right\}$$
and poses no difficulty in the practical implementation using a standard Newton method.
Figure 3 depicts the evolution of the numerical solution $\rho_{h,m}$ for $m=100$ and with a time step $h=0.005$.
We remark that the tumor first saturates the constraint ($\rho\nearrow 1$) in its initial support, and then starts diffusing outwards.
This is consistent with the qualitative behaviour described in [34].
6 A tumor growth model with nutrient
In this section we use the same approach for the following tumor growth model with nutrients, appearing e.g. in [34]
$$\displaystyle\left\{\begin{array}[]{l}\partial_{t}\rho-\operatorname*{div}(%
\rho\nabla p)=\rho\left((1-p)(c+c_{1})-c_{2}\right),\\
\partial_{t}c-\Delta c=-\rho c,\\
0\leqslant\rho\leqslant 1,\\
p\geqslant 0\mbox{ and }p(1-\rho)=0,\\
\rho_{|t=0}=\rho_{0},\,c_{|t=0}=c_{0}.\end{array}\right.$$
(6.1)
Here $c_{1}$ and $c_{2}$ are two positive constants, and the nutrient $c$ is now diffusing in $\Omega$ in addition to begin simply consumed by the tumor $\rho$, according to the second equation.
For technical convenience we work here on a convex bounded domain $\Omega\subset\mathbb{R}^{d}$, endowed with natural Neumann boundary conditions for both $\rho$ and $c$.
Contrarily to section 5 this is not a $\mathtt{WFR}$ gradient flow anymore, and we therefore introduce a semi-implicit splitting scheme.
Starting from the initial datum $\rho^{0}_{h,m}:=\rho_{0},c_{h,m}^{0}:=c_{0}$ we construct four sequences $\rho_{h,m}^{k+1/2},\rho_{h,m}^{k},c_{h,m}^{k+1/2},c_{h,m}^{k}$, defined recursively as
$$\displaystyle\left\{\begin{array}[]{l}\rho_{h,m}^{k+1/2}\in\operatorname*{%
argmin}\limits_{\rho\in\mathcal{M}^{+},|\rho|=|\rho_{h,m}^{k}|}\left\{\frac{1}%
{2h}\mathtt{W}^{2}(\rho,\rho_{h,m}^{k})+\mathcal{F}_{m}(\rho)\right\},\\
\\
\par c_{h,m}^{k+1/2}\in\operatorname*{argmin}\limits_{c\in\mathcal{M}^{+},|c|=%
|c_{h,m}^{k}|}\left\{\frac{1}{2h}\mathtt{W}^{2}(c,c_{h,m}^{k})+\mathcal{E}(%
\rho)\right\},\end{array}\right.$$
(6.2)
and
$$\displaystyle\left\{\begin{array}[]{l}\rho_{h,m}^{k+1}\in\operatorname*{argmin%
}\limits_{\rho\in\mathcal{M}^{+}}\left\{\frac{1}{2h}\mathtt{FR}^{2}(\rho,\rho_%
{h,m}^{k+1/2})+\mathcal{E}_{1,m}(\rho|c_{h,m}^{k+1/2})\right\},\\
\\
c_{h,m}^{k+1}\in\operatorname*{argmin}\limits_{c\in\mathcal{M}^{+}}\left\{%
\frac{1}{2h}\mathtt{FR}^{2}(c,c_{h,m}^{k+1/2})+\mathcal{E}_{2}(c|\rho_{h,m}^{k%
+1/2})\right\},\par\end{array}\right.$$
(6.3)
where
$$\mathcal{E}(\rho):=\int_{\Omega}\rho\log(\rho),$$
$$\mathcal{E}_{1,m}(\rho|c):=\int_{\Omega}\left(c+c_{1}\right)\frac{\rho^{m}}{m-%
1}+\int_{\Omega}(c_{2}-c-c_{1})\rho,$$
and
$$\mathcal{E}_{2}(c|\rho):=\int_{\Omega}\rho c.$$
As earlier it is easy to see that these sequences are well-defined (i-e there exists a unique minimizer for each step), and the pressures are defined as before as
$$p_{h,m}^{k+1/2}:=\frac{m}{m-1}(\rho_{h,m}^{k+1/2})^{m-1}\quad\mbox{and}\quad p%
_{h,m}^{k+1}:=\frac{m}{m-1}(\rho_{h,m}^{k+1})^{m-1}.$$
We denote again by $a_{h,m}(t),\tilde{a}_{h,m}(t)$ the piecewise constant interpolation of any discrete quantity $a^{k+1}_{h,m},a^{k+1/2}_{h,m}$ respectively.
Our main result reads:
Theorem 6.1.
Assume $\rho_{0}\in BV(\Omega)$ with $\rho_{0}\leqslant 1$ and $c_{0}\in L^{\infty}(\Omega)\cap BV(\Omega)$. Then $\rho_{h,m}$ and $\tilde{\rho}_{h,m}$ strongly converge to $\rho$ in $L^{1}((0,T)\times\Omega)$ and $c_{h,m}$ and $\tilde{c}_{h,m}$ strongly converge to $c$ in $L^{1}((0,T)\times\Omega)$ when $h\searrow 0$ and $m\nearrow+\infty$.
Moreover, if $mh\rightarrow 0$, then $p_{h,m},\tilde{p}_{h,m}$ converge weakly in $L^{2}((0,T),H^{1}(\Omega))$ to a unique $p$, and $(\rho,p,c)$ is a solution of (6.1).
Note that uniqueness of solutions would result in convergence of the whole sequence.
Uniqueness was proved in [34, thm. 4.2] for slightly more regular weak solutions, but we did not push in this direction for the sake of simplicity.
The method of proof is almost identical to section 5 so we only sketch the argument and emphasize the main differences.
We start by recalling the optimality conditions for the scheme (6.2)-(6.3).
The Euler-Lagrange equations for the tumor densities in the Wasserstein and Fisher-Rao steps are
$$\displaystyle\left\{\begin{array}[]{l}\rho_{h,m}^{k+1/2}\nabla p_{h,m}^{k+1/2}%
=\frac{\nabla\varphi}{h}\rho_{h,m}^{k+1/2},\\
\sqrt{\rho_{h,m}^{k+1}}-\sqrt{\rho_{h,m}^{k+1/2}}=\frac{h}{2}\sqrt{\rho_{h,m}^%
{k+1}}\left((1-p_{h,m}^{k+1})(c_{h,m}^{k+1/2}+c_{1})-c_{2}\right),\end{array}\right.$$
(6.4)
where $\varphi$ is a (backward) Kantorovich potential for $\mathtt{W}(\rho_{h,m}^{k+1/2},\rho_{h,m}^{k})$.
For the nutrient, the Euler-Lagrange equations are
$$\displaystyle\left\{\begin{array}[]{l}\nabla c_{h,m}^{k+1/2}=\frac{\nabla\psi}%
{h}c_{h,m}^{k+1/2},\\
\sqrt{c_{h,m}^{k+1}}-\sqrt{c_{h,m}^{k+1/2}}=-\frac{h}{2}\sqrt{c_{h,m}^{k+1}}%
\rho_{h,m}^{k+1/2},\end{array}\right.$$
(6.5)
with $\psi$ a Kantorovich potential for $\mathtt{W}(c_{h,m}^{k+1/2},c_{h,m}^{k})$.
Using the optimality conditions for the Fischer-Rao steps, we obtain directly the following $L^{\infty}$ bounds:
Lemma 6.2.
For all $k\geqslant 0$
$$\|c_{h,m}^{k+1}\|_{L^{\infty}(\Omega)}\leqslant\|c_{h,m}^{k+1/2}\|_{L^{\infty}%
(\Omega)}\leqslant\|c_{h,m}^{k}\|_{L^{\infty}(\Omega)},$$
and at the continuous level
$$\|c_{h,m}(t,\cdot)\|_{L^{\infty}(\Omega)},\|\tilde{c}_{h,m}(t,\cdot)\|_{L^{%
\infty}(\Omega)}\leqslant\|c_{0}\|_{L^{\infty}(\Omega)}\qquad\forall\,t\geq 0.$$
Moreover,
$$\|\rho_{h,m}(t,\cdot)\|_{\infty},\|\tilde{\rho}_{h,m}(t,\cdot)\|_{\infty}\leqslant
1$$
and there exists $c_{T}\equiv c_{T}(\|c_{0}\|_{L^{\infty}}),C_{T}\equiv C_{T}(\|c_{0}\|_{L^{%
\infty}})>0$ such that
$$\begin{array}[]{c}(1-c_{T}h)\rho_{h,m}^{k+1/2}(x)\leqslant\rho_{h,m}^{k+1}(x)%
\leqslant(1+C_{T}h)\rho_{h,m}^{k+1/2}(x)\qquad\mbox{a.e. in }\Omega.\\
(1-h)c_{h,m}^{k+1/2}(x)\leqslant c_{h,m}^{k+1}(x)\leqslant c_{h,m}^{k+1/2}(x)%
\qquad\mbox{a.e. in }\Omega.\end{array}$$
(6.6)
Démonstration.
The proof of the estimates on $c_{h,m}$ and $\tilde{c}_{h,m}$ is obvious because one step of Wasserstein gradient flow with the Boltzmann entropy decreases the $L^{\infty}$-norm in (6.2) (see [32, 1]), and, because the product $\sqrt{c_{h,m}^{k+1}}\rho_{h,m}^{k+1/2}$ is nonnegative in (6.5), the $L^{\infty}$-norm is also nonincreasing during the Fischer-Rao step.
The proof for $\rho_{h,m}$ and $\tilde{\rho}_{h,m}$ is the same as in lemma 5.2.
Using the fact that $\|\tilde{\rho}_{h,m}(t,\cdot)\|_{\infty}\leqslant 1$, we see that the term $\Phi(p_{h,m}^{k+1},c_{h,m}^{k+1/2}):=(1-p_{h,m}^{k+1})(c_{h,m}^{k+1/2}+c_{1})-%
c_{2}$ in (6.4) is bounded in $L^{\infty}$ uniformly in $k$.
This allows to argue exactly as in Lemma 3.5 to retrieve the estimate (6.6) and concludes the proof.
∎
With these bounds it is easy to prove as in proposition 3.15 that
$$\begin{array}[]{c}\mathcal{F}_{m}(\rho_{h,m}^{k+1})\leqslant\mathcal{F}_{m}(%
\rho_{h,m}^{k+1/2})+C_{T}h,\\
\mathcal{E}_{1,m}(\rho_{h,m}^{k+1/2}|c_{h,m}^{k+1/2})-\mathcal{E}_{1,m}(\rho_{%
h,m}^{k+1}|c_{h,m}^{k+1/2})\leqslant C_{T}h,\\
\mathcal{E}(c_{h,m}^{k+1})\leqslant\mathcal{E}(c_{h,m}^{k+1/2})+C_{T}h,\\
\mathcal{E}_{2}(c_{h,m}^{k+1/2}|\rho_{h,m}^{k+1/2})-\mathcal{E}_{2}(c_{h,m}^{k%
+1}|\rho_{h,m}^{k+1/2})\leqslant C_{T}h,\end{array}.$$
for some $C_{T}$ independent of $m$.
Then we obtain the usual $\frac{1}{2}$-Hölder estimates in time with respect to the $\mathtt{WFR}$ distance, which in turn implies that $\rho_{h,m},\tilde{\rho}_{h,m}$ converge to some $\rho\in L^{\infty}([0,T],L^{1}(\Omega))$ and $c_{h,m},\tilde{c}_{h,m}$ converge to some $c\in L^{\infty}([0,T],L^{1}(\Omega))$ pointwise in time with respect to $\mathtt{WFR}$, see (3.20), Proposition 3.8, and (3.22) for details.
As before we need to improve the convergence in order to pass to the limit in the nonlinear terms.
For $\rho_{h,m}$ and $\tilde{\rho}_{h,m}$, this follows from
Lemma 6.3.
For all $T>0$, if $\rho_{0},c_{0}\in BV(\Omega)$,
$$\begin{array}[]{c}\sup\limits_{t\in[0,T]}\left\{\|\rho_{h,m}(t,\cdot)\|_{BV(%
\Omega)}+\|c_{h,m}(t,\cdot)\|_{BV(\Omega)}\right\}\leqslant e^{C_{T}T}(\|\rho_%
{0}\|_{BV(\Omega)}+\|c_{0}\|_{BV(\Omega)})\\
\sup\limits_{t\in[0,T]}\left\{\|\tilde{\rho}_{h,m}(t,\cdot)\|_{BV(\Omega)}+\|%
\tilde{c}_{h,m}(t,\cdot)\|_{BV(\Omega)}\right\}\leqslant e^{C_{T}T}(\|\rho_{0}%
\|_{BV(\Omega)}+\|c_{0}\|_{BV(\Omega)}).\end{array}$$
Démonstration.
The argument is a generalization of Lemma 5.4, see [18, remark 5.1]. First, the $BV$-norm is nonincreasing during the Wasserstein step, [14, thm. 1.1],
$$\|\rho_{h,m}^{k+1/2}\|_{BV(\Omega)}\leqslant\|\rho_{h,m}^{k}\|_{BV(\Omega)}%
\text{ and }\|c_{h,m}^{k+1/2}\|_{BV(\Omega)}\leqslant\|c_{h,m}^{k}\|_{BV(%
\Omega)}.$$
Arguing as in Lemma 5.4, we observe that, inside $\mathop{\rm supp}\rho_{h,m}^{k+1/2}=\mathop{\rm supp}\rho_{h,m}^{k+1}$, the minimizer $\rho=\rho_{h,m}^{k+1}(x)$ is the unique positive solution of $f(\rho,\rho_{h,m}^{k+1/2}(x),c_{h,m}^{k+1/2}(x))=0$, with
$$f(\rho,\mu,c)=\sqrt{\rho}\left(1-\frac{h}{2}\left(\left(1-\frac{m}{m-1}\rho^{m%
-1}\right)(c+c_{1})-c_{2}\right)\right)-\sqrt{\mu}.$$
For $\mu>0$ the implicit function theorem gives as before a $\mathcal{C}^{1}$ map $R$ such that $f(\rho,\mu,c)=0\Leftrightarrow\rho=R(\mu,c)$.
An easy algebraic computation and (6.6) then gives $0<\partial_{\mu}R(\mu,c)\leqslant(1+C_{T}h)$ and $|\partial_{c}R(\mu,c)|\leqslant C_{T}h$ for some constant $C_{T}>0$ independent of $h,m,k$.
This implies that
$$\displaystyle\|\rho_{h,m}^{k+1}\|_{BV(\Omega)}$$
$$\displaystyle\leqslant$$
$$\displaystyle(1+C_{T}h)\|\rho_{h,m}^{k+1/2}\|_{BV(\Omega)}+C_{T}h\|c_{h,m}^{k+%
1/2}\|_{BV(\Omega)}$$
$$\displaystyle\leqslant$$
$$\displaystyle(1+C_{T}h)\|\rho_{h,m}^{k}\|_{BV(\Omega)}+C_{T}h\|c_{h,m}^{k}\|_{%
BV(\Omega)}.$$
The same argument shows that
$$\|c_{h,m}^{k+1}\|_{BV(\Omega)}\leqslant(1+C_{T}h)\|c_{h,m}^{k}\|_{BV(\Omega)}+%
C_{T}h\|\rho_{h,m}^{k}\|_{BV(\Omega)},$$
and a simple induction allows to conclude.
∎
Proposition 6.4.
Up to extraction of a discrete sequence $h\to 0,m\to+\infty$,
$$\rho_{h,m},\,\tilde{\rho}_{h,m}\to\rho\qquad\mbox{strongly in }L^{1}(Q_{T})$$
$$p_{h,m}\rightharpoonup p\mbox{ and }\tilde{p}_{h,m}\rightharpoonup\tilde{p}%
\qquad\mbox{weakly in all }L^{q}(Q_{T})$$
for all $T>0$.
If in addition $mh\to 0$ then $p=\tilde{p}\in L^{2}((0,T),H^{1}(\Omega))$ and $(\rho,p)$ satisfies
$$0\leqslant\rho,p\leqslant 1\quad\text{and}\quad p(1-\rho)=0\qquad\text{a.e. in%
}Q_{T}.$$
Démonstration.
The proof is the same as Proposition 5.5, Lemma 5.6, and Lemma 5.7.
∎
In order to conclude the proof of Theorem 6.1 we only need to check that $\rho,p,c$ satisfy the weak formulation of (6.1): the strong convergence of $\rho_{h,m},c_{h,m}$ and the weak convergence of $p_{h,m}$ are enough to take the limit in the nonlinear terms as in section 5.2, and we omit the details.
Acknowledgements
We warmly thank G. Carlier for fruitful discussions and suggesting us the problem in section 3
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On Hadamard’s global inverse function theorem
Michael Ruzhansky and Mitsuru Sugimoto
Michael Ruzhansky:
Department of Mathematics
Imperial College London
180 Queen’s Gate, London SW7 2AZ, UK
E-mail address [email protected]
Mitsuru Sugimoto:
Graduate School of Mathematics
Nagoya University
Furocho, Chikusa-ku, Nagoya 464-8602, Japan
E-mail address [email protected]
(Date:: November 26, 2020)
Abstract.
Hadamard’s global inverse theorem provides conditions for a function to be globally
invertible on ${\mathbb{R}}^{n}$. We show that for $n\geq 3$ the conditions are robust enough for the
conclusion to hold even if we
relax them by removing the assumption at a finite number of points.
As a consequence, we
get a global inverse function theorem for homogeneous functions.
Key words and phrases:Inverse function theorem
1991 Mathematics Subject Classification: Primary 26B10; Secondary 26B05, 26-01
The first author was supported by the EPSRC
Leadership Fellowship EP/G007233/1.
1. Introduction
A differentiable map between manifolds is called a $C^{1}$-diffeomorphism if
it is one-to-one and its inverse is also differentiable. We will mostly discuss
the mappings $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$,
in which case we denote its Jacobian by $\det Df:=\det(\partial f_{i}/\partial x_{j})$.
The Hadamard global
inverse function theorem states:
Theorem 1.1.
A $C^{1}$-map $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a $C^{1}$-diffeomorphism
if and only if
the Jacobian $\det Df(w)$ never vanishes and
$|f(y)|\to\infty$ whenever $|y|\to\infty$.
This theorem goes back to Hadamard
[Had06a, Had06b, Had].
In fact, in 1972 W. B. Gordon wrote
“This theorem goes back at least to Hadamard, but it does not appear to be
‘well-known’. Indeed, I have found that most people do not believe it when
they see it and that the skepticism of some persists until they see two proofs.”
The reason behind this is that while we know that the function is locally a $C^{1}$-diffeomorphism
by the usual local inverse function theorem,
the condition that $|f(y)|\to\infty$ as $|y|\to\infty$, guarantees that the function is
both injective and, more importantly, surjective on the whole of ${\mathbb{R}}^{n}$.
And indeed, W. B. Gordon proceeds in [Gor72]
by giving two different proofs for it, for
$C^{2}$ and for $C^{1}$ mappings.
Here we want to show that for dimensions $n\geq 3$
the Hadamard global inverse function theorem is, in fact, a remarkably
robust statement, in the sense that the conclusion remains still valid even if we
relax the assumption, in some sense rather
substantially, almost removing it at a finite number of points. This is very
often not the case in many statements, most notably, the famous one being the
‘hairy ball theorem’, which fails completely if we assume that the vector field
may be not differentiable at one point.
In fact, here we will show the following:
Theorem 1.2.
Let $n\geq 3$. Let $a\in{\mathbb{R}}^{n}$.
Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be such that
$f$ is $C^{1}$ on ${\mathbb{R}}^{n}\backslash\{a\}$, with
$\det Df\not=0$ on ${\mathbb{R}}^{n}\backslash\{a\}$, and that $f$ is
continuous at $a$.
Let $b:=f(a)$, and assume that
$f({\mathbb{R}}^{n}\backslash\{a\})\subset{\mathbb{R}}^{n}\backslash\{b\}$
and that $|y|\to\infty$ implies $|f(y)|\to\infty$.
Then
the mapping $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a global homeomorphism and its restriction
$f:{\mathbb{R}}^{n}\backslash\{a\}\to{\mathbb{R}}^{n}\backslash\{b\}$
is a global $C^{1}$-diffeomorphism.
We note that if we assume that $f$ is also differentiable at $a$, Theorem
1.2 together with the local inverse function
theorem applied at $a$ imply Theorem 1.1.
An important consequence of this is the global inverse function theorem for
homogeneous functions, important for a better understanding of
different phenomena, for example of the global invertibility of
the Hamiltonian flows. Here, we take $a=b=0$, and it is a natural example since
a homogeneous function fails to be smooth at the
origin, or fails to be in $C^{k}$ for $k$ depending on the homogeneity order.
Thus, we have the following for any $k\in{\mathbb{N}}\cup\{\infty\}$:
Corollary 1.3.
Let $n\geq 3$ and $1\leq k\leq\infty$.
Let $f:{\mathbb{R}}^{n}\backslash 0\to{\mathbb{R}}^{n}$
be a positively homogeneous mapping of order
$\varkappa>0$, i.e.
$$f(\tau\xi)=\tau^{\varkappa}f(\xi)\textrm{ for all }\tau>0,\ \xi\not=0.$$
Assume that $f\in C^{k}({\mathbb{R}}^{n}\backslash 0)$ and
that its Jacobian never vanishes on ${\mathbb{R}}^{n}\backslash 0$.
Then $f$ is bijective from ${\mathbb{R}}^{n}\backslash 0$ to
${\mathbb{R}}^{n}\backslash 0$, its global inverse satisfies
$f^{-1}\in C^{k}({\mathbb{R}}^{n}\backslash 0)$ and is positively
homogeneous of order $1/\varkappa$.
Moreover, if we extend $f$ to ${\mathbb{R}}^{n}$ by setting $f(0)=0$,
the extension is a global homeomorphism on ${\mathbb{R}}^{n}$.
We remark that we always assume $n\geq 3$ in
Theorem 1.2
and Corollary 1.3.
As for the case $n=2$, they are not always true
because the Jacobian of $f(x,y)=(x^{2}-y^{2},2xy)$ never vanishes
on ${\mathbb{R}}^{n}\backslash 0$ but $f$ is not globally invertible since
$f(x,y)=f(-x,-y)$.
We will prove Theorem 1.2 but note
that exactly the same proof also yields the following further
extension:
Theorem 1.4.
Let $n\geq 3$.
Let $A\subset{\mathbb{R}}^{n}$ be a closed set.
Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be such that
$f$ is $C^{1}$ on ${\mathbb{R}}^{n}\backslash A$, with
$\det Df\not=0$ on ${\mathbb{R}}^{n}\backslash A$, that $f$ is
continuous and injective on $A$, and that
${\mathbb{R}}^{n}\backslash f(A)$ is simply connected.
Assume that
$f({\mathbb{R}}^{n}\backslash A)\subset{\mathbb{R}}^{n}\backslash f(A)$ and that
$|y|\to\infty$ implies $|f(y)|\to\infty$.
Then
the mapping $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a global homeomorphism and its restriction
$f:{\mathbb{R}}^{n}\backslash A\to{\mathbb{R}}^{n}\backslash f(A)$
is a global $C^{1}$-diffeomorphism.
2. Proof
First we observe that by translation (by $a$) in $x$ and
by subtracting $b$ from $f$, we may assume without
loss of generality that $a=b=0$.
To prove Theorem 1.2,
we start with preliminary statements.
Lemma 2.1.
Let $F\subset{\mathbb{R}}^{n}\backslash 0$. Then $F$ is compact
in ${\mathbb{R}}^{n}\backslash 0$ if and only if it is compact in ${\mathbb{R}}^{n}$.
Proof.
Assume that $F\subset{\mathbb{R}}^{n}\backslash 0$ is compact in
${\mathbb{R}}^{n}\backslash 0$. Let $F\subset\bigcup_{\alpha}V_{\alpha}$ for a
family of sets $V_{\alpha}$ which are open in ${\mathbb{R}}^{n}$. Then
$$F\subset{\left({\bigcup_{\alpha}V_{\alpha}}\right)}\cap({\mathbb{R}}^{n}%
\backslash 0)=\bigcup_{\alpha}{\left({V_{\alpha}\cap({\mathbb{R}}^{n}%
\backslash 0)}\right)},$$
so that $F$ is covered by a family of sets
$V_{\alpha}\cap({\mathbb{R}}^{n}\backslash 0)$ which are open in ${\mathbb{R}}^{n}\backslash 0$.
Since $F$ is compact in
${\mathbb{R}}^{n}\backslash 0$, there is a finite subfamily $V_{j}$, $j=1,\cdots,m$,
such that
$$F\subset\bigcup_{j=1}^{m}{\left({V_{j}\cap({\mathbb{R}}^{n}\backslash 0)}%
\right)}={\left({\bigcup_{j=1}^{m}V_{j}}\right)}\cap({\mathbb{R}}^{n}%
\backslash 0).$$
Hence $F\subset\bigcup_{j=1}^{m}V_{j}$, so that $F$ is compact in
${\mathbb{R}}^{n}$.
Conversely, assume that $F\subset{\mathbb{R}}^{n}\backslash 0$ is compact in
${\mathbb{R}}^{n}$, and let
$F\subset\bigcup_{\alpha}U_{\alpha}$, for a family of sets
$U_{\alpha}\subset{\mathbb{R}}^{n}\backslash 0$ which are open in
${\mathbb{R}}^{n}\backslash 0$. Then
$U_{\alpha}=V_{\alpha}\cap({\mathbb{R}}^{n}\backslash 0)$, for
some $V_{\alpha}$ open in ${\mathbb{R}}^{n}$. Hence $F\subset\bigcup_{\alpha}V_{\alpha}$,
and by compactness of $F$ in ${\mathbb{R}}^{n}$, there is a finite subcovering
$F\subset\bigcup_{j=1}^{m}V_{j}$.
Since $F\subset{\mathbb{R}}^{n}\backslash 0$, we have
$$F\subset{\left({\bigcup_{j=1}^{m}V_{j}}\right)}\cap({\mathbb{R}}^{n}\backslash
0%
)=\bigcup_{j=1}^{m}{\left({V_{j}\cap({\mathbb{R}}^{n}\backslash 0)}\right)}=%
\bigcup_{j=1}^{m}U_{j},$$
which proves that $F$ is compact in ${\mathbb{R}}^{n}\backslash 0$.
∎
We recall that a mapping $f$ is called proper if $f^{-1}(K)$
is compact whenever $K$ is compact.
Corollary 2.2.
Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be proper and such that
$f(0)=0$ and $f({\mathbb{R}}^{n}\backslash 0)\subset{\mathbb{R}}^{n}\backslash 0$.
Then the restriction $f:{\mathbb{R}}^{n}\backslash 0\to{\mathbb{R}}^{n}\backslash 0$
is proper.
Proof.
Let $K\subset{\mathbb{R}}^{n}\backslash 0$ be compact in
${\mathbb{R}}^{n}\backslash 0$. By Lemma 2.1 it is
compact in ${\mathbb{R}}^{n}$, and, since $f$ is proper, the set
$f^{-1}(K)$ is compact in ${\mathbb{R}}^{n}$. We notice that
if $0\in f^{-1}(K)$ then we would have $0=f(0)\in K$, which
is impossible since $K\subset{\mathbb{R}}^{n}\backslash 0$. Hence
$f^{-1}(K)\subset{\mathbb{R}}^{n}\backslash 0$, and by
Lemma 2.1 again, the set
$f^{-1}(K)$ is compact in ${\mathbb{R}}^{n}\backslash 0$. Hence
the restriction $f:{\mathbb{R}}^{n}\backslash 0\to{\mathbb{R}}^{n}\backslash 0$
is proper.
∎
Lemma 2.3.
Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be continuous everywhere. Then
$f$ is proper if and only if
$|y|\to\infty$ implies $|f(y)|\to\infty$.
Proof.
We show the if part.
Let $K\subset{\mathbb{R}}^{n}$ be compact. Then it is closed and hence
$f^{-1}(K)\subset{\mathbb{R}}^{n}$ is closed. Suppose
$f^{-1}(K)$ is not bounded. Then there is a sequence
$y_{j}\in f^{-1}(K)$ such that $|y_{j}|\to\infty$. Hence
$f(y_{j})\in K$ and also $|f(y_{j})|\to\infty$ by the assumption
on $f$, which yields a contradiction with the boundedness
of $K$. The converse implication is clearly also true.
∎
Lemma 2.4.
Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be proper, injective, and continuous.
Then $f$ is an open map.
Proof.
Let us assume that $f(U)$ is not open
for an open subset $U\subset{\mathbb{R}}^{n}$.
Then there is a point $a\in U$ such that $f(a)$ is on the boundary
of $f(U)$, and we can construct a sequence $y_{j}\in{\mathbb{R}}^{n}\backslash U$
such that $f(y_{j})\to f(a)$.
Since $f$ is proper, there exists a subsequence $y_{j}^{\prime}$
which converges to some point $b\in{\mathbb{R}}^{n}\backslash U$.
Note that $b\not=a$.
Since $f$ is continuous, $f(y_{j}^{\prime})\to f(b)$,
but we also have $f(y_{j}^{\prime})\to f(a)$ which contradicts to the
fact that $f$ is injective.
∎
We quote here a generalised version of Theorem 1.1
(see [Gor72, Theorem B]):
Proposition 2.5.
Let $M$ and $N$ be connected, oriented, $d$-dimensional
$C^{1}$-manifolds, without boundary.
Let $f:M\to N$ be a proper $C^{1}$-map
such that the Jacobian $J(f)$ never vanishes.
Then $f$ is surjective.
If $N$ is simply connected in addition, then
$f$ is also injective.
Proof.
This fact was also known to Hadamard, but a rigorous proof
for surjectivity can been found in [NR62].
As for the injectivity, it is based on the fact that a
simply connected manifold is its own
universal covering space.
A precise proof
can be found in [Gor72, Section 3]).
∎
The following result is a straight forward consequence of Proposition
2.5.
Corollary 2.6.
Let $n\geq 2$.
Let $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be proper and such that
and $f(0)=0$ and
$f({\mathbb{R}}^{n}\backslash 0)\subset{\mathbb{R}}^{n}\backslash 0$.
Moreover, assume that $f$ is $C^{1}$ on ${\mathbb{R}}^{n}\backslash 0$, with
$\det Df\not=0$ on ${\mathbb{R}}^{n}\backslash 0$. Then
the restriction $f:{\mathbb{R}}^{n}\backslash 0\to{\mathbb{R}}^{n}\backslash 0$
is surjective.
If $n\geq 3$ in addition,
$f:{\mathbb{R}}^{n}\backslash 0\to{\mathbb{R}}^{n}\backslash 0$
is also injective.
Proof.
By Corollary 2.2, the restriction of $f$
is a proper map from $M={\mathbb{R}}^{n}\backslash 0$ to $N={\mathbb{R}}^{n}\backslash 0$.
Note that $N$ is simply connected if $n\geq 3$.
Then Proposition 2.5 implies the statement.
∎
With all these facts, Theorem 1.2 is immediate:
Proof of Theorem 1.2.
By Corollary 2.6 and Lemma 2.3,
the map $f:{\mathbb{R}}^{n}\backslash 0\to{\mathbb{R}}^{n}\backslash 0$ is bijective.
Hence it is a global $C^{1}$ diffeomorphism by the usual local inverse
function theorem.
Furthermore, the map $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is also bijective
since $f(0)=0$, hence the global inverse $f^{-1}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ exists
and continuous by Lemma 2.4.
Since $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is also continuous, it is a homeomorphism.
∎
Proof of Corollary 1.3.
First, let us extend $f$ to ${\mathbb{R}}^{n}$ by setting $f(0)=0$ and show
that $f$ is continuous at $0$.
We observe that $f({\mathbb{S}}^{n-1})$ has a finite maximum.
Let $\xi_{j}\to 0$, and $\xi_{j}\not=0$ for all $j$. Then
$$|f(\xi_{j})|=|\xi_{j}|^{\varkappa}{\left|{f{\left({\frac{\xi_{j}}{|\xi_{j}|}}%
\right)}}\right|}\leq C|\xi_{j}|^{\varkappa}\to 0,$$
so that $f$ is continuous at $0$.
Let us now check that other conditions of Theorem 1.2
are satisfied.
We observe that $f({\mathbb{S}}^{n-1})$ has a positive minimum
$\min_{|\xi|=1}|f(\xi)|=c_{0}>0$.
Indeed, if $f(\omega)=0$
for some $\omega\in{\mathbb{S}}^{n-1}$,
then $f(t\omega)(=t^{\kappa}f(\omega))=0$ for any $t>0$.
Differentiating it in $t$, we have $\omega=0$
since the Jacobian of $f$ never vanishes on ${\mathbb{R}}^{n}\backslash 0$,
which is a contradiction.
Then we have
$$|f(\xi)|=|\xi|^{\varkappa}{\left|{f{\left({\frac{\xi}{|\xi|}}\right)}}\right|}%
\geq c_{0}|\xi|^{\varkappa},\quad\xi\neq 0,$$
which induces that $f({\mathbb{R}}^{n}\backslash 0)\subset{\mathbb{R}}^{n}\backslash 0$
and that $|y|\to\infty$ implies $|f(y)|\to\infty$.
Therefore, by Theorem 1.2,
$f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a homeomorphism, and
$f^{-1}$ is $C^{k}$ on ${\mathbb{R}}^{n}\backslash 0$ by the usual local inverse
function theorem. Let us finally show that $f^{-1}$ is positively
homogeneous of order $1/\varkappa$. Indeed, for every
$\tau>0$ and $\xi\not=0$ we have
$f^{-1}(\tau^{\varkappa}f(\xi))=f^{-1}(f(\tau\xi))=\tau\xi$.
Since $f$ is invertible, $\eta=f(\xi)\not=0$, and we have
$f^{-1}(\tau^{\varkappa}\eta)=\tau f^{-1}(\eta)$, or
$f^{-1}(\tau\eta)=\tau^{1/\varkappa}f^{-1}(\eta)$.
∎
Acknowledgement. The authors would like to thank
Professor Adi Adimurthi for valuable remarks on the first version of
our manuscript, leading to its considerable improvement,
and for the correspondence concerning Theorem 1.4.
References
[Gor72]
W. B. Gordon.
On the diffeomorphisms of Euclidean space.
Amer. Math. Monthly, 79:755–759, 1972.
[Had]
J. Hadamard.
Sur les correspondances ponctuelles,
Oeuvres, pp. 383–384.
[Had06a]
J. Hadamard.
Sur les transformations planes.
C. R. Math. Acad. Sci. Paris, 142:74, 1906.
[Had06b]
J. Hadamard.
Sur les transformations ponctuelles.
Bull. Soc. Math. France, 34:71–84, 1906;
Oeuvres, pp. 349–363.
[NR62]
A. Nijenhuis and R. W. Richardson, Jr.
A theorem on maps with non-negative Jacobians.
Michigan Math. J., 9:173–176, 1962. |
AstroCloud, a Cyber-Infrastructure for Astronomy Research: Data Archiving and Quality Control
Boliang He${}^{1}$, Chenzhou Cui${}^{1}$, Dongwei Fan${}^{1}$, Changhua Li${}^{1}$, Jian Xiao${}^{2}$, Ce Yu${}^{2}$, Chuanjun Wang${}^{3}$, Zihuang Cao${}^{1}$, Junyi Chen${}^{2}$, Weimin Yi${}^{3}$, Shanshan Li${}^{1}$, Linying Mi${}^{1}$ and Sisi Yang${}^{1}$
Abstract
AstroCloud is a cyber-Infrastructure for Astronomy Research initiated by Chinese Virtual Observatory (China-VO) under funding support from NDRC (National Development and Reform commission) and CAS (Chinese Academy of Sciences)111http://astrocloud.china-vo.org(O8-5_Cui_adassxxiv). To archive the astronomical data in China, we present the implementation of the astronomical data archiving system (ADAS). Data archiving and quality control are the infrastructure for the AstroCloud. Throughout the data of the entire life cycle, data archiving system standardized data, transferring data, logging observational data, archiving ambient data, And storing these data and metadata in database. Quality control covers the whole process and all aspects of data archiving.
${}^{1}$National Astronomical Observatories, Chinese Academy of Sciences (CAS), 20A Datun Road, Beijing 100012, China
${}^{2}$Tianjin University, 92 Weijin Road, Tianjin 300072, China
${}^{3}$Yunnan Astronomical Observatory, CAS, P.0.Box110, Kunming 650011, China
1 Introduction
There are tens of telescopes running in China. Every night and day, they are producing several terabytes data. To archive these huge data and manage them, we present an implementation of an Astronomical Data Archiving System (ADAS). The data types which would be archived are the observation data and ambient data. The observation data such as image FITS, spectra FITS and observation log, are produced by telescope and data reduce pipeline. Ambient data are some environment data, such as weather, seeing data and allsky camera images.
Archived data is stored into the observatory¡¯ data center first, then Data transferred to AstroCloud data center via ADAS. In AstroCloud, we build a Data Access API For users and programs to access data. The following telescopes have been already using this archiving system to archive their data. These telescope are located in multiple sites in China: Guo Shoujing Telescope (LAMOST), Lijiang GMG 2.4m Telescope, Xinglong 2.16m Telescope, Delingha 50Bin Telescope, Huairou Solar Radio Telescope, Huairou Solar Multi-Channel Telescope and Fuxian 1m New Vacuum Solar Telescope (NVST).
2 Data Model
The type of ¡°raw¡± data include files and tables. FITS file mainly contain the raw data. FITS can be image, can be spectral, etc. The tables are catalog tables, ambient data tables, observational logs, etc.
Metadata consists of two types:
•
Schema Metadata: Schema Metadata stores all the databases, schemas, tables and columns information. The database-schema is similar to the IVOA TAP schemas(IVOA_TAP).
•
Archive Metadata: Archive Metadata stores the FITS files¡¯ header information. The must filed in database-schema is shown in Table 1. Usually, One telescope has one table in archive database.
3 Software archiving Architecture
The system consists of four submodules (2014PASP..126..674L):
1
Data Transfer System (DTS). Data transferring is via network. The network transfer is scheduled. In the central data center in NAOC, we set up a Transfer Server to accept data transfer. We choose rsync tools running this service. Because it is open source and has a very good performances. (2009ASPC..411..540Z)
2
Data Ingest System (DIS). DIS provides the data to database function. This procedure will parse the FITS header and choose the necessary filed to record into the database. We use the AstroPy(AstroPy) to manipulate the FITS file, which can collect the FITS file header easily.(2012SPIE.8451E..19D)
3
Logging System (LGS). All the operation will be logged into the database. LGS is the procedure to log the operation: data transfer, data ingest, database replication, etc.
4
Archive Backup System (BKS). BKS consists of files backup, database replication, and database backup. These operations are scheduled.
4 Archiving Pipeline
1
Data will be transferred to the data center in NAOC by DTS in schedule.
2
After the data is finished transferred. DIS will start running, DIS will check the files¡¯ checksum, collect the FITS files¡¯ header and insert it into the archive database.
3
All the files has been checked and record into the database, gather these information (file amount, transfer log, database log, etc) to email these information to the system administrator and telescope operator.
4
These FITS files and database will be backup by BKS in schedule.
5
Database replication: archive database is the write-only database, the SkyTools(SkyTools) replication procedure will replicate the database to the Query Databases for other user or system to access, such as Data Publish System222http://explore.china-vo.org(P1-3_Fan_adassxxiv).
5 Quality Control
Data quality can be controlled by the data archiving process. In DTS, every file has been made a MD5 checksum, before transferred and after transferred, transfer procedure will valid the checksum. Database is been checked and valid by schedule.
6 Conclusions
We developed and implemented an astronomical data archiving system that can be operated automatic. When the data is produced, the procedure will be running quietly. When the procedure is finished, the operator will receive the job detail email.
Acknowledgments
This paper is funded by National Natural Science Foundation of China (U1231108), Ministry of Science and Technology of China (2012FY120500), Chinese Academy of Sciences (XXH12503-05-05). Data resources are supported by Chinese Astronomical Data Center.
References |
Homology representations of unitary reflection groups
Justin Koonin
School of Mathematics and Statistics
University of Sydney, NSW 2006, Australia
E-mail address: [email protected]
(Date:: )
Abstract.
This paper continues the study of the poset of eigenspaces of elements of a unitary reflection group (for a fixed eigenvalue), which was commenced in [6] and [5]. The emphasis in this paper is on the representation theory of unitary reflection groups. The main tool is the theory of poset extensions due to Segev and Webb ([16]). The new results place the well-known representations of unitary reflection groups on the top homology of the lattice of intersections of hyperplanes into a natural family, parameterised by eigenvalue.
AMS subject classification (2010): 20F55, 05E18
Keywords: Poset topology, unitary reflection groups, homology representations
The author is supported by ARC Grant #DP110103451 at the University of Sydney.
1. Introduction
Let $V$ be a complex vector space of finite dimension, and $G\subseteq GL(V)$ a unitary reflection group in $V$. Denote by $\mathcal{A}(G)$ the set of
reflecting hyperplanes of all reflections in $G$, and
$\mathcal{M}_{\mathcal{A}(G)}$ the hyperplane complement – that is, the smooth
manifold which remains when all the reflecting hyperplanes are removed from
$V$. There is an extensive literature studying the topology of
$\mathcal{M}_{\mathcal{A}(G)}$ ([1], [3], [12],
[13], [9], [2]).
In particular, Orlik and Solomon [12, Corollary 5.7] showed that $H^{\ast}(\mathcal{M}_{\mathcal{A}(G)},\mathbb{C})$ is determined (as a graded representation of $G$) by the poset $\mathcal{L}(\mathcal{A}(G))$ of intersections of the hyperplanes in $\mathcal{A}(G)$.
The poset $\mathcal{L}(\mathcal{A}(G))$ is
known to coincide with the poset of fixed point subspaces (or
1-eigenspaces) of elements of $G$ (see [14, Theorem 6.27]). This paper is the third in a series (following [6] and [5]) which uses the eigenspace theory of Springer and Lehrer ([17], [10], [11]) to study generalisations of $\mathcal{L}(\mathcal{A}(G))$ for arbitrary eigenvalues.
Whereas the focus in the first two papers was on topological properties of the posets in question, the emphasis of this paper is on representation theory. The main tool is the theory of poset extensions due to Segev and Webb ([16]).
The papers [6] and [5] study the structure of a poset we call $\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$ in detail, whose elements are eigenspaces of elements of a reflection coset $\gamma G$ in $V$, for fixed eignenvalue $\zeta$, ordered by the reverse of inclusion. This poset is defined in §2.2.
The main theorem of [6] - Theorem 1.1 - states that in the case $\gamma=\operatorname{Id}$, the poset $\widetilde{\mathcal{S}}_{\zeta}^{V}(G)$ is Cohen-Macaulay over
$\mathbb{Z}$. Thus the homology of this poset is concentrated in top dimension. However the structure of the representation of $G$ on this top homology is difficult to understand. Indeed the most information we have at present is an exponential generating function for the dimension of the representation, when $G$ is an imprimitive reflection group (see [5]), as well as explicit computations for the dimensions in the other irreducible cases. (In fact, such exponential generating functions exist also in the more general setting of reflection cosets of imprimitive reflection groups.)
This paper suggests a slight modification of $\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$, which we call $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$. Again we define this poset in 2.2. Essentially, it consists of adjoining an additional element to the original poset which lies beneath all but the maximal eigenspaces.
The motivation for this modification comes from the theory of poset extensions due to Segev and Webb (see [16]), which will be explained in §3. The new poset $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is shown to be homotopy equivalent to a bouquet of spheres. The number of spheres can be expressed neatly in terms of the invariant theory of $G$, and the representation on $G$ is shown to be that induced from the action of the normaliser of a maximal eigenspace on that eigenspace (see Corollary 4.4).
2. Preliminaries
2.1. Homology
Representations
This section introduces the application of poset homology to representation
theory. Most of the material can be found in [4], and see also
[15], [19, §2.3], [18].
Definition 2.1.
$\;$
(i)
Suppose $(P,\leqslant)$ is a poset and $G$ a
group. Call $P$ a $G$-poset if $G$ acts on the elements of $P$, and if for all $g\in G$, and all $x,y\in P$,
$$x\leqslant_{P}y\quad\mbox{implies}\quad gx\leqslant_{p}gy.$$
That is, $G$ acts as a group of automorphisms of $P$.
(ii)
If $P$ and $Q$ are both $G$-posets and $\phi:P\rightarrow Q$ is an
order-preserving map such that for all $g\in G,x\in P$
$$\phi(gx)=g\phi(x),$$
then $\phi$ is said to be a map of $G$-posets.
(iii)
If $P$ and $Q$ are $G$-posets and $\phi:P\rightarrow Q$ is an
isomorphism of posets and also a $G$-poset map, then $\phi$ is said to be an
isomorphism of $G$-posets.
(iv)
If $P$ and $Q$ are $G$-posets and $\phi:P\rightarrow Q$ is a homotopy
equivalence of posets and also a $G$-poset map, then $\phi$ is said to be a
$G$-homotopy equivalence.
(v)
If $P$ is a $G$-poset which is $G$-homotopy equivalent to a point, then $P$ is
said to be $G$-contractible.
If $P$ is a $G$-poset then $G$ also acts on chains of $P$:
$$g(x_{0}<\cdots<x_{k})=(gx_{0}<\cdots<gx_{n}).$$
Note that there is no degeneracy as $G$ acts as a group of automorphisms of
$P$. This action of $G$ commutes with the boundary homomorphism. Thus $G$
acts on the homology modules $H_{i}(P,\mathbb{A})$ and $\widetilde{H}_{i}(P,\mathbb{A})$, for all $i$. When the ring $\mathbb{A}$ is a
field, this gives a representation of $G$ for each $i$. Such a representation is known as a homology representation of $G$. See [4, §1] for further details.
Suppose $P$ is a $G$-poset, and $Q$ is an $H$-poset. Then the product $P\times Q$ is a $(G\times H)$-poset, with $(G\times H)$-action given by
$$(g,h)(p.q)=(gp,hq),$$
where $g\in G,h\in H,p\in P$ and $q\in Q$.
Clearly this action generalises to finite products of groups, so that if
$P_{j}$ is a $G_{j}$-poset for $j=1,\ldots,n$ then $P_{1}\times\cdots\times P_{n}$ is a $(G_{1}\times\cdots\times G_{n})$-posets.
2.2. Taxonomy of Posets
The central theme of this paper is the study of homological properties of various posets of eigenspaces associated with unitary reflection groups, and associated homology representations of these groups. This section defines the posets we shall consider.
Let $V$ be a vector space over a field $\mathbb{F}$, and $\zeta\in\mathbb{F}$. If $x\in\operatorname{End}(V)$, recall that $V(x,\zeta)$ is the $\zeta$-eigenspace of
$x$ acting on $V$. That is, $V(x,\zeta):=\{v\in V\mid xv=\zeta v\}.$
Definition 2.2.
Let $\gamma G$ be a reflection coset in $V=\mathbb{C}^{n}$, and $\zeta\in\mathbb{C}^{\times}$ be a complex root of
unity. Define $\mathcal{S}_{\zeta}^{V}(\gamma G)$ to be the set $\{V(x,\zeta)\mid x\in\gamma G\}$, partially ordered by the reverse of
inclusion.
Remark 2.3.
There is a natural action of $G$ on
$\mathcal{S}_{\zeta}^{V}(\gamma G)$ which arises from the action of $G$ on
$V$. If $g\in G$ and $V(\gamma x,\zeta)\in\mathcal{S}_{\zeta}^{V}(\gamma G)$, then $g\cdot V(\gamma x,\zeta):=V(g\gamma xg^{-1},\zeta)=V(\gamma g^{\prime}xg^{-1},\zeta)$ for some $g^{\prime}\in G$, since $\gamma$
normalises $G$. This action clearly respects the order relation on
$\mathcal{S}_{\zeta}^{V}(\gamma G)$, and hence turns $\mathcal{S}_{\zeta}^{V}(\gamma G)$ into a $G$-poset.
It is known (see [6, Corollary 3.3]) that the poset
$\mathcal{S}_{\zeta}^{V}(\gamma G)$ always has a unique maximal element $\hat{1}$, and
it may or may not have a unique minimal element $\hat{0}$ as
well (the full space $V$, for example). This is important to remember in the definitions of the following
posets, which are modifications of $\mathcal{S}_{\zeta}^{V}(\gamma G)$:
Definition 2.4.
Define $\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$ to be the subposet of $\mathcal{S}_{\zeta}^{V}(\gamma G)$ obtained by
removing the unique maximal element, as well as the unique minimal element if it
exists.
Definition 2.5.
Define $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ to
be the poset $\mathcal{S}_{\zeta}^{V}(\gamma G)\backslash\{\hat{1}\}$.
The difference between
$\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$ and $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is that the former does not contain the unique minimal element of
$\mathcal{S}_{\zeta}^{V}(\gamma G)$ (if it exists), whereas the latter does.
Definition 2.6.
Define $\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$ to
be the subposet of $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ consisting of
eigenspaces which are not maximal.
That is, $\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is the subposet of $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ obtained by deleting elements of rank 0.
Definition 2.7.
Define the poset $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ as follows.
The elements of $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ are those of $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ together
with one additional element $\hat{0}_{S}$. The order relation on
$\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is the following. Given $x,y\in\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$, $x<y$ if and only if either $x,y\in\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ and $x<y$ in $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$, or $x=\hat{0}_{S}$ and $y\in\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$.
The purpose of this construction will become apparent in
§4.
In the special case $\zeta=1,\gamma=\operatorname{Id}$, the posets we have defined take the following form. The posets $\widetilde{\mathcal{S}}_{1}^{V}(G)$ and $\left.\mathcal{T}^{\prime}\right._{1}^{V}(G)$ are equal to the poset $\widetilde{\mathcal{L}(\mathcal{A}(G))}$ (the intersection lattice of the reflecting hyperplanes, with minimal and maximal elements removed), $\left.\mathcal{S}^{\prime}\right._{1}^{V}(G)$ is this same intersection lattice with only the maximal element removed, while $\left.\mathcal{U}^{\prime}\right.^{V}_{1}(G)$ is the suspension of $\widetilde{\mathcal{S}}_{1}^{V}(G)$ (see Definition 3.4 and Proposition 4.1).
It is clear that $\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$,
$\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ and $\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$
are subposets of $\mathcal{S}_{\zeta}^{V}(\gamma G)$ which are stable under the
action of $G$, and are therefore $G$-posets themselves. Define an action of
$G$ on $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ by letting $G$ act trivially on the
additional element $\hat{0}_{S}$. This makes $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$
into a $G$-poset as well.
Remark 2.8.
The posets which hold the most interest for us are
$\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$ and $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$. The others may be regarded as intermediary posets, whose
definition is necessary to facilitate the study of these two. The papers [6] and [5] study the structure of $\widetilde{\mathcal{S}}_{\zeta}^{V}(\gamma G)$ in detail. This paper deals with the poset $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$.
3. Poset Extensions
The background material on extensions of
$G$-posets comes from [16], and the exposition and notation follows that
paper closely. Proofs of the results in this section can be found in that paper, and in [7]. In this section, all homology is taken over $\mathbb{Z}$ unless otherwise stated.
Definition 3.1.
Let $Q$ be a subposet of $P$. The poset $P$ is said
to be an extension of $Q$ if $Q$ is an upper order ideal of $P$, and if for all $p\in P,$ $Q_{\geqslant p}\neq\emptyset$.
If $P$ is an extension of $Q$ such that for all $p\in P$ either $p\in Q$ or
$p$ is a minimal element of $P$, then $P$ is said to be an extension of $Q$ by
minimal elements.
Definition 3.2.
If $P$ is an extension of $Q$, define a new poset $P_{Q}$ as
follows.
The elements of $P_{Q}$ are those of $P$, together with one additional element
$\hat{0}_{Q}$. Given $x,y\in P_{Q}$, define $x<_{P_{Q}}y$ if and only
if either $x,y\in P$ and $x<_{P}y$, or $x=\hat{0}_{Q}$ and $y\in Q$.
Denote by $Q_{Q}\subseteq P_{Q}$ the poset with elements $Q\cup\hat{0}_{Q}$
and the same order relation as $P_{Q}$. Thus $Q_{Q}$ is just $Q$ with a
minimal element adjoined.
If $P$ is a $G$-poset and $Q$ is stable under the action of $G$, then $P_{Q}$
becomes a $G$-poset by letting $G$ act trivially on $\hat{0}_{Q}$.
Denote the
simiplicial chain group of a poset $P$ at dimension $n$ by $C_{n}(P)$ ($n\geqslant 0$), and $\widetilde{C}_{n}(P)$ the augmented simplicial chain group. Thus $\widetilde{C}_{n}(P)=C_{n}(P)$ for $n\geqslant 0$, while $\widetilde{C}_{-1}(P)=\mathbb{Z}$. As usual let $Z_{n}(P)$ ($\widetilde{Z}_{n}(P)$) be the group
of $n$-cycles, $B_{n}(P)$ ($\widetilde{B}_{n}(P))$ the group of $n$-boundaries.
Given a cycle $z\in\widetilde{Z}_{n}(P)$ denote by $[z]=z+\widetilde{B}_{n}(P)$
the corresponding element in $\widetilde{H}_{n}(P)$.
Proposition 3.3.
[16, Proposition 1.1] Suppose $P$ is an extension
of $Q$. Then
(i)
$\Delta P_{Q}=\Delta P\cup\Delta Q_{Q}$ and $\Delta P=\Delta Q_{Q}\cap\Delta Q$.
(ii)
There is a long exact Mayer-Vietoris sequence in reduced homology
given by
$$\cdots\rightarrow\;\widetilde{H}_{n}(Q)\;\xrightarrow{\iota_{*}}\;\widetilde{H%
}_{n}(P)\;\xrightarrow{\kappa_{*}}\;\widetilde{H}_{n}(P_{Q})\;\xrightarrow{r}%
\;\widetilde{H}_{n-1}(Q)\;\rightarrow\cdots$$
where $\iota_{\ast}$, $\kappa_{\ast}$ are the maps on homology
induced by the obvious inclusion maps $\iota$, $\kappa,$ and $r$ is given as
follows. If $\alpha\in\widetilde{C}_{n}(P)$ and $\beta\in\widetilde{C}_{n}(Q_{Q})$
are such that $\partial(\alpha+\beta)=0$, then $r([\alpha+\beta)]=[\partial\alpha]$, where $\partial$ is the differential map of $P_{Q}$. If
$P$ is a $G$-poset and $Q$ is stable under the action of $G$, then then
Mayer-Vietoris sequence is one of $\mathbb{Z}G$-modules.
When $P$ is an extension of $Q$ by miminal elements, it is possible to show
that $\Delta P_{Q}$ is homotopy equivalent to a wedge of suspensions of certain
other posets. Before describing how this is done, it is necessary to define
poset analogues for some common topological constructions.
Definition 3.4.
Let $R$ be a poset. Define the suspension of $R$,
denoted $\Sigma R$, as follows:
The elements of $\Sigma R$ are those of $R$, together with two additional
elements $\hat{0}_{R}$ and $\hat{0}_{R}^{\prime}$. Given $x,y\in\Sigma R$, define
$x<_{\Sigma R}y$ if and only if $x=\hat{0}_{R}$ and $y\neq\hat{0}_{R}^{\prime}$,
or $x=\hat{0}_{R}^{\prime}$ and $y\neq\hat{0}_{R}$, or $x,y\in R$ and $x<_{R}y$.
In order to describe the action of $G$ on the homology of $R$ and $\Sigma R$,
some more notation is needed. If $s=(r_{0}<r_{1}<\cdots<r_{n-1})$ is
an $(n-1)$-simplex of $R$ and $r<r_{0}$, define $r\ast s:=(r<r_{0}<r_{1}<\cdots<r_{n-1})$. If $z\in\widetilde{Z}_{n-1}(R)$, write $z=\sum_{i=1}^{m}n_{i}s_{i}$, where $s_{i}$ is an $(n-1)$-simplex of $R$. Define
$\hat{0}_{R}\ast z:=\sum_{i=1}^{m}n_{i}(\hat{0}_{R}\ast s_{i})\in\widetilde{C}_%
{n}(\Sigma R)$, and define $\hat{0}_{R}^{\prime}\ast z$ similarly. Also
define $\Sigma(z):=\hat{0}_{R}\ast z-\hat{0}^{\prime}_{R}\ast z.$
If $\Delta$ is an abstract
simplicial complex and $\Phi$ is the abstract simplicial complex consisting of
two distinct, isolated vertices $v_{1}$ and $v_{2}$, then the suspension of
$\Delta$, denoted $\Sigma(\Delta)$, is the join $\Phi\ast\Delta$.
Proposition 3.5.
[16, Proposition 2.1]) Let $R$ be a
$G$-poset. Then
(i)
There is a $G$-equivariant homeomorphism $\Delta(\Sigma R)\cong_{G}\Sigma(\Delta R).$
(ii)
For $n\geqslant 1$, if $z\in\widetilde{Z}_{n-1}(R)$, then
$\partial(\hat{0}_{R}\ast z)=\partial(\hat{0}_{R}^{\prime}\ast z)=z$, where
$\partial$ is the differential map of $\Sigma R$. Thus $\Sigma(z)\in\widetilde{Z}_{n}(\Sigma R).$
(iii)
The map $\widetilde{H}_{n-1}(R)\rightarrow\widetilde{H}_{n}(\Sigma R)$ given by $[z]\rightarrow[\Sigma(z)]$ is an isomorphism of
$\mathbb{Z}G$-modules.
It is also necessary to define a wedge of suspensions of of a set of posets.
Definition 3.6.
Suppose $\{R_{t}\mid t\in\mathcal{T}\}$ is a
family of posets indexed by some set $\mathcal{T}$. Define the wedge of
suspensions of the poset $R_{t}$, denoted $\bigvee_{t\in\mathcal{T}}\Sigma R_{t}$, as follows.
The elements of $\bigvee_{t\in T}\Sigma R_{t}$ are defined to be $\bigcup_{t\in\mathcal{T}}(R_{t}\times\{t\})\cup\mathcal{T}\cup\{\hat{0}\}$.
Define a partial order on this set as follows. For $t\in\mathcal{T}$
define $j_{t}:R_{t}\times\{t\}\rightarrow R_{t}$ by $j_{t}(r,t)=r$.
If $x,y\in\bigvee_{t\in T}\Sigma R_{t}$, define $x<y$ if and only if
one of the following holds:
(i)
there exists $t\in\mathcal{T}$ such that $x,y\in R_{t}\times\{t\}$ and $j_{t}(x)<j_{t}(y),$
(ii)
$x=t\in\mathcal{T}$ and $y\in R_{t}\times\{t\}$,
(iii)
$x=\hat{0}$ and $y\not\in\mathcal{T}\cup\{\hat{0}\}$.
Note that $R_{t}\times\{t\}$ can be identified with $R_{t}$. The use of $R_{t}\times\{t\}$ is to ensure that all sets are disjoint as $t$ runs through
$\mathcal{T}$. In the following proposition this identification is made.
Proposition 3.7.
[16, Proposition 2.2] Suppose $\{R_{t}\mid t\in\mathcal{T}\}$ is a family of
posets. Then:
(i)
$\Delta(\bigvee_{t\in T}\Sigma R_{t})\cong\bigvee_{t\in\mathcal{T}}\Sigma(%
\Delta R_{t}),$
(ii)
for $n\geqslant 1$ the map
$$\mu:\bigoplus_{t\in\mathcal{T}}\widetilde{H}_{n-1}(R_{t})\rightarrow\widetilde%
{H}_{n}(\bigvee_{t\in\mathcal{T}}\Sigma R_{t})$$
defined by $\mu(\sum_{t\in\mathcal{T}}[z_{t}])=\sum_{t\in\mathcal{T}}[t\ast z_{t}-\hat{0}%
\ast z_{t}]$ is an isomorphism, where for
all $t\in\mathcal{T}$, $z_{t}\in\widetilde{Z}_{n-1}(R_{t}).$
Of particular interest is the case when $P$ is an extension of $Q$ by minimal
elements. Let the indexing set $\mathcal{T}$ be the set $\mathcal{M}=P\backslash Q$
of elements in $P$ but not $Q$. By definition, this set consists of minimal
elements of $P$. Also take the posets $R_{t}$ to be the subsposets $P_{>m}$,
$m\in\mathcal{M}$. Recall that by
definition, $P_{>m};=\{x\in P\mid x>m\}.$
If $P$ is a $G$-poset and $Q$ is invariant under the action of $G$, then the
poset $\bigvee_{m\in\mathcal{M}}\Sigma P_{>m}$ admits an action of $G$,
defined by
$$\displaystyle g\cdot(p,m)$$
$$\displaystyle=(g\cdot p,g\cdot m)$$
$$\displaystyle\quad\mbox{ for $(p,m)\in P_{>m}\times\{m\}$},$$
$$\displaystyle g\cdot m$$
$$\displaystyle=m$$
$$\displaystyle\mbox{ for $m\in\mathcal{M}$},$$
$$\displaystyle g\cdot\hat{0}$$
$$\displaystyle=\hat{0}.$$
Hence the homology groups of $\bigvee_{m\in\mathcal{M}}\Sigma P_{>m}$
become $\mathbb{Z}G$-modules.
There is also an action of $G$ on the simplicial complex $\bigvee_{m\in\mathcal{M}}\Sigma(\Delta P_{>m})$. With this action, if $x\in\Sigma(\Delta P_{>m})$ and $g\in G$, then $g\cdot x\in\Sigma(\Delta(P_{>g\cdot m})$.
Proposition 3.8.
[16, Proposition 2.3] Suppose that $P$ is an
extension of $Q$ by minimal elements. Further, suppose that $P$ is a $G$-poset
and that $Q$ is stable under the action of $Q$. Let $\mathcal{M}=P\backslash Q$. Then
(i)
There is a $G$-equivariant homeomorphism
$$\Delta\Bigl{(}\bigvee_{m\in\mathcal{M}}\Sigma P_{>m}\Bigr{)}\cong_{G}\bigvee_{%
m\in\mathcal{M}}\Sigma(\Delta P_{>m}).$$
(ii)
For $n\geqslant 1$ the group $\oplus_{m\in\mathcal{M}}\widetilde{H}_{n-1}(P_{>m})$ acquires the structure of an induced
$\mathbb{Z}G$-module
$$\bigoplus_{m\in\mathcal{M}}\widetilde{H}_{n-1}(P_{>m})\simeq_{G}\bigoplus_{m%
\in[G\backslash\mathcal{M}]}\operatorname{Ind}_{G_{m}}^{G}(\widetilde{H}_{n-1}%
(P_{>m})),$$
where $G_{m}$ denotes the stabiliser of m in G, and $[G\backslash\mathcal{M}]$ denotes the set of G-orbits on $\mathcal{M}$. The
mapping
$$\mu:\bigoplus_{m\in\mathcal{M}}\widetilde{H}_{n-1}(P_{>m})\rightarrow%
\widetilde{H}_{n}\Bigl{(}\bigvee_{m\in\mathcal{M}}\Sigma P_{>m}\Bigr{)}$$
of Proposition 3.7 is an isomorphism of
$\mathbb{Z}G$-modules.
Now we describe how the wedge of suspensions construction is useful in the
case when $P$ is an extension of $Q$ by minimal elements. Set $\mathcal{M}=P\backslash Q$, and define $j:\bigvee_{m\in\mathcal{M}}\Sigma P_{>m}\rightarrow P_{Q}$ as follows. Define
$$\displaystyle j(x)$$
$$\displaystyle=j_{m}(x)$$
$$\displaystyle\quad\mbox{for $x\in P_{>m}\times\{m\}$, where $j_{m}$ was %
defined in Definition
\ref{definition:wedgeposet}},$$
$$\displaystyle j(m)$$
$$\displaystyle=m$$
for $$m\in\mathcal{M}$$
$$\displaystyle j(\hat{0})$$
$$\displaystyle=\hat{0}_{Q}.$$
Theorem 3.9.
[16, Theorem 2.4] Suppose $P$ is an extension
of $Q$ by minimal elements. Further, suppose that $P$ is a $G$-poset and that $Q$
is stable under the action of $G$. Let $\mathcal{M}=P\backslash Q$. Then
(i)
$j:\bigvee_{m\in\mathcal{M}}\Sigma P_{>m}\rightarrow P_{Q}$
is a $G$-homotopy equivalence.
(ii)
for $n\geqslant 1$ the map
$$\mu:\bigoplus_{m\in\mathcal{M}}\widetilde{H}_{n-1}(P_{>m})\rightarrow%
\widetilde{H}_{n}(P_{Q})$$
is an isomorphism of $\mathbb{Z}G$-modules, where
$$\mu(\sum_{m\in\mathcal{M}}[z_{m}])=\sum_{m\in\mathcal{M}}[m\ast z_{m}-\hat{0}_%
{Q}\ast z_{m}],$$
where for all $m\in\mathcal{M}$, $z_{m}\in\widetilde{Z}_{n-1}(P_{>m})$.
4. Extensions for $\mathcal{S}_{\zeta}^{V}(\gamma G)$
Adopt the notation of §2.2.
Note that if we set $P=\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ and $Q=\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$ then
$\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is an extension of
$\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$ by minimal elements, and the resulting
poset $P_{Q}=\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$. This construction
explains the motivation for the definition.
It is well known (see [12, Corollary 5.7]) that in the case $\zeta=1,\gamma=\operatorname{Id}$, the posets $\widetilde{\mathcal{S}}_{1}^{V}(G)=\widetilde{\mathcal{L}(\mathcal{A}(G))}$ play an important role in the theory
of hyperplane complements and in the representation theory of unitary
reflection groups. The poset $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is a
natural generalisation of $\widetilde{\mathcal{S}}_{1}^{V}(G)=\widetilde{\mathcal{L}(\mathcal{A}(G)}$ by virtue of the following
proposition:
Proposition 4.1.
Let $V$ be a finite dimensional complex vector space,
and $G$ a unitary reflection group in $V$. Then $\left.\mathcal{U}^{\prime}\right.^{V}_{1}(G)=\Sigma\widetilde{\mathcal{S}}_{1}%
^{V}(G)$.
Proof.
This follows from Definition 3.4, noting
that $\mathcal{S}_{1}^{V}(G)$ always has a unique minimal element $V$.
∎
Corollary 4.2.
Let $V$ be a finite dimensional complex vector space of dimension $n$, and $G$ a
unitary reflection group acting on $V$. Then $\widetilde{H}_{n-1}(\left.\mathcal{U}^{\prime}\right._{1}^{V}(G))\simeq_{G}%
\widetilde{H}_{n-2}(\widetilde{\mathcal{S}}_{1}^{V}(G)).$
Proof.
This follows immediately from Proposition 3.5(iii)
and Proposition 4.1.
∎
For general $\gamma$ and $\zeta$, we have the following theorem:
Theorem 4.3.
Suppose $\gamma G$ is a unitary reflection coset in $V=\mathbb{C}^{n}$. Let $\zeta\in\mathbb{C}^{n}$ and
set $\mathcal{M}=\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)\backslash%
\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)$. Thus $\mathcal{M}$ is the set of maximal eigenspaces of
$\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$. Then
(i)
There is a long exact sequence of $\mathbb{Z}G$-modules
$$\cdots\rightarrow\widetilde{H}_{n}(\left.\mathcal{S}^{\prime}\right._{\zeta}^{%
V}(\gamma G))\xrightarrow{\iota_{*}}\mathcal{S}_{\zeta}^{V}(\gamma G)%
\widetilde{H}_{{}_{n}}(\left.\mathcal{T}^{\prime}\right._{\zeta}^{V}(\gamma G)%
)\xrightarrow{\kappa_{*}}\widetilde{H}_{n}(\left.\mathcal{U}^{\prime}\right._{%
\zeta}^{V}(\gamma G))\xrightarrow{r}\widetilde{H}_{n-1}(\left.\mathcal{S}^{%
\prime}\right._{\zeta}^{V}(\gamma G))\rightarrow\cdots$$
where
$$\iota_{\ast}:\widetilde{H}_{n}(\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(%
\gamma G))\rightarrow\widetilde{H}_{{}_{n}}(\left.\mathcal{T}^{\prime}\right._%
{\zeta}^{V}(\gamma G))$$
and
$$\kappa_{\ast}:\widetilde{H}_{{}_{n}}(\left.\mathcal{T}^{\prime}\right._{\zeta}%
^{V}(\gamma G))\rightarrow\widetilde{H}_{n}(\left.\mathcal{U}^{\prime}\right._%
{\zeta}^{V}(\gamma G))$$
are the maps on homology
induced by the obvious inclusion maps, and $r$ is the map defined in Proposition 3.3 with $P={\mathcal{S}^{\prime}}_{\zeta}^{V}(\gamma G)$ and $Q={\mathcal{T}^{\prime}}_{\zeta}^{V}(\gamma G)$
(ii)
$\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)\simeq\bigvee_{m\in%
\mathcal{M}}\Sigma(\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)_{>m})$
(iii)
For all $n\geqslant 0$,
$$\displaystyle\widetilde{H}_{n}(\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(%
\gamma G))$$
$$\displaystyle\simeq_{G}\bigoplus_{m\in\mathcal{M}}\widetilde{H}_{n-1}(\left.%
\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)_{>m})$$
$$\displaystyle\simeq_{G}\bigoplus_{m\in[G\backslash\mathcal{M}]}\operatorname{%
Ind}_{G_{m}}^{G}\widetilde{H}_{n-1}(\left.\mathcal{S}^{\prime}\right._{\zeta}^%
{V}(\gamma G))_{>m}).$$
Proof.
Part (i) now follows
directly from Proposition 3.3(ii), part (ii) from Theorem
3.9(i), and part (iii) from Proposition
3.8(ii).
∎
Corollary 4.4.
Suppose $\gamma G$ is a unitary reflection coset acting on $V=\mathbb{C}^{n}.$ Let $\zeta$ be a complex $m$-th root of unity, and suppose
$E$ is a maximal $\zeta$-eigepnspace for $\gamma G$. Let $N(E)$ and $C(E)$ be the
normaliser and centraliser of $E$, respectively. Let
$\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ be defined as in Definition
2.7. Then
(i)
The poset $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$
is homotopy equivalent to a bouquet (wedge) of spheres of dimension $l(\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G))$. The number of spheres is equal to
$$\frac{1}{\left|C(E)\right|}\Bigl{(}\prod_{d_{i}:m\nmid d_{i}}d_{i}\Bigr{)}%
\Bigl{(}\prod_{{d^{\prime}_{i}}^{\ast}}({d^{\prime}_{i}}^{\ast}+1)\Bigr{)},$$
where the $\left.d^{\prime}\right._{i}^{\ast}$ are the codegrees of $N(E)/C(E)$.
(ii)
When $m$ is a regular number for $\gamma G$, this number is equal to
$$\Bigl{(}\prod_{d_{i}:m\nmid d_{i}}d_{i}\Bigr{)}\Bigl{(}\prod_{d_{i}^{\ast}:m%
\mid d_{i}^{\ast}}(d_{i}^{\ast}+1)\Bigr{)}.$$
(iii)
$\widetilde{H}_{\operatorname{top}}(\left.\mathcal{U}^{\prime}\right._{\zeta}^{%
V}(\gamma G))\simeq_{G}\operatorname{Ind}_{N(E)}^{G}\widetilde{H}_{%
\operatorname{top}}(\widetilde{\mathcal{S}}^{E}_{1}(\gamma G))$.
Proof.
For (i), we use (ii) of Theorem 4.3. We have shown in
[6, THeorem 3.1] that $\mathcal{S}_{\zeta}^{V}(\gamma G)_{\geqslant m}\cong\mathcal{S}_{1}^{E}(N(E)/C%
(E))$, and
so $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)_{>m}\cong\widetilde{%
\mathcal{S}}_{1}^{E}(N(E)/C(E))$. It is known that the latter is homotopy equivalent to a
bouquet of spheres in dimension $l(\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G))-1$,
and that the number of such spheres is equal to the product of the
coexponents of $N(E)/C(E)$.
Hence $\Sigma(\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)_{>m})$ is homotopy
equivalent to the same number of spheres, but in dimension $l(\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G))=l(\left.\mathcal{U}%
\right.^{\prime V}_{\zeta}(\gamma G))$. To complete the proof of (i) it therefore suffices to count
the number of maximal eigenspaces in $\gamma G.$ Recall that $G$ acts
transitively on the set of maximal eigenspaces of $\gamma G$ (see [8, Theorem 12.19]). The stabiliser of a maximal eigenspace $E$
is $N:=N(E)$. Hence the number of maximal eigenspaces is
$$\frac{\left|G\right|}{\left|N(E)\right|}=\frac{\left|G\right|/\left|C(E)\right%
|}{\left|N(E)\right|/\left|C(E)\right|}=\frac{1}{\left|C(E)\right|}\Bigl{(}%
\prod_{d_{i}:m\nmid d_{i}}d_{i}\Bigr{)}$$
since it is known ([8, Corollary 11.17]) that the
degrees of $N/C$ are precisely those degrees of $G$ which are divisible by
$m$, and that the order of a unitary reflection group is equal to the
product of its degrees ([17, Theorem 2.4]). The
statement in (i) now follows.
For (ii), note that by [8, Lemma 11.22], $m$ is regular for $\gamma G$ precisely when $C=\{1\}$. Note that this lemma is stated for
reflection groups, but applies equally to reflection cosets. Furthermore,
in this case the codegrees of $N/C$ are precisely those codegrees of $G$
which are divisible by $m$ (see [8, Theorem 11.39]).
To prove (iii) we use Theorem 4.3(iii). We need only
note again that the maximal eigenspaces are all conjugate under the action of $G$
([8, Theorem 12.19]), so that there is only one term
in the direct sum.
∎
Remark 4.5.
This corollary places the well-known representation of $G$ on the top homology of the lattice of interesting hyperlanes (see [12]) into a natural family of representations,
depending on $m$, the order of $\zeta$. This representation is the case $\gamma=\operatorname{Id}$ and $\zeta=1$. One of the advantages of working
with $\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ rather than $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is that the latter may or may not have a unique minimal element. Since it is necessary to remove any unique minimal element before computing
homology, the posets $\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ must be treated in
a non-uniform manner. By contrast, the construction of
$\left.\mathcal{U}^{\prime}\right._{\zeta}^{V}(\gamma G)$ is exactly the same whether or not
$\left.\mathcal{S}^{\prime}\right._{\zeta}^{V}(\gamma G)$ has a unique minimal element.
Example 4.6.
Consider the case $G=E_{8}=G_{37}$, $\gamma=\operatorname{Id}$, $\zeta$ a primitive
3rd root of unity. The degrees of $G$ are 2, 8, 12, 14, 18, 20, 24, 30,
and the corresponding codegrees are 0, 6, 10, 12, 16, 18, 22, 28 (see for
example [8, Table D.3, p.275]. Suppose $E$ is a maximal eigenspace
among $\{V(g,\zeta)\mid g\in E_{8}\}$. By [8, Corollary 11.17], the degrees of $N(E)/C(E)$ are
precisely the degrees of $G$ which are divisible by 3 – namely 12, 18, 24 and 30. Now $N(E)/C(E)$ acts irreducibly on $E$ (by [8, Theorem 11.38]), and hence an inspection of the list
of irreducible reflection groups reveals that the only possibility is $N(E)/C(E)\simeq L_{4}=G_{32}$.
By [8, Proposition 11.14], the maximal eigenspaces all
have dimension equal to the number of degrees divisible by 3. In this
case, $\dim(E)=4.$ Hence $l(\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E_{8}))=3$, and so $\widetilde{H}_{j}(\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E%
_{8}))=0$ for $j\neq 3$, and in particular
$\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E_{8})$ is homotopy equivalent to a
bouquet of spheres in dimension 3. Now 3 is a regular number for $E_{8}$, by
[8, Theorem 11.28]. Hence by Corollary
4.4(ii), the number of spheres in the bouquet is equal to
$(2\ast 8\ast 14\ast 20)\ast(1\ast 7\ast 13\ast 19)=7\,745\,920$. Also, by Corollary 4.4(iii), $\widetilde{H}_{3}(\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E%
_{8}))\simeq\operatorname{Ind}_{L_{4}}^{E_{8}}\widetilde{H}_{2}(\widetilde{%
\mathcal{S}}^{E}_{1}(L_{4})).$
Similarly consider the case $G=E_{8}$, $\gamma=\operatorname{Id}$, $\zeta$ a
primitive 4th root of unity. If $E$ is a maximal eigenspace then $N(E)/C(E)\simeq O_{4}=G_{31}$. Again, $\widetilde{H}_{i}(\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E%
_{8}))=0$ for $i\neq 3$, and in particular
$\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E_{8}))$ is homotopy equivalent to a bouquet
of spheres in dimension 3. The number of spheres is equal to $(2\ast 14\ast 18\ast 30)\ast(1\ast 13\ast 17\ast 29)=63\,488\,880$, and $\widetilde{H}_{3}(\left.\mathcal{U}^{\prime}\right.^{\mathbb{C}^{8}}_{\zeta}(E%
_{8}))\simeq\operatorname{Ind}_{O_{4}}^{E_{8}}\widetilde{H}_{2}(\widetilde{%
\mathcal{S}}^{E}_{1}(O_{4}))$.
5. Acknowledgements
The author would like to acknowledge his supervisor Gus Lehrer and associate supervisor Anthony Henderson for their encouragement, patience, generosity and enthusiasm.
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A spin chain model with non-Hermitian interaction: The Ising quantum
spin chain in an imaginary field
Olalla A. Castro-Alvaredo and Andreas Fring
Centre for Mathematical Science, City University London,
Northampton Square, London EC1V 0HB, UK
E-mail: [email protected], [email protected]
Abstract:
We investigate a lattice version of the
Yang-Lee model which is characterized by a non-Hermitian quantum
spin chain Hamiltonian. We propose a new way to implement
$\mathcal{PT}$-symmetry on the lattice, which serves to guarantee the reality of the
spectrum in certain regions of values of the coupling constants.
In that region of unbroken $\mathcal{PT}$-symmetry we construct
a Dyson map, a metric operator and find the Hermitian counterpart
of the Hamiltonian for small values of the number of sites, both
exactly and perturbatively. Besides the standard perturbation
theory about the Hermitian part of the Hamiltonian, we also carry
out an expansion in the second coupling constant of the model.
Our constructions turns out to be unique with the sole assumption
that the Dyson map is Hermitian.
Finally we compute the magnetization of the chain in the $z$ and
$x$ direction.
††conference: The Ising quantum spin chain in an imaginary field
1 Introduction
It is known for about thirty years that ordinary second order phase
transitions can be described by the Yang-Lee model [1, 2, 3]. This
model admits a quantum field theoretical description in form of a
Landau-Ginzburg Hamiltonian for a scalar field $\phi$ with an additional $\phi^{3}$-interaction and a term linear in the scalar field with an
imaginary coupling constant. The model has been identified [4] as a perturbation of the $\mathcal{M}_{5,2}$-model in the $\mathcal{M}_{p,q}$-series of minimal conformal field theories [5]. It
is the simplest non-unitary model in this infinite class of models, which
are all characterized by the condition $p-q>1$ and whose corresponding
Hamiltonians are all expected to be non-Hermitian.
Here we shall investigate a discretised lattice version of the Yang-Lee
model considered by von Gehlen [6, 7], which is an Ising
quantum spin chain in the presence of a magnetic field in the $z$-direction
as well as a longitudinal imaginary field in the $x$-direction. The
corresponding Hamiltonian for a chain of length $N$ is given by
$$H(\lambda,\kappa)=-\frac{1}{2}\sum_{j=1}^{N}(\sigma_{j}^{z}+\lambda\sigma_{j}^%
{x}\sigma_{j+1}^{x}+i\kappa\sigma_{j}^{x}),\qquad\lambda,\kappa\in\mathbb{R}.$$
(1)
It acts on a Hilbert space of the form $(\mathbb{C}^{2})^{\otimes N}$ where
we employed the standard notation for the $2^{N}\times 2^{N}$-matrices $\sigma_{i}^{x,y,z}=\mathbb{I\otimes I\otimes\ldots\otimes}\sigma^{x,y,z}%
\otimes\ldots\otimes\mathbb{I\otimes I}$ with Pauli matrices
describing spin 1/2 particles
$$\sigma^{x}=\left(\begin{array}[]{cc}0&1\\
1&0\end{array}\right),\qquad\sigma^{y}=\left(\begin{array}[]{cc}0&-i\\
i&0\end{array}\right),\qquad\sigma^{z}=\left(\begin{array}[]{cc}1&0\\
0&-1\end{array}\right),$$
(2)
as $i$th factor acting on the site $i$ of the chain. Their commutation
relations are direct sums of su(2) algebras
$$[\sigma_{j}^{x},\sigma_{k}^{y}]=2i\sigma_{j}^{z}\delta_{jk},\quad[\sigma_{j}^{%
z},\sigma_{k}^{x}]=2i\sigma_{j}^{y}\delta_{jk},\quad[\sigma_{j}^{y},\sigma_{k}%
^{z}]=2i\sigma_{j}^{x}\delta_{jk},\quad\text{with }j,k=1,\ldots,N$$
(3)
A further real parameter $\beta$ may be introduced into the model by
allowing different types of boundary conditions $\sigma_{N+1}^{x,y,z}=\beta\sigma_{1}^{x,y,z}$, albeit here we will only consider the case of periodic
boundary conditions and take $\beta=1$.
Since all Pauli matrices are Hermitian it is obvious that $H(\lambda,\kappa)$ is non-Hermitian
$$H^{\dagger}(\lambda,\kappa)=H(\lambda,-\kappa)\neq H(\lambda,\kappa).$$
(4)
This poses immediately two questions: First of all, is the spectrum still
real, despite the fact that the vital property of Hermiticity which
guarantees this is given up and second is it still possible to formulate a
meaningful quantum mechanical description associated to this type of
Hamiltonians? These issues have attracted a considerable amount of attention
in the last ten years, since the seminal paper by Bender and Boettcher [8] and meanwhile many satisfying answers have been found to most of them;
for recent reviews see [9, 10, 11].
Our manuscript is organised as follows: In section 2 we present various
alternatives about how $\mathcal{PT}$-symmetry can be implemented for
quantum spin chains. In section 3 we establish our notation and recall some
of the well known facts concerning a consistent quantum mechanical framework
for $\mathcal{PT}$-symmetric systems. We analyze the model (1) in
section 4 and section 5, where the former is devoted to non-perturbative and
the latter to perturbative results. In section 6 we compute the
magnetization for the model (1) and we state our conclusions in
section 7.
2 $\mathcal{PT}$-symmetry for spin chains
Preceding the above mentioned recent activities von Gehlen found numerically
[6, 7] that for certain values of the dimensional parameters
$\lambda$ and $\kappa$ the eigenvalues for $H(\lambda,\kappa)$ are all
real, whereas for the remaining values they occur in complex conjugate
pairs. He provided an easy explanation for this feature: Acting adjointly on
the Hamiltonian with a spin rotation operator
$$\mathcal{R}=e^{\frac{i\pi}{4}S_{z}^{N}}=\prod\limits_{i=1}^{N}\frac{1}{\sqrt{2%
}}(\mathbb{I}+i\sigma^{z})_{i},\quad\text{with }\quad S_{z}^{N}=\sum\limits_{i%
=1}^{N}\sigma_{i}^{z},\quad\mathbb{I}=\left(\begin{array}[]{cc}1&0\\
0&1\end{array}\right),$$
(5)
has the effect of rotating the spins at each site clockwise by $\pi/2$ in
the $xy$-plane, such that the corresponding map acts as $\mathcal{R}:(\sigma_{i}^{x},\sigma_{i}^{y},\sigma_{i}^{z})\rightarrow(-\sigma_%
{i}^{y},\sigma_{i}^{x},\sigma_{i}^{z})$. The resulting Hamiltonian is a $2^{N}\times 2^{N}$ non-symmetric matrix with real entries given by
$$\hat{H}(\lambda,\kappa)=\mathcal{R}H(\lambda,\kappa)\mathcal{R}^{-1}=-\frac{1}%
{2}\sum_{i=1}^{N}(\sigma_{i}^{z}+\lambda\sigma_{i}^{y}\sigma_{i+1}^{y}-i\kappa%
\sigma_{i}^{y}).$$
(6)
Its eigenvalues and those of $H(\lambda,\kappa)$ are therefore either all
real or occur in complex conjugate pairs. This is precisely the well known
behaviour one finds when $H(\lambda,\kappa)$ is symmetric with respect to
an anti-linear operator [12, 13, 14, 15, 16], which as mentioned above
has recently attracted a lot of attention. In quantum mechanical or field
theoretical models the anti-linear operator is commonly taken to be the $\mathcal{PT}$-operator, which carries out a simultaneous parity
transformation $\mathcal{P}:x\rightarrow-x$ and time reversal $\mathcal{T}:t\rightarrow-t$. When acting on complex valued functions the anti-linear
operator $\mathcal{T}$ is understood to act as complex conjugation. Real
eigenvalues are then found for unbroken $\mathcal{PT}$-symmetry, meaning
that both the Hamiltonian and the eigenfunctions remain invariant
under $\mathcal{PT}$-symmetry, whereas broken $\mathcal{PT}$-symmetry leads
to complex conjugate pairs of eigenvalues.
We will now argue that $\mathcal{PT}$-symmetry on the lattice can be
interpreted in various ways. One may for instance reflect the chain across
its midpoint via the map $\mathcal{P}^{\prime}:\sigma_{i}^{x,y,z}\rightarrow\sigma_{N+1-i}^{x,y,z}$ as suggested by Korff and
Weston [17] and used thereafter in [18]. It is obvious
that the Hamiltonian (1) is invariant with regard to this symmetry.
However, when keeping the interpretation of $\mathcal{T}$ as a complex
conjugation, and thus ensuring that the $\mathcal{P}^{\prime}\mathcal{T}$-operator is anti-linear, one easily observes that this type of
transformation does not leave the Hamiltonian (1) invariant, i.e. we
have $\left[\mathcal{P}^{\prime}\mathcal{T},H\right]\neq 0$.
Therefore we need to implement $\mathcal{PT}$-symmetry in a different way
for $H(\lambda,\kappa)$ to be able to analyze its properties along the
lines proposed in [12, 13, 14, 15, 16]. We
propose here that one carries out a parity transformation at each individual
site and reflect every spin for instance in the $xy$-plane on $y=-x$. This
is obviously achieved by $\mathcal{R}^{2}$. As $\mathcal{R}^{4}=\prod_{i=1}^{N}(-\mathbb{I})_{i}=(-1)^{N}\mathbb{I}^{\otimes N}$ and
not the desired identity operator, we take here
$$\mathcal{P}=-i\mathcal{R}^{2}=e^{\frac{i\pi}{2}(S^{z}-\mathbb{I})}=\prod_{i=1}%
^{N}\sigma_{i}^{z},\quad\text{with }\quad\mathcal{P}^{2}=\mathbb{I}^{\otimes N},$$
(7)
as our parity operator. Consequently this transformation acts as
$$\mathcal{P}:(\sigma_{i}^{x},\sigma_{i}^{y},\sigma_{i}^{z})\rightarrow(-\sigma_%
{i}^{x},-\sigma_{i}^{y},\sigma_{i}^{z}).$$
(8)
Thus with $\mathcal{T}$ being the usual complex conjugation, which acts on
the Pauli matrices as
$$\mathcal{T}:(\sigma_{i}^{x},\sigma_{i}^{y},\sigma_{i}^{z})\rightarrow(\sigma_{%
i}^{x},-\sigma_{i}^{y},\sigma_{i}^{z}),$$
(9)
we have identified an anti-linear operator constituting a symmetry of the
Hamiltonian (1)
$$\left[\mathcal{PT},H\right]=0.$$
(10)
This operator provides more information than the transformation (6),
because we have now in addition a concrete criterium, which distinguishes
the regimes of real and complex eigenvalues. We can precisely separate the
two domains $U_{\mathcal{PT}}$ and $U_{b\mathcal{PT}}$ in the parameter
space of $\lambda$ and $\kappa$ defined by the action on the eigenstates $\Phi(\lambda,\kappa)$ of $H(\lambda,\kappa)$
$$\mathcal{PT}\Phi(\lambda,\kappa)\left\{\begin{array}[]{l}=\Phi(\lambda,\kappa)%
\text{ \quad for }(\lambda,\kappa)\in U_{\mathcal{PT}}\\
\neq\Phi(\lambda,\kappa)\text{ \quad for }(\lambda,\kappa)\in U_{b\mathcal{PT}%
.}\end{array}\right.$$
(11)
According to the general reasoning provided in [12, 13, 14, 15, 16], simultaneous
eigenfunctions of $\mathcal{PT}$ and $H(\lambda,\kappa)$, that is for $(\lambda,\kappa)\in U_{\mathcal{PT}}$, are then associated with real
eigenvalues whereas in the regime of broken $\mathcal{PT}$-symmetry, that is
$(\lambda,\kappa)\in U_{b\mathcal{PT}}$, the eigenvalues emerge in complex
conjugate pairs.
From the above it is clear that we may define equally well different types
of $\mathcal{PT}$-operators closely related to the one introduced in (7). For instance we can define
$$\mathcal{P}_{x}:=\prod_{i=1}^{N}\sigma_{i}^{x}\qquad\text{and\qquad}\mathcal{P%
}_{y}:=\prod_{i=1}^{N}\sigma_{i}^{y},$$
(12)
which obviously act as
$$\mathcal{P}_{x}:(\sigma_{i}^{x},\sigma_{i}^{y},\sigma_{i}^{z})\rightarrow(%
\sigma_{i}^{x},-\sigma_{i}^{y},-\sigma_{i}^{z})\qquad\text{and\qquad}\mathcal{%
P}_{y}:(\sigma_{i}^{x},\sigma_{i}^{y},\sigma_{i}^{z})\rightarrow(-\sigma_{i}^{%
x},\sigma_{i}^{y},-\sigma_{i}^{z}).$$
(13)
Clearly these parity operators can not be used in the same way as $\mathcal{P}$ in (7) to introduce a $\mathcal{PT}$-symmetry for $H(\lambda,\kappa)$ when keeping $\mathcal{T}$ unchanged. However, they serve to
treat non-Hermitian Hamiltonians of a different kind, such as obvious
modifications of $H(\lambda,\kappa)$ and also to allow for alternative
treatments of non-Hermitian spin chains, such as the XXZ-spin-chain in a
magnetic field [19]
$$H_{XXZ}=\frac{1}{2}\sum_{i=1}^{N-1}\left[(\sigma_{i}^{x}\sigma_{i+1}^{x}+%
\sigma_{i}^{y}\sigma_{i+1}^{y}+\Delta_{+}(\sigma_{i}^{z}\sigma_{i+1}^{z}-1)%
\right]+\frac{\Delta_{-}}{2}(\sigma_{1}^{z}-\sigma_{N}^{z}),$$
(14)
with $\Delta_{\pm}=(q\pm q^{-1})/2$ previously studied in [17, 18]. Obviously when $q\notin\mathbb{R}$ this Hamiltonian is
non-Hermitian, but we observe that it is $\mathcal{PT}$-symmetric when using
any of the parity operators defined in (12) and keeping $\mathcal{T}$
to be the usual complex conjugation
$$\left[\mathcal{P}_{x}\mathcal{T},H_{XXZ}\right]=0\qquad\text{and\qquad}\left[%
\mathcal{P}_{y}\mathcal{T},H_{XXZ}\right]=0.$$
(15)
Thus besides reflecting the chain across its midpoint in form of a
“macro-reflections”, as suggested in [17], we may also carry out the parity transformations on each
individual side. It appears that these “micro-reflections” (7), (12) allow for a
wider range of possibilities, such as for instance Hamiltonians of the type $H(\lambda,\kappa)$ in (1), which could not be tackled with $\mathcal{P}^{\prime}:\sigma_{i}^{x,y,z}\rightarrow\sigma_{N+1-i}^{x,y,z}$. The
different possibilities are simply manifestations of the well known
ambiguities non-Hermitian Hamiltonians possess with regard to their operator
content [21]. This also means that the symmetries (15) will
lead to a different kind of physical systems than those identified in [17].
It is well known that $H_{XXZ}$ can be expressed in terms of generators of a
Temperley-Lieb algebra $E_{i}$, i.e. simply by writing the Hamiltonian
alternatively as $H_{XXZ}=$ $\sum_{i=1}^{N-1}E_{i}$. It is then trivial to
see that the algebra remains invariant under a $\mathcal{PT}$-transformation
when realized as (12): $\mathcal{T}:E_{i}\rightarrow E_{i}^{\ast}$, $\mathcal{P}_{x,y}:E_{i}\rightarrow E_{i}^{\ast}$, such that $\mathcal{P}_{x,y}\mathcal{T}:E_{i}\rightarrow E_{i}$. On the other hand when
implementing the “macro-reflection” on
the entire chain, the $\mathcal{P}^{\prime}\mathcal{T}$-symmetry on the
generators is broken, i.e. $\mathcal{P}^{\prime}\mathcal{T}:E_{i}\rightarrow E_{N+1-i}$, as was found in [17].
A further interesting non-Hermitian quantum spin chain has recently been
investigated by Deguchi and Ghosh [20]
$$H_{DG}=\sum_{i=1}^{N}\kappa_{zz}\sigma_{i}^{z}\sigma_{i+1}^{z}+\kappa_{x}%
\sigma_{i}^{x}+\kappa_{y}\sigma_{i}^{y},$$
(16)
with $\kappa_{zz}\in\mathbb{R}$ and $\kappa_{x}$, $\kappa_{y}\in\mathbb{C}$. Clearly when $\kappa_{x}$ or $\kappa_{y}\notin\mathbb{R}$ the
Hamiltonian $H_{DG}$ is not Hermitian, which is the case we will consider.
As the previous model also the quasi-Hermitian transverse Ising model allows
for different types of realizations for the $\mathcal{PT}$-symmetry. We
easily observe that the macro-reflections can not be implemented
$$\left[\mathcal{P}^{\prime}\mathcal{T},H\right]\neq 0,$$
(17)
whereas all the micro-reflections can be realized
$$\left[\mathcal{PT},H\right]=0\text{ \ \ for }\kappa_{x},\kappa_{y}\in i\mathbb%
{R},\quad\left[\mathcal{P}_{x/y}\mathcal{T},H\right]=0\text{ \ \ for }\kappa_{%
x/y}\in\mathbb{R},\kappa_{y/x}\in i\mathbb{R}\text{.}$$
(18)
Once again these different possibilities raise the question about the unique
of the operator content in the model.
Having an explanation for the nature of the eigenvalue spectra, it is left
to show that one may in addition construct a meaningful metric for this
Hamiltonian with well defined quantum mechanical observables associated to
it. As already indicated, the metric is not even expected to be unique so
that, unlike as for the Hermitian case, the observables are no longer
defined by the Hamiltonian alone [21]. It remains therefore
ambiguous what Hamiltonians of the type $H(\lambda,\kappa)$ describe in
terms of physical observables. Having constructed a metric one may often
also compute an isospectral Hermitian counterpart for $H(\lambda,\kappa)$
for which the physical observables have the standard meaning.
One of the main purposes of this manuscript is that of finding the Hermitian
counterparts of the Hamiltonian (1) and studying in some detail (at
least for small $N$) how many such Hermitian Hamiltonians can be constructed.
3 Generalities
3.1 A new metric and an isospectral Hermitian partner from $\mathcal{PT}$-symmetry
For the sake of self-consistency, we briefly recall the well known procedure
[12, 13, 14, 15, 16] of how to construct a
meaningful metric and isospectral Hermitian counterpart, $h$, for a
non-Hermitian Hamiltonian, $H$. We assume the Hamiltonian to be
diagonalizable and to possess a discrete spectrum. Being non-Hermitian the
Hamiltonian has non identical left $\left|\Phi\right\rangle$ and right
eigenvectors $\left|\Psi\right\rangle$ with eigenvalue equations
$$H\left|\Phi_{n}\right\rangle=\varepsilon_{n}\left|\Phi_{n}\right\rangle\qquad%
\text{and\qquad}H^{\dagger}\left|\Psi_{n}\right\rangle=\epsilon_{n}\left|\Psi_%
{n}\right\rangle\text{\qquad for }n\in\mathbb{N}.$$
(19)
The eigenvectors are in general not orthogonal $\left\langle\Phi_{n}\right.\left|\Phi_{m}\right\rangle\neq\delta_{nm}$, but form a
biorthonormal basis
$$\left\langle\Psi_{n}\right.\left|\Phi_{m}\right\rangle=\delta_{nm},\qquad\sum_%
{n}\left|\Psi_{n}\right\rangle\left\langle\Phi_{n}\right|=\mathbb{I}.$$
(20)
We assume the existence of a selfadjoint, but not necessarily positive,
parity operator $\mathcal{P}$ whose adjoint action conjugates the
Hamiltonian
$$H^{\dagger}=\mathcal{P}H\mathcal{P}\qquad\text{with}\qquad\mathcal{P}^{2}=%
\mathbb{I}.$$
(21)
The action of this operator on the eigenvectors
$$\mathcal{P}\left|\Phi_{n}\right\rangle=s_{n}\left|\Psi_{n}\right\rangle\text{%
\qquad with }s_{n}=\pm 1$$
(22)
defines the signature $s=(s_{1},s_{2},\ldots,s_{n})$, which serves to
introduce the so-called $\mathcal{C}$-operator111The is an unfortunate notation and it should be pointed out that the
operator is not related to the standard charge conjugation operator in
quantum field theory.
$$\mathcal{C}:=\sum_{n}s_{n}\left|\Phi_{n}\right\rangle\left\langle\Psi_{n}%
\right|,$$
(23)
satisfying
$$\left[\mathcal{C},H\right]=0,\qquad\left[\mathcal{C},\mathcal{PT}\right]=0,%
\qquad\mathcal{C}^{2}=\mathbb{I}.$$
(24)
Next we employ this operator to define a new operator $\rho$, which also
relates the Hamiltonian to its conjugate
$$\rho:=\mathcal{PC},\mathcal{\qquad}H^{\dagger}\rho=\rho H.$$
(25)
Depending now on the assumptions made for $\rho$, such systems allow for
different types of conclusions. When $\rho$ is positive and Hermitian, but
not necessarily invertible, the system is referred to as quasi-Hermitian
[22, 21]. In this case the existence of a definite metric is
guaranteed and the eigenvalues are real. In turn when $\rho$ is invertible
and Hermitian, but not necessarily positive, the system is called
pseudo-Hermitian [23, 24, 25]. For this type
of scenario the eigenvalues are always real but no definite conclusions can
be made with regard to the existence of a definite metric. Here we will
identify operators $\rho$ which are quasi-Hermitian as well as
pseudo-Hermitian.
Finally we may factorize $\rho$ into a new operator222When $\eta$ is Hermitian, it just corresponds to a Dyson transformation
[26] employed in the so-called Holstein-Primakov method [27]. For practical purposes it is useful to have a name for this
operator and therefore we refer to $\eta$ from now on as the Dyson map. $\eta$ and use it to construct an isospectral Hermitian counterpart for $H$
$$h=\eta H\eta^{-1}=h^{\dagger}\text{ \qquad}\Leftrightarrow\qquad H^{\dagger}=%
\rho H\rho^{-1}\quad\text{with }\rho=\eta^{\dagger}\eta\text{.}$$
(26)
In other words assuming the existence of an inverse for $\rho$ and its
factorization in form of (26) one can derive a Hermitian counterpart $h$
for $H$ and vice versa.
3.2 Expectation values of local observables
As discussed above, when dealing with non-Hermitian Hamiltonians
the standard metric is generally indefinite and therefore a new, physically
sensible, metric needs to be defined by means of the construction described
before. This amounts to introducing a new inner product $\langle\quad|\quad\rangle_{\rho}$ which is defined in terms of the standard inner
product $\langle\quad|\quad\rangle$ as
$$\langle\Phi|\Psi\rangle_{\rho}:=\langle\Phi|\rho\Psi\rangle,$$
(27)
for arbitrary states, $\langle\Phi|$ and $|\Psi\rangle$. Assuming that
all local operators $\mathcal{O}$ in the non-Hermitian theory are related to
their counterparts $o$ in the Hermitian theory in the same manner as the
corresponding Hamiltonians
$$\eta\mathcal{O}\eta^{-1}=o,$$
(28)
one finds that a generic matrix element of the operator $\mathcal{O}$ has
the form,
$$\langle\Phi|\rho\mathcal{O}|\Psi\rangle=\langle\Phi|\eta^{\dagger}o\eta|\Psi%
\rangle=\langle{\phi}|o|\psi\rangle,$$
(29)
where $|\Psi\rangle$ and $\langle\Phi|$ are eigenstates of the
non-Hermitian Hamiltonian and its conjugate, respectively. The states $|\psi\rangle$ and $\langle\phi|$ are related to the previous two states by $|\psi\rangle=\eta|\Psi\rangle$ and $\langle\phi|=\langle\Phi|\eta^{\dagger}$, that is, they are eigenstates of the Hermitian Hamiltonian
corresponding to the same eigenvalues. Equation (29) will be used
later on in this paper for the computation of various kinds of expectation
values.
3.3 Perturbation theory
In most cases the above mentioned operators can not be computed exactly and
one has to resort to a perturbative analysis. Let us recall the main
features of such a treatment. To start with it is convenient to separate the
Hamiltonian into its Hermitian and non-Hermitian part as $H(\lambda,\kappa)=h_{0}(\lambda)+i\kappa h_{1}$, where $h_{0}$ and $h_{1}$ are both
Hermitian with $\kappa$ being a real coupling constant. The latter term may
then be treated as the perturbing term. For our concrete case (1) the
individual components are
$$h_{0}(\lambda)=-\frac{1}{2}\sum_{i=1}^{N}(\sigma_{i}^{z}+\lambda\sigma_{i}^{x}%
\sigma_{i+1}^{x}),\qquad\text{and}\qquad h_{1}=-\frac{1}{2}\sum_{i=1}^{N}%
\sigma_{i}^{x},$$
(30)
such that $h_{0}(\lambda)$ corresponds to the Ising spin chain coupled to a
magnetic field in the $z$ direction and the perturbing term is an imaginary
magnetic field in the $x$-direction. In order to determine $\eta$, $\rho$
and $h$ we can now solve either of the two equations in (26). Here we
decide to commence with the latter. Making the further assumption that $\eta$ is Hermitian and of the form $\eta=e^{q/2}$ this amounts to solving
$$H^{\dagger}=e^{q}He^{-q}=H+[q,H]+\frac{1}{2}[q,[q,H]]+\frac{1}{3!}[q,[q,[q,H]]%
]+\cdots$$
(31)
where we have employed the Backer-Campbell-Hausdorff identity. Writing $H$
and $H^{\dagger}$ in terms of $h_{0}$ and $h_{1}$ equation (31)
becomes
$$2i\kappa h_{1}+i\kappa[q,h_{1}]+\frac{i\kappa}{2}[q,[q,h_{1}]]+\cdots=[h_{0},q%
]+\frac{1}{2}[q,[h_{0},q]]+\cdots$$
(32)
For most non-Hermitian Hamiltonians, such as for our model (1), this
equation is very difficult to solve for $q$. When the $(\ell+1)$-fold
commutator of $q$ with $h_{0}$, denoted by $c_{q}^{(\ell+1)}(h_{0})$
vanishes, closed formulae were found in [28]
$$h=h_{0}+\sum\limits_{n=1}^{[\frac{\ell}{2}]}\frac{(-1)^{n}E_{n}}{4^{n}(2n)!}c_%
{q}^{(2n)}(h_{0}),\quad H=h_{0}-\sum\limits_{n=1}^{[\frac{\ell+1}{2}]}\frac{%
\kappa_{2n-1}}{(2n-1)!}c_{q}^{(2n-1)}(h_{0}),$$
(33)
where $\left[x\right]$ denotes the integer part of a number $x$ and $E_{n}$
are Euler’s numbers, e.g. $E_{1}=1$, $E_{2}=5$, $E_{3}=61$, $E_{4}=1385,\ldots$ The coefficients $\kappa_{2n-1}$ were determined by
means of a recursive equation, which was solved by
$$\kappa_{n}=\frac{1}{2^{n}}\sum\limits_{m=1}^{\left[(n+1)/2\right]}(-1)^{n+m}%
\binom{n}{2m}E_{m},$$
(34)
such that $\kappa_{1}=1/2,\kappa_{3}=-1/4,\kappa_{5}=1/2,\kappa_{7}=-17/8,\ldots$
One may also impose some further structure on $q$ and expand it as
$$q=\sum_{k=1}^{\infty}\kappa^{2k-1}q_{2k-1},$$
(35)
so that each perturbative contribution $q_{2k-1}$ is a $\kappa$-independent
matrix. For models of the form considered here only odd powers of $\kappa$
appear in the perturbative expansion. This is essentially due to the fact
that $H$ and $H^{\dagger}$ are related to each other by $\kappa\rightarrow-\kappa$. Substituting the expansion (35) into the equation (32) one finds a set of equations for $q_{1},q_{3},q_{5},\ldots$ by
equating those terms in (32) which are of the same order in
perturbation theory in $\kappa$. The first few equations are given by
$$\displaystyle[h_{0},q_{1}]=2ih_{1},$$
(36)
$$\displaystyle[h_{0},q_{3}]=\frac{i}{6}[q_{1},[q_{1},h_{1}]],$$
(37)
$$\displaystyle[h_{0},q_{5}]=\frac{i}{6}[q_{1},[q_{3},h_{1}]]+\frac{i}{6}[q_{3},%
[q_{1},h_{1}]]-\frac{i}{360}[q_{1},[q_{1},[q_{1},[q_{1},h_{1}]]]].$$
(38)
As we can see easily, they can be solved recursively, namely once $q_{1}$ is
known, one case solve for $q_{3}$ and so on. A closed expression for the
commutator $[h_{0},q_{n}]$ in terms of commutators $[q_{m},h_{1}]$ with $m<n$
was derived in [18]. Perturbation theory has been carried out in the
past for various non-Hermitian models, e.g. [16, 28, 29, 30, 31, 18].
The model at hand is special in the sense that it involves two coupling
constants, i.e. $\kappa$ and $\lambda$, such that it allows for an
alternative perturbative expansion in terms of the latter. Indeed we will
demonstrate below that the case $\lambda=0$ can be solved exactly and we
can therefore expand around that solution. Proceeding similarly as for the $\kappa$-perturbation theory we separate the Hamiltonian into its single
spin contribution and into the nearest neighbour interaction term $H(\lambda,\kappa)=\tilde{H}_{0}(\kappa)+\lambda\tilde{h}_{1}$with
$$\tilde{H}_{0}(\kappa)=-\frac{1}{2}\sum_{i=1}^{N}(\sigma_{i}^{z}+i\kappa\sigma_%
{i}^{x})\quad\text{and\quad}\tilde{h}_{1}=-\frac{1}{2}\sum_{i=1}^{N}\sigma_{i}%
^{x}\sigma_{i+1}^{x}.$$
(39)
We stress that the counterparts of (36)-(38) in the well known $\kappa$-expansion explained above differ substantially in the $\lambda$-expansion. The details will be explained in the main part of the manuscript
below. Having the option to construct two perturbative series, we in
principle have in addition the possibility to combine them in a manner that
has proved to be very successful in the context of high intensity laser
physics [32].
3.4 Ambiguities in the physical observables
As mentioned previously, one can argue that the metric $\rho$ is not
unique. In the perturbation theory framework, this can be easily seen from
the fact that the equations (36)-(38) (and any other equations
arising at higher orders in perturbation theory) admit many different
solutions. The non-uniqueness of $\eta$ or, equivalently, the fact that
several independent Hermitian Hamiltonians $h$ may exist which are all
related to the same non-Hermitian Hamiltonian by different unitary
transformations is well known in the literature. Indeed, this fact has been
noticed already in the past [21, 33, 30, 34, 35, 36]
and is currently still object of debate [37, 38].
Assuming now the Dyson map $\eta$ in (26) to be Hermitian and related
to the operators $\mathcal{P}$, $\mathcal{C}$ and $\rho$ as defined in (25) we simply obtain
$$\eta=\eta^{\dagger}\qquad\Rightarrow\qquad\eta^{2}=\rho=\mathcal{PC}.$$
(40)
Writing $\eta=e^{q/2}$, it is obvious that we can always add to $q$ any
matrix $b$ that commutes with the full Hamiltonian $[H,b]=0$ and with $q$, $[q,b]=0$
$$h=e^{q/2+b}He^{-q/2-b}=e^{q/2}He^{-q/2},$$
(41)
and still solve equations (26). This kind of ambiguity is not very
interesting, as it will not change $h$ and therefore not lead to new
physics. A somewhat less trivial ambiguity was pointed out in [30],
which will generate different types of Hermitian counter-parts to $H$. It
originates from the fact that we can always add to $q_{1},q_{3},q_{5},\ldots$ any matrix commuting with $h_{0}$ as we may easily observe in equations (36)-(38). Below we will see that in principle for specific
examples many such matrices can be found.
However, by relating $\eta$ to the operators $\mathcal{C}$ and $\mathcal{P}$
as in (40) we are introducing further constraints on the form of
$\eta$. These constraints follow from the equations (24),
particularly the last two equations there. Using the explicit form (40) they can be rewritten as
$$\mathcal{PT}e^{q}\mathcal{PT}=e^{q},\qquad\mathcal{P}e^{q}\mathcal{P}=e^{-q}.$$
(42)
by employing the equality $\mathcal{C}=\eta^{2}\mathcal{P}=e^{q}\mathcal{P}$. In order for (42) to be satisfied, it is required that
$$\mathcal{P}q\mathcal{P}=\mathcal{T}q\mathcal{T}=-q,$$
(43)
and consequently
$$\mathcal{P}q_{2k-1}\mathcal{P}=\mathcal{T}q_{2k-1}\mathcal{T}=-q_{2k-1},\qquad%
\forall\quad k\in\mathbb{Z}^{+}.$$
(44)
Below, we will show that these constraints are sufficient in many cases to
fix the operator $\eta$ and therefore the metric completely. However, it
should be noted that these arguments are based on the assumption that $\rho$
acquires the form (40) and furthermore that the parity operator
is unique, which as we exemplified (12) is not always the case.
4 The Yang-Lee quantum chain: non perturbative results
We will now employ the general ideas and definitions introduced in the
previous subsection for the quantum spin chain Hamiltonian (1). In
particular, we will show how to obtain exact solutions for the operators $\eta$, $\rho$ and $h$ in the two particular situations: i) $\lambda$ or $\kappa$ are vanishing and $N$ is generic and ii) $\lambda$ and $\kappa$
are arbitrary and $N$ is taken to be small.
For large values of $N$ it will be convenient to use the following
abbreviation
$$S_{a_{1}a_{2}\ldots a_{p}}^{N}:=\sum_{k=1}^{N}\sigma_{k}^{a_{1}}\sigma_{k+1}^{%
a_{2}}\ldots\sigma_{k+p-1}^{a_{p}},\qquad\text{for}\quad a_{i}=x,y,z,u;\quad i%
=1,\ldots,p\leq N.$$
(45)
We denote here $\sigma^{u}=\mathbb{I}$ to allow for non-local, i.e. not
nearest neighbour, interactions. In this notation the Hamiltonian (1)
reads
$$H(\lambda,\kappa)=h_{0}(\lambda)+i\kappa h_{1},\quad\text{with}\quad h_{0}(%
\lambda)=-\frac{1}{2}(S_{z}^{N}+\lambda S_{xx}^{N}),\qquad h_{1}=-\frac{1}{2}S%
_{x}^{N}.$$
(46)
In what follows it will also be important to use the adjoint action of $\mathcal{P}$, $\mathcal{T}$ and $\mathcal{PT}$ on the generators $S_{a_{1}a_{2}\ldots a_{p}}^{N}$. It is easy to compute
$$\displaystyle\mathcal{P}S_{a_{1}a_{2}\ldots a_{p}}^{N}\mathcal{P}$$
$$\displaystyle=$$
$$\displaystyle(-1)^{n_{y}+n_{x}}S_{a_{1}a_{2}\ldots a_{p}}^{N},$$
(47)
$$\displaystyle\mathcal{T}S_{a_{1}a_{2}\ldots a_{p}}^{N}\mathcal{T}$$
$$\displaystyle=$$
$$\displaystyle(-1)^{n_{y}}S_{a_{1}a_{2}\ldots a_{p}}^{N},$$
(48)
$$\displaystyle\mathcal{PT}S_{a_{1}a_{2}\ldots a_{p}}^{N}\mathcal{PT}$$
$$\displaystyle=$$
$$\displaystyle(-1)^{n_{x}}S_{a_{1}a_{2}\ldots a_{p}}^{N},$$
(49)
where $n_{x}$, $n_{y}$ are the numbers of indices $a_{i}$ equal to $x$, $y$,
respectively. These identities follow directly from the definitions (8) and (9).
4.1 Limiting cases: $\lambda=0$ or $\kappa=0$
Let us start by considering the special case $\lambda=0$ for
which
$$h_{0}(0)=-\frac{1}{2}S_{z}^{N}\quad\text{and}\quad h_{1}=-\frac{1}{2}S_{x}^{N}.$$
(50)
Although the Hamiltonian is extremely simple, it is still non-Hermitian, and
thus serves as a benchmark to illustrate the above mentioned notions. For
example, a matrix $\eta$ that relates $H(0,\kappa)$ to its Hermitian
counterpart $h(0,\kappa)$ is easily found to be
$$\eta=e^{q/2}=e^{-\frac{1}{2}\text{arctanh}(\kappa)S_{y}^{N}}.$$
(51)
Its adjoint action on $S_{x}^{N}$ and $S_{z}^{N}$ is simply
$$\eta S_{x}^{N}\eta^{-1}=\frac{1}{\sqrt{1-\kappa^{2}}}(i\kappa S_{z}^{N}+S_{x}^%
{N}),\qquad\eta S_{z}^{N}\eta^{-1}=\frac{1}{\sqrt{1-\kappa^{2}}}(S_{z}^{N}-i%
\kappa S_{x}^{N}),$$
(52)
which when we evaluate (26) yields the Hermitian counterpart to $\tilde{H}_{0}(\kappa)$ in (39)
$$h(0,\kappa)=-\frac{1}{2}\sqrt{1-\kappa^{2}}S_{z}^{N}.$$
(53)
This Hamiltonian describes a spin chain for which no mutual interaction
between spins along the chain occurs. An external magnetic field is applied
at each site of the chain, whose intensity is governed by the value of $\kappa$ and is the same at every site. The constraint $-1<\kappa<1$
ensures the Hamiltonian $h(0,\kappa)$ and $\eta$ to be Hermitian. Given
the simplicity of $h(0,\kappa)$ we can easily find its full set of
eigenstates and eigenvalues, hence those of $H(0,\kappa)$. The operator $S_{z}^{N}$ is a diagonal matrix with entries
$$S_{z}^{N}=\text{diag}(N,N-2,\ldots,-N+2,-N).$$
(54)
The entries in the diagonal (eigenvalues) are $N-2p$ with $p=0,\ldots,N$.
They are not necessarily in decreasing order and, except for $N$ and $-N$,
all other eigenvalues are degenerate. For example, the eigenvalues $N-2$ and
$2-N$ are always $N$ times degenerate. This means that there is a single
ground state with minimum energy,
$$E_{g}(\kappa)=-\frac{N}{2}\sqrt{1-\kappa^{2}},$$
(55)
and the corresponding eigenstate is simply
$$|\psi_{g}\rangle=\bigotimes_{i=1}^{N}\left(\begin{array}[]{c}1\\
0\end{array}\right)_{i},$$
(56)
associated to a configuration with all spins “up”, hence aligned with the magnetic field that is being
applied at each site of the chain.
The situation when $\kappa=0$ and $\lambda$ is arbitrary corresponds to
the Hermitian Hamiltonian given by $h_{0}(\lambda)$, that is the Ising spin
chain with a magnetic field in the $z$-direction. In this case, $\eta=\mathbb{I}$, which is automatically ensured when using perturbation theory.
The eigenstates and eigenvalues of this Hamiltonian have been studied in the
literature by using the Bethe ansatz approach, see e.g. [39, 40]. In particular, the ground state can not be written in such as simple form
as (56), as it will depend on the value of $\lambda$. One does know
however, that, for finite $N$, it will interpolate between the $\lambda=0$
case, in which the ground state is (56) and the $\lambda\rightarrow\infty$ case, in which the ground state will correspond to alternating
up-down spins.
4.1.1 Uniqueness of the Dyson operator
In light of the discussion in section 3.4 it is also
interesting to investigate the uniqueness of (51). Indeed, we will now
show that (51) is the only solution to (26) which is consistent
with (42) for the Hamiltonian $H(0,\kappa)$. This can be proven in
two steps: firstly we will characterize the subset of matrices and linear
combinations thereof that satisfy (43) and secondly, we will show
that none of these matrices can be in the kernel of $h_{0}(0)$. Let us define
the matrices, which provide a basis for the set of $2^{N}\otimes 2^{N}$-Hermitian matrices,
$$M_{a_{1}\ldots a_{N}}=\sigma_{1}^{a_{1}}\otimes\cdots\otimes\sigma_{N}^{a_{N}}%
,\quad\text{with}\quad a_{i}=x,y,z,\,\,\text{or}\,\,u\quad\forall\,\,i=1,%
\ldots,N.$$
(57)
Recall the definition $\sigma_{i}^{u}=\mathbb{I}_{i}$. Let us consider an
arbitrary linear combination of the matrices (57). The action of parity
and time reversal on such a linear combination is analogous to (47)
and (48). From this it follows that, in order for any linear
combination of matrices $M_{a_{1},\ldots,a_{p}}$ to transform as $q$ does in
equations (43) it must be such that for all matrices in the linear
combination $n_{y}$ is odd and $n_{x}$ is even ($n_{x}$ and $n_{y}$ as defined after
equation (49)).
We will now argue that no matrix in the kernel of $h_{0}(0)$ is of this form.
There are various ways of having a vanishing commutator $[h_{0}(0),B]=0$. The
most obvious solution is for $B$ to be a diagonal matrix, as $h_{0}(0)$ is
itself diagonal. In terms of the matrices (57), this means selecting
out those that are tensor products of $\sigma^{z}$ and $\mathbb{I}$ only.
There are overall $2^{N}$ such matrices and obviously none of them has $n_{y}$
odd. This would be sufficient to conclude that the solution (51) is
unique if only the kernel of $h_{0}(0)$ had dimension $2^{N}$. This is not so
because $h_{0}(0)$ has degenerate eigenvalues.
Any additional matrices in the kernel will be some linear combination of
matrices (57) involving at least one index $x$ or $y$. Employing the
commutation relations (3), it is easy to see that there are
basically two kinds of additional matrices that are in the kernel of $h_{0}(0)$: firstly, the matrices $M_{xyu\ldots u}-M_{yxu\ldots u}$ and
generalizations thereof , which are antisymmetric under the exchange of
indices $x\leftrightarrow y$ and violate the condition $n_{x}$ even and
secondly, the matrices $M_{xxu\ldots u}+M_{yyu\ldots u}$ and
generalizations thereof, which violate the condition $n_{y}$ odd and are
symmetric under the exchange of indices $x\leftrightarrow y$.
Generalizations of these matrices are those obtained by replacing any number
of indices $u$ by $z$ and/or permuting indices, as well as other matrices of
similar characteristics, such as $M_{xxxxu\ldots u}+M_{yyyyu\ldots u}+M_{xxyyu\ldots u}+M_{yyxxu\ldots u}$ and so on. Since this is more an
argument than a proof, we would like to support it with two examples. For $N=2$
$$h_{0}(0)=\text{diag}(-1,0,0,1),$$
(58)
and the kernel has dimensions 6, as one eigenvalue is twice degenerate. It
is generated by the matrices
$$M_{xy}-M_{yx},\quad M_{xx}+M_{yy},\quad M_{zz},\quad M_{zu},\quad M_{uz}\quad%
\text{and}\quad M_{uu}=\mathbb{I}.$$
(59)
For $N=3$ we have that $h_{0}(0)$ has four different eigenvalues, two of which
are three times degenerate,
$$h_{0}(0)=\text{diag}\left(-\frac{3}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-%
\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{2}\right),$$
(60)
The dimension of the kernel then becomes 20. Its generators are the matrices
$$\displaystyle M_{xyu}-M_{yxu},\quad M_{xuy}-M_{yux},\quad M_{uxy}-M_{uyx},$$
$$\displaystyle M_{xxu}+M_{yyu},\quad M_{xux}+M_{yuy},\quad M_{uxx}+M_{uyy},$$
$$\displaystyle M_{xyz}-M_{yxz},\quad M_{xzy}-M_{yzx},\quad M_{zxy}-M_{zyx},$$
$$\displaystyle M_{xxz}+M_{yyz},\quad M_{xzx}+M_{yzy},\quad M_{zxx}+M_{zyy},$$
$$\displaystyle M_{zzz},\quad M_{zzu},\quad M_{zuz},\quad M_{uzz},\quad M_{zuu},%
\quad M_{uzu},\quad M_{uuz},\quad M_{uuu},$$
(61)
As shown before, these examples confirm once more that no element in the
kernel of $h_{0}(0)$ can fulfill the conditions (43) and therefore
could not be added to $q$, whilst fulfilling such conditions. Thus no
matrices in the kernel of $h_{0}(0)$ satisfy the conditions (43) and
the solution (51) is unique if the operator $\eta=e^{q/2}$ is to be
Hermitian.
4.2 The $N=2$ case: two sites
We have already identified the $\mathcal{PT}$-symmetry for the
Hamiltonian (1) with $\mathcal{P}$ given as specified in (7)
satisfying (21). Let us now take the length of the spin chain to be $N=2$ and compute the quantities as outlined in the previous section.
For two sites we may chose without loss of generality the boundary
conditions to be periodic $\sigma_{N+1}^{x}=\sigma_{1}^{x}$ as any other
choice may be achieved simply by a re-definition of $\lambda$. In this case
the Hamiltonian (1) acquires the simple form of a non-Hermitian $4\times 4$-matrix. In order to make notations clear, we will write this
matrix here in the various notations introduced so far,
$$\displaystyle H(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}\left[\sigma^{z}\otimes\mathbb{I}+\mathbb{I}\otimes%
\sigma^{z}+2\lambda\sigma^{x}\otimes\sigma^{x}+i\kappa\left(\mathbb{I}\otimes%
\sigma^{x}+\sigma^{x}\otimes\mathbb{I}\right)\right],$$
(62)
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}\left[\sigma_{1}^{z}+\sigma_{2}^{z}+2\lambda\sigma_{1%
}^{x}\sigma_{2}^{x}+i\kappa\left(\sigma_{2}^{x}+\sigma_{1}^{x}\right)\right],$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}[S_{z}^{2}+\lambda S_{xx}^{2}+i\kappa S_{z}^{2}]=-%
\left(\begin{array}[]{rrrr}-1&\frac{i\kappa}{2}&\frac{i\kappa}{2}&\lambda\\
\frac{i\kappa}{2}&0&\lambda&\frac{i\kappa}{2}\\
\frac{i\kappa}{2}&\lambda&0&\frac{i\kappa}{2}\\
\lambda&\frac{i\kappa}{2}&\frac{i\kappa}{2}&-1\end{array}\right),$$
where the first line shows the most explicit way of writing the Hamiltonian,
the second line shows a simplified version, were the tensor products are
omitted and absorbed into the $\sigma$s as specified after (1). The
last line uses the notation introduced in (45).
Figure 1: Domains of broken and unbroken $\mathcal{PT}$-symmetry
At first we shall be concerned with the spectral properties of this
Hamiltonian. The two subdomains $U_{\mathcal{PT}}$ and $U_{b\mathcal{PT}}$
, as introduced in (11), have already been identified numerically in
[6] for spin chain lengths up to $N=19$, that is for matrices up
to the remarkable size of $524288\times 524288$. For $N=2$ the eigenvalues
for (62) are easily computed analytically as the characteristic
polynomial factorizes into a third and first order polynomial. The
discriminant $\Delta$ of the third order polynomial is computed by
$$\Delta=r^{2}-q^{3}\text{\qquad with \ }q=\frac{1}{9}\left(-3\kappa^{2}+4%
\lambda^{2}+3\right),\quad r=\frac{\lambda}{27}\left(18\kappa^{2}+8\lambda^{2}%
+9\right).$$
(63)
The eigenvalues are guaranteed to be real when the discriminant is smaller
or equal to zero, such that $U_{\mathcal{PT}}$ is defined as
$$U_{\mathcal{PT}}=\left\{\lambda,\kappa:\Delta=\kappa^{6}+8\lambda^{2}\kappa^{4%
}-3\kappa^{4}+16\lambda^{4}\kappa^{2}+20\lambda^{2}\kappa^{2}+3\kappa^{2}-%
\lambda^{2}-1\leq 0\right\}.$$
(64)
The regions $U_{\mathcal{PT}}$ and $U_{b\mathcal{PT}}$ are depicted in
Figure 1, from which we note that in order to have a real eigenvalue
spectrum $\kappa$ is restricted to take values between $0$ and $1$, whereas
$\lambda$ is left unbounded $\lambda\in[0,\infty)$.
The four real eigenvalues are then computed to
$$\begin{array}[]{ccc}\varepsilon_{1}=\lambda,&\varepsilon_{2}=2q^{\frac{1}{2}}%
\cos\left(\frac{\theta}{3}\right)-\frac{\lambda}{3},&\varepsilon_{3,4}=2q^{%
\frac{1}{2}}\cos\left(\frac{\theta}{3}+\pi\mp\frac{2\pi}{3}\right)-\frac{%
\lambda}{3},\end{array}$$
(65)
where the additional abbreviation $\theta=\arccos\left(r/q^{3/2}\right)$
has been introduced. We depict these eigenvalues in Figure 2,
Figure 2: Avoided level crossing: eigenvalues as
functions of $\lambda$ ($\kappa$) for fixed $\kappa$ ($\lambda$).
where we observe the typical avoided level crossing behaviour of
the eigenvalues as a function of the parameters [41], i.e. the
eigenvalues $\varepsilon_{3}$ and $\varepsilon_{4}$ only meet in the
exceptional point when they simultaneously become complex.
For the computations of physical observables, which we will carry out below,
it is important to identify the lowest eigenvalue, which turns out to be
always $\varepsilon_{4}$.
Next we compute the right eigenvectors of $H(\lambda,\kappa)$ to
$$\left|\Phi_{1}\right\rangle=(0,-1,-1,0),\quad\left|\Phi_{n}\right\rangle=(%
\gamma_{n},-\alpha_{n},-\alpha_{n},\beta_{n}),\quad n=2,3,4,$$
(66)
with $\alpha_{n}=i\kappa\left(\lambda-\varepsilon_{n}+1\right)$, $\beta_{n}=\kappa^{2}+2\lambda^{2}+2\lambda\varepsilon_{n}$ and $\gamma_{n}=-\kappa^{2}-2\varepsilon_{n}^{2}+2\lambda-2\lambda\varepsilon{}_{n}%
+2\varepsilon_{n}$. We verify that left and right eigenvectors are
related via a conjugation $\left|\Psi_{n}\right\rangle=\left\langle\Phi_{n}\right|$ and compute the signature as defined in (22) to $s=(+,-,+,-)$ for the parity operator (7). Normalizing the vectors in
(66) by dividing with $N_{1}=\sqrt{2}$, $N_{n}=(2\alpha_{n}^{2}+\beta_{n}^{2}+\gamma_{n}^{2})^{1/2}$ for $n=2,3,4$ we compute the $\mathcal{C}$-operator according to (23) to
$$\mathcal{C}=\left(\begin{array}[]{cccc}C_{5}&-C_{3}&-C_{3}&C_{4}\\
-C_{3}&-C_{1}-1&-C_{1}&C_{2}\\
-C_{3}&-C_{1}&-C_{1}-1&C_{2}\\
C_{4}&C_{2}&C_{2}&2(C_{1}+1)-C_{5}\end{array}\right)$$
(67)
where the matrix entries are
$$\begin{array}[]{lll}C_{1}=\frac{\alpha_{4}^{2}}{N_{4}^{2}}-\frac{\alpha_{2}^{2%
}}{N_{2}^{2}}-\frac{\alpha_{3}^{2}}{N_{3}^{2}}-\frac{1}{2},&C_{2}=\frac{\alpha%
_{4}\beta_{4}}{N_{4}^{2}}-\frac{\alpha_{2}\beta_{2}}{N_{2}^{2}}-\frac{\alpha_{%
3}\beta_{3}}{N_{3}^{2}},&C_{3}=\frac{\alpha_{2}\gamma_{2}}{N_{2}^{2}}+\frac{%
\alpha_{3}\gamma_{3}}{N_{3}^{2}}-\frac{\alpha_{4}\gamma_{4}}{N_{4}^{2}},\\
C_{4}=\frac{\beta_{2}\gamma_{2}}{N_{2}^{2}}+\frac{\beta_{3}\gamma_{3}}{N_{3}^{%
2}}-\frac{\beta_{4}\gamma_{4}}{N_{4}^{2}},&C_{5}=\frac{\gamma_{2}^{2}}{N_{2}^{%
2}}+\frac{\gamma_{3}^{2}}{N_{3}^{2}}-\frac{\gamma_{4}^{2}}{N_{4}^{2}}.&\end{array}$$
(68)
We may now verify that $\mathcal{C}$ indeed satisfied the properties (24) upon the use of the identities
$$\begin{array}[]{lll}C_{2}=C_{2}C_{5}-C_{3}C_{4},&C_{3}=C_{5}C_{3}-C_{2}C_{4}-2%
C_{1}C_{3},&C_{4}=C_{2}C_{3}-C_{1}C_{4},\\
1=2C_{3}^{2}+C_{4}^{2}+C_{5}^{2},&0=C_{2}^{2}+C_{3}^{2}+2C_{1}(C_{1}+1).&\end{array}$$
(69)
Next we compute the metric operator in the form $\rho=\mathcal{PC}$
simply from (7) and (67) to
$$\rho=\left(\begin{array}[]{cccc}C_{5}&-C_{3}&-C_{3}&C_{4}\\
C_{3}&1+C_{1}&C_{1}&-C_{2}\\
C_{3}&C_{1}&1+C_{1}&-C_{2}\\
C_{4}&C_{2}&C_{2}&2(1+C_{1})-C_{5}\end{array}\right)$$
(70)
Since $i\alpha_{i},\beta_{i},\gamma_{i}\in\mathbb{R}$ it follows that $C_{1},iC_{2},iC_{3},C_{4},C_{5}\in\mathbb{R}$ and therefore we conclude
immediately that $\rho$ is Hermitian. To see whether $\rho$ is also
positive, as it ought to be, we compute its eigenvalues
$$y_{1}=y_{2}=1\qquad\text{and\qquad}y_{3/4}=1+2C_{1}\pm 2\sqrt{C_{1}(1+C_{1})}.$$
(71)
Since $C_{1}>0$ all eigenvalues of $\rho$ are obviously guaranteed to be
positive.
Next we determine the corresponding eigenstates to
$$\left|r_{1}\right\rangle=(0,-1,1,0),\quad\left|r_{2}\right\rangle=(C_{4},0,0,1%
-C_{5}),\quad\left|r_{3/4}\right\rangle=(\tilde{\gamma}_{3/4},\tilde{\alpha}_{%
3/4},\tilde{\alpha}_{3/4},\tilde{\beta}_{3/4})$$
(72)
with $\tilde{\alpha}_{3/4}=y_{3/4}(C_{3}C_{4}+C_{2}(-4C_{1}+C_{5}-1))/2-C_{3}C_{4}$, $\tilde{\beta}_{3/4}=-C_{3}^{2}-C_{1}-C_{1}C_{5}+\left(C_{3}^{2}+C_{1}(4C_{1}-C%
_{5}+3)\right)y_{3/4}$ and $\tilde{\gamma}_{3/4}=C_{1}C_{4}-C_{2}C_{3}+(C_{2}C_{3}+C_{1}C_{4})y_{3/4}$. Defining now
the matrix $U=\{r_{1},r_{2},r_{3},r_{4}\}$, whose column vectors are the
eigenvectors of $\rho$, we may take the square root of $\rho$, such that $\eta=\rho^{1/2}=UD^{1/2}U^{-1}$, where $D=\mathop{\mathrm{d}iag}(y_{1,}y_{2,},y_{3,},y_{4})$. The isospectral Hermitian counterpart of $H$
results from (26) to an $XYZ$ spin chain (with just two sites) in a
magnetic field
$$\displaystyle h(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\eta H\eta^{-1}=UD^{1/2}U^{-1}HUD^{-1/2}U^{-1}$$
(73)
$$\displaystyle=$$
$$\displaystyle\mu_{xx}^{2}(\lambda,\kappa)S_{xx}^{2}+\mu_{yy}^{2}(\lambda,%
\kappa)S_{yy}^{2}+\mu_{zz}^{2}(\lambda,\kappa)S_{zz}^{2}+\mu_{z}^{2}(\lambda,%
\kappa)S_{z}^{2}.$$
(74)
It is clear that the coefficients $\mu_{xx}^{2}$, $\mu_{yy}^{2}$, $\mu_{zz}^{2}$, $\mu_{z}^{2}$ can be computed explicitly, but the expressions are rather
lengthy and we will therefore not present them here. They are all real
functions of $\lambda$ and $\kappa$. Their explicit form can be found in
appendix A in terms of functions of $\lambda$ and $\kappa$ (86)
which will be introduced in section 5, in the context of perturbation
theory. In the next section we wish to compare this exact result with a
perturbative expansion. Let us therefore report two numerical examples for
some isospectral Hermitian counterpart of $H(\lambda,\kappa)$
$$h(0.1,0.5)=\left(\begin{array}[]{cccc}-0.829536&0&0&-0.0606492\\
0&-0.0341687&-0.1341687&0\\
0&-0.1341687&-0.0341687&0\\
-0.0606492&0&0&0.897873\end{array}\right),$$
(75)
and
$$h(0.9,0.1)=\left(\begin{array}[]{cccc}-0.985439&0&0&-0.890532\\
0&-0.0094167&-0.909417&0\\
0&-0.909417&-0.0094167&0\\
-0.890532&0&0&1.00427\end{array}\right).$$
(76)
Notice that $h_{23}=h_{32}=h_{22}-\lambda=h_{33}-\lambda$.
We have carried out a similar analysis for the chain with three sites
explicitly, albeit the resulting formulae are rather cumbersome to present.
In any case for longer chains one has to resort to more sophisticated and
less transparent techniques as for instance the Bethe ansatz. Alternatively,
we may employ perturbation theory.
5 The Yang-Lee quantum chain: perturbative results
In this section we want to address the problem of obtaining the matrices $\eta$, $\rho$ and $h$ from a perturbative analysis as described in section 3.3. We will study the $N=2,3$ and $4$ cases in detail and draw some
conclusions concerning the analytic expressions of $\eta$, $\rho$ and $h$
for generic $N$.
5.1 The $N=2$ case: perturbation theory in $\kappa$
Despite the fact that $H(\lambda,\kappa)$ is just the $4\times 4$-matrix (62), it is actually not easy to find the matrix $q$ in (31) exactly. As discussed in section 3.4, it is clear that the
equations (36) to (38) as well as the equations that would be
obtained for higher orders in perturbation theory, admit many solutions. Any
solution $q_{2k-1}$ can be modified by adding a matrix that commutes with $h_{0}(\lambda)$. However, not all solutions obtained in this manner would
be valid solutions if the equations (42) are to hold. For the
particular case $N=2$, we are about to show that these constraints actually
select out a unique Hermitian counterpart to the Hamiltonian $H(\lambda,\kappa)$. We will start by finding the most general matrix $q_{1}(\lambda)$ which solves the identity (36). It is quite clear that given one
solution $q_{1}(\lambda)$, any matrix of the form $q_{1}(\lambda)+B(\lambda)$ with $[h_{0}(\lambda),B(\lambda)]=0$ will also be a
solution, so we may start by finding all such matrices. In this simple case,
there are four basic independent solutions to the equation $[h_{0}(\lambda),B(\lambda)]=0$
$$B_{1}=\mathbb{I}\text{,\quad}B_{2}=S_{zz}^{2}\text{,\quad}B_{3}=S_{xx}^{2}+S_{%
yy}^{2}\quad\text{and\quad}B_{4}=S_{z}^{2}-\lambda S_{yy}^{2}\text{.}$$
(77)
Since $h_{0}(\lambda)$ is a $4\times 4$-diagonalizable matrix, with
non-degenerate eigenvalues, there can be at most four independent matrices
that commute with it, namely those shown above or combinations thereof. On
the other hand, it is clear that any polynomial function of the Hamiltonian $h_{0}(\lambda)$ would also commute with $h_{0}(\lambda)$. As the four
matrices in (77) constitute a basis, we expect to be able to express
any power of $h_{0}$ as linear combinations of them. Indeed, we find
$$\displaystyle h_{0}(\lambda)^{2n}$$
$$\displaystyle=$$
$$\displaystyle\frac{(1+\lambda^{2})^{n}}{2}(B_{1}+\frac{1}{2}B_{2})+\frac{%
\lambda^{2n}}{2}(B_{1}-\frac{1}{2}B_{2}),$$
(78)
$$\displaystyle h_{0}(\lambda)^{2n+1}$$
$$\displaystyle=$$
$$\displaystyle(1+\lambda^{2})^{n}h_{0}(\lambda)+\frac{\lambda(1+\lambda^{2})^{n%
}-\lambda^{2n+1}}{4}B_{3},$$
(79)
for $n\in\mathbb{N}_{0}$. Therefore, the most general solution to the first
order equation (36) for the present model is
$$q_{1}(\lambda)=-S_{y}^{2}-\lambda(S_{yz}^{2}+S_{zy}^{2})+\sum\limits_{i=1}^{4}%
{{f_{i}(\lambda)}}B_{i},$$
(80)
where the $f_{i}(\lambda)$, $i=1,2,3,4$ are arbitrary functions of $\lambda$.
Before we proceed to determine $q_{3}(\lambda)$ by solving (37) let
us comment on the ambiguities and answer the question of whether all
solutions (80) are compatible with the equations (42).
Specializing equations (47) and (48) for the matrices in (80) we find
$$\mathcal{P}X\mathcal{P=\mathcal{T}}X\mathcal{\mathcal{T=}-}X,\qquad\text{for }%
X=S_{y}^{2},S_{yz}^{2},S_{zy}^{2}$$
(81)
whereas
$$\mathcal{P}B_{i}\mathcal{P}=\mathcal{T}B_{i}\mathcal{T}=B_{i},\qquad\text{for %
}i=1,2,3,4.$$
(82)
These equations imply that the equalities (42) can only be satisfied
if the functions $f_{i}(\lambda)=0$ for $i=1,2,3,4$. Thus we have selected
out a unique solution for $q_{1}(\lambda)$, namely
$$q_{1}(\lambda)=-S_{y}^{2}-\lambda(S_{yz}^{2}+S_{zy}^{2}).$$
(83)
More generally, the conditions (42) together with the properties (47) and (48) imply that
•
any solutions $q_{2k-1}$ must be linear combinations of matrices (45) with $n_{y}$ odd,
•
any solutions $q_{2k-1}$ must be linear combinations of matrices (45) with $n_{y}+n_{x}$ odd,
•
or, combining the two conditions above, any solutions $q_{2k-1}$ must
be linear combinations of matrices (45) with $n_{y}$ odd and $n_{x}$
even,
as anticipated in subsection 4.1.1. These conditions then
automatically guarantee the validity of the $\mathcal{PT}$-properties (44) for the $q_{2k-1}$. For $N=2$, this singles out the matrices $S_{y}^{2}$ and $S_{yz}^{2}=S_{zy}^{2}$ in (83), so that, even before
attempting to solve (36) we would already know that it can only be a
linear combination of those two matrices. As indicated above, these
constraints apply for all other $q_{2k-1}(\lambda)$, with $k>1$ so that we
can safely claim that, at all orders in perturbation theory, the matrices $q_{2k-1}(\lambda)$ must be linear combinations of the form,
$$q_{2k-1}(\lambda)=a_{2k-1}(\lambda)S_{y}^{2}+b_{2k-1}(\lambda)(S_{yz}^{2}+S_{%
zy}^{2}),$$
(84)
where $a_{2k-1}(\lambda),b_{2k-1}(\lambda)$ are real functions of $\lambda$. In other words, all the terms in the perturbative expansion of $q$ are
linear combinations of the same two matrices. Hence, we can write
$$e^{q}=e^{\alpha(\lambda,\kappa)S_{y}^{2}+\beta(\lambda,\kappa)(S_{yz}^{2}+S_{%
zy}^{2})},$$
(85)
which, after computing the exponential becomes
$$\left(\begin{array}[]{cccc}\frac{\rho(\lambda,\kappa)^{2}+\epsilon(\lambda,%
\kappa)^{2}\cosh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda,\kappa)^{2}}&-\frac%
{i\epsilon(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda,%
\kappa)}&-\frac{i\epsilon(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2%
\gamma(\lambda,\kappa)}&-\frac{\delta(\lambda,\kappa)\sinh^{2}[\gamma(\lambda,%
\kappa)]}{\gamma(\lambda,\kappa)^{2}}\\
\frac{i\epsilon(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda%
,\kappa)}&\cosh^{2}\gamma(\lambda,\kappa)&\sinh^{2}\gamma(\lambda,\kappa)&-%
\frac{i\rho(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda,%
\kappa)}\\
\frac{i\epsilon(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda%
,\kappa)}&\sinh^{2}\gamma(\lambda,\kappa)&\cosh^{2}\gamma(\lambda,\kappa)&-%
\frac{i\rho(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda,%
\kappa)}\\
\frac{\delta(\lambda,\kappa)\sinh^{2}\gamma(\lambda,\kappa)}{\gamma(\lambda,%
\kappa)^{2}}&\frac{i\rho(\lambda,\kappa)\sinh[2\gamma(\lambda,\kappa)]}{2%
\gamma(\lambda,\kappa)}&\frac{i\rho(\lambda,\kappa)\sinh[2\gamma(\lambda,%
\kappa)]}{2\gamma(\lambda,\kappa)}&\frac{\epsilon(\lambda,\kappa)^{2}+\rho(%
\lambda,\kappa)^{2}\cosh[2\gamma(\lambda,\kappa)]}{2\gamma(\lambda,\kappa)^{2}%
}\end{array}\right)$$
where
$$\alpha(\lambda,\kappa)=\sum_{k=0}^{\infty}\kappa^{2k+1}a_{2k+1}(\lambda),%
\qquad\beta(\lambda,\kappa)=\sum_{k=0}^{\infty}\kappa^{2k+1}b_{2k+1}(\lambda),$$
(86)
and
$$\gamma(\lambda,\kappa)=\sqrt{\alpha(\lambda,\kappa)^{2}+4\beta(\lambda,\kappa)%
^{2}},\quad\delta(\lambda,\kappa)=\alpha(\lambda,\kappa)^{2}-4\beta(\lambda,%
\kappa)^{2}.$$
(87)
$$\epsilon(\lambda,\kappa)=\alpha(\lambda,\kappa)+2\beta(\lambda,\kappa),\quad%
\rho(\lambda,\kappa)=\alpha(\lambda,\kappa)-2\beta(\lambda,\kappa).$$
(88)
Notice that, for $\alpha(\lambda,\kappa)$ and $\beta(\lambda,\kappa)$
real, the matrix above is explicitly Hermitian, as it should be. Once the
coefficients $\alpha(\lambda,\kappa)$ and $\beta(\lambda,\kappa)$ have
been obtained, the Hermitian Hamiltonian (26) can be easily computed.
The difficulty here is however that general formulae for the coefficients $a_{2k+1}(\lambda)$ and $b_{2k+1}(\lambda)$ are very difficult to obtain.
Nonetheless, perturbation theory allows us to compute these coefficients up
to very high orders in powers of $\kappa$. In order to solve for such high
orders, we have resorted to the use of the algebraic manipulation software
Mathematica. It allows us to find the entries of the matrix (85) as
perturbative series in $\kappa$ and to fix the coefficients $a_{2k+1}(\lambda)$ and $b_{2k+1}(\lambda)$ by matching the entries of $H^{\dagger}(\lambda,\kappa)$ and $\eta^{2}H(\lambda,\kappa)\eta^{-2}$, order by order in perturbation theory, as expected from (26). For
numerical computations and sufficiently small values of $\kappa$ this gives
results which are very close to the exact values. In tables 1 and 2 we
present the coefficients $a_{2k+1}(\lambda)$ and $b_{2k+1}(\lambda)$ up to
$k=7$.
$$-\lambda^{0}$$
$$-\lambda^{2}$$
$$-\lambda^{4}$$
$$-\lambda^{6}$$
$$-\lambda^{8}$$
$$-\lambda^{10}$$
$$-\lambda^{12}$$
$$-\lambda^{14}$$
$$a_{1}(\lambda)$$
1
0
0
0
0
0
0
0
$$a_{3}(\lambda)$$
$$\frac{1}{3}$$
$$\frac{2^{4}}{3}$$
0
0
0
0
0
0
$$a_{5}(\lambda)$$
$$\frac{1}{5}$$
$$\frac{244}{15}$$
$$\frac{2^{8}}{5}$$
0
0
0
0
0
$$a_{7}(\lambda)$$
$$\frac{1}{7}$$
$$\frac{1152}{35}$$
$$\frac{35104}{105}$$
$$\frac{2^{12}}{7}$$
0
0
0
0
$$a_{9}(\lambda)$$
$$\frac{1}{9}$$
$$\frac{17432}{315}$$
$$\frac{43408}{35}$$
$$\frac{1890368}{315}$$
$$\frac{2^{16}}{9}$$
0
0
0
$$a_{11}(\lambda)$$
$$\frac{1}{11}$$
$$\frac{289616}{3465}$$
$$\frac{797296}{231}$$
$$\frac{38228224}{1155}$$
$$\frac{355526144}{3465}$$
$$\frac{2^{20}}{11}$$
0
0
$$a_{13}(\lambda)$$
$$\frac{1}{13}$$
$$\frac{353372}{3003}$$
$$\frac{72293440}{9009}$$
$$\frac{655729408}{5005}$$
$$\frac{2275245568}{3003}$$
$$\frac{15442769920}{9009}$$
$$\frac{2^{24}}{13}$$
0
$$a_{15}(\lambda)$$
$$\frac{1}{15}$$
$$\frac{7100416}{45045}$$
$$\frac{67453952}{4095}$$
$$\frac{896579072}{2145}$$
$$\frac{58903814656}{15015}$$
$$\frac{717363822592}{45045}$$
$$\frac{1273503367168}{45045}$$
$$\frac{2^{28}}{15}$$
Table 1: The coefficients ${a}_{2k+1}(\lambda)$
for $k<8$.
$$-\lambda$$
$$-\lambda^{3}$$
$$-\lambda^{5}$$
$$-\lambda^{7}$$
$$-\lambda^{9}$$
$$-\lambda^{11}$$
$$-\lambda^{13}$$
$$-\lambda^{15}$$
$$b_{1}(\lambda)$$
1
0
0
0
0
0
0
0
$$b_{3}(\lambda)$$
$$\frac{4}{3}$$
$$\frac{2^{4}}{3}$$
0
0
0
0
0
0
$$b_{5}(\lambda)$$
$$\frac{23}{15}$$
$$\frac{2^{7}}{5}$$
$$\frac{2^{8}}{5}$$
0
0
0
0
0
$$b_{7}(\lambda)$$
$$\frac{176}{105}$$
$$\frac{7544}{105}$$
$$\frac{2^{10}(3)}{7}$$
$$\frac{2^{12}}{7}$$
0
0
0
0
$$b_{9}(\lambda)$$
$$\frac{563}{315}$$
$$\frac{49136}{315}$$
$$\frac{212816}{105}$$
$$\frac{2^{16}}{9}$$
$$\frac{2^{16}}{9}$$
0
0
0
$$b_{11}(\lambda)$$
$$\frac{6508}{3465}$$
$$\frac{335576}{1155}$$
$$\frac{7827328}{1155}$$
$$\frac{164005504}{3465}$$
$$\frac{2^{18}(5)}{11}$$
$$\frac{2^{20}}{11}$$
0
0
$$b_{13}(\lambda)$$
$$\frac{88069}{45045}$$
$$\frac{4400960}{9009}$$
$$\frac{39578944}{2145}$$
$$\frac{1947324416}{9009}$$
$$\frac{9037578752}{9009}$$
$$\frac{2^{23}(3)}{13}$$
$$\frac{2^{24}}{13}$$
0
$$b_{15}(\lambda)$$
$$\frac{91072}{45045}$$
$$\frac{34381136}{45045}$$
$$\frac{178162048}{4095}$$
$$\frac{1068366848}{1365}$$
$$\frac{37570428928}{6435}$$
$$\frac{903387164672}{45045}$$
$$\frac{2^{26}(7)}{15}$$
$$\frac{2^{28}}{15}$$
Table 2: The coefficients ${b}_{2k+1}(\lambda)$
for $k<8$.
These tables should be understood as follows: in order to obtain
the corresponding coefficient the numbers in a given row are to be
multiplied by the power of $\lambda$ (with a minus sign added) at the top
of the same column and added up. For example:
$$a_{5}(\lambda)=-\frac{1}{5}-\frac{244{\lambda}^{2}}{15}-\frac{2^{8}{\lambda}^{%
4}}{5}.$$
(89)
The only case for which it is easy to conjecture the expressions of $a_{2k+1}(\lambda),b_{2k+1}(\lambda)$ for generic values of $k$ corresponds
to $\lambda=0$. Then $a_{2k+1}(0)=-1/(2k+1)$ and $b_{2k+1}(0)=0$, which
gives the already known result $\alpha(0,\kappa)=-\text{arctanh}(\kappa)$
and $\beta(0,\kappa)=0$, see section 4.1. Having found $\eta$, it
is straightforward using (26) to determine the Hermitian counterpart of
$H(\lambda,\kappa)$. In general, we find
$$\displaystyle h(\lambda,\kappa)=e^{q/2}H(\lambda,\kappa)e^{-q/2}=\left(\begin{%
array}[]{cccc}h_{11}(\lambda,\kappa)&0&0&h_{14}(\lambda,\kappa)\\
0&h_{22}(\lambda,\kappa)&h_{22}(\lambda,\kappa)-\lambda&0\\
0&h_{22}(\lambda,\kappa)-\lambda&h_{22}(\lambda,\kappa)&0\\
h_{14}(\lambda,\kappa)&0&0&h_{44}(\lambda,\kappa)\end{array}\right)$$
(90)
$$\displaystyle=$$
$$\displaystyle\frac{h_{22}(\lambda,\kappa)-\lambda+h_{14}(\lambda,\kappa)}{4}S_%
{xx}^{2}+\frac{h_{22}(\lambda,\kappa)-\lambda-h_{14}(\lambda,\kappa)}{4}S_{yy}%
^{2}$$
$$\displaystyle+\frac{h_{11}(\lambda,\kappa)+h_{44}(\lambda,\kappa)-2h_{22}(%
\lambda,\kappa)}{8}S_{zz}^{2}+\frac{h_{11}(\lambda,\kappa)-h_{44}(\lambda,%
\kappa)}{4}S_{z}^{2}$$
$$\displaystyle+\frac{h_{11}(\lambda,\kappa)+h_{44}(\lambda,\kappa)+2h_{22}(%
\lambda,\kappa)}{4},$$
which is the same kind of structure found in (74). The functions $h_{11}(\lambda,\kappa)$, $h_{22}(\lambda,\kappa)$, $h_{14}(\lambda,\kappa)$ and $h_{44}(\lambda,\kappa)$ are real functions of the coupling
constants which can be evaluated very accurately for fixed values of $\lambda$ and $\kappa$ by using the perturbative results above. In fact,
the remaining entries of the matrix are not explicitly zero as functions of $\alpha(\lambda,\kappa)$ and $\beta(\lambda,\kappa)$. They are
complicated functions of the latter which when carrying out the perturbation
theory result to be zero up order $\kappa^{15}$. This is consistent with
the exact results obtained before. The explicit expressions of the entries
of $h(\lambda,\kappa)$ in terms of the functions (87) and (88) can be found in appendix A. Here, we will just present their
expression as a series expansion in $\kappa$ up to order $\kappa^{4}$ (for
higher orders, expression become too cumbersome),
$$\displaystyle h_{11}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle-1+\left(\frac{1}{2}+\lambda\right){\kappa}^{2}+\left(\frac{1}{8}%
+\lambda+\frac{3{\lambda}^{2}}{2}+4{\lambda}^{3}\right){\kappa}^{4}+\mathcal{O%
}(\kappa^{6}),$$
(91)
$$\displaystyle h_{22}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle-\lambda{\kappa}^{2}-\lambda\left(1+4{\lambda}^{2}\right){\kappa}%
^{4}+\mathcal{O}(\kappa^{6}),$$
(92)
$$\displaystyle h_{44}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle 1+\left(-\frac{1}{2}+\lambda\right){\kappa}^{2}+\left(-\frac{1}{%
8}+\lambda-\frac{3{\lambda}^{2}}{2}+4{\lambda}^{3}\right){\kappa}^{4}+\mathcal%
{O}(\kappa^{6}),$$
(93)
$$\displaystyle h_{14}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle-\lambda+\lambda{\kappa}^{2}+\left(\frac{3\lambda}{2}+4{\lambda}^%
{3}\right){\kappa}^{4}+\mathcal{O}(\kappa^{6}).$$
(94)
From this expansions we can deduce some interesting features which also
extend to higher orders in perturbation theory
$$h_{11}(-\lambda,\kappa)=-h_{44}(\lambda,\kappa),\quad h_{22}(-\lambda,\kappa)=%
-h_{22}(\lambda,\kappa),\quad h_{14}(-\lambda,\kappa)=-h_{14}(\lambda,\kappa).$$
(95)
Furthermore, we note that the Hermitian Hamiltonian $h(\lambda,\kappa)$ is
an even function of $\kappa$, so that the series expansion of its
components involves only even powers of the coupling. Finally, as it should
be, the Hamiltonian $h(\lambda,\kappa)$ is also $\mathcal{PT}$-symmetric,
which follows from the fact that all matrices involved ($S_{xx}^{2},S_{yy}^{2},S_{zz}^{2}$ and $S_{z}^{2}$) are invariant under the
adjoint action of the operator $\mathcal{PT}$. These are in fact the only
matrices that are both $\mathcal{PT}$-symmetric and real. In fact we could
have known a priori before carrying any computations that $h(\lambda,\kappa)$ has to be some linear combination of $S_{xx}^{2},S_{yy}^{2},S_{zz}^{2}$
and $S_{z}^{2}$. Notice that the reality of $h(\lambda,\kappa)$ can be
expressed by saying that any matrices (45) involved must have $n_{y}$ even, as defined in the paragraph after equation (49).
In order to compare with the results obtained in section 4 we give
below the numerical values of the entries of the Hermitian Hamiltonian $h(\lambda,\kappa)$ for fixed values of the couplings
$$h(0.1,0.5)=\left(\begin{array}[]{cccc}-0.82953\underline{4}&0&0&-0.0606%
\underline{716}\\
0&-0.034168\underline{8}&-0.13416\underline{9}&0\\
0&-0.13416\underline{9}&-0.034168\underline{8}&0\\
-0.0606\underline{716}&0&0&0.89787\underline{2}\end{array}\right),$$
(96)
and
$$h(0.9,0.1)=\left(\begin{array}[]{cccc}-0.985439&0&0&-0.890532\\
0&-0.00941674&-0.909417&0\\
0&-0.909417&-0.00941674&0\\
-0.890532&0&0&1.00427\end{array}\right).$$
(97)
We underlined the digits which differ from the exact values computed in (75) and (76) and note that the perturbative expressions for $h(0.1,0.5)$ and $h(0.9,0.1)$ agree extremely well with them, especially for
smaller values of $\kappa$, as is expected.
In order to see how fast this precision is reached in the perturbation
theory we report in table 3 the relative error for the entry $h_{11}$ order
by order up to $15$
$$\lambda,\kappa\backslash\mathcal{O}(\kappa)$$
$$2$$
$$4$$
$$6$$
$$8$$
$$10$$
$$12$$
$$14$$
$$0.9,0.1$$
$$5.7~{}10^{-4}$$
$$4.6~{}10^{-5}$$
$$4.7~{}10^{-6}$$
$$5.3~{}10^{-7}$$
$$6.4~{}10^{-8}$$
$$8.2~{}10^{-9}$$
$$1.1~{}10^{-9}$$
$$0.1,0.5$$
$$2.5~{}10^{-2}$$
$$6.3~{}10^{-3}$$
$$2.1~{}10^{-3}$$
$$7.5~{}10^{-4}$$
$$2.9~{}10^{-4}$$
$$1.6~{}10^{-4}$$
$$4.7~{}10^{-5}$$
Table 3: Relative error = —(perturbative value -
exact value) / exact value— for $h_{11}$ order by order.
We observe that the convergence is fairly fast, which allows to
extract useful information from the perturbation theory even at low order.
We shall not be concerned here with more rigorous mathematical arguments
regarding the summability and convergence in general.
5.2 The $N=2$ case: perturbation theory in $\lambda$
In the previous section we have employed the standard version of
perturbation theory when dealing with non-Hermitian Hamiltonians of the type
(1), that is decomposing the Hamiltonian into a Hermitian and a
non-Hermitian part as in (30) and then treating the non-Hermitian
part as the perturbation. Since the Hamiltonian (1) depends on two
independent coupling constants, $\kappa$ and $\lambda$, it is also
natural, albeit less standard, to consider perturbation theory in the
coupling constant $\lambda$ rather than in $\kappa$. In other words we
expand around the exact solution for $\lambda=0$ provided in section $4.1$
and treat the nearest neighbour interaction term as perturbation. As
announced already in section 3.2., we decompose $H(\lambda,\kappa)$ into
$$H(\lambda,\kappa)=\tilde{H}_{0}(\kappa)+\lambda\tilde{h}_{1},\quad\text{where}%
\quad\tilde{H}_{0}(\kappa)=-\frac{1}{2}(S_{z}^{N}+i\kappa S_{x}^{N}),\qquad%
\tilde{h}_{1}=-\frac{1}{2}S_{xx}^{N}.$$
(98)
We wish now once again to solve the equations (26) for the Dyson map $\eta$, that is
$$H^{\dagger}(\lambda,\kappa)=e^{w}H(\lambda,\kappa)e^{-w},$$
(99)
where we have assumed that $\eta$ admits the exponential form
$$\eta=e^{w/2}\quad\qquad\text{with}\quad\qquad w=\sum_{a=0}^{\infty}\lambda^{a}%
w_{a}(\kappa).$$
(100)
At order $\lambda^{0}$ equation (99) becomes simply
$$\tilde{H}_{0}^{\dagger}(\kappa)=e^{w_{0}(\kappa)}\tilde{H}_{0}(\kappa)e^{-w_{0%
}(\kappa)}.$$
(101)
The solution to this equation for all $N$ was found in subsection 4.1
and corresponds to the Dyson map identified in equation (51). For $N=2$
this means that
$$w_{0}(\kappa)=-\text{arctanh}(\kappa)S_{y}^{2}.$$
(102)
Employing the once again the Backer-Campbell-Hausdorff identity to select $\mathcal{O}(\lambda)$ terms in (99) we find the condition
$$\tilde{h}_{1}=e^{w_{0}(\kappa)}\tilde{h}_{1}e^{-w_{0}(\kappa)}+\sum_{k=1}^{%
\infty}\sum_{i=1}^{k}\sum_{\begin{subarray}{c}a_{i}=1,\\
a_{j\neq i}=0\end{subarray}}\frac{1}{k!}\left[w_{a_{1}}(\kappa),\left[w_{a_{2}%
}(\kappa),\cdots,\left[w_{a_{k}}(\kappa),H_{0}(\kappa)\right]\cdots\right]\right]$$
(103)
Notice that, because of the presence of the zeroth order term $w_{0}(\kappa)$, the equation (103) involves a sum of infinitely many
contributions, as would equations corresponding to higher orders in
perturbation theory. Because of this, it would in general be difficult to
solve (99) using perturbation theory in $\lambda$. However, for $N=2$
we can solve up to high orders in $\lambda$ by exploiting the fact that $\eta$ must have the structure identified in the previous section. This
means that $\eta$ is a matrix of the form (85) with
$$\alpha(\lambda,\kappa)=\sum_{a=0}^{\infty}\lambda^{a}y_{a}(\kappa),\qquad\beta%
(\lambda,\kappa)=\sum_{a=0}^{\infty}\lambda^{a}z_{a}(\kappa).$$
(104)
It is then possible to find the real functions $y_{a}(\kappa)$ and $z_{a}(\kappa)$ which solve equation (99) order by order in $\lambda$
by employing Mathematica, as explained in the previous subsection. In this
way, we have obtained the functions $y_{a}(\kappa)$ and $z_{a}(\kappa)$
above up to order $\lambda^{15}$. Here we will just report the first five
orders,
$$\displaystyle y_{0}(\kappa)$$
$$\displaystyle=$$
$$\displaystyle-\text{arctanh}(\kappa),$$
(105)
$$\displaystyle z_{1}(\kappa)$$
$$\displaystyle=$$
$$\displaystyle\frac{y_{0}(\kappa)}{1-\kappa^{2}},$$
(106)
$$\displaystyle y_{2}(\kappa)$$
$$\displaystyle=$$
$$\displaystyle-\frac{2(\kappa+2{\kappa}^{3}+\left(1-{\kappa}^{2}\right)y_{0}(%
\kappa))}{{\left(1-{\kappa}^{2}\right)}^{3}},$$
(107)
$$\displaystyle z_{3}(\kappa)$$
$$\displaystyle=$$
$$\displaystyle-\frac{2(\kappa+2{\kappa}^{3}+\left(1-{\kappa}^{2}-2{\kappa}^{4}%
\right)y_{0}(\kappa))}{{\left(1-{\kappa}^{2}\right)}^{4}},$$
(108)
$$\displaystyle y_{4}(\kappa)$$
$$\displaystyle=$$
$$\displaystyle\frac{2\left(\kappa\left(3-5{\kappa}^{2}-32{\kappa}^{4}-8{\kappa}%
^{6}\right)+\left(3-6{\kappa}^{2}-5{\kappa}^{4}+8{\kappa}^{6}\right)y_{0}(%
\kappa)\right)}{{\left(1-{\kappa}^{2}\right)}^{6}},$$
(109)
$$\displaystyle z_{5}(\kappa)$$
$$\displaystyle=$$
$$\displaystyle\frac{2\left(\kappa\left(3-5{\kappa}^{2}-36{\kappa}^{4}-16{\kappa%
}^{6}\right)+\left(3-6{\kappa}^{2}-9{\kappa}^{4}+28{\kappa}^{6}+8{\kappa}^{8}%
\right)y_{0}(\kappa)\right)}{{\left(1-{\kappa}^{2}\right)}^{7}},$$
(110)
and $y_{2a+1}(\kappa)=z_{2a}(\kappa)=0$ for all $a=0,1,\ldots$ From these
formulae, it is possible to find an expression for the Hermitian Hamiltonian
$h(\lambda,\kappa)$ as a perturbative series in $\lambda$. As it should
be, one finds the same structure (90) with
$$\displaystyle h_{11}(\lambda,\kappa)=-\sqrt{1-{\kappa}^{2}}+\frac{{\kappa}^{2}%
\lambda}{1-{\kappa}^{2}}-\frac{6\left(-2+{\kappa}^{2}+2\sqrt{1-{\kappa}^{2}}%
\right){\lambda}^{2}}{{\left(1-{\kappa}^{2}\right)}^{\frac{5}{2}}}+\frac{4{%
\kappa}^{4}{\lambda}^{3}}{{\left(1-{\kappa}^{2}\right)}^{4}}$$
$$\displaystyle-\frac{2\left(40-44{\kappa}^{2}-57{\kappa}^{4}+28{\kappa}^{6}+8%
\sqrt{1-{\kappa}^{2}}\left(-5+3{\kappa}^{2}+8{\kappa}^{4}\right)\right){%
\lambda}^{4}}{{\left(1-{\kappa}^{2}\right)}^{\frac{11}{2}}}+\mathcal{O}(%
\lambda^{5}),$$
(111)
$$\displaystyle h_{22}(\lambda,\kappa)=-\frac{{\kappa}^{2}\lambda}{1-{\kappa}^{2%
}}-\frac{4{\kappa}^{4}{\lambda}^{3}}{{\left(1-{\kappa}^{2}\right)}^{4}}+%
\mathcal{O}(\lambda^{5}),$$
(112)
$$\displaystyle h_{44}(\lambda,\kappa)=\sqrt{1-{\kappa}^{2}}+\frac{{\kappa}^{2}%
\lambda}{1-{\kappa}^{2}}+\frac{6\left(-2+{\kappa}^{2}+2\sqrt{1-{\kappa}^{2}}%
\right){\lambda}^{2}}{{\left(1-{\kappa}^{2}\right)}^{\frac{5}{2}}}+\frac{4{%
\kappa}^{4}{\lambda}^{3}}{{\left(1-{\kappa}^{2}\right)}^{4}}$$
$$\displaystyle+\frac{2\left(40-44{\kappa}^{2}-57{\kappa}^{4}+28{\kappa}^{6}+8%
\sqrt{1-{\kappa}^{2}}\left(-5+3{\kappa}^{2}+8{\kappa}^{4}\right)\right){%
\lambda}^{4}}{{\left(1-{\kappa}^{2}\right)}^{\frac{11}{2}}}+\mathcal{O}(%
\lambda^{5}),$$
(113)
$$\displaystyle h_{14}(\lambda,\kappa)=\frac{\left(-4+4{\kappa}^{2}+3\sqrt{1-{%
\kappa}^{2}}\right)\lambda}{{\left(1-{\kappa}^{2}\right)}^{\frac{3}{2}}}$$
$$\displaystyle+\frac{4\left(8-10{\kappa}^{2}-2{\kappa}^{4}+4{\kappa}^{6}+\sqrt{%
1-{\kappa}^{2}}\left(2+{\kappa}^{2}\right)\left(-4+5{\kappa}^{2}\right)\right)%
{\lambda}^{3}}{{\left(1-{\kappa}^{2}\right)}^{\frac{9}{2}}}+\mathcal{O}(%
\lambda^{5}).$$
(114)
Notice that the same symmetries (95) are also found here. We also
see once again that $h(\lambda,\kappa)$ is an even function of $\kappa$,
as only even powers are involved. Computing again numerical values for $h(0.1,0.5)$ and $h(0.9,0.1)$ we find almost perfect agreement with the exact
results. There is extremely good agreement both with the exact results (75) and (76) and with the result from perturbation theory in $\kappa$ (96) and (97). In order to see how fast this precision is
reached in the perturbation theory we report in table 4 the relative error
for the entry $h_{11}$order by order up to $15$, omitting the odd orders
despite the fact that they occur in the $\lambda$-perturbation theory
$$\lambda,\kappa\backslash\mathcal{O(\lambda)}$$
$$2$$
$$4$$
$$6$$
$$8$$
$$10$$
$$12$$
$$14$$
$$0.9,0.1$$
$$3.4~{}10^{-3}$$
$$2.3~{}10^{-5}$$
$$1.9~{}10^{-6}$$
$$1.8~{}10^{-7}$$
$$1.9~{}10^{-8}$$
$$2.0~{}10^{-9}$$
$$2.2~{}10^{-10}$$
$$0.1,0.5$$
$$1.1~{}10^{-3}$$
$$6.3~{}10^{-5}$$
$$4.9~{}10^{-6}$$
$$3.6~{}10^{-7}$$
$$3.1~{}10^{-8}$$
$$2.8~{}10^{-9}$$
$$2.6~{}10^{-10}$$
Table 4: Relative error = —(perturbative value -
exact value) / exact value— for $h_{11}$ order by order.
We note that the perturbation theory converges extremely fast,
even for large values of $\lambda$, for which one would not expect such a
behaviour. This can be explained as follows: In the domain of unbroken $\mathcal{PT}$-symmetry $U_{\mathcal{PT}}$ the allowed values for $\kappa$
become very small as $\lambda$ increases. As we note from the expressions (111)-(114) the order of $\kappa$ increases with the order
of $\lambda$ term by term.
5.3 The $N=3$ case
We will now carry out an analogous perturbative study in $\kappa$ for the
three sites case. We keep the choice of periodic boundary condition, even
though for sites more than two this means some loss of generality.
Proceeding as before, we will try to obtain the matrix $q$ perturbatively,
by solving the consistency conditions (36)-(38). Now we have to
solve the problem for $8\times 8$-matrices. We commence by computing the
kernel of $h_{0}$
$$\begin{array}[]{llll}B_{1}=\mathbb{I},&B_{2}=S_{zz}^{3}-{\lambda}S_{yyz}^{3},&%
B_{3}=\lambda S_{yy}^{3}-(1-\lambda^{2})S_{yyz}^{3}-S_{xxz}^{3},&B_{4}=S_{xy}^%
{3}-S_{yx}^{3},\\
B_{5}=S_{zzz}^{3},&B_{6}=S_{xyz}^{3}-S_{yxz}^{3},&B_{7}=\lambda S_{xx}^{3}+S_{%
z}^{3}=-2h_{0}(\lambda),&B_{8}=S_{xx}^{3}+S_{yy}^{3}+\lambda S_{yyz}^{3},\end{array}$$
in addition to this eight matrices, there are another four, due to the fact
that two of the eigenvalues of $h_{0}(\lambda)$ are degenerate. Hence the
dimension of the kernel is 12,
$$\displaystyle B_{9}$$
$$\displaystyle=$$
$$\displaystyle S_{z}^{3}-\lambda(S_{yy}^{3}+S_{zz}^{3}-\sigma_{1}^{y}\sigma_{3}%
^{y}-\sigma_{1}^{z}\sigma_{3}^{z}-\sigma_{1}^{x}\sigma_{3}^{x}),\quad B_{10}=%
\sigma_{2}^{y}\sigma_{3}^{y}+\sigma_{2}^{z}\sigma_{3}^{z}+\sigma_{2}^{x}\sigma%
_{3}^{x},$$
(115)
$$\displaystyle B_{11}$$
$$\displaystyle=$$
$$\displaystyle S_{zz}^{3}+\lambda S_{xxz}^{3}-\lambda(\sigma_{1}^{z}+\sigma_{3}%
^{z}+\sigma_{1}^{x}\sigma_{2}^{z}\sigma_{3}^{x}+\sigma_{1}^{y}\sigma_{2}^{z}%
\sigma_{3}^{y}),\quad B_{12}=\sigma_{3}^{z}-\sigma_{1}^{x}\sigma_{2}^{x}\sigma%
_{3}^{z}-\sigma_{1}^{y}\sigma_{2}^{y}\sigma_{3}^{z},$$
with $[B_{i},h_{0}(\lambda)]=0$ for $i=1,\ldots,12$. Similarly as in the
case $N=2$ we find that all of these matrices are parity invariant
$$\mathcal{P}B_{i}\mathcal{P}=B_{i},\quad\forall\quad i=1,\ldots,8,$$
(116)
which from equations (44) means that no linear combination of the
matrices $B_{i}$ can be added to $q_{2k-1}$ that would be compatible with
the constraints (42). Therefore, with such constraints, there is a
unique solution to (36) which has the form,
$$q_{1}(\lambda)=-S_{y}^{3}-\lambda(S_{yz}^{3}+S_{zy}^{3})+2\lambda^{2}(S_{yyy}^%
{3}-S_{zzy}^{3}).$$
(117)
As we can see, the two first terms in $q_{1}(\lambda)$ are a direct
generalization of the result for two sites, which hints at the existence of
a general pattern. As for the $N=2$ case we find once again, that even
before attempting to solve (36), we could have predicted from (44) that the matrices $q_{2k-1}(\lambda)$ can only be linear
combinations of $S_{y}^{3},S_{yz}^{3},S_{zy}^{3},S_{yyy}^{3},S_{zzy}^{3}$
and $S_{xxy}^{3}$ (for $k=1$, equation (77) tells us though that the
coefficient of $S_{xxy}^{3}$ is zero. This will change for higher orders in
perturbation theory). We can therefore write,
$$q=\hat{\alpha}(\lambda,\kappa)S_{y}^{3}+\hat{\beta}(\lambda,\kappa)(S_{yz}^{3}%
+S_{zy}^{3})+\hat{\gamma}(\lambda,\kappa)S_{yyy}^{3}+\hat{\delta}(\lambda,%
\kappa)S_{xxy}^{3}+\hat{\epsilon}(\lambda,\kappa)S_{zzy}^{3},$$
(118)
where
$$\displaystyle\hat{\alpha}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\sum\nolimits_{k=1}^{\infty}\hat{a}_{2k-1}(\lambda)\kappa^{2k-1},%
\quad\hat{\beta}(\lambda,\kappa)=\sum\nolimits_{k=1}^{\infty}\hat{b}_{2k-1}(%
\lambda)\kappa^{2k-1},$$
(119)
$$\displaystyle\hat{\gamma}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\sum\nolimits_{k=1}^{\infty}\hat{s}_{2k-1}(\lambda)\kappa^{2k-1},%
\quad\hat{\delta}(\lambda,\kappa)=\sum\nolimits_{k=1}^{\infty}\hat{d}_{2k-1}(%
\lambda)\kappa^{2k-1},$$
(120)
$$\displaystyle\hat{\epsilon}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\sum\nolimits_{k=1}^{\infty}\hat{e}_{2k-1}(\lambda)\kappa^{2k-1}.$$
(121)
Computing coefficients up to order $\kappa^{7}$ we find the results in
tables 5-7.
$$-\lambda^{0}$$
$$-\lambda^{2}$$
$$-\lambda^{4}$$
$$-\lambda^{6}$$
$$-\lambda^{8}$$
$$-\lambda^{10}$$
$$-\lambda^{12}$$
$$\hat{a}_{1}(\lambda)$$
1
0
0
0
0
0
0
$$\hat{a}_{3}(\lambda)$$
$$\frac{1}{3}$$
$$\frac{8}{3}$$
16
0
0
0
0
$$\hat{a}_{5}(\lambda)$$
$$\frac{1}{5}$$
$$\frac{122}{15}$$
144
$$\frac{5024}{15}$$
$$\frac{3072}{5}$$
0
0
$$\hat{a}_{7}(\lambda)$$
$$\frac{1}{7}$$
$$\frac{576}{35}$$
$$\frac{9616}{15}$$
$$\frac{432832}{105}$$
$$\frac{1755136}{105}$$
$$\frac{2720768}{105}$$
$$\frac{196608}{7}$$
$$\hat{d}_{1}(\lambda)$$
0
0
0
0
0
0
0
$$\hat{d}_{3}(\lambda)$$
0
0
$$\frac{2^{4}}{3}$$
0
0
0
0
$$\hat{d}_{5}(\lambda)$$
0
$$\frac{2}{3}$$
$$\frac{496}{15}$$
$$\frac{1184}{15}$$
$$\frac{2^{10}}{5}$$
0
0
$$\hat{d}_{7}(\lambda)$$
0
$$\frac{2^{5}}{15}$$
$$\frac{4432}{35}$$
$$\frac{86848}{105}$$
$$\frac{65024}{15}$$
$$\frac{754688}{105}$$
$$\frac{2^{16}}{7}$$
Table 5: The coefficients $\hat{a}_{2k+1}(\lambda)$ and $\hat{d}_{2k+1}(\lambda)$ for $k<4$.
$$-\lambda$$
$$-\lambda^{3}$$
$$-\lambda^{5}$$
$$-\lambda^{7}$$
$$-\lambda^{9}$$
$$-\lambda^{11}$$
$$-\lambda^{13}$$
$$\hat{b}_{1}(\lambda)$$
1
0
0
0
0
0
0,
$$\hat{b}_{3}(\lambda)$$
$$\frac{4}{3}$$
$$\frac{28}{3}$$
$$\frac{2^{6}}{3}$$
0
0
0
0
$$\hat{b}_{5}(\lambda)$$
$$\frac{23}{15}$$
$$\frac{664}{15}$$
$$\frac{1568}{5}$$
$$\frac{3328}{5}$$
$$\frac{2^{12}}{5}$$
0
0,
$$\hat{b}_{7}(\lambda)$$
$$\frac{176}{105}$$
$$\frac{4344}{35}$$
$$\frac{13536}{7}$$
$$\frac{52416}{5}$$
$$\frac{1104384}{35}$$
$$\frac{311296}{7}$$
$$\frac{2^{18}}{7}$$
Table 6: The coefficients $\hat{b}_{2k+1}(\lambda)$ for $k<4$.
$$-\lambda^{2}$$
$$-\lambda^{4}$$
$$-\lambda^{6}$$
$$-\lambda^{8}$$
$$-\lambda^{10}$$
$$-\lambda^{12}$$
$$-\lambda^{14}$$
$$\hat{s}_{1}(\lambda)$$
-2
0
0
0
0
0
0
$$\hat{s}_{3}(\lambda)$$
-4
-8
$$-\frac{2^{7}}{3}$$
0
0
0
0
$$\hat{s}_{5}(\lambda)$$
$$-\frac{28}{5}$$
$$-\frac{112}{5}$$
$$-\frac{2592}{5}$$
$$-\frac{4608}{5}$$
$$-\frac{2^{13}}{5}$$
0
0
$$\hat{s}_{7}(\lambda)$$
$$-\frac{232}{35}$$
$$\frac{288}{35}$$
$$-\frac{91008}{35}$$
$$-\frac{452224}{35}$$
$$-\frac{356352}{7}$$
$$-\frac{491520}{7}$$
$$\frac{2^{19}}{7}$$
$$\hat{e}_{1}(\lambda)$$
2
0
0
0
0
0
0
$$\hat{e}_{3}(\lambda)$$
$$\frac{20}{3}$$
$$\frac{56}{3}$$
$$\frac{2^{7}}{3}$$
0
0
0
0
$$\hat{e}_{5}(\lambda)$$
$$\frac{196}{15}$$
$$\frac{400}{3}$$
$$\frac{3872}{5}$$
$$\frac{6656}{5}$$
$$\frac{2^{13}}{5}$$
0
0
$$\hat{e}_{7}(\lambda)$$
$$\frac{440}{21}$$
$$\frac{53152}{105}$$
$$\frac{206336}{35}$$
$$\frac{75904}{3}$$
$${69632}$$
$$\frac{622592}{7}$$
$$\frac{2^{19}}{7}$$
Table 7: The coefficients $\hat{s}_{2k+1}(\lambda)$ and $\hat{e}_{2k+1}(\lambda)$ for $k<4$.
It is now possible to use these perturbative results to compute $h(\lambda,\kappa)$ for particular values of $\lambda$ and $\kappa$. We find that
the structure of the Hermitian counterpart of the original Hamiltonian is:
$$\displaystyle h(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\mu_{xx}^{3}(\lambda,\kappa)S_{xx}^{3}+\mu_{yy}^{3}(\lambda,%
\kappa)S_{yy}^{3}+\mu_{zz}^{3}(\lambda,\kappa)S_{zz}^{3}+\mu_{z}^{3}(\lambda,%
\kappa)S_{z}^{3}$$
(122)
$$\displaystyle+\mu_{xxz}^{3}(\lambda,\kappa)S_{xxz}^{3}+\mu_{yyz}^{3}(\lambda,%
\kappa)S_{yyz}^{3}+\mu_{zzz}^{3}(\lambda,\kappa)S_{zzz}^{3},$$
which resembles the result for two sites, but includes few extra terms that
couple all three sites. The functions $\mu_{xx}^{3},\ldots,\mu_{zzz}^{3}$
are all real functions of the couplings. As for $N=2$, the Hamiltonian above
is $\mathcal{PT}$-symmetric, which follows from the fact that all matrices
involved are invariant under the adjoint action of the operator $\mathcal{PT}$ (see equation (49)). As for $N=2$ also, these are the only
matrices that are both $\mathcal{PT}$ symmetric and real (notice that, from
the definition (45) for $N=3$, it holds that $S_{xxz}^{3}=S_{zxx}^{3}=S_{xzx}^{3}$ and $S_{yyz}^{3}=S_{zyy}^{3}=S_{yzy}^{3}$).
5.4 The $N=4$ case
It is interesting to investigate how the perturbative results generalize as
we increase the number of sites. The $N=4$ case is especially interesting as
it is the simplest example for which we may see non local interaction terms
in the Hermitian Hamiltonian. There is again only one solution for $q_{1}(\lambda)$ which is compatible with the conditions (44), that
is
$$\displaystyle q_{1}(\lambda)$$
$$\displaystyle=$$
$$\displaystyle-S_{y}^{4}-\lambda(S_{yz}^{4}+S_{zy}^{4})-\frac{6\lambda^{3}(S_{%
yuz}^{4}-S_{yz}^{4}-S_{zy}^{4})}{40\lambda^{2}-9}$$
(123)
$$\displaystyle+\frac{1}{40\lambda^{2}-9}\left[(9-32\lambda^{2})\lambda^{2}(S_{%
yzz}^{4}+S_{zzy}^{4})-32\lambda^{4}S_{zyz}^{4}-2\lambda^{2}(3-16\lambda^{2})S_%
{yyy}^{4}\right.$$
$$\displaystyle-\left.3\lambda^{2}(S_{xxy}^{4}-2S_{xyx}^{4}+S_{yxx}^{4})+2%
\lambda^{3}(S_{xxyz}^{4}-5S_{xyxz}^{4}+S_{xxzy}^{4})\right.$$
$$\displaystyle+\left.2\lambda^{3}(9S_{yzzz}^{4}-7S_{yyyz}^{4})+64\lambda^{5}(S_%
{yyyz}^{4}-S_{zzzy}^{4})\right].$$
In many ways, this is a simple generalization of the results of two and
three sites. The matrices that enter the expression are to a large extent
the same we find for less sites, but we have now extra contributions
involving Pauli matrices sitting at all four sites of the chain, which was
to be expected. There are however two major changes
•
the dependence on $\lambda$ of the coefficients is not polynomial
anymore,
•
the first occurrence of non-local interactions appears through the
matrix $S_{yuz}^{4}$.
As for lower values of $N$, it is not difficult to argue that the matrices (45) entering the linear combination (123) are the only
ones that are compatible with (42). Hence, as expected, the same
structure extends to higher orders in perturbation theory, although
expressions become extremely involved. The table below gives $q_{3}(\lambda)$ as a sum of terms given by the matrices on the first column multiplied
by the corresponding coefficients in the second column,
$$q_{3}(\lambda)$$
Coefficients
$$S_{y}^{4}$$
$$\frac{-81+72{\lambda}^{2}+1892{\lambda}^{4}-4224{\lambda}^{6}+28672{\lambda}^{%
8}-131072{\lambda}^{10}}{3{\left(-9+40{\lambda}^{2}\right)}^{2}}$$
$$S_{yz}^{4}+S_{zy}^{4}$$
$$\frac{2916\lambda-22842{\lambda}^{3}+27216{\lambda}^{5}+81152{\lambda}^{7}+251%
904{\lambda}^{9}+786432{\lambda}^{11}-6291456{\lambda}^{13}}{3{\left(-9+40{%
\lambda}^{2}\right)}^{3}}$$
$$S_{yuz}^{4}$$
$$-\frac{64{\lambda}^{5}\left(351+276{\lambda}^{2}-8352{\lambda}^{4}-4096{%
\lambda}^{6}+65536{\lambda}^{8}\right)}{3{\left(-9+40{\lambda}^{2}\right)}^{3}}$$
$$S_{yzz}^{4}+S_{zzy}^{4}$$
$$-\frac{4{\lambda}^{2}\left(-1215+7722{\lambda}^{2}+12432{\lambda}^{4}-151808{%
\lambda}^{6}+131072{\lambda}^{8}-262144{\lambda}^{10}+2097152{\lambda}^{12}%
\right)}{3{\left(-9+40{\lambda}^{2}\right)}^{3}}$$
$$S_{zyz}^{4}$$
$$\frac{4{\lambda}^{2}\left(1215-16065{\lambda}^{2}+26952{\lambda}^{4}+72448{%
\lambda}^{6}-2097152{\lambda}^{12}\right)}{3{\left(-9+40{\lambda}^{2}\right)}^%
{3}}$$
$$S_{yyy}^{4}$$
$$\frac{4{\lambda}^{2}\left(-729+8667{\lambda}^{2}-14040{\lambda}^{4}-97024{%
\lambda}^{6}+393216{\lambda}^{8}-1048576{\lambda}^{10}+2097152{\lambda}^{12}%
\right)}{3{\left(-9+40{\lambda}^{2}\right)}^{3}}$$
$$S_{xxy}^{4}+S_{yxx}^{4}$$
$$-\frac{8{\lambda}^{2}\left(243-2079{\lambda}^{2}+7752{\lambda}^{4}+16768{%
\lambda}^{6}-159744{\lambda}^{8}+262144{\lambda}^{10}\right)}{3{\left(-9+40{%
\lambda}^{2}\right)}^{3}}$$
$$S_{xyx}^{4}$$
$$-\frac{4{\lambda}^{2}\left(-972+6939{\lambda}^{2}-31512{\lambda}^{4}+82176{%
\lambda}^{6}-188416{\lambda}^{8}+262144{\lambda}^{10}\right)}{3{\left(-9+40{%
\lambda}^{2}\right)}^{3}}$$
$$S_{xxyz}^{4}+S_{xxzy}^{4}$$
$$-\frac{2{\lambda}^{3}\left(405-19368{\lambda}^{2}+146048{\lambda}^{4}-349184{%
\lambda}^{6}-131072{\lambda}^{8}+1048576{\lambda}^{10}\right)}{3{\left(-9+40{%
\lambda}^{2}\right)}^{3}}$$
$$S_{xyxz}^{4}$$
$$-\frac{64{\lambda}^{3}\left(81-99{\lambda}^{2}-692{\lambda}^{4}+672{\lambda}^{%
6}+4096{\lambda}^{8}\right)}{3{\left(-9+40{\lambda}^{2}\right)}^{3}}$$
$$S_{zyyy}^{4}$$
$$\frac{32{\lambda}^{3}\left(-567+6390{\lambda}^{2}-21448{\lambda}^{4}+12096{%
\lambda}^{6}+49152{\lambda}^{8}-196608{\lambda}^{10}+524288{\lambda}^{12}%
\right)}{3{\left(-9+40{\lambda}^{2}\right)}^{3}}$$
$$S_{yzzz}^{4}$$
$$-\frac{32{\lambda}^{3}\left(-729+7290{\lambda}^{2}-19512{\lambda}^{4}-5952{%
\lambda}^{6}+32768{\lambda}^{8}-65536{\lambda}^{10}+524288{\lambda}^{12}\right%
)}{3{\left(-9+40{\lambda}^{2}\right)}^{3}}$$
In general we have,
$$\displaystyle q=\zeta(\lambda,\kappa)S_{y}^{4}+\theta(\lambda,\kappa)(S_{zy}^{%
4}+S_{yz}^{4})+\vartheta(\lambda,\kappa)S_{yuz}^{4}$$
$$\displaystyle+\mu(\lambda,\kappa)(S_{yzz}^{4}+S_{zzy}^{4})+\nu(\lambda,\kappa)%
S_{zyz}^{4}+\xi(\lambda,\kappa)S_{yyy}^{4}+\varpi(\lambda,\kappa)(S_{xxy}^{4}+%
S_{yxx}^{4})$$
(124)
$$\displaystyle+\varrho(\lambda,\kappa)S_{xyx}^{4}+\varsigma(\lambda,\kappa)(S_{%
xxyz}^{4}+S_{xxzy}^{4})+\tau(\lambda,\kappa)S_{xyxz}^{4}+\upsilon(\lambda,%
\kappa)S_{zyyy}^{4}+\chi(\lambda,\kappa)S_{yzzz}^{4},$$
where all coefficients $\zeta(\lambda,\kappa),\theta(\lambda,\kappa),\ldots,\chi(\lambda,\kappa)$ can be expressed as expansions of the
form (119) and are real functions of the couplings. Perturbation theory
results show that the Hermitian Hamiltonian $h(\lambda,\kappa)$ has the
following structure:
$$\displaystyle h(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\mu_{xx}^{4}(\lambda,\kappa)S_{xx}^{4}+\nu_{xx}^{4}(\lambda,%
\kappa)S_{xux}^{4}+\mu_{yy}^{4}(\lambda,\kappa)S_{yy}^{4}+\nu_{yy}^{4}(\lambda%
,\kappa)S_{yuy}^{4}$$
(125)
$$\displaystyle+\mu_{zz}^{4}(\lambda,\kappa)S_{zz}^{4}+\nu_{zz}^{4}(\lambda,%
\kappa)S_{zuz}^{4}+\mu_{z}^{4}(\lambda,\kappa)S_{z}^{4}+\mu_{xxz}^{4}(\lambda,%
\kappa)(S_{xxz}^{4}+S_{zxx}^{4})$$
$$\displaystyle+\mu_{xzx}^{4}(\lambda,\kappa)S_{xzx}^{4}+\mu_{yyz}^{4}(\lambda,%
\kappa)(S_{yyz}^{4}+S_{zyy}^{4})+\mu_{yzy}^{4}(\lambda,\kappa)S_{yzy}^{4}+\mu_%
{zzz}^{4}(\lambda,\kappa)S_{zzz}^{4}$$
$$\displaystyle+\mu_{xxxx}^{4}(\lambda,\kappa)S_{xxxx}^{4}+\mu_{yyyy}^{4}(%
\lambda,\kappa)S_{yyyy}^{4}+\mu_{zzzz}^{4}(\lambda,\kappa)S_{zzzz}^{4}+\mu_{%
xxyy}^{4}(\lambda,\kappa)S_{xxyy}^{4}$$
$$\displaystyle+\mu_{xyxy}^{4}(\lambda,\kappa)S_{xyxy}^{4}+\mu_{zzyy}^{4}(%
\lambda,\kappa)S_{zzyy}^{4}+\mu_{zyzy}^{4}(\lambda,\kappa)S_{zyzy}^{4}+\mu_{%
xxzz}^{4}(\lambda,\kappa)S_{xxzz}^{4}$$
$$\displaystyle+\mu_{xzxz}^{4}(\lambda,\kappa)S_{xzxz}^{4}.$$
As expected from the expression of $q$, we find that $h(\lambda,\kappa)$
involves non-local interaction terms proportional to $S_{xux}^{4}$, $S_{yuy}^{4}$ and $S_{zuz}^{4}$. The remaining terms are the natural
generalization of the those appearing for the $N=2,3$ cases plus additional
terms corresponding to interactions that couple all four sites of the chain.
Once again, all coefficients $\mu_{xx}^{4},\ldots,\mu_{xzxz}^{4}$ are
real functions of the couplings. As for previous cases, it turns out that
matrices appearing in the linear combination (125) are exactly those
that are both invariant under $\mathcal{PT}$-symmetry, according to equation
(49), and real.
5.5 Some general features from perturbation theory
We would like to end this section by summarizing the main results that we
have obtained from our perturbative analysis. Since we have only solved for $2,3$ and 4 sites, our conclusions are based on a case-by-case analysis
rather than rigorous proofs. However, we believe that the consistent
occurrence of certain features across the various examples that we have
studied provides strong support for these conclusions.
Firstly we found that the combination of perturbation theory and the
assumption of Hermiticity of the Dyson operator $\eta=e^{q/2}$ fix the
metric $\rho$ and therefore the Hermitian Hamiltonian $h(\lambda,\kappa)$
with its corresponding observables uniquely. We have established
this for $N=2,3,4$ and arbitrary values of both coupling constants as well
as for arbitrary $N$ if $\lambda=0$.
Secondly, concerning the specific algebraic structure of the Hermitian
Hamiltonian, we have seen that it becomes more involved for higher values of
$N$. For $N>2$ it generally includes interaction terms that couple two or
more adjacent sites, as well as non-local terms that couple non-adjacent
sites. In addition, this structure is entirely dictated by $\mathcal{PT}$
symmetry, which selects out which tensor products of Pauli and identity
matrices the Hamiltonian will be a linear combination of. Combining the
requirement of $\mathcal{PT}$ symmetry with the requirement of $h(\lambda,\kappa)$ being real completely fixes the general structure of $h(\lambda,\kappa)$, although not the specific dependence on the coupling constants $\lambda$ and $\kappa$, which is fixed by perturbation theory. All examples
studied indicate that for a given value of $N$, all solutions $q_{2k-1}(\lambda)$, with $k\geq 0$ at different perturbative orders, share
a common structure, namely they are all linear combinations of the same set
of matrices, with coefficients that increase in complexity with increasing
values of $k$.
Finally, concerning the numerical accuracy of perturbation theory, we have
demonstrated in detail that it converges very quickly for $N=2$. For $N=2,3$
and 4 it becomes very difficult to perform computations up to such high
orders of perturbation theory reached for $N=2$ and the rate of convergence
has not been analysed in detail for such cases. An interesting aspect of the
model studied here is the dependence of the Hamiltonian on two coupling
constants. For $N=2$, we have carried out perturbation theory in both such
couplings and found quick convergence in both cases. All our perturbation
theory results, suggest that the entries of the Hermitian Hamiltonian $h(\lambda,\kappa)$ can generally be expressed as a double Taylor series in
$\lambda$ and $\kappa$.
6 Expectation values of local operators: form factors
In this section we want to employ our general formulae in order to compute
the expectation values of certain local operators of the chain. In
particular, we will be looking at the expectation values of the total spin
in the $x$ and $z$ directions in the ground state of the chain. These
expectation values are commonly known as the magnetization in the $x$ and $z$
directions. Recalling the results from section 3.2, we define
$$\displaystyle M_{z}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\langle\Psi_{g}|\eta S_{z}^{N}\eta|\Psi_{g}\rangle=%
\frac{1}{2}\langle\psi_{g}|S_{z}^{N}|\psi_{g}\rangle,$$
(126)
$$\displaystyle M_{x}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\langle\Psi_{g}|\eta S_{x}^{N}\eta|\Psi_{g}\rangle=%
\frac{1}{2}\langle\psi_{g}|S_{x}^{N}|\psi_{g}\rangle.$$
(127)
where $|\psi_{g}\rangle$ is the ground state of the Hermitian Hamiltonian
and $|\Psi_{g}\rangle$ is the ground state of the non-Hermitian one. We
assume that the states are normalized to $\langle\psi_{g}|\psi_{g}\rangle=\langle\Psi_{g}|\Psi_{g}\rangle=1$. In the following sections, we will
carry out this computation for $\lambda=0$ with generic $N$ and for $\lambda\neq 0$ for small values of $N$.
6.1 General solutions for $\lambda=0$
In section 4.1 we described in detail how for $\lambda=0$
the original Hamiltonian and its Hermitian counterpart simplify greatly.
Indeed, the latter can be found in all generality, for any number of sites,
resulting in the expression (53). Taking (54) and (56) into account, it is very easy to show that
$$M_{z}(0,\kappa)=\frac{N}{2},$$
(128)
which is nothing but the total spin of the chain and does not depend on the
particular value of the coupling $\kappa$. This result is to be expected
for a Hamiltonian like (53). Naturally, the spins of the chain tend
to align in the direction of the field, and will all be up so that the
magnetization is just the total spin of the chain and maximal. A similar
computation can be performed for $M_{x}(0,\kappa)$ for each particular
value of $N$. In all cases one finds
$$M_{x}(0,\kappa)=0,$$
(129)
which is also what one would expect for this model, as the Hamiltonian (53) does not favour any particular direction of the spin $\sigma_{x}$.
6.2 General solutions for $\kappa=0$
For $\kappa=0$ the Hamiltonian (1) is Hermitian and
therefore computations of the magnetization simplify, as $\eta=\mathbb{I}$.
The ground state will nonetheless still depend on the value of $\lambda$.
For example, for $N=2$ it is
$$|\psi_{g}\rangle=\frac{1}{\sqrt{2(1+{\lambda}^{2}+\sqrt{1+{\lambda}^{2}})}}%
\left(\begin{array}[]{c}{1+\sqrt{1+{\lambda}^{2}}}\\
0\\
0\\
{\lambda}\end{array}\right)$$
(130)
with energy $E_{g}=-\sqrt{1+\lambda^{2}}$ and the magnetizations becomes
simply
$$M_{z}(\lambda,0)=\frac{1}{\sqrt{1+\lambda^{2}}}\qquad\text{and\qquad}M_{x}(%
\lambda,0)=0.$$
(131)
The function $M_{z}(\lambda,0)$ flows between the value $M_{z}(0,0)=1$, as
seen in the previous section, and $M_{z}(\infty,0)\rightarrow 0$. This is
simply because for $\kappa=0$ our model is nothing but the Ising chain with
a magnetic field of intensity $1/\lambda$ in the $z$-direction. Therefore,
as $\lambda\rightarrow\infty$ the intensity of the perturbing field tends
to zero, and the ground state of the chain has zero magnetization, as
consecutive spins align in opposite directions to minimize energy. This is a
general feature that will also hold for higher values of $N$. For example,
we find
$$\displaystyle\mu(\lambda,0)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}+\frac{2-\lambda}{2\sqrt{1+\left(-1+\lambda\right)%
\lambda}}\quad\text{for $N=3$},$$
(132)
$$\displaystyle\mu(\lambda,0)$$
$$\displaystyle=$$
$$\displaystyle\frac{\left(1-{\lambda}^{2}+\sqrt{1+{\lambda}^{4}}\right)\sqrt{1+%
{\lambda}^{2}+\sqrt{1+{\lambda}^{4}}}}{\sqrt{2}\sqrt{1+{\lambda}^{4}}}\quad%
\text{for $N=4$}.$$
(133)
In both cases we recover the expected behaviour: $\mu(0,0)=N/2$ and $\mu(\infty,0)=0$. The $\kappa=0$ curve in figure 4 is precisely a plot of the
function (131) for $N=2$.
The second equation in (131) can also be explained easily as a
consequence of the symmetry of the Hamiltonian $H(\lambda,0)$. Such
Hamiltonian is invariant under the transformation $\sigma_{i}^{x}\rightarrow-\sigma_{i}^{x}$ at each site $i$ of the chain. This
means that any form factor involving the operators $\sigma_{i}^{x}$ must
have the same symmetry. Therefore,
$$M_{x}(\lambda,0)=-M_{x}(\lambda,0),$$
(134)
which implies $M_{x}(\lambda,0)=0$ for all values of $N$.
6.3 The $N=2$ case for $\kappa,\lambda\neq 0$
Let us now compute $M_{z}(\lambda,\kappa)$ and $M_{x}(\lambda,\kappa)$
in the more generic situation when both coupling $\lambda$ and $\kappa$ are
non vanishing. We will start by analyzing the magnetization in the $z$-direction. In this case ($\lambda\neq 0$), the form of the ground state of
the Hermitian chain is not particularly simple and therefore we will work
with the first equality in (127) and employ the properly normalized
ground state of the non-Hermitian Hamiltonian. As figure 3 shows, the
magnetization is maximal at $\lambda=0$ with value 1, and exhibits different
kinds of behaviour as $\lambda$ increases, depending of the value of $\kappa$
under consideration.
For every fixed value of $\kappa$, the corresponding graph in figure 3
generally only covers a small region of values of $\lambda$. These are
precisely the values that lie in the region $U_{\mathcal{PT}}$ of figure 1,
namely those values for which all eigenvalues of $H(\lambda,\kappa)$ are
real. As shown in figure 3, the smaller the value of $\kappa$ the larger
this region becomes in $\lambda$. Depending on the value of $\kappa$ the
magnetization exhibits a rich structure: for $\kappa\geq 0.7$ it is a
strictly decreasing function, whereas for $\kappa\leq 0.6$ it has a
minimum. This minimum is located near the critical value of $\lambda$ above
which some eigenvalues of the Hamiltonian become complex, except for $\kappa=0.6$, where the minimum of the magnetization shifts to a smaller value of $\lambda$.
With regard to the magnetization in the $x$-direction we find that it
vanishes for all values of $\lambda$ and $\kappa$. This is so because the
Hermitian counter-part of $H(\lambda,\kappa)$ with $N=2$ has the form (73) and therefore the Hamiltonian $h(\lambda,\kappa)$ has the same
symmetry described at the end of the previous section.
Figure 3: The magnetization in the $z$-direction
for $N=2$ as a function of $\lambda$ and $\kappa$.
It is also interesting to analyze how the presence of an imaginary magnetic
field in the $x$-direction in (1), as opposed to a real one really
changes the physics of the model. Figure 4 precisely shows the magnitude of
that change for the magnetization when $\kappa$ is an imaginary number. The
Hamiltonian (1) is now that of the Ising spin chain with both a
perpendicular and longitudinal fields applied at each site of the chain. The
competition between these two fields will determine the values of the
magnetization in both the $x$ and $z$-directions.
Figure 4: The magnetization in the $x$ and $z$
directions for $N=2$ and $\kappa$ imaginary.
We also observe that the magnetization is strictly smaller than 1, as it
should be. Computing the expressions (126) and (127) in the
standard metric $\rho=\mathbb{I}$, i.e. disregarding the fact that the
Hamiltonian is non-Hermitian, leads to non-physical values larger than one.
7 Conclusions
We have demonstrated that there are various possibilities to implement $\mathcal{PT}$-symmetry for quantum spin chains, either as a
“macro-reflection” by reflection across
the entire chain or as “micro-reflection” by reflecting at individual sites. These new possibilities constitute
symmetries for the model $H(\lambda,\kappa)$ in (1) we focussed on,
i.e. Ising quantum spin chain in the presence of a magnetic field in the $z$-direction as well as a longitudinal imaginary field in the $x$-direction.
However, there are also implications for other Hamiltonians such as $H_{XXZ}$
in (14) and $H_{DG}$ in (16). Due to the various
possibilities to implement parity the corresponding metric and therefore the
underlying physical model is more ambiguous and it requires further
clarification as to which physical system it describes. Remarkably the
non-Hermitian Hamiltonian $H(\lambda,\kappa)$ fixes the underlying physics
uniquely under the sole assumption the Dyson map $\eta$ is Hermitian. As
pointed out above this uniqueness is not obtained in general. One might
conjecture that this is due to the finite dimensionality of the Hilbert
space, as opposed to continuous models studied for instance in [16, 29], but our comments on $H_{XXZ}$ and $H_{DG}$ suggest
this is not the case. The explanation lies surely in the different types of
symmetries a Hamiltonian might possess, which is supported by the fact that
two different types of metric operators, say $\rho$ and $\hat{\rho}$, can
always be used to define a new non-unitary symmetry operator $S=\hat{\rho}\rho^{-1}$ [34, 35].
We have shown that all these possibilities serve to define anti-linear
operators, which can not only be used to explain the reality of the spectra
and identify the corresponding domains in the coupling constants, but can
also be employed to define a consistent quantum mechanical framework.
Regarding the technical feasibility of this programme, we have demonstrated
for two sites that the perturbation theory, in $\kappa$ as well as in $\lambda$, converges very fast by comparing it with the exact result. We
took this as encouragement to tackle also three and four sites, albeit up to
not as high orders of perturbation theory. Our perturbative analysis has
allowed us to demonstrate for specific examples that the combination of
perturbation theory and Hermiticity of the Dyson operator are sufficient to
uniquely fix $\eta,\rho$ and $h(\lambda,\kappa)$. In fact, for the model at
hand, the constraint of Hermiticity of $\eta$ appears to be sufficient to
entirely fix the algebraic structure of these quantities, even before any
perturbative analysis is carried out.
Clearly there are various open issues and follow up problems associated to
our investigations. Firstly one may try to complete the analysis for the
Hamiltonian $H(\lambda,\kappa)$ by carrying out further numerical studies,
perturbative computations for more sites and ultimately obtain a complete
analytic understanding for instance by means of the Bethe ansatz. Special
attention should be given to the values of $\kappa$ and $\lambda$
corresponding to the exceptional points, when the usual analysis is expected
to break down. Secondly one may consider the model for higher spin values as
for instance studied in [7]. Finally it would be also very
interesting to investigate some other members of the class belonging to the
perturbed $\mathcal{M}_{p,q}$-series of minimal conformal field theories.
Acknowledgments: A.F. is grateful to Günther
von Gehlen for bringing the papers [6, 7] to our attention.
O.C.A. would like to thank Benjamin Doyon for helpful discussions and
suggestions. We are grateful to Pijush K. Ghosh for bringing reference [20] to our attention and Vincent Caudrelier for comments on the
manuscript.
Appendix A Exact Hermitian Hamiltonian for $N=2$
As demonstrated in section 5 perturbation theory, both in $\kappa$ and $\lambda$, agrees numerically very well with the exact results
of section 4. We showed that the Hermitian counterpart to the
Hamiltonian (1) for $N=2$ can be obtained by computing
$$h(\lambda,\kappa)=e^{q/2}H(\lambda,\kappa)e^{-q/2},\quad\text{with}\quad q=%
\alpha(\lambda,\kappa)S_{y}^{2}+\beta(\lambda,\kappa)(S_{yz}^{2}+S_{zy}^{2}),$$
(135)
where the functions $\alpha(\lambda,\kappa)$ and $\beta(\lambda,\kappa)$ have been evaluated perturbatively employing (86) in section 5. In terms of these functions and combinations thereof defined in (87) and (88), the entries of the Hamiltonian (135) are
given by,
$$\displaystyle h_{11}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle-(2{\gamma(\lambda,\kappa)}^{4})^{-1}\left[\alpha(\lambda,\kappa)%
\beta(\lambda,\kappa)\left(6\alpha(\lambda,\kappa)\beta(\lambda,\kappa)-{%
\gamma(\lambda,\kappa)}^{2}\right)\right.$$
(136)
$$\displaystyle\left.+2\kappa\sinh(\gamma(\lambda,\kappa))\alpha(\lambda,\kappa)%
\gamma(\lambda,\kappa)\delta(\lambda,\kappa)\right.$$
$$\displaystyle\left.+2\cosh(\gamma(\lambda,\kappa))\delta(\lambda,\kappa)\left(%
2\lambda\alpha(\lambda,\kappa)\beta(\lambda,\kappa)+\delta(\lambda,\kappa)%
\right)\right.$$
$$\displaystyle\left.+\lambda\alpha(\lambda,\kappa)\left({\alpha(\lambda,\kappa)%
}^{2}-3\alpha(\lambda,\kappa)\beta(\lambda,\kappa)+4{\beta(\lambda,\kappa)}^{2%
}\right)\epsilon(\lambda,\kappa)\right.$$
$$\displaystyle\left.+\cosh(2\gamma(\lambda,\kappa))\alpha(\lambda,\kappa)\left(%
\beta(\lambda,\kappa)-\lambda\alpha(\lambda,\kappa)\right){\epsilon(\lambda,%
\kappa)}^{2}\right.$$
$$\displaystyle\left.+\kappa\sinh(2\gamma(\lambda,\kappa))\beta(\lambda,\kappa)%
\gamma(\lambda,\kappa){\epsilon(\lambda,\kappa)}^{2}\right],$$
$$\displaystyle h_{22}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle\frac{\sinh(\gamma(\lambda,\kappa))}{\gamma(\lambda,\kappa)^{2}}%
\left[\sinh(\gamma(\lambda,\kappa))\alpha(\lambda,\kappa)\left(\beta(\lambda,%
\kappa)-\lambda\alpha(\lambda,\kappa)\right)\right.$$
(137)
$$\displaystyle\left.\qquad\qquad\qquad+{\kappa}\cosh(\gamma(\lambda,\kappa))%
\beta(\lambda,\kappa)\gamma(\lambda,\kappa)\right],$$
$$\displaystyle h_{44}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle({2{\gamma(\lambda,\kappa)}^{4}})^{-1}\left[\alpha(\lambda,\kappa%
)\left((6-\lambda)\alpha(\lambda,\kappa){\beta(\lambda,\kappa)}^{2}+\left(1+4%
\lambda\right){\beta(\lambda,\kappa)}^{3}\right.\right.$$
(138)
$$\displaystyle\left.\left.+{\alpha(\lambda,\kappa)}^{2}\beta(\lambda,\kappa)%
\left(1-2\lambda\right)-\lambda{\alpha(\lambda,\kappa)}^{3}\right)+2\kappa%
\sinh(\gamma(\lambda,\kappa))\alpha(\lambda,\kappa)\gamma(\lambda,\kappa)%
\delta(\lambda,\kappa)\right.$$
$$\displaystyle\left.+2\cosh(\gamma(\lambda,\kappa))\delta(\lambda,\kappa)\left(%
2\lambda\alpha(\lambda,\kappa)\beta(\lambda,\kappa)+\delta(\lambda,\kappa)%
\right)\right.$$
$$\displaystyle\left.+\cosh(2\gamma(\lambda,\kappa))\alpha(\lambda,\kappa)\left(%
\lambda\alpha(\lambda,\kappa)-\beta(\lambda,\kappa)\right){\rho(\lambda,\kappa%
)}^{2}\right.$$
$$\displaystyle\left.-\kappa\sinh(2\gamma(\lambda,\kappa))\beta(\lambda,\kappa)%
\gamma(\lambda,\kappa){\rho(\lambda,\kappa)}^{2}\right],$$
$$\displaystyle h_{14}(\lambda,\kappa)$$
$$\displaystyle=$$
$$\displaystyle(2\gamma(\lambda,\kappa)^{4})^{-1}\left[-4\cosh(\gamma(\lambda,%
\kappa))\alpha(\lambda,\kappa)\beta(\lambda,\kappa)\left(2\lambda\alpha(%
\lambda,\kappa)\beta(\lambda,\kappa)+\delta(\lambda,\kappa)\right)\right.$$
(139)
$$\displaystyle\left.+2\kappa\sinh(\gamma(\lambda,\kappa))\beta(\lambda,\kappa)%
\gamma(\lambda,\kappa)\left(-2{\alpha(\lambda,\kappa)}^{2}+\cosh(\gamma(%
\lambda,\kappa))\delta(\lambda,\kappa)\right)\right.$$
$$\displaystyle\left.-\rho(\lambda,\kappa)\left(\lambda{\alpha(\lambda,\kappa)}^%
{2}-3\alpha(\lambda,\kappa)\beta(\lambda,\kappa)-2\lambda{\beta(\lambda,\kappa%
)}^{2}\right)\epsilon(\lambda,\kappa)\right.$$
$$\displaystyle\left.-\cosh(2\gamma(\lambda,\kappa))\alpha(\lambda,\kappa)\left(%
\lambda\alpha(\lambda,\kappa)-\beta(\lambda,\kappa)\right)\epsilon(\lambda,%
\kappa)\rho(\lambda,\kappa)\right]$$
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Quantum Mechanics as a Space-Time Theory
Abstract.
We show how quantum mechanics can be understood as a space-time
theory provided that its spatial continuum is modelled by a
variable real number (qrumber) continuum. Such a continuum can be
constructed using only standard Hilbert space entities. The geometry of
atoms and subatomic objects differs from that of classical objects.
The systems that are non-local when measured in the classical
space-time continuum may be localized in the quantum continuum. We
compare this new description of space-time with the Bohmian picture of
quantum mechanics.
John Corbett111Mathematics Department,
Macquarie University, Sydney, NSW 2109, Australia, email: [email protected], Thomas Durt222TENA-TONA
Free University of
Brussels, Pleinlaan 2, B-1050 Brussels, Belgium. email:
[email protected]
¥
November 20, 2020
November 20, 2020
1. What is quantum space-time?
Both modern mathematics and modern physics underwent serious foundational crises during the 20th century.
The crisis in mathematics occured at the beginning of the century and the main problem was
to deal with certain infinities that are directly related to the concept of real number.
Poincaré poincare explained this crisis in terms of different attitudes to infinity, related to Aristotle’s
actual infinity and the potential infinity (the first attitude believes that the actual infinity
exists, we begin with the collection in which we find the pre-existing objects,
the second holds that a collection is formed by successively adding new members,
it is infinite because we can see no reason why this process should stop). It led finally to the emergence of new, non-standard definitions of real numbers.
The crisis in physics concerns the interpretation of the quantum theory, the measurement problem and the question of non-locality.
In previous works we showed how in principle certain paradoxes of the quantum theory can be explained provided we enlarge our conception of number durt .Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of non-standard real numbers that we call qrumbers. It is our goal in the present paper to analyze non-locality and the concept of space-time at the light of the new conceptual tools that we developed in the past.
Our main motivation is that most discussions of quantum mechanics use a background space-time that is
the same as classical space-time, usually without any supportive arguments
and even sometimes denying that quantum mechanics is a space-time theory. And yet many of the difficulties in understanding quantum phenomena derive from the use of classical space-time. We claim in the present paper that the space-time of quantum phenomena differs from that of classical phenomena in the nature of its continuum. According to our theory durt , the description of quantum phenomena requires a real number continuum that is not the classical continuum. It is not even a fixed element of the theory but varies with the quantum system in a way similar to the way the metric geometry of Einstein’s general relativity varies with the physical system adelman2 . This is not
part of the usual paradigm of quantum theory but adopting it enables us to reformulate the paradoxes of the standard interpretation when each quantum system has its own real number continuum.
The points of quantum space are identified with triplets of quantum real numbers, which we call qrumbers to help distinguish them from quantum numbers of standard quantum theory. Qrumbers are real numbers that are taken as numerical values by quantum systems adelman2 . One important way in which they differ from standard real numbers is that each qrumber has a non-trivial extent to which it is valid. This extent depends upon the condition of the system. Moreover a copy of the standard reals is embedded in the qrumbers, in fact, to every extent, the standard rationals are dense in the qrumbers.
In this approach, a non-standard qrumber is never obtained as the output from the measurement of a quantity because a measurement is a process in which a standard rational number is obtained as an approximation to the qrumber value of the quantity whose extent is conditioned by the measurement durt . The empirical fact behind this is that measurements can only produce standard rational numbers.
If we accept the identification of classical points in standard Euclidean space with triplets of standard real numbers, then a single point of quantum space may be approximated by different classical points depending on how the measurements are made. For example, it is possible that a quantum particle localized in the vicinity of a point in quantum space is not localized in the neighbourhood of a single classical point. Furthermore a quantum particle with qrumber values for its position and momentum has a trajectory in qrumber space adelman2 . This does not contradict the Heisenberg uncertainty relations which only restrict the product of the ranges of standard real number values of positions and momenta that can be prepared or measured.durt The trajectories in qrumber space are obtained as solutions to equations of motion that are discussed in Ref.adelman2 ; durt . They have the form reminiscent of the classical Hamiltonian equations of motion but instead they are formulated in terms of the qrumber values of position and momentum. In a sense this result evokes a picture reminiscent of the Bohmian picture in which trajectories in a generalised configuration space are associated with a quantum system. However the ontologies of the two pictures are different at the level of kinematics and dynamics.
In this paper the standard Hilbert space formulation of quantum mechanics
is interpreted as a space-time theory by using an approach to the
coordination problem similar to that suggested by Dieks dieks .
In his approach the numerical values of the position of a particle are taken
to be attributes of the particle in the same way as its mass or charge. Then
the assignment of numbers to the physical properties of a particle is made
in such a way that the standard form of the physical laws governing the
motion of the particle is maintained. We use a real number continuum given by
the sheaf of Dedekind reals $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ in the topos of sheaves on the quantum state space $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$. A qrumber
is a local section of the sheaf $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ defined in section 7.2 maclane , adelman2 .
Accordingly, the ontology of quantum particles is that of classical particles except that the numerical values of their attributes are given by qrumbers.
This definition mixes standard concepts of the Hilbert space but also sophisticated concepts that were developed in the framework of non-standard analysis. Nevertheless it is sufficient for the comprehension of the nextcoming results to visualise qrumbers associated to a set of quantum observable as a cloud of standard real numbers that we obtain by computing the average values of these observables while the density matrix associated to the system varies in a certain open set or extent. In this view, sharp values for the observables that are characteristic of the classical picture (in terms of standard real numbers) gets replaced by ”fuzzy” values.
Accordingly, the ontology of quantum particles is that of classical particles except that the numerical values of their attributes are given by qrumbers.
In this paper we focus on the changes in the concept of localisation. A particle that
is localised in terms of the qrumber space may be non-localised when viewed
from the classical real number space. For example, we consider Bell’s experiment
for two massive identical spin-1/2 particles bell . When the standard description is
expressed in terms of qrumbers we can show that the qrumber distance
between the identical particles is zero at the time that the spin measurements
in the classically separated Stern-Gerlach apparatuses are carried out.
We compare this with the Bohmian picture bohm2 in which the separation is zero only in average durtbohm .
2. What is a space-time theory?
In this paper, we restrict our attention to space-time theories of particles the
prototype of which was given by Newton in $\it{Principia}$ newton .
In it the motion of a material point particle is described by giving the values of
its spatial coordinates as functions of time. The space consists of the set of values of the spatial coordinates available to the material particle. The space of a
Newtonian particle is three-dimensional Euclidean space, that is, the spatial
coordinates of a classical particle at a particular instant are given by a triplet
of three standard real numbers. Each triplet is identified with a point in a three dimensional Euclidean geometry. The properties of the points are defined
abstractly through Euclid’s postulates which impose restrictions on the classes
of real number continua that can be used to label the points, but the restrictions are
not sufficient to uniquely define the real number continuum. Once a real number continuum is determined for the theory, then the numerical values of the spatial coordinates at an instant depend upon a frame of reference. The set of permissible frames of reference is invariant under the relativity group of the theory.
We define a space-time theory of a Galilean relativistic quantum system to be
a theory of that system in which the particles have spatial coordinates that take
numerical values at each instant of time. The set of values of the spatial
coordinates available to the particles gives the spatial continuum, the set of
instants available to the particles gives the time continuum. Mathematically
these identifications are only valid up to an isomorphism.
Newton in Ref.newton introduced the concepts of absolute and relative space,
”Absolute space, in its own nature, without relation to anything
external, remains always similar and immovable. Relative space is some
movable dimension or measure of the absolute space, which our senses
determine by its position to bodies; and which is commonly taken for
immovable space.”
We understand this distinction in the following way. When we have a physical
theory which is expressed in mathematical terms we can understand absolute
space as being just the abstract mathematical structure of space. For example,
Newton’s absolute space is the purely axiomatic Euclidean geometry used in $\it{Principia}$. Then the distinction between absolute and relative space
merely denotes the difference between an axiomatic geometry for space,
which is a purely abstract mathematical construction, and what Einstein
called the ”practical” geometry of space that is obtained when the ”empty
conceptual schemata” of axiomatic geometry are coordinated with ”real
objects of experience”. The recognition and identification of the ”real objects
of experience” depends upon the structures that the theorist has imagined
previously. In 1930 Einstein einstein1 noted,
”It seems that the human mind has first to construct forms independently
before we can find them in things. Kepler’s marvelous achievement is a
particularly fine example of the truth that knowledge cannot spring from
experience alone but only from the comparison of the inventions of the
intellect with observed fact.”
The purely abstract mathematical construction that lies behind the present
approach is a special case of the idea of a topos first developed
by Grothendieck, then by Lawvere and Tierney to replace set theory as
the proper framework for mathematics maclane .
We will use a spatial topos, $\mathop{\mathrm{Shv}}(X)$, the category of sheaves
on a topological space $X$. The standard real number
continuum is the prototype of a class of real number continua, which we
call the Dedekind reals $\mathord{{\mathbb{R}}_{\mathrm{D}}}(X)$, that are given by the sheaf of
germs of continuous real valued functions on the topological space $X$.
The open subsets of $X$ are the extents to which the numbers exist,
they give the truth values. Hence the internal logic is intuitionistic in
general but is Boolean when $X$ is the one point space or when the
topology on $X$ is trivial.
In the following paragraphs we assume that a physical attribute of a
physical system is a quantity that would yield a single-valued, standard or classical real number if
it were measured. Thus each of the three components of a position
vector is a physical attribute.
In the standard formulations of Galilean relativistic quantum mechanics,
in a given reference frame, at each instant of time, each physical attribute
of a quantum system is represented by a self-adjoint operator $\hat{A}\in\mathcal{A}$, an $O^{\ast}$-algebra inoue . The states of the quantum
system are normalized positive linear functionals on $\mathcal{A}$. When
restricted to the real algebra $\mathcal{A}_{sa}$ of self-adjoint operators
the functionals map $\mathcal{A}_{sa}$ to ${\mathbb{R}}$. The set of functionals
is called the quantum state space, $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$, of the system. The qrumbers of a system are the Dedekind real numbers $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ on its quantum state space.
The classical mechanical analogue of quantum state
space, $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$, is isomorphic to the one point space $\{\ast\}$, so that
the Dedekind real numbers for a classical system are $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\{\ast\})={\mathbb{R}}$, the standard real numbers. To see this, consider a system of massive
particles in the Hamiltonian formalism of classical mechanics. In a given reference frame, at each instant of time, each physical attribute of the classical particles is represented by a real number, its value at that instant of time in
that reference frame. Therefore the algebra $\mathcal{A}_{cl}$ of the
representatives of all physical attributes at one time in a given reference
frame is isomorphic to the algebra of real numbers ${\mathbb{R}}$. Then it follows
that the set of normalized positive linear functionals on $\mathcal{A}_{cl}$
contains only the identity map. Therefore the space of “states” for a
classical mechanical system is the one point space and hence
$\mathord{{\mathbb{R}}_{\mathrm{D}}}(\{\ast\})={\mathbb{R}}$.
The concept ’state’ employed in the preceding paragraph is different from that
normally used in classical mechanics. Here, as in standard quantum mechanics, a state is a normalized positive linear functional on the algebra of physical
quantities in a given reference frame at a given time. In classical mechanics,
a state is given by a point $(q,p)$ in phase space. The latter is both
determinative, because $(q,p)$ is the initial condition for Hamilton’s equations
of motion that uniquely determines the future states of the system, and generative,
because every physical quantity is given by a real-valued function of $(q,p)$. The quantum states do not have the generative property and therefore do not play the same role as the classical mechanical states. If the classical mechanical states are taken to be ontic, the quantum states can be taken to be epistemic.
2.1. Arguments that QM is not a space-time theory
The following is a summary of the argument that
quantum mechanics is not a space-time theory dieks .
1. The Hilbert space formalism is self-sufficient and does not need a
space-time manifold as a background. There is no special role for
position, all physical quantities have the same status; they are all
represented by ’observables’, i.e. by self-adjoint operators.
2. Furthermore, the position and momentum quantities are represented by
operators that do not commute. It then follows from the standard
interpretation that they cannot both have well-defined values at the same time.
If the eigenvalue-eigenstate link is accepted (this is the rule which says that a quantity has a definite value only if its system is in an eigenstate of the corresponding observable) a particle cannot have both a well-defined
position and a well-defined momentum at any instant and hence cannot have
a trajectory because non-commuting operators don’t have common eigenstates.
3. The nub of the argument is that ”the standard mathematical formalism of
quantum theory strongly deviates from its classical counterpart in that
physical magnitudes are not represented by functions on space (or on
phase space)”. The operators used for the representation of physical
magnitudes in quantum theory are not $\it{automatically}$ associated with
real number values.
We think that these arguments do not prove that quantum
mechanics is not a space-time theory, rather they show that a
particular interpretation of the Hilbert space formalism of
quantum mechanics does not have a space-time picture.
1. The first point is dubious because many of the problems of the
standard interpretation of QM, such as the problems of locality, can be
related to the absence of quantum space. It has been asserted of this
interpretation that “all the problems centre around the relation between -
on the one hand - the values of physical quantities, and - on the other
hand - the results of measurements”isham . These values and results
are usually taken to be the same type of entities, by the precedent of
classical physics, but the ancient Greek “measurement problem” - that the
results of measurements should be rational numbers, but the value of the
ratio of the diagonal to the side of a square is not - should caution
us. The values of physical quantities can be a type of real number more
general than the numbers obtained as the results of measurements.
Moreover the standard interpretation does not adequately distinguish the
representation of a physical quantity from the values that the physical
quantity may take. This appears to be a prejudice based on their
confluence in classical mechanics.
2. The second point re-emphasizes the problems of the standard
interpretation. The eigenvalue - eigenstate link hypothesis is
ambiguous333This hypothesis is often
not listed in the axioms for QM, e.g. it is not in Ref.cohen-tannoudji , because
operators don’t always have eigenstates. In particular the position and momentum operators have no eigenvalues, although we can construct states that yield approximate eigenvalues vonneumann .; it is not clear whether it refers only to measured values.
Dirac states that “a measurement always causes the system to jump
into an eigenstate of the dynamical variable that is being measured”dirac
but does not say from where the state has jumped, c.f., the
discussion in Bell bell .
We will re-interpret the standard mathematical formalism of QM to reveal an underlying space that generalises the space of classical mechanics. From this point of view, the main error of the standard interpretation was to not examine the
mathematical structure of quantum mechanics with sufficient care to
discover the spatial structure existing within it. It is worth noting that in the Bohmian approach bohm2 where the privileged observable (beable) is the position, the eigenvalue-eigenstate link hypothesis is not fulfilled in general, even when the observable that we consider is the position. Indeed if this hypothesis was fulfilled, the fact that trajectories are supposed to exist would imply that wave functions are not spread in space. The eigenvalue-eigenstate link is also not true when we consider velocities. This does not lead to any inconsistency in the Bohmian picture
where instantaneous velocities are not directly measurable bohm2 .
3. Again this comment is true only if by ”space” we mean the space of
standard real numbers. It shows that quantum mechanics cannot
be represented as a space-time theory with standard real numbers. In
classical mechanics, the algebra of the physical quantities is
represented by real numbers that are identified with the entities that
represent their values. This leads to the identification of physical
quantities, such as the energy $H(p,q)$, that are given by functions on
phase space with the values that their functions take at points on
phase space. It is this identification that is not maintained in quantum
mechanics, but we claim that classical space (or phase space) does not
provide the underlying continuum for quantum mechanics. The qr-number
values of physical quantities in quantum mechanics are given by continuous functions on the standard quantum state space (defined in section 7.2).
A brief review of the development of the real number concept is relevant,
especially in view of the identification of real numbers with points on
a straight line, initiated by Descartes, that opened the way to the use of
functions to describe the relations between geometrical objects and
points. The ancient Greeks, who developed what we call Euclidean geometry,
had not made that identification hartshorne ; magnitudes such as
lengths, angles, areas and volumes were measured using geometrical
motions of translations and rotations of line segments, triangles etc
aided by concepts of similarity and congruence that depended upon the
Archimedean principle that we discuss in section 4. It is not surprising that they only used ratios of natural numbers for if we only admit numbers that arise as the outputs from measurements we only admit rational numbers.
It is not until we have a mathematical theory that describes what happens
between the outcomes and inputs that we need numbers more general
than rational numbers to ensure that the equations of the theory have
numerical solutions. The properties of the equations and their solutions
are used to deepen our understanding of the physical world. It is in this
sense that Dedekind’s statement, “Numbers are
free creations of the human intellect, they serve as a means of grasping
more easily and more sharply the diversity of things. ”dedekind ,
may be understood.
3. Qrumbers; Real numbers for quantum mechanics
The numerical values taken by physical quantities in quantum mechanics
differ from standard real numbers in much the same way as variable functions differ from constants. Like functions they have domains of definition over which they may vary. Their definition comes from topos theory.
Definition 1.
The qr-number continuum for a quantum particle is the continuum of Dedekind reals $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ given by the sheaf $\mathcal{C}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}})$ of continuous
real-valued functions in the topos $\mathop{\mathrm{Shv}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}})$ of sheaves on
$\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$.
To apply this definition to quantum systems,
we start from the standard Hilbert space formulation of quantum
mechanics as given by von Neumann vonneumann . In it the physical
quantities are represented by the self-adjoint operators defined on dense
subsets of Hilbert space or, in the case of bounded operators,
on the whole Hilbert space. These self-adjoint operators
belong to an $O^{*}$ algebra $\mathcal{A}$, called the algebra
of observables, if the symmetric product is used they form a real
algebra $\mathcal{A}_{sa}$ inoue .
The states are the other main ingredient of the standard Hilbert space
formulation of quantum mechanics. In the initial formulation the pure
states are given by elements (vectors) belonging to the Hilbert
space. The concept of state was later generalised to include impure
states which are given by positive bounded self-adjoint trace class
operators of trace 1. As noted above, the general concept of state is that of
positive continuous linear functionals on $\mathcal{A}$ (see also section 7.2). The pure
states are given by projection operators onto one dimensional subspaces
of the Hilbert space, which connects them to the original definition
in terms of Hilbert space vectors, at least up to a complex
multiplicative factor of modulus 1. We denote by $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ the
Schwartz subspace of the state space described in the appendix.
In the standard theory $\mathop{\mathrm{Tr}}\hat{\rho}{\hat{A}}$ is the average or expectation
value of the quantity represented by ${\hat{A}}$ in the state given by
$\hat{\rho}$. In our theory for each operator ${\hat{A}}\in\mathcal{A}_{sa}$ and
each open subset $W\in\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$, $\mathop{\mathrm{Tr}}\hat{\rho}{\hat{A}}$ defines a real valued function with domain $W$. In the topos of sheaves on the topological space $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$, the sheaf of Dedekind reals $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ is isomorphic to the
sheaf of continuous real-valued functions on $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$, maclane , so that a
qrumber defined to extent $W$ is a continuous function defined on $W$.
With the appropriate topology on $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ the functions defined using
$\mathop{\mathrm{Tr}}\hat{\rho}{\hat{A}}$ are continuous and thus
define qrumbers adelman2 .
It is worth noting that, beside the fact that it shows how non-standard numbers
can be used in order to reformulate the quantum theory,
our approach also makes it possible to reformulate the postulates of quantum mechanics
in terms of average (expectation) values only. This must be put in parallel with the results
of S. Weigert weigert who showed that one obtains,
using the expectation values
of a quorum of quantities, a closed system of linear
differential equations that describes the quantum evolution of a system of spin $s$. ¥
For a Galilean relativistic quantum mechanical system, $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ is the set of
all normalised, strongly positive linear functionals on the enveloping algebra
of the irreducible representation of the Lie algebra of the extended
Galilean group. The irreducible representation of the enveloping Lie
algebra of the extended Galilean group, labelled by $(m,U,s)$ with
central element $\hbar I$, is unitarily equivalent to the tensor product
$M\otimes M_{s}$ of the $Schr\ddot{o}dinger$ representation $M$ of the
algebra of the Canonical Commutation Relations (CCR-algebra) generated
by the operators $\{\vec{P},\vec{X},\hbar I\}$ with $M_{s}$ the
irreducible representation of dimension $2s+1$ of the Lie algebra of
the rotation group $SO(3)$ adelman2 . We choose the topology on $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ to be the weakest that makes continuous all the functions of the form $a_{Q}(\hat{\rho})=\mathop{\mathrm{Tr}}(\hat{\rho}\hat{A})$ for self-adjoint operators
$\hat{A}\in M\otimes M_{s}$. Then the functions $a_{Q}$ form a subobject ${\mathbb{A}}$
of $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ on $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ adelman2 . Each qr-number defined to extent $W$ is either a continuous function of the $a_{Q}(W)$ or a constant real valued function on $W$. The sheaf $\mathcal{C}(W)$ of continuous
real-valued functions over the open set $W\subset\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ can extended to a
sheaf over $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ by prolongation by zero swan .
4. Physical Interpretation .
A physical interpretation of these non-standard real numbers may be
linked, via the interpretation of a quantum state $\hat{\rho}$ as
representing an ideal preparation process, to the requirement that the whole
experimental arrangement must be included in the determination of
physical quantities bohr . However our model differs from this
requirement in two important ways: firstly, the physical processes that
constitute the preparation process may occur naturally without the
intervention of experimentalists and, secondly, the preparation
procedures are represented by open sets of states.
If a physical attribute of a particle is represented by the
self-adjoint operator $\hat{A}$ then the qrumber values that the
quantity can take are given by functions $a_{Q}(W)$ defined on open sets
$W$ by $a_{Q}(\hat{\rho})=\mathop{\mathrm{Tr}}\hat{\rho}\hat{A}$ for all $\hat{\rho}\in W$.
The open set $W$ is the extent to which the quantum particle exists; we
say that $W$ is the ontic state of the particle. If the
ontic state of the particle is $W$ then the qrumber value of the quantity
represented by $\hat{B}$ is given by a function $b_{Q}(W)$ defined on $W$.
In this sense the qrumber values of any quantity are determined by
the ontic state $W$. An ontic state $W$ is specified by the set of all
non-empty open subsets of $W$.
Since all conditions are defined pointwise, if $U\neq\emptyset$ and $U\subseteq W$ then a qrumber $a_{Q}(U)$
will satisfy any condition satisfied by $a_{Q}(W)$. For example,
$\epsilon$ sharp collimation, see durt , is a property of the ontic state $W$
because if it holds on $W$ it holds on all $V\subset W$. If $\vec{\hat{X}}$ is the triplet of self-adjoint operators that represent the position of a particle then its qrumber positions are given by the continuous functions $\vec{x}_{Q}(W)$ on open sets $W$. The set of all its qrumber positions constitute the quantum particle’s three dimensional space, whose geometric properties depend upon the structure of the underlying continuum of $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ stout .
$\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ is a field that is partially ordered but not totally
ordered, in particular, trichotomy does not always hold.
Also it is not Archimedean in the sense that there are qrumbers
$a\in\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{%
\mathcal{S}}}})$ such that $a>0$ but there is no natural number
$n\in\mathbb{N}$ such that $n>a$. $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ is a complete
metric space with respect to a distance function derived from the norm
function $|\cdot|:\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{%
\mathcal{S}}}})-->\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{%
\mathord{\mathcal{S}}}})$ that takes a to
$max(a,-a)$ stout . With this we can define a distance function between particle positions in $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})^{3}$ that is used to define localisation.
Quantum localisation is different from classical localisation.
For example, suppose that the z-coordinate of a particle has the qrumber
value $z_{Q}(W)$, where $W$ is the union of disjoint open
subsets, $W=U\cup V$ with $U\cap V=\emptyset$.
$z_{Q}(W)$ is a single qrumber value and hence represents a single
point on the z-axis in quantum space. But if there are standard real
numbers $z_{1}<z_{2}<z_{3}<z_{4}$, with $z_{1}<Z_{Q}(U)<z_{2}$,
$z_{3}<Z_{Q}(V)<z_{4}$ and $z_{2}\ll z_{3}$ then the single qrumber
value may be viewed classically as a pair of separated intervals of
standard real numbers and hence non-localised. This example of the
difference between quantum and classical localisation can be used to
understand the 2-slit experiments. A particle may be localised
in terms of the qrumber values of its position but not localised in
terms of the classical standard real number values. The qrumber
distance between a pair of particles may be small even though the
classical distance between them is large.
In the Bohmian approach, non-locality epr ; bohm2 ; bell is related to the fact that
the configuration space to which particles belong is not 3-dimensional but 3N dimensional. In our approach we generalise the concept of real number on which the concept of spatial point relies.
4.1. What is a quantum particle?
A Galilean relativistic quantum particle is an entity localised in the qrumber
space with the following characteristics which generalize those of a
classical particle bitbol :
(i) it has permanent properties which always possess qrumber values that may be approximated by standard real number values which are revealed in observations expressed by counterfactual empirical propositions (such as ’if I had performed such and such experiments, I would have obtained such and such standard real number outcomes’),
(ii) it has individuality,
(iii) it can be re-identified through time.
We must recognize the distinction between
intrinsic properties that are independent of the condition of the
particle, how it was prepared or what interactions it is undergoing and
the extrinsic properties that may depend upon the condition of the
particle. Particles with the same intrinsic properties are normally said
to be identical jauch . For Galilean relativistic particles in
quantum mechanics, the intrinsic properties are the mass $m$, a positive
standard real, the internal energy $U$, a standard real, and the
spin $s$, a natural number or half a natural number. $(m,U,s)$ label
the irreducible projective unitary representations of the Galilean
group $\mathcal{G}$ levy . The basic properties of the irreducible
representations of $\mathcal{G}$ are given an appendix.
It turns out that the
concepts of indistinguishability (:= two entities are indistinguishable
if they agree with respect to all their attributes) and identity (:= two
entities are identical if they are the same object)
are not equivalent in general dacosta .
Nevertheless, in quantum mechanics it is true that for all fundamental particles identity
implies indistinguishability and vice versa.
We will now study two particle systems in our approach.
5. Two particle systems
5.1. Two different particles
In the standard mathematical framework the Hilbert space for the quantum
mechanics of two different particles of unequal masses $m_{1}$ and
$m_{2}$, internal energies $U_{1}$ and $U_{2}$ and arbitrary spins $s_{1}$ and
$s_{2}$ is given by the tensor product
$\mathord{\mathcal{H}}(1)\otimes\mathord{\mathcal{H}}(2)$ of the Hilbert spaces $\mathord{\mathcal{H}}(1)$ and $\mathord{\mathcal{H}}(2)$ that are respectively the carrier spaces for the irreducible
projective unitary representations of the Galilean group $\mathcal{G}$
labelled by $(m_{1},U_{1},s_{1})$ and $(m_{2},U_{2},s_{2})$. For each
$j=1,2$, $\mathord{\mathcal{H}}(j)=\mathord{\mathcal{H}}_{j}(ccr)\otimes\mathbb{C}^{2s_{%
j}+1}$
where $\mathord{\mathcal{H}}_{j}(ccr)=\mathcal{L}^{2}(\mathbb{R}^{3})$ for both
$j=1,2$.
We will write
(1)
$$\mathord{\mathcal{H}}(1,2)=\mathord{\mathcal{H}}(1)\otimes\mathord{\mathcal{H}%
}(2).$$
The physical quantities associated with particle 1 include
operators, representing elements of the enveloping algebra of the Lie
algebra of $\mathcal{G}$, that are essentially self-adjoint on the
Schwartz subspace $\mathord{\mathcal{S}}$ of the spatial Hilbert space
$\mathcal{L}^{2}(\mathbb{R}^{3})$ tensored with self-adjoint spin matrices
acting on $\mathbb{C}^{2s_{1}+1}$. To simplify the notation we will usually
denote a tensor product of operators associated with particle 1 by a
single symbol $A(1)$. Then the vector operators for particle $1$ are
position $\vec{X}(1)=\vec{X}\otimes I_{s_{1}}$, momentum $\vec{P}(1)=\vec{P}\otimes I_{s_{1}}$, angular momentum $\vec{L}(1)=\vec{L}\otimes I_{s_{1}}$
and spin
$\vec{s}(1)=I_{1}\otimes\vec{s}$, where $I_{s_{1}}$ is the identity
matrix on
$\mathbb{C}^{2s_{1}+1}$ and $I_{1}$ is the identity operator on
$\mathcal{L}^{2}(\mathbb{R}^{3})$. We take the Schwartz space $\mathord{\mathcal{S}}(1)$ to
be the tensor product $\mathord{\mathcal{S}}\otimes\mathbb{C}^{2s_{1}+1}$ . The physical
quantities associated with particle 2 have a similar tensor
product composition and will be simply denoted by operators $B(2)$ acting
on $\mathord{\mathcal{S}}(2)=\mathord{\mathcal{S}}\otimes\mathbb{C}^{2s_{2}+1}%
\subset\mathcal{L}^{2}(\mathbb{R}^{3})\otimes\mathbb{C}^{2s_{2}+1}$.
They include the vector operators for position $\vec{X}(2)=\vec{X}\otimes I_{s_{2}}$, momentum $\vec{P}(2)=\vec{P}\otimes I_{s_{2}}$, angular
momentum
$\vec{L}(2)=\vec{L}\otimes I_{s_{2}}$ and spin $\vec{s}(2)=I_{2}\otimes\vec{s}$, where $I_{s_{2}}$ is the identity matrix on $\mathbb{C}^{2s_{2}+1}$
and
$I_{2}$ is the identity operator on $\mathcal{L}^{2}(\mathbb{R}^{3})$.
Because the tensor product $\mathord{\mathcal{S}}\otimes\mathbb{C}^{2s_{j}+1}$ is
denoted by
$\mathord{\mathcal{S}}(j)$ for $j=1,2$, then $\mathord{\mathcal{S}}(1,2)=\mathord{\mathcal{S}}(1)\otimes\mathord{\mathcal{S}%
}(2)$ represents the two particle Schwartz space.
The algebra
$\mathcal{A}_{sa}(1,2)$ of operators representing physical quantities
associated with the compound system contains elements of the form
$A(1)\otimes B(2)$ as well as other operators, such the total energy
$H(1,2)=\Sigma_{j=1}^{3}P_{j}(1)^{2}/(2m_{1})\otimes I(2)+I(1)\otimes\Sigma_{j=%
1}^{3}P_{j}(2)^{2}/(2m_{2})+V(1,2)$ where the interaction
operator $V(1,2)$ is generally not of the form $A(1)\otimes B(2)$.
When considered as part of the compound system the properties of
particle 1 are represented by operators of the form $A(1)\otimes I(2)$ and
the properties of particle 2 are represented by operators of the form
$I(1)\otimes B(2)$, with $I(j)=I_{j}\otimes I_{s_{j}},j=1,2$.
Schwartzian state space $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$ is the state space of the compound
system. $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$ is contained in the convex
hull of projections $\mathord{\mathcal{P}}(1,2)$ onto the one-dimensional subspaces of
$\mathcal{H}(1,2)$ spanned by unit vectors $\psi(1,2)$ belonging to the two particle Schwartz space $\mathord{\mathcal{S}}(1,2)$.
Each state $\hat{\rho}\in\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$ is a trace class positive
bounded operator on $\mathord{\mathcal{H}}(1,2)$ with trace $1$ which can be written as
$\hat{\rho}=\sum\lambda_{n}\mathord{\mathcal{P}}_{n}(1,2)$. For all
$n$, $\lambda_{n}\geq{0}$, $\sum\lambda_{n}=1$ and the $\mathord{\mathcal{P}}_{n}(1,2)$
are orthogonal projections onto one-dimensional subspaces in
$\mathord{\mathcal{S}}(1,2)$. For each $A(1,2)\in\mathcal{A}_{sa}(1,2)$ we
define $a(1,2)_{Q}(\hat{\rho})=\mathop{\mathrm{Tr}}\hat{\rho}A(1,2)$ for $\hat{\rho}\in\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$. $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$ is given the weakest topology that makes all the functions $a(1,2)_{Q}$ continuous.
Physical quantities that are represented by operators of the form
$A(1)\otimes B(2)$ have qrumber values $(a(1)\otimes b(2))_{Q}(W)$ defined on open subsets $W\in\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$. Even though
as operators $A(1)\otimes B(2)=(A(1)\otimes I(2))(I(1)\otimes B(2))$
it is generally not true that $(a(1)\otimes b(2))_{Q}(W)=(a(1)\otimes 1(2))_{Q}(W))((1(1)\otimes b(2))_{Q}(W))$ because any open set $W$ will
contain non-product states from $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$. Of course we can rewrite
$(1(1)\otimes b(2))_{Q}(W)=b(2)_{Q}(W(2))$ where $W(2)$ is the open
subset of $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(2)$ that is obtained by partial tracing $W\in O(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))$ over an orthonormal basis in $\mathord{\mathcal{H}}(1)$. In a similar
fashion $(a(1)\otimes 1(2))_{Q})(W)=a(1)_{Q}(W(1))$ where $W(1)$ is the
open subset of
$\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1)$ that is obtained by partial tracing $W\in O(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))$
over an orthonormal basis in $\mathord{\mathcal{H}}(2)$. Therefore $(a(1)\otimes b(2))_{Q}(W)\neq a(1)_{Q}(W(1))\times b(2)_{Q}(W(2))$ unless for every
$\hat{\rho}\in W,\hat{\rho}=\hat{\rho}(1)\otimes\hat{\rho}(2)$
where $\hat{\rho}(j)\in W(j),j=1,2$. Because quantum systems can be entangled, the latter condition is not satisfied in general and qrumbers do not always factorize.
Of particular interest are the qrumber values of the
kinematical variables of either of the two particles. For example, on an
open set
$W\in\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$; the position vector of particle 1 will have
qrumber value $\vec{x}(1)_{Q}(\tilde{W}(1))$,
the momentum vector of particle 1 will have qrumber value
$\vec{p}(1)_{Q}(\tilde{W}(1))$, while the
position vector of particle 2 will have qrumber value
$\vec{x}(2)_{Q}(\tilde{W}(2))$ and the
momentum vector of particle 2 will have qrumber value
$\vec{p}(2)_{Q}(\tilde{W}(2))$, where for $j=1,2$, $\tilde{W}(j)$ is an open
subset of $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(j)$ that is obtained by partial tracing $W(j)$ over an orthonormal basis in $\mathbb{C}^{2s_{j}+1}$, where $W(j)$ was obtained
by tracing over an orthonormal basis in $\mathord{\mathcal{H}}(k),k\neq j$. The qrumber distance between the two particles when
the compound system is in the open set $W\in\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$ is
$d(1,2)_{Q}(W)$ given by, where $\vec{x}(1)=\vec{X}(1)\otimes I_{s_{j}}$ etc.,
(2)
$$\|(\vec{x}(1)\otimes I(2)-(I(1)\otimes\vec{x}(2))_{Q}(W))\|_{3}=\|\vec{x}(1)_{%
Q}(\tilde{W}(1))-\vec{x}(2)_{Q}(\tilde{W}(2))\|_{3}.$$
That is, for each $\hat{\rho}(1,2)\in W$,
(3)
$$d(1,2)_{Q}(\hat{\rho}(1,2))=\|\mathop{\mathrm{Tr}}\hat{\rho}(1)\vec{X}(1)-%
\mathop{\mathrm{Tr}}\hat{\rho}(2)\vec{X}(2)\|_{3}.$$
where $\hat{\rho}(j)$ is obtained by partially tracing
$\hat{\rho}(1,2)$ over
an orthonormal basis in $\mathbb{C}^{2s_{j}+1}$ after having partially
traced over an orthonormal basis in $\mathord{\mathcal{H}}(k),k\neq j$. $\|\vec{v}\|_{3}=(\vec{v}\cdot\vec{v})^{1/2}$ is the standard Euclidean norm of the
vector $\vec{v}\in{\mathbb{R}}^{3}$.
Consider the following situation; at $t=0$ particles 1 and 2 have the same
qr-number position and their qr-number momenta are of
the same magnitude but in opposite directions, that is,
$(\vec{x}(1)\otimes I(2))_{Q}(W)=(I(1)\otimes\vec{x}(2))_{Q}(W)$, and
$(\vec{p}(1)\otimes I(2))_{Q}(W)=-(I(1)\otimes\vec{p}(2))_{Q}(W)$. If
the particles move freely then at any time $t>0$, durt
$(\vec{x}(1)\otimes I(2))_{Q}(W)(t)=(\vec{x}(1)\otimes I(2))_{Q}(W)+{t\over m_{%
1}}(\vec{p}(1)\otimes I(2))_{Q}(W)$ and $(I(1)\otimes\vec{x}(2))_{Q}(W)(t)=(I(1)\otimes\vec{x}(2))_{Q}(W)+{t\over m_{2}%
}(I(1)\otimes\vec{p}(2))_{Q}(W)$. Therefore at $t>0$ the qrumber
distance between the particles is
(4)
$$d(1,2)_{Q}(W)(t)={(m_{1}+m_{2})\over(m_{1}m_{2})}t\|(\vec{p}(1)\otimes I(2))_{%
Q}(W)\|_{3}.$$
where $\|(\vec{p}(1)\otimes I(2))_{Q}(W)\|_{3}=\|(I(1)\otimes\vec{p}(2))_{Q}(W)\|_{3}$.
Each particle has its own continuous qr-number trajectory, viz., the
two particles move in opposite directions along a straight line in
qr-number space. But they do not have trajectories in standard real
number space. However, at any instant of time, we can show that one of
the pair of particles is in a region of classical space while the other
is in a second classical region, well-separated from the first. This
is done by finding standard real numbers that approximate the
qrumbers $(\vec{x}(1)\otimes I(2))_{Q}(W)(t)$ and $(I(1)\otimes\vec{x}(2))_{Q}(W)(t)$. That is, we can, in principle, determine an approximate
classical trajectory for each particle from its qrumber trajectory.This
contrasts with the standard interpretation in which Heisenberg’s
uncertainty relations prevents us from determining a continuous trajectory
in classical space by which a particular particle could be labelled and
its path followed over time.
If the particles move freely then the total momentum is conserved so
that for all times $t$,
(5)
$$(\vec{p}(1)\otimes I(2))_{Q}(W)(t)=-(I(1)\otimes\vec{p}(2))_{Q}(W)(t).$$
Thus each particle has both a definite qrumber value for its position
and momentum for all times, these values remain correlated until some
interaction with the particles breaks the symmetry. Of course
measurement of the position and momentum of either particle will yield
approximate standard real number values which are in agreement with
Heisenberg’s uncertainty relations durt .
5.2. Two identical particles
When the two particles are identical then the standard quantum theory does not distinguish between them.
In the standard mathematical framework the Hilbert spaces for the quantum
mechanics of two identical particles of mass $m\neq 0$, internal energy
$V$ and spin $s$ is given by the symmetric subspace (bosons), or
the anti-symmetric subspace (fermions), of the tensor product
$\mathord{\mathcal{H}}(1,2)=\mathord{\mathcal{H}}(1)\otimes\mathord{\mathcal{H}%
}(2)$ of the Hilbert spaces $\mathord{\mathcal{H}}(1)$ and $\mathord{\mathcal{H}}(2)$ that are identical copies of the carrier space for the
irreducible projective unitary representation $\mathcal{U}$ of the
Galilean group. $\mathcal{U}$ has labels $(m,V,s)$. For each
$j=1,2$, $\mathord{\mathcal{H}}(j)=\mathord{\mathcal{H}}(ccr)\otimes\mathbb{C}^{2s+1}$
where $\mathord{\mathcal{H}}(ccr)=\mathcal{L}^{2}(\mathbb{R}^{3})$. The symmetric
tensor product is written $(\mathord{\mathcal{H}}(1)\otimes\mathord{\mathcal{H}}(2))_{+}$ and the
antisymmetric tensor product is written $(\mathord{\mathcal{H}}(1)\otimes\mathord{\mathcal{H}}(2))_{-}$. We will write
(6)
$$\mathord{\mathcal{H}}(1,2)_{\pm}=(\mathord{\mathcal{H}}(1)\otimes\mathord{%
\mathcal{H}}(2))_{\pm}.$$
If $\Pi_{\pm}$ represent, respectively, the orthogonal projection
operators from $\mathord{\mathcal{H}}(1,2)$ to $\mathord{\mathcal{H}}(1,2)_{+}$ or to $\mathord{\mathcal{H}}(1,2)_{-}$
then for any vector
$u\otimes v\in\mathord{\mathcal{H}}(1,2)$, $\Pi_{\pm}(u\otimes v)={1\over 2}(u\otimes v\pm v\otimes u)$. It is easy to show that the operators
that represent physical quantities associated with systems of identical
particles must be invariant under permutation of the particles
jauch . For if we let $\mathcal{P}$ denote the unitary operator on
$\mathord{\mathcal{H}}(1,2)$ that interchanges the vector states of the two particles,
$\mathcal{P}(u\otimes v)=v\otimes u$, then
$\mathcal{P}=2\Pi_{+}-I$ where $I$ is the identity operator on
$\mathord{\mathcal{H}}(1,2)$. We then observe that $\mathord{\mathcal{H}}(1,2)_{\pm}$ are the eigenspaces
of $\mathcal{P}$ corresponding to the eigenvalues $\pm 1$. An
operator respects the bosonic or fermionic identity of the particles if it
sends vectors in one of these eigenspaces to the same eigenspace,
therefore it must commute with $\mathcal{P}$. Hence the operators
that represent bosonic or fermionic physical quantities must be invariant
under permutations.
That is the physical quantities associated with a system of two
identical particles, bosons or fermions, are represented by
operators
$A(1,2)$ that are symmetric under interchange of the labels of the particles.
The set of such operators form an algebra $(\mathcal{A}(1,2))_{+}$.
Examples of operators $A(1,2)\in(\mathcal{A}(1,2))_{+}$ are $A(1)\otimes A(2)$ and $A(1)\otimes I(2)+I(1)\otimes A(2)$ where the $A(j)$
are operators, built from products of operators representing elements of
the enveloping algebra of the Lie algebra of $\mathcal{G}$, that are
essentially self-adjoint on the Schwartz subspace $\mathord{\mathcal{S}}$ of the
configuration Hilbert space $\mathcal{L}^{2}(\mathbb{R}^{3})$ tensored
with symmetric spin matrices acting on $\mathbb{C}^{2s+1}$. The $I(j)=I_{j}\otimes I_{s_{j}}$ are the identity operators on
$\mathord{\mathcal{H}}(j)=\mathord{\mathcal{H}}(ccr)\otimes\mathbb{C}^{2s_{j}+1%
},j=1,2$.
It follows that all states $\hat{\rho}(1,2)$ of a system of two identical
particles must also be symmetric under the interchange of the particles’
labels. This holds for both bosons and fermions. Therefore state space
of a system of two identical particles is contained in the symmetric Schwarzian
state space $(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$. The open subsets of $(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$
are the restrictions of the open subsets of $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2)$ to
$(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$. They give the weakest topology that makes
continuous all the functions $a(1,2)_{Q}$ from $(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$ to
$\mathbb{R}$;
(7)
$$a(1,2)_{Q}(\hat{\rho}(1,2))=\mathop{\mathrm{Tr}}\hat{\rho}(1,2)A(1,2)$$
for
$A(1,2)\in(\mathcal{A}(1,2))_{+}$, when
$\hat{\rho}(1,2)\in(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$. In particular, these functions are
continuous with respect to the topology on $(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$ given by
the restriction of the trace norm topology
$\nu(\hat{\rho}(1,2))=\mathop{\mathrm{Tr}}|\hat{\rho}(1,2)|$ to $(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$.
The kinematical quantities associated with a system of two identical
Galilean particles of mass $m\neq 0$ and spin $s$ have qrumber values
defined to extents $W\in O((\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+})$ by
continuous functions;
$\vec{x}(1,2)_{Q}$,$\vec{p}(1,2)_{Q}$,$\vec{L}(1,2)_{Q}$, defined for
$\hat{\rho}(1,2)\in W$
by $\vec{x}(1,2)_{Q}(\hat{\rho}(1,2))=\mathop{\mathrm{Tr}}\hat{\rho}(1,2)\vec{X}(1%
,2)$,
where $\vec{X}(1,2)=\vec{X}(1)\otimes I(2)+I(1)\otimes\vec{X}(2)$,
etc. The two particle system has a trajectory in qrumber space that is defined by
the time evolution of these values durt . To obtain trajectories for the
individual particles inside this system we define their kinematical
quantities following the method used when the particles were not
identical.
Proposition 2.
In a system of two identical particles, the qrumber value of
any property of particle 1 is the same as the qrumber value of the corresponding
property of particle 2 when both qrumbers are defined on the
same open subset $W(1,2)\subset(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$.
Proof.
Suppose a physical property of a single particle of type $(m,V,s)$ is
represented by the operator $A$, so that when considered as part of the
compound system the property of particle 1 is represented by an operator
of the form $A(1)\otimes I(2)$ and the property of particle 2 is
represented by an operator of the form $I(1)\otimes A(2)$. Either
$A(1)=A_{1}\otimes I_{s_{1}},A(2)=A_{2}\otimes I_{s_{2}}$ when $A$
represents a kinematical quantity or $A(1)=I_{1}\otimes A_{s_{1}},A(2)=I_{2}\otimes A_{s_{2}}$ , when A represents a spin variable.
Consider the operator $B(1,2)=A(1)\otimes I(2)-I(1)\otimes A(2)$.
$B(1,2)=-B(2,1)$ is skew-symmetric with respect to the interchange of
particle labels. Therefore for any state $\hat{\rho}(1,2)\in(\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}(1,2))_{+}$,
(8)
$$\mathop{\mathrm{Tr}}\hat{\rho}(1,2)B(1,2)=-\mathop{\mathrm{Tr}}\hat{\rho}(1,2)%
B(1,2)$$
because $\mathop{\mathrm{Tr}}\mathcal{P}^{-1}\hat{\rho}(1,2)\mathcal{P}\mathcal{P}^{-1}%
B(1,2)\mathcal{P}=\mathop{\mathrm{Tr}}\hat{\rho}(2,1)B(2,1)$.
Therefore
(9)
$$\mathop{\mathrm{Tr}}\hat{\rho}(1,2)(A(1)\otimes I(2))=\mathop{\mathrm{Tr}}\hat%
{\rho}(1,2)(I(1)\otimes A(2))$$
∎
It is important to be clear that this result says only that when
particles 1 and 2 are identical then the qrumber values of the
physical quantities of particle 1 are the same as those of particle 2.
This does not imply that the measured values of quantities associated
with particle 1 must be the same as the measured values of the quantities
associated with particle 2. Nevertheless this result has considerable
consequences for the concept of locality for identical particles, because
it says that for each $\hat{\rho}(1,2)\in W(1,2)$,
(10)
$$\mathop{\mathrm{Tr}}\hat{\rho}(1,2)(\vec{X}(1)\otimes I_{s_{1}})\otimes I(2)=%
\mathop{\mathrm{Tr}}\hat{\rho}(1,2)(I(1)\otimes\vec{X}(2)\otimes I_{s_{2}}).$$
Therefore the qrumber distance, $d(1,2)_{Q}(W(1,2))=\|(\vec{X}(1))_{Q}(W(1,2))-(\vec{X}(2))_{Q}(W(1,2))\|_{3}$, between identical particles is
zero.
5.2.1. EPR-type experiment
Let us now consider an experiment which uses two identical massive particles that are produced in such a way that the sum of their momenta is zero. The experimenter prepares the momenta to be along the line of a standard real number vector $\vec{a}$. The two particle system is defined to an extent that is given by an $\epsilon$-neighbourhood, $W(1,2)$, of the symmetrised pure state $\Pi_{\pm}$ that projects onto the vector ${1\over 2}(u_{L}(1)\otimes v_{R}(2)\pm v_{R}(1)\otimes u_{L}(2))$ where $L$ and $R$ refer to wave packets propagating along opposite directions following the line parallel to $\vec{a}$.
If $\epsilon$ is small enough, the reduced states of the single particles belong to open sets that are a fifty-fifty convex combination of an $\epsilon$-neighbourhood of the projector onto $u_{L}$ and an $\epsilon$-neighbourhood of the projector onto $v_{R}$ (see section 7.2). The qrumber value of the position and velocity of an individual particle are then close to zero for all times for each particle (both 1 and 2). We physically interpret this property as follows: as a consequence of indistinguishability, the individual particle trajectories reduce to those of the centre of mass of the system. If now we consider two particles trajectories in the qrumber approach, this is no longer true. Indeed, in our approach, we can describe the position of the pair by the qrumbers ${1\over 2}(\hat{x}^{i}(1))\otimes\hat{x}^{j}(2))_{Q}(W(1,2))$, ($i,j=1,2,3$). If we compute their value, up to negligible corrections that are proportional to $\epsilon$, using the appropriate system of reference axes we obtain the value zero for all except
$((\vec{a}.\vec{x})(1)\otimes(\vec{a}.\vec{x})(2))_{Q}(W(1,2))$ which has a qrumber value on $W(1,2)$ which is close to the standard real value $-(v.t)^{2}$. $v$ is the absolute value of the velocity of each component along the $\vec{a}$ direction and $t$ the time that has elapsed since their emission from the source.
Combining this information about the trajectories with our previous observation of the fact that the center of mass does not depart from the origin, we can infer the correct picture for the pair trajectory, i.e. that one particle moves to the left and the other moves to the right, with equal speed $v$.
It is interesting to note that, similar to what happens in quantum optics, the pictures that we get about one and two particle trajectories are, in a sense, complementary and, to some extent, independent. This is not true in the Bohmian approach where individual trajectories are supposed to contain all the relevant information about the physical reality of quantum systems. Actually, in the exemple that we considered above, the two particles are entangled which explains why the knowledge of the pair does not reduce to the knowledge of its single components. The explanation of non-local features of the system is also different in the Bohmian approach and in ours because in the former it is due to the non-locality of the quantum potential and undirectly to the existence of an absolute spatio-temporal reference frame while in the latter non-locality is a property of conventional space-time, not of the quantum space. Moreover the measurement process itself is non-local as we shall now discuss with the help of the example provided by the EPR-Bohm-Bell experiment that we shall discuss now.
5.2.2. EPR-Bohm-Bell experiment for identical particles
Consider an EPR-Bohm-Bell experiment which uses
two identical spin one-half massive particles.
We can obtain the usual quantum
mechanical results for the experiment while maintaining that the two
identical particles always have a qrumber position and hence are localised
in qrumber space.
The two particle system is now defined to an extent that consists of an $\epsilon$-neighbourhood, $W(1,2;s_{1},s_{2})$ of the symmetrised pure state $\Pi_{\pm}\otimes\pi(s_{1},s_{2})$ that projects onto the vector ${1\over 2}(u_{L}(1)\otimes v_{R}(2)\pm v_{R}(1)\otimes u_{L}(2))\otimes(|+_{s_%
{1}}>\otimes|-_{s_{2}}>+|-_{s_{1}}>\otimes|+_{s_{2}}>)$ where $L$ and $R$ label the opposite directions along the line parallel to $\vec{a}$ and $|\pm_{s_{j}}>$ represents a spin up (down) polarisation state along the directions $\vec{b}$ orthogonal to $\vec{a}$ for particle $j$. If we trace over the spin degrees of freedom the pure state $\Pi_{\pm}\otimes\pi(s_{1},s_{2})$ reduces to the pure spatial state $\Pi_{\pm}$ which is of the same form as that used for spinless particles. Therefore before the spins interact with the magnetic fields, the discussion of the previous section applies; the individual particle trajectories reduce to those of the centre of mass of the system and the qrumber values of position and velocity of the individual particles are then close to zero for all times for each particle. It is only when the two particle trajectories are calculated that we see that one particle travels to the left and the other to the right, but we cannot tell which.
In order to carry out an EPR-Bohm-Bell experiment we must check the spin correlations of the particles. This can be done by letting them pass through Stern-Gerlach devices with magnetic fields orthogonal to the direction $\vec{a}$. The positions of the particles are then measured after their passage through the Stern-Gerlach devices.
As we discuss with greater detail in Ref.durt there exist several possible ways to describe the evolution of the system in the absence of measurement. For instance one could let it evolve according to the standard unitary (Schroedinger) evolution, or one could define an Hamilton-like evolution at the level of the qrumbers. In the present case, taking account of the gyromagnetic coupling of the spins, the two evolutions would be
equivalent up to $\epsilon$ when the system is defined to an extent that consists of an $\epsilon$ neighbourhood $W(1,2;s_{1},s_{2})$ around the symmetrised pure state $\Pi_{\pm}\otimes\pi(s_{1},s_{2})$.
By choosing $\epsilon$ small enough the trajectories will be enclosed in regions of volume comparable to that due to the spread of the wave function. In well conceived experiments
we measure the positions of the particles with detectors larger than this spread durtthesis , so that it does not really matter which type of evolution we adopt in our description of the evolution of the system, provided that the time of passage through the magnetic region of the Stern-Gerlach devices is short enough.
After the passage through these regions and before reaching the detectors, the two particle system is defined to an extent given by an $\epsilon$-neighbourhood of the projection $\Pi_{\pm}^{s_{1},s_{2}}(1,2)$ onto the vector
$\Psi_{\pm}^{s_{1},s_{2}}(1,2)$
$={1\over 2}(u^{+}_{L}(1)\otimes v^{-}_{R}(2)\otimes|+s_{L}(1)>\otimes|-s_{R}(2%
)>(\pm)\ v^{-}_{R}(1)\otimes u^{+}_{L}(2))\otimes|-s_{R}(1)>\otimes|+s_{L}(2)>%
+u^{-}_{L}(1)\otimes v^{+}_{R}(2)\otimes|-s_{L}(1)>\otimes|+s_{R}(2)>(\pm)\ v^%
{+}_{R}(1)\otimes u^{-}_{L}(2))\otimes|+s_{R}(1)>\otimes|-s_{L}(2)>)$
where $L^{\pm}$ and $R^{\pm}$ refer to wave packets propagating to the upper (lower) left and right parts of the plane spanned by $\vec{a}$ and $\vec{b}_{K},K=L,R$, the direction of the magnetic field on the left or right. $\pm s_{K}(j)$ denotes spin up/down in the direction of $\vec{b}_{K},K=L,R$ for the $jth$ particle.
Tracing over the spin variables we obtain a reduced spatial state $P_{\pm}(1,2)$ given by ${1\over 4}(P_{u_{L}^{+}(1)}\otimes P_{v_{R}^{-}(2)}+P_{u_{L}^{-}(1)}\otimes P_%
{v_{R}^{+}(2)}+P_{v_{R}^{+}(1)}\otimes P_{u_{L}^{-}(2)}+P_{v_{R}^{-}(1)}%
\otimes P_{u_{L}^{+}(2)})$. If the wave packets are assumed to be non-overlapping narrow Gaussians then the single particle reduced state for the $jth$ particle is $\rho_{\pm}(j)={1\over 4}(P_{u_{L}^{+}(j)}+P_{u_{L}^{-}(j)}+P_{v_{R}^{+}(j)}+P_%
{v_{R}^{-}(j)})$, which is the same state for both bosons and fermions so we’ll suppress the $\pm$.The
extent that the $jth$ particle exists is given by an $\epsilon$-neighbourhood of $\rho(j)$.
If $\epsilon$ is small enough, the $\epsilon$-neighbourhood of $\rho(j)$ is a convex combination with equal weights of $\epsilon$-neighbourhoods of the pure state projectors onto the four vectors $u^{\pm}_{L},v^{\pm}_{R}$.
In each reduced spatial state $P_{\pm}(1,2)$, the pure states for particles 1 and 2 are paired. If we associate a segment of a qrumber trajectory with a single particle pure state via its $\epsilon$-neighbourhood then the qrumber trajectory for the identical particles is composed of a pair of distinct trajectories in the crossing diagram $(>--<)$. In the first of the pair, one of the particles, we cannot tell which, is at the top of the descending branch of the cross on the left and the other particle is at its bottom on the right; in the second of the pair, one of the particles, we cannot tell which, is at the top of the ascending branch of the cross on the right and the other particle is at its bottom on the left.
This situation clearly differs from the Bohmian picture in which either the ascending branch is selected or the descending one, but not both at the same time durtbohm .
As we discuss in Ref.durt , it is at the moment of the measurement, when detectors placed beyong the region of the magnetic field click, one of the two branches of the qrumber trajectory is selected in a process equivalent to the collapse of a wave-function. We show in that paper that the probability associated to the respective branches necessarily obeys the Born rule in order to derive a self-consistent formulation of the qrumber approach. Clearly the interaction with the measuring apparatus is non-local in the usual sense when the branches of the cross on the left and right are separated by spacelike distances (where we refer here to distances measured in standard Euclidean space).
In a sense the problem of non-locality for identical particles can be seen
to disappear when qrumbers are used because the qr-number distance
between the particles 1 and 2 is zero, provided that we accept that space-time of
quantum systems is described by its qrumber continuum rather than the classical real number continuum.
6. Conclusions
We have shown that if we are willing to accept that the spatial continuum is
not given a priori but is an artefact of the theory then Galilean relativistic quantum mechanics has a space-time that is as real as the standard space-time of classical
physics. Galilean relativistic atoms and sub-atomic particles exist in a space
whose points are labelled by qrumbers rather than standard real numbers, they
move along trajectories in the space of qrumbers. Their properties have
qrumber values always. In this setting the ontology of Galilean
relativistic atoms and sub-atomic particles is similar to that of classical
particles except that the values taken by their attributes are qrumbers
not standard real numbers. Accepting that we can have a non-standard spatial continuum for a physical theory allows us to better understand the theory, for
example the problems of non-locality in the standard theory of quantum mechanics
disappear when the qrumber spatial distance between quantum particles is used.
For example, we showed in an explicit example of a Bell-type experiment how identical particles have zero qrumber spatial separation and hence their
interactions are in a sense not non-local. The mystery of the two slit experiment is removed
because the qrumber position of a quantum particle can be such that the one qrumber position can cover more than one standard spatial position of a classical
particle.
Nevertheless, the quantum wholeness remains present because, due to the presence of entanglement, 2 particle behavior does not reduce to 1 particle behavior.
That the quantum space of a quantum system is different from standard Euclidean
space does not reduce the efficacy of the space-time view for understanding physics.
The standard real number continuum of classical physics presents much the same class of difficulties as the qrumber continuum: we never can observe all the points on the trajectory of a moving particle, there are many more points in the continuum than we can ever observe. There is one important difference, the points of quantum space are faithfully represented by qrumbers which exist to varying extents reflecting the conditions on the physical system under which the quantities have a value. The dependence of the condition of the physical system raises new possibilities. The
universality of the numerical values taken by quantities is an idealisation that has
real limitations; even in classical physics the length of a metal rod depends upon
the ambient temperature.
This paper only discusses Galilean relativistic particle theories. we think that will be possible to extend this approach to different relativistic quantum field theories when
their symmetry groups are used in place of the Galilean group.
7. Appendices.
7.1. The standard mathematical formalism
We review the mathematical formalism of standard quantum
mechanics.
A Galilean relativistic particle of mass $m$, internal energy $U$ and
spin $s$ is associated with an irreducible projective unitary
representation of the Galilean group $\mathcal{G}$ levy .
$\mathcal{G}$ is parameterised as follows
$g=(b,\vec{a},\vec{v},R)$:
$b\in{\mathbb{R}}$ for time translations, $\vec{a}\in{\mathbb{R}}^{3}$ for space
translations, $\vec{v}\in{\mathbb{R}}^{3}$ for pure Galilean transformations or
velocity translations, $R\in SO(3)$ for rotations about a point.
Classically $g\in\mathcal{G}$ acts upon the classical space-time
coordinates sending $(\vec{x},t)$ to $(\vec{x}^{\prime},t^{\prime})=(R\vec{x}+\vec{a}+t\vec{v},t+b)$ then
(11)
$$(b^{\prime},\vec{a}^{\prime},\vec{v}^{\prime},R^{\prime})(b,\vec{a},\vec{v},R)%
=(b^{\prime}+b,\vec{a}^{\prime}+R^{\prime}\vec{a}+b\vec{v}^{\prime},\vec{v}^{%
\prime}+R^{\prime}\vec{v},R^{\prime}R)$$
The irreducible projective unitary representation $g\in\mathcal{G}\mapsto\mathcal{U}(g)$, labelled by $(m,U,s)$ where $m$
is a positive real number, $U$ is a standard real number and $s$ is a
natural number or half a natural number, acts on the Hilbert
space $\mathord{\mathcal{H}}:=\mathcal{L}^{2}(\mathbb{R}^{3})\otimes\mathbb{C}^{2s+1}$. The elements of
$\mathord{\mathcal{H}}$ are $(2s+1)$-component vectors of square integrable functions
$\{\psi_{i}(\vec{x}):\vec{x}\in{\mathbb{R}}^{3}\}_{i=-s}^{s}$. The
corresponding space-time functions $\psi_{i}(\vec{x},t)$ are defined
using the generator $H={1\over 2m}\vec{P}\cdot\vec{P}+U$ of the time
translations,
$\psi_{i}(\vec{x},t):=(exp(-iHt/\hbar)\psi_{i})(\vec{x})$. Then
(12)
$$\mathcal{U}(g)\psi_{i}(\vec{x},t)=exp(i\alpha_{m})\Sigma_{j}D_{ij}^{s}(R)\psi_%
{j}(R^{-1}(\vec{x}-\vec{v}(t-b)-\vec{a}),t-b)$$
where $\alpha_{m}=\alpha_{m}(b,\vec{a},\vec{v},R;\vec{x},t):=[-{1\over 2}m\vec{v}%
\cdot\vec{v}(t-b)+m\vec{v}\cdot(\vec{x}-\vec{a})]/\hbar$ and
$D_{ij}^{s}(R)$ are the matrix elements of the irreducible projective
representation of $SO(3)$ on $\mathbb{C}^{2s+1}$.
If $H$ is the infinitesimal generator of the subgroup of time
translations, $P_{1},P_{2},P_{3}$ are the infinitesimal generators of the
subgroup of spatial translations along the three orthogonal axes of the
standard basis, $K_{1},K_{2},K_{3}$ are the infinitesimal generators of the
subgroup of velocity translations along those axes and
$J_{1},J_{2},J_{3}$ are the infinitesimal generators of the
subgroup of rotations around those axes then
the Lie algebra of the extended Galilean group, with elements
$(\theta,g);\theta\in{\mathbb{R}},g\in\mathcal{G}$, is generated by $H,\vec{P},\vec{K},\vec{J}$ and the central element $\hbar I$ whose Lie
brackets with all the other 10 elements vanish. The other Lie brackets
are
(13)
$$[J_{i},A_{j}]=\epsilon_{ijk}A_{k},\vec{A}=\vec{P},\vec{K},\vec{J};\vskip 3.0pt%
plus 1.0pt minus 1.0pt[H,B_{i}]=0,\vec{B}=\vec{P},\vec{J};\vskip 3.0pt plus 1%
.0pt minus 1.0pt[H,K_{i}]=-P_{i}.$$
and
(14)
$$[K_{i},K_{j}]=0,[P_{i},P_{j}]=0,[K_{i},P_{j}]=m\hbar I\delta_{ij}.$$
As well as $\hbar I$, the element $U:=H-{1\over 2m}\vec{P}\cdot\vec{P}$ of the enveloping algebra commutes with all the infinitesimal
generators.
$U$ may be identified as the internal energy of the particle. We usually
take $U=0$. The vector position operator for the particle is $\vec{X}={1\over m}\vec{K}$, the vector operator $\vec{L}=\vec{X}\times\vec{P}$
is the orbital angular momentum. The spin vector operator is $\vec{S}=\vec{J}-\vec{L}$. The operator $\vec{S}\cdot\vec{S}$ commutes with all the
elements of the Lie algebra and in any irreducible unitary
representations $\vec{S}\cdot\vec{S}=s(s+1)$ with $s$ an integer or
half-integer.
The irreducible representation of the enveloping Lie algebra of the
extended Galilean group labelled by $(m,U,s)$ with central element
$\hbar I$ is unitarily equivalent to the tensor product $M\otimes M_{s}$
of the $Schr\ddot{o}dinger$ representation $M$ of the algebra of the
Canonical Commutation Relations (CCR-algebra) generated by the operators
$\{\vec{P},\vec{X},\hbar I\}$ with the irreducible matrix
representation $M_{s}$, of dimension $2s+1$, of the Lie algebra of the
rotation group $SO(3)$.
The $Schr\ddot{o}dinger$ representation of the CCR-algebra $M$ is the
representation in which the Hilbert space is $\mathord{\mathcal{H}}(ccr)=\mathcal{L}^{2}(\mathbb{R}^{3})$.
$X_{j}$ is represented by multiplication by the real variable $x_{j}$
and $P_{j}$ by $(1/i)$ times the operator of differentiation with
respect to $x_{j}$. Let $\mathord{\mathcal{S}}({{\mathbb{R}}^{3}})$ denote the Schwartz
space of infinitely differentiable functions of rapid
decrease on ${\mathbb{R}}^{3}$. Then the physical quantities are
represented by self-adjoint elements in the closure ${\mathord{\bar{M}}}$
of the CCR-algebra $M$, where $\bar{M}$ is the smallest closed
extension of $M$,
${\mathord{\bar{M}}}=\bigl{\{}\tilde{X}\bigm{|}X\in M\;\allowbreak\text{and}%
\allowbreak\;\tilde{X}\allowbreak\text{is the restriction to }\mathord{%
\mathcal{S}}({{\mathbb{R}}^{3}})\text{ of the Hilbert space closure of}%
\allowbreak\;X\bigr{\}}$. Following the definitions of Powers inoue , $M$ is
essentially self-adjoint because the adjoint $M^{*}$ of $M$ equals the
closure $\bar{M}$ of $M$.
The irreducible representations of the Lie algebra of $SO(3)$ are
labelled by the half integer s, $s=0,{1\over 2},1,{3\over 2},2,...$,
and the dimension of the representation is $(2s+1)$. The elements of
the Lie algebra are represented by self-adjoint matrices acting on
$\mathbb{C}^{2s+1}$, they form the matrix algebra $M_{s}$.
The physical quantities associated with a particle are represented by
operators $\hat{A}$ that are the tensor product of operators, $\hat{A}=\hat{C}\otimes\hat{S}$. $\hat{C}$ represents an element of the enveloping
algebra of the Lie algebra of
$\mathcal{G}$ and is essentially self-adjoint on the Schwartz subspace
$\mathord{\mathcal{S}}({{\mathbb{R}}^{3}})$ of the Hilbert space
$\mathcal{L}^{2}(\mathbb{R}^{3})$.
$\hat{S}$ is a self-adjoint spin matrix acting on $\mathbb{C}^{2s+1}$.
7.2. Qrumbers
We use a real number continuum given by
the sheaf of Dedekind reals $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ in the topos of sheaves on the quantum state space $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$. A qrumber
is a local section of the sheaf $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ maclane , adelman2
$\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ is the Schwartz subspace of the state space, consisting of those states $\hat{\rho}$ that are rapidly decreasing convex combinations of projection operators
$\hat{P}_{j}$ onto one dimensional subspaces spanned by vectors in Schwartz
space $\mathord{\mathcal{S}}$, that is, $\hat{\rho}=\Sigma_{j=1}^{\infty}\lambda_{j}\hat{P}_{j}$ with $\lambda_{j}\geq 0$ and $\lim_{j\to\infty}\lambda_{j}j^{n}=0$
for all positive integers $n$ adelman2 .
The topology on $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ is the weakest that makes continuous all the functions of the form $a_{Q}(\hat{\rho})=\mathop{\mathrm{Tr}}(\hat{\rho}\cdot\hat{A})$ for self-adjoint operators
$\hat{A}\in M\otimes M_{s}$. Then the functions $a_{Q}$ form a subobject ${\mathbb{A}}$
of $\mathord{{\mathbb{R}}_{\mathrm{D}}}(\mathord{\mathcal{E}_{\mathord{\mathcal{S}%
}}})$ which is the sheaf of locally linear functions
on $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ adelman2 . Each qr-number defined to extent $W$ is either a continuous function of the $a_{Q}(W)$ or a constant real valued function on $W$. The sheaf $\mathcal{C}(W)$ of continuous
real-valued functions over the open set $W\subset\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ can extended to a
sheaf over $\mathord{\mathcal{E}_{\mathord{\mathcal{S}}}}$ by prolongation by zero swan .
One useful type of open set is the $\epsilon$-neighbourhood of a state $\rho_{0}$.
Definition 3.
The $\epsilon$-neighbourhood of a state $\rho_{0}$ is $\nu(\rho_{0};\epsilon)=\{\rho:Tr|\rho-\rho_{0}|<\epsilon\}$. That is, it is an open ball, in the trace norm topology, of radius $\epsilon$ centred on $\rho_{0}$.
When $\epsilon$ is small, we can get a good idea of the properties of $a_{Q}(\nu(\rho_{0};\epsilon))$ from those of $Tr(\rho_{0}\cdot\hat{A})$.
The following result is used in the discussions of identical particles.
Let $\Psi(1,2)={1\over\sqrt{2}}(\psi_{+}(1)\psi_{-}(2)\pm\psi_{-}(1)\psi_{+}(2))$ be an entangled wave function of particles 1 and 2, the normalised single particle wave functions being orthogonal,
$<\psi_{+}(j),\psi_{-}(j)>=0,j=1,2$.
The corresponding pure state is $P_{\Psi(1,2)}={1\over 2}(P_{\psi_{+}(1)}\otimes P_{\psi_{-}(2)}+P_{\psi_{-}(1)%
}\otimes P_{\psi_{+}(2)}\pm V_{+,-}(1)\otimes V_{-,+}(2)\pm V_{-,+}(1)\otimes V%
_{+,-}(2))$, where the partial isometries $V_{+,-}(j)=|\psi_{+}(j)><\psi_{-}(j)|=V_{-,+}(j)^{\ast},j=1,2$. The reduced state of particle j is the mixed state,
$\rho_{0}(j)={1\over 2}(P_{\psi_{+}(j)}+P_{\psi_{-}(j)})$, for both bosons and fermions.
Lemma 4.
If $\epsilon<1$ then the $\epsilon$-neighbourhood of $\rho_{0}(j)$ does not contain any pure state.
If $\epsilon$ is small enough it essentially only contains fifty-fifty convex combination of the $\epsilon$-neighbourhoods of $P_{\psi_{+}(j)}$ and
$P_{\psi_{-}(j)}$, that is, $\nu(\rho_{0}(j);\epsilon)={1\over 2}\nu(P_{\psi_{+}(j)};\epsilon)+{1\over 2}%
\nu(P_{\psi_{-}(j)};\epsilon),j=1,2$.
Proof.
We will prove the result for particle 1.
If $\chi(1)$ is a unit vector with $<\chi(1)|\psi_{\pm}(1)>=\gamma_{\pm}$ then the vectors $\{\psi_{+}(1),\psi_{-}(1),\chi(1)\}$ span a subspace of dimension at most 3. Then it is easy to calculate $Tr|P_{\chi(1)}-\rho_{0}(1)|=1+(1-|\gamma_{+}|^{2}-|\gamma_{-}|^{2})^{{1\over 2}}$ which is less than $\epsilon$ only if $\epsilon>1$. Therefore, when $\epsilon<1$ every state $\rho\in\nu(\rho_{0}(1);\epsilon)$ is impure.
Now ${1\over 2}\nu(P_{\psi_{+}(1)};\epsilon)+{1\over 2}\nu(P_{\psi_{-}(1)};\epsilon)$
is contained in $\nu(\rho_{0}(1);\epsilon)$. Because if $\rho={1\over 2}\rho_{+}+{1\over 2}\rho_{-}$
with $\rho_{j}\in\nu(P_{\psi_{j}(1)};\epsilon)$ for $j=+,-$, then since $\nu$ determines a norm on the space of trace class operators,
(15)
$$Tr|\rho-\rho_{0}(1)|\leq{1\over 2}(Tr|\rho_{+}-P_{\psi_{+}(1)}|+Tr|\rho_{-}-P_%
{\psi_{-}(1)}|)<\epsilon.$$
There are other states in the $\epsilon$-neighbourhood of $\rho_{0}(1)$. They are of the form, $\rho=\lambda P_{\psi_{+}(1)}+(1-\lambda)P_{\psi_{-}(1)}$ where ${1\over 2}-{\epsilon\over 2}<\lambda<{1\over 2}+{\epsilon\over 2}$. They are impure states
which are convex combinations of the same two states as $\rho_{0}(1)$ is. When $\epsilon$
is small they behave like $\rho_{0}(1)$ itself.
∎
Therefore if a system of two identical particles exists to the extent $\nu(P_{\Psi(1,2)};\epsilon)$ then the jth particle exists to the extent $\nu(\rho_{0}(j);\epsilon)={1\over 2}\nu(P_{\psi_{+}(j)};\epsilon)+{1\over 2}%
\nu(P_{\psi_{-}(j)};\epsilon)$.
Therefore the $ith$ coordinate qrumber value of the jth particle’s position is ${1\over 2}x^{i}_{Q}(\nu(P_{\psi_{+}(j)};\epsilon))+{1\over 2}x^{i}_{Q}(\nu(P_{%
\psi_{-}(j)};\epsilon))$.
In the EPR - type experiment, we used $\psi_{+}=u_{L}$ and $\psi_{-}=v_{R}$ which are assumed to be non-overlapping narrow Gaussians, $u_{L}$ moving with speed $v$ to the left along the axis $\vec{a}$, $v_{R}$ moving with speed $v$ to the right along $\vec{a}$. Then the qrumber value of $\vec{a}\cdot\vec{x}(j)$ for the jth particle is approximately zero.
On the other hand when we measure $\vec{a}\cdot\vec{x}(j)$ for the jth particle in the vicinity of either of the states $P_{\psi_{+}(j)}$ or $P_{\psi_{-}(j)}$ we will get a value, which leads us to deduce that the $jth$ particle has a $\vec{a}\cdot\vec{x}(j)$ qrumber value to both the extents $\nu(P_{\psi_{+}(j)};\epsilon)$ and $\nu(P_{\psi_{-}(j)};\epsilon)$. It follows from the orthogonality of the wave functions and $\epsilon<1$ that the extents are disjoint.
The calculation for the two particles trajectories is based on the assumption that the particles are moving freely. Since we have assumed that for each particle the single particle wave functions are non-overlapping it is easy to show that the terms
$TrP_{\Psi(1,2)}({\hat{P}_{i}(1)\over m_{1}}\otimes\hat{X}_{j}(2)+\hat{X}_{i}(1%
)\otimes{\hat{P}_{j}(2)\over m_{2}})$, $TrP_{\Psi(1,2)}(\hat{X}_{i}(1)\otimes\hat{X}_{j}(2))$ and $TrP_{\Psi(1,2)}(\hat{P}_{i}(1)\otimes\hat{P}_{j}(2))$ vanish for all $i,j$ except for the component $(a,a)$. For instance, $TrP_{\Psi(1,2)}(\hat{X}_{a}(1)\otimes\hat{X}_{a}(2))=-(v.t)^{2}$, where $X_{a}=\vec{a}\cdot\vec{X}/(\vec{a}\cdot\vec{a})$. These results extend to the open set
$W=\nu(P_{\Psi(1,2)};\epsilon)$ by continuity.
The extension of these results to an EPR-Bohm-Bell experiment with two identical spin one-half massive particles is straightforward: the qrumber trajectories obtained for spinless particles in the EPR-experiment are split by the interaction between the spin and the magnetic field. The pair of qrumber trajectories become four qrumber trajectories.
8. Acknowledgments
Parts of this work has benefitted from comments and criticisms of members of the
Centre for Time at Sydney University, where JVC is an Honorary Associate.
T.D. is a Postdoctoral Fellow of the Fonds voor
Wetenschappelijke Onderzoek, Vlaanderen and also thanks supports of
Inter-University Attraction Pole Program of the Belgian government
under grant V-18, International Solvay Institutes for Physics and Chemistry, the Concerted Research Action Photonics in
Computing and the research council (OZR) of the VUB. The work was realised during a 6 months visit of T.D. at Macquarie’s university supported by a F.W.O. mobility grant.
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Optimizing Guided Traversal for Fast Learned Sparse Retrieval
Yifan Qiao, Yingrui Yang, Haixin Lin, Tao Yang
Department of Computer Science, University of CaliforniaSanta BarbaraCalifornia93106USA
yifanqiao, yingruiyang, haixinlin, [email protected]
(2023)
Abstract.
Recent studies show that BM25-driven dynamic index skipping
can greatly accelerate MaxScore-based document retrieval based on the learned sparse representation derived by DeepImpact.
This paper investigates the effectiveness of such a traversal guidance strategy during top $k$
retrieval when using other models such as SPLADE and uniCOIL, and
finds that unconstrained BM25-driven skipping could have a visible relevance degradation
when the BM25 model is not well aligned with a learned weight model or when retrieval depth $k$ is small.
This paper generalizes the previous work and
optimizes the BM25 guided index traversal with a two-level pruning control scheme and model alignment
for fast retrieval using a sparse representation.
Although there can be a cost of increased latency, the proposed scheme
is much faster than the original MaxScore method without BM25 guidance while retaining the relevance effectiveness.
This paper analyzes the competitiveness of this two-level pruning scheme,
and evaluates its tradeoff in ranking relevance and
time efficiency when searching several test datasets.
††journalyear: 2023††copyright: rightsretained††conference: Proceedings of the ACM Web Conference 2023; May 1–5, 2023; Austin, TX, USA††booktitle: Proceedings of the ACM Web Conference 2023 (WWW ’23), May 1–5, 2023, Austin, TX, USA††doi: 10.1145/3543507.3583497††isbn: 978-1-4503-9416-1/23/04
1. Introduction
Document retrieval for searching a large dataset
often uses a sparse representation of document feature vectors implemented as
an inverted index which associating each search term with a list of documents containing such a term.
Recently learned sparse representations have been developed to
compute term weights using a neural model such as transformer based retriever (Dai and Callan, 2020; Bai et al., 2020; Formal et al., 2021b, a; Mallia et al., 2021; Lin and Ma, 2021) and
deliver strong relevance results,
together with document expansion (e.g. (Cheriton, 2019)).
A downside is that top $k$
document retrieval latency using a learned sparse representation is much large than using the BM25 model
as discussed in (Mallia et al., 2021; Mackenzie et al., 2021).
In the traditional BM25-based document retrieval with additive ranking,
a dynamic index pruning strategy based on top $k$ threshold is very effective by computing
the rank score upper bound on the fly for each visited document
during index traversal in order to skip low-scoring documents that are unable to appear in the final top $k$ list.
Well known traversal algorithms with such dynamic pruning strategies include MaxScore (Turtle and Flood, 1995) and WAND (Broder et al., 2003), and
their block-based versions Block-Max WAND (BMW) (Ding and Suel, 2011) and Block-Max MaxScore (BMM) (Chakrabarti et al., 2011; Dimopoulos et al., 2013).
Mallia et al. (Mallia et al., 2022) propose to use BM25 to guide traversal, called GT,
for fast learned sparse retrieval because the distribution of learned weights results in less pruning opportunities
and they conducted an evaluation with retrieval model DeepImpact (Mallia et al., 2021). One variation they propose is to compute the final rank scoring as a linear
combination of the learned weights and BM25 weights, denoted as GTI.
GT is a special case of GTI and this paper treats GTI as the main baseline.
Since the BM25 weight for a document term pair may not exist in a learned sparse index,
zero filling is used in Mallia et al. (Mallia et al., 2022) to align the BM25 and learned weight models.
During our evaluation using GT for SPLADE v2 and its revision SPLADE++ (Formal et al., 2021a, 2022),
we find that
as retrieval depth $k$ decreases, BM25 driven skipping becomes too aggressive in dropping
documents desired by top $k$ ranking based on learned term weights, which
can cause a significant relevance degradation.
In addition, there is still some room to further improve index alignment of GTI for more accurate
BM25 driven pruning.
To address the above issues,
we improve our earlier pruning study on dual guidance with combined BM25 and learned weights (Qiao et al., 2022).
Our work generalizes GTI by constraining the pruning influence of BM25
and providing an alternative smoothing method to align the BM25 index with learned weights.
In Section 4, we
propose a two-level parameterized guidance scheme with index alignment,
called 2GTI, to manage pruning decisions during MaxScore based traversal.
We analyze some formal properties of 2GTI on its relevance behaviors and configuration conditions when 2GTI
outperforms a two-stage top $k$ search algorithm for a query in relevance.
Section 5 and Appendix A
present an evaluation of
2GTI with SPLADE++ (Formal et al., 2021b, a, 2022)
and uniCOIL (Lin and Ma, 2021; Gao et al., 2021a) in addition to DeepImpact (Mallia et al., 2021) when using MaxScore
on the MS MARCO datasets.
This evaluation shows that when retrieval depth $k$ is small,
or when the BM25 index is not well aligned with the underlying learned sparse representation,
2GTI can outperform GTI and retain relevance more effectively.
In some cases, there is a tradeoff that 2GTI based retrieval may be slower than that of GTI while 2GTI is still much faster than the original MaxScore method without BM25 guidance.
2GTI is also effective for the BEIR datasets in terms of the zero-shot relevance and retrieval latency.
In Appendix B, we have
extended the use of 2GTI for a BMW-based algorithm such as VBMW (Mallia et al., 2017).
We demonstrate that 2GTI with VBMW can be useful for a class of short queries and when $k$ is small.
2. Background and Related Work
The top-$k$ document retrieval problem identifies top ranked results in matching a query.
A document representation uses a feature vector to capture the semantics of a document.
If these vectors contain much more zeros than non-zero entries, then such a representation is considered sparse.
For a large dataset, document retrieval often uses a
simple additive formula as the first stage of search and it computes the rank score of each document $d$ as:
(1)
$$\sum_{t\in Q}w_{t}\cdot w(t,d),$$
where $Q$ is the set of all search terms,
$w(t,d)$ is a weight contribution of term $t$ in document $d$,
and $w_{t}$ is a document-independent or query-specific term weight.
Assume that $w(t,d)$ can be statically or dynamically scaled,
this paper views $w_{t}=1$ for simplicity of presentation.
An example of such formula is BM25 (Jones et al., 2000) which is widely used.
For a sparse representation, a retrieval algorithm often uses
an inverted index with a set of terms, and a document posting list of each term.
A posting record in this list contains document ID and its weight for the corresponding term.
Threshold-based skipping. During the traversal of posting lists in document retrieval,
the previous studies have advocated dynamic pruning strategies
to skip low-scoring documents, which cannot appear on the final top-$k$ list
(Broder et al., 2003; Strohman and Croft, 2007).
To skip the scoring of a document, a pruning strategy computes the upper bound rank score of a candidate document $d$,
referred to as $Bound(d)$.
If $Bound(d)\leq\theta$ where $\theta$ is the rank score threshold in the top final $k$ list, this document can be skipped.
For example, WAND (Broder et al., 2003) uses the maximum term weights of documents of each posting list to determine the rank score upper bound of
a pivot document while BMW (Ding and Suel, 2011) and its variants (e.g. (Mallia et al., 2017)) optimize WAND
use block-based maximum weights to compute the score upper bounds. MaxScore (Turtle and Flood, 1995) uses
term partitioning and the top-$k$ threshold to skip unnecessary index visitation and scoring computation.
Previous work has also pursued a “rank-unsafe” skipping strategy by deliberately over-estimating the current top-$k$ threshold
by a factor (Broder et al., 2003; Macdonald et al., 2012; Tonellotto et al., 2013; Crane et al., 2017).
Learned sparse representations.
Earlier sparse representation studies are conducted in (Zamani et al., 2018),
DeepCT (Dai and Callan, 2020), and SparTerm (Bai et al., 2020).
Recent work on this subject includes
SPLADE (Formal et al., 2021b, a, 2022), which learns token importance for document expansion with sparsity control.
DeepImpact (Mallia et al., 2021) learns neural term weights on documents expanded by DocT5Query (Cheriton, 2019).
Similarly, uniCOIL (Lin and Ma, 2021) extends the work of COIL (Gao et al., 2021a) for contextualized term weights.
Document retrieval with term weights learned from a transformer has been found slow in (Mallia et al., 2022; Mackenzie et al., 2021).
Mallia et al. (Mallia et al., 2022) state that the MaxScore retrieval algorithm does not efficiently exploit the DeepImpact scores.
Mackenzie et al. (Mackenzie et al., 2021) view that the learned sparse term weights are “wacky” as they affect document skipping during retrieval thus they
advocate ranking approximation with score-at-a-time traversal.
Our scheme uses a hybrid combination of BM25 and learned term weights,
motivated by the previous work in composing lexical and neural ranking (Yang et al., 2022; Lin et al., 2021b; Gao et al., 2021b; Ma et al., 2021; Li et al., 2022). GTI adopts that for final ranking.
A key difference in our work is that hybrid scoring is used for two-level pruning control and its formula can be different from final ranking. The multi-level hybrid scoring difference provides an opportunity for additional pruning and its quality control.
Thus the outcome of 2GTI is not a simple linear ranking combination of BM25 and learned weights and
two-level guided pruning yields a non-linear ensemble effect
to improve time efficiency while retaining relevance.
Our evaluation will include a relevance and efficiency comparison with MaxScore using a simple linear combination.
This paper mainly focuses on MaxScore because it has been shown more effective for relatively longer queries (Mallia et al., 2019a).
We also consider VBMW (Mallia et al., 2017) because
it is generally acknowledged to represent the state of the art (Mackenzie et al., 2021) for many cases,
especially when $k$ is small and the query length is short (Mallia et al., 2019a).
3. Design Considerations
Figure 1 shows the performance of the original MaxScore retrieval algorithm without BM25 guidance, GTI, and the proposed 2GTI scheme in terms of MRR@10 and recall@k when varying top $k$ in searching MS MARCO passages on Dev query set.
Here $k$ is the targeted number of top documents to retrieve and it is also called retrieval depth sometime in the literature.
Section 5 has more detailed dataset and index information.
For both SPLADE++ and uniCOIL, we build the BM25 model following (Mallia et al., 2022) to expand passages first using DocT5Query,
and then use the BERT’s Word Piece tokenizer to tokenize the text, and align the token choices of BM25 with these learned models.
From Figure 1, there are significant recall and MRR drops with GTI
when $k$ varies from 1,000 to 10. There are two reasons contributing to the relevance drops.
[leftmargin=*]
(1)
When the number of top documents $k$ is relatively small, the relevance drops significantly.
As $k$ is small, dynamically-updated top $k$ score threshold becomes closer to the maximum rank score of the best document.
Fewer documents fall into top $k$ positions and more documents below the updated top $k$ score threshold
would be removed earlier. Then the accuracy of skipping becomes more sensitive.
The discrepancy of BM25 scoring and learned weight scoring can cause good candidates
to be removed inappropriately by BM25 guided pruning, which can lead to a significant relevance drop for small $k$.
(2)
The relevance drop for SPLADE++ with BM25 guided pruning is noticeably much more significant than uniCOIL.
That can be related to the fact that SPLADE++ expands tokens of each document tokens differently and much more aggressively than uniCOIL.
As a result, 98.6% of term document pairs in SPLADE++ index does not exist in the
BM25 index even after docT5Query document expansion while this number is 1.4% for uniCOIL.
Thus, BM25 guidance can become less accurate and improperly skip more good documents.
With the above consideration, our objective is to control the influence of BM25 weights in a constrained manner for safeguarding relevance prudently,
and to develop better weight alignment when the BM25 index is not well aligned with the learned sparse index.
In Figure 1, the recall@k number of 2GTI marked with blue squares is similar to that of the original method
without BM25 guidance. Their MRR@10 numbers overlapped with each other, forming a nearly-flat lines, which indicates
their MRR@10 numbers are similar even $k$ decreases.
The following two sections present our solutions in addressing the above two issues respectively.
4. Two-level Guided Traversal
4.1. Two-level guidance for MaxScore
We assume the posting list of each term is sorted in an increasing order of document IDs in the list.
The MaxScore algorithm (Turtle and Flood, 1995)
can be viewed to conduct a sequence of traversal steps and at each traversal step, it conducts term partitioning and then
examines if scoring of a selected document should be skipped.
We differentiate pruning-oriented actions in two levels as follows.
[leftmargin=*]
•
Global level.
MaxScore
uses the maximum scores (upper bounds) of each term and the current known top $k$ threshold
to partition terms into two lists at each index traversal step: the essential list and non-essential list.
The documents that do not contain essential terms are impossible to appear in top $k$ results and thus can be eliminated.
In the next step of index traversal, it will start with the minimum unvisited document ID
only from the posting lists of essential terms.
Thus index visitation is driven by moving such a minimum document ID pointer from the essential list.
We consider this level of pruning as global because
it guides skipping of multiple documents and explores inter-document relationship implied by maximum term weights.
Figure 2(a) depicts an example of global pruning flow in MaxScore with 4 terms and each posting list maintains a
pointer to the current document being visited at a traversal step. The term partitioning identifies two essential terms $t_{3}$ and $t_{4}$.
The minimum document ID among the current document pointers in these essential terms
is $d_{3}$, and any document ID smaller than $d_{3}$ is skipped from further consideration during this traversal step.
The current visitation pointer of the posting list of non-essential lists also moves to the smallest document ID
equal to or bigger than $d_{3}$.
•
Local level.
Once a document is selected for possible full evaluation,
the ranking score upper bound of this document can be estimated and gradually tightened using
maximum weight contribution or the actual weight of each query term for this document.
This incrementally refined score upper bound is compared against the dynamically updated top $k$ threshold,
which provides another opportunity to fully or partially skip the evaluation of this document.
We differentiate this level of skipping decision as local because this pruning is localized towards a specific document selected.
Figure 2(b) illustrates an example of local pruning in MaxScore.
$d_{3}$ is the document selected after term partitioning and the maximum or actual weights contributed from all posting lists for document $d_{3}$
are utilized for the local pruning decision.
Instead of directly using BM25 to guide pruning at the global and local levels,
we propose to use a linear combination of BM25 weights and learned weights to guide skipping at each level as follows,
which allows a parameterizable control of their influence.
[leftmargin=*]
•
We incrementally maintain three accumulated scores for each document $Global(d)$, $Local(d)$, and $RankScore(d)$.
$Global(d)$ is for global pruning, $Local(d)$ is for local pruning, and $RankScore(d)$ is for final ranking.
$$\displaystyle Global(d)$$
$$\displaystyle=\alpha RankScore_{B}(d)+(1-\alpha)RankScore_{L}(d)$$
$$\displaystyle Local(d)$$
$$\displaystyle=\beta RankScore_{B}(d)+(1-\beta)RankScore_{L}(d)$$
$$\displaystyle RankScore(d)$$
$$\displaystyle=\gamma RankScore_{B}(d)+(1-\gamma)RankScore_{L}(d)$$
where $0\leq\alpha,\beta,\gamma\leq 1$,
$RankScore_{B}(d)$ follows Expression 1 using BM25 weights, and
$RankScore_{L}(d)$ follows Expression 1 using learned weights.
The RankScore formula follows the GTI setting in (Mallia et al., 2022), and 2GTI with $\alpha=\beta=1$ behaves like GTI.
2GTI with $\alpha=\beta=\gamma$ is the same as MaxScore retrieval and
it uses learned neural weights only when $\gamma=0$.
•
With the above three scores for each evaluated document, we maintain three separate queues:
$Q_{Gl}$,
$Q_{Lo}$,
$Q_{Rk}$ for documents with the $k$ largest scores in terms of $Global(d)$, $Local(d)$, and $RankScore(d)$ respectively.
The lowest-scoring document in each queue is removed separately without inter-queue coordination.
These queues are maintained for different purposes:
the first two queues regulate global and local pruning while the last queue is to produce the final top $k$ results.
When a document based on local pruning is eliminated for further consideration, this document is not added to global and local queues $Q_{Gl}$ and $Q_{Lo}$.
But this document may have some partial score accumulated for its $RankScore(d)$, and it is still added to $Q_{Rk}$ in case this document with the partial
score may qualify in the top $k$ results based on the latest $RankScore(d)$ value.
These three queues yield three dynamic top-$k$ thresholds $\theta_{Gl}$, $\theta_{Lo}$, and $\theta_{Rk}$.
They can be used for a pruning decision to avoid any further scoring effort to obtain or refine $RankScore(d)$.
Revised MaxScore pruning control flow:
Figure 2(c) illustrates the extra control flow added for the revised MaxScore algorithm.
Let $N$ be the number of query terms. We define:
[leftmargin=*]
•
Given $N$ posting lists corresponding to $N$ query terms,
each $i$-th posting list contains a sequence of posting records and each record contains document ID $d$, its BM25 weight $w_{B}(i,d)$ and learned weight $w_{L}(i,d)$.
Posting records are sorted in an increasing order of their document IDs.
•
An array $\sigma_{L}$ of $N$ where $\sigma_{L}[i]$ is the maximum contribution of the learned weight to any document for $i$-th term.
•
An array $\sigma_{B}$ of $N$ where $\sigma_{B}[i]$ is the maximum contribution of the BM25 weight to any document for $i$-th term.
•
$N$ search terms are presorted so that
$\alpha\sigma_{B}[i]+(1-\alpha)\sigma_{L}[i]\leq\alpha\sigma_{B}[i+1]+(1-\alpha)\sigma_{L}[i+1]$ where $1\leq i\leq N-1$.
Global pruning with term partitioning.
For each query term $1\leq i\leq N$, we find the largest integer $pivot$ from 1 to $N$ so that
$\sum_{j=1}^{pivot-1}(\alpha\sigma_{B}[j]+(1-\alpha)\sigma_{L}[j])\leq\theta_{Gl}$.
All terms from $pivot$ to $N$ are considered as essential.
If a document $d$ does not contain any essential term, the upper bound of
$Global(d)\leq\sum_{j=1}^{pivot-1}\alpha\sigma_{B}[j]+(1-\alpha)\sigma_{L}[j]\leq\theta_{Gl}$.
This document cannot appear in the final top $k$ list based on the global score.
Then this document is skipped without appearing in any of the three queues.
Once the essential term list above the $pivot$ position is determined, let the next minimum document ID among the current position
pointers in the posting lists of all essential terms be document $d$. We also call it the $pivot$ document.
Local pruning. Next we check if the detailed scoring of the selected pivot document $d$ can be avoided fully or partially.
Following an implementation in (Tonellotto et al., 2018), we describe this procedure with a modification to use hybrid scoring as follows
and it repeats the following three steps with the initial value of term position $x$ as the $pivot$ position and $x$ decreases by 1 at each loop iteration.
[leftmargin=*]
•
Let $PartialScore_{Local}(d)$ be the sum of all term weights of document $d$ in the posting lists from position $x$ to $N$ after linear combination.
Namely $PartialScore_{Local}(d)=\sum_{i=x}^{N-1}\beta w_{B}(i,d)+(1-\beta)w_{L}(i,d)$ when $i$-th posting list contains $d$, and otherwise this value is 0.
As $x$ decreases, the term weight of pivot document $d$ is extracted from the posting list of $x$-th term if available.
•
Let $PartialBound_{Local}(d)$ be the bound for partial local score of document $d$ in the posting lists of the first to $x$-th query terms.
$$PartialBound_{Local}(d)=\sum_{j=1}^{x}\beta\sigma_{B}[j]+(1-\beta)\sigma_{L}[j].$$
•
At any time during the above calculation, if
$$PartialBound_{Local}(d)+PartialScore_{Local}(d)\leq\theta_{Lo},$$
further rank scoring for $pivot$ document $d$ is skipped and
this document will not appear in any of the three queues.
Figure 2(b) depicts that the partial bound and partial score of $Local(d_{3})$ for pivot document $d_{3}$
are computed to assist a pruning decision.
Complexity. 2GTI’s complexity is the same as MaxScore and GTI.
The in-memory space cost includes the space
to host the inverted index involved for this query and the three queues.
The time complexity is proportional to the total number of posting records involved for a query
multiplied by $\log k$ for queue updating.
A posting list may be divided and compressed in a block-wise manner and Block MaxScore can use 2GT similarly while
a previous study (Mallia et al., 2019a) shows Block-Max MaxScore is actually slower than MaxScore under several compression schemes.
We will discuss the use of 2GT in block-based BMW in Appendix B.
4.2. Relevance properties of 2GTI
2GTI ensembles BM25 and learned weights for pruning in addition to rank score composition,
producing a top $k$ ranked list which can be different than additive ranking with
learned weights or their linear combination of BM25 weights.
Thus 2GTI is not rank-safe compared to any of such baselines. Two-level pruning is driven by
different combination coefficients $\alpha$, $\beta$, and $\gamma$ configured in 2GTI
and their value gap provides an opportunity for additional pruning while 2GTI tries to retain relevance effectiveness.
Is there a relevance guarantee 2GTI can offer in case such pruning skips relevant documents erroneously sometimes?
To address this question analytically, this subsection
presents three properties regarding the relevance outcome and competitiveness of the 2GTI based retrieval.
Our analysis will use the following terms.
Given query $Q$, let $R_{x}$ be a ranked list of all documents of the given dataset
sorted in a descend order of their rank scores
based on a linear combination of their BM25 weights and learned weights with coefficient $x$,
namely $\sum_{t\in Q}x*w_{B}(t,d)+(1-x)w_{L}(t,d)$ for document $d$.
Specifically, there are three ranked lists: $R_{\alpha}$, $R_{\beta}$, and $R_{\gamma}$.
2GTI maintains 3 queues $Q_{Gl}$, $Q_{Lo}$, and $Q_{Rk}$
with 3 dynamically updated top $k$ thresholds, $\theta_{Gl}$, $\theta_{Lo}$, $\theta_{Rk}$.
Let $\Theta_{Gl}$, $\Theta_{Lo}$, $\Theta_{Rk}$
be the final top $k$ threshold of these 3 queues at the end of 2GTI.
Namely it is the rank score of $k$-th document in the corresponding queue.
The following fact is true:
$$\theta_{Gl}\leq\Theta_{Gl},\ \theta_{Lo}\leq\Theta_{Lo},\ \mbox{ and }\theta_{Rk}\leq\Theta_{Rk}.$$
Proposition 1 ().
Assume the subset of top $k$ documents in each of $R_{\alpha}$,$R_{\beta}$, and $R_{\gamma}$ is unique
after arbitrarily swapping rank positions of documents with the same score.
Then any document that appears in top-$k$ positions of $R_{\alpha}$, $R_{\beta}$, and $R_{\gamma}$ is in the top-$k$ outcome of
2GTI.
Proof.
For any document $d$ that appears in the top $k$ positions of all three ranked lists,
$Global(d)\geq\Theta_{Gl}\geq\theta_{Gl}$, $Local(d)\geq\Theta_{Lo}\geq\theta_{Lo}$ and $RankScore(d)\geq\Theta_{Rk}\geq\theta_{Rk}.$
If document $d$ is eliminated by global pruning during 2GTI retrieval, $Global(d)=\Theta_{Gl}=\theta_{Gl}$
and
the $R_{\alpha}$-based rank score
of both document $d$ and $(k+1)$-th document in ranked list $R_{\alpha}$ has to be $\Theta_{Gl}$.
Then the subset of top $k$ documents in $R_{\alpha}$ is not unique after arbitrarily
swapping rank positions of documents with the same score, which is a contradiction.
With the same reason, we can argue that document $d$ cannot be eliminated by
local pruning or rejected by $\theta_{Rk}$ when being added to $Q_{Rk}$
during 2GTI retrieval.
Then this document has to appear in the final outcome of 2GTI.
∎
The following two propositions analyze when 2GTI performs better in relevance than
a two-stage search algorithm called $R2_{\alpha,\gamma}$
which fetches top $k$ results from list $R_{\alpha}$, and then
re-ranks using the scoring formula of $R_{\gamma}$.
Proposition 2 ().
Assume the subset of top $k$ documents in each of $R_{\alpha}$,$R_{\beta}$, and $R_{\gamma}$ is unique
after arbitrarily swapping rank positions of documents with the same score.
If 2GTI is configured with $\alpha=\beta$ or $\beta=\gamma$,
the average $R_{\gamma}$-based rank score of the top $k$ documents produced
by 2GTI is no less than that of two-stage algorithm
$R2_{\alpha,\gamma}$.
Proof.
We let $R2[k]$ denote the top $k$ document subset in the outcome of $R2_{\alpha,\gamma}$.
To prove this proposition, we compare the average $R_{\gamma}$-based rank score
of documents in $R2[k]$ and that in $Q_{Rk}$ at the end of 2GTI.
Notice that
for any document $d$ satisfying $d\in R2[k]$,
it is in the top $k$ results of ranked list $R_{\alpha}$ and this top $k$ subset is deterministic
based on the assumption of this proposition. Then $d$ cannot be eliminated by global pruning in 2GTI.
Given any document $d$ satisfying $d\in R2[k]$ and $d\not\in Q_{Rk}$ at the end of 2GTI,
it is either eliminated by local pruning with threshold $\Theta_{Lo}$
or by top $k$ thresholding of Queue $Q_{Rk}$ with threshold $\theta_{Rk}$.
In the later case, $RankScore(d)\leq\theta_{Rk}\leq\Theta_{Rk}$.
When $d$ is eliminated by local pruning, global pruning has to use a different formula because $d$ is not eliminated by global pruning,
and then 2GTI has to be configured with $\beta=\gamma$ instead of $\alpha=\beta$.
In that case local pruning is identical to elimination with top $k$ threshold of $Q_{Rk}$.
Then $RankScore(d)\leq\theta_{Rk}\leq\Theta_{Rk}$.
Since the size of both $R2[k]$ and $Q_{Rk}$ is $k$,
$|R2[k]-R2[k]\cap Q_{Rk}|=$ $|Q_{Rk}-R2[k]\cap Q_{Rk}|$. We can derive:
$$\displaystyle\sum_{d\in R2[k]}RankScore(d)$$
$$\displaystyle=$$
$$\displaystyle\sum_{d\in R2[k]\cap Q_{Rk}}RankScore(d)+\sum_{d\in R2[k],d\not\in Q_{Rk}}RankScore(d)$$
$$\displaystyle\leq$$
$$\displaystyle\sum_{d\in R2[k]\cap Q_{Rk}}RankScore(d)+\sum_{d\in R2[k],d\not\in Q_{Rk}}\Theta_{Rk}$$
$$\displaystyle=$$
$$\displaystyle\sum_{d\in R2[k]\cap Q_{Rk}}RankScore(d)+\sum_{d\not\in R2[k],d\in Q_{Rk}}\Theta_{Rk}$$
$$\displaystyle\leq$$
$$\displaystyle\sum_{d\in R2[k]\cap Q_{Rk}}RankScore(d)+\sum_{d\not\in R2[k],d\in Q_{Rk}}RankScore(d)$$
$$\displaystyle=$$
$$\displaystyle\sum_{d\in Q_{Rk}}RankScore(d).$$
Thus
$$\frac{1}{k}\sum_{d\in R2[k]}RankScore(d)\leq\frac{1}{k}\sum_{d\in Q_{Rk}}RankScore(d).$$
∎
Definition 1.
For a dataset in which documents are only labeled relevant or irrelevant
for any test query, we call ranked list $R_{x}$ outmatches $R_{y}$
if whenever $R_{y}$ orders a pair of relevant and irrelevant documents correctly for a query, $R_{x}$ also orders them correctly.
Proposition 3 ().
Assume documents in a dataset are only labeled as relevant or irrelevant for a test query.
Given a query, when $R_{\gamma}$ outmatches $R_{\beta}$, which outmatches $R_{\alpha}$,
2GTI retrieves equal or more relevant documents in top-$k$ positions
than two-stage algorithm $R2_{\alpha,\gamma}$.
Proof.
When 2GTI completes its retrieval for a query,
we count the number of relevant documents in top $k$ positions
of list $R_{\alpha}$,
queue $Q_{Lo}$, and queue $Q_{Rk}$ as $c_{\alpha},c_{\beta}$, and $c_{\gamma}$,
respectively. To show $c_{\alpha}\leq c_{\beta}$,
we initialize them as 0 first and run the following loop to compute $c_{\alpha}$ and $c_{\beta}$ iteratively.
The loop index variable $i$ varies from $k$, $k-1$, until $1$, and at each iteration
we look at document $x$ at Position $i$ of $R_{\alpha}$,
and document $y$ at Position $i$ of $Q_{Lo}$.
Let $L_{x}$ and $L_{y}$ be their binary label by which value 1 means relevant and 0 means irrelevant.
•
If $L_{x}=L_{y}$, we add $L_{x}$ to both $c_{\alpha}$ and $c_{\beta}$. Continue this loop.
•
Now $L_{x}\neq L_{y}$.
If $L_{x}=0$, $L_{y}=1$, we add $1$ to $c_{\beta}$, and continue the loop.
If $L_{x}=1$, $L_{y}=0$, there are two cases:
–
If $x$ is within top $i$ positions of current $Q_{Lo}$,
we add $1$ to both $c_{\alpha}$ and $c_{\beta}$. Swap the positions of documents $x$ and $y$ in $Q_{Lo}$. Continue the loop.
–
If $x$ is not within top $i$ positions of $Q_{Lo}$, since $x$ is in the top
$k$ of $R_{\alpha}$, it cannot be globally pruned and it will be evaluated by 2GTI for a possibility of entering $Q_{Lo}$.
If $x$ is ranked before $y$ in list $R_{\alpha}$, and
since $R_{\beta}$ outmatches $R_{\alpha}$, $x$ has to be ranked before $y$ in both $R_{\beta}$ and $Q_{Lo}$. That is a contradiction.
If $x$ is ranked after $y$ in $R_{\alpha}$, we swap the positions of $x$ and $y$ in $R_{\alpha}$. Continue the loop.
The above process repeats and moves to a higher position until $i=1$.
When $i=1$,
with top-1 document $x$ in
$R_{\alpha}$
and top-1 $y$ in $Q_{Lo}$, the only possible cases are $L_{x}=L_{y}$ or $L_{x}=0$ and $L_{y}=1$.
Therefore, at the end of the above process, $c_{\beta}\geq c_{\alpha}$.
Similarly, we can verify that
$c_{\gamma}\geq c_{\beta}$ since $R_{\gamma}$ outmatches $R_{\beta}$.
Therefore $c_{\gamma}\geq c_{\beta}\geq c_{\alpha}$.
The number of relevant documents up to position $k$ retrieved for 2GTI is $c_{\gamma}$
while the number of relevant documents up to position $k$ retrieved for $R2_{\alpha,\gamma}$
is $c_{\alpha}$. Thus this proposition is true.
∎
The above analysis indicates that the top documents agreed by three rankings $R_{\alpha}$,
$R_{\beta}$, and $R_{\gamma}$ are always kept on the top by 2GTI,
and a properly configured 2GTI algorithm could outperform a two-stage retrieval and re-ranking
algorithm in relevance, especially
when ranking $R_{\gamma}$ outmatches $R_{\beta}$ and $R_{\beta}$ outmatches $R_{\alpha}$ for a query.
Since two-stage search with neural re-ranking conducted after BM25 retrieval is well adopted in the literature,
this analysis provides useful insight into the “worst-case” relevance competitiveness of 2GTI with two-level pruning.
GTI can be considered as
a special case of 2GTI with $\alpha=\beta=1$ when the same index is used, and the above three propositions are true for GTI.
2GTI provides more flexibility in pruning with quality control than GTI and Section 5 further
evaluates their relevance difference.
4.3. Alignment of tokens and weights
The BM25 model is usually built on word-level tokenization on the original or expanded document sets and the popular expansion method uses DocT5Query
with the same tokenization method. When a learned representation uses a different tokenization method such as BERT’s WordPiece based on subwords from BERT vocabulary,
we need to align it with BM25 for a consistent term reference.
For example, when using BM25 to guide the traversal of SPLADE index,
the WordPiece tokenizer is used for a document expanded with DocT5Query before BM25 weighting is applied to each token.
Once tokens are aligned, from the index point of view, the same token has two different posting lists based on BM25 weights and based on SPLADE.
To merge them when postings do not align one-to-one, the missing weight is set to zero as proposed in (Mallia et al., 2022).
We call this zero-filling alignment. As alternatives, we propose two more methods to fill missing weights with better weight smoothness.
[leftmargin=*]
•
One-filling alignment. We assign 1 as term frequency for a missing token in the BM25 model
while this token appears in the learned token list of a document. The justification is that a zero weight is to be too abrupt when such a term is considered to be useful for a document
based on a learned neural model. Having term frequency one means that this token is present in the document, even with the lowest value.
•
Scaled alignment.
This alternative replaces the missing weights in the BM25 model based on a scaled learned score by
using the ratio of mean values of non-zero weights in both models.
For document ID $d$ that contains term $t$, let its BM25 weight be
$w_{B}(t,d)$ and its learned weight be $w_{L}(t,d)$.
Let $w^{*}_{B}(t,d)$ be an adjusted BM25 weight. Set
$P_{B}$ contains all posting records with nonzero BM25 weights. Set $P_{L}$ contains posting records with non-zero learned weights. Then $w_{B}^{*}(t,d)$ is defined as:
$$w_{B}^{*}(t,d)=\left\{\begin{aligned} w_{B}(t,d)&,&w_{B}(t,d)\neq 0,\\
\frac{\sum_{(t^{\prime},d^{\prime})\in P_{B}}w_{B}(t^{\prime},d^{\prime})/|P_{B}|}{\sum_{(t^{\prime},d^{\prime})\in P_{L}}w_{L}(t^{\prime},d^{\prime})/|P_{L}|}w_{L}(t,d)&,&w_{B}(t,d)=0.\end{aligned}\right.$$
5. Evaluations
Datasets and settings.
Our evaluation uses
the MS MARCO document and passage collections (Craswell et al., 2020; Campos et al., 2016),
and 13 publicly available BEIR datasets (Thakur et al., 2021).
The results for the BEIR datasets are described in Appendix A.
For MS MARCO, the contents in the document collections are segmented during indexing and re-grouped after retrieval using “max-passage” strategy following (Lin et al., 2021a).
There are 8.8M passages with an average length of 55 words, and 3.2M documents with an average length of 1131 words before segmentation.
The Dev query set for passage and document ranking has 6980 and 5193 queries respectively with about one judgment label per query.
Each of the passage/document ranking task of TREC Deep Learning (DL) 2019 and 2020 tracks provides
43 and 54 queries respectively with many judgment labels per query.
In producing an inverted index, all words use lower case letters. Following GT, we packed the learned score and the term frequency in the same integer. For DeepImpact, we adopt GT’s index111https://github.com/DI4IR/dual-score directly. The BM25-T5’s index is dumped from the DeepImpact index. Both BM25-T5 and DeepImpact are using natural words tokenization.
SPLADE and uniCOIL
use the BERT’s Word Piece tokenizer. In order to align with them, the BM25-T5-B index reported in the following tables uses the same tokenizer as well.
The impact scores of uniCOIL is obtained from Pyserini (Lin et al., 2021a) 222https://github.com/castorini/pyserini/blob/master/docs/experiments-unicoil.md.
For SPLADE, in order to achieve the best performance, we retrained the model following the setup in SPLADE++ (Formal et al., 2022).
We start from the pretrained model coCondenser (coCondenser, 2021)
and
distill using the sentenceBERT hard negatives 333https://huggingface.co/datasets/sentence-transformers/msmarco-hard-negatives from a cross-encoder
teacher (MiniLM-L-6-v2, 2022)
with MarginMSE loss. For FLOP regularization, we use 0.01 and 0.008 for query and documents respectively. We construct the inverted indexes, convert them to the PISA format,
and compress them using SIMD-BP128 (Lemire and Boytsov, 2015) following (Mallia et al., 2019a, 2022).
Table 1 shows the dataset and index characteristics of the different weighting models
on the MS MARCO Dev dataset.
Following (Mackenzie et al., 2021), we assume that a query can be pre-processed with a ”pseudo-document” trick that
assigns custom weights to query terms in uniCOIL and SPLADE.
Therefore, there may be token repetition in each query to reflect token weighting.
Column 1 is the mean query length in tokens without or with counting duplicates.
Column 3 is the inverted index size while the last column is the size after merging BM25 and learned weights in
the index.
The C++ implementation of 2GTI with the modified MaxScore and VBMW algorithms
are embedded in PISA (Mallia et al., 2019b), and the code will be released in https://github.com/Qiaoyf96/2GTI.
Our evaluation using this implementation
runs as a single thread on a Linux server with Intel i5-8259U 2.3GHz and 32GB memory.
Weights are chosen by sampling queries from the MS MARCO training dataset.
Metrics.
For MS MARCO Dev set, we report the relevance in terms of
mean reciprocal rank (MRR@10 on passages and MRR@100 on documents), following the official leader-board standard.
We also report the recall@k ratio which is the percentage of relevant-labeled results appeared in the final top-$k$ results.
For TREC DL test sets, we report normalized discounted cumulative
gain (nDCG@10) (Järvelin and Kekäläinen, 2002). The above reporting follows the common practice of the previous
work (e.g. (Mallia et al., 2021; Gao et al., 2021a, b; Formal et al., 2021a)).
Before timing queries, all compressed posting lists and metadata for tested queries are pre-loaded into memory,
following the same assumption in (Khattab et al., 2020; Mallia et al., 2017).
Retrieval mean response times (MRT) are reported in milliseconds.
The 99th percentile time ($P_{99}$) is reported within parentheses in the tables below,
corresponding to the time occurring in the 99th percentile denoted as tail latency in (Mackenzie et al., 2018).
Statistical significance.
For the reported numbers on MS MARCO passage and document Dev sets in the rest of this section,
we have performed a pairwise t-test on relevance difference between
2GTI and a GTI baseline, and between 2GTI and the original learned sparse retrieval without BM25 guidance.
No statistically significant degradation has been observed at the 95% confidence level.
We have also performed a pairwise t-test comparing the reported relevance numbers of 2GTI and GTI and
mark ‘${}^{\dagger}$’ in the evaluation tables if there is a statistically significant improvement by 2GTI over GTI at the
95% confidence level.
We do not perform a t-test on DL’19 and DL’20 query sets as the number of queries in these sets is small.
Overall results with MS MACRO.
Table 2 lists a comparison of 2GTI with the baseline
using three sparse representations for retrieval on MS MARCO and TREC DL datasets.
2GTI uses scaled filling alignment as default while GTI uses zero filling as specified in (Mallia et al., 2022).
The $\gamma$ value is chosen the same for GTI and 2GTI for each representation, which is the best for most of cases.
The “accurate” configuration denotes the one that reaches the highest relevance score.
The “fast” configuration denotes the one that reaches a relevance score within 1% of the accurate configuration
while being much more faster.
2GTI vs. GTI in SPLADE++. Table 2 shows
2GTI with default scaled filling significantly outperforms GTI with default zero filling for
SPLADE++, where BM25 index is not well aligned.
“SPLADE++-Org” denotes the original MaxScore retrieval performance using SPLADE++ model trained by ourselves
and its MRR@10 number is higher than what has been reported in (Formal et al., 2022).
When $k=1,000$, GT is slightly better than GTI,
and with the fast configuration, MRR@10 of 2GTI is 32.4% higher than that of GT while 2GTI is 7.8x faster than GT for the Dev set.
The significant increase in nDCG@10 and decrease in the MRT are also observed in DL’19 and DL’20.
When $k=10$, there is also a large relevance increase and time reduction
from GTI or GT to 2GTI for all three test sets.
For example, the relevance is 46.4% or 44.6% higher and the mean latency is 5.2x or 5.3x faster for the Dev set.
Compared to the original MaxScore method,
2GTI has about the same relevance score for both $k=10$ and $k=1,000$
while having much smaller latency. For example, 6.5x reduction (278ms vs. 43.1ms) for the Dev passage set when $k=1,000$
and 5.3x reduction when $k=10$ (121ms vs 22.7ms) with the 2GTI-fast configuration.
2GTI vs. GTI in DeepImpact and uniCOIL. As shown in Table 2,
GTI (or GT) performs
very well for $k=1,000$ in both DeepImpact and uniCOIL in speeding up retrieval while maintaining a relevance similar as the original retrieval.
The two-level differentiation for dynamic index pruning does not improve relevance or shorten retrieval time.
This can be explained as BM25-T5 index is well aligned with the DeepImpact index and with the uniCOIL index.
Also because of this reason, filling to address index alignment is not needed with no improvement in these two cases.
When $k$ decreases from 1,000 to 10, as shown in Figure 1 discussed in Section 3,
the recall ratio starts to drop, and relevance effectiveness degrades.
When $k=10$ as shown in Table 2, DeepImpact-2GTI-fast
can increase MRR@10 from 0.3375 by GTI to 0.3395
for the Dev set and deliver slightly higher MRR@10 or nDCG@10 scores than GTI in DL’19 and DL’19 sets.
For uniCOIL, 2GTI-fast increases MRR@10 from 0.3384 by GTI to 0.3548 for the Dev set and increases nDCG@10 from 0.6959 to 0.7135 for DL’19.
There is also a modest relevance increase for DL’20 passages with $k=10$ and a similar trend is observed for the document retrieval task.
The price paid for 2GTI is its retrieval latency increase while its latency is still much smaller than the original retrieval time.
Design options
with weight alignment and threshold over-estimation.
Table 3
examines the impact of weight alignment and a design alternative based on threshold over-estimation
for MS MARCO passage Dev set using SPLADE++ when $k=10$.
In the top portion of this table,
threshold over-estimation by a factor of $F$ (1.1, 1.3, and 1.5)
is used in the original retrieval algorithm without BM25 guidance, and these factor choices are similar as ones in
(Macdonald et al., 2012; Tonellotto et al., 2013; Crane et al., 2017).
That essentially sets $\alpha=0$, $\beta=0$, and $\gamma=0$ while multiplying $\theta_{Gl}$ and $\theta_{Lo}$ by the above factor in 2GTI.
The result shows that even threshold over-estimation can reduce the retrieval time, relevance reduction is significant, meaning
that the aggressive threshold used causes incorrect dropping of some desired documents.
The second portion of Table 3 examines the impact of
different weight filling methods described in Section 4.3 for alignment
when they are applied to GTI and 2GTI, respectively.
In both cases, scaled filling marked as “/s”
is most effective while one-filling marked as “/1” outperforms zero-filling marked as “/0” also.
The MRT of 2GTI/s becomes 10.5x smaller than 2GTI/0
while there is no negative impact to its MRR@10.
The MRT of GTI/s is about 13.0x smaller than GTI/0 while there is a large MRR@10 number increase.
A validation on 2GTI’s properties.
To corroborate the competitiveness analysis in Section 4.2,
Table 4 gives MRR@10 scores and retrieval times in milliseconds
of the algorithms with different configurations
on the Dev set of MS MARCO passages with $k=10$ and SPLADE++ weights.
The result shows that the listed configurations of 2GTI
have a higher MRR@10 number than
2-stage search $R2_{\alpha,\gamma}$, and
2GTI with $\alpha=\beta=1$ that behaves as GTI.
MRR@10 of ranking with a simple linear combination of BM25 and learned weights
is only slightly higher than 2GTI, but it is much slower.
Sensitivity on weight distribution.
We have distorted the SPLADE++ weight distribution in several ways to examine the sensitivity of 2GTI
and found that 2GTI is still effective.
For example, we apply a square root function to the neural weight of every token in MS MARCO passages,
the relevance score of both original retrieval and 2GTI drops to 0.356 MRR@10 due to weight distortion,
while 2GTI is 5.0x faster than the original MaxScore when $k=10$.
Efficient SPLADE model.
Table 5 shows the application of 2GTI in a recently published
efficient SPLADE model (Lassance and Clinchant, 2022) which has made several improvements in retrieval speed.
We have used the released checkpoint of this efficient model called BT-SPLADE-L,
which has a weaker MRR@10 score, but significantly faster than our trained SPLADE baseline reported in Table 2.
When used with this new SPLADE model,
2GTI/s-Fast version results in a 2.2x retrieval time
speedup over MaxScore. Its MRR@10 is higher than GTI/s and
has less than 1% degradation compared to the original MaxScore.
6. Concluding Remarks
The contribution of this paper is a two-level parameterized guidance scheme with index alignment to optimize
retrieval traversal with a learned sparse representation.
Our formal analysis shows that a properly configured 2GTI algorithm including GTI
can outperform a two-stage retrieval and re-ranking algorithm in relevance.
Our evaluation
shows that the proposed 2GTI scheme can make the BM25 pruning guidance more accurate
to retain the relevance.
For MaxScore with SPLADE++ on MS MARCO passages, 2GTI can lift relevance by up-to 32.4% and is 7.8x faster than GTI when $k=1,000$,
and by up-to 46.4% more accurate and 5.2x faster when $k=10$.
In all evaluated cases, 2GTI is much faster than the original retrieval without BM25 guidance.
For example, up-to 6.5x faster than MaxScore on SPLADE++ when $k=10$.
We have also observed similar performance patterns on BEIR datasets when comparing 2GTI with GTI and
the original MaxScore using SPLADE++ learned weights.
Compared to other options such as threshold underestimation to reduce the influence of BM25 weights,
the two-level control is more accurate in maintaining the strong relevance with a much lower time cost.
While our study is mainly centered with MaxScore-based retrieval,
2GTI can be used for VBMW and our evaluation shows that VBMW-2GTI can be a preferred choice for
a class of short queries without stop words when $k$ is small.
Acknowledgments. We thank Wentai Xie, Xiyue Wang, Tianbo Xiong, and anonymous referees
for their valuable comments and/or help.
This work is supported in part by NSF IIS-2225942
and has used the computing resource of the XSEDE and ACCESS programs supported by NSF.
Any opinions, findings, conclusions or recommendations expressed in this material
are those of the authors and do not necessarily reflect the views of the NSF.
Appendix A Additional Evaluation Results
Impact of $\alpha$ and $\beta$ adjustment on 2GTI.
Figure 3 examines the impact of adjusting parameters $\alpha$ and
$\beta$ on global and local pruning for the MS MARCO Dev passage test set when $k=10$
in controlling the influence of BM25 weights
for SPLADE++ (left)
and uniCOIL (right).
The $x$ axis corresponds to the latency increase
while $y$ axis corresponds to the MRR@10 or nDCG@10 increase.
The results for MS MARCO DL’19, and DL’20 are similar.
The red curve connected with dots fixes $\beta=1$ and varies $\alpha$ from 1 at the left end to 0 at the right end.
As $\alpha$ decreases from 1 to 0, the latency increases because BM25 influences diminish at the global pruning level and
fewer documents are skipped. The relevance for this curve is relatively flat in general
and lower than that of the blue curve, representing the global level BM25 guidance reduces time significantly, while having less impact on the relevance.
The blue curve connected with squares fixes $\alpha=1$ at the global level and varies $\beta$ from 1 at the left bottom end to 0 at the right top end.
Decreasing $\beta$ value is positive in general for relevance towards some point as BM25 influence decreases gradually at the local level
and after such a point, the relevance gain becomes much smaller or negative. For example, after $\beta$ in the blue curve in SPLADE++
becomes 0.3 for the Dev set, its additional decrease does not lift MRR@10 visibly anymore while the latency continues to increase, which
indicates the relevance benefit has reached the peak at that point.
Our experience with the tested datasets is that the parameter setting for 2GTI can reach a relevance peak typically when $\alpha$ is close to 1 and $\beta$ varies between 0.3 and 1.
Note that even the above result advocates that
$\alpha$ is close to 1, $\alpha$ and $\beta$ still have different values to be more effective for the tested data,
reflecting the usefulness
of two-level pruning control.
Threshold under-estimation.
In Figure 3, the brown curve connected with triangles fixes $\alpha=\beta=1$
and under-estimates the skipping threshold by a factor of $F$ at the local and global levels.
That behaves like GTI coupled with scaled weight filling as a special case of 2GTI.
$F$ varies from 1 at the left bottom end to 0.7 at the right top end of this brown curve.
As $F$ decreases, the skipping threshold becomes very loose and there is less chance that desired documents are skipped.
Then retrieval relevance can improve while retrieval time can increase substantially. Comparing with the blue curve that adjusts $\beta$,
retrieval takes a much longer time in the brown curve to reach the peak relevance, as shown in this figure, and
the brown curve is generally placed on the right side of the blue curve. For example on the Dev set with uniCOIL,
the brown curve with threshold under-estimation reaches the best relevance at mean latency 3.7ms
while the blue curve with $\beta$ adjustment reaches the same peak at mean latency 2.3ms, which is 1.6x faster.
Zero-shot performance on the BEIR datasets.
We evaluate the zero-shot ranking effectiveness and response time of 2GTI using the 13 search and semantic relatedness
datasets from the BEIR
collection.
Our training of SPLADE++ model is only based on MS MARCO data without using any BEIR data.
Table 6 lists the nDCG@10 scores
of original MaxScore on SPLADE++,
2GTI/s-Fast ($\alpha$=1, $\beta$=0.3, $\gamma$=0.05)
and GTI ($\alpha$=$\beta$=1, $\gamma$=0.05).
The retrieval depth is $k=10$ and $k=1000$.
This table also reports mean response time of retrieval in milliseconds.
The SPLADE++ model trained by ourself has an average nDCG@10 score 0.500
close to 0.507 reported in the SPLADE++ paper (Formal et al., 2022).
The original MaxScore’s nDCG@10 score does not change when $k=10$ and $k=1000$.
When $k=10$, 2GTI has almost identical nDCG@10 scores as the original MaxScore
while 2GTI is on average 2.0x faster than MaxScore for these BEIR datasets.
When GTI runs on the same index data, its average nDCG@10 score is 0.43 MRR@10 and
it is faster than 2GTI
with an average 6.1x speedup over the original MaxScore for these datasets.
Two-level pruning in 2GTI can preserve relevance better than GTI and
this is consistent with what we have observed for searching MS MARCO passages.
When $k=1000$, the guided traversal algorithms have a better chance to retain relevance.
2GTI has a slightly higher average relevance of 0.501 MRR@10 than that with $k=10$
and it is about 2.5x faster on average than the original MaxScore.
For GTI running on the same index with the same alignment,
the average MRR@10 is 0.496 whil average speedup 2.7x over MaxScore.
Its relevance score is close to that of 2GTI as BM25-driven pruning under a large $k$ value can
still keep a good recall ratio.
Appendix B Two-level guidance for BMW
Two-level guidance can be adopted to control index traversal of a BMW based algorithm such as VBMW as well
because we can also view that such an algorithm conducts a sequence of index traversal steps,
and can differentiate its index pruning of each traversal step at the global inter-document and local intra-document levels.
We use the same symbol notations as in the previous subsection,
assuming the posting lists are sorted by an increasing order of their document IDs.
We still keep a position pointer in each posting list of search terms to track the current document ID $d_{t_{i}}$ being handled for each term $t_{i}$,
incrementally accumulate three scores $Global(d)$, $Local(d)$, and $RankScore(d)$ for each document $d$ visited,
and maintain three separate score-sorted queues $Q_{Gl}$, $Q_{Lo}$, and $Q_{Rk}$.
[leftmargin=*]
•
Pruning at the global inter-document level with pivot identification.
BMW (Ding and Suel, 2011) keeps a sorted search term list in each traversal step so that
$d_{t_{i}}\leq d_{t_{i+1}}$ with $1\leq i\leq N-1$.
The pivot position that partitions these current document pointers is the smallest integer called $pivot$ such
that
$\sum_{i=1}^{pivot}\alpha\sigma_{B}[i]+(1-\alpha)\sigma_{L}[i]>\theta_{Gl}.$
This inequality means that any document ID $d$ where $d_{t_{0}}\leq d<d_{t_{pivot}}$ does not qualify for being
in the final top $k$ list based on score $Global(d)$.
Then with the above pivot detection, for $1\leq i<pivot$,
the current visitation pointer of the $i$-th posting list moves to the closest block that
contains a document ID equal to or bigger than $d_{t_{pivot}}$.
Figure 4 illustrates an example of global pruning in BMW with 4 terms and each posting list maintains a
pointer to the current document being visited at a traversal step. Documents in each posting list are covered by a curved rectangle,
representing these lists are stored and compressed in a block-wise manner.
In the figure, the pivot identification at one traversal step locates document $d_{3}$, and document IDs smaller than $d_{3}$ are skipped for any further consideration in
this traversal step.
•
Local pruning. Let $d$ be the corresponding pivot document in pivot term $t_{pivot}$.
In Figure 4, pivot term $t_{pivot}=t_{3}$ and $d=d_{3}$.
A traversal procedure is executed to check if detailed scoring of document $d$
can be avoided fully or partially and this procedure can be similar as the one in the revised MaxScore algorithm described earlier.
As each posting list is packaged in a block manner in BMW, let $\Delta_{B}[x]$ and $\Delta_{L}[x]$ be the
BM25 and learned block maximum weights of the block in the $x$-th posting list that contains $d$, respectively, and they are 0 if no such a block exists in this list.
The upper bound of $Local(d)$ can be tightened using the block-wise maximum weight instead of the list-wise maximum weight contributed by
each term as:
$\sum_{i=1}^{N}\beta\Delta_{B}[i]+(1-\beta)\Delta_{L}[i].$
When decompressing the needed block of a posting list,
the block-max contribution from the corresponding term in the above expression can be replaced by the actual BM25 and learned weights for
document $d$. Then the upper bound of $Local(d)$ is further tightened, which can be directly compared with $\theta_{Lo}$ after every downward adjustment.
Evaluations on effectiveness of 2GTI on VBMW.
We choose uniCOIL to study the usefulness of VBMW-2GTI in searching the MS MARCO Dev set.
SPLADE++ is not chosen because the test queries
are long on average and MaxScore is faster than VBMW for such queries.
Table 7 reports the performance for VBMW-2GTI, VBMW-GTI,
and MaxScore-2GTI for passage retrieval with uniCOIL when varying $k$.
Each entry has a report format of $x,y(z)$ where $x$ is MRR@10 for Dev or NDCG@10 for DL’19 and DL’20.
$y$ is the MRT in ms, and $z$ is the $P_{99}$ latency in ms.
2GTI uses the fast setting with $\alpha=1$, $\beta=0.3$. For both 2GTI and GTI, $\gamma=0.1$.
The result shows 2GTI provides a positive boost in relevance for VBMW compared to GTI when $k$ is 10 and 20.
For $k=100$, the relevance difference is negligible.
MaxScore-2GTI is still faster than VBMW-2GTI on average for all tested queries while their relevance difference is small.
we examine below if VBMW-2GTI can be useful for a subset of queries.
Table 8 reports the relevance and time of these three algorithms
in the passage Dev set for queries subdivided based on their lengths and if a query contains a stop word or not.
That is for uniCOIL with $k=10$, $\alpha=1$, $\beta=0.3$, and $\gamma=0.1$.
Each entry has the same report format as in Table 7.
The result shows that
VBMW-2GTI is much faster than MaxScore-2GTI for short queries ($k\leq 5$) that do not contain stop words and
VBMW-2GTI has an edge in relevance over VBMW-GTI while being very close to MaxScore-2GTI for this class of queries.
The above result suggests that a fusion method can do well by
switching the algorithm choice based on query characteristics and VBMW-2GTI can be used for a class of queries.
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[
[
Abstract
Planar Navier-Stokes Equations; Vorticity; Stream Function; Non-linearity; Laminar Flow; Transition; Turbulence; Diffusion
In this technical note, we demonstrate the robustness of our numerical scheme of vorticity iteration in dealing with the dipole-wall interaction at small viscosity, with emphasis on mesh convergence, boundary vorticity as well as wall viscous dissipation. In particular, it is found that, among the four different dipole configurations, the processes of vortex-wall collision at no-slip surfaces are exceedingly complex and are functions of the initial conditions. The critical issue direct numerical simulations is to establish mesh convergence which appears to be case-dependent. Roughly speaking, converged $2D$ meshes are found to be inversely proportional to viscosity at uniform spacings. Essentially, we have ruled out the possibility of anomalous energy dissipation in the limit of small viscosity. Our computational results show that the rate of the energy degradation follows the predictive trend of the well-known Prandtl scaling.
Planar Navier-Stokes Equations in Bounded Domain]Viscous flow regimes in a square. Part 2. Impact and rebound process of vortex dipole-wall interaction
F. Lam]F. Lam
1 Background
The computation of viscous flows in the presence of solid surfaces is a delicate matter in numerical analysis. Consider the equations of motion of incompressible flow in $2$ space dimensions:
$$\nabla.{\bf u}=0,\;\;\;\partial_{t}{\bf u}+\nu\nabla{\times}\zeta=-({\bf u}.%
\nabla){\bf u}-\nabla p,$$
(1)
where ${\bf u}{=}(u,v)$ denotes the velocity, and $p$ the pressure (unit density) which are treated as a continuum. All symbols have their usual meanings in fluid dynamics. In the vorticity-stream function formulation, the dynamics is described by
$$\Delta\psi=-\zeta,\;\;\;\partial_{t}\zeta-\nu\Delta\zeta=-{\partial_{y}\psi}\;%
{\partial_{x}\zeta}+{\partial_{x}\psi}\;{\partial_{y}\zeta}.$$
(2)
The no-slip condition ${\bf u}_{{\partial\Sigma}}=0$ applies for $t\geq 0$ on the four sides of the unit square, and this boundary condition implies
$$\psi_{{\partial\Sigma}}=0.$$
Once the stream function $\psi$ is calculated for known $\zeta$, the velocity is recovered as $(u,v)=(\partial_{y}\psi,\;-\partial_{x}\psi)$. We are mainly interested in the transient Navier-Stokes dynamics from given initial solenoidal data ${\bf u}_{0}$ or in terms of vorticity $\zeta_{0}=\nabla{\times}{\bf u}_{0}$. At $t=0$, the solenoidal ${\bf u}_{0}$ may also be recovered from $\psi_{0}$ or $\zeta_{0}$.
The pressure Poisson equation is solved to obtain the pressure gradients
$$\Delta p=2\>\big{(}\partial_{x}u\>\partial_{y}v-\partial_{y}u\>\partial_{x}v%
\big{)},$$
(3)
subject to the Neumann boundary conditions, $\partial_{x}p$ and $\partial_{y}p$, which are obtained from (1) for known vorticity and velocity for $t>0$.
In practice, the pressure at the start $t=0$ is somehow unspecified because $\partial_{t}{\bf u}_{0}$ is not available (unless assumed otherwise). Nevertheless, the initial data may be mathematically assigned as a step function ${\bf u}=0,\;t<0$, and ${\bf u}={\bf u}_{0},\;t\geq 0$. In this theoretical setting, the initial pressure gradients may be fixed in terms of generalised functions according to the momentum equations.
It is known that the vorticity evolves in a self-contained manner. Our numerical procedure is to determine the fixed-point solutions ($\zeta,{\bf u}$) at given time $t$. This can be done efficiently by an iteration procedure (Lam 2018). To solve the vorticity dynamics numerically, the unit square is subdivided into equally-spaced grids, denoted by $n$, and the grid points by ($i,j$). We use the implicit Euler scheme for time discretisation and a semi-implicit scheme for the non-linear term. Let $k$ denote the time step. The discretised vorticity matrix (size $n^{2}$) is iterated until the error difference satisfies a prescribed convergence criterion
$$\delta\Pi=\sum_{ij}\Big{|}\;\Pi^{k+1}-\Pi^{k}\;\Big{|}<\epsilon.$$
(4)
The difference $\delta\Pi$ may be scaled by the previous error size if $|\Pi^{k}|>1$. Throughout the present calculations, we set the tolerance $\epsilon=10^{-8}$. As a general tool, no symmetry conditions have been imposed in our implementation. In what follows we will make an effort to examine the problem of mesh convergence with attention to the simulations of dipole-wall head-on impact. We will hence investigate how flow energy is redistributed and dissipated at different viscosities.
2 Energy dissipation
The energy and enstrophy are defined by
$$E(t)=\frac{1}{2}\int_{{\Sigma}}\big{(}u^{2}+v^{2}\big{)}({\bf x},t)\;\rd{\bf x%
},\;\;\;\mbox{and}\;\;\;\Omega(t)=\int_{{\Sigma}}\zeta^{2}({\bf x},t)\;\rd{\bf
x},$$
respectively. The momentum equation (1) gives the rate of the energy dissipation
$$\frac{\rd E}{\rd t}=-\nu\>\Omega.$$
(5)
Thus the principle of energy conservation is expressed in
$$E(t)+\nu\int_{0}^{t}\!\Omega(\tau)\>\rd\tau=E(0).$$
(6)
Because of the term $\nabla{\times}\zeta$ in (1), we must examine the function palinstropgy
$$Z(t)=\int_{{\Sigma}}\Big{(}\>\big{(}\partial_{x}\zeta\big{)}^{2}+\big{(}%
\partial_{y}\zeta\big{)}^{2}\>\Big{)}({\bf x},t)\;\rd{\bf x},$$
(7)
as it is related to the rate of change in the enstrophy
$$\frac{\rd\Omega}{\rd t}=-2\nu\>Z+2\nu\>\Big{(}\big{[}\zeta\>\zeta_{y}\big{]}_{%
x=0,1}+\big{[}\zeta\>\zeta_{x}\big{]}_{y=0,1}\Big{)},$$
(8)
where the square brackets in the last term refer to the wall values. In general, the size of $Z$ is much larger than the last term so that the enstrophy is being consumed by viscous effects over flow evolution.
An anomalous energy dissipation is a mathematical argument which conjectures the rate of energy dissipation $\rd E/\rd t$ would be independent of viscosity $\nu$ when viscosity becomes vanishingly small. In particular, the anomaly is assumed to occur in thin boundary layers in the vicinity of solid surfaces. In other words, should there exist a flow in which $\Omega\propto\nu^{-1}$, the energy of the flow remains to be dissipated and thus is an inviscid process. There has been an expectation, largely unjustified, that the Euler equations ($\nu=0$ in (1)) are capable of describing fluid motions, even for turbulence.
The anomaly hypothesis is established on the agreement of the energy norm between viscous and inviscid equations in the limit $\nu\rightarrow 0$. The mathematical analyses of a possible anomalous dissipation might be valid over short time intervals. There are no particular physical reasons which categorically exclude such momentary instances; it is a matter of the detailed local vorticity dynamics. In a flow having an arbitrary initial condition, the conservation law (6) does not say that the energy inside the square must continuously decrease from the start $t=0$. In fact, vorticity theory and numerical experiment suggest that the critical quantity is not the energy but the di-vorticity. Whether the conjecture is genuine or not depends on demonstrating, at least, both $\Omega(t)\;and\;Z(t)\propto\nu^{-1}$ over a sufficiently long period of time.
3 Dipole topology and dynamics
For the present demanding problem of dipole-wall interaction, the vorticity field will undergo drastic changes over $\Delta t$. The errors of the incorrectly-imposed wall vorticity will contaminate the numerical solutions. For the incompressible flows, many previous studies show that numerical meshes $\sim O(1/\nu)$ have to be used in order to properly resolve various fine-scale motions. Such stringent requirements seem method-independent. For instance, Clerxc & Bruneau (2006) made use of a pseudo-spectral scheme and a finite difference approximation to model dipole collisions. They have found that mode-convergence could hardly be achieved by coarse grids even at low Reynolds numbers of few thousands. Since the solution of the vorticity equation is unique and regular, we have to accept the reality that fairly dense meshes are a must for satisfactory numerical simulations of fluids.
Twin-core dipoles
A localised shear concentration at the centre of the square is given by
$$\zeta_{0}({\bf x})=325\>\Big{(}\>\re^{-r_{1}^{2}}\;\big{(}1-r_{1}^{2}/b^{2}%
\big{)}-\;\re^{-r_{2}^{2}}\big{(}1-r_{2}^{2}/b^{2}\big{)}\>\Big{)},$$
(9)
where $r_{1}=(2x{-}1{-}a)^{2}+(2y{-}1)^{2}$, $r_{2}=(2x{-}1{+}a)^{2}+(2y{-}1)^{2}$, $a=b=0.05$, and $0\leq x,y\leq 1$ (see figure 1). The constant $325$ is chosen so that the initial energy $E_{0}\approx 1$ though this choice is not essential.
Our initial vorticity dipole is derived from that of Clercx & Bruneau (2006) or Kramer et al. (2007)
$$\zeta_{0}({\bf x})=300\>\Big{(}\>\re^{-s_{1}^{2}}\;\big{(}1-s_{1}^{2}/0.01\big%
{)}-\;\re^{-s_{2}^{2}}\big{(}1-s_{2}^{2}/0.01\big{)}\>\Big{)},$$
(10)
where $s_{1}^{2}=(x-0.1)^{2}+y^{2}$, $s_{2}^{2}=(x+0.1)^{2}+y^{2}$, for $-1\leq x,y\leq 1$ (figure 2).
The initial downward velocity at the core is greater than that of the preceding case.
Kirchhoff pair
There are other simple functions that define vortex pairs. For example, the following algebraic expression resembles a couple of Kirchhoff vortices:
$$\zeta_{0}({\bf x})=\exp(-r)\>\Big{(}\>\frac{x_{f}}{\kappa{+}(x_{f}{-}1/4)^{2}{%
+}y_{f}^{2}/2}+\frac{x_{f}}{\kappa{+}(x_{f}{+}1/4)^{2}{+}y_{f}^{2}/2}\>\Big{)},$$
(11)
where $\kappa=10^{-4}$, $x_{f}=4x{-}2$, $y_{f}=4y{-}2$, and $r=x_{f}^{2}+y_{f}^{2}$, see figure 3.
Lamb dipole
The following exponential function gives a Lamb-dipole
$$\zeta_{0}({\bf x})=(2\pi)^{4}\>r^{2}\>\exp\big{(}-2\pi\>r^{2}\big{)}\>\cos(2\theta)$$
(12)
where $\theta=\tan^{-1}(y_{f}/x_{f})$, see figure 4. The rotational symmetry parameter in the cosine function determines the number of vortex pairs.
4 Discussion and outlook
For the dipoles having certain symmetry, it is useful to examine the circulation inside the square
$$\Gamma(t)=\int_{{\Sigma}}\zeta({\bf x},t)\;\rd{\bf x}=\int_{{\partial\Sigma}}%
\big{(}\;u\>\rd x+v\>\rd y\;\big{)}=0.$$
There are no boundary conditions for the vorticity. As a definite routine, we choose to specify an arbitrary initial vorticity which may not produce a compatible velocity field. Thus the iterative method is first to solve the Poisson equation $\Delta\psi_{0}=-\zeta_{0}$ subject to the no-slip condition. Figure 5 illustrates how quickly the compatible solutions can be found. Within the viscosity range of the present note, the time increment $\Delta t$ is usually chosen as $O(10^{-4}\sim 10^{-5})$, then the numerical method finds the solenoidal velocity within $5$ to $10$ iterations.
Our numerical procedures are first validated against the results of Clercx & Bruneau (2006) (see figure 2). In figure 6, we show one example that demonstrates the convergence of our iterative procedures. The results of mesh convergence and small-scale flow fields are given in figure 7 to figure 11. The flow developments of dipole data (10) are shown in figure 12 for $Re=2500$. Tables 1 and 2 list the numerical values of the comparison.
The Kirchhoff dipole of simple algebraic description (11) undergoes a much milder collision, see figure 13 and figure 14. The evolution of the Lamb dipole (12) is summarised in figure 15 and figure 16.
As our revised initial dipole (9) has a modest initial strength, its evolution is relatively easy to compute. We present a set of snap-shots in figure 17 at $\nu=10^{-4}$. Figure 18 displays the variations of the integral quantities over time. It is hardly surprising to see the complexity of the detailed flow solution (figure 19). Figure 20 summarises the characters of the energy dissipation. Briefly, the conclusion of the present calculations is in line with the opinions of Sutherland et al. (2013).
The numerical formulation of the vorticity and stream-function has been further validated. We find that the iterative method is robust in the simulations of vortex-wall interaction. In practice, it is essential to ensure adequate temporal and spatial resolutions of the flow fields so as to avoid spurious solutions.
We must admit that the notion of anomalous energy dissipation is obscured in physics.
In brief, viscous effects, as dominated by fluids’ microscopic structures, instigate vorticity gradients that are responsible for the energy degradation. There are other successful continuum systems, such as diffusion and heat transfer, that are formulated on the macroscopic scale on the premise of averaged microscopic contributions.
The present solutions of vorticity dynamics clearly show that the unsteady separation of viscous-layer from the walls of the square is a consequence of the non-linearity in the full Navier-Stokes equation, characterised by $({\bf u}.\nabla)\zeta$. Specifically, the pressure merely plays an auxiliary role in the dynamics. Large vorticity gradients can exist not only within the wall layers but also in the regions away square’s boundaries. The integral (7) effects on di-vorticity $\nabla\zeta$ of the attached as well as the separated shears. With the advent of modern computational techniques, we should avoid the use of the boundary layer approximations. Versatile numerical solutions are bound to be instrumental to our understanding of complex flows at arbitrarily small viscosity.
References
[1]
Clercx, H.J.H. & Bruneau, C.H. 2006 The normal and oblique collision of a dipole with a no-slip boundary. Computers & Fluids 35, 245â279.
[2]
Kramer, W., Clercx, H.J.H. & van Heijst, G.J.F. 2007 Vorticity dynamics of a dipole colliding with a no-slip wall. Phys. Fluids 19, 126603.
[3]
Lam, F. 2018 Viscous flow regimes in a square. arXiv:1804.04041v1.
[4]
Orlandi, P. 1990 Vortex dipole rebound from a wall. Phys. Fluids A 2(8), 1429-1436.
[5]
Sutherland, D., Macaskill, C. & Dritschel, D.G. 2013 The effect of slip length on vortex rebound from a rigid boundary. Phys. Fluids 25, 093104.
Acknowledgements.
25 April 2018
[email protected] |
Failure of $n$-uniqueness: a family of examples
Elisabetta Pastori
Dipartimento di Matematica,
Università degli Studi di Torino,
Via Carlo Alberto, 10 10123
Torino, Italy
[email protected]
Pablo Spiga
Centre for Mathematics of Symmetry and Computation
The University of Western Australia
35 Stirling Highway
Crawley WA 6009
Australia
[email protected]
()
Abstract
In this paper, the connections between model theory and the
theory of infinite permutation groups (see [11]) are used to study the
$n$-existence and the $n$-uniqueness for
$n$-amalgamation problems of stable
theories. We show that, for any
$n\geq 2$, there exists a stable theory having $(k+1)$-existence
and $k$-uniqueness, for every $k\leq n$, but has neither
$(n+2)$-existence nor $(n+1)$-uniqueness.
In particular, this generalizes the example, for $n=2$, due to
E.Hrushovski given
in [3].
1 Introduction
Considerable work (e.g. [1], [3], [4], [9],
[13]) has explored higher amalgamation properties for stable and
simple theories. In this paper we analyze uniqueness and existence
properties for a countable family of stable theories. In contrast to
previous methods our approach uses
group-theoretic techniques. We begin by giving some basic
definitions.
Let $T$ be a complete and simple $L$-theory with quantifier
elimination. We denote by $\mathcal{C}_{T}$ the category of algebraically
closed substructures of models of $T$ with embeddings as
morphisms. Also, given $n\in\mathbb{N}$, we denote by $P(n)$ the
partially ordered
set of all subsets of $\{1,\dots,n\}$ and by
$P(n)^{-}$ the set $P(n)\setminus\{1,\dots,n\}$.
An $n$-amalgamation problem over
$\operatorname{acl}(\emptyset)$ is a functor $a:P(n)^{-}\rightarrow\mathcal{C}_{T}$ such
that
$(i)$
$a(\emptyset)=\operatorname{acl}(\emptyset)$;
$(ii)$
whenever $s_{1},s_{2},s_{3}\in P(n)^{-}$ and $(s_{1}\cap s_{2})\subset s_{3}$, the algebraically closed sets
$a(s_{1}),a(s_{2})$ are independent over $a(s_{1}\cap s_{2})$ within $a(s_{3})$;
$(iii)$
$a(s)=\operatorname{acl}\{a(i)\,|\,i\in s\}$, for every $s\in P(n)^{-}$.
In here we denote by $\operatorname{acl}(A)$ the algebraic closure of $A$ in
$T^{\textrm{eq}}$. We recall that the objects of $P(n)^{-}$ (viewed as
a category) are simply the elements of $P(n)^{-}$. Also, the
morphisms of $P(n)^{-}$ are the inclusions $\iota_{s,t}:s\hookrightarrow t$,
for every $s,t\in P(n)^{-}$ with $s\subseteq t$. In particular, an
$n$-amalgamation problem assigns a morphism
$$a_{s,t}:a(s)\to a(t),$$
to every $s,t\in P(n)^{-}$ with $s\subseteq t$. The morphism
$a_{s,t}$ is called transition map and, by functoriality, we have
$$a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=a_{s_{1},s_{3}},$$
for every $s_{1},s_{2},s_{3}\in P(n)^{-}$ with $s_{1}\subseteq s_{2}\subseteq s_{3}$. By definition, the morphisms
in $\mathcal{C}_{T}$ are the embeddings, that is,
$a_{s,t}$ is the restriction of an automorphism to the algebraically
closed substructure $a(s)$.
A solution of $a$ is a functor $\bar{a}:P(n)\rightarrow\mathcal{C}_{T}$ extending $a$ to the full power set $P(n)$ and satisfying
the conditions $(i),(ii),(iii)$ (i.e. including the case
$s=\{1,\dots,n\}$). In particular, in order to find a solution of
$a$, we need to determine $n$ embeddings
$$f_{i}:a(\{1,\ldots,n\}\setminus\{i\})\longrightarrow a(\{1,\ldots,n\})=%
\operatorname{acl}(\{a(i)\mid i\in\{1,\ldots,n\}\}),$$
(for $1\leq i\leq n$) compatible with $a$, that
is,
$$f_{i}\circ a_{s,\{1,\ldots,n\}\setminus\{i\}}=f_{j}\circ a_{s,\{1,\ldots,n\}%
\setminus\{j\}}$$
for every $i,j\in\{1,\ldots,n\}$ and $s\subseteq\{1,\ldots,n\}\setminus\{i,j\}$.
The theory
$T$ is said to have $n$-existence (over $\operatorname{acl}(\emptyset)$) if every
$n$-amalgamation problem over $\operatorname{acl}(\emptyset)$ has at least one
solution. Similarly, we shall
say that the theory $T$ has $n$-uniqueness (over
$\operatorname{acl}(\emptyset)$) if every
$n$-amalgamation problem over $\operatorname{acl}(\emptyset)$ has at most one solution up to
isomorphism (for more details see [9] and [12]).
It is a well known fact that every simple
theory has $2$-existence, by the presence of non-forking
extensions. Moreover, if the theory is stable, then, by stationarity of
strong types, $2$-uniqueness holds. Consequentially, also
$3$-existence holds (for a proof see Lemma $3.1$ of
[9]). However, $3$-uniqueness and $4$-existence can fail
for a general stable theory. Indeed, in
[3], the authors thank E. Hrushovski for supplying an example
of a stable theory which has neither
$4$-existence nor $3$-uniqueness. The example is the following. Its construction involves a finite cover (for more details about finite covers see [5]).
Example 1
Let $\Omega$ be a countable set,
$[\Omega]^{2}$ the set of $2$-subsets of $\Omega$, and
$C=[\Omega]^{2}\times{\mathbb{Z}}/2{\mathbb{Z}}$. Also let
$E\subseteq\Omega\times[\Omega]^{2}$ be the membership relation, and let
$P$ be the subset of $C^{3}$ such that
$((w_{1},\delta_{1}),(w_{2},\delta_{2}),(w_{3},\delta_{3}))$ lies in $P$ if and only if
there are distinct $c_{1},c_{2},c_{3}\in\Omega$ such that
$w_{1}=\{c_{2},c_{3}\},w_{2}=\{c_{1},c_{3}\},w_{3}=\{c_{1},c_{2}\}$ and
$\delta_{1}+\delta_{2}+\delta_{3}=0$. Now let $M$ be the model with the
$3$-sorted universe $\Omega,[\Omega]^{2},C$ and equipped with relations $E,P$
and projection on the first coordinate $\pi:C\rightarrow[\Omega]^{2}$. Since $M$ is a reduct of $(\Omega,{\mathbb{Z}}/2{\mathbb{Z}})^{\textrm{eq}}$, we get that $T=\operatorname{Th}(M)$ is stable. It is
shown in [3]
that $T$ has neither $4$-existence nor $3$-uniqueness.
In this paper we generalize this
example. We summarize our main results in the following theorem.
Theorem 2
For any
$n\geq 2$, there exists a stable theory $T_{n}$ such that $T_{n}$ has
$(k+1)$-existence and $k$-uniqueness for any $k\leq n$, but $T_{n}$ has
neither $(n+2)$-existence nor $(n+1)$-uniqueness.
Also in Proposition 29 we prove that, for $n=2$, the
stable theory $T_{2}$ given in Theorem 2 coincides with
the theory in Example 1.
All the material we present is expressed in a purely algebraic
terminology. Indeed, the problem of $n$-uniqueness for a theory has also a
natural formulation in terms of permutation groups, as is shown
in [9, Proposition $3.5$]. We adopt this approach here.
In Section 2, we introduce certain permutation modules which will be
used to construct the automorphism groups of the countable
$\aleph_{0}$-categorical structures $M_{n}$ on which is based Theorem 2.
As is clear from the definition, the study of amalgamation problems
requires a precise understanding of the algebraic closure in
$T^{\textrm{eq}}$. Since the structures $M_{n}$ are countable and
$\aleph_{0}$-categorical, the algebraic closure can
be rephrased with group theoretic terminology: it can be determined
by studying certain closed subgroups of the automorphism group of
$M_{n}$. This is done in Section 3 and Section 4.
2 The $\operatorname{Sym}(\Omega)$-submodule structure of
$\mathbb{F}^{[\Omega]^{n}}$
We begin by reviewing some definitions and basic facts about
permutation groups and permutation modules.
If $C$ is a set, then the symmetric group $\operatorname{Sym}(C)$ on $C$
can be considered as a topological group. The
open sets in this topology are arbitrary unions of cosets of pointwise
stabilizers of finite subsets of $C$. A subgroup $\Gamma$ of
$\operatorname{Sym}(C)$ is closed if and only if each element of
$\operatorname{Sym}(C)$ which preserves all the orbits of $\Gamma$ on $C^{n}$, for all
$n\in{\mathbb{N}}$, is in $\Gamma$. It is well known that closed subgroups in
this topology are precisely automorphism groups of first-order
structures on $C$, see [2, Theorem $5.7$] or [11].
Throughout the sequel we denote by ${\mathbb{F}}$ a field, ${\mathbb{F}}_{2}$
the integers modulo $2$, $\Omega$ a countable set and $[\Omega]^{n}$ the
set of $n$-subsets of $\Omega$.
The natural action of the symmetric group $\operatorname{Sym}(\Omega)$ on $[\Omega]^{n}$ turns
$\mathbb{F}[\Omega]^{n}$, the vector
space over ${\mathbb{F}}$ with basis consisting of the elements of
$[\Omega]^{n}$, into a $\operatorname{Sym}(\Omega)$-module. We will characterize the
submodules of $\mathbb{F}[\Omega]^{n}$ in terms of certain
$\operatorname{Sym}(\Omega)$-homomorphisms. The following definition is based on
concepts first introduced in [10].
Definition 3 ([6], Def. $3.4$)
If $0\leq j\leq n$, then the map $\beta_{n,j}:\mathbb{F}[\Omega]^{n}\to\mathbb{F}[\Omega]^{j}$, given by
$$\beta_{n,j}(\omega)=\sum_{\omega^{\prime}\in[\omega]^{j}}\omega^{\prime}\qquad%
(\textrm{for }\omega\in[\Omega]^{n})$$
and extended linearly to $\mathbb{F}[\Omega]^{n}$, is a
$\operatorname{Sym}(\Omega)$-homomorphism (in here we
denote by $[\omega]^{j}$ the set of $j$-subsets of $\omega$).
It is shown in
[6] (see also [10]) that the submodules of
$\mathbb{F}[\Omega]^{n}$ are completely determined by the maps $\beta_{n,j}$.
Indeed, it is proved in [6, Corollary $3.17$] that every
submodule $U$ of $\mathbb{F}[\Omega]^{n}$ is an intersection of kernels of
$\beta$-maps, i.e. $U=\cap_{j\in S}\ker\beta_{n,j}$ for some subset
$S$ of $\{0,\ldots,n\}$.
Using the controvariant Pontriagin duality we have that the dual module of
$\mathbb{F}[\Omega]^{n}$ is ${\mathbb{F}}^{[\Omega]^{n}}$, i.e. the set of functions
from $[\Omega]^{n}$ to
${\mathbb{F}}$. We recall that ${\mathbb{F}}^{[\Omega]^{n}}$ has a natural faithful
action on $[\Omega]^{n}\times{\mathbb{F}}$ given by
$(w,\delta)^{f}=(w,f(w)+\delta)$. Hence, ${\mathbb{F}}^{[\Omega]^{n}}$, endowed
with the relative topology,
becomes a topological $\operatorname{Sym}(\Omega)$-module and a profinite subgroup
of $\operatorname{Sym}([\Omega]^{n}\times{\mathbb{F}})$. Also, given any map
$\beta_{n,j}:\mathbb{F}[\Omega]^{n}\rightarrow\mathbb{F}[\Omega]^{j}$, there is a natural dual continuous $\operatorname{Sym}(\Omega)$-homomorphism
$\beta^{\ast}_{n,j}:\mathbb{F}^{[\Omega]^{j}}\rightarrow\mathbb{F}^{[\Omega]^{n}}$
defined by
$$(\beta^{\ast}_{n,j}f)(\omega)=\sum_{x\in[\omega]^{j}}f(x).$$
Now, the lattice of the closed submodules of $\mathbb{F}^{[\Omega]^{n}}$ is the
dual of the lattice of the submodules of $\mathbb{F}[\Omega]^{n}$. We
point out that using the algorithm
described in [6, Section $5$], the lattice of the closed submodules of
${\mathbb{F}}^{[\Omega]^{n}}$ can be easily computed. Here we record the
following fact that we are frequently going to use.
Proposition 4
For $n\geq 1$, $\mathbb{F}=\mathbb{F}_{p}$ with $p>0$, we have $\operatorname{im}\beta_{n,n-1}^{\ast}=\operatorname{ker}\beta_{n+1,n}^{\ast}$.
Proof. The submodule $\operatorname{im}\beta_{n+1,n}$ of
$\mathbb{F}[\Omega]^{n}$ is of the form $\cap_{j\in S}\operatorname{ker}\beta_{n,j}$, for some subset $S$ of
$\{0,\ldots,n\}$. By [6, Proposition $3.19$], we have that $\operatorname{im}\beta_{n+1,n}\subseteq\operatorname{ker}\beta_{n,j}$ if and only if $2$ divides
$n+1-j$. Therefore $S=\{j\mid 2\textrm{ divides }n+1-j\}$.
Also by [6, Proposition $4.1$], we have that if $2$ divides
$n+1-j$, then $\operatorname{ker}\beta_{n,n-1}\subseteq\operatorname{ker}\beta_{n,j}$. This yields
$\operatorname{im}\beta_{n+1,n}=\cap_{j\in S}\operatorname{ker}\beta_{n,j}=%
\operatorname{ker}\beta_{n,n-1}$. In particular, the sequence
$$\xymatrix{\mathbb{F}[\Omega]^{n+1}\ar@{->}[rr]^{\beta_{n+1,n}}&&\mathbb{F}[%
\Omega]^{n}\ar@{->}[rr]^{\beta_{n,n-1}}&&\mathbb{F}[\Omega]^{n-1}}$$
is exact.
Now the Pontriagin duality is an exact controvariant functor on the
sequences of the
form $A\to B\to C$. This says that $\operatorname{im}\beta_{n,n-1}^{\ast}=\operatorname{ker}\beta_{n+1,n}^{\ast}$.
3 Closed submodules of finite index in
$\mathbb{F}_{2}^{[\Omega]^{n}}$
If $A$ is a finite subset of $\Omega$, then we write simply
$\operatorname{Sym}(\Omega\setminus A)$ for the subgroup of $\operatorname{Sym}(\Omega)$ fixing
pointwise $A$. In this section we study the closed
$\operatorname{Sym}(\Omega\setminus A)$-submodules
of $\mathbb{F}_{2}^{[\Omega]^{n-1}}$ of finite index. We start by
considering the case $A=\emptyset$.
Lemma 5
If $n\geq 1$, then $\mathbb{F}_{2}^{[\Omega]^{n}}$ has no
proper closed $\operatorname{Sym}(\Omega)$-submodule of finite index.
Proof.Let $K$ be a closed submodule of
${\mathbb{F}}_{2}^{[\Omega]^{n}}$ of finite index. Then,
${\mathbb{F}}_{2}^{[\Omega]^{n}}/K$ is a finite $\operatorname{Sym}(\Omega)$-module. Since
$\operatorname{Sym}(\Omega)$ has no proper subgroup of finite index, we get that
$\operatorname{Sym}(\Omega)$ centralizes ${\mathbb{F}}_{2}^{[\Omega]^{n}}/K$. It follows that
$f^{\sigma}-f\in K$, for every $\sigma\in\operatorname{Sym}(\Omega)$.
Let $L$ be the annihilator of $K$ in $\mathbb{F}_{2}[\Omega]^{n}$,
i.e. $L=\{w\in\mathbb{F}_{2}[\Omega]^{n}\mid g(w)=0\textrm{ for every
}g\in K\}$. Since $K$ is a closed $\operatorname{Sym}(\Omega)$-submodule, the set $L$
is a $\operatorname{Sym}(\Omega)$-submodule of
$\mathbb{F}_{2}[\Omega]^{n}$. Now, let $f$ be in
$\mathbb{F}_{2}^{[\Omega]^{n}}$, $\sigma$ in $\operatorname{Sym}(\Omega)$ and $w$ in
$L$. We
get
$$0=(f^{\sigma}-f)(w)=f^{\sigma}(w)-f(w)=f(w^{\sigma^{-1}}-w).$$
This says that $w^{\sigma^{-1}}-w$ is annihilated by every
element of $\mathbb{F}_{2}^{[\Omega]^{n}}$. Therefore,
$w^{\sigma^{-1}}-w=0$ and $\sigma$ centralizes $w$. This shows
that $\operatorname{Sym}(\Omega)$ centralizes $L$. Since $n\geq 1$, the only element
of $\mathbb{F}_{2}[\Omega]^{n}$ centralized by $\operatorname{Sym}(\Omega)$ is the zero
vector. Hence $L=0$ and, by the Pontriagin duality,
$K=\mathbb{F}_{2}^{[\Omega]^{n}}$.
In the forthcoming analysis we shall denote finite subsets of $\Omega$
by capital letters, while the elements of $[\Omega]^{n}$ will be generally
denoted by lower cases.
Now, let $A$ be a finite subset of $\Omega$. To
describe the closed $\operatorname{Sym}(\Omega\setminus A)$-submodules
of
$\mathbb{F}_{2}^{[\Omega]^{n-1}}$ of finite
index we have to
introduce some notation. Let $B$ be a subset of $A$. We denote by
$V_{B,A}$ the $\operatorname{Sym}(\Omega\setminus A)$-submodule of
${\mathbb{F}}_{2}^{[\Omega]^{n-1}}$ defined by
$$V_{B,A}=\{f\in{\mathbb{F}}_{2}^{[\Omega]^{n-1}}\mid f(w)=0\,\,\forall\,w\in[%
\Omega]^{n-1}\textrm{ with }w\cap A\neq B\}$$
(1)
and we denote by $V_{A}$ the $\operatorname{Sym}(\Omega\setminus A)$-submodule of
${\mathbb{F}}_{2}^{[\Omega]^{n-1}}$ defined by
$$\displaystyle V_{A}$$
$$\displaystyle=$$
$$\displaystyle\bigoplus_{\small B\subseteq A,|B|<n-1\normalsize}V_{B,A}.$$
(2)
In the following lemma we describe the elements of $V_{A}$.
Lemma 6
Let $A$ be a finite subset of
$\Omega$. Then
$$V_{A}=\{f\in{\mathbb{F}}_{2}^{[\Omega]^{n-1}}\mid f(w)=0\textrm{
for every }w\in[A]^{n-1}\}.$$
(3)
Proof.
We denote by $W$ the vector space on the right hand side of
Equation $(\ref{eq:3})$. We start by proving that $V_{A}\subseteq W$.
Let $B$ be a subset of $A$ with $|B|<n-1$ and $f$ be in
$V_{B,A}$. Consider $w$ in $[A]^{n-1}$. Since $|B|<n-1$, $|w|=n-1$ and
$w\subseteq A$, we have $w\cap A=w\neq B$. By Equation $(\ref{eq:1})$, we get $f(w)=0$.
This implies $f\in W$ and so
$V_{B,A}\subseteq W$. Thence, by Equation $(\ref{eq:2})$, we obtain
$V_{A}\subseteq W$.
Conversely, we prove that $W\subseteq V_{A}$. Let $f$ be in $W$. For every subset
$B$ of $A$ with $|B|<n-1$ define
$$f_{B}(w)=\left\{\begin{array}[]{ccl}f(w)&&\textrm{if }w\cap A=B,\\
0&&\textrm{if }w\cap A\neq B.\end{array}\right.$$
Clearly, $f_{B}\in{\mathbb{F}}_{2}^{[\Omega]^{n-1}}$ and, by Equation $(\ref{eq:1})$,
$f_{B}\in V_{B,A}$. Let $w$ be in $[\Omega]^{n-1}$ with $w\nsubseteq A$. Since $|w\cap A|<n-1$, we have
$$\displaystyle\left(\sum_{\small B\subseteq A,|B|<n-1\normalsize}f_{B}\right)(w)$$
$$\displaystyle=$$
$$\displaystyle\sum_{\small B\subseteq A,|B|<n-1\normalsize}f_{B}(w)=f_{w\cap A}%
(w)=f(w).$$
Similarly, let $w$ be in $[\Omega]^{n-1}$
with $w\subseteq A$ (that is, $w\in[A]^{n-1}$). As $f\in W$, we have
$f(w)=0$. Also, by definition of $f_{B}$, we obtain $f_{B}(w)=0$. This
shows that $f=\sum_{B\subseteq A,|B|<n-1}f_{B}$. By Equation $(\ref{eq:2})$, it follows that
$f\in V_{A}$.
Lemma 7
Let $A$ be a finite subset of $\Omega$. For each $B\subseteq A$, the
$\operatorname{Sym}(\Omega\setminus A)$-modules $V_{B,A}$ are closed submodules of
${\mathbb{F}}_{2}^{[\Omega]^{n-1}}$. Moreover,
$${\mathbb{F}}_{2}^{[\Omega]^{n-1}}=\bigoplus_{\small B\subseteq A,|B|\leq n-1%
\normalsize}V_{B,A}$$
(4)
and each $V_{B,A}$ is $\operatorname{Sym}(\Omega\setminus A)$-isomorphic to
${\mathbb{F}}_{2}^{[\Omega\setminus A]^{n-1-|B|}}$.
Proof.Since $V_{B,A}$ is an intersection of
pointwise stabilizers of finite sets of $[\Omega]^{n-1}\times{\mathbb{F}}_{2}$, it is closed in
${\mathbb{F}}_{2}^{[\Omega]^{n-1}}$. It is
straightforward to verify the remaining statements.
Lemma 8
Let $A$ be a finite subset of $\Omega$.
The module $V_{A}$ has finite index in
$\mathbb{F}_{2}^{[\Omega]^{n-1}}$. Also, if
$V$ is a closed $\operatorname{Sym}(\Omega\setminus A)$-submodule
of ${\mathbb{F}}_{2}^{[\Omega]^{n-1}}$ of finite index, then $V_{A}\subseteq V$.
Proof. By Equations $(\ref{eq:2})$ and $(\ref{eq:1111})$,
we have that $\mathbb{F}_{2}^{[\Omega]^{n-1}}/V_{A}$ is isomorphic to
$\oplus_{|B|=n-1}V_{B,A}$, which has dimension ${|A|\choose n-1}$. Therefore $V_{A}$ has finite index in
$\mathbb{F}_{2}^{[\Omega]^{n-1}}$.
Let $V$ be a closed $\operatorname{Sym}(\Omega\setminus A)$-submodule
of $\mathbb{F}_{2}^{[\Omega]^{n-1}}$ of finite index. Let $B\subseteq A$ with
$|B|<n-1$. By Lemma
7, $V_{B,A}$ is
$\operatorname{Sym}(\Omega\setminus A)$-isomorphic to ${\mathbb{F}}_{2}^{[\Omega\setminus A]^{n-1-|B|}}$. Since $[V_{B,A}:V_{B,A}\cap V]=[V_{B,A}+V:V]$ is finite, we have that $V_{B,A}\cap V$ has
finite index in
$V_{B,A}$. Now, by Lemma 5, the module $V_{B,A}$
does not have
any proper closed $\operatorname{Sym}(\Omega\setminus A)$-submodule of finite
index. Therefore $V_{B,A}=V_{B,A}\cap V$ and $V_{B,A}\subseteq V$. By
definition of $V_{A}$ in Equation $(\ref{eq:2})$, we get $V_{A}\subseteq V$.
In the following lemma we describe the elements of $V_{A}+\operatorname{ker}\beta_{n,n-1}^{\ast}$.
Lemma 9
Let $A$ be a finite subset of $\Omega$. We
have $V_{A}+\operatorname{ker}\beta_{n,n-1}^{\ast}=\{f\in\mathbb{F}_{2}^{[\Omega]^{n%
-1}}\mid(\beta_{n,n-1}^{\ast}f)(w)=0\textrm{ for every }w\in[A]^{n}\}$.
Proof.If $n=1$, then the equality is clear. So assume $n\geq 2$.
By Lemma 6, the elements of $V_{A}$ are the
functions $f\in\mathbb{F}_{2}^{[\Omega]^{n-1}}$ vanishing on each element of $[A]^{n-1}$. Now, if $f_{1}\in V_{A}$, $f_{2}\in\operatorname{ker}\beta_{n,n-1}^{\ast}$ and
$w\in[A]^{n}$,
then
$$(\beta_{n,n-1}^{\ast}(f_{1}+f_{2}))(w)=(\beta_{n,n-1}^{\ast}f_{1})(w)=\sum_{w^%
{\prime}\in[w]^{n-1}}f_{1}(w^{\prime})=0.$$
Therefore, it remains to prove that if $f\in\mathbb{F}_{2}^{[\Omega]^{n-1}}$ and $(\beta_{n,n-1}^{\ast}f)(w)=0$ for
every $w\in[A]^{n}$, then $f\in V_{A}+\operatorname{ker}\beta_{n,n-1}^{\ast}$. Let $a$
be a fixed element of $A$ and let
$g\in\mathbb{F}_{2}^{[\Omega]^{n-2}}$ be the function defined by
$$g(\omega)=\left\{\begin{array}[]{cll}f(\omega\cup\{a\})&&\textrm{if }\omega%
\subseteq A\textrm{ and
}a\notin\omega,\\
0&&\textrm{otherwise }.\end{array}\right.$$
Set $f_{2}=\beta_{n-1,n-2}^{\ast}g$. By Proposition 4, we have that
$f_{2}\in\operatorname{im}\beta_{n-1,n-2}^{\ast}=\operatorname{ker}\beta_{n,n-1%
}^{\ast}$. Set
$f_{1}=f-f_{2}$. We claim that $f_{1}$
lies in $V_{A}$, from which the lemma follows. By Lemma 6, it
suffices to prove that
$f_{1}(w^{\prime})=0$ for every $w^{\prime}\in[A]^{n-1}$. Let $w^{\prime}$ be in
$[A]^{n-1}$. Assume $a\in w^{\prime}$. By the definition of $g$, we have
$$f_{2}(w^{\prime})=(\beta_{n-1,n-2}^{\ast}g)(w^{\prime})=\sum_{\omega\in[w^{%
\prime}]^{n-2}}g(\omega)=g(w^{\prime}\setminus\{a\})=f(w^{\prime})$$
and $f_{1}(w^{\prime})=0$. Now assume $a\notin w^{\prime}$. By the definition of $g$ and by
the hypothesis on $f$, we have
$$\displaystyle f_{2}(w^{\prime})$$
$$\displaystyle=$$
$$\displaystyle(\beta_{n-1,n-2}^{\ast}g)(w^{\prime})=\sum_{\omega\in[w^{\prime}]%
^{n-2}}g(\omega)=\sum_{\omega\in[w^{\prime}]^{n-2}}f(\omega\cup\{a\})$$
$$\displaystyle=$$
$$\displaystyle\sum_{x\in[w^{\prime}\cup\{a\}]^{n-1}}\!\!\!\!\!\!f(x)+f(w^{%
\prime})=(\beta_{n,n-1}^{\ast}f)(w^{\prime}\cup\{a\})+f(w^{\prime})=f(w^{%
\prime}),$$
and $f_{1}(w^{\prime})=0$.
Definition 10
We write $W_{A}$ for $\beta_{n,n-1}^{\ast}(V_{A})$, with $V_{A}$ as in
Equation $(\ref{eq:2})$.
Now, using the previous
lemmas we describe the
closed $\operatorname{Sym}(\Omega\setminus A)$-submodules of
$\operatorname{im}\beta_{n,n-1}^{\ast}$ of finite index.
Proposition 11
Let $A$ be a finite subset of
$\Omega$. The module $W_{A}$ is the unique minimal
closed $\operatorname{Sym}(\Omega\setminus A)$-submodule of
$\operatorname{im}\beta_{n,n-1}^{\ast}$ of finite index. Furthermore,
$W_{A}=\{g\in\operatorname{im}\beta_{n,n-1}^{\ast}\mid g(w)=0\textrm{ for every%
}w\in[A]^{n}\}$.
Proof.Let $W$ be a closed $\operatorname{Sym}(\Omega\setminus A)$-submodule
of $\operatorname{im}\beta_{n,n-1}^{\ast}$ of finite index. By the first
isomorphism theorem $W$ is the image via $\beta_{n,n-1}^{\ast}$ of some closed
$\operatorname{Sym}(\Omega\setminus A)$-submodule $V$ of
$\mathbb{F}_{2}^{[\Omega]^{n-1}}$ of finite index. Now, by
Lemma 8, we get
$V_{A}\subseteq V$. So
$\beta_{n,n-1}^{\ast}(V_{A})\subseteq\beta_{n,n-1}^{\ast}(V)=W$. Hence, $W_{A}=\beta_{n,n-1}^{\ast}(V_{A})$ is the unique minimal
closed $\operatorname{Sym}(\Omega\setminus A)$-submodule of
$\operatorname{im}\beta_{n,n-1}^{\ast}$ of finite index.
Now, from Lemma 9 the rest of the proposition is immediate.
4 The infinite family of examples
Before introducing our examples, we need to set some
auxiliary notation.
Definition 12
Let $M$ be a structure and $A,B$ subsets
of $M$. We denote
by $\overline{\operatorname{Aut}(A/B)}$ the subgroup of $\operatorname{Aut}(M)$
fixing setwise $A$ and fixing pointwise $B$. The setwise stabilizer of $A$ in $\operatorname{Aut}(M)$ will be denoted by $\operatorname{Aut}(M)_{\{A\}}$, while the permutation group
induced by $\overline{\operatorname{Aut}(A/B)}$ on $A$ will be denoted by $\operatorname{Aut}(A/B)$.
Let $n\geq 2$ be an integer and $\Omega$ be a countable set.
Definition 13
We
consider $M_{n}$
the multisorted structure with
sorts $\Omega$, $[\Omega]^{n}$ and $[\Omega]^{n}\times\mathbb{F}_{2}$ and with
automorphism group $\operatorname{im}\beta^{\ast}_{n,n-1}\rtimes\operatorname{Sym}(\Omega)$. Note
that this is well-defined as $\operatorname{im}\beta_{n,n-1}^{\ast}$ is a closed
submodule of $\mathbb{F}_{2}^{[\Omega]^{n}}$.
Moreover, the theory $T_{n}=\operatorname{Th}(M_{n})$ is stable (see Section 6).
In the next paragraph we introduce some notation that would
be useful to describe the algebraically closed sets of $M_{n}$.
Denote by
$\pi:[\Omega]^{n}\times\mathbb{F}_{2}\to[\Omega]^{n}$ the projection on
the first coordinate. Given $A$ a finite subset of $M_{n}$, we have
that $A$ is of the form $A_{1}\cup A_{2}\cup A_{3}$, where $A_{1}$ belongs to
the sort $\Omega$, $A_{2}$ belongs to the sort $[\Omega]^{n}$ and $A_{3}$
belongs to the sort $[\Omega]^{n}\times\mathbb{F}_{2}$. Consider
$\tilde{A_{2}}\subseteq\Omega$ the union of the elements in $A_{2}$ and $\tilde{A_{3}}\subseteq\Omega$ the union of the elements in $\pi(A_{3})$. We define
the support of $A$, written $\mathop{\mathrm{supp}}(A)$, to be the subset $A_{1}\cup\tilde{A_{2}}\cup\tilde{A_{3}}$ of $\Omega$. Finally, we define $\operatorname{cl}(A)$ to be the subset of $M_{n}$
$$\operatorname{cl}(A):=\mathop{\mathrm{supp}}(A)\cup[\mathop{\mathrm{supp}}(A)]%
^{n}\cup([\mathop{\mathrm{supp}}(A)]^{n}\times\mathbb{F}_{2})$$
In the rest of this section we describe the algebraically closed sets in the
structure $M_{n}$. Here we consider structures up to
interdefinability, which allows us to identify an
$\aleph_{0}$-categorical structure with its automorphism group. So we
identify two substructures $A_{1},A_{2}$ of a structure $M$, if
$\operatorname{Aut}(A_{1})=\operatorname{Aut}(A_{2})$. If $M$ is an $\aleph_{0}$-categorical structure and
$A\subset M$, we denote the algebraic closure $\operatorname{acl}^{\rm{eq}}(A)$ of
$A$ simply by $\operatorname{acl}(A)$, i.e.
the union of the finite $\operatorname{Aut}(M/A)$-invariant sets of
$M^{\rm{eq}}$. We recall that definable subsets of $\operatorname{acl}(A)$
correspond, up to interdefinability, to closed subgroups of
$\operatorname{Aut}(M/A)$ of finite
index, see [8, Section $4.1$] or Theorem $4.1$ in the article
“The structure of totally categorical structures” by
W. Hodges [11, page $116$].
Similarly, if $A\subset M$, we denote the definable closure $\operatorname{dcl}^{\rm{eq}}(A)$ of
$A$ simply by $\operatorname{dcl}(A)$, i.e.
the set of the points of $M^{\rm{eq}}$ fixed by $\operatorname{Aut}(M/A)$.
Lemma 14
Let $A$ be a finite set of $M_{n}$. Then
$$\operatorname{Aut}(M_{n}/\operatorname{cl}(A))=W_{\mathop{\mathrm{supp}}(A)}%
\rtimes\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$$
(where $W_{\mathop{\mathrm{supp}}(A)}$ is the closed $\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$-submodule of
$\operatorname{im}\beta_{n,n-1}^{\ast}$ in Definition 10). Moreover, $\operatorname{Aut}(M_{n}/\operatorname{cl}(A))$ is the unique
minimal closed subgroup of finite index of $\operatorname{Aut}(M_{n}/A)$.
Proof.
Set $\Gamma=\operatorname{Aut}(M_{n}/\operatorname{cl}(A))$. We first prove that
$\Gamma=W_{\mathop{\mathrm{supp}}(A)}\rtimes\operatorname{Sym}(\Omega\setminus%
\mathop{\mathrm{supp}}(A))$. By
definition of the multisorted structure $M_{n}$, we have $\operatorname{Aut}M_{n}=\operatorname{im}\beta_{n,n-1}^{*}\rtimes\operatorname%
{Sym}(\Omega)$. Therefore, an element of $\Gamma$ is
an ordered pair of the form $g\sigma$, where $g\in\operatorname{im}\beta_{n,n-1}^{*}$ and $\sigma\in\operatorname{Sym}(\Omega)$. The action of $g\sigma$ on the elements belonging
to the sorts $\Omega$ and $[\Omega]^{n}$ is given by the permutation
$\sigma$. Also, the action of $g\sigma$ on the element $(w,x)$ belonging to
the sort $[\Omega]^{n}\times{\mathbb{F}}_{2}$ is given
by
$$(w,x)^{g\sigma}=(w^{\sigma},x+g(w)).$$
This implies that the automorphism $g\sigma$ fixes the elements in
$\mathop{\mathrm{supp}}(A)$ and in $[\mathop{\mathrm{supp}}(A)]^{n}$ (in the sorts $\Omega$ and
$[\Omega]^{n}$) if and only if
$\sigma\in\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$. Also, the automorphism
$g\sigma$ fixes the
elements in $[\mathop{\mathrm{supp}}(A)]^{n}\times{\mathbb{F}}_{2}$ (in the sort
$[\Omega]^{n}\times{\mathbb{F}}_{2}$) if and only if $g(w)=0$ for
every $w\in[\mathop{\mathrm{supp}}(A)]^{n}$. Hence, by the description of the elements of
$W_{\mathop{\mathrm{supp}}(A)}$ in Proposition 11, we have $g\sigma\in\Gamma$ if and only if $g\sigma\in W_{\mathop{\mathrm{supp}}(A)}\rtimes\operatorname{Sym}(\Omega%
\setminus\mathop{\mathrm{supp}}(A))$.
We claim that $\Gamma$ is the unique
minimal closed subgroup of $\operatorname{Aut}(M_{n}/A)$ of finite index. Note that $\Gamma$ is a closed subgroup of
$\operatorname{Aut}(M_{n}/A)$ of finite index.
Now, let $H$ be
a closed subgroup of
$\operatorname{Aut}(M_{n}/A)$
of finite index. Up to
replacing $H$ with $H\cap\Gamma$, we may assume that $H\subseteq\Gamma$. Let
$\mu:\Gamma\to\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$ be the
natural projection. Since $\mu$ is a surjective continuous closed map
and $\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$ has no proper subgroup of finite
index, we get that $\mu(H)=\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$. This
yields that $H\cap W_{\mathop{\mathrm{supp}}(A)}$ is a closed
$\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(A))$-submodule of $W_{\mathop{\mathrm{supp}}(A)}$ of finite
index. Now Proposition 11 shows that $H\cap W_{\mathop{\mathrm{supp}}(A)}=W_{\mathop{\mathrm{supp}}(A)}$. So $W_{\mathop{\mathrm{supp}}(A)}\subseteq H$ and $H=\Gamma$.
In the following we denote by $\operatorname{acl}_{M_{n}}$ the $\operatorname{acl}$ in $M_{n}$.
Proposition 15
Let $A$ be a finite set of $M_{n}$. Then
$\operatorname{acl}_{M_{n}}(A)=\operatorname{cl}(A)$.
Proof. Let $\overline{b}$ be an $m$-tuple in $M_{n}$ and $A$ be a finite set of $M_{n}$. We first claim that $\operatorname{Aut}(M_{n}/\overline{b})\geq\operatorname{Aut}(M_{n}/%
\operatorname{cl}(A))$ if and only if the underlying set of $\overline{b}$ is conteined in $\operatorname{cl}(A)$ . One direction is obvious. Suppose that $\operatorname{Aut}(M_{n}/\overline{b})\geq\operatorname{Aut}(M_{n}/%
\operatorname{cl}(A))$ for some finite $A\subset M_{n}$. Then by Lemma 14 we have that $\operatorname{Aut}(M_{n}/\operatorname{cl}(\operatorname{cl}(A),\overline{b})$ is a closed subgroup of finite index in $\operatorname{Aut}(M_{n}/\operatorname{cl}(A),\overline{b})=\operatorname{Aut}%
(M_{n}/\operatorname{cl}(A))$. Hence $\operatorname{Aut}(M_{n}/\operatorname{cl}(\operatorname{cl}(A),\overline{b})$ is a closed subgroup of finite index in $\operatorname{Aut}(M_{n}/A)$. By uniqueness of the minimal closed subgroup of finite index of $\operatorname{Aut}(M_{n}/A)$ we get that $W_{\mathop{\mathrm{supp}}(A)}\rtimes\operatorname{Sym}(\Omega\setminus\mathop{%
\mathrm{supp}}(A))$ is equal to $W_{\mathop{\mathrm{supp}}(\operatorname{cl}(A),\overline{b})}\rtimes%
\operatorname{Sym}(\Omega\setminus\mathop{\mathrm{supp}}(\operatorname{cl}(A),%
\overline{b}))$ and, since $\mathop{\mathrm{supp}}(\operatorname{cl}(A),\overline{b})=\mathop{\mathrm{supp%
}}(A,\overline{b})$, this is possible if and only if $\mathop{\mathrm{supp}}(\overline{b})\subseteq\mathop{\mathrm{supp}}(A)$, which proves the claim.
By definition, $\operatorname{acl}_{M_{n}}(A)$ is the union of the finite orbits on $M_{n}$ of $\operatorname{Aut}(M_{n}/A)$. Let $c\in\operatorname{acl}_{M_{n}}(A)$. Then $\operatorname{Aut}(M_{n}/A,c)$ is a closed subgroup of finite index in $\operatorname{Aut}(M_{n}/A)$. Hence, by Lemma 14, $\operatorname{Aut}(M_{n}/A,c)\geq\operatorname{Aut}(M_{n}/\operatorname{cl}(A)$. By the above argument we have that $c\in\operatorname{cl}(A)$.
Let $c\in\operatorname{cl}(A)$, then $\operatorname{Aut}(M_{n}/A)\geq\operatorname{Aut}(M_{n}/A,c)\geq\operatorname{%
Aut}(M_{n}/\operatorname{cl}(A))$. Hence the index of $\operatorname{Aut}(M_{n}/A,c)$ in $\operatorname{Aut}(M_{n}/A)$ is finite.
Let $c^{\rm{eq}}\in M_{n}^{\rm{eq}}$. Then $c^{\rm{eq}}$ is a $0$-definable equivalence class of a tuple $b$ of elements in $M_{n}$. We denote by $\mathcal{s}(c^{\rm{eq}})$ the union of elements in $M_{n}$ of $c^{\rm{eq}}$. Similarly if $A\subseteq M_{n}^{\rm{eq}}$, we denote by $\mathcal{s}(A)$ the set of elements in $M_{n}$ $\bigcup_{c^{\rm{eq}}\in A}\mathcal{s}(c^{\rm{eq}})$.
Proposition 16
Let $A$ be a finite set of $M_{n}$. Then $\mathcal{s}(\operatorname{acl}(A))=\operatorname{cl}(A)$. In particular
$\operatorname{acl}(\emptyset)=\emptyset$.
Proof. Fix an enumeration $\overline{b}$ of $\operatorname{acl}_{M_{n}}(A)$ and set $\Gamma=\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))$. Consider the trivial relation $R=\{(b^{\alpha},b^{\alpha}):\alpha\in\operatorname{Aut}(M_{n})\}$. Since $R$ is an $\operatorname{Aut}(M_{n})$-orbit, $R$ is a $0$-definable equivalence relation in $M_{n}$. Consider the $R$-equivalence class of $\overline{b}$. The pointwise stabilizer of $\overline{b}$ in $\operatorname{Aut}(M_{n})$ is $\Gamma$ which, by Lemma 14 and Proposition 15, has finite index in $\operatorname{Aut}(M_{n}/A)$ and so $\overline{b}\in\operatorname{acl}(A)$.
Let $c^{\rm{eq}}\in\operatorname{acl}(A)$, then $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$ is a closed subgroup of finite index of $\operatorname{Aut}(M_{n}/A)$. By Lemma 14 $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$ contains $\Gamma$. Being $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$ also open in $\operatorname{Aut}(M_{n}/A)$ there exists a finite tuple $\overline{b}$ in $M_{n}$ such that $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$ contains the basic open subgroup $\operatorname{Aut}(M_{n}/A,\overline{b})$. Moreover $c^{\rm{eq}}=\overline{b}^{\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})}$. By $\aleph_{0}$-categoricity the index of $\operatorname{Aut}(M_{n}/A,\overline{b})$ in $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$ is finite. Then, the index of $\operatorname{Aut}(M_{n}/A,\overline{b})$ in $\operatorname{Aut}(M_{n}/A)$ is finite and so $\Gamma\leq\operatorname{Aut}(M_{n}/A,\overline{b})$. Hence by the same argument used in Proposition 15, we get that the underlying set in $M_{n}$ of $\overline{b}$ is contained in $\operatorname{cl}(A)=\operatorname{acl}_{M_{n}}(A)$. From the fact that $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})\leq\operatorname{Aut}(M_{n}/A)$ and $\overline{b}\in\operatorname{acl}_{M_{n}}(A)$ it follows immediately that also the underlying set of the $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$-orbit $\overline{b}^{\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})}$ is contained in $\operatorname{acl}_{M_{n}}(A)$.
Corollary 17
Let $A$ be a finite set of $M_{n}$. Then,
$$\operatorname{Aut}(M_{n})_{\{\operatorname{acl}_{M_{n}}(A)\}}=\operatorname{%
Aut}(M_{n})_{\{\operatorname{acl}(A)\}}.$$
Proof. From Proposition 15 and Proposition 16 it follows that $\operatorname{Aut}(M_{n})_{\{\operatorname{acl}(A)\}}\leq\operatorname{Aut}(M_%
{n})_{\{\operatorname{acl}_{M_{n}}(A)\}}$. Now, let $g\in\operatorname{Aut}(M_{n})_{\{\operatorname{acl}_{M_{n}}(A)\}}$. Note that $\operatorname{acl}_{M_{n}}(A^{g})=\operatorname{acl}_{M_{n}}(A)$. Consequently, $\operatorname{acl}(A^{g})=\operatorname{acl}(A)$. If $c^{\rm{eq}}\in\operatorname{acl}(A)$, then the index of $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$ in $\operatorname{Aut}(M_{n}/A)$ is finite. Therefore, $\operatorname{Aut}(M_{n}/A^{g},(c^{\rm{eq}})^{g})=g^{-1}\operatorname{Aut}(M_{%
n}/A,c^{\rm{eq}})g$ has finite index in $\operatorname{Aut}(M_{n}/A^{g})=g^{-1}\operatorname{Aut}(M_{n}/A)g$, which implies that $(c^{\rm{eq}})^{g}\in\operatorname{acl}(A^{g})=\operatorname{acl}(A)$.
Proposition 18
Let $A$ be a finite subset of $M_{n}$. Then,
$\operatorname{dcl}(\operatorname{acl}_{M_{n}}(A))=\operatorname{acl}(A).$
Proof. Let $c^{\rm{eq}}\in\operatorname{acl}(A)$, i.e. the stabilizer of $c^{\rm{eq}}$ in $\operatorname{Aut}(M_{n}/A)$ has finite index in $\operatorname{Aut}(M_{n}/A)$. We need to show that the stabilizer of $c^{\rm{eq}}$ in $\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))$ is equal to $\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))$. We have the following disequality:
$$|\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A)):\operatorname{Aut}(M_%
{n}/\operatorname{acl}_{M_{n}}(A),c^{\rm{eq}})|\leq|\operatorname{Aut}(M_{n}/A%
):\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})|$$
Then $|\operatorname{Aut}(M_{n}/A):\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n%
}}(A),c^{\rm{eq}})|$ is finite. By Lemma 14 and Proposition 15 it follows that $\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A),c^{\rm{eq}})$, is equal to $\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))$, i.e. $c^{\rm{eq}}\in\operatorname{dcl}(\operatorname{acl}_{M_{n}}(A))$.
Let $c^{\rm{eq}}\in\operatorname{dcl}(\operatorname{acl}_{M_{n}}(A))$. We need to show that $\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})$, has finite index in $\operatorname{Aut}(M_{n}/A)$. We have that
$$\begin{array}[]{c}|\operatorname{Aut}(M_{n}/A):\operatorname{Aut}(M_{n}/%
\operatorname{cl}(A)),c^{\rm{eq}})|=\\
|\operatorname{Aut}(M_{n}/A):\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})||%
\operatorname{Aut}(M_{n}/A,c^{\rm{eq}}):\operatorname{Aut}(M_{n}/\operatorname%
{cl}(A),c^{\rm{eq}})|\end{array}$$
(5)
Since $c^{\rm{eq}}\in\operatorname{dcl}(\operatorname{acl}_{M_{n}}(A))$ we have that $\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A),c^{\rm{eq}})=%
\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))$. Lemma 14 and the equality (5) imply that $|\operatorname{Aut}(M_{n}/A):\operatorname{Aut}(M_{n}/A,c^{\rm{eq}})|$ is finite. This proves that $c^{\rm{eq}}\in\operatorname{acl}(A)$ and the proof is complete.
Corollary 19
Let $A$ be a finite subset of $M_{n}$. Then
$$\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))=\operatorname{Aut}(M_{%
n}/\operatorname{acl}(A)).$$
Proof.
Let $g\in\operatorname{Aut}(M_{n}/\operatorname{acl}_{M_{n}}(A))$ and $c^{\rm{eq}}\in\operatorname{acl}(A)$. Proposition 18 yields that $(c^{\rm{eq}})^{g}=c^{\rm{eq}}$, which means that $g\in\operatorname{Aut}(M_{n}/\operatorname{acl}(A))$.
It remains to prove that $\operatorname{Aut}(M_{n}/\operatorname{acl}(A))\leq\operatorname{Aut}(M_{n}/%
\operatorname{acl}_{M_{n}}(A))$. Consider the trivial relation $R$ given by $R=\{(b,b):b\in M_{n}\}$. This is a 0-definable relation. Let $a\in\operatorname{acl}_{M_{n}}(A)$. Then $\{a\}\in M_{n}^{\rm{eq}}$ and $\operatorname{Aut}(M_{n}/A,\{a\})=\operatorname{Aut}(M_{n}/A,a)$ is a closed subgroup of finite index in $\operatorname{Aut}(M_{n}/A)$. Hence, we can consider that $\operatorname{acl}_{M_{n}}(A)\subseteq\operatorname{acl}(A)$ and the thesis follows at once.
Remark 20
Proposition 15 yields that if $A$ is a finite set of $M_{n}$, then $\operatorname{acl}_{M_{n}}(A)=\operatorname{acl}_{M_{n}}(\mathop{\mathrm{supp}%
}(A))$. Therefore, from Proposition 18 it follows that $\operatorname{acl}(A)=\operatorname{acl}(\mathop{\mathrm{supp}}(A))$.
Proposition 21
Let $A_{1},\dots,A_{n}$ be finite subsets in the sort $\Omega$. Then
$$\operatorname{acl}(\operatorname{acl}(A_{1}),\dots,\operatorname{acl}(A_{n}))=%
\operatorname{acl}(\bigcup_{i=1}^{n}A_{i}).$$
Proof.
Obviously, $\operatorname{acl}(\bigcup_{k=1}^{n}A_{k})\subseteq\operatorname{acl}(%
\operatorname{acl}(A_{1}),\dots,\operatorname{acl}(A_{n}))$.
Let $c^{\rm{eq}}\in\operatorname{acl}(\operatorname{acl}(A_{1}),\dots,\operatorname%
{acl}(A_{n}))$ and set $G=\operatorname{Aut}(M_{n}/\operatorname{acl}(A_{1}),\dots,\operatorname{acl}(%
A_{n}))$. Then, the pointwise stabilizer $G_{c^{\rm{eq}}}$ has finite index in $G$. By Corollary 19 we have that
$$G=\bigcap_{i=1}^{n}W_{A_{i}}\rtimes\operatorname{Sym}(\Omega\setminus A_{i}).$$
Moreover, $G\geq W_{\bigcup_{i=1}^{n}A_{i}}\rtimes\operatorname{Sym}(\Omega\setminus%
\bigcup_{i=1}^{n}A_{i})$
and $G$ is a closed subgroup in $\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i})$. So, $G$ is a closed subgroup of finite index in $\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i})$ which implies that also $G_{c^{\rm{eq}}}$ is of finite index in $\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i})$. Now, $G_{c^{\rm{eq}}}=G\cap\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i},c^{\rm{eq%
}})$ and
$$\begin{array}[]{c}|\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i}):%
\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i},c^{\rm{eq}})|=\\
|\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i}):G_{c^{\rm{eq}}}|/|%
\operatorname{Aut}(M_{n}/\bigcup_{i=1}^{n}A_{i},c^{\rm{eq}}):G_{c^{\rm{eq}}}|,%
\end{array}$$
i.e. $c^{\rm{eq}}\in\operatorname{acl}(\bigcup_{i=1}^{n}A_{i})$.
5 $k$-existence and $k$-uniqueness for $M_{n}$
In this section we prove Theorem 2. Note that, up to
renaming the elements of $\Omega$, we may assume that
$\Omega=\mathbb{N}$. In the sequel we denote by $[k]$
the subset $\{1,\ldots,k\}$ of ${\mathbb{N}}$. Also, given $i\in[k]$, we
denote by
$[k]-i$ the set $\{1,\ldots,k\}\setminus\{i\}$. Finally, we denote
the theory $\operatorname{Th}(M_{n})$ by $T_{n}$.
We start by studying $k$-uniqueness in $T_{n}$. We first single out
the following technical lemma which would be used in Proposition 23.
Lemma 22
Let $k$ and $n$ be integers, with $k<n$, and
$A_{1},\ldots,A_{k}$ be subsets of $\Omega$. Then
$$({\dagger})\qquad\bigcap_{i=1}^{k}\left(V_{A_{i}}+\operatorname{ker}\beta_{n,n%
-1}^{\ast}\right)=\left(\bigcap_{i=1}^{k}V_{A_{i}}\right)+\operatorname{ker}%
\beta_{n,n-1}^{\ast}.$$
Proof.
We denote the left-hand-side of $({\dagger})$ by $V_{1,k}$ and the
right-hand-side of $({\dagger})$ by $V_{2,k}$ (where the label $k$ is used
in order to remember
the number of intersections).
We argue by induction on $k$. Note that if $k=0$ or $k=1$, then there
is nothing to prove. Assume
$({\dagger})$ holds for $k$ intersections (where $k\geq 1$)
and that $k+1<n$. In particular, we point out that $n>2$. We prove
that $({\dagger})$ holds for $k+1$ intersections.
Clearly, $V_{2,k+1}\subseteq V_{1,k+1}$. Let $g$ be in $V_{1,k+1}$. We
need to show that $g\in V_{2,k+1}$. By induction hypothesis (on the
sets $A_{1},\ldots,A_{k}$),
we
have
$$V_{1,k+1}=\left(\left(\bigcap_{i=1}^{k}V_{A_{i}}\right)+\operatorname{ker}%
\beta_{n,n-1}^{\ast}\right)\cap(V_{A_{k+1}}+\operatorname{ker}\beta_{n,n-1}^{%
\ast}).$$
(6)
By Equation (6) and Proposition 4, we have
$$g=g_{1}+\beta_{n-1,n-2}^{\ast}h_{1}=g_{2}+\beta_{n-1,n-2}^{\ast}h_{2},$$
(7)
where
$g_{1}\in\cap_{i=1}^{k}V_{A_{i}}$, $g_{2}\in V_{A_{k+1}}$ and $h_{1},h_{2}\in{\mathbb{F}}_{2}^{[\Omega]^{n-2}}$. We claim that (up to replacing $h_{1}$ by
$h_{1}+l$, where $l\in\operatorname{ker}\beta_{n-1,n-2}^{\ast}$), we may assume that
$h_{1}-h_{2}\in\cap_{i=1}^{k}V_{A_{i}\cap A_{k+1}}$.
Let $w$ be an $(n-1)$-subset of $\Omega$ contained in $A_{i}\cap A_{k+1}$
for some $i=1,\ldots,k$. Since $g_{1}\in V_{A_{i}}$ and $g_{2}\in V_{A_{k+1}}$, we see that $g_{1}(w)=g_{2}(w)=0$. So, from
Equation (7) we obtain
$$g(w)=(\beta_{n-1,n-2}^{\ast}h_{1})(w)=(\beta_{n-1,n-2}^{\ast}h_{2})(w),$$
that is,
$(\beta_{n-1,n-2}^{\ast}(h_{1}-h_{2}))(w)=0$. As $w$ is an arbitrary $(n-1)$-subset of
$A_{i}\cap A_{k+1}$, Lemma 9
yields $h_{1}-h_{2}\in V_{A_{i}\cap A_{k+1}}+\operatorname{ker}\beta_{n-1,n-2}^{\ast}$. As
$i$ is an arbitrary element in $\{1,\ldots,k\}$, we get
$$h_{1}-h_{2}\in\bigcap_{i=1}^{k}(V_{A_{i}\cap A_{k+1}}+\operatorname{ker}\beta_%
{n-1,n-2}^{\ast}).$$
Since $k+1<n$, we have $k<n-1$ and so we may now apply our
inductive hypothesis on the sets $A_{1}\cap A_{k+1},\ldots,A_{k}\cap A_{k+1}$. We have
$$h_{1}-h_{2}\in\left(\bigcap_{i=1}^{k}V_{A_{i}\cap A_{k+1}}\right)+%
\operatorname{ker}\beta_{n-1,n-2}^{\ast}.$$
(8)
From Equation (8), we get
$h_{1}-h_{2}=h+l$, where $h\in\cap_{i=1}^{k}V_{A_{i}\cap A_{k+1}}$ and $l\in\operatorname{ker}\beta_{n-1,n-2}^{\ast}$. Set $h_{1}^{\prime}=h_{1}+l$. We have
$$h_{1}^{\prime}-h_{2}=h_{1}+l-h_{2}=h\in\cap_{i=1}^{k}V_{A_{i}\cap A_{k+1}}$$
and our claim is proved.
Let $t$ be the element of ${\mathbb{F}}_{2}^{[\Omega]^{n-2}}$ defined by
$$t(w)=\left\{\begin{array}[]{ccl}h_{1}(w)&&\textrm{if }w\subseteq A_{i}\textrm{%
for some }i=1,\ldots,k,\\
h_{2}(w)&&\textrm{if }w\subseteq A_{k+1},\\
0&&\textrm{otherwise}.\end{array}\right.$$
Note that the function $t$ is well-defined. Indeed, recall that $n>2$
and note that if $w$ is an
$(n-2)$-subset of $\Omega$ with $w\subseteq A_{i}\cap A_{k+1}$ (for some
$i=1,\ldots,k$), then $h_{1}(w)=h_{2}(w)$ as $h_{1}-h_{2}\in V_{A_{i}\cap A_{k+1}}$.
We claim that $g+\beta_{n-1,n-2}^{\ast}t\in\cap_{i=1}^{k+1}V_{A_{i}}$. We have to show that $g+\beta_{n-1,n-2}^{\ast}t$ vanishes in $[A_{i}]^{n-1}$, for $i=1,\ldots,k+1$. Let
$w$ be an $(n-1)$-subset of $\Omega$ with $w\subseteq A_{i}$, for some
$i=1,\ldots,k+1$. If $i\leq k$, then we have
$$(g+\beta_{n-1,n-2}^{\ast}t)(w)=(g_{1}(w)+\beta_{n-1,n-2}^{\ast}h_{1}(w))+\beta%
_{n-1,n-2}h_{1}(w)=0,$$
where in the first equality we used Equation (7) and the fact
that $t$ and $h_{1}$ coincide in $[A_{i}]^{n-2}$, and in the second
equality we used that $g_{1}\in V_{A_{i}}$.
Similarly, if $i=k+1$, then
$$(g+\beta_{n-1,n-2}^{\ast}t)(w)=(g_{2}(w)+\beta_{n-1,n-2}^{\ast}h_{2}(w))+\beta%
_{n-1,n-2}h_{2}(w)=0,$$
where in the first equality we used Equation (7) and the fact
that $t$ and $h_{2}$ coincide in $[A_{k+1}]^{n-2}$, and in the second
equality we used that $g_{2}\in V_{A_{k+1}}$.
Finally, as $\beta_{n-1,n-2}^{\ast}t\in\operatorname{ker}\beta_{n,n-1}^{\ast}$, we get that
$g\in V_{2,k+1}$.
Proposition 23
The theory $T_{n}$ has $k$-uniqueness for every $k\leq n$.
Proof.Let $k$ be an integer with $k\leq n$ and
$a:P(k)^{-}\to\mathcal{C}_{T_{n}}$ be a $k$-amalgamation problem. We need
to show
that $a$ has at most one solution up to
isomorphism. Since every stable theory has $1$- and $2$-uniqueness, we
may assume that $k\geq 3$. Set
$\Gamma_{1}=\operatorname{Aut}(a([k-1])/\cup_{i=1}^{k-1}a([k]-i))$
and $\Gamma_{2}=\operatorname{Aut}(a([k-1])/\cup_{i=1}^{k-1}a([k-1]-i))$. By [9, Proposition $3.5$], it is
enough to prove
that
$$\Gamma_{1}=\Gamma_{2},$$
(9)
i.e. $\overline{\Gamma_{1}},\overline{\Gamma_{2}}$ give rise to the same action on
$a([k-1])$ (see Definition 12).
By Remark 20, the algebraically closed
sets of finite subsets of $M_{n}$ are of the form $\operatorname{acl}(A)$, for some finite subset $A$ of the sort
$\Omega$. By Corollary 17 the setwise stabilizer of $\operatorname{acl}(A)$ in
$\operatorname{Aut}(M_{n})$ is simply $(\operatorname{Sym}(\Omega\setminus A)\times\operatorname{Sym}(A))\ltimes%
\operatorname{im}\beta_{n,n-1}^{\ast}$. Using
Corollary 19, we get that the
pointwise stabilizer of
$\operatorname{acl}(A)$ in $\operatorname{Aut}(M_{n})$ is $\operatorname{Sym}(\Omega\setminus A)\ltimes W_{A}$.
Let $a(i)=\operatorname{acl}(B_{i})$, where $B_{i}$ are finite subsets of $M_{n}$ for $1\leq i\leq k$. Set $A_{i}=\mathop{\mathrm{supp}}(B_{i})$, for $1\leq i\leq k$, and
$A=\cup_{i=1}^{k-1}A_{i}$. Note that by definition of amalgamation problem and by Proposition 21, we
have $a([k-1])=\operatorname{acl}(A)$. Therefore, by the previous paragraph, as
$k\geq 3$, we
get that $\overline{\Gamma_{1}}$ is equal to
$$((\operatorname{Sym}(\Omega\setminus A)\times\operatorname{Sym}(A))\ltimes%
\operatorname{im}\beta_{n,n-1}^{\ast})\cap\bigcap_{i=1}^{k-1}(\operatorname{%
Sym}(\Omega\setminus((A\cup A_{k})\setminus A_{i}))\ltimes W_{(A\cup A_{k})%
\setminus A_{i}})$$
i.e.
$$\overline{\Gamma_{1}}=\operatorname{Sym}(\Omega\setminus(A\cup A_{k}))\ltimes%
\bigcap_{i=1}^{k-1}W_{(A\cup A_{k})\setminus A_{i}}$$
(10)
and $\overline{\Gamma_{2}}$ is equal to
$$((\operatorname{Sym}(\Omega\setminus A)\times\operatorname{Sym}(A))\ltimes%
\operatorname{im}\beta_{n,n-1}^{\ast})\cap\bigcap_{i=1}^{k-1}(\operatorname{%
Sym}(\Omega\setminus(A\setminus A_{i}))\ltimes W_{A\setminus A_{i}})$$
i.e.
$$\overline{\Gamma_{2}}=\operatorname{Sym}(\Omega\setminus A)\ltimes\bigcap_{i=1%
}^{k-1}W_{A\setminus A_{i}}.$$
(11)
As $\operatorname{Sym}(\Omega\setminus(A\cup A_{k}))$ and $\operatorname{Sym}(\Omega\setminus A)$
act trivially on the elements of $\operatorname{acl}(A)$, by
Equations $(\ref{eq:G1})$ and $(\ref{eq:G2})$, in order to prove that
$\Gamma_{1}=\Gamma_{2}$ it suffices to show that
$$W_{1}=\bigcap_{i=1}^{k-1}W_{(A\cup A_{k})\setminus A_{i}}\quad\textrm{and}%
\quad W_{2}=\bigcap_{i=1}^{k-1}W_{A\setminus A_{i}}$$
induce the same action on $\operatorname{acl}(A)$. Also, $W_{1}$ and $W_{2}$ act
trivially on the elements belonging to the sorts $\Omega$ and
$[\Omega]^{n}$ of $M_{n}$. Thus, it suffices to study the action of $W_{1}$
and $W_{2}$ on the elements of $\operatorname{acl}(A)$ belonging to the sort
$[\Omega]^{n}\times{\mathbb{F}}_{2}$, that is, on
$[A]^{n}$. Clearly,
$W_{1}\subseteq W_{2}$. Therefore, it remains to show that for every element
$f$ of $W_{2}$ there exists an element $\overline{f}$ of $W_{1}$ such that
$f$ and $\overline{f}$ induce the same action on $[A]^{n}$.
Let $f$ be in $W_{2}$. By
Definition 10, we get that $f=\beta_{n,n-1}^{\ast}g$, for some
$g\in\cap_{i=1}^{k-1}(V_{A\setminus A_{i}}+\operatorname{ker}\beta_{n,n-1}^{%
\ast})$.
Lemma 22 (applied to $k-1,n$ and $(A\setminus A_{1}),\ldots,(A\setminus A_{k-1})$) yields
$$\bigcap_{i=1}^{k-1}\left(V_{A\setminus A_{i}}+\operatorname{ker}\beta_{n,n-1}^%
{\ast}\right)=\left(\bigcap_{i=1}^{k-1}V_{A\setminus A_{i}}\right)+%
\operatorname{ker}\beta_{n,n-1}^{\ast}.$$
Thence, up to replacing $g$ by $g+l$ (for some $l\in\operatorname{ker}\beta_{n,n-1}^{\ast}$), we may assume that $g\in\cap_{i=1}^{k-1}V_{A\setminus A_{i}}$. Let $\overline{g}$ be the
function in ${\mathbb{F}}_{2}^{[\Omega]^{n-1}}$ defined
by
$$\overline{g}(w)=\left\{\begin{array}[]{ccl}g(w)&&\textrm{if }w\subseteq A,\\
0&&\textrm{otherwise}.\end{array}\right.$$
Set $\overline{f}=\beta_{n,n-1}^{\ast}\overline{g}$. By construction,
$f$ and $\overline{f}$ coincide in $[A]^{n}$, that is,
$f$ and $\overline{f}$ induce the same action on $[A]^{n}$.
Thus, it remains to prove that $\overline{f}\in W_{1}$, that is,
$\overline{f}$ vanishes on every $n$-subset $L$ of $(A\cap A_{i})\setminus A_{i}$, for $i=1,\ldots,k$. Let $L$ be an
$n$-subset of $(A\cup A_{k})\setminus A_{i}$. We consider three cases
$L\subseteq A$, $|L\cap A_{k}|\geq 2$ and $|L\cap A_{k}|=1$.
If $L\subseteq A$, then
$\overline{f}(L)=f(L)=0$ (because $f$ and
$\overline{f}$ coincide on $[A]^{n}$).
If $|L\cap A_{k}|\geq 2$, then $(L\setminus\{x\})\nsubseteq A$, for every
$x$ in $L$. By definition of $\overline{g}$, we have
$\overline{g}(L\setminus\{x\})=0$ and $\overline{f}(L)=\sum_{x\in L}\overline{g}(L\setminus\{x\})=0$.
If $|L\cap A_{k}|=1$ and $L\cap A_{k}=\{\overline{x}\}$, then (arguing as
in the previous paragraph)
$\overline{f}(L)=\sum_{x\in L}\overline{g}(L\setminus\{x\})=g(L\setminus\{%
\overline{x}\})$. As
$L\subseteq(A\cup A_{k})\setminus A_{i}$, we have that
$L\setminus\{\overline{x}\}\subseteq A\setminus A_{i}$. Since $g\in V_{A\setminus A_{i}}$, we get that
$\overline{g}(L\setminus\{\overline{x}\})=g(L\setminus\{\overline{x}\})=0$.
J.Goodrick and A.Kolesnikov recently proved that if a complete stable theory
$T$ has $k$-uniqueness for every $2\leq k\leq n$, then $T$ has
$n+1$-existence [7]. For completeness we report the proof of
their result.
Theorem 24
Let $T$ be a complete stable theory. If $T$ has
$k$-uniqueness for
every $2\leq k\leq n$, then $T$ has $n+1$-existence.
Proof.
Note that the existence and the uniqueness of nonforking extensions of
types in a stable theory yields that any stable theory has both
$2$-existence and $2$-uniqueness.
Since $T$ is a
complete stable theory, for every regular cardinal $k$, there exists a
saturated model of cardinality $k$. In the sequel we shall consider
the objects of $\mathcal{C}_{T}$ lying inside a very large saturated
“monster model” $\mathfrak{C}$ of $T$.
Suppose $a$ is an $(n+1)$-amalgamation problem. We have to prove that
$a$ has a solution $a^{\prime}$. First, let $B_{0}$ and $B_{1}$ be sets of
$\mathfrak{C}$ such that
$\textup{tp}(B_{0}/a(\emptyset))=\textup{tp}(a([n])/a(\emptyset))$, $\textup{tp}(B_{1}/a(\emptyset))=\textup{tp}(a(\{n+1\})/a(\emptyset))$, and
$$B_{0}\mathop{\mathchoice{\kern 7.499886pt\hbox to 0.0pt{\hss$\mid$\hss}\lower 6%
.299904pt\hbox to 0.0pt{\hss$\smile$\hss}\kern 7.499886pt\displaystyle{}}{%
\kern 7.499886pt\hbox to 0.0pt{\hss$\mid$\hss}\lower 6.299904pt\hbox to 0.0pt{%
\hss$\smile$\hss}\kern 7.499886pt\textstyle{}}{\kern 7.499886pt\hbox to 0.0pt{%
\hss$\mid$\hss}\lower 6.299904pt\hbox to 0.0pt{\hss$\smile$\hss}\kern 7.499886%
pt\scriptstyle{}}{\kern 7.499886pt\hbox to 0.0pt{\hss$\mid$\hss}\lower 6.29990%
4pt\hbox to 0.0pt{\hss$\smile$\hss}\kern 7.499886pt\scriptscriptstyle{}}}_{a(%
\emptyset)}B_{1}.$$
Let $\sigma_{0}$ and $\sigma_{1}$ be two
automorphisms of $\mathfrak{C}$ fixing pointwise $a(\emptyset)$ and
such
that $B_{0}=\sigma_{0}(a([n]))$, $B_{1}=\sigma_{1}(a(\{n+1\}))$.
Define $a^{\prime}([n+1])$ to be the algebraic closure of
$B_{0}\cup B_{1}$. To determine the solution $a^{\prime}$ of $a$, it remains to define
the transition maps $a^{\prime}_{s,[n+1]}:a^{\prime}(s)\to a^{\prime}([n+1])$, for all subsets
$s$ of
$[n+1]$. The map $a^{\prime}_{\emptyset,[n+1]}$ must be the
identity on $a(\emptyset)$. For $i$ in $[n]$, we let
$a^{\prime}_{\{i\},[n+1]}:a(\{i\})\to a^{\prime}([n+1])$ be the map $\sigma_{0}\circ a_{\{i\},[n]}$, and we let $a^{\prime}_{\{n+1\},[n+1]}$ be the map $\sigma_{1}$. Now, the following claim concludes the proof of the theorem.
Claim:
For every proper non-empty subset $s$ of $[n+1]$, there is a way to define the
transition maps
$a^{\prime}_{s,[n+1]}$, which is
consistent with $a$ and the definition of
$a^{\prime}_{\{i\},[n+1]}$ given above, and such that
$$a^{\prime}_{s,[n+1]}(a(s))=\operatorname{acl}\left(\bigcup_{i\in s}a(\{i\})%
\right).$$
We argue by induction on the size $k$ of the set $s$. If $k=1$, then
there is nothing to prove.
Suppose we have defined $a^{\prime}_{s,[n+1]}$ as in the claim, for all $s\subseteq[n+1]$ such that $|s|<k$. Let $s$ be a subset of
$[n+1]$ such that $|s|=k$. The family of
sets $\{a(t)\mid t\subsetneq s\}$ forms a
$k$-amalgamation problem with the same transition maps as $a$. Call $a^{1}$
this amalgamation problem. By the induction hypothesis, the
family of sets
$\{a^{\prime}_{t,[n+1]}(a(t))\mid t\subsetneq s\}$ forms
another $k$-amalgamation problem with the transition maps
given by set inclusions. Call $a^{2}$ this amalgamation problem. Notice
that $a^{1}$ and $a^{2}$
are isomorphic, and that both have independent solutions. Namely, $a^{1}$ can be
completed to $a(s)$ using the transition maps in $a$, and $a^{2}$ has a
natural solution $(a^{2})^{\prime}$ such that
$$(a^{2})^{\prime}(s)=\operatorname{acl}\left(\bigcup_{i\in s}a(\{i\})\right),$$
where the transition maps are again given by set
inclusions. So, by the $k$-uniqueness property, there is an
isomorphism of these solutions, which yields the desired transition
map $a^{\prime}_{s,[n+1]}$ from $a(s)$ to $\operatorname{acl}(\bigcup_{i\in s}a(\{i\}))$.
Now we are ready to prove that $T_{n}$ has $k$-existence for every $k\leq n+1$.
Proposition 25
The theory $T_{n}$ has $k$-existence for every
$k\leq n+1$.
Proof.
By definition, $T_{n}=\operatorname{Th}(M_{n})$ is complete. Since $T_{n}$ is a stable
theory, the proof of this proposition follows at
once from Proposition 23 and Theorem 24.
Next, we show that $T_{n}$ does not have $n+1$-uniqueness.
Proposition 26
The theory $T_{n}$ does not have $n+1$-uniqueness.
Proof. Recall that by construction $n\geq 2$. Let
$a:P(n+1)^{-}\to\mathcal{C}_{T_{n}}$ be the $(n+1)$-amalgamation problem
defined on the objects by $a(s)=\operatorname{acl}(s)$ and where the morphisms are
inclusions. In order to
prove this proposition we show the following equations:
$$\displaystyle|\operatorname{Aut}(\operatorname{acl}([n])/\cup_{i=1}^{n}%
\operatorname{acl}([n+1]-i))|$$
$$\displaystyle=$$
$$\displaystyle 1,$$
(12)
$$\displaystyle|\operatorname{Aut}(\operatorname{acl}([n])/\cup_{i=1}^{n}%
\operatorname{acl}([n]-i))|$$
$$\displaystyle=$$
$$\displaystyle 2.$$
(13)
In fact, by [9, Proposition $3.5$],
Equations (12), (13) yield that $a$ has more than one
solution up to isomorphism, i.e. $T_{n}$ does
not have $n+1$-uniqueness.
We start by proving Equation (12). Since $[n],[n+1]-i$ have
size $n$, Proposition 15 yields
$\operatorname{acl}_{M_{n}}([n])=[n]\cup\{[n]\}\cup\{([n],0),([n],1)\}$ and
$\operatorname{acl}_{M_{n}}([n+1]-i)=([n+1]-i)\cup\{[n+1]-i\}\cup\{([n+1]-i,0),%
([n+1]-i,1)\}$.
By the description given in the previous paragraph, every permutation
in $\operatorname{Sym}(\Omega)$ fixing
pointwise the elements in $\cup_{i=1}^{n}\operatorname{acl}([n+1]-i)$ also fixes
pointwise every element in $\operatorname{acl}([n])$. Therefore, it suffices to
consider the elements in $\operatorname{im}\beta_{n,n-1}^{\ast}$. Let $f$ be in
$\operatorname{im}\beta^{\ast}_{n,n-1}$ and suppose that $f$ fixes every element in
$\cup_{i=1}^{n}\operatorname{acl}([n+1]-i)$, i.e. $f([n+1]-i)=0$, for $1\leq i\leq n$. Let $g\in\mathbb{F}_{2}^{[\Omega]^{n-1}}$ such that
$f=\beta_{n,n-1}^{\ast}g$. We have
$$0=\sum_{i=1}^{n}f([n+1]-i)=\sum_{i=1}^{n}\sum_{j\neq i}^{n+1}g([n+1]\setminus%
\{i,j\}).$$
(14)
Now, for $j\neq n+1$, the summand $g([n+1]\setminus\{i,j\})$ appears
twice in Equation (14) and therefore over $\mathbb{F}_{2}$
their sum is zero. Hence
$$0=\sum_{i=1}^{n}f([n+1]-i)=\sum_{i=1}^{n}g([n]-i)=(\beta_{n,n-1}^{\ast}g)([n])%
=f([n]).$$
This yields that $f$ fixes $([n],0),([n],1)$. Hence
Equation (12) follows.
We now prove Equation (13). Since $[n]-i$ has size $n-1$,
Proposition 15 implies $\operatorname{acl}_{M_{n}}([n]-i)=[n]-i$. Therefore,
$$\cup_{i=1}^{n}\operatorname{acl}_{M_{n}}([n]-i)=\cup_{i=1}^{n}([n]-i)=[n].$$
Also,
$\operatorname{acl}_{M_{n}}([n])=[n]\cup\{[n]\}\cup\{([n],0),([n],1)\}$. Corollary 17 and Corollary 19 yield that
every element of $\operatorname{Aut}(\operatorname{acl}([n])/\cup_{i=1}^{n}\operatorname{acl}([n%
]-i))$ fixes the
elements belonging to the sorts $\Omega$ and $[\Omega]^{n}$ of
$\operatorname{acl}_{M_{n}}([n])$. Hence, in order to prove
Equation (13), it suffices to find an automorphism of
$\operatorname{acl}_{M_{n}}([n])$ mapping $([n],0)$ into $([n],1)$. Let
$g\in{\mathbb{F}}_{2}^{[\Omega]^{n-1}}$ with $g([n-1])=1$ and $g(w)=0$ for
$w\neq[n-1]$. Set $f=\beta_{n,n-1}^{\ast}g$ and note that $f([n])=1$. As
$\operatorname{Aut}(M_{n})=\operatorname{im}\beta_{n,n-1}^{\ast}\rtimes%
\operatorname{Sym}(\Omega)$, the map $f$
is an automorphism of $M_{n}$. By construction $f$ is an automorphism
of $\operatorname{acl}_{M_{n}}([n])$ and $([n],0)^{f}=([n],0+f([n]))=([n],1)$.
Finally, we show that $T_{n}$ does not have $n+2$-existence.
Proposition 27
The theory $T_{n}$ does not have $n+2$-existence.
Proof.
We construct an $n+2$-amalgamation problem $a$ over $\emptyset$ (that
is, $a(\emptyset)=\emptyset$) for $T_{n}$
with no solution.
Let $g$ be the element of $\mathbb{F}_{2}^{[\Omega]^{n-1}}$ defined by
$$g(w)=\left\{\begin{array}[]{ccl}1&&\textrm{if }w=[n-1],\\
0&&\textrm{if }w\neq[n-1].\end{array}\right.$$
Consider
$f=\beta_{n,n-1}^{\ast}g$ and note that, as
$\operatorname{Aut}(M_{n})=\operatorname{im}\beta_{n,n-1}^{\ast}\rtimes%
\operatorname{Sym}(\Omega)$, the element
$f$ is an automorphism of $M_{n}$.
Let $a$ be the functor
$a:P(n+2)^{-}\to\mathcal{C}_{T_{n}}$ defined on
the objects by $a(s)=\operatorname{acl}(s)$ and with morphisms defined by
$$\displaystyle a_{s,s^{\prime}}=\left\{\begin{array}[]{ccl}f|_{a(s)}&&\textrm{%
if }s=[n]\textrm{ and }s^{\prime}=[n+1],\\
\textrm{inclusion}&&\textrm{otherwise},\end{array}\right.$$
(15)
where $f|_{a(s)}$ denotes the restriction of the automorphism $f$ to
$a(s)$. It is not obvious from Equation $(\ref{eq:6})$
that $a$ is a functor. Therefore, in the following paragraph, we prove
that $a$ is
well-defined, that is, $a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=a_{s_{1},s_{3}}$ for every
$s_{1},s_{2},s_{3}$ in $P(n+2)^{-}$ with $s_{1}\subseteq s_{2}\subseteq s_{3}$.
If $s_{2}\neq[n+1]$ and $s_{3}\neq[n+1]$, then (by Equation $(\ref{eq:6})$)
the morphisms
$a_{s_{1},s_{2}},a_{s_{2},s_{3}}$ and
$a_{s_{1},s_{3}}$ are inclusions and so clearly $a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=a_{s_{1},s_{3}}$. If $s_{2}=[n+1]$, then $s_{2}$ is a maximal
element of the partially ordered set $P(n+2)^{-}$. Thence
$s_{3}=s_{2}$ and, by Equation $(\ref{eq:6})$,
$a_{s_{2},s_{3}}$ is the identity map. Thus $a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=a_{s_{1},s_{3}}$. In particular, from now on we may assume that
$s_{3}=[n+1]$ and $s_{2}\neq[n+1]$. As $s_{1}\subseteq s_{2}$, if $s_{2}\neq[n]$, then $s_{1}\neq[n]$ and so, by
Equation $(\ref{eq:6})$, the morphisms
$a_{s_{1},s_{2}},a_{s_{2},s_{3}}$ and
$a_{s_{1},s_{3}}$ are inclusions and $a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=a_{s_{1},s_{3}}$. If $s_{2}=s_{1}=[n]$, then $a_{s_{1},s_{2}}$ is the
identity map and $a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=a_{s_{1},s_{3}}$. The only case that remains to consider is
$s_{3}=[n+1]$, $s_{2}=[n]$ and $s_{1}\neq[n]$. Thence $a_{s_{1},s_{2}}$ and
$a_{s_{1},s_{3}}$ are inclusion maps and $a_{s_{2},s_{3}}=f|_{a(s_{2})}$. Since
$s_{1}\subseteq s_{2}=[n]$ and $s_{1}\neq[n]$, we have
$|s_{1}|<n-1$. Therefore,
$a(s_{1})=\operatorname{acl}(s_{1})$ and by Proposition 15 $\operatorname{acl}_{M_{n}}(s_{1})=s_{1}$ consists only of elements belonging to the sort
$\Omega$ of $M_{n}$. As $f$ acts trivially on the elements belonging to
the sort $\Omega$, by Proposition 16 we obtain $a_{s_{2},s_{3}}\circ a_{s_{1},s_{2}}=(f|_{a(s_{2})})|_{a(s_{1})}=f|_{a(s_{1})}%
=a_{s_{1},s_{3}}$.
Finally, this proves that $a:P(n+2)^{-}\to\mathcal{C}_{T_{n}}$ is a functor.
By Proposition 14,
$a(\emptyset)=\operatorname{acl}(\emptyset)=\emptyset$. Therefore, the functor $a$ is an
$n+2$-amalgamation problem over $\emptyset$ for $M_{n}$.
We claim that
$a$ cannot be extended to
$P(n+2)$. We argue by contradiction. Let $\overline{a}:P(n+2)\to\mathcal{C}_{T_{n}}$ be a solution of $a$. In particular, $\overline{a}$
is an extension of $a$ to the whole of $P(n+2)$.
Denote by $x_{i}$ the morphisms $\overline{a}_{[n+2]-i,[n+2]}$, for
$1\leq i\leq n+2$. So, by definition of morphism, $x_{i}$ is the
restriction to $\operatorname{acl}([n+2]-i)$ of an automorphism $f_{i}\sigma_{i}$ of
$M_{n}$, where $f_{i}\in\operatorname{im}\beta_{n,n-1}^{\ast}$ and $\sigma_{i}\in\operatorname{Sym}(\Omega)$.
Since $\overline{a}$ is a functor and $\overline{a}$ extends $a$, we get
$$\displaystyle x_{i}\circ a_{[n+2]\setminus\{i,j\},[n+2]-i}$$
$$\displaystyle=$$
$$\displaystyle\overline{a}_{[n+2]-i,[n+2]}\circ\overline{a}_{[n+2]\setminus\{i,%
j\},[n+2]-i}$$
$$\displaystyle=$$
$$\displaystyle\overline{a}_{[n+2]-j,[n+2]}\circ\overline{a}_{[n+2]\setminus\{i,%
j\},[n+2]-j}$$
$$\displaystyle=$$
$$\displaystyle x_{j}\circ a_{[n+2]\setminus\{i,j\},[n+2]-j}.$$
Let $i$ and $j$ be in $[n+2]$ with $i\neq j$. Fix an enumeration of $\operatorname{acl}_{M_{n}}([n+2]\setminus\{i,j\})$ and denote it as $\overline{b_{ij}}=(b_{ij_{1},},\dots)$. Then, as it is shown in Proposition 16 $\overline{b_{ij}}\in\operatorname{acl}([n+2]\setminus\{i,j\})$ and, of course, also in $\operatorname{acl}([n+2]\setminus\{i\})$. By
Proposition 15 the ordered pair $([n+2]\setminus\{i,j\},0)$ belongs to the sort
$[\Omega]^{n}\times{\mathbb{F}}_{2}$ of $M_{n}$ and lies in $\operatorname{acl}_{M_{n}}([n+2]\setminus\{i,j\})$. Set $b_{ij_{1}}=([n+2]\setminus\{i,j\},0)$. We have
$$\displaystyle x_{i}(\overline{b_{ij}})$$
$$\displaystyle=$$
$$\displaystyle x_{i}(([n+2]\setminus\{i,j\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle((([n+2]\setminus\{i,j\})^{\sigma_{i}},0+f_{i}([n+2]\setminus\{i,%
j\})),\dots)$$
$$\displaystyle=$$
$$\displaystyle((([n+2]\setminus\{i,j\})^{\sigma_{i}},m_{ij}),\dots),$$
where
$$m_{ij}=f_{i}([n+2]\setminus\{i,j\}).$$
(18)
Consider
the matrix $M=(m_{ij})_{ij}$, with $m_{ii}=0$.
Let $i$ and $j$ be in $[n+2]$ with $i\neq j$ and $\{i,j\}\neq\{n+1,n+2\}$. By
Equation $(\ref{eq:6})$ and by hypothesis on $\{i,j\}$, the morphism
$a_{[n+2]\setminus\{i,j\},[n+2]-i}$ is an inclusion map and so it fixes
$([n+2]\setminus\{i,j\},0)$. Therefore,
$$\displaystyle x_{i}\circ a_{[n+2]\setminus\{i,j\},[n+2]-i}(\overline{b_{ij}})$$
$$\displaystyle=$$
$$\displaystyle x_{i}\circ a_{[n+2]\setminus\{i,j\},[n+2]-i}(([n+2]\setminus\{i,%
j\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle x_{i}(([n+2]\setminus\{i,j\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle((([n+2]\setminus\{i,j\})^{\sigma_{i}},m_{ij}),\dots),$$
where in the last equality we used
Equations $(\ref{eq:1212})$ and $(\ref{eq:1010})$. Similarly,
replacing $i$ with $j$, we obtain
$$\displaystyle x_{i}\circ a_{[n+2]\setminus\{i,j\},[n+2]-i}(\overline{b_{ij}})$$
$$\displaystyle=$$
$$\displaystyle x_{j}\circ a_{[n+2]\setminus\{i,j\},[n+2]-j}(([n+2]\setminus\{i,%
j\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle x_{j}(([n+2]\setminus\{i,j\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle((([n+2]\setminus\{i,j\})^{\sigma_{j}},m_{ji}),\dots).$$
Now, by Equation $(\ref{eq4})$, we have
$$\displaystyle x_{i}\circ a_{[n+2]\setminus\{i,j\},[n+2]-i}(\overline{b_{ij}})$$
$$\displaystyle=$$
$$\displaystyle x_{i}\circ a_{[n+2]\setminus\{i,j\},[n+2]-i}(([n+2]\setminus\{i,%
j\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle x_{j}\circ a_{[n+2]\setminus\{i,j\},[n+2]-j}(([n+2]\setminus\{i,%
j\},0),\dots).$$
In particular,
$$m_{ij}=m_{ji},\qquad\textrm{for every }i,j\textrm{ with }\{i,j\}\neq\{n+1,n+2\}.$$
(19)
By Equation $(\ref{eq:6})$ the morphism
$a_{[n+2]\setminus\{n+1,n+2\},[n+2]-(n+1)}$ is an inclusion map and so
it fixes
$([n+2]\setminus\{n+1,n+2\},0)$. Therefore,
$$\displaystyle x_{n+1}\circ a_{[n+2]\setminus\{n+1,n+2\},[n+2]-(n+1)}(\overline%
{b}_{n+1,n+2})$$
$$\displaystyle=$$
$$\displaystyle x_{n+1}\circ a_{[n+2]\setminus\{n+1,n+2\},[n+2]-(n+1)}(([n+2]%
\setminus\{n+1,n+2\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle x_{n+1}(([n+2]\setminus\{n+1,n+2\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle((([n+2]\setminus\{n+1,n+2\})^{\sigma_{n+1}},m_{(n+1)(n+2)}),%
\dots).$$
By Equation $(\ref{eq:6})$ the morphism
$f|_{a([n])}=a_{[n],[n+1]}=a_{[n+2]\setminus\{n+1,n+2\},[n+2]-(n+2)}$
maps $([n+2]\setminus\{n+1,n+2\},0)$ to
$([n+2]\setminus\{n+1,n+2\},1)$. Therefore,
$$\displaystyle x_{n+2}\circ a_{[n+2]\setminus\{n+1,n+2\},[n+2]-(n+2)}(\overline%
{b}_{n+1,n+2})$$
$$\displaystyle=$$
$$\displaystyle x_{n+2}\circ a_{[n+2]\setminus\{n+1,n+2\},[n+2]-(n+2)}(([n+2]%
\setminus\{n+1,n+2\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle x_{n+2}\circ f|_{a([n])}(([n+2]\setminus\{n+1,n+2\},0),\dots)$$
$$\displaystyle=$$
$$\displaystyle x_{n+2}(([n+2]\setminus\{n+1,n+2\},1),\dots)$$
$$\displaystyle=$$
$$\displaystyle((([n+2]\setminus\{n+1,n+2\})^{\sigma_{n+2}},m_{(n+2)(n+1)}+1),%
\dots).$$
By Equation $(\ref{eq4})$ (applied to $i=n+1$ and $j=n+2$), we have
$$\displaystyle(([n+2]\setminus\{n+1,n+2\})^{\sigma_{n+1}},m_{(n+1)(n+2)})$$
$$\displaystyle=$$
$$\displaystyle(([n+2]\setminus\{n+1,n+2\})^{\sigma_{n+2}},m_{(n+2)(n+1)}+1)$$
and
$$m_{(n+1)(n+2)}=m_{(n+2)(n+1)}+1.$$
(20)
Now, we are ready to get a contradiction. We claim that each row of
$M$ adds up to zero. We have
$$\displaystyle\sum_{j=1}^{n+2}m_{ij}$$
$$\displaystyle=$$
$$\displaystyle\sum_{j\in([n+2]-i)}m_{ij}=\sum_{j\in([n+2]-i)}f_{i}([n+2]%
\setminus\{i,j\})$$
$$\displaystyle=$$
$$\displaystyle(\beta_{n+1,n}^{\ast}f_{i})([n+2]-i)=0,$$
where in the first equality we used that $m_{ii}=0$, in the second
equality we used Equation $(\ref{eq:1010})$ and in the last equality
we used that $f_{i}\in\operatorname{im}\beta_{n,n-1}^{\ast}=\operatorname{ker}\beta_{n+1,n}^%
{*}$. In particular,
the sum of all the entries of
$M$ is zero. Hence
$$0=\sum_{ij}m_{ij}=\sum_{i<j}(m_{ij}+m_{ji}).$$
By Equation $(\ref{eq:yyy})$,
$m_{ij}=m_{ji}$ if $\{i,j\}\neq\{n+1,n+2\}$. So, in the previous sum there is
only one non-zero summand. Namely,
$m_{(n+1)(n+2)}+m_{(n+2)(n+1)}=0$. Now, Equation $(\ref{eq:1414})$
yields
$$m_{(n+1)(n+2)}+m_{(n+2)(n+1)}=m_{(n+1)(n+2)}+m_{(n+1)(n+2)}+1=1,$$
a
contradiction. This contradiction finally
proves that the extension $\overline{a}$ does not exist.
Now, Theorem 2 follows at once from
Proposition 23, 25, 26, 27. Finally,
we point out that Proposition 26 also follows from
Theorem 24 and Proposition 27.
6 Extension of Example 1
In this section we remark that for every $n\geq 2$ the theories $T_{n}$ are stable and that the family of examples $\{M_{n}\}_{n\geq 2}$
generalizes the example due to E.Hrushovski given in [3], see
Example 1 in Section 1.
Definition 28
Let $\Omega$ be a countable set, and
$C=[\Omega]^{n}\times{\mathbb{Z}}/2{\mathbb{Z}}$. Also let
$E\subseteq\Omega\times[\Omega]^{2}$ be the membership relation, and let
$P$ be the subset of $C^{n+1}$ such that
$((w_{1},\delta_{1}),\dots,(w_{n+1},\delta_{n+1}))\in P$ if and only if
there are distinct $c_{1},\dots,c_{n+1}\in\Omega$ such that
$w_{i}=\{c_{1},\dots,c_{n+1}\}\setminus c_{i}$ and
$\delta_{1}+\cdots+\delta_{n+1}=0$. Now let $\overline{M}_{n}$ be the model with the
$3$-sorted universe $\Omega,[\Omega]^{n},C$ and equipped with relations $E,P$
and projection on the first coordinate $\pi:C\rightarrow[\Omega]^{n}$. Since $\overline{M}_{n}$ is a reduct of $(\Omega,{\mathbb{Z}}/2{\mathbb{Z}})^{\textrm{eq}}$, we get that $\operatorname{Th}(\overline{M}_{n})$ is stable.
Proposition 29
Let $\overline{M}_{n}$ be the structures described in Definition 28. Then
$\operatorname{Aut}(\overline{M}_{n})=\operatorname{im}\beta^{\ast}_{n,n-1}%
\rtimes\operatorname{Sym}(\Omega)$. In particular, $\overline{M}_{n}$
and $M_{n}$ are interdefinable.
Proof. First we show that $\operatorname{Sym}(\Omega)$ is a subgroup of
$\operatorname{Aut}(\overline{M}_{n})$. Indeed, the group $\operatorname{Sym}(\Omega)$ acts with its natural action
on the sorts $\Omega$ and $[\Omega]^{n}$ of $\overline{M}_{n}$. Also,
if $g\in\operatorname{Sym}(\Omega)$ and $(\{a_{1},\dots,a_{n}\},\delta)\in C$, then we set
$(\{a_{1},\dots,a_{n}\},\delta)^{g}=(\{a_{1}^{g},\dots,a_{n}^{g}\},\delta)$. This defines an
action of $\operatorname{Sym}(\Omega)$ on $\overline{M}_{n}$. It is
straightforward to see that the relations $E,P$ and the partition
given by the fibers of $\pi$ are preserved by
$\operatorname{Sym}(\Omega)$.
Hence, $\operatorname{Sym}(\Omega)\leq\operatorname{Aut}(\overline{M}_{n})$.
Let $\mu:\operatorname{Aut}(\overline{M}_{n})\rightarrow\operatorname{Sym}(\Omega)$ be the map given by
restriction on the sort $\Omega$ of $\overline{M}_{n}$. Since $\mu$ is a surjective
homomorphism, we
have that $\operatorname{Aut}(\overline{M}_{n})$ is a split extension of $\operatorname{ker}\mu$ by
$\operatorname{Sym}(\Omega)$. Every element of $\operatorname{ker}\mu$ preserves the fibres of
$\pi$ and fixes all the elements of
$[\Omega]^{n}$. So $\operatorname{ker}\mu$ is a closed $\operatorname{Sym}(\Omega)$-submodule of
${\mathbb{F}}_{2}^{[\Omega]^{n}}$.
Let $((w_{1},\delta_{1}),\dots,(w_{n+1},\delta_{n+1}))$ be in $P$ and
$f$ be in
$\operatorname{ker}\mu$. Since $\operatorname{ker}\mu$ preserves $P$, we have
$$f(w_{1})+\delta_{1}+\cdots+f(w_{n+1})+\delta_{n+1}=0.$$
From the definition of $P$ and $\beta_{n+1,n}^{\ast}$, we get
$$\operatorname{ker}\mu=\{f\in{\mathbb{F}}_{2}^{[\Omega]^{n}}\mid\sum_{x\in[w]^{%
n}}f(x)=0\textrm{ for every }w\in[\Omega]^{n+1}\}=\operatorname{ker}\beta_{n+1%
,n}^{\ast}.$$
By Proposition 4, we have
that $\operatorname{ker}\beta^{\ast}_{n+1,n}=\operatorname{im}\beta^{\ast}_{n,n-1}$. Therefore
$\operatorname{Aut}(\overline{M}_{n})=\operatorname{Aut}(M_{n})$ and $\overline{M}_{n},M_{n}$ are interdefinable.
Acknowledgements
The authors thank J.Goodrick and A.Kolesnikov for
the proof of Theorem 24 and for
giving their permission to include their proof in our paper. We are grateful to D. M. Evans for his stimulating suggestions and we thank
the anonymous referee for the very valuable comments and remarks on an earlier draft of the paper.
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46, (1983), 241–273. |
Spin current injection by intersubband transitions in quantum wells.
E. Ya. Sherman, Ali Najmaie, and J.E. Sipe
Department of Physics, University of Toronto,
60 St. George Street, Toronto, ON M5S 1A7, Canada
Abstract
We show that a pure spin current can be injected in quantum wells by
the absorption of linearly polarized infrared radiation, leading to transitions
between subbands. The magnitude and the direction
of the spin current depend on the Dresselhaus and Rashba spin-orbit
coupling constants and light frequency and, therefore, can be
manipulated by changing the light frequency and/or applying an external bias
across the quantum well. The injected spin current should be observable
either as a voltage generated via the anomalous spin-Hall effect, or by spatially resolved pump-probe
optical spectroscopy.
Spin current is an interesting physical phenomenon in its own right, and
could have application in the
delivery and transfer of electron spins in spintronics devices. From a
fundamental point of view, various issues raised in the theory of this
effect are far from being satisfactorily settled. As was shown by
Rashba Rashba03 , a spin current exists even in the equilibrium state
of a two-dimensional (2D) electron gas with spin-orbit (SO) coupling. The application
of an external electric field has been suggested as a strategy for driving the
system out of equilibrium and inducing a spin current exhibiting transport
effects. Mal’shukov et al. Malshukov03 and
Governale et al. Governale03 suggested
applying a time-dependent bias across a semiconducting heterostructure,
thus modulating the strength of the SO coupling and generating a spin
current. Murakami et al. Murakami03 and
Sinova et al. Sinova04 have shown that an in-plane electric
field can cause a spin current, leading to the ”intrinsic spin-Hall effect”.
Another possibility
for the injection of spin current is coherently controlled optical
excitations between the valence and the conduction band, as predicted by
Bhat and Sipe Bhat00 ; Bhat04 and observed experimentally in bulk crystals Stevens02 ; Hubner03 and quantum wells (QWs) Stevens03 .
Here we show that a spin
current can be injected in QWs by infrared (IR) light absorption, driving
transitions between different subbands. The injection of spin-polarized electric current
in QWs due to intersubband
transitions caused by circularly polarized radiation
has already been observed by Ganichev et al. Ganichev03 .
In contrast, here we investigate a pure spin current, where electrons
moving in opposite directions have opposite orientations of spins, not accompanied by a net
electrical current. We show that the
strength and direction of this pure spin current can be manipulated by
modulating the SO coupling strength via applied bias Nitta97
and/or adjusting the light frequency.
As an example we consider the (011) GaAs QW,
where the electron spins have a
considerable out-of plane component, thus making possible the observation of the pure
spin current by detecting the voltage generated via the anomalous spin-Hall effect Abakumov72 ; Bakun85 .
The first two subbands in the well are typically separated by the energy
$\hbar\omega_{0}\approx 100$ meV; the exact value depends on the width of the QW, dopant
concentration, and the boundary conditions. The SO Hamiltonian for the (011) QW,
${H}_{\mathrm{SO}}={H}_{D}+{H}_{R},$ is the sum of a Dresselhaus
term Dyakonov86 , ${H}_{D},$ originating from the unit cell
inversion asymmetry, and a Rashba term Rashba84 , ${H}_{R},$
originating from the asymmetric doping and/or a bias applied across the well:
$${H}_{D}^{[n]}=\alpha_{D}^{[n]}k_{y}{\sigma}^{{}_{z}}F_{[n]}(\mathbf{k)},\qquad%
{H}_{R}^{[n]}=\alpha_{R}^{[n]}\left({\sigma}^{x}k_{y}-{\sigma}^{y}k_{x}\right),$$
(1)
where $n$ is the subband index, $\mathbf{k}=(k_{x},k_{y})$ is the in-plane wavevector of
the electron envelope function,
$F_{[n]}(\mathbf{k)=}1-\left(k_{y}^{2}-2k_{x}^{2}\right)\lambda_{[n]}^{2}$, where
$\lambda_{[n]}$ depends on the QW width $w$, and the ${\sigma}^{i}$ are the
Pauli matrices. The $z-$axis is
perpendicular to the QW plane and the in-plane axes are: $x=[100]$ and
$y=[0\overline{1}1]$. The parameters $\alpha_{D}^{[n]}$ and
$\alpha_{R}^{[n]}$ depend on $n$; in the model of
rigid QW walls one has
$\alpha_{D}^{[n]}=-\alpha_{0}n^{2}\left(\pi/w\right)^{2}/2$, where $\alpha_{0}$ is the
Dresselhaus constant for the bulk, and $\lambda_{[n]}=w/n\pi$ Dyakonov86 .
The deviation of $F_{[n]}(\mathbf{k)}$ from unity becomes important at electron concentrations
$N_{\mathrm{el}}\approx 10^{12}$ cm${}^{-2}$.
The spin-related energy is given by
$E_{\mathrm{SO}}^{[n]}\left(\mathbf{k}\right)=\sqrt{\left(\alpha_{D}^{[n]}k_{y}%
F_{[n]}(\mathbf{k)}\right)^{2}+\left(\alpha_{R}^{[n]}k\right)^{2}},$ with ”up” $(u)$
and ”down” $(d)$ states having energies
$E_{u,d}^{[n]}\left(\mathbf{k}\right)=\pm E_{\mathrm{SO}}^{[n]}\left(\mathbf{k}\right)$,
and leads to the subband spectra:
$$\varepsilon_{s_{1}}=\frac{\hbar^{2}k^{2}}{2m}\pm E_{\mathrm{SO}}^{[1]}\left(%
\mathbf{k}\right),\qquad\varepsilon_{s_{2}}=\hbar\omega_{0}+\frac{\hbar^{2}k^{%
2}}{2m}\pm E_{\mathrm{SO}}^{[2]}\left(\mathbf{k}\right).$$
(2)
where $m$ is the electron effective mass and the indices $s_{1},s_{2}$
describe the $u(+)$ and $d(-)$ spin states in the subbands $n=1$ and $n=2$,
respectively. The corresponding spin eigenstates $\phi^{s_{n}}_{\bf k}$
result in expectation values of the spin components:
$$\langle\left.\phi^{s_{n}}_{\bf k}\right|\sigma^{z}\left|\phi^{s_{n}}_{\bf k}%
\right.\rangle=\pm\frac{\alpha_{D}^{[n]}k_{y}F_{[n]}(\mathbf{k)}}{E_{\mathrm{%
SO}}^{[n]}\left(\mathbf{k}\right)},\qquad\langle\left.\phi^{s_{n}}_{\bf k}%
\right|{\sigma}_{\|}\left|\phi^{s_{n}}_{\bf k}\right.\rangle=\pm\frac{\alpha_{%
R}^{[n]}}{E_{\mathrm{SO}}^{[n]}\left(\mathbf{k}\right)}(k_{y},-k_{x}),$$
(3)
where upper(lower) sign corresponds to the $u(d)$ state and $\mathbf{\sigma}_{\|}=(\sigma^{x},\sigma^{y})$.
There is not yet consensus in the literature on the fundamental description of spin current,
and the effect of disorder on it, as discussed e.g., in Ref.SHE ; spin current is not a ”true” current, in that its density
does not satisfy a continuity equation describing the evolution of a
spin density Rashba03 . Nonetheless, we introduce a ”physical” definition of spin current
per electron as:
$$j_{\mu}^{\beta}\left(\mathbf{k,}s_{n}\right)=\frac{\hbar}{4}\cdot\langle\left.%
\phi^{s_{n}}_{\bf k}\right|v_{\mu}{\sigma}^{\beta}+{\sigma}^{\beta}v_{\mu}%
\left|\phi^{s_{n}}_{\bf k}\right.\rangle,$$
(4)
where $\mu$ and $\beta$ are Cartesian indices.
Velocity components
${v}_{i}=\partial{H}/\hbar\partial k_{i}$ are the sums of normal
${v}_{i,{\rm n}}=\hbar k_{i}/m$ and anomalous terms given in our
model (Eq.(1)) by:
$$v_{x,{\rm an}}^{[n]}=-\frac{\alpha_{R}^{[n]}}{\hbar}{\sigma}^{y}+4k_{y}k_{x}%
\frac{\alpha_{D}^{[n]}}{\hbar}{\sigma}^{z}\lambda_{[n]}^{2},\qquad v_{y,{\rm an%
}}^{[n]}=\frac{\alpha_{D}^{[n]}}{\hbar}\left(1-\left(3k_{y}^{2}-2k_{x}^{2}%
\right)\lambda_{[n]}^{2}\right){\sigma}^{z}.$$
(5)
Below we consider only the spin current components associated with the $z-$axis spin projection. First we calculate the equilibrium spin current at
typical experimental conditions, where only the first subband is occupied,
and then find the changes induced by the
intersubband excitations. For this purpose we introduce the equilibrium
Fermi distribution function for two spin projections in the first subband:
$$f_{\pm}(\mathbf{k})=\frac{1}{\exp\left[\left(\hbar^{2}k^{2}/2m\pm E^{[1]}_{%
\mathrm{SO}}(\mathbf{k})-\mu\right)/k_{B}T\right]+1},$$
(6)
where $\mu$ is the chemical potential for a given $N_{\mathrm{el}}$, and $k_{B}$
is the Boltzmann’s constant. The spin current density component
$J_{y}^{z}$ is the sum of the normal, $J_{y,{\rm n}}^{z}$, and the anomalous,
$J_{y,{\rm an}}^{z}$, parts. By
integrating $j_{\mu}^{\beta}\left(\mathbf{k,}s_{1}\right)$ over the equilibrium state we obtain:
$$J_{y}^{z}=\frac{\hbar}{2}\left\{\frac{\alpha_{D}^{[1]}}{\hbar}\int\left(f_{+}%
\left(\mathbf{k}\right)+f_{-}\left(\mathbf{k}\right)\right)\left(1-\left(3k_{y%
}^{2}-2k_{x}^{2}\right)\lambda_{1}^{2}\right)\frac{d^{2}k}{\left(2\pi\right)^{%
2}}+\frac{\hbar}{m}\int\left(f_{+}\left(\mathbf{k}\right)-f_{-}\left(\mathbf{k%
}\right)\right)k_{y}\left.\langle\phi^{u_{1}}_{\bf k}\right|\sigma^{z}\left|%
\phi^{u_{1}}_{\bf k}\right.\rangle\frac{d^{2}k}{\left(2\pi\right)^{2}}\right\},$$
(7)
where $\left.\langle\phi^{u_{1}}_{\bf k}\right|\sigma^{z}\left|\phi^{u_{1}}_{\bf k}%
\right.\rangle$
is defined in Eq.(3), and $J_{x}^{z}=0$ by symmetry.
The contributions $J_{y,{\rm an}}^{z}$ and $J_{y,{\rm n}}^{z}$ (first and second term in Eq.(7), respectively)
almost cancel each other.
At $T=0$ each of them is close in absolute value to
$\left|\alpha_{D}^{[1]}\right|N_{\mathrm{el}}/2\left(1-\pi N_{\mathrm{el}}%
\lambda_{[n]}^{2}/2\right)$,
and
$J_{y}^{z}/J_{y,\mathrm{an}}^{z}\approx\left(m\alpha_{R}^{[1]}/\hbar^{2}k_{F}%
\right)^{2}\ll 1$
where $\hbar k_{F}$ is the Fermi momentum (see also Rashba Rashba03 ).
We show $J_{y}^{z}/J_{y,\mathrm{an}}^{z}$ as a function of $N_{\rm el}$ in Fig. 1.
Now we can investigate the spin current injection by linearly-polarized IR radiation due
to the intersubband transitions, as shown in Fig. 2a.
The external field is a pulse
$\mathbf{E}(t)=\mathcal{E}(t)\exp\left(-i\omega t\right)+c.c.$
with the carrier frequency $\omega$, and
slowly varying amplitude $\mathcal{E}(t)$ of duration $\tau$.
We consider oblique incidence with $\mathcal{E}(t)$ lying in the plane
of incidence. The radiation frequency $\omega$ is close to $\omega_{0}$,
with a detuning $\Omega=\omega-\omega_{0}$,
such that it can cause transitions between the subbands, with
$\hbar\Omega$ being of the order of few meV.
For $\tau\gg\omega^{-1}$ the exact shape of the pulse has no influence on our results; however, to
have the possibility of momentum-selective excitations as shown in Fig.2a one
needs sufficiently long pulses, with $\tau>\hbar/\alpha_{D}k_{F}\approx 1$
ps, for $\alpha_{D}$ $\approx 10^{-9}$ eV$\cdot$cm, a typical value of the
Dresselhaus coupling Dyakonov86 . This condition also implies applicability of Fermi’s
Golden Rule, since the pulse contains many periods of the field oscillations.
Since $\alpha_{D}^{[n]},$ $\alpha_{R}^{[n]}$ and, in turn, the
spin states and anomalous velocities depend on the subband, the intersubband
transitions can cause the injection of a spin current. The ratio
$\alpha_{D}^{[n]}F_{[n]}(\mathbf{k})/\alpha_{R}^{[n]}$,
which determines the direction of the effective SO field acting on the spin, depends on the subband.
Therefore, the spin states in different subbands are not mutually orthogonal, so
$\left.\langle\phi^{s_{1}}_{\bf k}\right|\phi^{s_{2}}_{\bf k}\rangle\neq 0$, and,
”spin-flip” transitions $u\longleftrightarrow d$ are allowed with linearly
polarized IR light absorption.
The transitions $\phi^{s_{1}}_{\bf k}\rightarrow\phi^{s_{2}}_{\bf k}$
occur in the vicinity of the resonance curves in the momentum space,
determined by the $\mathbf{k}=\mathbf{k}_{r}^{s_{2},s_{1}}(\Omega)$ where
$\mathbf{k}_{r}^{s_{2},s_{1}}(\Omega)$ is specified by the constraint
of energy conservation. For a given $\Omega$ there are in fact two such curves.
In our case $E_{\mathrm{SO}}^{[2]}\left(\mathbf{k}\right)>E_{\mathrm{SO}}^{[1]}\left(%
\mathbf{k}\right)$
for all $k$. Therefore, for $\Omega>0$
the transitions $d\rightarrow u$ and $u\rightarrow u$ are allowed, while for
$\Omega<0$ we obtain $d\rightarrow d$ and $u\rightarrow d$ transitions.
As one can see in Figs. 2a and 2b, $k_{r}^{s_{2},s_{1}}(\Omega)$
is larger for the ”spin-conserving” than for the ”spin-flip”-transitions.
The transition matrix elements depend on the spin states in both subbands,
and can be factorized in the dipole approximation as:
$$M\left(\mathbf{k}_{r}^{s_{2},s_{1}}(\Omega)\right)=\mathcal{E}e\frac{\sin 2%
\theta_{0}}{\epsilon\cos\theta_{0}+\sqrt{\epsilon}\cos\theta_{1}}\left\langle%
\varphi^{(2)}(z)\right|z\left|\varphi^{(1)}(z)\right\rangle\left\langle\phi^{s%
_{2}}_{\bf k}\right|\left.\phi^{s_{1}}_{\bf k}\right\rangle,\qquad$$
(8)
where $\theta_{0}$ and $\theta_{1}$ are, respectively, the incidence and refraction angles,
$\sin\theta_{1}=\sin\theta_{0}/\sqrt{\epsilon}$, $\epsilon$ is the dielectric constant, $e$ is the electron charge,
and $\varphi^{(1)}(z)$, $\varphi^{(2)}(z)$ are the envelope electron wavefunctions in the
subbands $n=1$ and $n=2,$ respectively.
A transfer of one electron to the second subband injects a spin current:
$$\Delta j_{y}^{z}\left(\mathbf{k;}s_{2},s_{1}\right)=j_{y}^{z}\left(\mathbf{k,}%
s_{2}\right)-j_{y}^{z}\left(\mathbf{k,}s_{1}\right),$$
(9)
where we neglect the small photon momentum. The incident radiation injects the concentration
of electrons in the second subband $N_{2}$, with a rate $dN_{2}\left(\Omega\right)/dt$ and,
correspondingly, drives the spin current density
component with the rate $d\Delta J_{y}^{z}\left(\Omega\right)/dt.$
The injection rates can be written as:
$$\frac{d\Delta J_{y}^{z}\left(\Omega\right)}{dt}=\frac{\hbar}{2}\zeta(\Omega)%
\frac{dN_{2}\left(\Omega\right)}{dt},\qquad\frac{dN_{2}\left(\Omega\right)}{dt%
}=\frac{\xi(\Omega)}{\hbar\omega}\left\langle S\right\rangle,$$
(10)
where $\zeta(\Omega)$ characterizes the effective speed of electrons forming the
pure spin current, $\left\langle S\right\rangle=(c/2\pi)\mathcal{E}^{2}$ is
the radiation power per unit area, and $\xi(\Omega)$ is a dimensionless function.
Within Fermi’s Golden Rule the speed characterizing the spin injection is obtained
as:
$$\frac{\hbar}{2}\zeta(\Omega)=\frac{\sum\limits_{s_{1},s_{2}}\displaystyle{\int%
}f_{s_{1}}\left(\mathbf{k}\right)\left|\langle\phi_{\bf k}^{s_{2}}\right|\left%
.\phi_{\bf k}^{s_{1}}\rangle\right|^{2}\Delta j_{y}^{z}\left(\mathbf{k;}s_{2},%
s_{1}\right)dk_{r}^{s_{2},s_{1}}(\Omega)/v^{s_{2},s_{1}}_{\mathbf{k}}}{\sum%
\limits_{s_{1},s_{2}}\displaystyle{\int}f_{s_{1}}\left(\mathbf{k}\right)\left|%
\langle\phi_{\bf k}^{s_{2}}\right|\left.\phi_{\bf k}^{s_{1}}\rangle\right|^{2}%
dk_{r}^{s_{2},s_{1}}(\Omega)/v^{s_{2},s_{1}}_{\mathbf{k}}},$$
(11)
with the velocity associated with the joint density of states given by:
$$\mathbf{v}^{s_{2},s_{1}}_{\mathbf{k}}=\frac{\partial}{\hbar\partial\mathbf{k}}%
\left(\varepsilon_{s_{2}}-\varepsilon_{s_{1}}\right).$$
(12)
The integration in Eq.(11) is performed along the resonance curves.
With the increase of $\left|\Omega\right|$,
$k_{r}^{s_{2},s_{1}}(\Omega)$ increases and eventually arrives at regions of small electron occupancy,
as can be seen in Fig. 2b. Hence, $d\Delta J_{y}^{z}\left(\Omega\right)/dt$ and
$dN_{2}/dt$ become small at $\hbar\left|\Omega\right|$ larger than some
critical $\hbar\Omega_{\rm c}$ (a few meV) determined by the condition
$\min k_{r}^{s_{2},s_{1}}(\Omega_{\rm c})>k_{0}$,
where $k_{0}=k_{F}$ or $k_{0}=\sqrt{mk_{B}T}/\hbar$ in the
degenerate and non-degenerate gas, respectively.
The photoinduced spin current is the sum of normal
$\Delta J_{y,\mathrm{n}}^{z}$ and anomalous $\Delta J_{y,\mathrm{an}}^{z}$ contributions,
each containing spin-flip $\left(s_{1}\neq s_{2}\right)$ and spin
conserving $\left(s_{1}=s_{2}\right)$ terms. The anomalous spin-conserving
term is of the order of
$\left(\alpha_{D}^{[2]}-\alpha_{D}^{[1]}\right)N_{2},$
while the other terms depend on the difference of the ratio
$\alpha_{R}^{[2]}/\alpha_{D}^{[2]}-\alpha_{R}^{[1]}/\alpha_{D}^{[1]}$
and $\lambda_{[n]}$. An estimate of the relative contributions is:
$$\frac{\Delta J_{y,\mathrm{n}}^{z}\left(s_{1}=s_{2}\right)}{\Delta J_{y,\mathrm%
{an}}^{z}\left(s_{1}=s_{2}\right)}\approx\frac{\hbar^{2}k_{F}}{m\alpha_{D}}%
\left[\left(\frac{\alpha_{R}^{[2]}}{\alpha_{D}^{[2]}}\right)^{2}-\left(\frac{%
\alpha_{R}^{[1]}}{\alpha_{D}^{[1]}}\right)^{2}\right].$$
(13)
Due to a large prefactor $\hbar^{2}k_{F}/m\alpha_{D},$ which is the ratio
of the normal and anomalous velocities, the normal term can be large
and lead to a change in the sign of the spin current
at particular light frequencies, as seen in Figs.3a and 3b.
In Fig.3a we present the speed $\zeta(\Omega)$, while
in Fig.3b we show the normal and anomalous parts of the injected spin current density.
The spin-flip contribution in both the normal and anomalous terms is much smaller than the
”spin-conserving” one. Recently, Golub
Golub03 demonstrated that
the direction of electric current induced by interband light absorption in QWs
can depend on the light frequency. In his scenario the change occurs as
new subbands are accessed, and thus appears on a scale of 100 meV. In our scenario
for pure spin current injection, the change occurs on a much smaller scale.
Now we estimate the magnitude of the injected spin current assuming that the
contributions of the anomalous and normal terms are of the same order of
magnitude. Fig.4 presents the efficiency of the energy absorption
$\xi(\Omega)$ (Eq.(10)). The concentration of the electrons excited to the second subband
can be estimated from Eqs.(8)-(10) as
$N_{2}\approx 2\pi(e^{2}w^{2}k_{F}/\epsilon^{2}\hbar c\alpha_{D})\langle S\rangle\tau$.
At $\epsilon=12$, $k_{F}\approx 10^{6}$ cm${}^{-1},w=100$ Å,
$\theta_{0}$ close to $\pi/4$ and $N_{\mathrm{el}}\approx 10^{12}$
cm${}^{-2}$ we obtain:
$N_{2}/N_{\mathrm{el}}\approx 10^{-6}$ $\left(\langle S\rangle/(\mathrm{W/cm}^{2})\right)\cdot\left(\tau/\mathrm{ps}\right)$.
Under excitation of a 1% fraction of electrons,
achieved at $\langle S\rangle\approx 10$ $\mathrm{kW/cm}^{2}$ and
$\tau\approx 1$ $\mathrm{ps,}$ the corresponding effective current density
$e\Delta J_{y}^{z}/\hbar\approx 1$ $\mathrm{mAmp/cm.}$ This is of the same magnitude
as would be generated by the ac spin pumping in the $n=1$ subband,
as proposed by Mal’shukov et al. Malshukov03 , but the
effect would operate on a nanosecond time scale, as opposed to the
picosecond time scale relevant here.
Having found the magnitude of the spin current, we discuss
its experimental observation. A possible technique is the measurement
of the voltage generated by the anomalous spin-Hall effect due to
scattering of electrons by impurities. The spin current
$\Delta J_{y}^{z}$ causes a spin-Hall bias $V_{sH}$ along the $x$ axis. Its
magnitude can be estimated as $V_{sH}\approx\tan(\theta_{sH})V_{\mathrm{eff}}$,
where $\theta_{sH}$ is the spin-Hall angle and
$V_{\mathrm{eff}}$ is the effective lateral bias that would cause a current
density $e\Delta J_{y}^{z}/\hbar$.
As follows from the discussion preceding Eq.(13), the
corresponding current density is
of the order of $eN_{2}(\alpha_{D}/\hbar)$.
The bias $V_{\mathrm{eff}}$ that would cause this current density is:
$V_{\mathrm{eff}}\approx L\left(N_{2}/N_{\mathrm{el}}\right)\alpha_{D}/\hbar\mu$,
where $\mu$ is the mobility, and $L\approx 1$ cm is the lateral size of the system.
At $\mu\approx 10^{5}$ $\mathrm{cm}^{2}/(\mathrm{Vs})$, $N_{2}/N_{\mathrm{el}}\approx 10^{-3}$, and
$\alpha_{D}/\hbar\approx 10^{6}$ cm/s we obtain:
$V_{\mathrm{eff}}/L\approx 10^{-2}$ $\mathrm{V/cm.}$ The spin-Hall angle was
estimated by Huang et al. Huang04 as $\theta_{sH}\approx 10^{-3}$, which would lead
to $V_{sH}\approx 10^{-5}$ V. Their model assumed charged dopants embedded directly in the QW,
which considerably overestimates the magnitude of the
effect when only a remote doping is present. For this reason, $10^{-5}$ V is clearly
an upper estimate of the spin Hall bias. Nonetheless, even a bias smaller by two orders of magnitude than
this would be experimentally accessible Bakun85 .
Another possibility for observing the pure spin current is spatially resolved
pump-probe spectroscopy, as applied by Hübner et al. Hubner03 and
Stevens et al. Stevens03 to
investigate the spin current injected by interband transitions. In those
experiments the centers of the spin-up and spin-down of excited electron
distribution were
separated by approximately 20 nm. In the experimental situation considered
here, the spin-polarized spots can be separated by distances of the order of the electron free path
$\ell\approx\left(\hbar k_{F}/m\right)\tau_{k},$ with $\tau_{k}$ being the momentum
relaxation time. At mobility
$\mu\approx 10^{5}$ $\mathrm{cm}^{2}/(\mathrm{Vs}),$
one obtains $\ell\approx 10^{3}$ nm, and so a possible approach would be
to observe this separation experimentally by using a linearly
polarized IR light as a pump and circularly polarized light as a probe
of the spin-dependent transmission. In a real sample, of course, we have to expect
some inhomogeneity in the spin-orbit interaction due to quantum well thickness variations,
dopant fluctuations, inhomogeneous strain, and the like Sherman03 .
We are currently investigating the consequences of such inhomogeneity, and
will return to it in a later communication.
To conclude, we have shown that a pure spin current can be injected in QWs
by IR intersubband absorption, calculated its magnitude, and found that
it could be measured experimentally. The dependence of the spin current on the
light frequency, and on the Rashba SO coupling parameter, opens the
possibility of its manipulation applying an external bias and by changing the
light frequency. The spin current should be observable by anomalous spin-Hall
effect measurements or by pump-probe optical spectroscopy.
E.Y.S is grateful to the Austrian Science Fund for financial support. A.N. acknowledges
support from an Ontario Graduate Scholarship. This work was supported in part by the National
Science and Engineering Research Council or Canada (NSERC) and the DARPA SpinS program.
We thank P. Marsden, H. van Driel, and J. Sinova for useful discussions.
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Model Calibration via Distributionally Robust Optimization: On the NASA Langley Uncertainty Quantification Challenge
Yuanlu Bai
[email protected]
Department of Industrial Engineering & Operations Research, Columbia University, USA.
Zhiyuan Huang
[email protected]
Department of Management Science & Engineering, Tongji University, Shanghai, China.
Henry Lam
[email protected]
Department of Industrial Engineering & Operations Research, Columbia University, USA.
Abstract
We study a methodology to tackle the NASA Langley Uncertainty Quantification Challenge, a model calibration problem under both aleatory and epistemic uncertainties. Our methodology is based on an integration of robust optimization, more specifically a recent line of research known as distributionally robust optimization, and importance sampling in Monte Carlo simulation. The main computation machinery in this integrated methodology amounts to solving sampled linear programs. We present theoretical statistical guarantees of our approach via connections to nonparametric hypothesis testing, and numerical performances including parameter calibration and downstream decision and risk evaluation tasks.
keywords:
uncertainty quantification, model calibration, distributionally robust optimization, importance sampling, linear programming, nonparametric.
We consider the NASA Langley Uncertainty Quantification (UQ) Challenge problem Crespo and
Kenny (2020) where, given a set of “output” data and under both aleatory and epistemic uncertainties, we aim to infer a region that contains the true values of the associated variables. These steps allow us to investigate the reduction of uncertainty by obtaining further information and estimate the failure probabilities of related systems. To tackle these challenges, we study a methodology based on an integration of robust optimization (RO), more specifically, a recent line of research known as distributionally robust optimization (DRO), and importance sampling in Monte Carlo simulation. We will see that the main computation machinery in this integrated methodology boils down to solving sampled linear programs (LPs). In this paper, we will explain our methodology, introduce theoretical statistical guarantees via connections to nonparametric hypothesis testing, and present the numerical results on this UQ Challenge.
We briefly introduce the Challenge and notations, where details can be found in Crespo and
Kenny (2020). The uncertainty model in the Challenge is given by $\langle f_{a},E\rangle$, where $a\sim f_{a}$ is an aleatory variable following a probability density $f_{a}$ and probability distribution function $F_{a}$, and $e\in E$ is an epistemic variable inside the deterministic set $E$. Both the true distribution of $a$ and the true value of $e$ are unknown. Initially, we are given $E_{0}\supset E$ and data $D_{1}=\{y^{(i)}(t)\},i=1,\ldots,n_{1}$ in the form of a discrete-time trajectory $t=0,\ldots,T$. We have the computational capability to simulate $y(a,e,t)$ for given values of $a\in A,e\in E_{0}$. The task is to calibrate the distribution of $a$ and value of $e$ with uncertainty quantification, as well as using them to conduct downstream decision and risk evaluation tasks.
1 Overview of Our Methodology (Problem A)
We first give a high-level overview of our methodology in extracting a region $E$ that contains the true epistemic variables. For convenience, we call this region an “eligibility set” of $e$. For each value of $e$ inside $E$, we also have a set (in the space of probability distributions) that contains “eligible” distributions for the random variable $a$. For the sake of computational tractability (as we will see shortly), the eligibility set of $e$ is represented by a set of sampled points in $E_{0}$ that approximate its shape, whereas the eligibility set of $a$ is represented by probability weights on sampled points on $A$. The eligibility set $E$ and the corresponding eligibility set of distributions for $a$ are obtained by solving an array of LPs that are constructed from these properly sampled points, and then deciding eligibility by checking the LP optimal values against a threshold that resembles the “$p$-value” approach in hypothesis testing. As another key ingredient, this methodology involves a dimension-collapsing transformation $\mathbf{S}$, applied on the raw data, which ultimately allows using the Kolgomorov-Smirnov (KS) statistic to endow rigorous statistical guarantees.
Algorithm 1 is a procedural description of our approach to construct the eligibility set $E$, which also gives as a side product an eligibility set of the distributions of $a$ for each $e$, represented by weights in the set (16). In the following, we explain the elements and terminologies in this algorithm in detail.
2 A DRO Perspective
Our starting idea is to approximate the set
$$E=\{e\in E_{0}:\text{there exists\ }P_{e}\text{\ s.t.\ }d(P_{e},\hat{P})\leq\eta\}$$
(1)
where $P_{e}$ is the probability distribution of $\{y(a,e,t)\}_{t=0,\ldots,T}$, namely the outputs of the simulation model $\{y(a,e,t)\}_{t=0,\ldots,T}$ at a fixed $e$ but random $a$. $\hat{P}$ denotes the empirical distribution of $D_{1}$, more concretely the distribution given by
$$\hat{P}(\cdot)=\frac{1}{n_{1}}\sum_{i=1}^{n_{1}}\delta_{(y^{(i)}(t))_{t=0,\ldots,T}}(\cdot)$$
where $\delta_{(y^{(i)}(t))_{t=0,\ldots,T}}(\cdot)$ denotes the Dirac measure at $(y^{(i)}(t))_{t=0,\ldots,T}$. $d(\cdot,\cdot)$ denotes a discrepancy between two probability distributions, and $\eta\in\mathbb{R}_{+}$ is a suitable constant. Intuitively, $E$ in Eq. (1) is the set of $e$ such that there exists a distribution for the outputs that is close enough to the empirical distribution from the data. If for a given $e$ there does not exist any possible output distribution that is close to $\hat{P}$, then $e$ is likely not the truth. The following gives a theoretical justification for using Eq. (1):
Theorem 1.
Suppose that the true distribution of the output $(y(t))_{t=0,\ldots,T}$, called $P_{true}$, satisfies $d(P_{true},\hat{P})\leq\eta$ with confidence level $1-\alpha$, i.e., we have
$$\mathbb{P}(d(P_{true},\hat{P})\leq\eta)\geq 1-\alpha$$
(2)
where $\mathbb{P}$ denotes the probability with respect to the data. Then the set $E$ in Eq. (1) satisfies $\mathbb{P}(e_{true}\in E)\geq 1-\alpha$, where $e_{true}$ denotes the true value of $e$. Similar deduction holds if Eq. (2) holds asymptotically (as the data size grows), in which case the same asymptotic modification holds for the conclusion.
The proof of Theorem 1 comes from a straightforward set inclusion.
Proof.
Note that $d(P_{true},\hat{P})\leq\eta$ implies $e_{true}\in E$. Thus we have $\mathbb{P}(e_{true}\in E)\geq\mathbb{P}(d(P_{true},\hat{P})\leq\eta)\geq 1-\alpha$. Similar derivation holds for the asymptotic version.
∎
In Eq. (1), the set of distributions $\{P_{e}:d(P_{e},\hat{P})\leq\eta\}$ is analogous to the so-called uncertainty set or ambiguity set in the RO literature (e.g., Bertsimas
et al. (2011); Ben-Tal and
Nemirovski (2002)), which is a set postulated to contain the true values of uncertain parameters in a model. RO generally advocates decision-making under uncertainty that hedges against the worst-case scenario, where the worst case is over the uncertainty set (and thus often leads to a minimax optimization problem). DRO, in particular, focuses on problems where the uncertainty is on the probability distribution of an underlying random variable (e.g., Wiesemann
et al. (2014); Delage and
Ye (2010)). This is the perspective that we are taking here, where $a$ has a distribution that is unknown, in addition to the uncertainty on $e$. Moreover, we also take a generalized view of RO or DRO here as attempting to construct an eligibility set of $e$ instead of finding a robust decision via a minimax optimization.
Theorem 1 focuses on the situation where the uncertainty set is constructed and calibrated from data, which is known as data-driven RO or DRO (Bertsimas
et al. (2018a); Hong
et al. (2020)). If such an uncertainty set has the property of being a confidence region for the uncertain parameters or distributions, then by solving RO or DRO, the confidence guarantee can be translated to the resulting decision, or the eligibility set in our case. Here we have taken a nonparametric and frequentist approach, as opposed to other potential Bayesian methods.
In implementation we choose $\alpha=0.05$, so that the eligibility set $E$ has the interpretation of approximating a $95\%$ confidence set for $e$. In the above developments, $d(P_{e},\hat{P})\leq\eta$ can in fact be replaced with more general set $P_{e}\in\mathcal{U}$ where $\mathcal{U}$ is calibrated from the data. Nonetheless, the distance-based set (or “ball”) surrounding the empirical distribution is intuitive to understand, and our specific choice of the set below falls into such a representation.
To use Eq. (1), there are two immediate questions:
1.
What $d(\cdot,\cdot)$ should and can we use, and how do we calibrate $\eta$?
2.
How do we determine whether there exists $P_{e}$ that satisfies $d(P_{e},\hat{P})\leq\eta$ for a given $e$?
In the following two sections, we address the above two questions respectively which would then lead us to Algorithm 1.
3 Constructing Discrepancy Measures
For the first question, we first point out that in theory many choices of $d$ could be used (basically, any $d$ that satisfies the confidence property in Theorem 1). But, a poor choice of $d$ would lead to a more conservative result, i.e., larger $E$, than others. A natural choice of $d$ should capture the discrepancy of the distributions efficiently. Moreover, the choice of $d$ should also account for the difficulty in calibrating $\eta$ such that the assumption in Theorem 1 can be satisfied, as well as the computational tractability in solving the eligibility determination problem in Eq. (1).
Based on the above considerations, we construct $d$ and calibrate $\eta$ as follows. First, we “summarize” the data $D_{1}$ into a lower-dimensional representation, say $\{s_{1}^{(i)},\ldots,s_{m}^{(i)}\},i=1,\ldots,n_{1}$, where $s_{v}^{(i)}=S_{v}({y^{(i)}(t)}_{t=0,\ldots,T})$ for some function $S_{v}(\cdot)$. For convenience, we denote $\mathbf{S}(\cdot)=(S_{1}(\cdot),\ldots,S_{m}(\cdot)):\mathbb{R}^{n_{t}+1}\to\mathbb{R}^{m}$, and $\mathbf{s}^{(i)}=(s_{1}^{(i)},\ldots,s_{m}^{(i)})$. We call $\mathbf{S}(\cdot)$ the “summary function” and $\mathbf{s}^{(i)}$ the “summaries” of the $i$-th output. $\mathbf{S}(\cdot)$ attempts to capture important characteristics of the raw data (we will see later that we use the positions and values of the peaks extracted from Fourier analysis). Also, the low dimensionality of $\mathbf{s}^{(i)}$ is important to calibrate $\eta$ well.
Next, we define
$$d(P_{e},\hat{P})=\max_{v=1,\ldots,m}\sup_{x\in\mathbb{R}}\left|F_{e,v}(x)-\hat{F}_{v}(x)\right|$$
(3)
where $\hat{F}_{v}(x)=\frac{1}{n_{1}}\sum_{i=1}^{n_{1}}I(s_{v}^{(i)}\leq x)$, with $I(\cdot)$ denoting the indicator function, is the empirical distribution function of $s_{v}^{(i)}$ (i.e., the distribution function of $\hat{P}$ projected onto the $v$-th summary). $F_{e,v}(x)$ is the probability distribution function of the $v$-th summary of the simulation model output $S_{v}(y(a,e,t))_{t=0,\ldots,T}$ (i.e., the distribution function of the projection of $P_{e}$ onto the $v$-th summary). We then choose $\eta=q_{1-\alpha/m}/\sqrt{n_{1}}$ as the $(1-\alpha/m)$-quantile of the Kolmogorov-Smirnov (KS) statistic, namely that $q_{1-\alpha/m}$ is the $(1-\alpha/m)$-quantile of $\sup_{x\in[0,1]}|BB(x)|$ where $BB(\cdot)$ denotes a standard Brownian bridge.
To understand Eq. (3), note that the set of $P_{e}$ that satisfies $d(P_{e},\hat{P})\leq\eta$ is equivalent to $P_{e}$ that satisfies
$$\sup_{x\in\mathbb{R}}\left|F_{e,v}(x)-\hat{F}_{v}(x)\right|\leq\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}},\ \ v=1,\ldots,m$$
(4)
Here, $\sup_{x\in\mathbb{R}}\left|F_{e,v}(x)-\hat{F}_{v}(x)\right|$ is the KS-statistic for a goodness-of-fit test against the distribution $F_{e,v}(x)$, using the data on the $v$-th summary. Since we have $m$ summaries and hence $m$ tests, we use a Bonferroni correction and deduce that
$$\displaystyle\liminf_{n_{1}\to\infty}\mathbb{P}\bigg{(}$$
$$\displaystyle\sup_{x\in\mathbb{R}}\left|F_{true,v}(x)-\hat{F}_{v}(x)\right|\leq\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}}\text{ for }v=1,\ldots,m\bigg{)}\geq 1-\alpha\ $$
where $F_{true,v}$ denotes the true distribution function of the $v$-th summary. Thus, the (asymptotic version of the) assumption in Theorem 1 holds.
Note that here the quality of the summaries does not affect the statistical correctness of our method (in terms of overfitting), but it does affect crucially the resulting conservativeness (in the sense of getting a larger $E$). Moreover, in choosing the number of summaries $m$, there is a tradeoff between the conservativeness coming from representativeness and simultaneous estimation. On one end, using more summaries means more knowledge we impose on $P_{e}$, which translates into a smaller feasible set for $P_{e}$ and ultimately a smaller eligibility set $E$. This relation, however, is true only if there is no statistical noise coming from the data. In the case of finite data size $n_{1}$, then more summaries also means that constructing the feasible set for $P_{e}$ requires more simultaneous estimations in calibrating its size, which is manifested in the Bonferroni correction whose degree increments with each additional summary. In our implementation (see Section 7), we find that using 12 summaries seems to balance well this representativeness versus simultaneous estimation error tradeoff.
4 Determining Existence of an Aleatory Distribution
Now we address the second question on how we can decide, for a given $e$, whether a $P_{e}$ exists such that $d(P_{e},\hat{P})\leq\eta$.
We first rephrase the representation
with a change of measure. Consider a “baseline” probability distribution, say $P_{0}$, that is chosen by us in advance. A reasonable choice, for instance, is the uniform distribution over $A$, the support of $a$. Then we can write $d(P_{e},\hat{P})\leq\eta$
as
$$\sup_{x\in\mathbb{R}}\left|\int_{S_{v}(u)\leq x}W_{e}(u)dP_{0}(u)-\hat{F}_{v}(x)\right|\leq\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}}$$
(5)
for $v=1,\ldots,m$ where $W_{e}(\cdot)=dP_{e}/dP_{0}$ is the Radon-Nikodym derivative of $P_{e}$ with respect to $P_{0}$, and we have used the change-of-measure representation $F_{e,v}(x)=\int_{S_{v}(u)\leq x}W_{e}(u)dP_{0}(u)$. Here we have assumed that $P_{0}$ is suitably chosen such that absolute continuity of $P_{e}$ with respect to $P_{0}$ holds. Eq. (5) turns the determination of the existence of eligible $P_{e}$ into the existence of an eligible Radon-Nikodym derivative $W_{e}(\cdot)$.
The next step is to utilize Monte Carlo simulation to approximate $P_{0}$. More specifically, given $e$, we run $k$ simulation runs under $P_{0}$ to generate $(y(a^{(j)},e,t))_{t=0,\ldots,T}$ for $j=1,\ldots,k$. Then Eq. (5) can be approximated by
$$\sup_{x\in\mathbb{R}}\Bigg{|}\sum_{j=1}^{k}W_{j}I(S_{v}((y(a^{(j)},e,t))_{t=0,\ldots,T})\leq x)-\hat{F}_{v}(x)\Bigg{|}\leq\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}},\ r=1,\ldots,m$$
(6)
where $W_{j}=(1/k)(dP_{e}/dP_{0}((y(a^{(j)},e,t))))$ represents the (unknown) sampled likelihood ratio from the view of importance sampling Juneja and
Shahabuddin (2006); Blanchet and
Lam (2012). Our task is to find a set of weights, $W_{j},j=1,\ldots,k$, such that Eq. (6) holds. These weights should approximately satisfy the properties of the Radon-Nikodym derivative, namely positivity and integrating to one. Thus, we seek for $W_{j},j=1,\ldots,k$ such that
$$\displaystyle\sup_{x\in\mathbb{R}}\Bigg{|}\sum_{j=1}^{k}W_{j}I(S_{v}((y(a^{(j)},e,t))_{t=0,\ldots,T})\leq x)-\hat{F}_{v}(x)\Bigg{|}\leq\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}},\ r=1,\ldots,m$$
(7)
$$\displaystyle\sum_{j=1}^{k}W_{j}=1,\ W_{j}\geq 0\ \text{\ for\ }j=1,\ldots,k$$
(8)
where Eq. (8) enforces the weights to lie in a probability simplex. If
$k$ is much larger than
$n_{1}$, then the existence of $W_{j},j=1,\ldots,k$ satisfying Eq. (7) and Eq. (8) would determine that the considered $e$ is in $E$.
To summarize, we have:
Theorem 2.
Suppose $k=\omega(n_{1})$, and $P_{true}$ is absolutely continuous with respect to $P_{0}$ and that $\|dP_{true}/dP_{0}\|_{\infty}\leq C$ for some constant $C>0$ and $\|\cdot\|_{\infty}$ denotes the essential supremum. Suppose, for each $e$, we generate $k$ simulation replications to get $(y(a^{(j)},e,t))_{t=0,\ldots,T}),j=1,\ldots,k$, where $a^{(j)}$ are drawn from $P_{0}$ in an i.i.d. fashion. Then the set
$$E=\Big{\{}e:\text{there exists }W_{j},j=1,\ldots,k\text{ such that Eq.~{}\eqref{elaborate constraint KS} and Eq.~{}\eqref{elaborate constraint3} hold}\Big{\}}$$
will satisfy
$$\liminf_{n_{1}\to\infty,k/n_{1}\to\infty}\mathbb{P}(e_{true}\in E)\geq 1-\alpha$$
The proof is in the appendix. Note that in Theorem 2, $W_{j}$’s represent the unknown sampled likelihood ratios such that, together with the $a^{(j)}$’s generated from $P_{0}$, the function
$$\sum_{j=1}^{k}W_{j}I(S_{v}((y(a^{(j)},e,t))_{t=0,\ldots,T})\leq\cdot)$$
approximates the unknown true $v$-th summary distribution function $F_{true,v}$.
To use the above $E$ and elicit the guarantee in Theorem 2, we still need some steps in order to conduct feasible numerical implementation. First, we need to discretize or sufficiently sample $e$’s over $E_{0}$, since checking the existence of eligible $W_{j}$’s for all $e$ is computationally infeasible. In our implementation we draw $n_{2}=1000$ $e$’s uniformly over $E_{0}$, call them $e^{(1)},\ldots,e^{(n_{2})}$, and then put together the geometry of $E$ from the eligible $e^{(l)}$’s. Second, the current representation of the KS constraint Eq. (7)
involves entire distribution functions.
We can write Eq. (7) as a finite number of linear constraints, given by
$$\displaystyle\hat{F}_{v}(s_{v}^{(i)}+)-\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}}$$
(9)
$$\displaystyle\leq$$
$$\displaystyle\sum_{j=1}^{k}W_{j}I(S_{v}((y(a^{(j)},e,t))_{t=0,\ldots,T})\leq s_{v}^{(i)})$$
$$\displaystyle\leq$$
$$\displaystyle\hat{F}_{v}(s_{v}^{(i)}-)+\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}}$$
for $i=1,\ldots,n_{1},r=1,\ldots,m$ where $s_{v}^{(i)},i=1,\ldots,n_{1}$ are the $v$-th summary of the $i$-th data point, and $s_{v}^{(i)}+$ and $s_{v}^{(i)}-$ denote the right and left limits of the empirical distribution at
$s_{v}^{(i)}$.
Thus, putting everything together, we solve, for each $e^{(l)},l=1,\ldots,n_{2}$, the feasibility problem:
Find $W_{j},j=1,\ldots,k$ such that Eq. (9) and Eq. (8) hold.
If there exists feasible $W_{j},j=1,\ldots,k$, then $e^{(l)}$ is eligible. The set $\{e^{(l)}:e^{(l)}\text{\ is eligible}\}$ is an approximation of $E$. Note that this is a “sampled” subset of $E$. In general, without running the simulation at the other points of $E$, there is no guarantee whether these other points are eligible or not. However, if the distribution of $\{y(a,e,t)\}_{t=0,\ldots,T}$ is continuous in $e$ in some suitable sense, then it is reasonable to believe that the neighborhood of an eligible point $e^{(l)}$ is also eligible (and vice versa). In this case, we can “smooth” the discrete set of $\{e^{(l)}:e^{(l)}\text{\ is eligible\ }\}$ if needed (e.g., by doing some clustering and taking the convex hull of each cluster). Finally, note that the feasibility problem above is a linear problem in the decision variables $W_{j}$’s.
5 Towards the Main Procedure
To link to our main Algorithm 1, we offer an equivalent approach to the above feasibility-problem-based procedure that allows further flexibility in choosing the threshold $q_{1-\alpha/m}$, which currently is set as the Bonferroni-adjusted KS critical value. This equivalent approach leaves this choice of threshold open and can determine the set of eligible $e^{(l)}$ as a function of the threshold, thus giving some room to improve conservativeness should the formed approximate $E$ turns out to be too loose according to other expert opinion. Here, we solve, for each $e^{(l)},l=1,\ldots,n_{2}$, the optimization problem
$$\displaystyle\begin{array}[]{lll}q_{l}^{*}=&\min&q\\
&\text{s.t.}&\hat{F}_{v}(s_{v}^{(i)}+)-\frac{q}{\sqrt{n_{1}}}\\
&&\leq\sum_{j=1}^{k}W_{j}I(S_{v}((y(a^{(j)},e^{(l)},t))_{t=0,\ldots,T})\leq s_{v}^{(i)})\\
&&\leq\hat{F}_{v}(s_{v}^{(i)}-)+\frac{q}{\sqrt{n_{1}}}\text{ for }i=1,\ldots,n_{1},v=1,\ldots,m;\\
&&\sum_{j=1}^{k}W_{j}=1,\ W_{j}\geq 0\text{ for\ }j=1,\ldots,k\end{array}$$
(15)
where the decision variables are $W_{j},j=1,\ldots,k$ and $q$. If the optimal value $q_{l}^{*}$ satisfies $q_{l}^{*}\leq q_{1-\alpha/m}$, then $e^{(l)}$ is eligible (This can be seen by checking its equivalence to the feasibility problem via the monotonicity of the feasible region for $W_{j}$’s in Eq. (15) as $q$ increases). The rest then follows as above that $\{e^{(l)}:e^{(l)}\text{\ is eligible}\}$ is an approximation of $E$. Like before, Eq. (15) is an LP. Moreover, here $q^{*}_{l}$ captures in a sense the “degree of eligibility” of $e^{(l)}$, and allows convenient visualization by plotting $q_{l}^{*}$ against $e^{(l)}$ to assess the geometry of $E$. For these reasons we prefer to use Eq. (15) over the feasibility problem before. These give the full procedure in Algorithm 1. Note that Algorithm 1 has a variant where we re-generate a sample of $a^{(j)}$’s for each different $e^{(l)}$. It is clear that the correctness guarantee (Theorem 2) still holds in this case.
Moreover, we also present how to find eligible distributions of $a$ for an eligible $e^{(l)}$. The set of eligible distributions of $a$ is approximated by the weights $W_{j}$’s that satisfy Eq. (9) and Eq. (8), namely
$$\displaystyle\Bigg{\{}W_{j},j=1,\ldots,k:\hat{F}_{v}(s_{v}^{(i)}+)-\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}}$$
$$\displaystyle\leq\sum_{j=1}^{k}W_{j}I(S_{v}((y(a^{(j)},e^{(l)},t))_{t=0,\ldots,T})\leq s_{v}^{(i)})$$
$$\displaystyle\leq\hat{F}_{v}(s_{v}^{(i)}-)+\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}},\text{ for }i=1,\ldots,n_{1},v=1,\ldots,m;$$
$$\displaystyle\sum_{j=1}^{k}W_{j}=1,\ W_{j}\geq 0\ \text{\ for\ }j=1,\ldots,k\Bigg{\}}$$
(16)
where $W_{j}$ is the probability weight on $a^{(j)}$. From this, one could also obtain approximate bounds for quantities related to the distribution of $a$. For instance, to get approximate bounds for the mean of $a$, we can maximize and minimize $\sum_{j}W_{j}a^{(j)}$ subject to constraint (16).
6 Related Literature
Before we discuss our numerical findings, we discuss some related literature on the problem setting and our proposed methodology.
The model calibration problem that infers input from output data has been studied across different disciplines. In scientific areas it is viewed as an inverse problem Tarantola (2005), in which Bayesian methodologies are predominantly used (e.g., Currin et al. (1991); Craig
et al. (2001); Kennedy and
O’Hagan (2001); Bayarri et al. (2007); Higdon et al. (2008)). Our presented approach is an alternative to Bayesian methods that aim to provide frequentist guarantees in the form of confidence regions. This approach is motivated from the need of sophisticated techniques such as approximate Bayesian computation Marjoram et al. (2003) in the Bayesian framework, and that the DRO methodology that we develop appears to be well-suited to the UQ Challenge setup. In addition to Bayesian approaches, other alternative methods include entropy maximization Kraan and
Bedford (2005) that use the entropy as a criterion to select the “best” distribution, but it does not have the frequentist guarantee in recovering the true distribution that we provide in this UQ Challenge.
We point out that model calibration has also been investigated in the stochastic simulation community Sargent (2010); Kleijnen (1995). In this setting, model calibration is often viewed together with model validation. To validate a model, the conventional approach is to use statistical tests such as the two-sample mean-difference tests Balci and
Sargent (1982) or others like the Schruben-Turing test Schruben (1980) that decides whether the simulated output data and historical real output data are close enough. If not, then the simulation model is re-calibrated, and this process is repeated until the gap between simulation and real data is sufficiently close. Though having a long history, the development of rigorous frameworks to conduct model calibration and validation has been quite open with relatively few elaborate discussions in the literature Nelson (2016).
In terms of methodology, our approach is closely related to RO, which is an established method for optimization under uncertainty that advocates the representation of unknown or uncertain parameters in the model as a (deterministic) set (e.g., Bertsimas
et al. (2011); Ben-Tal and
Nemirovski (2002)). This set is often called an uncertainty set or an ambiguity set. In the face of decision-making, RO optimizes the decision over the worst-case scenario within the uncertainty set, which usually comes in the form of a minimax problem with the outer optimization on the decision while the inner optimization on the worst case scenario. DRO, a recently active branch of RO, considers stochastic optimization where the underlying probability distribution is uncertain (e.g., Goh and
Sim (2010); Wiesemann
et al. (2014); Delage and
Ye (2010)). In this case, the uncertainty set lies in the space of probability distributions and one attempts to make decisions under the worst-case distribution. In this paper we take a generalized view of RO or DRO as attempting to find a set of eligible “decisions”, namely the $e$, so it does not necessarily involve a minimax problem but instead a set construction.
In data-driven RO or DRO, the uncertainty set is constructed or calibrated from data. If such a set has the property of being a confidence region for the uncertain parameters or distributions, then by solving the RO or DRO, the confidence guarantee can be translated to bounds on the resulting decision, and in our case the eligibility set. This approach of constructing uncertainty sets, by viewing them as confidence regions or via hypothesis testing, has been the main approach in data-driven RO or DRO Bertsimas
et al. (2018b). Recently, alternate approaches have been studied to reduce the conservativeness in set calibration, by utilizing techniques from empirical likelihood Lam and
Zhou (2017); Lam (2019); Duchi
et al. , Wasserstein profile function Blanchet
et al. (2019), Bayesian perspectives Gupta (2019) and data splitting Hong
et al. (2020); Lam and
Qian (2019).
In our development, we have constructed an uncertainty set for the unknown distribution $P_{e}$ via a confidence region associated with the KS goodness-of-fit test. This uncertainty set has been proposed in Bertsimas
et al. (2018c). Other distance-based uncertainty sets, including $\phi$-divergence Petersen
et al. (2000); Ben-Tal et al. (2013); Glasserman and
Xu (2014); Lam (2016, 2018); Bayraksan and
Love (2015) and Wasserstein distance Esfahani and
Kuhn (2018); Blanchet and
Murthy (2016); Gao and
Kleywegt (2016), have also been used, as well as sets based on moment Delage and
Ye (2010); Ghaoui
et al. (2003); Hu
et al. (2012) or distributional shape information Popescu (2005); Li
et al. (2017); Van Parys
et al. (2016). We use a simultaneous group of KS statistics with Bonferroni correction, motivated by the tractability in the resulting integration with the importance weighting. The closest work to our framework is the stochastic simulation inverse calibration problem studied in Goeva
et al. (2019), but they consider single-dimensional output and parameter to calibrate the input distributions, in contrast to our “summary” approach via Fourier analysis and the multi-dimensional settings we face.
Another important ingredient in our approach is importance sampling. This is often used as a variance reduction tool (e.g., Siegmund (1976); Glynn and
Iglehart (1989); Asmussen and
Glynn (2007) Chapter 5; Glasserman (2013) Chapter 4) and is shown to be particularly effective in rare-event simulation (e.g., Bucklew (2013); Rubinstein and
Kroese (2016); Juneja and
Shahabuddin (2006); Blanchet and
Lam (2012)). It operates by sampling a random variable from a different distribution from the true underlying distribution, and applies a so-called likelihood ratio to de-bias the resulting estimate. Other than variance reduction, importance sampling is also used in risk quantification in operations research and mathematical finance that uses a robust optimization perspective (e.g., Glasserman and
Xu (2014); Ghosh and
Lam (2019); Lam (2016)), which is more closely related to our use in this paper. Additionally, it is used in Bayesian computation Liu (2008), and more recently in machine learning contexts such as covariate shift estimation Pan and
Yang (2009); Sugiyama et al. (2007) and off-policy evaluation in reinforcement learning Precup (2000); Schlegel et al. (2019).
In the remainder of this paper, we illustrate the use of our methodology and report our numerical results on the UQ Challenge.
7 Summarizing Discrete-Time Histories using Fourier Transform
By observing the plot of the outputs $y^{(i)},i=1,\dots,n_{1}$ (see Fig. 1), we judge that these time series are highly seasonal. Naturally, we choose to use Fourier transform to summarize $(y(t))_{t=0,\dots,T}$, and we may write $y(t)$ in the form $y(t)=\sum_{k=-\infty}^{\infty}C_{k}e^{-ik\omega_{0}t}.$
First we apply Fourier transform to $y^{(i)},i=1,\dots,n_{1}$. For each $y^{(i)}$, we compute the $C_{k}$’s. Fig. 2 shows the real part and the imaginary part of $C_{k}$’s against the corresponding frequencies.
For the real part, we see that there is a large positive peak, a large negative peak, a small positive peak and a small negative peak. After testing, we confirm that for any $i$, the large peaks lie in the first 14 terms (from 0Hz to 1.59Hz), while the small peaks lie between the 15th term and the 50th term (from 1.71Hz to 5.98Hz). For the imaginary part, we see that there is a large negative peak and a small positive peak. The large peak is also located in the first 14 terms and the small peak between the 15th term and the 50th one.
Therefore, we choose to use the following method to summarize $y$ (i.e., construct the function $\textbf{S}(\cdot)$): first, we apply the Fourier transform to compute $C_{k}$’s and the corresponding frequencies; second, we compute the real part and the imaginary part of $C_{k}$’s; third, for the real part, we find the maximum value and the minimum value over $[0Hz,1.59Hz]$ and $[1.71Hz,5.98Hz]$, as well as their corresponding frequencies; fourth, for the imaginary part, we find the minimum value over $[0Hz,1.59Hz]$ and the maximum value over $[1.71Hz,5.98Hz]$ as well as their corresponding frequencies. Then we use these 12 parameters as the summaries of $y$.
To illustrate how well these summaries fit $y$, Fig. 3(a) shows the comparison for $y^{(1)}$. The fit qualities of other time series are similar to this example. Though they are not extremely close to each other, the fitted curves do resemble the original curves. Note that it is entirely possible to improve the fitting if we keep more frequencies even if they are not as significant as the main peaks. For instance, Fig. 3(b) shows the improved fitting curve if for both real part and imaginary part, we respectively keep the 20 frequencies with the largest values. It can be seen that now the fit quality is quite good. On the other hand, as discussed in Section 2, using a larger number of summaries both represents more knowledge of $P_{e}$ (better fitting) but also leads to more simultaneous estimation error when using the Bonferroni correction needed in calibrating the set for $P_{e}$. To balance the conservativeness of our approach coming from representativeness versus simultaneous estimation, we choose to use the 12-parameter summaries depicted before.
8 Uncertainty Reduction (Problem B)
8.1 Ranking Epistemic Parameters (B.1 and B.2)
Now we implement Algorithm 1 with $n_{2}=k=1000$ and the summary function $\mathbf{S}(\cdot)$ defined in the previous section. The dimension of the summary function is $m=12$. We choose $\alpha$ to be 0.05. Thus, following the algorithm, for each $l=1,\dots,n_{2}$, we compute $q_{l}^{*}$ and then compare it with $q_{1-\alpha/m}=q_{1-0.05/12}=1.76$.
In Fig. 4, we plot the $q_{l}^{*}$’s against each dimension of $e$. The red horizontal lines in the graphs correspond to $q_{1-\alpha/m}=1.76$. Thus the dots below the red lines constitute the eligible $e$’s. We rank the epistemic parameters according to these graphs, namely we rank higher the parameter whose range can potentially be reduced the most. Note that this ranking scheme can be summarized using more rigorous metrics related to the expected amount of eligible $e$’s after range shrinkage, but since there are only four dimensions, using the graphs directly seem sufficient for our purpose here.
We find that the values of $e_{2}$ and $e_{4}$ of the eligible $e$’s broadly range from 0 to 2, which implies that reducing the ranges of these two dimensions could hardly reduce our uncertainty. By contrast, the values of $e_{1}$ and $e_{3}$ of the eligible $e$’s are both concentrated in the lower part of $[0,2]$. Thus, our ranking of the epistemic parameters according to their ability to improve the predictive ability is $e_{3}>e_{1}>e_{2}>e_{4}$.
Chances are that the true values of $e_{1}$ and $e_{3}$ are relatively small. In order to further pinpoint the true values of $e_{1}$ and $e_{3}$, we choose to make two uncertainty reductions: increase the lower limits of the bounding interval of $e_{1}$ and $e_{3}$.
8.2 Impact of the value of $n_{1}$ (A.2)
To investigate the impact of the value of $n_{1}$,
for different values of $n_{1}$ we randomly sample $n_{1}$ outputs without replacement. Then we take these outputs as the new data set. By repeatedly implementing Alg. 1, we find that the larger is $n_{1}$, the smaller is the proportion of eligible $e$’s. It is intuitive that as the data size grows, $e$ can be better pinpointed. Moreover, except for $e_{4}$, the range of each epistemic variable of eligible $e$’s obviously shrinks as $n_{1}$ increases, which further confirms that $e_{4}$ is the least important epistemic variable.
8.3 Updated Parameter Ranking (B.3)
After the epistemic space is reduced, we repeat the process in Section 8.1 but now $e$’s are generated uniformly from $E_{1}$. From the associated scatter plots (Fig. 5), the updated ranking of the epistemic parameters is $e_{2}>e_{3}>e_{1}>e_{4}$.
9 Reliability of Baseline Design (Problem C)
9.1 Failure Probabilities and Severity (C.1, C.2 and C.5)
Combining the refined range of $e$ provided by the host with our Algorithm 1, we construct $E\subset E_{1}$. To estimate
$\min_{e\in E}/\max_{e\in E}\mathbb{P}(g_{i}(a,e,\theta)\geq 0)$, we run simulations to respectively solve
$$\begin{split}\min/\max\ &\sum_{j=1}^{k}W_{j}I(g_{i}(a^{(j)},e,\theta)\geq 0)\\
\text{s.t. }&e\in E,W\in U\end{split}$$
(17)
where $U$ is the set of $(W_{1},\cdots,W_{k})$ in Eq. (16). These give the range of $R_{i}(\theta)$. We use the same method to approximate $R(\theta)$, the failure probability for any requirement. Note that in our implementation the $E$ in the formulations above is represented by discrete points $e^{(l)}$’s. As discussed previously, under additional smoothness assumptions, we could “smooth” these points to obtain a continuum. Nonetheless, under sufficient sampling of $e^{(l)}$, the discretized set should be a good enough approximation in the sense that the optimal values from the “discretized” problems are close to those using the continuum.
Using the above method, we get that the ranges of $R_{1}(\theta)$, $R_{2}(\theta)$, $R_{3}(\theta)$ and $R(\theta)$ are approximately $[0,0.6235]$, $[0,0.7320]$, $[0,0.5270]$ and $[0,0.8217]$. Though the ranges seem to be quite wide, they can provide us useful information to be utilized next.
To evaluate $s_{i}(\theta)$, the severity of each individual requirement violation, similarly we simulate
$$\max_{e\in E}\max_{W\in U}\sum_{j=1}^{k}W_{j}g_{i}(a^{(j)},e,\theta)I(g_{i}(a^{(j)},e,\theta)\geq 0).$$
The results for $s_{1}(\theta)$, $s_{2}(\theta)$ and $s_{3}(\theta)$ are respectively 0.1464, 0.0493 and 3.5989. Clearly the violation of $g_{3}$ is the most severe one while the violation of $g_{2}$ is the least.
9.2 Rank for Uncertainties (C.3)
Our analysis on the rank for epistemic uncertainties is based on the range of $R(\theta)$ obtained above. In our computation, we obtain
$$\min_{W\in U}/\max_{W\in U}\sum_{j=1}^{k}W_{j}I(g_{i}(a^{(j)},e,\theta)\geq 0\text{\ for some\ }i=1,2,3)$$
for each eligible $e\in E$. For simplicity, we use $R_{min}$ and $R_{max}$ to denote these two values for each eligible $e\in E$ respectively.
Our approach is to scrutinize the plots of $R_{min}$ and $R_{max}$ against each epistemic variable (Fig. 6 and 7). For $R_{min}$, large value is notable, since it means that any distribution that provides similarity to the original data is going to fail with large probability. Therefore the most ideal reduction is to avoid the region of $e$ such that all $R_{min}$’s are large. For $R_{max}$, the largest $R_{max}$ for the region denotes the maximum failure probability that one can have. So we pay attention to the epistemic variables that could potentially reduce the “worst-case” failure probability. Based on these considerations, we conclude that the rank for epistemic uncertainties is $e_{3}>e_{1}>e_{2}>e_{4}$.
9.3 Representative Realizations (C.4)
Since the distribution of $a$ in our approach is defined as an ambiguity set that depends on $e$, the failure domain would also be based on each eligible $e$. We classify an eligible $e$ to be notable if its corresponding $R_{min}$ is relatively large (e.g., $>0.1$). For convenience, we denote the “best-case”distribution corresponding to $R_{min}$ as $w_{min}$, where
$$w_{min}=\arg\min_{W\in U}\sum_{j=1}^{k}W_{j}I(g_{i}(a^{(j)},e,\theta)\geq 0).$$
We consider the representative realizations of uncertainties as those $a$’s with large value of $w_{min}$ (in our case we consider $>0.05$).
From our observation, we find that these representative realizations have a clear pattern on the scatter plot with $a_{1}$ and $a_{3}$ as the coordinates (as in Fig. 8). We also provide some example responses of cases in each group. We observe that there is a clear similarity in the responses within each group, which can be interpreted as different failure patterns.
10 Reliability-Based Design (Problem D)
To find a reliability-optimal design point $\theta_{new}$, we minimize
$$\max_{e\in E}\min_{W\in U}\sum_{j=1}^{k}W_{j}I(g(a^{(j)},e,\theta)\geq 0).$$
(18)
Here is the reason why we choose this function as the objective. For an eligible $e\in E$, if $\min_{W\in U}\sum_{j=1}^{k}W_{j}I(g(a^{(j)},e,\theta)\geq 0)$ is large, then the true probability in which the system fails must be even larger than this “best-case” estimate, which implies that this point $e$ has a considerable failure likelihood. The objective above thus aims to find a design point to minimize this best-case estimate, but taking the worst-case among all the eligible $e$’s. Arguably, one can use other criteria such as minimizing $\max_{e\in E}\max_{W\in U}\sum_{j=1}^{k}W_{j}I(g(a^{(j)},e,\theta)\geq 0)$, but this could make our procedure more conservative.
The optimization problem (18) is of a “black-box” nature since the function $g$ is only observed through simulation, and the problem is easily non-convex. Our approach is to use a gradient descent to guide us towards a better $\theta_{new}$, with a goal of finding a reasonably good $\theta_{new}$ (instead of insisting on full optimality which could be difficult to achieve in this problem). Note that we need to sample $a^{(j)}$ when we land at a new $\theta$ during our iterations, and hence our approach takes the form of a stochastic gradient descent or stochastic approximation Fu (2015); Jian and
Henderson (2015). Moreover, the gradient cannot be estimated in an unbiased fashion as we only have black-box function evaluation, and thus we need to resort to the use of finite-difference. This results in a zeroth-order or the so-called Kiefer-Wolfowitz (KW) algorithm Kushner and
Yin (2003); Ghadimi and
Lan (2013). As we have a nine-dimensional design variable, we choose to update $\theta$ via a coordinate descent, namely at each iteration we choose one of the dimensions and run a central finite-difference along that dimension, followed by a movement of $\theta$ guided by this gradient estimate with a suitable step size. The updates are done in a round-about fashion over the dimensions. The perturbation size in the finite-difference is chosen of order $1/n^{1/4}$ here as it appears to perform well empirically (though theoretically other scaling could be better).
Algorithm 2 shows the details of our optimization procedure. Considering that the components of $\theta_{baseline}$ are of very different magnitudes, we first perform a normalization to ease this difference. The quantity $x_{now}$ encodes the position of the normalized $\theta_{now}$, and $1_{9}$ denotes a nine-dimensional vector of $1$’s that is set as the initial normalized design point. We set $c_{0}=a_{0}=0.1$ and $N_{max}=8$.
After running the algorithm, we arrive at a new design point, $\theta_{new}$: ($-0.1999$, $-0.6975$, $315.31$, $4525.3$, $4924.2$, $1.0358$, $280.32$, $14.171$, $132.52$). Compared with the baseline design, the objective function decreases from 0.3656 to 0.2732. Note that this means that the best-case estimate of the failure probability, among the worst possible of all eligible $e$’s, is 0.2732.
For $\theta_{new}$, the ranges of $R_{1}(\theta)$, $R_{2}(\theta)$, $R_{3}(\theta)$ and $R(\theta)$ (defined in Section 9.1) are approximately $[0,0.5935]$, $[0,0.7469]$, $[0,0.5465]$ and $[0,0.8205]$.
We could observe from the plots of $R_{min}$ and $R_{max}$ that $e_{2}$ has significant different patterns on high values in both plots. According to the trends shown in the plots, we rank the epistemic variables as $e_{2}>e_{3}>e_{1}>e_{4}$.
11 Design Tuning (Problem E)
With data sequence $D_{2}=\{z^{(i)}(t)\}$ for $i=1,\dots,n_{2}$, we may incorporate the additional information to update our model as before. Similar to Section 7, we use Fourier transform to summarize the highly seasonal responses. In particular, we represent $(z^{(i)}_{1}(t))_{t=0,...,T}$ and $(z^{(i)}_{2}(t))_{t=0,...,T}$ as $z^{(i)}_{1}(t)=\sum_{k=-\infty}^{\infty}C^{1}_{k}e^{-ik\omega_{0}t}$ and $z^{(i)}_{2}(t)=\sum_{k=-\infty}^{\infty}C^{2}_{k}e^{-ik\omega_{0}t}$ respectively. As shown in Figures 9 and 10, the responses in frequency domain have common patterns in the positive and negative peaks. Again we use the values of these peaks and their corresponding frequencies to summarize $z_{1}$ and $z_{2}$, which leads to 20 extra parameters adding to the 12 parameters extracted from $D_{1}$.
With the extracted parameters from both $D_{1}$ and $D_{2}$, we now update our eligibility set for $E$ by computing $q_{l}^{*}$’s. We determine eligible $e$’s with the new threshold $q_{1-0.05/32}=1.89$. The values of $q_{l}^{*}$’s are presented in Figure 11. Compared with Figure 5, we observe that the trend in $e_{2}$ changes slightly. The $q_{l}^{*}$’s with high value in $e_{2}$ become higher after introducing the information from $D_{2}$, which indicates that $e$ with higher $e_{2}$ is less eligible. Based on the stronger trend in $e_{2}$ and the observation in Section 10, we determine to refine $e_{2}$ on both ends.
In Figure 12, we present $q_{l}^{*}$’s of samples of $e$ for determining the final eligibility set, $E_{2}$. With these updated information, the final design $\theta_{final}$ is obtained using Algo. 2, where $\theta_{final}$: ($-0.21762$, $-0.66706$, $295.61$, $4410.3$, $4394.1$, $1.1968$, $264.49$, $16.444$, $127.18$). The ranges of $R_{1}(\theta_{final})$, $R_{2}(\theta_{final})$, $R_{3}(\theta_{final})$ and $R(\theta_{final})$ are [0,0.1676], [0,0.1620], [0,0.046] and [0,0.2551] respectively. Compared to $\theta_{baseline}$ and $\theta_{new}$, the worst-case reliability performance is significantly improved.
12 Risk-Based Design (Problem F)
Recall that we create an eligibility set for $e$ in the form of $\{e^{(l)}:q^{*}_{l}\leq q_{1-\alpha/m}\}$, which provides us $(1-\alpha)$ confidence for covering the truth asymptotically. To reduce $r\%$ volume of the eligibility set, we remove $r\%$ number of eligible points in the set with the larger $q^{*}_{l}$’s. Since larger $q^{*}_{l}$ indicates less similarity with the true response, the reduced eligibility set maintains more important $e^{(l)}$’s.
In our setting for $e$, taking risks is equivalent to reducing the confidence level for covering the truth. Let us assume the $r\%$ upper quantile of $q^{*}_{l}$ is $q_{r\%}$. Then the reduced eligibility set can be represented as $\{e^{(l)}:q^{*}_{l}\leq q_{r\%}\}$. By finding the $\tilde{\alpha}$ such that $q_{r\%}=q_{1-\tilde{\alpha}/m}$, we can find the confidence level $1-\tilde{\alpha}$ that corresponds to each choice of $r\%$. In later discussion, the reduced eligibility set corresponding to risk level $r\%$ is denoted as $E_{r\%}$.
In our experiment, we use $E_{2}$ in Section 11 as the baseline. Table 1 shows the risk levels and their corresponding confidence levels. The relation between $r\%$ and $\tilde{\alpha}$ highly depends on the value of $q_{l}^{*}$’s. In our case, we observe that a large portion of $q_{l}^{*}$’s are close to $q_{1-\alpha/m}$. As a consequence, the reduction in the volume of the set does not lead to a similar extent of reduction in the confidence level. Since the confidence level is almost not changed, we can anticipate that the design results with different $r\%$ in the range of $(0,10)$ will perform similarly.
With different $r\%$’s, we construct $E_{r\%}$’s using the above approach and implement Algo. 2 to obtain risk-based designs $\theta_{r\%}$’s. Then we evaluate the $\theta_{r\%}$’s by computing the reliability and severity metrics based on their corresponding eligibility set $E_{r\%}$ and also $E_{2}$. The evaluation results using $E_{2}$ are shown in Figure 13. We observe from Figure 13 that both the reliability or severity metrics are insensitive to the change of $r\%$. In fact, the results are also insensitive to
whether using $E_{2}$ or $E_{r\%}$ to compute the metrics. Since the difference can be neglected (the largest difference is smaller than 0.01), we omit the results using $E_{r\%}$. From these results, we confirm our conjecture that taking risks would not make much difference in our design approach.
13 Discussion
In this UQ Challenge, we propose a methodology to calibrate model parameters and quantify calibration errors from output data under both aleatory and epistemic uncertainties. The approach utilizes a framework based on an integration of distributionally robust optimization and importance sampling, and operates computationally by solving sampled linear programs. It provides theoretical confidence guarantees on the coverage of the ground truth parameters and distributions. We apply and illustrate our approach to the model calibration and downstream risk analysis tasks in the UQ Challenge. Our approach is drastically different from established Bayesian methodologies, both in the type of guarantee (frequentist versus Bayesian) and computation method (optimization versus posterior sampling). We anticipate much further work in the future in expanding our methodology to more general problems as well as comparing with the established approaches.
We discuss some immediate future improvement in our implementation in this UQ Challenge. Our procedure relies on several configurations that warrant further explorations.
First, the eligibility set geometry is dictated by the choices of the distance metric between distributions and the summary function. Our choice of KS-distance is motivated from nonparametric hypothesis testing that provides asymptotic guarantees. However, since only a finite number of samples is available in practice, its performance can be problem dependent, and other nonparametric test statistics could be considered. Regarding summary functions, we have chosen them based on the visualization of Fourier transform and justify their number via a balance of representativeness and conservativeness in simultaneous estimation. Our refinement results indicate that our eligibility set performs well in locating $e$, which validates our configurations. Nonetheless, a more rigorous approach to choose both the distance metric and the summary functions is desirable.
Our approach requires sampling a number of $a$ and $e$ for eligibility set and aleatory distribution construction. Since a limited size of naive (uniform) sample might miss important information in a large continuous space and cause high variance, we have used several variance reduction techniques including stratified sampling and common random numbers. We note that the samples for $a$ have a larger effects on designs, since they are used to construct the associated best- and worst-case distributions and the quality of samples can be crucial to correctly evaluating the design performances. Moreover, a good sampling scheme can also lead to higher stability of the stochastic gradient descent algorithm.
Lastly, we note that the conservative nature of our robust approach is reflected in the system design. While our robust approach performs well in locating the eligibility set and providing upper bounds on reliability, directly using these bounds as the objectives for optimizing designs appear over-conservative. Further work on improving the choice of eligibility sets and sampling on $a$ and $e$ could help improve these design performances.
Appendix A Proof of Theorem 2
This proof is adapted from Goeva
et al. (2019). We denote $L=dP_{true}/dP_{0}$. Let $W_{j}=\frac{L(a^{(j)})}{\sum_{j=1}^{k}L(a^{(j)})}$. For simplicity, we use $\mathbf{y}(a)$ to denote $(y(a,e_{true},t))_{t=1,\dots,T}$ and use $\mathbf{y}_{j}$ to denote $(y(a^{(j)},e_{true},t))_{t=1,\dots,T}$. Then we have that
$$\displaystyle\sup_{x\in\mathbb{R}}\left|\sum_{j=1}^{k}W_{j}I(S_{v}(\mathbf{y}_{j})\leq x)-\hat{F}_{v}(x)\right|$$
$$\displaystyle\leq$$
$$\displaystyle\sup_{x\in\mathbb{R}}\left|\sum_{j=1}^{k}W_{j}I(S_{v}(\mathbf{y}_{j})\leq x)-\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)\right|+$$
$$\displaystyle\sup_{x\in\mathbb{R}}\left|\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)-E_{P_{0}}(L(a)I(S_{v}(\mathbf{y}(a))\leq x))\right|+$$
$$\displaystyle\sup_{x\in\mathbb{R}}\left|E_{P_{true}}(I(S_{v}(\mathbf{y}(a))\leq x))-\hat{F}_{v}(x)\right|.$$
For the first term, we have that
$$\displaystyle\sum_{j=1}^{k}W_{j}I(S_{v}(\mathbf{y}_{j})\leq x)-\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)\left(\frac{k}{\sum_{j=1}^{k}L(a^{(j)})}-1\right).$$
Since $\|dP_{true}/dP_{0}\|_{\infty}\leq C$, we get that $\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)\leq C$. Moreover, we know that $E_{P_{0}}(L)=1$ and $var_{P_{0}}(L)<\infty$, and thus
$$\sqrt{k}\left(\frac{k}{\sum_{j=1}^{k}L(a^{(j)})}-1\right)\Rightarrow N(0,var_{P_{0}}(L)).$$
Hence, we get that
$$\sup_{x\in\mathbb{R}}\left|\sum_{j=1}^{k}W_{j}I(S_{v}(\mathbf{y}_{j})\leq x)-\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)\right|=O_{p}(1/\sqrt{k}).$$
For the second term, following the proof in Goeva
et al. (2019), we know that
$$\left\{\sqrt{k}\left(\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)-E_{P_{0}}(L(a)I(S_{v}(\mathbf{y}(a))\leq x))\right)\right\}\Rightarrow\{G(x)\}$$
in $\ell^{\infty}\left(\left\{a\mapsto L(a)I(S_{v}(\mathbf{y}(a))\leq x):x\in\mathbb{R}\right\}\right)$ and $G$ is a Gaussian process. Therefore, we get that
$$\sup_{x\in\mathbb{R}}\left|\frac{1}{k}\sum_{j=1}^{k}L(a^{(j)})I(S_{v}(\mathbf{y}_{j})\leq x)-E_{P_{0}}(L(a)I(S_{v}(\mathbf{y}(a))\leq x))\right|=O_{p}(1/\sqrt{k}).$$
Finally, it is known that
$$\sqrt{n_{1}}\sup_{x\in\mathbb{R}}\left|E_{P_{true}}(I(S_{v}(\mathbf{y}(a))\leq x))-\hat{F}_{v}(x)\right|\Rightarrow\sup_{x\in[0,1]}|BB(F_{true,r}(x))|.$$
Combining the above results, we get that for each $v=1,\dots,m$,
$$\limsup_{n_{1}\rightarrow\infty,k/n_{1}\rightarrow\infty}\mathbb{P}\left(\sup_{x\in\mathbb{R}}\left|\sum_{j=1}^{k}W_{j}I(S_{v}(\mathbf{y}_{j})\leq x)-\hat{F}_{v}(x)\right|>\frac{q_{1-\alpha/m}}{\sqrt{n_{1}}}\right)\leq\frac{\alpha}{m}$$
and hence
$$\liminf_{n_{1}\rightarrow\infty,k/n_{1}\rightarrow\infty}\mathbb{P}(e_{true}\in E)\geq 1-\alpha.$$
Acknowledgements
We gratefully acknowledge support from the National Science Foundation under grants CAREER CMMI-1834710 and IIS-1849280. A preliminary conference version of this paper has appeared in the Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference (ESREL-PSAM) 2020.
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Dimensions of Automorphic Representations, $L$-Functions and Liftings
Solomon Friedberg and David Ginzburg
Friedberg: Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, USA
[email protected]
Giinzburg: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 6997801, Israel
[email protected]
Abstract.
There are many Rankin-Selberg integrals representing Langlands $L$-functions, and it is not apparent what the limits of the Rankin-Selberg
method are. The Dimension Equation is an equality satisfied by many such integrals that suggests a priority for further investigations.
However there are also Rankin-Selberg integrals that do not satisfy this equation. Here we propose an extension and reformulation
of the dimension equation that includes many additional cases. We explain some of these cases, including the new doubling integrals
of the authors, Cai and Kaplan. We then show how this same
equation can be used to understand theta liftings, and how doubling integrals fit into a lifting framework. We give an example of a new type of
lift that is natural from this point of view.
Key words and phrases:Rankin-Selberg integral, Langlands $L$-function, dimension equation, unipotent orbit, Gelfand-Kirillov dimension, doubling integral,
theta correspondence
2010 Mathematics Subject Classification: Primary 11F66; Secondary 11F27, 11F70, 17B08, 22E50, 22E55
This research was supported by the US-Israel Binational Science foundation, grant number 2016000, and by the NSF, grant number DMS-1801497 (Friedberg).
1. Introduction
Two broad classes of integrals appear frequently in the theory of automorphic forms.
Let $G$ be a reductive group defined over a number field $F$, $\rho$ be a complex analytic representation of the $L$-group of $G$, and $\pi$ be
an irreducible automorphic representation of $G(\mathbb{A})$. First,
one may sometimes represent the Langlands $L$-function $L(s,\pi,\rho)$ (for $\Re(s)\gg 0$) as an integral, and the desired analytic properties
of this $L$-function may then be deduced from the integral representation. Such constructions, often called Rankin-Selberg integrals, have a long history with many examples (Google Scholar
lists 3,370 results for the phrase “Rankin-Selberg integral”), with Eisenstein
series and their Fourier coefficients appearing in many of the integrals. See Bump [2] for an engaging survey.
Second, given two reductive groups $G,H$ there is sometimes
a function $\theta(g,h)$ on $G(\mathbb{A})\times H(\mathbb{A})$ that is $G(F)\times H(F)$ invariant and that may be used as an integral kernel
to transport automorphic representations on $G(\mathbb{A})$ to $H(\mathbb{A})$. One considers the functions
$$f_{\varphi}(h):=\int_{G(F)\backslash G(\mathbb{A})}\varphi(g)\,\theta(g,h)\,dg$$
as $\varphi$ runs over the functions in $\pi$ and lets $\sigma$ be the representation of $H(F)\backslash H(\mathbb{A})$ generated by the functions $f_{\varphi}$. The most familiar example is the classical theta correspondence, where the integral kernel $\theta$ is constructed from the Weil representation (see Gan [15] for recent progress), while other classes of such integrals
are given by the authors and Bump [7] and by Leslie [33].
We pose two natural questions. First, the construction of Rankin-Selberg integrals representing $L$-functions is often quite involved, and it may take a great deal of work
to see that a specific integral is Eulerian. Is there any commonality among the known Rankin-Selberg integrals that can be used
to decide whether a specific integral is a worthy candidate for investigation, or to say it differently, to rule out
integrals as being unlikely to represent an $L$-function? Second, is there any way to know whether an automorphic function of two variables $\theta(g,h)$ is likely to be a
useful kernel function, and if so, can one predict the properties of the representation $\sigma$ from those of $\pi$ and $\theta$? For example, given $\pi$ and $\theta$,
when is it reasonable to think that $\sigma$ might be generic?
An answer to the first question was proposed by the second named author in [18]: the dimension equation describes an equality
between the dimensions of groups and the dimensions of the representations that is satisfied by many integrals that represent $L$-functions.
Below we develop the dimension question in some detail and illustrate it in many cases, including cases of integrals that represent the product of two or more distinct
Langlands $L$-functions in separate complex variables. In fact a refinement of the dimension equation expands its applicability to additional cases.
We shall explain this refinement, and show how this
allows us to include doubling integrals including the new doubling integrals of the authors, Cai and Kaplan [11]. Then we shall use the dimension equation to discuss integral kernels,
showing how the equation gives an indication of what to expect in integral kernel constructions, and
explain how doubling integrals may be used to bridge these two classes of constructions. Last, we shall pursue the dimension equation farther in specific cases, providing new examples
for further study.
To conclude our introduction, we describe the contents by section.
In Section 2 we introduce unipotent orbits and state the dimension equation, following [18].
Though this paper may be read independently from [18], it is a natural continuation of that work. Then, in Section 3 we give many examples of Rankin-Selberg integrals
that satisfy the dimension equation. We conclude the section by presenting an exotic example—an integral of Rankin-Selberg type that is Eulerian by two different choices of Eisenstein series, one satisfying the dimension equation
but the other not. This motivates the need to extend the dimension equation. This extension is described in Section 4, and examples
of the extended dimension equation are presented in Section 5. Next, in Section 6 integral kernels are connected to the dimension equation,
and the use of the dimension equation to predict aspects of the resulting correspondence is illustrated. Section 7 revisits doubling integrals from the perspective of integral
kernels, and formulates a general classification question. Then in Section 8 a new low rank example is presented and its analysis is described in brief.
The global theory has a local counterpart, and the final Section illustrates the local properties that appear in this context.
Part of the preparation of this paper occurred when the first-named author was a visitor at the Simons Center for Geometry and Physics, and he expresses his appreciation for this opportunity.
While the paper was in the final stages of preparation, Aaron Pollack communicated to us that Shrenik Shah has independently suggested an extension of the dimension equation.
2. The Dimension Equation
Fix a number field $F$ and let $\mathbb{A}$ be its ring of adeles. If $G$ is an algebraic group defined over $F$, then we write $[G]$ for the quotient $G(F)\backslash G(\mathbb{A})$.
We begin by recalling some facts about unipotent orbits. References for this material are Collingwood and McGovern [12] and Ginzburg [17].
Let $G$ be a reductive group defined over $F$, $\overline{F}$ be an algebraic closure of $F$, and let $\mathcal{O}$ be a unipotent orbit of $G(\overline{F})$. If $G\subset GL(V)$ is a classical
group then these orbits are indexed by certain partitions of $\dim(V)$. For example, if $G=GL_{n}$ then they are indexed by all partitions of $n$, while if
$G=Sp_{2n}$ then they are indexed by the partitions of $2n$ such that each odd part occurs an even
number of times. When $\mathcal{O}$ is indexed by a partition $\mathcal{P}$, we simply write $\mathcal{O}=\mathcal{P}$.
For convenience suppose that $G$ is a split classical group.
Fix a Borel subgroup $B=TN$ with unipotent radical $N$, let $\Phi$ denote the positive roots with respect to $B$, and for $\alpha\in\Phi$ let $x_{\alpha}(t)$ denote the corresponding one-parameter
subgroup of $N$.
Then one attaches a unipotent subgroup $U_{\mathcal{O}}\subseteq N$ to $\mathcal{O}$.
This group may be described as follows. If $\mathcal{O}$ is given in partition form by
$\mathcal{O}=(p_{1}^{e_{1}}\dots p_{k}^{e_{k}})$ (we show repeated terms in a partition using exponential notation) let $h_{\mathcal{O}}(t)$ be the diagonal matrix whose entries are $\{t^{p_{i}-2j-1}\mid 0\leq j\leq p_{i}-1\}$ repeated $e_{i}$ times, with the entries
arranged in non-increasing order in terms of power of $t$. This gives rise to a filtration $N\supset N_{1}\supset N_{2}\supset\cdots$ of $N$, where $N_{i}=N_{i,\mathcal{O}}$ is given by
(1)
$$N_{i,\mathcal{O}}(F)=\{x_{\alpha}(r)\in N(F)\mid\alpha\in\Phi~{}\text{and}~{}h%
_{\mathcal{O}}(t)x_{\alpha}(r)h_{\mathcal{O}}(t)^{-1}=x_{\alpha}(t^{j}r)~{}%
\text{for some $j$ with~{}}j\geq i\}.$$
Also, let
$G_{\mathcal{O}}$ be the stabilizer of $h_{\mathcal{O}}(t)$ in $G$.
Fix a nontrivial additive character $\psi$ of $F\backslash\mathbb{A}$, and let $L_{2,\mathcal{O}}=N_{2,\mathcal{O}}/[N_{2,\mathcal{O}},N_{2,\mathcal{O}}]$.
Then the characters of the abelian group $L_{2,\mathcal{O}}(\mathbb{A})$ may be identified with $L_{2,\mathcal{O}}(F)$. Also $G_{\mathcal{O}}(F)$ acts on the characters of $L_{2,\mathcal{O}}(\mathbb{A})$ and hence on $L_{2,\mathcal{O}}(F)$ by conjugation.
Over the algebraically closed field $\overline{F}$, the action of $G_{\mathcal{O}}(\overline{F})$ on $L_{2,\mathcal{O}}(\overline{F})$ has an open orbit, and the stabilizer in $G_{\mathcal{O}}(\overline{F})$
of a representative for this orbit is a reductive group whose Cartan type is uniquely determined by the orbit. A character $\psi_{\mathcal{O}}$ of $L_{2,\mathcal{O}}(F)$ will be called a generic
character associated to $\mathcal{O}$ if
the connected component of its stabilizer in $G_{\mathcal{O}}(F)$ has, after base change to $\overline{F}$, the same Cartan type. We caution the reader that there may be infinitely many
$G_{\mathcal{O}}(F)$-orbits of generic characters associated to a single $\mathcal{O}$. (For an example, see [18], p. 162.)
Suppose that $G$ is a reductive group, $\mathcal{O}$ is a unipotent orbit of $G$, and $\psi_{\mathcal{O}}$ is a generic character. Let $U_{\mathcal{O}}=N_{2,\mathcal{O}}$ and regard
$\psi_{\mathcal{O}}$ as a character of $U_{\mathcal{O}}(\mathbb{A})$ in the canonical way.
If $\varphi$ is an automorphic form on $G(\mathbb{A})$, its Fourier coefficient
with respect to $\mathcal{O}$ is defined to be
$$\varphi^{U_{\mathcal{O}},\psi_{\mathcal{O}}}(g)=\int_{[U_{\mathcal{O}}]}%
\varphi(ug)\psi_{\mathcal{O}}(u)\,du,\qquad g\in G(\mathbb{A}).$$
If $\pi$ is an automorphic representation of $G(\mathbb{A})$, we say that $\pi$ has nonzero Fourier coefficients with respect to $\mathcal{O}$ if the set of functions $\varphi^{U_{\mathcal{O}},\psi_{\mathcal{O}}}(g)$
is not identically zero as $\varphi$ runs over the space of $\pi$ and $\psi_{\mathcal{O}}$ runs over set of generic characters associated to $\mathcal{O}$.
We recall that the set of unipotent orbits has a natural partial ordering, which corresponds to inclusion of Zariski closures and corresponds to the dominance order for the associated partitions.
There is a unique maximal unipotent orbit $\mathcal{O}_{\text{max}}$ and for this orbit the group $U_{\mathcal{O}_{\text{max}}}$ is $N$,the unipotent
radical of the Borel subgroup.
If $\pi$ is an automorphic representation of $G(\mathbb{A})$ we let $\mathcal{O}(\pi)$ denote the set of maximal unipotent orbits for which $\pi$ has nonzero Fourier coefficients. For example,
the automorphic representation $\pi$ is generic if and only if $\mathcal{O}(\pi)=\{\mathcal{O}_{\text{max}}\}$.
We remark that these same definitions apply without change in the covering group case. Indeed, if $\widetilde{G}$ is a cover of $G(\mathbb{A})$, then $N(\mathbb{A})$ embeds canonically in $\widetilde{G}(\mathbb{A})$
and so the same definitions apply. For example, in [6] the authors construct a representation $\Theta_{2n+1}$ of the double cover of $SO_{2n+1}(\mathbb{A})$ for $n\geq 4$ such
that $\mathcal{O}(\Theta_{2n+1})=(2^{2m}1^{2j+1})$, where $n=2m+j$ with $j=0$ or $1$. Here and below we omit the set notation when the set
$\mathcal{O}(\pi)$ is a singleton and identify $\mathcal{O}(\pi)$ with the partition.
Recall that each nilpotent orbit in a Lie algebra has a dimension. In fact, these dimensions are computed for classical groups in terms of partitions in [12], Corollary 6.1.4.
We use this to discuss and work with the dimensions of the unipotent orbits under consideration here.
To illustrate, on the symplectic group $Sp_{2n}$, let ${\mathcal{O}}$ be the unipotent orbit associated to the partition $(n_{1}n_{2}\ldots n_{r})$ of $2n$.
For such a partition to be a symplectic partition it is required that all odd numbers in the partition occur with even multiplicity. Then,
(2)
$$\dim{\mathcal{O}}=2n^{2}+n-\frac{1}{2}\sum_{i=1}^{r}(2i-1)n_{i}-\frac{1}{2}a,$$
where $a$ is the number of odd integers in the partition $(n_{1}n_{2}\ldots n_{r})$.
Importantly, there is also a connection between the dimension of an orbit $\mathcal{O}$ and the Fourier coefficients above. That is:
(3)
$$\tfrac{1}{2}\dim{\mathcal{O}}=\dim N_{2,\mathcal{O}}+\tfrac{1}{2}\dim N_{1,%
\mathcal{O}}/N_{2,\mathcal{O}}.$$
As this suggests, when $N_{1,\mathcal{O}}\neq N_{2,\mathcal{O}}$, there is another natural coefficient of Fourier-Jacobi type (that is, involving a theta series);
see [18], equation (7), for details, and the discussion of (15) at the end of Section 4 below.
Throughout the rest of this paper we make the following assumption:
Assumption 1.
For all automorphic representations $\pi$ under consideration, the dimension of each orbit in the set $\mathcal{O}(\pi)$ is the same.
We know of no examples where Assumption 1 does not hold, and it is expected that it is always true ([17], Conjecture 5.10). In fact, we know of no examples where $\mathcal{O}(\pi)$ is not a singleton.
Under Assumption 1, we write $\dim\mathcal{O}(\pi)$ for the dimension of any orbit in the set $\mathcal{O}(\pi)$.
Given a representation for which Assumption 1 holds, define (following [17], Definition 5.15) the Gelfand-Kirillov dimension of $\pi$:
$$\dim(\pi)=\tfrac{1}{2}\dim\mathcal{O(\pi)},$$
(compare [1], (3.2.6) and [31] (2.4.2), Remark (iii)). By definition, if $\chi$ is an idele class character then $\dim(\chi)=0$. Similarly if $\psi$ is an additive
character of $[U]$ where $U$ is a unipotent group then $\dim(\psi)=0$.
It is expected that it is possible to compute the unipotent orbit attached to an Eisenstein series from the unipotent orbits of its inducing data. See [18], Section 4.3.
This has been confirmed in the degenerate case [10]. More generally, suppose that $P$ is a parabolic subgroup of $G$ with Levi decomposition $P=MN$, and $\tau$ is an automorphic
representation of $M(\mathbb{A})$ such that Assumption 1 holds for $\tau$, and consider the Eisenstein series $E_{\tau}(g,s)$ on $G(\mathbb{A})$ corresponding to the induced representation
$\text{Ind}_{P(\mathbb{A})}^{G(\mathbb{A})}\tau\delta_{P}^{s}.$ Then one expects that
(4)
$$\dim E_{\tau}(g,s)=\dim\tau+\dim U.$$
This formula can be checked in many cases (to give one example, if $\tau$ is generic then $E_{\tau}$ is too, and so (4) holds)
and we shall make use of it below.
We now have the information needed to explain the Dimension Equation of [18], Definition 3.
Suppose one has a Rankin-Selberg integral over the various groups $G_{j}$
and involving various automorphic representations $\pi_{i}$, and the integral unfolds to unique functionals which are factorizable.
Here both the adelic modulo rational quotients in the integral and the automorphic representations are with respect to the same field $F$ and ring of adeles $\mathbb{A}$.
Then the expectation formulated in [18] is
that the sum of the dimensions of the representations is (generally) equal to the sum of the dimensions of the groups.
Definition 1.
The Dimension Equation is the equality
$$\sum_{i}\dim\pi_{i}=\sum_{j}\dim G_{j}.$$
A number of examples of this equation in the context of Rankin-Selberg integrals are presented in [18]. To clarify the extent of applicability of this equation, we will give additional examples in the next Section.
3. Examples of the Dimension Equation
We begin with classical examples of the dimension equation. Riemann’s second proof of the analytic continuation of the zeta function represents it as a
Mellin transform of the Jacobi theta function, and this integral satisfies the dimension equation (both sides are 1). Similarly, the Hecke integral
(5)
$$\int_{[\mathbb{G}_{m}]}\varphi\begin{pmatrix}a&\cr&1\end{pmatrix}|a|^{s}\,d^{%
\times}a$$
satisfies the equation (both sides are again 1), and the classical Rankin-Selberg integral
(6)
$$\int_{[PGL_{2}]}\varphi_{1}(g)\varphi_{2}(g)E(g,s)\,dg$$
has group of dimension 3 and three representations that are each of dimension 1.
More generally, the Rankin-Selberg integrals of Jacquet, Piatetski-Shapiro and Shalika [29] on $GL_{n}\times GL_{k}$ satisfy the dimension equation. Suppose that
$\pi_{1}$, $\pi_{2}$ are irreducible cuspidal automorphic representations of $GL_{n}(\mathbb{A})$, $GL_{k}(\mathbb{A})$ resp.
Since all cuspidal automorphic representations of general linear groups are generic, we have $\dim(\pi_{1})=n(n-1)/2$ and $\dim(\pi_{2})=k(k-1)/2$.
Let $\phi_{i}\in\pi_{i}$ for $i=1,2$.
If $k=n$ then the Ranklin-Selberg integral representing $L(s,\pi_{1}\times\pi_{2})$, generalizing (6), is given by
(7)
$$\int_{[PGL_{n}]}\varphi_{1}(g)\varphi_{2}(g)E(g,s)\,dg$$
where $E$ is the (“mirabolic”) Eisenstein series induced from the standard parabolic subgroup $P$ of type $(n-1,1)$ and character $\delta_{P}^{s}\eta^{-1}$, where
$\delta_{P}$ is the modular character of $P$ and
$\eta$ is the product of the central
characters of $\pi_{1}$ and $\pi_{2}$.
In this integral the dimension of the group is $n^{2}-1$. As for the representations,
the orbit of the Eisenstein series $E$ is $(21^{n-2})$, so it is of dimension $n-1$ (compare (4)). The dimension equation is the equality
$$n^{2}-1=2(n(n-1)/2)+n-1.$$
If $k\neq n$, suppose $k<n$ without loss. When $k=n-1$ the integral is a direct generalization of (5):
$$\int_{[GL_{n-1}]}\varphi_{1}\begin{pmatrix}g&\cr&1\end{pmatrix}\varphi_{2}(g)|%
\det g|^{s}\,dg.$$
Here the group is of dimension $(n-1)^{2}$ and the representations $\pi_{1},\pi_{2}$ are of dimension $n(n-1)/2$ and $(n-1)(n-2)/2$ resp. Since
$$(n-1)^{2}=n(n-1)/2+(n-1)(n-2)/2$$
the dimension equation holds. However, if $k<n-1$ then a similar integral
$$\int_{[GL_{k}]}\varphi_{1}\begin{pmatrix}g&\cr&I_{n-k}\end{pmatrix}\varphi_{2}%
(g)|\det g|^{s}\,dg$$
does not satisfy the dimension equation: the group is of dimension $k^{2}$ while the representations are of total dimension
$n(n-1)/2+k(k-1)/2>k^{2}$. One way to satisfy the equation is to introduce an additional integration over a group of dimension $(n^{2}-n-k^{2}-k)/2$.
And indeed the integral of Jacquet, Piatetski-Shapiro and Shalika
$$\int_{[GL_{k}]}\int_{[Y_{n,k}]}\varphi_{1}\left(y\begin{pmatrix}g&\cr&I_{n-k}%
\end{pmatrix}\right)\varphi_{2}(g)\psi(y)|\det g|^{s}\,dg$$
includes an integration over the group
$Y_{n,k}$ consisting of upper triangular $n\times n$ unipotent matrices whose upper left $(k+1)\times(k+1)$ corner is the
identity matrix, a group which has dimension $n(n-1)/2-k(k+1)/2$. To be sure, the integral of [29]
requires an additive character $\psi$ of this group (the restriction of the standard generic character to $Y_{n,k}$), and this finer level of detail is not seen by the dimension equation,
but the dimension equation already makes the introduction of the group $Y_{n,k}$ natural.
Another example of the dimension equation is practically tautological. If $E(g,s)$ is an Eisenstein series on a reductive group $G$ formed from generic inducing data, then a straightforward
calculation shows that it too is generic. If $E(g,s)$ is generic with respect to the
unipotent subgroup $N$ and character $\psi_{N}$, then its Whittaker coefficient
$$\int_{[N]}E(n,s)\psi_{N}(n)\,dn$$
satisfies the dimension equation, since both the group and the representation have dimension $\dim(N)$. The Langlands-Shahidi method studies such coefficients
systematically and uses them to get information about $L$-functions. See Shahidi [37] and the references there.
Our next example of the dimension equation is given by the integral representing the Asai $L$-function for $GL_{n}$ (Flicker [13]).
Suppose $K/F$ is a quadratic extension, and for this example let $\mathbb{A}$ denote the adeles of $F$ and $\mathbb{A}_{K}$ the adeles of $K$.
Suppose that $\Pi$ is an irreducible cuspidal automorphic representation on $GL_{n}(\mathbb{A}_{K})$, and $\varphi\in\Pi$. Then this integral is of the form
$$\int_{Z(\mathbb{A})GL_{n}(F)\backslash GL_{n}(\mathbb{A})}\varphi(g)E(g,s)\,dg,$$
where $E(g,s)$ is again a mirabolic Eisenstein series on $GL_{n}(\mathbb{A})$ (the central character of $\Pi$ is built into $E$ so the product is invariant under the center $Z(\mathbb{A})$ of $GL_{n}(\mathbb{A}))$.
Here the group has dimension $n^{2}-1$. This time the automorphic form is on the $\mathbb{A}$-points of the restriction of scalars $\text{Res}_{K/F}GL_{n}$. Thus the
dimension of $\pi$ viewed as an automorphic representation over $F$ is twice its dimension over $K$. That is, in this context $\dim(\Pi)=2\times\frac{n(n-1)}{2}$. The dimension equation is satisfied
since
$$n^{2}-1=2\times\frac{n(n-1)}{2}+n-1.$$
There are also integrals that represent the product of two different Langlands $L$-functions
in two separate complex variables. These also satisfy the dimension equation. We give a number of examples.
The first instance of such an integral, representing the product of the
standard and exterior square $L$-functions, was provided by Bump and Friedberg [3]. Suppose
that $\pi$ is a cuspidal automorphic representation of $GL_{n}$. If $n=2k$ is even, then the integral is of the form
$$\int_{[GL_{1}\backslash(GL_{k}\times GL_{k})]}\varphi(\iota(g_{1},g_{2}))E(g_{%
1},s_{1})\left|\frac{\det g_{2}}{\det g_{1}}\right|^{s_{2}-1/2}\,dg_{1}\,dg_{2},$$
where $\iota:GL_{k}\times GL_{k}\to GL_{2k}$ is a certain embedding, $\varphi$ is in $\pi$, and $E$ is the mirabolic Eisenstein series
on $GL_{k}$. In the quotient $GL_{1}\backslash(GL_{k}\times GL_{k})$, the group $GL_{1}$ acts diagonally. This integral satisfies the dimension equation since the group is of dimension
$2k^{2}-1$ and the representations are of dimensions $(2k)(2k-1)/2$ and $k-1$. If $n=2k+1$ is odd, then the integral is similar, but is now over
$[GL_{1}\backslash(GL_{k+1}\times GL_{k})]$ with the Eisenstein series on $GL_{k+1}$. The dimension equation is satisfied in the case that $n$ is odd since
$$(k+1)^{2}+k^{2}-1=(2k+1)(2k)/2+((k+1)-1).$$
Three additional examples of Rankin-Selberg integrals representing the product of Langlands $L$-functions attached to the standard and spin representations
in separate complex variables
are found in Bump, Friedberg and Ginzburg [5]. Suppose $\pi$ on $GSp_{4}$ is generic with trivial central character and $\varphi$ is in the space
of $\pi$. Let $P$, $Q$
be the two non-conjugate standard maximal parabolic subgroups of $GSp_{4}$ and let $E_{P}$, $E_{Q}$ be the Eisenstein series induced from
from the modular functions $\delta_{P}^{s_{1}}$, $\delta_{Q}^{s_{2}}$ resp. Then the integral is of the form
$$\int_{[GL_{1}\backslash GSp_{4}]}\varphi(g)E_{P}(g,s_{1})E_{Q}(g,s_{2})\,dg.$$
Here the group is of dimension $10$ and the representations are of dimensions $4,3,3$ resp., so the dimension equation is satisfied.
Suppose next that $\pi$ is on $GSp_{6}$, generic, and has trivial central character, and let $H$ be the subgroup
of $GL_{2}\times GSp_{4}$ of pairs of group elements $(h_{1},h_{2})$ with equal similitude factors. Then, for a certain embedding $\iota:H\to GSp_{6}$, the integral given there is of the form
$$\int_{[GL_{1}\backslash H]}\varphi(\iota(h_{1},h_{2}))E_{1}(h_{1},s_{1})E_{2}(%
h_{2},s_{2})\,dh_{1}\,dh_{2}$$
where $E_{1}$ (resp. $E_{2}$) is a Borel (resp. Siegel) Eisenstein series on $GL_{2}$ (resp. $GSp_{4}$).
Here the group is of dimension $13$ and the three representations in the integral
are of dimensions 9, 1, 3 (resp.), as the Siegel Eisenstein series has unipotent orbit $(2^{2})$.
Third suppose that $\pi$ is on $GSp_{8}$ and is once again generic. Introduce the subgroup $H$ of $GL_{2}\times GSp_{6}$ of pairs with equal similitude factors
and let $\iota:H\to GSp_{8}$ be a certain embedding given in [4].
Then the authors and Bump prove that the integral
$$\int_{[GL_{1}\backslash H]}\varphi(\iota(h_{1},h_{2}))E(h_{2},s_{1},s_{2})\,dh%
_{1}\,dh_{2}$$
is once again a product of two different Langlands $L$-functions, where $E$ is the two-variable Eisenstein series on $GSp_{6}$ induced from the
standard parabolic with Levi factor $GL_{1}\times GL_{2}$. The group has dimension 24, $\pi$ has dimension 16, and indeed $E$ has dimension 8.
To present one further example,
Pollack and Shah [36] give an integral representing the product of three Langlands L-functions in three distinct complex variables. If $\pi$ is on $PGL_{4}$, then this is of the form
$$\int_{[PGL_{4}]}\varphi(g)E_{1}(g,w)E_{2}(g,s_{1},s_{2})\,dg$$
where $E_{1}$ and $E_{2}$ are the Eisenstein series with Levi factors $GL_{2}\times GL_{2}$ and $GL_{1}^{2}\times GL_{2}$, resp. Here the dimension of the group in the integral
is 15 and the representations are of dimensions 6,4,5 resp. Those authors also present a related integral on $GU(2,2)$ and the same dimension count holds.
The dimension equation holds as well when covering groups are involved. For example, the symmetric square integrals of Bump-Ginzburg [9] and Takeda [38] may be checked
to satisfy this.
Nonetheless, not all Rankin-Selberg integrals satisfy the dimension equation in the form presented in [18, 19].
Here is a simple, striking example.
Suppose $\pi$ is a cuspidal automorphic representation of $PGL_{4}(\mathbb{A})$, $\varphi$ is in the space of $\pi$, and $E(g,s)$ is an Eisenstein series on
the group $PGSp_{4}(\mathbb{A})$. Consider the integral
(8)
$$\int_{[PGSp_{4}]}\varphi(g)E(g,s)\,dg.$$
The basic dimension equation states that
$$\dim(\pi)+\dim(E)=\dim(PGSp_{4})=10.$$
Since $\pi$ is generic, it is of dimension 6, so this equation requires an Eisenstein series of dimension 4, that is, a generic Eisenstein series.
In fact there is an Eulerian integral with this data. Using the $GSp_{4}$ Eisenstein series with trivial central character
induced from an automorphic representation $\tau$ on $GL_{2}$
via the parabolic with Levi $GL_{1}\times GL_{2}$ (the Klingen parabolic), the integral unfolds to the degree 12
$L$-function $L(s,\pi\times\tau,\wedge^{2}\times\text{standard})$. Indeed, after using low rank isogenies one sees that this integral is essentially the same as the
integral for the $SO_{n}\times GL_{k}$ standard $L$-function obtained by Ginzburg [16] in the case $n=6$, $k=2$. For these parameters, the $L$-function
is realized as an integral of an $SO_{6}$ automorphic form against an Eisenstein series on $SO_{5}$.
However, there is another Rankin-Selberg integral of the form (8) that is also Eulerian!
That is the case that $E$ is an Eisenstein series of dimension 3 induced from the modular function $\delta_{P}^{s}$ of the Siegel parabolic $P$ of $Sp_{4}$ (one may also twist by
a character of $GL_{1}$).
Indeed, after using low rank isogenies to again regard this as an integral of an $SO_{6}$ automorphic form against an Eisenstein series on
$SO_{5}$, this is the integral treated by the authors and Bump [4] with $n=2,m=1$ (see (1.4) there). The integral is zero unless
the automorphic representation $\pi$ is a lift from $Sp_{4}$, and in that case it represents a degree 5 $L$-function. However
the dimension equation does not hold, since the Siegel Eisenstein series is of dimension 3, not 4.
There are other important examples of integrals that represent $L$-functions but that do not satisfy the dimension equation of [18].
One class of such integrals are the doubling integrals, a class first constructed by Piatetski-Shapiro and Rallis [34], which represent the standard $L$-function of an automorphic
representation on a classical group $G$, twisted by $GL_{1}$. The original doubling integral was for split classical groups but the construction has been extended or used by a number of authors including
Yamana [39], Gan [14], Lapid and Rallis [32], and Ginzburg and Hundley [20].
The doubling construction was extended to the tensor product $L$-function for $G\times GL_{k}$ by Cai, Friedberg, Kaplan and Ginzburg [11].
(This extension is particularly helpful as it allows one to use the converse theorem to prove lifting results.)
Another class of integrals representing $L$-functions that does not satisfy the original dimension equation is the “new way” integrals, a class also initiated by Piatetski-Shapiro and Rallis [35].
Once again this class has been extended, in particular there are the integrals of Bump, Furusawa and Ginzburg [8] and of Gurevich-Segal [28].
A third type of integral that is outside the scope of the dimension equation is the Godement-Jacquet integral [27].
Finally, the WO-model integrals of [4] do not satisfy the dimension equation.
These last examples all raise the question: is it possible to modify the dimension equation so that it encompasses these examples? In fact, the answer is yes.
We explain this in the next Section below.
4. Extending the Dimension Equation
Let us begin with the general form of a Rankin-Selberg integral, following [19]. Let $G_{i}$, $1\leq i\leq l$ be reductive
groups defined over $F$, $\pi_{i}$ be irreducible automorphic representations of $G_{i}(\mathbb{A})$, and let $\varphi_{i}\in\pi_{i}$ be automorphic forms.
Suppose that at least one automorphic form, say $\varphi_{1}\in\pi_{1}$, is cuspidal so that the integral to be considered converges.
Let
$U_{i}\subset G_{i}$ be subgroups attached to some unipotent orbits of $G_{i}$, and let $\psi_{i}$ be characters of $[U_{i}]$.
Here the groups $U_{i}$ may be trivial, and the characters $\psi_{i}$ need not be in ‘general position’ for $U_{i}$;
in particular we do not assume that $U_{i}$ is attached to the unipotent orbit of $\pi_{i}$.
We suppose that there is a reductive group $G$ such that for each $i$, the stabilizer of $\psi_{i}$ inside a suitable Levi subgroup of $G_{i}$ contains
$G$ (up to isomorphism). In this case if $\varphi_{i}$ is an automorphic form on $G_{i}$ then the Fourier coefficient $\varphi_{i}^{U_{i},\psi_{i}}$
is automorphic as a function of $G(\mathbb{A})$.
Then we consider the integral
(9)
$$\int_{[Z\backslash G]}\varphi_{1}^{U_{1},\psi_{1}}(g)\varphi_{2}^{U_{2},\psi_{%
2}}(g)\dots\varphi_{l}^{U_{l},\psi_{l}}(g)\,dg,$$
where $Z$ is the center of $G$ and the central characters are chosen compatibly so that the integrand is $Z(\mathbb{A})$ invariant.
We are concerned with the case that one automorphic form, say $\varphi_{l}$, is an Eisenstein series induced from a Levi subgroup $P=MN$ and an automorphic
representation $\tau$ of $M(\mathbb{A})$. Write $\varphi_{l}$ as $\varphi_{l}(g,s)$ and write the associated
section $f_{\tau}(g,s)$, so for $\Re(s)\gg 0$ we have
$$\varphi_{l}(g,s)=\sum_{\gamma\in P(F)\backslash G(F)}f_{\tau}(\gamma g,s).$$
Then the integral (9) is a function of a complex variable $s$, defined for $\Re(s)\gg 0$ and with analytic continuation (possibly with poles) and functional equation by
virtue of the corresponding properties for the Eisenstein series $\varphi_{l}(g,s)$.
Such integrals may represent $L$-functions when they can be shown to be equal to adelic integrals of some factorizable coefficients of the functions in the integrand. To write this generally,
suppose that for each $i$, $1\leq i\leq l-1$, the automorphic representation $\pi_{i}$ attached to $\varphi_{i}$
has the property that $\mathcal{O}(\pi_{i})$ consists of a single unipotent orbit $\mathcal{O}_{i}$ with unipotent group $V_{i}$, and that $\varphi_{i}$ has nonzero Fourier coefficients with
respect to the generic character $\psi_{V_{i}}$ of $[V_{i}]$. Write the associated Fourier coefficient
(10)
$$L_{i}(\varphi_{i},g)=\int_{[V_{i}]}\varphi_{i}(vg)\psi_{V_{i}}(v)\,dv,\qquad g%
\in G_{i}(\mathbb{A}).$$
Since the first step in analyzing an integral of the form (9) is to unfold the
Eisenstein series $\varphi_{l}(g,s)$, let $V_{l}=V_{\tau}\,N$ where $V_{\tau}$ is the unipotent subgroup of $M\subseteq G_{l}$
corresponding to the maximal
unipotent orbit $\mathcal{O}_{\tau}$ attached to $\tau$ (again assumed unique), and let $\psi_{V_{l}}$ be a corresponding character of the unipotent subgroup $V_{\tau}$
extended trivially on $N$. Then we write
(11)
$$L_{l}(f_{\tau},g,s)=\int_{[V_{l}]}f_{\tau}(vg,s)\psi_{V_{l}}(v)\,dv,\qquad g%
\in G_{l}(\mathbb{A}).$$
To describe an unfolding of the integral (9), for $1\leq i\leq l$, let $R_{i}$ be a unipotent subgroup of $G_{i}$, and $\psi_{R_{i}}$ be a character of $R_{i}(\mathbb{A})$.
For $1\leq i\leq l-1$ write
$$\varphi_{i}^{R_{i}}(g)=\int_{R_{i}(\mathbb{A})}L_{i}(\varphi_{i},rg)\psi_{R_{i%
}}(r)\,dr,\qquad g\in G_{i}(\mathbb{A})$$
and similarly let
$$f_{\tau}^{R_{l}}(g,s)=\int_{R_{l}(\mathbb{A})}L_{l}(f_{\tau},rg,s)\psi_{R_{l}}%
(r)\,dr,\qquad g\in G_{l}(\mathbb{A}).$$
Then we suppose that for $\Re(s)\gg 0$ the integral (9) unfolds to
(12)
$$\int_{Z(\mathbb{A})M(\mathbb{A})\backslash G(\mathbb{A})}\varphi_{1}^{R_{1}}(g%
)\varphi_{2}^{R_{2}}(g)\dots\varphi_{l-1}^{R_{l-1}}(g)f_{\tau}^{R_{l}}(w_{0}g,%
s)\,dg,$$
where $M$ is a subgroup of $G$ and $w_{0}$ is a Weyl group element.
Such an integral is called a unipotent global integral.
When the functionals
$$\mathcal{L}_{i}(\varphi_{i})=\varphi_{i}^{R_{i}}(e),1\leq i\leq l-1;\qquad%
\mathcal{L}_{l,s}(f_{\tau})=f_{\tau}^{R_{l}}(e,s)$$
are each factorizable, then the unfolded integral is Eulerian. Such an integral is called a Eulerian
unipotent integral in [19]. In fact, this broad class of integrals includes all the Rankin-Selberg integrals presented above
which satisfy the original form of the dimension equation. More generally,
the dimension equation
is expected to hold for all Eulerian unipotent integrals, and a classification of this class of integrals is initiated in [19].
Our goal now is to extend the dimension equation to include many of the examples noted at the last section, that is, Rankin-Selberg integrals
that do not satisfy the dimension equation. To do so, we begin
with the same integral (9) but we extend the notation so that each $\varphi_{i}$ is now either
a single automorphic form in $\pi_{i}$ or a pair of automorphic forms, one in $\pi_{i}$ and the other in its contragredient $\widetilde{\pi}_{i}$. We also
relax the description of the unfolding above in two ways.
We suppose once again that the
integral unfolds to an Eulerian expression (12). However, in this expression we no longer assume that $L_{i}$ is a Fourier coefficient given by an integral
of the form (10) or (11) over a
unipotent subgroup $V_{i}$
against a character of the maximal unipotent orbit attached to $\pi_{i}$ or $\tau$. Instead we allow the integrals in (10), (11) to be over arbitrary subgroups of $G_{i}$.
For example, $L_{i}$ might be an integral realizing a unique functional such as the Shalika functional
or the spherical functional. In this case, we do not use $\dim{\pi_{i}}$ in the dimension equation. Instead, we replace this term by the dimension
of the full group that realizes the unique functional.
To be specific, for $1\leq i\leq l-1$ suppose that there is an algebraic group $X_{i}\subset G_{i}$, not necessarily unipotent, such that
$$L_{i}(\varphi_{i})(g)=\int_{[X_{i}]}\varphi_{i}(xg)\,\psi_{X_{i}}(x)\,dx,$$
where $\psi_{X_{i}}$ is a character of $[X_{i}]$, and let $\mathcal{L}_{i}$ be the associated functional on $\pi_{i}$.
In this case we define the dimension of $\mathcal{L}_{i}$ to be the dimension of the algebraic group $X_{i}$. For example, if $\varphi_{i}=(\phi,\phi^{\prime})$ is a pair of
automorphic functions with $\phi\in\pi_{i}$, $,\phi^{\prime}\in\widetilde{\pi}_{i}$, we may consider $\mathcal{L}_{i}$ to be the functional that assigns to $\varphi_{i}$ the global matrix coefficient
(13)
$$\mathcal{L}_{i}(\varphi_{i})=\int_{[G_{i}]}\phi(g)\phi^{\prime}(g)\,dg.$$
In this case we define $\dim(\mathcal{L}_{i})=\dim G_{i}$. Similarly, for the Eisenstein series induced from $P$, we consider
$$L_{l}(f_{\tau},g,s)=\int_{[X_{l}]}f_{\tau}(xg,s)\psi_{X_{l}}(v)\,dv,\qquad g%
\in G_{l}(\mathbb{A}),$$
and we define
$\dim(\mathcal{L}_{l})=\dim X_{l}+\dim N$.
Note that we include the dimension of $N$ in the dimension of the
functional $\mathcal{L}_{l}$. This gives, by definition, an extension of (4).
This definition of the dimension of a functional requires a coda.
To explain why, suppose that $E(g,s)$ is the mirabolic Eisenstein series on $GL_{3}(\mathbb{A})$. This function has unipotent orbit $(21)$ (in fact, it generates the
automorphic minimal representation for $GL_{3}(\mathbb{A})$), so it has dimension 2. However, if $e_{i,j}$ denotes the matrix with $1$ in position $(i,j)$ and $0$ elsewhere, then it is easy to prove that
$$\int_{(F\backslash\mathbb{A})^{2}}E(I_{3}+re_{1,2}+me_{1,3},s)\,\psi(r)\,dr\,%
dm=\int_{(F\backslash\mathbb{A})^{3}}E(I_{3}+re_{1,2}+me_{1,3}+ne_{2,3},s)\,%
\psi(r)\,dr\,dm\,dn$$
and in fact both corresponding functionals are nonzero and unique. More generally, when an orbit $\mathcal{O}$ is small the integral defining the Fourier coefficient with respect to that orbit
will have additional invariance properties (there is a nontrivial group that normalizes $U_{\mathcal{O}}$ and stabilizes $\psi_{\mathcal{O}}$) and so can be enlarged in a similar way. Hence to define the dimension of a functional $\mathcal{L}$
we must specify that if a functional over a smaller group also realizes $\mathcal{L}$
then we use that smaller group in defining the dimension of $\mathcal{L}$.
We broaden the dimension equation to the following extended dimension equation.
Definition 2.
The Extended Dimension Equation is the equality
(14)
$$\dim(G)+\sum_{i=1}^{l}\dim(U_{i})=\sum_{i=1}^{l}\dim(\mathcal{L}_{i}).$$
That is, the sum of the dimensions of the groups in the Rankin-Selberg integral (9) is equal to the sum of dimensions of the functionals that are obtained
after unfolding. The key point is that we are using the dimensions of functionals in place of the Gelfand-Kirillov dimensions of the representations $\pi_{i}$.
We conclude this section by comparing the extended dimension equation with the original form of the dimension equation,
where $\dim(\mathcal{L}_{i})$ is replaced by $\dim(\pi_{i})$, a quantity which is computed by (3). Suppose first that
$\pi$ is an automorphic representation such that $\mathcal{O}(\pi)$
is a single unipotent orbit $\mathcal{O}$. Recall that there is an associated filtration $N_{i,\mathcal{O}}$ of $N$ given by (1).
Let $\varphi_{\pi}$ be an automorphic form in the space of $\pi$.
If $N_{1,\mathcal{O}}=N_{2,\mathcal{O}}$, and if an integral involving $\varphi_{\pi}$
unfolds to the Fourier coefficient $\varphi_{\pi}^{U_{\mathcal{O}},\psi_{\mathcal{O}}}$ i.e. to the functional (10) given by integration over $[N_{2,\mathcal{O}}]$, then since
$\dim(\pi)=\dim(N_{2,\mathcal{O}})$ (by (3)), in this situation $\dim{\mathcal{L}}=\dim\pi$. If instead $N_{1,\mathcal{O}}\neq N_{2,\mathcal{O}}$, then
$N_{1,\mathcal{O}}/N_{2,\mathcal{O}}$ has the structure of a Heisenberg group. In this situation, it is often the case that
an integral involving $\varphi_{\pi}$ unfolds to a Fourier-Jacobi coefficient of the form
(15)
$${L}(\varphi_{\pi})(h):=\int\limits_{[N_{1,\mathcal{O}}]}\theta_{Sp}(l(v)h)%
\varphi_{\pi}(vh)\,\psi_{N_{1}}(v)\,dv,$$
where $\theta_{Sp}$ is a certain theta function obtained via the Weil representation. (Here we might need to involve covering groups.)
See for example [24], Section 3.2; the notation is given in detail there. If $\mathcal{L}$ is the functional obtained by composing $L$ with evaluation
at the identity, then in this case we would have
$\dim{\mathcal{L}}=\dim{N_{1,\mathcal{O}}}.$
However, if $\Theta_{Sp}$ is the representation corresponding to $\theta_{Sp}$, then
it is known that $\dim\Theta_{Sp}=\tfrac{1}{2}\dim(N_{1,\mathcal{O}}/N_{2,\mathcal{O}})$.
Since $\dim\pi=\dim N_{2,\mathcal{O}}+\tfrac{1}{2}\dim(N_{1,\mathcal{O}}/N_{2,%
\mathcal{O}})$ (see (3)), we obtain
the equality
$\dim\Theta_{Sp}+\dim\pi=\dim{N_{1,\mathcal{O}}}.$
Thus the definition of $\dim{\mathcal{L}}$
is in fact consistent with original dimension equation in this case. We conclude that when Fourier-Jacobi coefficients are used to construct Eulerian integrals (see for example [21]),
the different versions of the dimension equation we have presented are consistent, and it is accurate to label (14) an extension of the original dimension equation.
Last, for the Eisenstein series, if the integral in the inducing data $\tau$ unfolds to
the Whittaker functional then the definition of $\dim(\mathcal{L}_{l})$ is consistent with (4) and with the original Dimension Equation.
5. Examples of the Extended Dimension Equation
In this section we offer examples of the extended dimension equation, and explain how other classes of integrals representing $L$-functions fit into the picture.
To begin, both integrals of the form (8) satisfy this extended dimension equation. Indeed, if $E$ is the Siegel Eisenstein series whose unipotent orbit
has dimension 3, then this integral unfolds to a WO model for $\pi$
in the sense of [4]. This model involves an integration over a 3-dimensional reductive group (a form of $SO_{3}$) as well as a 4-dimensional unipotent group so in this case the dimension of the functional
applied to $\pi$ is 7.
The modified dimension equation does indeed hold, in the form $10=7+3$ (in contrast to the integral involving the Klingen Eisenstein series, where the contributions from
the two functions in the integrand were $6$ and $4$).
Next we discuss doubling integrals. This is a class of integrals introduced by Piatetski-Shapiro and Rallis [34], of the form
(16)
$$\int\limits_{G(F)\times G(F)\backslash G({\mathbb{A}})\times G({\mathbb{A}})}%
\varphi_{\pi}(g)\varphi_{\sigma}(h)E(\iota(g,h),s)\,dg\,dh$$
where $G$ is a symplectic or orthogonal group, $\pi$ and $\sigma$ are two irreducible cuspidal automorphic representations of $G({\mathbb{A}})$,
and $\varphi_{\pi}$, $\varphi_{\sigma}$ are in the corresponding spaces of automorphic forms. The Eisenstein series is defined on
an auxilliary group $H({\mathbb{A}})$
and $\iota:G\times G\to H$ is an injection.
They show that after unfolding the Eisenstein series, the open orbit representative involves the inner product
(17)
$$<\varphi_{\pi},\varphi_{\sigma}>=\int\limits_{G(F)\backslash G({\mathbb{A}})}%
\varphi_{\pi}(g)\varphi_{\sigma}(g)\,dg$$
as inner integration. This integral is nonzero unless $\sigma$ is the contragredient of $\pi$, and in that case,
the integral involves the functional (13), that is, the matrix coefficient, which is factorizable. It is readily checked that
the original form of the dimension equation does not hold.
The dimension of the functional (13) is equal to $\dim(G)$. Thus the extended dimension equation attached to the integral (16) is
$$2\,\dim(G)=\dim(G)+\dim(E),$$
that is,
$$\dim(G)=\dim(E).$$
Moreover, this equation is satisfied by the integrals in [34]. For example,
consider integral (16) with $G=Sp_{2n}$. In this case the Eisenstein series $E(\cdot,s)$ is defined on the group $H({\mathbb{A}})$ with $H=Sp_{4n}$. It is the maximal parabolic Eisenstein
series attached to the parabolic $P$ with Levi factor $GL_{2n}$ obtained by inducing the modular character $\delta_{P}^{s}$.
Thus $\dim(E)=\dim(U(P))$, where $U(P)$ is the unipotent radical of $P$. This is given by
$$\dim(U(P))=\frac{1}{2}(1+2+\cdots+2n)=n(2n+1)=\dim(Sp_{2n}).$$
Thus the extended dimension equation indeed holds for the doubling integral (16). We remark that $E(\cdot,s)$ is attached to
the unipotent orbit $(2^{2n})$.
It may similarly be confirmed that the dimension equation holds for the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan [11],
that represent the Rankin-Selberg $L$-function on $G\times GL_{k}$, where $G$ is a classical group, attached to the tensor product of the standard representations.
These integrals are of the form
(18)
$$\int\limits_{G(F)\times G(F)\backslash G({\mathbb{A}})\times G({\mathbb{A}})}%
\varphi_{\pi}(g)\,\varphi_{\sigma}(h)\,E_{\tau}^{U,\psi_{U}}(\iota(g,h),s)\,dg%
\,dh$$
where now $E_{\tau}$ is an Eisenstein series on a larger group $H$ whose construction depends on $\tau$ and the superscripts on $E$ denote a Fourier coefficient
with respect to a unipotent group $U$ and character $\psi_{U}$ such that $\iota(G\times G)$ is contained in the stabilizer of $\psi_{U}$ inside the normalizer of $U$ in $H$.
Suppose that $G=Sp_{2n}$. Then $H=Sp_{4kn}$, the Eisenstein series is induced from a generalized Speh represntation on $GL_{2kn}$ which has unipotent
orbit $(k^{2n})$, and
the $(U,\psi_{U})$ coefficient is one corresponding to the unipotent orbit $((2k-1)^{2n}1^{2n})$ in $H$.
The integral once again unfolds to an integral involving matrix coefficients.
Due to the Fourier coefficient with respect to $(U,\psi_{U})$, the extended dimension equation in this case becomes
(19)
$$2\dim(Sp_{2n})+\dim(U)=\dim(Sp_{2n})+\dim(E_{\tau}).$$
To show that this is true, it follows from [17] that
$$\dim(E_{\tau})=\frac{1}{2}\dim((k)^{2n})+\dim(U(P)),$$
where $\dim((k)^{2n})$ is the dimension of the unipotent orbit of the inducing data,
and $U(P)$ is the unipotent radical of the maximal parabolic $P$ inside $H$ whose Levi factor is $GL_{2kn}$. (That is, (4) holds in this situation.)
The number $\frac{1}{2}\dim((k)^{2n}))$ is equal to
the dimension of the unipotent radical of the parabolic subgroup of $GL_{2nk}$ whose Levi
part is $GL_{2n}^{k}$, that is, $2n^{2}k(k-1)$. It is then easy to check that (19) holds.
There are other classes of Eulerian integrals. The ‘new way’ integrals of [35] unfold to functionals that are not unique. They satisfy the extended dimension equation,
albeit tautologically. The integrals of Godement-Jacquet type may be obtained from doubling integrals after unfolding. Hence they should be regarded as belonging
to this paradigm. Whether or not this is helpful for efforts to extend the method (the Braverman-Kazhdan-Ng$\hat{\textrm{o}}$ program) remains to be seen.
6. Integral Kernels and the Dimension Equation
Integral kernels appear often in the theory of automorphic forms as a way to relate automorphic forms on one group to automorphic
forms on a different group. In this brief Section, we explain the connection between such constructions and the dimension equation and illustrate the use
of this equation to detect properties of such a correspondence. We follow [18] and provide an additional example of interest.
Then in the next Section we turn to a similar analysis using the extended dimension equation.
Suppose that $G,H,L$ are reductive groups and there is an embedding $\iota:G\times H\to L$ such that the images of $G$ and $H$ in $L$ commute.
Let $U$ be a unipotent subgroup of $L$ and $\psi_{U}$ be a character of $[U]$ whose stabilizer in $L$ contains $\iota(G(\mathbb{A}),H(\mathbb{A}))$.
Let $\Theta$ denote an
automorphic representation of $L(\mathbb{A})$. Then one may seek to construct a lifting from automorphic representations of $G(\mathbb{A})$ to $H(\mathbb{A})$ as follows.
Let $\pi$ denote an irreducible cuspidal representation of $G({\mathbb{A}})$. Let $\sigma$ be the representation of $H({\mathbb{A}})$ generated by all functions of the form
(20)
$$f(h)=\int\limits_{[G]}\ \int\limits_{[U]}\varphi_{\pi}(g)\,\theta(u(g,h))\,%
\psi_{U}(u)\,du\,dg,$$
with $\theta$ in the space of $\Theta$ and $\varphi_{\pi}$ in the space of $\pi$. Notice that from the properties of $\Theta$,
the functions $f(h)$ are $H(F)$-invariant functions on $H(\mathbb{A})$. As a first case, suppose that $\sigma$ is an irreducible automorphic representation of $H(\mathbb{A})$.
In that case, the dimension equation attached to this construction is
(21)
$$\dim(G)+\dim(U)+\dim(\sigma)=\dim(\pi)+\dim(\Theta).$$
That is, since the integral (20) gives a representation on $H(\mathbb{A})$ instead of an $L$-function, we include the dimension of the lift, $\sigma$, with the dimensions of the groups.
See [18], Section 6. More generally, suppose that $\sigma$ is in the discrete part of the space $L^{2}(H(F)Z_{H}(\mathbb{A})\backslash H(\mathbb{A}),\omega)$, where
$Z_{H}$ is the center of $H$ and $\omega$ is a central character (this is true, for example, when $\sigma$ is cuspidal). In that case we expect that at least one of the summands of $\sigma$
will satisfy equation (21).
This simple equation turns out to be quite powerful. We illustrate with an example.
Suppose that $G=Sp_{2n}$, $H=SO_{2k}$, $L=Sp_{4nk}$, and $\Theta$ is the classical theta representation.
Note that the unipotent orbit attached to $\Theta$ is $(21^{4nk-2})$, and $\dim(\Theta)=2nk$. Suppose also that $\pi$ is generic, so
$\dim(\pi)=n^{2}$. Since $\dim(G)=2n^{2}+n$,
the dimension equation (21) becomes
$$\dim(\sigma)=2nk-n-n^{2}.$$
As a first consequence, if $k<\tfrac{n+1}{2}$ then the equation would assert that $\dim(\sigma)<0$. So we expect that the lift must be zero.
Second, let us ask for which $k$ the lift $\sigma$ can be generic. In that case, we would have $\dim(\sigma)=k^{2}-k$. We conclude that a necessary condition for $\sigma$ to be
generic is the condition
$$k^{2}-k=2nk-n-n^{2}.$$
For a fixed $n$, there are two solutions to this equation, namely $k=n+1$ and $k=n$. And indeed, both these consequences of the dimension equation are true.
The lift does vanish if $k<\tfrac{n+1}{2}$. And
the lift to $SO_{2k}$ with $k=n+1$ is always generic while the lift with $k=n$ is sometimes generic, and these are the only cases where the lift of a generic cuspidal automorphic
representation is generic. See [22], Cor. 2.3 and the last two paragraphs in Section 2;
for the analogous local result see Proposition 2.4 there.
7. Doubling Integrals and Integral Kernels
In this Section we connect doubling integrals and integral kernels. First let us describe such doubling integrals in general.
Suppose that $H$ is a group, $U\subseteq H$ is a unipotent subgroup, and $\psi_{U}$ is a character of $[U]$. Suppose that there is an embedding $\iota:G\times G\to N_{H}(U)$ whose
image fixes $\psi_{U}$ (under conjugation). Let $\pi$, $\sigma$ be automorphic representations of a group $G(\mathbb{A})$.
Then
we consider integrals of the form
(22)
$$\int\limits_{[G\times G]}\ \int\limits_{[U]}\varphi_{\pi}(g)\,\varphi_{\sigma}%
(h)\,E(u\iota(g,h),s,f_{s})\psi_{U}(u)\,du\,dg\,dh.$$
Note that this includes the integrals (16) considered by Piatetski-Shapiro and Rallis but that
we allow an extra unipotent integration, as in (18).
We say that the integral (22) is a doubling integral if for $\Re(s)$ large it is equal to
(23)
$$\int\limits_{G({\mathbb{A}})}\ \int\limits_{U_{0}({\mathbb{A}})}<\varphi_{\pi}%
,\sigma(g)\varphi_{\sigma}>f_{W}(\delta u_{0}(g,1),s)\psi_{U}(u_{0})\,du_{0}\,dg,$$
for some subgroup $U_{0}$ of $U$, some Fourier coefficient $f_{W}$ of the section $f_{s}$, and some
$\delta\in H(F)$. Due to the matrix coefficient, this integral is zero unless $\sigma$ is the contragredient of $\pi$, so we
suppose this from now on.
The extended dimension equation attached to the integral (22) is
(24)
$$\dim(G)+\dim(U)=\dim(E).$$
We emphasize that if a certain integral satisfies the extended dimension equation (24) it does not necessarily means that the integral will be non-zero or Eulerian. This can only be determined after the unfolding process. The advantage of the equation (24) is that it eliminates unlikely candidates.
Suppose that (22) is a doubling integral. Then we may make an integral kernel as follows.
Fix a cuspidal automorphic representation $\pi$ of $G(\mathbb{A})$, and consider the functions
(25)
$$f(h)=\int\limits_{[G]}\int\limits_{[U]}\varphi_{\pi}(g)E(u(g,h),s)\psi_{U}(u)%
\,du\,dg.$$
Arguing as in Theorem 1 of Ginzburg and Soudry [25] we deduce that the representation $\sigma$
generated by these functions is a certain twist of the representation $\pi$.
In particular we have $\dim(\sigma)=\dim(\pi)$. Hence the dimension equation (21) is the same as the extended dimension equation (24).
Accortdingly, one can view the construction given by (20) as a generalization of the construction of doubling integrals.
This discussion leads to the following Classification Problem, which illustrates the kind of question
that these constructions raise. Let $\pi$ denote a cuspidal representation of $G({\mathbb{A}})$, and $\sigma$ denote a cuspidal representation of $H({\mathbb{A}})$.
Find examples of representations $\Theta$ defined on a group $L({\mathbb{A}})$ as above which satisfy the following two conditions:
(1)
The dimension equation (21) holds.
(2)
Suppose that the integral
(26)
$$\int\limits_{[G\times H]}\int\limits_{[U]}\varphi_{\sigma}(h)\varphi_{\pi}(g)%
\theta(u(g,h))\,\psi_{U}(u)\,du\,dg\,dh$$
is not zero for some choice of data. Then the representation $\pi$ determines the representation $\sigma$ uniquely.
This classification problem has many solutions as stated. We present an example of how it may be approached in the next Section.
We remark that a similar analysis based on the dimension equation may be applied to the descent integrals of Ginzburg, Rallis and Soudry [24].
8. An Example
In this Section we will show how the above considerations can help find possible global integrals of the form given by equation
(26). To do so, in this Section we will work with the case $G=H=Sp_{2m}$.
The first step is to consider Fourier coefficients whose stabilizer contains the group $Sp_{2m}\times Sp_{2m}$.
From the theory of nilpotent orbits, the partition
$((2k-1)^{2m}(2r-1)^{2m})$ has this property (see [12]).
This leads us to look for a representation $\Theta$ defined on the group
$Sp_{4m(k+r-1)}({\mathbb{A}})$ which is $Sp_{4m(k+r-1)}(F)$ invariant and which satisfies the dimension equation given in (21).
Thus, we are seeking representations $\Theta$ such that $\mathcal{O}(\Theta)=\mathcal{O}$ with
(27)
$$\dim(Sp_{2m})+\tfrac{1}{2}\dim((2k-1)^{2m}(2r-1)^{2m})=\tfrac{1}{2}\dim({%
\mathcal{O}}).$$
Here $\mathcal{O}$ corresponds to a partition of the number $4m(k+r-1)$. There are many solutions.
For example, if we begin with the orbit $(5^{2}3^{2})$, that is $m=1$, $k=3$ and $r=2$, then the orbits $\mathcal{O}$ equal to
$(65^{2})$, $(83^{2}2)$, $(6^{2}2^{2}),$ and $(8421^{2})$ all satisfy condition (27).
Since this low rank case already offers so many possibilities,
this suggests that it is not expeditious
to classify all orbits $\mathcal{O}$ which satisfy (27).
Experience suggests that a good place to begin a further analysis is to focus on orbits of the form $(n_{1}^{2l_{1}}n_{2}^{2l_{2}}\ldots n_{p}^{2l_{p}})$ such that $p$ is minimal. For example, in the case above, if we begin with the orbit $(5^{2}3^{2})$,
we would seek $\Theta$ such that $\mathcal{O}(\Theta)=(6^{2}2^{2})$.
In general we have
Lemma 1.
We have
$$\dim(Sp_{2m})+\tfrac{1}{2}\dim((2k-1)^{2m}(2r-1)^{2m})=\tfrac{1}{2}\dim((2k)^{%
2m}(2r-2)^{2m}).$$
Proof.
Using equation (2), one may compute the difference
$$\tfrac{1}{2}\dim((2k)^{2m}(2r-2)^{2m})-\tfrac{1}{2}\dim((2k-1)^{2m}(2r-1)^{2m})$$
and to show that it is equal to $2m^{2}+m=\dim(Sp_{2m})$.
We omit the details.
∎
We observe that the doubling construction of the authors, Cai and Kaplan [11] fits this rubric, indeed, it provides an example
of such an integral with $r=1$.
More precisely, the representation $\Theta$ used in [11] is an Eisenstein series
$E(\cdot,s)$ defined on $Sp_{4mk}({\mathbb{A}})$, and it satisfies ${\mathcal{O}}(E(\cdot,s))=((2k)^{2m})$.
Then a Fourier coefficient of $E(\cdot,s)$ is taken with respect to a unipotent group $U$ and generic character corresponding to the partition
$((2k-1)^{2m}1^{2m})$ of $4km$.
Returning to the general case, the next step is to find an automorphic representation $\Theta$ defined on the group $Sp_{4m(k+r-1)}({\mathbb{A}})$
which satisfies ${\mathcal{O}}(\Theta)=((2k)^{2m}(2r-2)^{2m})$. The main source of examples for such representations are Eisenstein series and their residues.
If we have a candidate for $\Theta$ which is an Eisenstein series, then by unfolding the Eisenstein series it is often
possible to check if the non-vanishing of
equation (26) implies that $\pi$ determines
$\sigma$ uniquely. See [11] for an example. However, if the representation $\Theta$ is obtained as a residue of an Eisenstein series, an unfolding process is
not readily available to us unless we attempt to unfold first and then take the residue, a strategy that is often problematic, and in this case, typically only a weaker statement can be checked.
In the rest of this Section we will describe a simple case where the representation $\Theta$ is not an Eisenstein series. We will only give the flavor
of the construction here; we plan to present
the details in a
separate paper.
Let $\tau$ denote an irreducible cuspidal representation of the group $GL_{n}({\mathbb{A}})$. Suppose that $n>2$ is even and that the partial $L$-function
$L^{S}(\tau,\wedge^{2},s)$ has a simple pole at $s=1$. Let ${\mathcal{E}}_{\tau}$ denote the generalized Speh representation defined on the group
$GL_{3n}({\mathbb{A}})$. See [11]. Let $P(GL_{3n})$ be the maximal parabolic subgroup of $Sp_{6n}$ whose Levi part is $GL_{3n}$ (the so-called
Siegel parabolic).
Let $E_{\tau}(\cdot,s)$ denote the Eisenstein series defined on the group $Sp_{6n}({\mathbb{A}})$ attached to the induced space $Ind_{P(GL_{3n})({\mathbb{A}})}^{Sp_{6n}({\mathbb{A}})}{\mathcal{E}}_{\tau}%
\delta_{P}^{s}$. The poles of this Eisenstein series are determined by the poles of $L^{S}(\tau,\wedge^{2},s)$ and $L^{S}(\tau,\vee^{2},s)$. See [30]. It follows from that reference that if $L^{S}(\tau,\wedge^{2},s)$ has a simple pole at $s=1$, then the Eisenstein series $E_{\tau}(\cdot,s)$ has a simple pole at $s_{0}=(3n+2)/(6n+2)$. Denote $\Theta^{\prime}_{\tau}=\text{Res}_{s=s_{0}}E_{\tau}(\cdot,s)$. That is, $\Theta^{\prime}_{\tau}$ is the automorphic representation spanned by the residues of this family of Eisenstein series at the point $s_{0}$.
Then, it follows from [26] that there is an irreducible constituent of $\Theta_{\tau}$ of $\Theta^{\prime}_{\tau}$ such that
(28)
$${\mathcal{O}}(\Theta_{\tau})=((2n)^{2}n^{2}).$$
Choosing $k=n$, $m=1$ and $r=(n+2)/2$ in Lemma 1 we conclude that the dimension equation (21) is satisfied,
and we may form the global integral (26).
More generally, suppose that the representation $\Theta$ in (26) satisfies ${\mathcal{O}}(\Theta)=((2n)^{2}n^{2})$.
Suppose moreover that $\Theta=\otimes^{\prime}_{\nu}\Theta_{\nu}$ (restricted tensor product) where for almost all $\nu$ the unipotent orbit attached to $(\Theta)_{\nu}$ is also $((2n)^{2}n^{2})$.
This is true if we take $\Theta$ to
be $\Theta_{\tau}$. Indeed, in this case the corresponding local statement may be proved by arguing
analogously to the global case, i.e. replacing Fourier expansions by the geometric lemma, global root
exchange by local root exchange, etc.
Suppose that $\pi$ is an irreducible cuspidal automorphic representation of the group $SL_{2}(\mathbb{A})$ whose image under the ‘lift’ corresponding to $\Theta$
contains an irreducible cuspidal representation $\sigma$, also on the group $SL_{2}({\mathbb{A}})$.
We seek to establish the relation between $\pi$ and $\sigma$ assuming that integral (26) is not zero for some choice of data.
Since the representation $\Theta$ is not an Eisenstein series we cannot simply carry out an unfolding process. However, in this case we can show that the representations $\pi$ and $\sigma$ are nearly equivalent.
We sketch a local argument for representations in general position, omitting the details.
Suppose that $\pi=\otimes^{\prime}_{\nu}\pi_{\nu}$, $\sigma=\otimes^{\prime}_{\nu}\sigma_{\nu}$. At unramified places suppose that
$\pi_{\nu}=\text{Ind}_{B}^{SL_{2}}\chi_{\nu}\delta_{B}^{1/2}$ and that $\sigma_{\nu}=\text{Ind}_{B}^{SL_{2}}\mu_{\nu}\delta_{B}^{1/2}$,
where $B$ is the standard Borel subgroup of $SL_{2}$ and $\chi_{\nu}$ and $\mu_{\nu}$ are unramified characters. Suppose that
$\chi_{\nu}$ is in general position. If the integral (26) is not zero for some choice of data, then the space
(29)
$$\text{Hom}_{SL_{2}\times SL_{2}}(\text{Ind}_{B}^{SL_{2}}\chi_{\nu}\delta_{B}^{%
1/2}\times\text{Ind}_{B}^{SL_{2}}\mu_{\nu}\delta_{B}^{1/2},J_{U,\psi_{U}}(%
\Theta_{\nu}))$$
is not zero. Here $J_{U,\psi_{U}}$ is the local twisted Jacquet module
which corresponds to the Fourier coefficient over the group $U$ and the character $\psi_{U}$
of integral (26). It follows from Frobenius reciprocity that the space (29) is equal to
(30)
$$\text{Hom}_{GL_{1}\times GL_{1}}(\chi_{\nu}\delta_{B}^{1/2}\mu_{\nu}\delta_{B}%
^{1/2},J_{U_{1},\psi_{U}}(\Theta_{\nu})).$$
Here $U_{1}$ is a certain unipotent subgroup which contains $U$ and the character $\psi_{U}$ is the trivial extension from $U$ to $U_{1}$.
Then performing root exchanges and using that the unipotent orbit attached to $(\Theta)_{\nu}$ is $((2n)^{2}n^{2})$,
one can prove
that the nonvanishing of the space (30) implies that the space
(31)
$$\text{Hom}_{GL_{1}}(\chi_{\nu}\delta_{B}^{1/2}\mu_{\nu}\delta_{B}^{1/2},J_{U_{%
2},\psi_{U_{2}}}(\Theta_{\nu}))$$
is also nonzero. Here $GL_{1}$ is embedded in
$GL_{1}\times GL_{1}$ diagonally, and $J_{U_{2},\psi_{U_{2}}}$ is the twisted Jacquet module attached to a certain unipotent group
$U_{2}$ and character $\psi_{U_{2}}$. This last space may then be analyzed by using properties of the representation $\Theta_{\nu}$,
and to deduce that $\mu_{\nu}=\chi_{\nu}^{\pm 1}$.
This example illustrates how the dimension equation may be used to suggest new integral kernels.
9. Local Analogues
The unfolding of a Rankin-Selberg integral typically has a local analogue, so
when the dimension equation or extended dimension equation is satisfied, it is natural to seek local statements that hold. Once again we view the equation as necessary but not sufficient.
In this Section we illustrate the local statements that arise. Let $K$ be a non-archimedean local field whose residue field has cardinality $q$.
A first example is given by the Rankin-Selberg integrals of Jacquet, Piatetski-Shapiro and Shalika [29].
Let $\pi_{1}$, $\pi_{2}$ be irreducible admissible generic representations of $GL_{n}(K)$, $GL_{k}(K)$, resp.
We keep the notation of Section 3 above.
Consider $\pi_{2}$ to be a module for $Y_{n,k}(K)$ via $\psi$.
If $k<n$, then ([29], paragraph (2.11), Proposition) the space of $(GL_{k}\ltimes Y_{n,k})(K)$-equivariant bilinear forms
$$\text{Bil}_{(GL_{k}\ltimes Y_{n,k})(K)}(\pi\otimes|\det(\cdot)|^{s},\pi^{%
\prime})$$
has dimension at most 1, except for finitely many values of $q^{-s}$.
Similarly, if $n=k$,
the space
$$\text{Bil}_{GL_{n}(K)}(\pi\otimes\pi^{\prime}\otimes|\det(\cdot)|^{s},\text{%
Ind}_{P(K)}^{GL_{n}(K)}\delta_{P}^{-1/2})$$
has dimension at most 1, except for finitely many values of $q^{-s}$
([29], paragraph (2.10), Eqn. (5); see also the Proposition in (2.10) for an equivalent formulation in terms of trilinear forms).
We caution the reader that in the literature this is presented using $GL_{n}(K)$-equivariance rather than $PGL_{n}(K)$-equivariance, but unless the central characters are chosen compatibly the
space is zero. We need to use $PGL_{n}$ to satisfy the dimension equation. This is comparable to insisting that we choose the group of smallest possible dimension in assigning a dimension to a functional.
As an additional example, in the situation of the generalized doubling integrals of Cai, Kaplan and the authors [11], the study of the global integral (18) leads to the local Hom space
$$\text{Hom}_{G(K)\times G(K)}(J_{U,\psi_{U}^{-1}}(\text{Ind}_{P(K)}^{H(K)}(W_{c%
}(\tau)\delta_{P}^{s}),\pi^{\vee}\otimes\pi)$$
where $J_{U,\psi_{U}^{-1}}$ denotes a twisted Jacquet module with respect to the group $U(K)$ and character $\psi_{U}^{-1}$ and the remaining notation is given in [11], see especially (3.2) there.
Once again, this space is at most one dimensional except for finitely many values of $q^{-s}$
(see the proof of [11], Theorem 21).
As explained above, a dimension equation holds
but only if we use the extended dimension equation and treat $\pi^{\vee}\otimes\pi$ as having dimension equal to $\dim(G)$.
Finally, when the dimension equation appears in the context of a lifting result, then one may hope to prove the existence of a local correspondence similar to the Howe correspondence for the
classical theta representation. The local concerns that arise are illustrated by the treatment of (29) above. In particular, it is natural to seek to extend
such a correspondence beyond a matching of the unramified principal series.
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Nature of the charge-density wave excitations in cuprates
J. Q. lin
[email protected]
Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA
School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China
H. Miao
[
Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA
D. G. Mazzone
G. D. Gu
Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA
A. Nag
A. C. Walters
M. García-Fernández
Diamond Light Source, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom
A. Barbour
J. Pelliciari
I. Jarrige
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
M. Oda
K. Kurosawa
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
N. Momono
Department of Sciences and Informatics, Muroran Institute of Technology, Muroran 050-8585, Japan
K. Zhou
Diamond Light Source, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom
V. Bisogni
National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY 11973, USA
X. Liu
[email protected]
School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China
M. P. M. Dean
[email protected]
Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA
(January 16, 2021)
Abstract
The discovery of charge-density wave (CDW)-related effects in the resonant inelastic x-ray scattering (RIXS) spectra of cuprates holds the tantalizing promise of clarifying the interactions that stabilize the electronic order. Here, we report a comprehensive RIXS study of La${}_{2-x}$Sr${}_{x}$CuO${}_{4}$ (LSCO) finding that CDW effects persist up to a remarkably high doping level of $x=0.21$ before disappearing at $x=0.25$. The inelastic excitation spectra remain essentially unchanged with doping despite crossing a topological transition in the Fermi surface. This indicates that the spectra contain little or no direct coupling to electronic excitations near the Fermi surface, rather they are dominated by the resonant cross-section for phonons and CDW-induced phonon-softening. We interpret our results in terms of a CDW that is generated by strong correlations and a phonon response that is driven by the CDW-induced modification of the lattice.
Present address: ]Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Thirty years after the discovery of high-temperature superconductivity in the cuprates, there is still no consensus regarding the minimal set of interactions needed to describe the “normal state” from which superconductivity emerges Keimer et al. (2015). Popular Hubbard and ‘$t-J$’ theoretical models suggest that superconducting, charge-density wave (CDW), and spin-density wave (SDW) states have similar ground state energies Zaanen and Gunnarsson (1989); Poilblanc and Rice (1989); Emery et al. (1990); Castellani et al. (1995); Corboz et al. (2014); Huang et al. (2017); Zheng et al. (2017), which is consistent with experiments that reveal widespread interplay between all three states Tranquada et al. (1995); Ghiringhelli et al. (2012); Blanco-Canosa et al. (2014); Tabis et al. (2017); Comin et al. (2014); Croft et al. (2014); Thampy et al. (2014). This complexity motivates experimental efforts to measure collective excitations associated with CDW order that should clarify the pertinent interactions. Resonant inelastic x-ray scattering (RIXS), as illustrated in Fig. 1(a), has an enhanced sensitivity to valence charge and phonon excitations Ament et al. (2011a). Several experiments indeed reported anomalies in cuprate RIXS spectra at the CDW wavevector ($\textbf{\em Q}_{\text{CDW}}$), opening new routes to understand cuprate CDW s Dean et al. (2013a); Miao et al. (2017); Chaix et al. (2017); Arpaia et al. (2019); Yu et al. (2019); Peng et al. (2019). Uniquely isolating and interpreting CDW-effects in RIXS is, however, complicated as CDW s inevitably modify their host crystal lattice and thus the phonons. Compounding this problem, RIXS spectra near $\textbf{\em Q}_{\text{CDW}}$ have been conceptualized in several different ways including charge excitations Miao et al. (2019a); Arpaia et al. (2019); Yu et al. (2019), momentum-dependent electron-phonon coupling (EPC) Peng et al. (2019) and Fano-effects Chaix et al. (2017).
In this Letter, we use ultrahigh energy-resolution RIXS to examine the nature of the CDW in La${}_{2-x}$Sr${}_{x}$CuO${}_{4}$ (LSCO). We observe that CDW correlations and associated CDW-induced phonon-softening persist up to a strikingly high doping level of $x=0.21$ before both effects disappear at $x=0.25$. The large doping range $x=0.12\rightarrow 0.25$ traverses a topological transition in the Fermi surface Yoshida et al. (2006); Horio et al. (2018); Miao et al. (2019b), allowing us to empirically test the relative importance of charge and lattice effects in the excitation spectra. We find that the data can be described entirely in terms of CDW-induced phonon softening and wavevector-dependent changes in the phonon displacement, without invoking any more complex electronic excitations or CDW-related modification of the EPC. Overall, our results suggest a ‘real-space’ picture in which the CDW emerges due to strong electronic correlations and modifies the underlying phonons.
LSCO single crystals with $x=0.12$, $0.17$, $0.21$, and $0.25$ were grown using the floating zone method and are denoted as LSCO$n$ where $n=12$, $17$, $21$, and $25$, respectively sup . Structural and electronic characterization of the samples indicates excellent quality sup ; Miao et al. (2019b). High energy-resolution RIXS measurements were performed at 2-ID at the National Synchrotron Light Source II, Brookhaven National Laboratory with a resolution of $\Delta E=30$ meV full-width at half-maximum (FWHM) and at I21 at Diamond Light Source featuring a resolution of $\Delta E=55$ meV. The RIXS process is shown in Fig. 1(a). X-rays were tuned to the Cu $L_{3}$-edge and measurements were taken with $\sigma$ x-ray polarization perpendicular to the scattering plane (unless otherwise specified). Reciprocal lattice units (r.l.u.) are defined in terms of $(H,K,L)$ with lattice constants $a=b=3.76$ Å, $c=13.28$ Å. Different values of $H$ and $K$ were accessed by rotating the sample about the $\theta$ and $\chi$ axes without changing the scattering angle $2\theta$. Intensities were normalized to the intensity of the $dd$ excitations similar to previous works Ghiringhelli et al. (2012); Miao et al. (2017); Chaix et al. (2017); Miao et al. (2019a). Grazing incidence geometry (defined as negative $H$) was chosen to enhance the intensity of charge and lattice (phonon) excitations, and to suppress spin excitations, which are, in any case, $>200$ meV and outside of the energy window we focus on in this paper Dean et al. (2013b); Meyers et al. (2017).
Figure 1(b) plots RIXS data of LSCO at the Cu $L_{3}$-edge illustrating the main spectral features studied here: (i) a quasi-elastic peak and (ii) a dispersive feature around $50-65$ meV. Feature (i) contains a component of trivial elastically scattered x-rays due to the finite disorder (defects) in the sample and surface scattering. Quasi-elastic scattering is further enhanced by static or quasi-static CDW correlations and displays a peak at $\textbf{\em Q}_{\text{CDW}}$ whenever such correlations are present. The inelastic feature (ii) has been seen in several other cuprate RIXS experiments Chaix et al. (2017); Peng et al. (2019); Braicovich et al. (2019); Rossi et al. (2019); Yu et al. (2019); Miao et al. (2019a) and is assigned to the in-plane Cu-O bond-stretching (BS)phonon mode in agreement with early inelastic x-ray and neutron works Fukuda et al. (2005); Reznik et al. (2006); Graf et al. (2007); Park et al. (2014); Reznik et al. (2007); Pintschovius et al. (2006); Pintschovius and Braden (1999). The spectra were fitted with a Pseudo-Voigt function for the elastic peak, an anti-symmetric Lorentzian function for the BS mode and a linear background. All components of the fit were convoluted with the energy resolution function.
Having assigned the basic spectral features, we use the ultra-high throughput of the I21 beamline to comprehensively map out the momentum and doping dependence of LSCO$n$ $12\leq n\leq 25$, (see Fig. 2). This doping range crosses over from $1/8$ doping, where the CDW correlations are strongest, into the overdoped Fermi-liquid-like phase where the CDW disappears Miao et al. (2019b); Yoshida et al. (2006); Horio et al. (2018). Importantly, this traverses a topological transition in the electronic structure where hole-pocket/arc-type states transform into an electron-like Fermi surface with a more Fermi-liquid-like scattering rate Miao et al. (2019b); Yoshida et al. (2006); Horio et al. (2018); Cooper et al. (2009). This allows us to investigate the relationship of the RIXS spectra with the changes in electronic structure.
We first discuss the quasi-elastic CDW feature, which is summarized in Fig. 2(i) showing the integrated intensity in the $\pm 20$ meV energy range of Fig. 2(a)-(h). The intensity enhancement of the resonant process combined with the background suppression attained by energy-resolving the scattered beam make RIXS very sensitive to even very short correlation length CDW s. A CDW around an in-plane wavevector of $(-0.23,0)$ is not only observed for LSCO12, where it was seen several times previously Thampy et al. (2014); Croft et al. (2014); Wu et al. (2012); Wen et al. (2019), but also up to far higher dopings of LSCO21 111These RIXS results are confirmed by bulk sensitive hard (8 keV) x-ray diffraction measurement under review Miao et al. (2019b). The $H$-width of the quasi-elastic scattering is consistent with correlation lengths (calculated as 2/FWHM) of $25-45$ Å with shorter values for $x=0.17$. Within error, the peaks exist at the same $H_{\text{CDW}}=-0.231\pm 0.005$ r.l.u., consistent with the stripe phenomenology where $H_{\text{CDW}}$ saturates for $x>1/8$ Yamada et al. (1998). As expected, a substantial fraction of the CDW intensity is suppressed upon warming to 100 K, leaving only a much weaker and diffuse signal Thampy et al. (2014); Croft et al. (2014); Miao et al. (2019b). We further observe a non-monotonic intensity dependence as function of doping, with LSCO17 being weaker than LSCO12 and LSCO21. There are multiple potential explanations for such a behavior. Perhaps most plausible is to note that the quasi-static CDW intensity in cuprates tends to compete with superconductivity, so the reduced CDW intensity at $x=0.17$ may be associated with the enhanced superconducting correlations that are known to exist at this near-optimal doping level.
Next we discuss the inelastic component of the spectra in Fig. 2(a-h). In the 100-200 meV energy window, above the maximum phonon energy, flat, structure-less intensity arising from the charge continuum and the tail of the higher energy paramagnon excitations is present over all Q. This intensity shows no clear changes around $\textbf{\em Q}_{\text{CDW}}$ and minimal changes with doping. In fact, only a slight increase in the overall intensity is found with increasing doping, which is expected as overdoped samples are more metallic. The inelastic intensity below 100 meV is dominated by the BS phonon, which shows clear energy dispersion and very strong intensity varaiation. Superficial inspection of the raw intensity in Fig. 2(a) may appear as if a soft mode is dispersing to zero energy at $\textbf{\em Q}_{\text{CDW}}$. This can be examined through a systematic study of samples with different doping levels. To separate the phonon from charge and quasielastic intensity, we fit the data using the previously described model and display the results as red circles in Fig. 2(a-h). Since the phonon intensity drops strongly as $|H|$ decreases, we focus on $|H|>0.18$ where the BS phonon can be fitted with good precision.
Figure 3 summarizes the evolution of the BS phonon parameters. Although the phonon softening, shown in panel (a), is appreciable ($13\pm 4$ meV or $19\pm 6\%$), it never shows full soft-mode (i.e. zero energy) behavior. The simultaneous disappearance of both the elastic peak and the phonon softening in LSCO25 makes a strong case that the softening is intimately related to the CDW correlations. Inelastic x-ray and neutron scattering measurements of phonon softening in underdoped cuprates have also assigned the phonon
softening to CDW correlations Pintschovius and Braden (1999); Fukuda et al. (2005); Graf et al. (2007); Reznik et al. (2006); Park et al. (2014); Reznik et al. (2007); Pintschovius et al. (2006). It is noted that the phonon energy in LSCO25 is slightly higher than other samples, which might be linked to lattice contraction associated with large Sr concentrations.
The phonon intensity dispersion is plotted in Fig. 3(c). Within error, no phonon intensity anomalies are seen around $\textbf{\em Q}_{\text{CDW}}$, instead the clearest feature is a strong increase with $|H|$. Since RIXS excites phonons via the EPC process, the measured intensity reflects this interaction strength and scales with
$g^{2}$ where $g$ is the EPC Ament et al. (2011a); Devereaux et al. (2016); Meyers et al. (2018); Braicovich et al. (2019); Rossi et al. (2019). As a function of Q, the “breathing-type” Cu-O bond displacement involved in the BS mode changes. Assuming a well-defined Madelung energy change associated with Cu-O bond stretching, one can predict RIXS intensity scaling $I\propto g_{\text{br}}^{2}=\sin^{2}(\pi H)+\sin^{2}(\pi K)$ Johnston et al. (2010); Devereaux et al. (2016); Braicovich et al. (2019); Rossi et al. (2019). The comparison in Fig. 3(c) shows that this simple model is sufficient to describe the intensity behavior of our data, without invoking any more complex phenomenology. It is worth adding that definitively distinguishing $\sin^{2}(\pi H)$ scaling from other scaling forms is somewhat challenging. The photon energy corresponding to the Cu $L_{3}$ edge intrinsically limits the highest $|H|$ we can reach, and at low $|H|$ leakage of specular scattering intensity overwhelms the low-energy region of the RIXS spectra. The slight flattening of the dispersion at $x=0.25$ might arise from some leakage of additional background at high doping levels. With these considered, the agreement with $\sin^{2}(\pi H)$ scaling holds well in the reciprocal-space range measured.
Discussion of the CDW.— Both the quasielastic RIXS intensity and the phonon softening demonstrate the existence of CDW correlations up to a remarkably high doping level of $x=0.21$, traversing the topological transition in the electronic structure Yoshida et al. (2006); Horio et al. (2018); Miao et al. (2019b), in which arc or hole-pocket like states centered around the Billouin zone corner transform into an electron-like Fermi surface at the Brillouin zone center. This result confirms very recent non-resonant diffraction measurements and shows, due to the resonant nature of the RIXS probe, that the correlations involve the electronically active Cu states Miao et al. (2019b). The persistence of the CDW correlations, despite very substantial Fermi surface changes, provides a vivid demonstration that the LSCO CDW cannot be described using any type of weak-coupling Fermi surface nesting picture, as previously suggested for some other cuprate materials Comin and Damascelli (2016). Instead, the nearly doping-independent CDW wavevector supports mechanisms in which the periodicity of the CDW is set by the short-range electronic interactions. Here, doped holes can save super-exchange energy by clustering together and breaking fewer magnetic bonds, but by doing so, they pay a cost of increased kinetic and Coulomb energy. It has been proposed that a CDW is the optimal compromise between these two tendencies Emery et al. (1990). However, a fascinating question remains regarding why the CDW is so stable against increasing doping and electron itineracy. Furthermore, our results motivate a reexamination of the anomalous transport properties of the cuprates, which are often discussed in terms of strange metal physics below a critical doping level of $x_{c}\approx 0.19$, where the systems becomes increasingly more Fermi-liquid-like Keimer et al. (2015). Since CDW correlations can exist over a more extended doping range than previously thought, it is interesting to consider their influence on transport properties Seibold et al. (2019). These results, however, argues against a quantum critical point that is associated with a CDW transition as similar as CDW correlations exist in LSCO17 and LSCO21, either side of the putative critical doping of $x_{c}\approx 0.19$. In terms of the CDW fluctuations, we note that the size of the phonon softening is reduced when warming to $T=100$ K, but the magnitude of the reduction is considerably less than the reduction in the quasi-elastic CDW intensity. This suggests that only a relatively small fraction of the total CDW correlations are nucleated into the CDW-order, as otherwise the magnitude of the phonon softening would be expected to scale with the quasi-elastic intensity.
An intriguing feature of the CDW effects is that the Q-vector with the largest phonon softening does not coincide with the peak in the quasi-elastic scattering at 0.235 r.l.u., but occurs at a larger wavevector of $H=0.275$ r.l.u. We can consider two candidate explanations for this. The first is that the interactions causing the CDW instability are not the same as the interactions that pin the CDW. Electronic interactions, for instance, may potentially stabilize a wide-range of CDW wavevectors, but lattice- or SDW-coupling might play the final role in setting the final CDW wavevector. This concept was proposed several years ago Zachar et al. (1998) and is further supported by the previously observed change in CDW wavevector with temperature Miao et al. (2017, 2018, 2019a). The phonon softening does not show any obvious difference between $T_{SC}$ and $100$ K in Fig. 3(a)&(b), in contrast to the CDW peak, which also supports distinct mechanisms for CDW-formation and CDW-pinning. Another potential explanation for the shift is that the effect is due to the Q dependence of the EPC. Since EPC increases with $H$, the softening at higher $H$ will be enhanced, which would displace the point of maximum softening.
Discussion of electronic CDW excitations. — A strength of the current dataset is the opportunity to compare LSCO12 and LSCO21, which exhibit comparable levels of CDW order, despite their very different doping levels. Any electronic CDW excitations that may be present in the RIXS spectra would be expected to increase with electronic density of states and change substantially with doping. Since the overall form of the spectra is rather similar over this large doping range, our results argue against the presence of electronic CDW excitations, as suggested previously, as a ubiquitous, intrinsic feature of RIXS spectra of the cuprates Chaix et al. (2017). It is, however, possible that the x-ray polarization is important to observing electronic CDW excitations 222Data in Ref. Chaix et al. (2017) is taken with positive $H$ (grazing exit geometry), district from the negative $H$ (grazing incidence geometry)..
Electron phonon coupling. — The concept of using RIXS to extract EPC in cuprates has generated considerable excitement recently Ament et al. (2011b); Devereaux et al. (2016); Rossi et al. (2019); Braicovich et al. (2019); Peng et al. (2019). Our results support this, in the sense that we observe intensity scaling as $I\propto\sin^{2}(\pi H)$. However, no phonon intensity anomalies related to the CDW are observed. A recent preprint reports measurements of La${}_{1.8-x}$Eu${}_{0.2}$Sr${}_{x}$CuO${}_{4+\delta}$ which used an incident energy detuning method, formulated in Refs. Ament et al. (2011b); Rossi et al. (2019); Braicovich et al. (2019), to suggest a very large CDW-induced modification of the EPC from $g=0.30\rightarrow 0.35$ upon cooling into the CDW phase Peng et al. (2019). Assuming $I\propto g^{2}$, this would imply a phonon intensity change of $(0.35/0.3)^{2}\sim 1.4$. As such, it is very difficult to justify the absence of a discernible $\sim 40$% temperate-induced change in the phonon dispersion at $\textbf{\em Q}_{\text{CDW}}$ on resonance in our data or that in Ref. Peng et al. (2019).
RIXS as a probe of cuprate CDWs. — Overall, our results support a real-space picture of an electronically driven CDW, without needing to invoke nesting or a van Hole singularity. RIXS measures the phonon softening occuring due to the charge modulation, but the overall doping dependence is fully explicable without invoking more complex phason, Fano, or CDW-enhanced EPC effects that have generated considerable excitement recently Dean et al. (2013a); Miao et al. (2017); Chaix et al. (2017); Arpaia et al. (2019); Peng et al. (2019). Our observation is backed, by improved energy resolution ($\Delta E=30$ meV) and more extensive doping dependence compared with what has been done previously Dean et al. (2013a); Miao et al. (2017); Chaix et al. (2017); Arpaia et al. (2019); Peng et al. (2019). Excluding these exciting effects is at some level disappointing in terms of novel excitations, but is, however, highly important in view of the more-and-more extensive use of RIXS. Although RIXS, in this case, provides information similar to inelastic x-ray and neutron scattering, compelling applications for RIXS remain in cases where the x-ray penetration depth and resonant mode selectivity is important Meyers et al. (2018).
In conclusion, we report RIXS measurements of CDW correlations in LSCO over an extensive doping range. CDW-related quasi-elastic scattering and phonon softening is observed from $x=0.12$ to $x=0.21$, traversing a topological transition in the Fermi-surface, before disappearing at $x=0.25$. Based on these results, we conclude that the spectra have little or no direct coupling to electronic excitations. Instead the spectra are dominated by CDW-driven phonon softening and phonon intensity variations arising from changes in the phonon displacement as a function of Q. Overall, our results support a senario in which the CDW is driven by strong correlations and clarify that the low-energy RIXS response in cuprates are driven by the CDW modifying the lattice, invoking more complex interactions.
Acknowledgements.
This material is based upon work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Early Career Award Program under Award No. 1047478. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. X. L. and J.Q.L. were supported by the ShanghaiTech University startup fund, MOST of China under Grant No. 2016YFA0401000, NSFC under Grant No. 11934017 and the Chinese Academy of Sciences under Grant No. 112111KYSB20170059. This research used resources at the Soft Inelastic X-Ray beamline of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. We acknowledge Diamond Light Source for time on Beamline I21 under Proposal 22261.
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Modified Gravity (MOG), Cosmology and Black Holes
J. W. Moffat
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
and
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
Abstract
A covariant modified gravity (MOG) is formulated by adding to general relativity two new degrees of freedom, a scalar field gravitational coupling strength $G=1/\chi$ and a gravitational spin 1 vector field $\phi_{\mu}$. The $G$ is written as $G=G_{N}(1+\alpha)$ where $G_{N}$ is Newton’s constant, and the gravitational source charge for the vector field is $Q_{g}=\sqrt{\alpha G_{N}}M$, where $M$ is the mass of a body. Cosmological solutions of the theory are derived in a homogeneous and isotropic cosmology. Black holes in MOG are stationary as the end product of gravitational collapse and are axisymmetric solutions with spherical topology. It is shown that the scalar field $\chi$ is constant everywhere for an isolated black hole with asymptotic flat boundary condition. A consequence of this is that the scalar field loses its monopole moment radiation.
1 Introduction
Dark matter was introduced to explain the stable dynamics of galaxies and galaxy clusters. General relativity (GR) with only ordinary baryon matter cannot explain the present accumulation of astrophysical and cosmological data without dark matter. However, dark matter has not been observed in laboratory experiments [1]. Therefore, it is important to consider a modified gravitational theory. The difference between standard dark matter models and modified gravity is that dark matter models assume that GR is the correct theory of gravity and a dark matter particle such as WIMPS, axions and fuzzy dark matter are postulated to belong to the standard particle model. The present work introduces a simplified formulation of modified gravity (MOG), also called Scalar-Tensor-Vector-Gravity (STVG), that avoids unused generality of the original version [2]. The MOG is described by a fully covariant action and field equations, extending GR by the addition of two gravitational degrees of freedom. The first is $G=1/\chi$, where $G$ is the coupling strength of gravity and $\chi$ is a scalar field. The scalar field $\chi$ is motivated by the Brans-Dicke gravity theory [3, 4]. The second degree of freedom is a massive gravitational vector field $\phi_{\mu}$. The gravitational coupling of the vector graviton to matter is universal with the gravitational charge $Q_{g}=\sqrt{\alpha G_{N}}M$, where $\alpha$ is a dimensionless scalar field, $G_{N}$ is Newton’s gravitational constant and $M$ is the mass of a body.
We write $G=1/\chi$ as $G=G_{N}(1+\alpha)$ and $\chi$ is the only scalar field in the theory. The effective running mass of the spin 1 vector graviton is determined by the parameter $\mu$, which fits galaxy rotation curves and cluster dynamics without exotic dark matter [6, 7, 8, 9]. It has the value $\mu\sim 0.01-0.04\,{\rm kpc}^{-1}$, corresponding to $\mu^{-1}\sim 25-100$ kpc and an effective mass $m_{\phi}\sim 10^{-26}-10^{-28}\,{\rm eV}$.
We derive generalized Friedmann equations and field equations for the scalar field $\chi$ in a homogeneous and isotropic universe with the Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) line element and the energy-momentum tensor of a perfect fluid.
We will consider black holes in MOG as the end point of gravitational collapse, which must be stationary axisymmetric and spherical solutions of the field equations. Following Hawking [5], the scalar field $\chi$ is constant everywhere in a stationary black hole solution and the black holes are described by the Schwarzschild-MOG and Kerr-MOG metric solutions [10, 11]. An important consequence is that a MOG black hole will not have a scalar monopole moment. Moreover, due to the positivity of mass $M$ in the gravitational charge of the vector field, $Q_{g}=\sqrt{\alpha G_{N}}M$, a MOG black hole will not have a dipole moment.
In Section 2, we present the simplified MOG field equations and in Section 3, we review the equations of motion of a particle and the weak field approximation of the theory. In Section 4, we derive cosmological solutions and in Section 5, we investigate MOG black holes. We summarize the results in Section 6.
2 The MOG Field Equations
We will formulate the MOG action and field equations in a simpler and less general way than that first published in [2]. We introduce $\chi=1/G$ where $\chi$ is a scalar field and $G$ is the coupling strength of gravity [3, 4]. The MOG action is given by (we use the metric signature (+,-,-,-) and units with $c=1$):
$$S=S_{G}+S_{\phi}+S_{M},$$
(1)
where
$$S_{G}=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}\biggl{(}\chi R+\frac{\omega_{M}}{%
\chi}\nabla^{\mu}\chi\nabla_{\mu}\chi+2\Lambda\biggr{)},$$
(2)
and
$$S_{\phi}=\int d^{4}x\sqrt{-g}\biggl{(}-\frac{1}{4}B^{\mu\nu}B_{\mu\nu}+\frac{1%
}{2}\mu^{2}\phi^{\mu}\phi_{\mu}-\phi_{\mu}J^{\mu}\biggr{)},$$
(3)
and $S_{M}$ is the matter action. $\nabla_{\mu}$ denotes the covariant derivative with respect to the metric $g_{\mu\nu}$,
Varying the action with respect to $g_{\mu\nu}$, $\chi$ and $\phi_{\mu}$, we obtain the field equations:
$$G_{\mu\nu}=-\frac{\omega_{M}}{\chi^{2}}\biggl{(}\nabla_{\mu}\chi\nabla_{\nu}%
\chi-\frac{1}{2}g_{\mu\nu}\nabla^{\alpha}\chi\nabla_{\alpha}\chi\biggr{)}\\
-\frac{1}{\chi}(\nabla_{\mu}\chi\nabla_{\nu}\chi-g_{\mu\nu}\Box\chi)+\frac{8%
\pi}{\chi}T_{\mu\nu},$$
(4)
$$\nabla_{\nu}B^{\mu\nu}+\mu^{2}\phi^{\mu}=J^{\mu},$$
(5)
$$\Box\chi=\frac{8\pi}{(2\omega_{M}+3)}T,$$
(6)
where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R$, $\Box=\nabla^{\mu}\nabla_{\mu}$, $J^{\mu}=\kappa\rho u^{\mu}$, $\kappa=\sqrt{G_{N}\alpha}$, $\rho$ is the density of matter and field energy and $u^{\mu}=dx^{\mu}/ds$. We expand $G$ by $G=G_{N}(1+\alpha)$, $\Lambda$ is the cosmological constant and $\mu$ is the effective running mass of the spin 1 graviton vector field. The energy-momentum tensor is
$$T_{\mu\nu}=T^{M}_{\mu\nu}+T^{\phi}_{\mu\nu}+g_{\mu\nu}\frac{\chi\Lambda}{8\pi},$$
(7)
where
$$T^{\phi}_{\mu\nu}=-\biggl{(}{B_{\mu}}^{\alpha}B_{\alpha\nu}-\frac{1}{4}g_{\mu%
\nu}B^{\alpha\beta}B_{\alpha\beta}+\mu^{2}\phi_{\mu}\phi_{\nu}-\frac{1}{2}g_{%
\mu\nu}\phi^{\alpha}\phi_{\alpha}\biggr{)},$$
(8)
and $T=g^{\mu\nu}T_{\mu\nu}$. Eq.(6) follows from the equation
$$2\chi\Box\chi-\nabla_{\mu}\chi\nabla^{\mu}\chi=\frac{R}{\omega_{M}},$$
(9)
by substituting for $R$ from the contracted form of (4).
We can perform a conformal transformation of the metric, ${\bar{g}}_{\mu\nu}=\Omega^{2}g_{\mu\nu}$, where $\Omega^{2}=\chi G_{0}$ and obtain the field equations in the Einstein frame:
$${\bar{G}}_{\mu\nu}=-\frac{(2\omega_{M}+3)}{16\pi G_{0}\chi^{2}}\biggl{(}{\bar{%
\nabla}}_{\mu}\chi{\bar{\nabla}}_{\nu}\chi-\frac{1}{2}{\bar{g}}_{\mu\nu}{\bar{%
\nabla}}_{\alpha}\chi{\bar{\nabla}}^{\alpha}\chi\biggr{)}+8\pi G_{0}{\bar{T}}_%
{\mu\nu},$$
(10)
$${\bar{\Box}}\chi=\frac{8\pi}{(2\omega_{M}+3)}{\bar{T}},$$
(11)
$$\bar{\nabla}_{\nu}{\bar{B}}^{\mu\nu}+\mu^{2}{\bar{\phi}}^{\mu}={\bar{J}}^{\mu},$$
(12)
where $G_{0}$ is the constant gravitational coupling strength, ${\bar{B}}_{\mu\nu}=\partial_{\mu}{\bar{\phi}}_{\nu}-\partial_{\nu}{\bar{\phi}}%
_{\mu}$, $\bar{\nabla}$ is the covariant derivative with respect to the metric ${\bar{g}}_{\mu\nu}$ and $\bar{\Box}=\bar{\nabla}^{\mu}\bar{\nabla}_{\mu}$.
3 Equations of Motion and Weak Field Approximation
The equation of motion for a massive test particle in MOG has the
covariant form [2, 12]:
$$m\biggl{(}\frac{du^{\mu}}{ds}+{\Gamma^{\mu}}_{\alpha\beta}u^{\alpha}u^{\beta}%
\biggr{)}=q_{g}{B^{\mu}}_{\nu}u^{\nu},$$
(13)
where $u^{\mu}=dx^{\mu}/ds$ with $s$ the proper time along the particle
trajectory and ${\Gamma^{\mu}}_{\alpha\beta}$ denote the Christoffel
symbols. Moreover, $m$ and $q_{g}$ denote the test particle mass $m$ and
gravitational charge $q_{g}=\sqrt{\alpha G_{N}}m$, respectively. For a
massless photon the gravitational charge vanishes,
$q_{\gamma}=\sqrt{\alpha G_{N}}m_{\gamma}=0$, so photons travel on null
geodesics $k^{\nu}\nabla_{\nu}k^{\mu}=0$ [13]:
$$\frac{dk^{\mu}}{ds}+{\Gamma^{\mu}}_{\alpha\beta}k^{\alpha}k^{\beta}=0,$$
(14)
where $k^{\mu}$ is the photon momentum and $k^{2}=k^{\mu}k_{\mu}=0$. We
note that for $q_{g}/m=\sqrt{\alpha G_{N}}$ the equation of motion for a
massive test particle (13) satisfies the (weak)
equivalence principle, leading to the free fall of particles in a
homogeneous gravitational field, although the free-falling particles
do not follow geodesics.
In the weak field region, $r\gg 2GM$, surrounding a stationary mass
$M$ centred at $r=0$ the spherically symmetric field $\phi_{\mu}$, with
effective mass $\mu$, is well approximated by the Yukawa potential:
$$\phi_{0}=-Q_{g}\frac{\exp(-\mu r)}{r},$$
(15)
where $Q_{g}=\sqrt{\alpha G_{N}}M$ is the gravitational charge of the source mass $M$.
The radial equation of motion of a non-relativistic test particle,
with mass $m$ and at radius $r$, in the field of $M$ is then given by
$$\frac{d^{2}r}{dt^{2}}+\frac{GM}{r^{2}}=\frac{q_{g}Q_{g}}{m}\frac{\exp(-\mu r)}%
{r^{2}}(1+\mu r).$$
(16)
The mass $\mu$ is tiny — comparable to the experimental bound on
the mass of the photon — giving a range $\mu^{-1}$ of the
repulsive exponential term the same order of magnitude as the size of
a galaxy. Since $q_{g}Q_{g}/m=\alpha G_{N}M$, the modified Newtonian
acceleration law for a point particle can be written
as [2]:
$$a_{\rm MOG}(r)=-\frac{G_{N}M}{r^{2}}[1+\alpha-\alpha\exp(-\mu r)(1+\mu r)].$$
(17)
This reduces to Newton’s gravitational acceleration in the limit
$\mu r\ll 1$.
In the limit that $r\rightarrow\infty$, we get from
(17) for approximately constant $\alpha$ and
$\mu$:
$$a_{\rm MOG}(r)\approx-\frac{G_{N}(1+\alpha)M}{r^{2}}.$$
(18)
The MOG acceleration has a Newtonian-Kepler behaviour for large $r$ with enhanced
gravitational strength $G=G_{N}(1+\alpha)$. The transition from
Newtonian acceleration behavior for small $r$ to non-Newtonian
behaviour for intermediate values of $r$ is due to the repulsive
Yukawa contribution in (17). This can also
result in the circular orbital rotation velocity $v_{c}$ having a
maximum value in the transition region.
For a distributed baryonic matter source, the MOG (weak field) acceleration law becomes:
$${a}_{\rm MOG}({\boldsymbol{\mathrm{x}}})=-G_{N}\int d^{3}{\boldsymbol{\mathrm{%
x}}}^{\prime}\frac{\rho_{\rm bar}({\boldsymbol{\mathrm{x}}}^{\prime})({%
\boldsymbol{\mathrm{x}}}-{\boldsymbol{\mathrm{x}}}^{\prime})}{|{\boldsymbol{%
\mathrm{x}}}-{\boldsymbol{\mathrm{x}}}^{\prime}|^{3}}[1+\alpha-\alpha\exp(-\mu%
|{\boldsymbol{\mathrm{x}}}-{\boldsymbol{\mathrm{x}}}^{\prime}|)(1+\mu|{%
\boldsymbol{\mathrm{x}}}-{\boldsymbol{\mathrm{x}}}^{\prime}|)],$$
(19)
where $\rho_{\rm bar}$ is the total baryon mass density.
A phenomenological formula for $\alpha$ for approximately constant $\alpha$ and weak gravitational field is [14]:
$$\alpha=\alpha_{\inf}\frac{M}{(\sqrt{M}+E)^{2}},$$
(20)
where $\alpha_{\inf}\sim{\cal O}(10)$ and $E=2.5\times 10^{4}\,M_{\odot}$. For $r\ll\mu^{-1}\sim 25-100$ kpc the MOG acceleration reduces to the Newtonian acceleration $a_{\rm Newt}$. This is consistent with $\alpha\sim 10^{-9}$ obtained from (20) for $M=1\,M_{\odot}$, guaranteeing that the solar system observational data is satisfied.
We have written the gravitational strength $G=G_{N}(1+\alpha)$, so we can relate the scalar field $\chi$ to this expansion of $G$ with $\alpha$ as $\chi\sim 1/G_{N}(1+\alpha)$. The constant $\omega_{M}$ can be set to $\omega_{M}=1$. In the weak gravitational acceleration formula (17), the acceleration of a particle identified with a planet in the solar system approaches the Newtonian acceleration law as $r$ approached the size of the solar system. This is consistent with the phenomenological formula (20) for $M\sim 1M_{\odot}$, yielding $\alpha\sim 10^{-9}$ and $\chi\sim 1/G_{N}$. In Brans-Dicke theory [3], the constant $\omega$ is chosen to be large so that the theory can agree with solar system measurements. A prominent difference between Brans-Dicke gravity and MOG is the additional degree of freedom of the gravitational vector field $\phi_{\mu}$, allowing us to obtain the weak field MOG acceleration formula (17). We have adopted a different approach for obtaining the solar system weak field limit by choosing the equivalent constant $\omega_{M}=1$ and the relation $\chi=1/G_{N}(1+\alpha)$. In the fitting of data when applying the field acceleration formula (17) to weak gravitational fields, the parameters $\alpha$ and $\mu$ are treated as running constants that are not universal constants. The magnitude of $\alpha$ depends on the physical length scale or averaging scale $\ell$ of the system. For the solar system, $\ell_{\odot}\sim 0.5$ pc and for a galaxy $\ell_{G}\sim 5-24$ kpc.
4 Cosmology
We base our cosmology on the homogeneous and isotropic FLRW background metric:
$$ds^{2}=dt^{2}-a^{2}(t)\biggl{[}\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+\sin^%
{2}\theta d\phi^{2})\biggr{]},$$
(21)
where $K=-1,0,1\,({\rm length}^{-2})$ for open, flat and closed universes, respectively. We use the energy-momentum tensor of a perfect fluid:
$$T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}-g^{\mu\nu}p.$$
(22)
We have ${T_{0}}^{0}=\rho$ and the density $\rho$ is
$$\rho=\rho_{M}+\rho_{r}+\rho_{\Lambda},$$
(23)
where $\rho_{\Lambda}=\Lambda/8\pi G_{N}$ and
$$\rho_{M}=\rho_{b}+\rho_{f}.$$
(24)
Moreover, $\rho_{f}=\rho_{\chi}+\rho_{\phi}$ and $\rho_{b},\rho_{\chi},\rho_{\phi}$ denote the densities of baryon matter, scalar field matter and the electrically neutral gravitational vector field $\phi_{\mu}$, respectively. The radiation density $\rho_{r}=\rho_{\gamma}+\rho_{\nu}$, where $\rho_{\gamma}$ and $\rho_{\nu}$ denote the densities of photons and neutrinos, respectively.
In the following, we assume a spatially flat universe and we obtain the equations:
$$\biggl{(}\frac{\dot{a}}{a}\biggr{)}^{2}=\frac{8\pi\rho}{3\chi}-\frac{{\dot{%
\chi}\dot{a}}}{\chi a}+\frac{\omega_{M}{\dot{\chi}}^{2}}{6\chi^{2}},$$
(25)
$$\frac{\ddot{a}}{a}+\frac{{\dot{a}}^{2}}{2a^{2}}=-\frac{4\pi p}{\chi}-\frac{%
\dot{\chi}\dot{a}}{\chi a}-\frac{\omega_{M}\dot{\chi}^{2}}{4{\chi}^{2}}-\frac{%
\ddot{\chi}}{2\chi},$$
(26)
$$\frac{1}{\chi^{3}}\frac{d}{dt}({\dot{\chi}a^{3}})=\frac{8\pi}{(2\omega_{M}+3)}%
(\rho-3p).$$
(27)
The energy conservation equation is
$$\dot{\rho}+3\frac{d\ln a}{dt}(\rho+p)=0.$$
(28)
The integral of (27) is given by [4]:
$${\dot{\chi}}a^{3}=\frac{8\pi}{(2\omega_{M}+3)}\int dt(\rho-3p)\chi^{3}+C,$$
(29)
where $C$ is a constant. We obtain two types of solutions depending on whether $C=0$ or $C\neq 0$ [4].
Let us consider the case $C=0$. The matter dominated epoch has $p=0$ and we write
$$a=a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{n_{1}},\quad\chi=\chi_{0}\biggl{(}%
\frac{t}{t_{0}}\biggr{)}^{n_{2}}.$$
(30)
Then we have $\rho_{M}\propto t^{-3n_{1}}$ and the field equations give
$$n_{1}=\frac{2\omega_{M}+2}{3\omega_{M}+4},\quad n_{2}=\frac{2}{3\omega_{M}+4},$$
(31)
and
$$\rho_{M0}=\frac{(2\omega_{M}+3)n_{2}\chi_{0}}{8\pi t_{0}^{2}}.$$
(32)
We have for $\omega_{M}=1$, $n_{1}=4/7$, so there is a small deviation from the behavior in the matter dominated era, $a=a_{0}\biggl{(}\frac{t}{t_{0}}\biggr{)}^{2/3}$ and $\rho_{M}\propto 1/t^{3}$ of the GR Einstein-de Sitter universe model. Because $G=1/\chi$, the time dependence of $G$ for $C=0$ is given by
$$\frac{\dot{G}}{G}=-\frac{2}{3\omega_{M}+4}\frac{1}{t}=-\frac{H}{\omega_{M}+1},$$
(33)
where $H$ is the Hubble parameter.
For the case $C\neq 0$, the $\chi$ contributions dominate the dynamics of the early universe. We can choose values of $C$ that can describe both the matter dominated era with $p=0$ and the radiation dominated era with $p=\frac{1}{3}\rho$. For a large enough $|C|$, the $\chi$ dominated solutions can differ significantly from the matter dominated solutions. Then, for $C$ large and negative $G$ can increase with time.
For the case of the flat Euclidean universe, $K=0$, $\rho_{0}=\rho_{\rm crit}$, and $\Omega_{i}=8\pi\rho_{i}/3\chi H^{2}$. We have
$$\Omega_{M}=\frac{8\pi\rho_{M}}{3\chi H^{2}},\quad\Omega_{\Lambda}=\frac{8\pi%
\rho_{\Lambda}}{3\chi H^{2}},\quad\Omega_{r}=\frac{8\pi\rho_{r}}{3\chi H^{2}}.$$
(34)
We have two possible models for the universe. In the first model, it is assumed that $\rho_{b}<\rho_{f}=\rho_{\chi}+\rho_{\phi}$ in the early universe, followed by a transition to $\rho_{b}>\rho_{f}$ when the reionization period in the expansion of the universe begins and stars and galaxies are formed. For this model, we can find values of $C\neq 0$ for the early universe and have $\rho_{f}>\rho_{b}$. In the late universe, following this period, MOG explains galaxy and galaxy cluster dynamics without dominant dark matter. In the second model, we assume that baryons dominate in the early and late time universe, $\rho_{b}>\rho_{f}$. For this model, we have $C=0$ and the baryons dominate the scalar and vector field contributions in both the early and late universe.
The first model can describe the cosmic microwave (CMB) data. The baryon sound wave oscillations due to the baryon-photon pressure prior to the decoupling time produce acoustical peaks in the angular power spectrum, ${\cal D}_{\ell}=\ell(\ell+1)C_{\ell}/2\pi$. We can match the $\Lambda$CDM calculation of the CMB angular acoustical power spectrum. The calculation of the power spectrum in the ${\Lambda}CDM$ model is duplicated in MOG, using the Planck 2018 Collaboration best-fit parameter values [15]: $\Omega_{b}h^{2}=0.0224\pm 0.0001$, $\Omega_{f}h^{2}=0.120\pm 0.001$, $\Omega_{\Lambda}=0.680\pm 0.013$, $n_{s}=0.965\pm 0.004$, $\sigma_{8}=0.811\pm 0.006$, $H_{0}=67.4\pm 0.5\,{\rm km}\,{\rm sec}^{-1}\,{\rm Mpc}^{-1}$, together with the remaining parameters in the fitting process. The second model has to fit the CMB data as well as the galaxy and cluster dynamics without exotic dark matter [16].
5 MOG Black Holes
For the matter free $\phi_{\mu}$ field-vacuum case with $\Lambda=0$, $T^{M}_{\mu\nu}=0$ and $J^{\mu}=0$, the field equations are given by
$$G_{\mu\nu}=-\frac{\omega_{M}}{\chi^{2}}\biggl{(}\nabla_{\mu}\chi\nabla_{\nu}%
\chi-\frac{1}{2}g_{\mu\nu}\nabla^{\alpha}\chi\nabla_{\alpha}\chi\biggr{)}\\
-\frac{1}{\chi}(\nabla_{\mu}\chi\nabla_{\nu}\chi-g_{\mu\nu}\Box\chi)+\frac{8%
\pi}{\chi}T^{\phi}_{\mu\nu},$$
(35)
$$\Box\chi=\frac{8\pi}{(2\omega_{M}+3)}T^{\phi},$$
(36)
$$\nabla_{\nu}B^{\mu\nu}=0,$$
(37)
where
$$T^{\phi}\equiv g^{\mu\nu}T^{\phi}_{\mu\nu}=\mu^{2}\phi^{\mu}\phi_{\mu}.$$
(38)
The field equations in the Einstein frame are given by
$${\bar{G}}_{\mu\nu}=-\frac{(2\omega_{M}+3)}{16\pi G_{0}\chi^{2}}({\bar{\nabla}}%
_{\mu}\chi{\bar{\nabla}}_{\nu}\chi-\frac{1}{2}{\bar{g}}_{\mu\nu}{\bar{\nabla}}%
_{\alpha}\chi{\bar{\nabla}}^{\alpha}\chi)+8\pi G_{0}{\bar{T}}^{\phi}_{\mu\nu},$$
(39)
$${\bar{\Box}}\chi=\frac{8\pi}{(2\omega_{M}+3)}{\bar{T}}^{\phi}$$
(40)
,
$$\bar{\nabla}_{\nu}{\bar{B}}^{\mu\nu}=0.$$
(41)
A stationary black hole must be axisymmetric or static. In the former case there will be two Killing vectors fields, $\xi^{\mu},\zeta^{\mu}$, the first is timelike and the second spacelike at infinity. A bivector $\xi^{[\alpha}\zeta^{\beta]}$ will be timelike at infinity and will have the magnitude
$f=\xi^{[\alpha}\zeta^{\beta]}\xi_{[\alpha}\zeta_{\beta]}$. An event horizon occurs when $f=0$. Outside the horizon at each point there exists a linear combination of $\xi^{\mu}$ and $\zeta^{\mu}$ which is timelike. According to Hawking [5] the scalar field $\chi$ must be constant along the directions of $\xi^{\mu},\zeta^{\mu}$, for they are Killing vectors. Therefore, in the exterior region the gradient of $\chi$ must be spacelike or zero everywhere. The same will be true in the static case, because there will be one Killing vector $\xi^{\mu}$ which is timelike everywhere in the exterior region. Let there be a partial Cauchy surface ${\cal S}$ for ${\bar{J}}^{+}({\cal J}^{-})\cup{\bar{J}}^{-}({\cal J}^{+})$ and a partial Cauchy surface ${\cal S}^{\prime}$ determined by moving each point of ${\cal S}$ a unit distance along the integral curves of $\xi^{\mu}$. Let ${\cal V}$ be the region bounded by ${\cal S},{\cal S}^{\prime}$, containing a portion of the event horizon and a timelike surface at infinity. The region exterior to the event horizon is empty apart from the energy-momentum tensor for the $\phi_{\mu}$ field.
We adopt the boundary condition $\chi=\chi_{0}$ where $\chi_{0}$ is the constant value at infinity. We now multiply (40) by ${\bar{T}}^{\phi}$ and integrate over ${\cal V}$. Integrating by parts, we derive a volume integral over ${\cal V}$ [5, 17]:
$$\int_{\cal V}d^{4}x\sqrt{-{\bar{g}}}{\bar{T}}^{\phi}{\bar{\Box}}\chi=\frac{8%
\pi}{(2\omega_{M}+3)}\int_{\cal V}d^{4}x\sqrt{-{\bar{g}}}({\bar{T}^{\phi}})^{2},$$
(42)
together with surface integrals. Because of the isometry group the surface integral over the Cauchy surface ${\cal S}^{\prime}$ cancels out the surface integral over ${\cal S}$. The surface integral at infinity is zero, because asymptotic flatness requires that the integral vanishes over the region at infinity for $\chi\rightarrow\chi_{0}$. The surface integral over the horizon is zero, because the gradient of $\chi$ is orthogonal to the null vector tangent to the horizon, which is a linear combination of the Killing vectors $\xi^{\mu}$ and $\zeta^{\mu}$. We obtain the result:
$$\frac{8\pi}{(2\omega_{M}+3)}\int_{\cal V}d^{4}x\sqrt{-{\bar{g}}}({\bar{T}}^{%
\phi})^{2}=\frac{8\pi}{(2\omega_{M}+3)}\int_{\partial{\cal V}}d^{3}x\sqrt{|h|}%
{\bar{T}}^{\phi}{\bar{\nabla}}_{\mu}\chi n^{\mu}=0,$$
(43)
where $\partial{\cal V}$ denotes the boundary of ${\cal V}$, $n^{\mu}$ is the normal to the boundary, and $h$ is the determinant of the induces metric $h_{\mu\nu}$ on the boundary. It follows that ${\bar{T}}^{\phi}$ is manifestly zero. We have ${\bar{T}}^{\phi}=\mu^{2}{\bar{\phi}}^{\mu}{\bar{\phi}}_{\mu}$, so for MOG black holes and $\phi_{\mu}\neq 0$ the effective mass $\mu=0$. This is consistent with the mass $\mu$ being tiny, $\mu\sim{\cal O}(10^{-28})$ eV, in the fitting of galaxy and galaxy cluster data, and $\mu$ can be neglected for an isolated astrophysical body such as a black hole.
The conformally invariant gravitational $\phi$-field energy momentum tensor is
$$T^{\phi}_{\mu\nu}=-\biggl{(}{B_{\mu}}^{\alpha}B_{\alpha\nu}-\frac{1}{4}g_{\mu%
\nu}B^{\alpha\beta}B_{\alpha\beta}\biggr{)},$$
(44)
and we have ${\bar{T}^{\phi}}={\bar{g}}^{\mu\nu}{\bar{T}}^{\phi}_{\mu\nu}=0$ giving the equation:
$${\bar{\Box}}\chi=0.$$
(45)
We now multiply (45) by $\chi$ and integrate over the volume ${\cal V}$. Integrating by parts, we derive the volume integral [5]:
$$\int_{\cal V}d^{4}x\sqrt{-{\bar{g}}}{\bar{\nabla}}^{\mu}\chi{\bar{\nabla}}_{%
\mu}\chi=\int_{\partial{\cal V}}d^{3}x\sqrt{|h|}{\bar{\nabla}}_{\mu}\chi n^{%
\mu}=0.$$
(46)
Because the volume integral (46) must be zero and the gradient of $\chi$ is either spacelike or zero, then the gradient of $\chi$ must be zero everywhere and $\chi$ must be constant. In the original papers on MOG black holes [10, 11], we assumed that the gradient of $G$ is zero, so that $G=G_{N}(1+\alpha)=1/\chi$ is constant. We can now demonstrate that for MOG black holes, it is justified to have $G$ constant.
The metric for a static spherically symmetric black hole is given by [10, 11]:
$$ds^{2}=\biggl{(}1-\frac{2G_{N}(1+\alpha)M}{r}+\frac{\alpha(1+\alpha)G_{N}^{2}M%
^{2}}{r^{2}}\biggr{)}dt^{2}-\biggl{(}1-\frac{2G_{N}(1+\alpha)M}{r}+\frac{%
\alpha(1+\alpha)G_{N}^{2}M^{2}}{r^{2}}\biggr{)}^{-1}dr^{2}-r^{2}d\Omega^{2},$$
(47)
where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$. The metric reduces to the Schwarzschild solution when $\alpha=0$. The axisymmetric rotating black hole has the metric solution including the spin angular momentum $J=Ma$:
$$ds^{2}=\frac{\Delta}{\rho^{2}}(dt-a\sin^{2}\theta d\phi)^{2}-\frac{\sin^{2}%
\theta}{\rho^{2}}[(r^{2}+a^{2})d\phi-adt]^{2}-\frac{\rho^{2}}{\Delta}dr^{2}-%
\rho^{2}d\theta^{2},$$
(48)
where
$$\Delta=r^{2}-2G_{N}(1+\alpha)Mr+a^{2}+\alpha(1+\alpha)G_{N}^{2}M^{2},\quad\rho%
^{2}=r^{2}+a^{2}\cos^{2}\theta.$$
(49)
Consider the timelike Killing vector $\xi^{\mu}$ in a stationary asymptotically flat solution of an uncollapsed body. The gravitational mass of the body measured at infinity is given by the $1/r$ and the $1/r^{2}$ contributions to the MOG metric in $\xi^{\mu}\xi^{\nu}g_{\mu\nu}$. We have
$$M=\frac{1}{4}\pi\int d\Sigma_{\nu\beta}\nabla_{\mu}xg^{\mu\nu}\xi^{\beta},$$
(50)
where $\Sigma_{\nu\beta}$ is the surface element of a spacelike 2-surface at infinity and $x^{2}=\xi^{\mu}\xi^{\nu}g_{\mu\nu}$. In the Einstein frame, $M_{E}$ denotes the gravitational mass calculated in the Einstein frame. The total mass $M_{E}$ is decomposed into the sum of the tensor component $M_{t}$ and the scalar and vector components $M_{s}$ and $M_{v}$. If the body collapses to form a MOG black hole, the field $\chi=1/G$ will become constant as will the parameter $\alpha$, and so for an isolated black hole $M_{s}=0$. Moreover, because the source mass $M$ in the gravitational charge of the vector field, $Q_{g}=\sqrt{\alpha G_{N}}M$, is positive the vector (spin 1 graviton) field does not produce a dipole moment, so $M_{v}=0$. A consequence is that for an isolated black hole there will not be any scalar monopole or dipole vector gravitational sources. In the Einstein frame, it can be demonstrated that for merging black holes $M_{E}$ will decrease by the amount of tensor (quadrupole) field energy radiated at infinity.
6 Conclusions
A simplified fully covariant modified gravity $MOG$ is formulated in terms of a scalar field $\chi=1/G$ and a gravitational massive vector field $\phi_{\mu}$. The field equations are given in both the Jordan-Brans-Dicke frame and the Einstein frame. The equations of motion of a particle satisfy the weak equivalence principle, because particles are in free fall but do not follow geodesics. In the weak field approximation, the MOG acceleration of a slowly moving particle has a repulsive Yukawa-type gravitational force in addition to the Newtonian gravitational force.
Cosmological solutions are derived for a homogeneous and isotropic universe. Two cosmological models are considered. The first model assumes that the density, $\rho_{f}=\rho_{\chi}+\rho_{\phi}$, mimics the role of dark matter in the early universe with $\rho_{b}<\rho_{f}$. A calculation of the CMB acoustical TT power spectrum is duplicated by MOG, using the basic six parameters in the $\Lambda CDM$ model [15]. A transition to a baryon dominated era, $\rho_{b}>\rho_{f}$, occurs at the time of reionization and the formation of the first stars and galaxies and MOG describes galaxy and galaxy cluster dynamics without a dominant dark matter. In the second model, it is assumed that baryons dominate in both the early and late universe, $\rho_{b}>\rho_{f}$ [16].
The black holes as the end point of the gravitational collapse of an astrophysical body are stationary axisymmetric or static solutions of the matter free $\phi_{\mu}$-field vacuum MOG field equations with an asymptotic flat boundary condition. It is shown that the stationary space containing a MOG black hole is a solution of the field equations in the Einstein frame. It follows that the scalar field $\chi$ must be a constant everywhere in a stationary MOG solution, and the gravitational coupling strength $G=G_{N}(1+\alpha)$ is constant. This has the consequence that as a body collapses, it loses its scalar monopole moment. The vector field $\phi_{\mu}$ dipole moment is also absent due to the positivity of the mass $M$ in the vector field gravitational source charge, $Q_{g}=\sqrt{\alpha G_{N}}M$. A mass loss results from the emission of tensor (quadrupole) radiation associated to the gravitational energy.
Acknowledgments
I thank Martin Green and Viktor Toth for helpful discussions. Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canada through industry Canada and by the Province of Ontario through the Ministry of Research and Innovation (MRI).
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Equivalence of wave-particle duality to entropic uncertainty
Patrick J. Coles
Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117543 Singapore
Jędrzej Kaniewski
Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117543 Singapore
Stephanie Wehner
Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117543 Singapore
(December 5, 2020; December 5, 2020; December 5, 2020)
Abstract
Interferometers capture a basic mystery of quantum mechanics: a single quantum particle can exhibit wave behavior, yet that wave behavior disappears when one tries to determine the particle’s path inside the interferometer. This idea has been formulated quantitively as an inequality, e.g., by Englert and Jaeger, Shimony, and Vaidman, which upper bounds the sum of the interference visibility and the path distinguishability. Such wave-particle duality relations (WPDRs) are often thought to be conceptually inequivalent to Heisenberg’s uncertainty principle, although this has been debated. Here we show that WPDRs correspond precisely to a modern formulation of the uncertainty principle in terms of entropies, namely the min- and max-entropies. This observation unifies two fundamental concepts in quantum mechanics. Furthermore, it leads to a robust framework for deriving novel WPDRs by applying entropic uncertainty relations to interferometric models. As an illustration, we derive a novel relation that captures the coherence in a quantum beam splitter.
pacs: 03.67.-a, 03.67.Hk
I Introduction
When Feynman discussed the two-path interferometer in his famous lectures Feynman (1970), he noted that quantum systems (quantons) display the behavior of both waves and particles and that there is a sort of competition between seeing the wave behavior versus the particle behavior. That is, when the observer tries harder to figure out which path of the interferometer the quanton takes, the wave-like interference becomes less visible. This tradeoff is commonly called wave-particle duality (WPD). Feynman further noted that this is "a phenomenon which is impossible … to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]."
Many quantitative statements of this idea, so-called wave-particle duality relations (WPDRs), have been formulated Englert (1996); Jaeger et al. (1995); Wootters and Zurek (1979); Greenberger and Yasin (1988); Englert et al. (2008); Liu et al. (2009); Li et al. (2012); Huang et al. (2013); Qureshi (2013); Banaszek et al. (2013); Jia et al. (2014). Such relations typically consider the Mach-Zehnder interferometer for single photons, see Fig. 1. For example, a well-known formulation proven independently by Englert Englert (1996) and Jaeger et al. Jaeger et al. (1995) quantifies the wave behavior by fringe visibility $\mathcal{V}$, and particle behavior by the distinguishability of the photon’s path, $\mathcal{D}$. (See below for precise definitions; the idea is that “waves" have a definite phase, while “particles" have a definite location, hence $\mathcal{V}$ and $\mathcal{D}$ respectively quantify how definite the phase and location are inside the interferometer.) They found the tradeoff:
$$\mathcal{D}^{2}+\mathcal{V}^{2}\leqslant 1$$
(1)
which implies $\mathcal{V}=0$ when $\mathcal{D}=1$ (full particle behavior means no wave behavior) and vice-versa, and also treats the intermediate case of partial distinguishability.
It has been debated, particularly around the mid-1990’s Englert et al. (1995); Storey et al. (1994); Wiseman and Harrison (1995), whether the WPD principle, also known as Bohr’s complementarity principle, is equivalent to another fundamental quantum idea with no classical analog: Heisenberg’s uncertainty principle Heisenberg (1927). The latter states that there are certain pairs of observables, such as position and momentum or two orthogonal components of spin angular momentum, that cannot simultaneously be known or jointly measured. Likewise there are many quantitative statements of this idea, so-called uncertainty relations (URs) (see, e.g., Kennard (1927); Robertson (1929); Wehner and Winter (2010); Renes and Boileau (2009); Berta et al. (2010); Coles et al. (2011a, 2012); Tomamichel and Renner (2011)), and modern formulations typically use entropy instead of standard deviation as the uncertainty measure. See Wehner and Winter (2010) for a survey of entropic uncertainty relations (EURs), and Deutsch (1983); Bialynicki-Birula and Rudnicki for reasons why entropy provides a more powerful framework for uncertainty.
At present the debate regarding wave-particle duality and uncertainty remains unresolved, to our knowledge. Yet Feynman’s quote seems to suggest a belief that quantum mechanics has but one mystery and not two separate ones. In this article we lend quantitative support to this belief by showing a connection between URs and WPDRs, demonstrating that URs and WPDRs capture the same underlying physics 111Some partial progress along this line was made in Busch and Shilladay (2006).. This may come as a surprise, since Englert Englert (1996) originally argued that (1) “does not make use of Heisenberg’s uncertainty relation in any form". To be fair, the uncertainty relation that we show is equivalent to (1) was not known at the time of Englert’s paper, and was only recently discovered Renes and Boileau (2009); Berta et al. (2010); Coles et al. (2011a, 2012); Tomamichel and Renner (2011). Specifically, we will consider EURs, where the particular entropies that are relevant to (1) are the so-called min- and max-entropies used in cryptography Konig et al. (2009).
In what follows we consider several different WPDRs from the literature and show that they are in fact particular examples of EURs. Making this sort of connection not only unifies two fundamental concepts in quantum mechanics, but also means that novel WPDRs can be derived simply by applying already-proven EURs. As an illustration, we derive a novel WPDR for an exotic scenario involving a “quantum beam splitter" Ionicioiu and Terno (2011); Kaiser et al. (2012); Tang et al. (2013), where testing our WPDR would allow the experimenter to verify the beam splitter’s quantum coherence.
Thus, in addition to unifying fundamental concepts, we provide a general framework for deriving and discussing WPDRs. We emphasize that the framework provided by EURs is highly robust, and entropies have well-characterized statistical meanings. Note that current approaches to deriving WPDRs often involve brute force calculation of the quantities one aims to bound; there is no general, elegant method currently in use. Our approach simply involves judicial application of the relevant uncertainty relation. What’s more, we emphasize that uncertainty relations can be applied to interferometers in two different ways. One involves the principle of preparation uncertainty, which says that a quantum state cannot be prepared having low uncertainty for two complementary observables, and it turns out this principle is the one relevant to (1). The other involves the principle of measurement uncertainty, which says that two complementary observables cannot be jointly measured, and we discuss why this principle is actually what was tested in some recent interferometry experiments Jacques et al. (2008); Kaiser et al. (2012). Joint measurability in the context of interferometers was also discussed in Busch and Shilladay (2006); Liu et al. (2009).
II Results
Our unified view associates a kind of behavior with the availability of a kind of information, or lack of behavior with missing information, as follows:
$$\displaystyle\text{lack of particle behavior:}\hskip 2.0ptH_{\min}(Z|J)$$
$$\displaystyle\text{lack of wave behavior:}\min_{W\in XY}H_{\max}(W|K)$$
where $H_{\min}$ and $H_{\max}$ are the min- and max-entropies (defined in Methods) which are commonly used in quantum information theory, $Z$ is the path observable identified with the standard qubit basis (see Fig. 1), $W$ is an orthonormal basis observable in the $XY$ plane of the Bloch sphere 222We use the same symbols ($Z$, $W$, $X$, $C$) for the observables as for the random variables they give rise to., and $J$ and $K$ are some other quantum systems that help to reveal the behavior (e.g., $J$ could be a which-path detector and $K$ could be the quanton’s internal degree of freedom). We formulate our general WPDR as
$$H_{\min}(Z|J)+\min_{W\in XY}H_{\max}(W|K)\geqslant 1$$
(2)
which states that, for a two-path interferometer for single quantons, the sum of the ignorances about the particle and wave behaviors is lower bounded by 1 (i.e., 1 bit).
To be clear, (2) is explicitly an entropic uncertainty relation. The fact that it can be thought of as a WPDR, and furthermore that it encompasses the majority of WPDRs found in the literature for two-path single-quanton interferometers, is our result.
III Discussion
To illustrate this, we consider the celebrated Mach-Zehnder interferometer, shown schematically in Fig. 1. In the simplest case one sends in a single photon towards a 50/50 (i.e., symmetric) beam splitter, $\textsf{BS}_{1}$, which results in the state $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$, where $Z=\{|0\rangle,|1\rangle\}$ is the which-path basis, then a phase $\phi$ is applied to the lower arm giving the state $(|0\rangle+e^{i\phi}|1\rangle)/\sqrt{2}$. Finally the two paths are recombined on a second 50/50 beam splitter $\textsf{BS}_{2}$ and the output modes are detected by detectors $D_{0}$ and $D_{1}$. Visibility is then defined as
$$\mathcal{V}:=\frac{p^{0}_{\max}-p^{0}_{\min}}{p^{0}_{\max}+p^{0}_{\min}}$$
(3)
where $p^{0}_{\max}=\max_{\phi}\Pr(C=0)$ and $p^{0}_{\min}=\min_{\phi}\Pr(C=0)$, where $C$ denotes the random variable revealing which detector $D_{C}$ clicks, with $C\in\{0,1\}$. In this trivial example one has $\mathcal{V}=1$. However many more complicated situations, for which the analysis is more interesting, have been considered in the extensive literature; we now illustrate how these situations fall under the umbrella of our framework with some illustrative examples.
III.1 Preparation uncertainty
$\mathcal{P}$-$\mathcal{V}$ relation.—As a warm-up, we begin with the simplest known WPDR, the so-called predictability-visibility tradeoff. Predictability $\mathcal{P}$ is defined as the prior knowledge, given the experimental setup, about which path the photon will take inside the interferometer. More precisely, $\mathcal{P}:=2p_{\text{guess}}(Z)-1$ where $p_{\text{guess}}(Z)$ is the probability of correctly guessing $Z$. Non-trivial predictability is typically obtained by choosing $\textsf{BS}_{1}$ to be asymmetric. In such situations, the following bound holds Wootters and Zurek (1979); Greenberger and Yasin (1988):
$$\mathcal{P}^{2}+\mathcal{V}^{2}\leqslant 1.$$
(4)
This particularly simple example is a special case of Robertson’s uncertainty relation involving standard deviations Busch and Shilladay (2006); Bosyk et al. (2013). However, Ref. Bosyk et al. (2013) argues that (4) is inequivalent to a family of EURs involving Rényi entropies, hence one gets the impression that entropic uncertainty is different from wave-particle duality, although Ref. Bosyk et al. (2013) did not consider the EUR (2) involving the min- and max-entropies. For an arbitrary probability distribution $P=\{p_{j}\}$, the unconditional min- and max-entropies are given by $H_{\min}(P)=-\log\max_{j}p_{j}$ and $H_{\max}(P)=2\log\sum_{j}\sqrt{p_{j}}$ 333All logarithms are base 2.. We find that Eq. (4) is equivalent to
$$H_{\min}(Z)+\min_{W\in XY}H_{\max}(W)\geqslant 1,$$
(5)
where the entropy terms are evaluated for the state at any time while the photon is inside the interferometer. It is straightforward to see that $H_{\min}(Z)=-\log\frac{1+\mathcal{P}}{2}$ and in the Methods we prove that
$$\min_{W\in XY}H_{\max}(W)=\log(1+\sqrt{1-\mathcal{V}^{2}})$$
(6)
Plugging these relations into (5) gives (4).
$\mathcal{D}$-$\mathcal{V}$ relation.—Let us move on to a more general and more interesting scenario where, in addition to prior which-path knowledge, one may obtain further which-path knowledge during the experiment due to the interaction of the photon with some environment $E$, which may act as a which-way detector. Most generally the interaction is given by a completely positive trace preserving (CPTP) map $\mathcal{E}$, with the input system being $S$ at time $t_{1}$ and output systems being $S$ and $E$ at time $t_{2}$, see Fig. 1. The final state is $\rho^{(2)}_{SE}=\mathcal{E}(\rho_{S}^{(1)})$, where we use the superscripts $(1)$ and $(2)$ to indicate the states at times $t_{1}$ and $t_{2}$. We do not require $\mathcal{E}$ to have any special form in order to derive our WPDR, so our treatment is slightly more general than Englert (1996), which derived (1) assuming the interaction is a path-preserving controlled unitary.
The path distinguishability is defined by $\mathcal{D}:=2p_{\text{guess}}(Z|E)-1$, where $p_{\text{guess}}(Z|E)$ is the probability for correctly guessing the photon’s path $Z$ at time $t_{2}$ given that the experimenter performs the optimally helpful measurement on $E$. We find that (1) is equivalent to
$$H_{\min}(Z|E)+\min_{W\in XY}H_{\max}(W)\geqslant 1,$$
(7)
where the entropy terms are evaluated for the state $\rho^{(2)}_{SE}$. First, it is obvious from the operational meaning of the conditional min-entropy Konig et al. (2009) that we have $H_{\min}(Z|E)=-\log p_{\text{guess}}(Z|E)=-\log\frac{1+\mathcal{D}}{2}$, and second we use our result (6) to rewrite (7) as (1). We remark that (1) and its entropic form (7) do not require $\textsf{BS}_{1}$ to be symmetric. This fact was emphasized in Englert (1996), hence we think of $\mathcal{D}$ as accounting for both the prior $Z$ knowledge associated with the asymmetry of $\textsf{BS}_{1}$ as well as the $Z$ information gained from $E$. Also, the power of our approach should be clear from the fact that, unlike some previous approaches, we did not have to explicitly state the form of $\mathcal{E}$ to derive the WPDR.
We now wish to make an important remark. The above analysis shows that Eqs. (1) and (4) correspond to applying the preparation uncertainty relation at time $t_{2}$ (just before the photon reaches $\textsf{BS}_{2}$). Preparation uncertainty restricts one’s ability to predict the outcomes of future measurements of complementary observables. Thus, to experimentally measure the quantity $\mathcal{P}$ or more generally $\mathcal{D}$, the experimenter removes $\textsf{BS}_{2}$ and sees how well he/she can guess which detector clicks, see Fig. 2A. ($\textsf{BS}_{2}$ can either be physically removed or effectively removed by exploiting another degree of freedom Ionicioiu and Terno (2011).) Of course, to then measure $\mathcal{V}$, the experimenter reinserts $\textsf{BS}_{2}$ in order to close the interferometer. (The fact that the apparatus must be modified to measure $\mathcal{V}$ versus $\mathcal{D}$ is one way of stating Bohr’s complementarity principle.) Our point of emphasis is that this procedure falls into the general framework of preparation uncertainty.
III.2 Measurement uncertainty
On the other hand, uncertainty relations can be applied in a conceptually different way. Instead of two complementary output measurements and a fixed input state, consider a fixed output measurement and two complementary sets of input states. Namely consider the input ensembles $Z_{i}=\{|0\rangle,|1\rangle\}$ and $W_{i}=\{|w_{\pm}\rangle\}$, where $i$ stands for “input" and $|w_{\pm}\rangle=(|0\rangle\pm e^{i\phi}|1\rangle)/\sqrt{2}$ (see Fig. 1 for definition of $|0\rangle$ and $|1\rangle$). The two $Z_{i}$ inputs are generated by blocking the opposite arm of the interferometer, as in Fig. 2B, while the $W_{i}$ states are generated by applying a phase (either 0 or $\pi$) to the lower arm. One can imagine this as a game, where Bob controls the input and Alice has control over both $E$ and the detectors, Bob flips a coin to determine which path he will block in the case of $Z_{i}$ (or which phase he will apply in the case of $W_{i}$) and Alice’s goal is to guess the outcome of Bob’s coin flip.
It may not be common knowledge that this scenario leads to a different class of WPDRs, therefore we illustrate the difference in Fig. 2. Furthermore, we refer to $\mathcal{D}$ introduced above as output distinguishability, whereas in the present scenario we use the symbol $\mathcal{D}_{i}$ and call this quantity input distinguishability, defined by
$$\mathcal{D}_{i}:=2p_{\text{guess}}(Z_{i}|EC)-1,$$
where $p_{\text{guess}}(Z_{i}|EC)=2^{-H_{\min}(Z_{i}|EC)}$ is Alice’s probability to correctly guess Bob’s $Z_{i}$ state given that she has access to $E$ and she knows which output detector clicks, stored in the random variable $C$ 444Even though $\mathcal{D}$ and $\mathcal{D}_{i}$ are relevant to different physical scenarios, they become mathematically equivalent $\mathcal{D}_{i}=\mathcal{D}$ in the special case when $C$ provides no which-path information and $\mathcal{E}$ is path preserving.. Likewise we define the notion of input visibility $\mathcal{V}_{i}$ via:
$$\mathcal{V}_{i}:=\big{[}1-\big{(}\min_{W\in XY}2^{H_{\max}(W_{i}|C)}-1\big{)}^%
{2}\big{]}^{1/2}$$
(8)
which quantifies how well Alice can determine $W_{i}$ using $C$.
Now the uncertainty principle says that there is a tradeoff: if Alice can guess the $Z_{i}$ states well then she cannot guess the $W_{i}$ states well, and vice-versa. In other words, Alice’s measurement apparatus, the apparatus to the right of the dashed line labeled $t_{1}$ in Fig. 1, cannot jointly measure Bob’s $Z$ and $W$ observables. EURs involving von Neumann entropy have previously been applied to the joint measurement scenario Coles et al. (2011b); Buscemi et al. (2013), we do the same for the min- and max-entropies to obtain
$$\displaystyle H_{\min}(Z_{i}|EC)+\min_{W\in XY}H_{\max}(W_{i}|C)\geqslant 1.$$
(9)
This can be rewritten as an explicit WPDR:
$$\mathcal{D}_{i}^{2}+\mathcal{V}_{i}^{2}\leqslant 1,$$
(10)
which can now be applied to a variety of situations.
Quantum $\textsf{BS}_{2}$.—As an interesting application of (9), we consider the scenario proposed in Ionicioiu and Terno (2011) and implemented in Kaiser et al. (2012); Tang et al. (2013), where the photon’s polarisation $P$ acts as a control system to determine whether or not $\textsf{BS}_{2}$ appears in the photon’s path and hence whether the interferometer is open or closed, see Fig. 3. Since $P$ can be prepared in an arbitrary input state $\rho^{(2)}_{P}$, such as a superposition, this effectively means that $\textsf{BS}_{2}$ is a “quantum beam splitter", i.e., it can be in a quantum superposition of being absent or present. The interaction coupling $P$ to $S$ is modelled as a controlled unitary as in Fig. 3. In this case we show (see Methods) that input and output visibility are equivalent:
$$\mathcal{V}_{i}=\mathcal{V}=2|\kappa|\sqrt{R(1-R)}\langle V|\rho^{(2)}_{P}|V\rangle$$
(11)
where we assume the dynamics are path-preserving, i.e., $\mathcal{E}^{S}(|0\rangle\!\langle 0|)=|0\rangle\!\langle 0|$ and $\mathcal{E}^{S}(|1\rangle\!\langle 1|)=|1\rangle\!\langle 1|$, where $\mathcal{E}^{S}={\rm Tr}_{E}\circ\mathcal{E}$ is the reduced channel on $S$, and we denote the action on off-diagonal elements by $\mathcal{E}^{S}(|0\rangle\!\langle 1|)=\kappa|0\rangle\!\langle 1|$ where $|\kappa|\leqslant 1$. In (11), $\mathcal{V}$ is evaluated for any pure state input $\rho^{(1)}_{S}$ from the $XY$ plane of the Bloch sphere (e.g., $|+\rangle$). Now we apply the joint measurement relation (10) to this scenario and use (11) to obtain:
$$\mathcal{D}_{i}^{2}+\mathcal{V}^{2}\leqslant 1,$$
(12)
which extends a recent result in Ref. Jia et al. (2014) to the case where $E$ is non-trivial. This general treatment includes the special case where $\rho^{(2)}_{P}=|V\rangle\!\langle V|$, corresponding to a closed interferometer with an asymmetric $\textsf{BS}_{2}$. Ref. Jacques et al. (2008) experimentally tested this special case, under the assumption that $E$ is trivial ($|\kappa|=1$), in which case our visibility formula becomes $\mathcal{V}_{i}=\mathcal{V}=2\sqrt{R(1-R)}$. We note that Ref. Jacques et al. (2008) did not remark that their experiment actually tested a relation different from (1), namely they tested a special case of (12).
Similarly, Ref. Kaiser et al. (2012) tested (12) (again neglecting $E$) rather than (1), but they allowed $\rho^{(2)}_{P}$ to be in a superposition. At first sight this seems to test the WPDR in the case of a quantum beam splitter, but note that Eq. (9), from which we derived (12), does not refer to the final polarisation $P$. The consequence is that neither the visibility $\mathcal{V}$ nor the distinguishability $\mathcal{D}_{i}$ depends on the phase coherence in $\rho^{(2)}_{P}$ and hence the data could be simulated by a classical mixture of $\textsf{BS}_{2}$ being absent or present 555Ref. Kaiser et al. (2012) used other means to check for coherence.. Nevertheless, our framework provides a WPDR that captures the coherence in $\rho^{(2)}_{P}$ by conditioning either of the entropy terms in (9) on $P$ after the photon exits the interferometer. For example, defining the polarisation-enhanced distinguishability: $\mathcal{D}_{i}^{P}:=2p_{\text{guess}}(Z_{i}|ECP)-1$ gives the novel WPDR:
$$(\mathcal{D}_{i}^{P})^{2}+\mathcal{V}^{2}\leqslant 1,$$
(13)
which captures the beam splitter’s coherence (see Appendix) and could be tested with the setup in Kaiser et al. (2012).
III.3 Conclusions
We have unified the wave-particle duality principle and the entropic uncertainty principle, showing that WPDRs are EURs in disguise. We leave it for future work to extend the connection of WPDRs to EURs for multiple photons Huang et al. (2013) and multiple interference pathways Englert et al. (2008). We believe the framework presented here can be applied fairly universally to the case of single quantons in two-path interferometers. Indeed we show in a forthcoming paper Coles et al. that the main results of both Ref. Li et al. (2012) (for asymmetric beam splitters) and Ref. Banaszek et al. (2013) (for an internal degree of freedom) fall under our entropic uncertainty framework.
Our framework makes it clear how to formulate novel WPDRs by simply applying known EURs to novel interferometer models, and these new WPDRs will likely inspire new interferometry experiments. We also note that all of our relations hold if one replaces both min- and max-entropy with the well-known von Neumann entropy. Alternatively, one can use smooth entropies Tomamichel and Renner (2011); Tomamichel et al. (2012), and the resulting smooth WPDRs may find application in the security analysis of interferometric quantum key distribution Ekert et al. (1992); Ekert and Palma (1994).
IV Methods
In what follows we first derive our preparation uncertainty WPDR, then our measurement uncertainty WPDR, and finally we discuss the QBS example. We begin by defining the relevant entropies. Let $\rho_{AB}$ be any quantum state and $X=\{X_{0},X_{1}\}$ be a binary POVM (positive operator valued measure) on $A$, then the classical-quantum min- and max-entropies are 666It can be shown that these definitions are equivalent to those given in, e.g., Konig et al. (2009); Tomamichel and Renner (2011).:
$$\displaystyle H_{\min}(X|B)$$
$$\displaystyle=1-\log(1+|\!|\sigma_{B}^{0}-\sigma_{B}^{1}|\!|_{1}),$$
(14)
$$\displaystyle H_{\max}(X|B)$$
$$\displaystyle=\log(1+2|\!|\sqrt{\sigma_{B}^{0}}\sqrt{\sigma_{B}^{1}}|\!|_{1}),$$
(15)
where the 1-norm is $\|K\|_{1}={\rm Tr}\sqrt{K^{\dagger}K}$, and $\sigma_{B}^{j}={\rm Tr}_{A}(X_{j}\rho_{AB})$ ††footnotemark: . For any tripartite state $\rho_{AB_{1}B_{2}}$ where $A$ is a qubit and letting $Z$ and $W$ be mutually unbiased bases on system $A$, we have Tomamichel and Renner (2011)
$$H_{\min}(Z|B_{1})+H_{\max}(W|B_{2})\geqslant 1.$$
(16)
Preparation WPDR.—We apply Eq. (16) at time $t_{2}$ in Fig. 1 to obtain a preparation uncertainty WPDR:
$$H_{\min}(Z|E)_{\rho^{(2)}}+\min_{W\in XY}H_{\max}(W)_{\rho^{(2)}}\geqslant 1,$$
(17)
where the subscript $\rho^{(2)}$ indicates the entropy is evaluated for the state $\rho^{(2)}_{SE}$. Note that we place no assumptions on the dynamics prior to $t_{2}$. We assume that after this time the photon reaches a symmetric (50/50) $\textsf{BS}_{2}$ and obtain the result:
$$\min_{W\in XY}H_{\max}(W)_{\rho^{(2)}}=\log(1+\sqrt{1-\mathcal{V}^{2}}).$$
(18)
To prove (18), we write $p^{0}_{\max}=\max_{W\in XY}\Pr(w_{+})$, where we define $\Pr(w_{\pm})=\langle w_{\pm}|\rho^{(2)}_{S}|w_{\pm}\rangle$, and $|w_{\pm}\rangle$ are the two orthonormal basis states associated with the $W$ observable. Suppose $\widetilde{W}$ is optimal such that $p^{0}_{\max}=\Pr(\widetilde{w}_{+})$, then due to the geometry of the Bloch sphere, we have $p^{0}_{\min}=\Pr(\widetilde{w}_{-})$. Thus, $p^{0}_{\max}+p^{0}_{\min}=1$ and $p^{0}_{\max}-p^{0}_{\min}={\rm Tr}(\widetilde{W}\rho^{(2)}_{S})$, which gives
$$\mathcal{V}^{2}=[{\rm Tr}(\widetilde{W}\rho^{(2)}_{S})]^{2}=\max_{W\in XY}%
\langle W\rangle^{2}_{\rho^{(2)}}.$$
Using $\Pr(w_{\pm})=(1\pm\langle W\rangle)/2$ we write
$$\displaystyle H_{\max}(W)_{\rho^{(2)}}$$
$$\displaystyle=\log(1+2\sqrt{\Pr(w_{+})\Pr(w_{-})})$$
$$\displaystyle=\log(1+\sqrt{1-\langle W\rangle^{2}_{\rho^{(2)}}}).$$
Thus we have
$$\min_{W\in XY}H_{\max}(W)_{\rho^{(2)}}=\log(1+\sqrt{1-\max_{W\in XY}\langle W%
\rangle^{2}_{\rho^{(2)}}})$$
giving the desired result (18).
As noted in the main text, (18) allows us to rewrite (17) as $\mathcal{D}^{2}+\mathcal{V}^{2}\leqslant 1$.
Measurement WPDR.—To prove our joint measurement relation in (9), which considers a fixed output measurement and complementary input ensembles, we proceed as follows. The input ensembles $Z_{i}=\{|0\rangle_{S},|1\rangle_{S}\}$ and $W_{i}=\{|w_{+}\rangle_{S},|w_{-}\rangle_{S}\}$ can be generated by performing the relevant measurements on a reference system $S^{\prime}$ that is initially entangled to system $S$. Associating state ensembles with measurements on a reference system is a useful trick for deriving (9), as we shall see. Thus, we introduce a copy $S^{\prime}$ of $S$ and consider a maximally entangled state $|\Phi\rangle_{S^{\prime}S}=(|00\rangle+|11\rangle)/\sqrt{2}$. Now we feed $S$ through the channel $\mathcal{E}$ to obtain the state $\rho_{S^{\prime}SE}=(\mathcal{I}\otimes\mathcal{E})(|\Phi\rangle\!\langle\Phi|)$. Finally, to model the measurement on the output modes, we consider a channel $\mathcal{C}$ that measures the binary POVM $C=\{C_{0},C_{1}\}$ on system $S$ and stores an extra copy in $C^{\prime}$, defined by $\mathcal{C}(\rho_{S})=\sum_{j}{\rm Tr}_{S}(C_{j}\rho_{S})|j\rangle\!\langle j|%
_{C}\otimes|j\rangle\!\langle j|_{C^{\prime}}$ ††footnotemark: . The final state is then:
$$\tau_{S^{\prime}CC^{\prime}E}=(\mathcal{I}\otimes\mathcal{C}\otimes\mathcal{I}%
)[(\mathcal{I}\otimes\mathcal{E})(|\Phi\rangle\!\langle\Phi|)],$$
(19)
which is often called the Choi-Jamiołkowski state. We apply (16) to this state to obtain
$$H_{\min}(Z_{S^{\prime}}|EC)_{\tau}+\min_{W\in XY}H_{\max}(W_{S^{\prime}}|C)_{%
\tau}\geqslant 1,$$
(20)
where $Z_{S^{\prime}}$ and $W_{S^{\prime}}$ are complementary observables on $S^{\prime}$, and by symmetry we replaced $C^{\prime}$ by $C$. Since $|\Phi\rangle$ is maximally entangled, measuring $Z_{S^{\prime}}$ corresponds to sending the states $\{|0\rangle_{S},|1\rangle_{S}\}$ with equal probability through the interferometer, and similarly for $W_{S^{\prime}}$ (with an inconsequential complication of taking the transpose of the $W$ basis states). Realizing this, (20) becomes (9).
QBS example.—We now solve for the visibility in special case of a quantum $\textsf{BS}_{2}$. In this case the process of applying the unitary $U_{PS}$ followed by measuring $S$ can be viewed as measuring a binary POVM $C$ on $S$, where the POVM elements are
$$\displaystyle C_{0}$$
$$\displaystyle={\rm Tr}_{P}[(\rho^{(2)}_{P}\otimes\openone)U_{PS}^{\dagger}(%
\openone\otimes|0\rangle\!\langle 0|)U_{PS}],$$
(21)
$$\displaystyle C_{1}$$
$$\displaystyle={\rm Tr}_{P}[(\rho^{(2)}_{P}\otimes\openone)U_{PS}^{\dagger}(%
\openone\otimes|1\rangle\!\langle 1|)U_{PS}].$$
(22)
Specialising $U_{PS}$ to be a controlled unitary implies that ${\rm Tr}C_{0}={\rm Tr}C_{1}=1$, and assuming the dynamics are path-preserving (see main text), we prove below that
$$\mathcal{V}_{i}=\mathcal{V}=|\kappa|r_{\bot}$$
(23)
where $\mathcal{V}$ is evaluated for any pure state input $\rho^{(1)}_{S}$ from the $XY$ plane of the Bloch sphere (e.g., $|+\rangle$), and $r_{\bot}:=\max_{W\in XY}{\rm Tr}(C_{0}W)$ where the maximization is over all observables $W$ in the $XY$ plane. For example suppose we choose $U_{PS}=|H\rangle\!\langle H|_{P}\otimes\openone_{S}+|V\rangle\!\langle V|_{P}%
\otimes U(R)$, where
$$U(R)=\begin{pmatrix}\sqrt{R}&\sqrt{1-R}\\
\sqrt{1-R}&-\sqrt{R}\end{pmatrix}$$
(24)
is the transformation for an asymmetric beam splitter. In this case we use Eq. (21) to obtain $r_{\bot}=2\sqrt{R(1-R)}\langle V|\rho^{(2)}_{P}|V\rangle$, giving the result in (11).
To prove (23) we first rewrite $p^{0}_{\max}$ appearing in $\mathcal{V}$ as
$$\displaystyle p^{0}_{\max}$$
$$\displaystyle=\max_{W\in XY}\Pr(C=0|W_{i}=|w_{+}\rangle)$$
$$\displaystyle=\Pr(C=0|\widetilde{W}_{i}=|\widetilde{w}_{+}\rangle),$$
which supposes that $\widetilde{W}_{i}\in XY$ achieves the optimisation. Then we have $p^{0}_{\min}=\Pr(C=0|\widetilde{W}_{i}=|\widetilde{w}_{-}\rangle)$. This is because we can think of $\Pr(C=0|W_{i}=|w_{+}\rangle)$ as the Hilbert-Schmidt inner product between $C_{0}$ and the density operator $\rho_{S}^{(2)}$, and the phase that minimizes this inner product is 180 degrees added to the phases that maximizes it. Thus the output visibility is:
$$\displaystyle\mathcal{V}$$
$$\displaystyle=\frac{\Pr(C=0|\widetilde{W}_{i}=|\widetilde{w}_{+}\rangle)-\Pr(C%
=0|\widetilde{W}_{i}=|\widetilde{w}_{-}\rangle)}{\Pr(C=0|\widetilde{W}_{i}=|%
\widetilde{w}_{+}\rangle)+\Pr(C=0|\widetilde{W}_{i}=|\widetilde{w}_{-}\rangle)}$$
$$\displaystyle=\Pr(\widetilde{W}_{i}=|\widetilde{w}_{+}\rangle|C=0)-\Pr(%
\widetilde{W}_{i}=|\widetilde{w}_{-}\rangle|C=0)$$
(25)
where the second line used Bayes’ rule:
$$\Pr(C=0|\widetilde{W}_{i}=|\widetilde{w}_{+}\rangle)=\frac{\Pr(\widetilde{W}_{%
i}=|\widetilde{w}_{+}\rangle|C=0)\Pr(C=0)}{\Pr(\widetilde{W}_{i}=|\widetilde{w%
}_{+}\rangle)}$$
and assumed that $\Pr(\widetilde{W}_{i}=|\widetilde{w}_{+}\rangle)=\Pr(\tilde{W}_{i}=|\widetilde%
{w}_{-}\rangle)=1/2$. Now we have
$$\displaystyle\sqrt{1-\mathcal{V}^{2}}$$
$$\displaystyle=\sqrt{4\Pr(\widetilde{W}_{i}=|\widetilde{w}_{+}\rangle|C=0)\Pr(%
\widetilde{W}_{i}=|\widetilde{w}_{-}\rangle|C=0)}$$
$$\displaystyle=\min_{W\in XY}\sqrt{4\Pr(W_{i}=|w_{+}\rangle|C=0)\Pr(W_{i}=|w_{-%
}\rangle|C=0)}$$
$$\displaystyle=\min_{W\in XY}2^{H_{\max}(W_{i}|C=0)}-1$$
$$\displaystyle=\min_{W\in XY}\frac{1}{2}2^{H_{\max}(W_{i}|C=0)}+\frac{1}{2}2^{H%
_{\max}(W_{i}|C=1)}-1$$
$$\displaystyle=\min_{W\in XY}2^{H_{\max}(W_{i}|C)}-1$$
$$\displaystyle=\sqrt{1-\mathcal{V}_{i}^{2}}$$
(26)
which notes that $H_{\max}(W_{i}|C=0)=H_{\max}(W_{i}|C=1)$. Hence this proves $\mathcal{V}=\mathcal{V}_{i}$. Finally, we have
$$\displaystyle\mathcal{V}$$
$$\displaystyle={\rm Tr}(\mathcal{E}^{S}(\widetilde{W})C_{0})={\rm Tr}(|\kappa|%
\widetilde{W}C_{0})=|\kappa|r_{\bot}.$$
(27)
V Acknowledgements
We thank S. Tanzilli for helpful correspondence. We acknowledge funding from the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant “Random numbers from quantum processes" (MOE2012-T3-1-009).
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Appendix A Appendix A: Testing coherence in a quantum beam splitter
A.0.1 Quantities sensitive to coherence
Here we further elaborate on the treatment of the quantum beam splitter (QBS) depicted in Fig. 3 of the main text. In particular we show that our novel WPDR stated in the main text:
$$(\mathcal{D}_{i}^{P})^{2}+\mathcal{V}^{2}\leqslant 1$$
(28)
captures the coherence of the beam splitter, whereas a weaker WPDR:
$$\mathcal{D}_{i}^{2}+\mathcal{V}^{2}\leqslant 1$$
(29)
does not. Here the different distinguishabilities are
$$\displaystyle\mathcal{D}_{i}$$
$$\displaystyle=2p_{\text{guess}}(Z_{i}|C)-1$$
(30)
$$\displaystyle\mathcal{D}_{i}^{P}$$
$$\displaystyle=2p_{\text{guess}}(Z_{i}|CP)-1$$
(31)
where here $P$ is the final polarisation after the QBS. For simplicity, we will neglect any interaction with an external environment $E$ in what follows, and hence conditioning these distinguishabilities on $E$ is not necessary. (Since we are interested in demonstrating that (28) can capture coherence, it suffices to demonstrate it for a special case where $E$ plays no role.)
For comparison, we will also define “decohered" versions of $\mathcal{D}_{i}$ and $\mathcal{D}_{i}^{P}$, $\mathcal{D}_{i}^{dec}$ and $\mathcal{D}_{i}^{P,dec}$ respectively, where the latter correspond to feeding in a decohered version of the polarisation state $\rho^{(2)}_{P}$, i.e., feeding in the corresponding classical mixture of $|H\rangle$ and $|V\rangle$ rather than a coherent superposition. Precisely this means replacing $\rho^{(2)}_{P}$ with $\mathcal{Z}(\rho^{(2)}_{P})$ where $\mathcal{Z}(\cdot)=|H\rangle\!\langle H|(\cdot)|H\rangle\!\langle H|+|V\rangle%
\!\langle V|(\cdot)|V\rangle\!\langle V|$ is the quantum channel that decoheres the polarisation state.
The first noteworthy point is that $\mathcal{D}_{i}=\mathcal{D}_{i}^{dec}$, hence measuring $\mathcal{D}_{i}$ does not reveal the coherence in the QBS. This is because $\mathcal{D}_{i}$ is not conditioned on $P$, so we evaluate it on the reduced state obtained from tracing over $P$, but tracing over $P$ removes any dependence on the off-diagonal elements of $\rho^{(2)}_{P}$ in $\{|H\rangle,|V\rangle\}$ basis, since the unitary $U_{PS}$ is controlled by the $\{|H\rangle,|V\rangle\}$ basis. Recall that the form of $U_{PS}$ is
$$U_{PS}=|H\rangle\!\langle H|_{P}\otimes\openone_{S}+|V\rangle\!\langle V|_{P}%
\otimes U(R).$$
(32)
where $U(R)$ was defined in (24).
On the other hand we show that, in general, $\mathcal{D}_{i}^{P}\neq\mathcal{D}_{i}^{P,dec}$, so $\mathcal{D}_{i}^{P}$ has the potential to reveal coherence. We also remark that the following hierarchy holds in general:
$$\mathcal{D}_{i}\leqslant\mathcal{D}_{i}^{P,dec}\leqslant\mathcal{D}_{i}^{P}$$
The first inequality holds because $\mathcal{D}_{i}=\mathcal{D}_{i}^{dec}$ and conditioning on $P$ can never decrease the guessing probability. The second inequality holds because the decoherence operation $\mathcal{Z}(\cdot)$ commutes with $U_{PS}$ and hence can be viewed as restricting the class of measurements over which one optimises to evaluate the guessing probability.
For simplicity, let us consider a one-parameter family of input states $\rho^{(2)}_{P}=|\psi^{(2)}_{P}\rangle\!\langle\psi^{(2)}_{P}|$ with $|\psi^{(2)}_{P}\rangle=\cos\alpha|H\rangle+\sin\alpha|V\rangle$. Thus, we have an open interferometer when $\alpha=0$ and a closed one when $\alpha=90\deg$. Note that in this case the visibility becomes
$$\mathcal{V}=2\sqrt{R(1-R)}\sin^{2}(\alpha).$$
Solving analytically for the different distinguishabilities (see below for derivation) gives:
$$\displaystyle\mathcal{D}_{i}$$
$$\displaystyle=|1-2(\sin^{2}\alpha)(1-R)|,$$
(33)
$$\displaystyle\mathcal{D}_{i}^{P,dec}$$
$$\displaystyle=1-(\sin^{2}\alpha)\cdot(1-|2R-1|),$$
(34)
$$\displaystyle\mathcal{D}_{i}^{P}$$
$$\displaystyle=\sqrt{1-4R(1-R)(\sin^{4}\alpha)}.$$
(35)
Clearly these formulas indicate that, in general, $\mathcal{D}_{i}^{P}\neq\mathcal{D}_{i}^{P,dec}$, hence showing that $\mathcal{D}_{i}^{P}$ reveals coherence. We can see a clear distinction between these distinguishabilities in Fig. 4, which considers the case of $R=0.4$.
A.0.2 Discussion of Ref. Kaiser et al. (2012)
We remark that $\mathcal{D}_{i}^{P}$ could be measured using polarisation-resolving detectors on the output modes, as in Fig. 3 of the main text, assuming one chooses the optimal polarisation basis to measure on each output mode. (See below where we explicitly solve for the optimal polarisation basis to measure.) We note that the setup in Ref. Kaiser et al. (2012) has polarisation-resolving detectors, and hence could measure $\mathcal{D}_{i}^{P}$. However, the procedure outlined in Kaiser et al. (2012) for measuring distinguishability corresponds to measuring our $\mathcal{D}_{i}$. Figure 5 shows our theoretical predictions for the situation in Kaiser et al. (2012), corresponding to $R=0.5$. At first sight our predictions appear to disagree with Kaiser et al. (2012) in the sense that our Fig. 5B, which plots $\mathcal{V}^{2}$, $\mathcal{D}_{i}^{2}$, and $\mathcal{V}^{2}+\mathcal{D}_{i}^{2}$, looks very different from the corresponding plot of these quantities in Fig. 4 of Kaiser et al. (2012). An explanation for the disagreement is that Kaiser et al. (2012) may have actually plotted $\mathcal{V}$, $\mathcal{D}_{i}$, and $\mathcal{V}+\mathcal{D}_{i}$ in their Fig. 4. Indeed, their Fig. 4 looks similar to our predictions for $\mathcal{V}$ and $\mathcal{D}_{i}$ in Fig. 5A, and we have $\mathcal{V}+\mathcal{D}_{i}=1$ which is consistent with their Fig. 4. The authors of Kaiser et al. (2012) have confirmed that their Fig. 4 plotted visibility and distinguishability as opposed to their squares Tanzilli . We emphasise that this minor issue with their plot does not affect the conclusions of Ref. Kaiser et al. (2012).
Since we predict that $\mathcal{V}^{2}+\mathcal{D}_{i}^{2}$ can be strictly less than 1, then testing our novel relation $\mathcal{V}^{2}+(\mathcal{D}_{i}^{P})^{2}\leqslant 1$ can give a more stringent test of wave-particle duality. Indeed we show that for the setup in Kaiser et al. (2012), this relation is as strong as possible, i.e., it is satisfied with equality $\mathcal{V}^{2}+(\mathcal{D}_{i}^{P})^{2}=1$. This is depicted in Fig. 5C.
A.0.3 Derivation of distinguishability formulas
The derivation of Eqs. (33)-(35) proceeds as follows. We start with a maximally entangled state on $SS^{\prime}$: $|\Phi\rangle_{SS^{\prime}}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ and the polarisation register in the state $|\psi\rangle_{P}=c|H\rangle+s|V\rangle$, where $c=\cos\alpha$ and $s=\sin\alpha$. Recall that $U_{PS}$ is given by (32), and its action results in the state:
$$\big{(}U_{PS}\otimes\openone_{S^{\prime}}\big{)}|\psi\rangle_{P}\otimes|\Phi%
\rangle_{SS^{\prime}}=c|H\rangle_{P}\otimes\frac{|0\rangle_{S}|0\rangle_{S^{%
\prime}}+|1\rangle_{S}|1\rangle_{S^{\prime}}}{\sqrt{2}}+s|V\rangle_{P}\otimes%
\frac{U(R)|0\rangle_{S}|0\rangle_{S^{\prime}}+U(R)|1\rangle_{S}|1\rangle_{S^{%
\prime}}}{\sqrt{2}}.$$
Measuring $S$ to obtain $C$ and $S^{\prime}$ to obtain $Z$ gives:
$$\displaystyle\rho_{ZCP}$$
$$\displaystyle=\frac{1}{2}|\hskip 1.0pt0\rangle\langle 0\hskip 1.0pt|_{Z}%
\otimes|\hskip 1.0pt0\rangle\langle 0\hskip 1.0pt|_{C}\otimes\left(\begin{%
array}[]{c c}c^{2}&sc\sqrt{R}\\
sc\sqrt{R}&s^{2}R\\
\end{array}\right)$$
$$\displaystyle+\frac{1}{2}|\hskip 1.0pt0\rangle\langle 0\hskip 1.0pt|_{Z}%
\otimes|\hskip 1.0pt1\rangle\langle 1\hskip 1.0pt|_{C}\otimes\left(\begin{%
array}[]{c c}0&0\\
0&s^{2}(1-R)\\
\end{array}\right)$$
$$\displaystyle+\frac{1}{2}|\hskip 1.0pt1\rangle\langle 1\hskip 1.0pt|_{Z}%
\otimes|\hskip 1.0pt0\rangle\langle 0\hskip 1.0pt|_{C}\otimes\left(\begin{%
array}[]{c c}0&0\\
0&s^{2}(1-R)\\
\end{array}\right)$$
$$\displaystyle+\frac{1}{2}|\hskip 1.0pt1\rangle\langle 1\hskip 1.0pt|_{Z}%
\otimes|\hskip 1.0pt1\rangle\langle 1\hskip 1.0pt|_{C}\otimes\left(\begin{%
array}[]{c c}c^{2}&-sc\sqrt{R}\\
-sc\sqrt{R}&s^{2}R\\
\end{array}\right).$$
(36)
From $\rho_{ZCP}$ we can compute the conditional states on $CP$ associated with the different values of $Z$:
$$\displaystyle\sigma_{CP}^{0}=\left(\begin{array}[]{c c c c}c^{2}&sc\sqrt{R}&&%
\\
sc\sqrt{R}&s^{2}R&&\\
&&0&0\\
&&0&s^{2}(1-R)\\
\end{array}\right),$$
$$\displaystyle\sigma_{CP}^{1}=\left(\begin{array}[]{c c c c}0&0&&\\
0&s^{2}(1-R)&&\\
&&c^{2}&-sc\sqrt{R}\\
&&-sc\sqrt{R}&s^{2}R\\
\end{array}\right).$$
Now we compute the distinguishability by using the fact that it is the trace distance between the conditional states. For the polarisation-enhanced distinguishability, this gives:
$$\mathcal{D}_{i}^{P}=\frac{1}{2}|\!|\sigma_{CP}^{0}-\sigma_{CP}^{1}|\!|_{1}=%
\sqrt{c^{4}+s^{4}(2R-1)^{2}+2(sc)^{2}}.$$
Decohering $\sigma_{CP}^{0}$ and $\sigma_{CP}^{1}$ before calculating the trace norm leads to the decohered distinguishability:
$$\mathcal{D}_{i}^{P,dec}=c^{2}+s^{2}\cdot|2R-1|.$$
Finally, to calculate the non-enhanced distinguishability we need to trace out the polarisation register:
$$\sigma_{C}^{0}=\left(\begin{array}[]{c c}c^{2}+s^{2}R&\\
&s^{2}(1-R)\\
\end{array}\right)\hskip 6.0pt\mbox{and}\hskip 6.0pt\sigma_{C}^{1}=\left(%
\begin{array}[]{c c}s^{2}(1-R)&\\
&c^{2}+s^{2}R\\
\end{array}\right)$$
and then
$$\mathcal{D}_{i}=\frac{1}{2}|\!|\sigma_{C}^{0}-\sigma_{C}^{1}|\!|_{1}=|c^{2}+s^%
{2}(2R-1)|.$$
A.0.4 Measuring $\mathcal{D}_{i}^{P}$
Measuring the polarisation-enhanced distinguishability $\mathcal{D}_{i}^{P}$ requires polarization-resolving detectors on the output modes of the interferometer, as in Fig. 3. As defined, $\mathcal{D}_{i}^{P}$ corresponds to measuring the optimal polarization basis on each output mode, i.e., optimally helpful for guessing which path the photon took. We now solve for this optimal polarization basis. We remark that varying the polarization measurement basis could be accomplished by varying the angle of a half-wave plate inserted just prior to the PBS’s in Fig. 3.
Let us begin by expanding the guessing probability as follows
$$p_{\text{guess}}(Z_{i}|CP)=\Pr(C=0)p_{\text{guess}}(Z_{i}|P,C=0)+\Pr(C=1)p_{%
\text{guess}}(Z_{i}|P,C=1)$$
where $p_{\text{guess}}(Z_{i}|CP)$ is related to $\mathcal{D}_{i}^{P}$ through Eq. (31). Since the probabilities $\Pr(C=0)$ and $\Pr(C=1)$ are experimentally accessible, we just need to find the measurements on $P$ that would allow the experimenter to compute $p_{\text{guess}}(Z_{i}|P,C=0)$ and $p_{\text{guess}}(Z_{i}|P,C=1)$.
This is simply a two-state discrimination problem on a qubit, and the optimal solution is well-known Herzog and Bergou (2004). Define the positive operators
$$\displaystyle\tau_{P}^{00}$$
$$\displaystyle={\rm Tr}_{ZC}((|0\rangle\!\langle 0|_{Z}\otimes|0\rangle\!%
\langle 0|_{C}\otimes\openone)\rho_{ZCP})/\Pr(C=0),$$
$$\displaystyle\tau_{P}^{10}$$
$$\displaystyle={\rm Tr}_{ZC}((|1\rangle\!\langle 1|_{Z}\otimes|0\rangle\!%
\langle 0|_{C}\otimes\openone)\rho_{ZCP})/\Pr(C=0),$$
$$\displaystyle\tau_{P}^{01}$$
$$\displaystyle={\rm Tr}_{ZC}((|0\rangle\!\langle 0|_{Z}\otimes|1\rangle\!%
\langle 1|_{C}\otimes\openone)\rho_{ZCP})/\Pr(C=1),$$
$$\displaystyle\tau_{P}^{11}$$
$$\displaystyle={\rm Tr}_{ZC}((|1\rangle\!\langle 1|_{Z}\otimes|1\rangle\!%
\langle 1|_{C}\otimes\openone)\rho_{ZCP})/\Pr(C=1).$$
where $\tau_{P}^{00}$ and $\tau_{P}^{10}$ are the (unnormalised) conditional states on $P$ associated $Z=0$ and $Z=1$ respectively, and both of which are conditioned on $C=0$. Likewise $\tau_{P}^{01}$ and $\tau_{P}^{11}$ are conditioned on $C=1$. From Herzog and Bergou (2004), the optimal polarisation bases to measure on the output modes 0 and 1 are, respectively, given by the eigenvectors of the following Hermitian operators:
$$\displaystyle\Lambda_{P}^{0}=\tau_{P}^{00}-\tau_{P}^{10},$$
$$\displaystyle\Lambda_{P}^{1}=\tau_{P}^{01}-\tau_{P}^{11}.$$
From the formula for $\rho_{ZCP}$ in (A.0.3), we compute that these correspond to the following polarisation observables (represented as matrices in the $\{|H\rangle,|V\rangle\}$ basis):
$$\displaystyle O_{P}^{0}$$
$$\displaystyle=\frac{1}{\mathcal{D}_{i}^{P}}\left(\begin{array}[]{c c}1-2s^{2}R%
&2sc\sqrt{R}\\
2sc\sqrt{R}&-(1-2s^{2}R)\\
\end{array}\right),$$
$$\displaystyle O_{P}^{1}$$
$$\displaystyle=\frac{1}{\mathcal{D}_{i}^{P}}\left(\begin{array}[]{c c}1-2s^{2}R%
&-2sc\sqrt{R}\\
-2sc\sqrt{R}&-(1-2s^{2}R)\\
\end{array}\right).$$
where we have normalised the observables such that they square to the identity. For example, choosing $R=0.5$ and $\alpha=45\deg$ (corresponding to an equal superposition of $\textsf{BS}_{2}$ being “absent" and “present") gives
$$\displaystyle O_{P}^{0}$$
$$\displaystyle=\frac{1}{\sqrt{3}}\left(\begin{array}[]{c c}1&\sqrt{2}\\
\sqrt{2}&-1\\
\end{array}\right),$$
$$\displaystyle O_{P}^{1}$$
$$\displaystyle=\frac{1}{\sqrt{3}}\left(\begin{array}[]{c c}1&-\sqrt{2}\\
-\sqrt{2}&-1\\
\end{array}\right).$$ |
Effective Potential for Emergent Majorana Fermions in Superconductor Systems
A. W. Teixeira
[email protected]
Departamento de Física, Universidade Federal de Viçosa, 36570-900,
Viçosa, Brazil
V. L. Carvalho-Santos
[email protected]
Departamento de Física, Universidade Federal de Viçosa, 36570-900,
Viçosa, Brazil
J. M. Fonseca
[email protected]
Departamento de Física, Universidade Federal de Viçosa, 36570-900,
Viçosa, Brazil
(November 25, 2020)
Abstract
Majorana fermions are particles that are its own antiparticles but they are not found in nature as a free fundamental
particle, however, in condensed matter systens they emerge as a collective excitation. In this work,
using functional integration techniques in complex time representation, we calculated the effective
potential for emergent Majorana fermions in the Kitaev chain and showed how the superconductor parameter
behave as function of temperature. We also found the particle number and
showed the existence of both electrons e holes in the topological phase of the system.
Using a surface induced superconductivity Hamiltonian in a Topological Insulator,
we have calculated the effective potential for emergent Majorana fermions and showed
the equivalence of the gap equation with the one in a quasi-two-dimensional
Dirac electronic system which is a candidate to explain high-T${}_{c}$ superconductivity.
Finally for the $p-$wave superconductor we have found a critical value of the electron-electron interaction
determining the existence or not of induced superconductivity in the surface of the Topological Insulator,
a remarkable result to guide experiments.
pacs:
I Introduction
Majorana Fermions [1, 2] are exotic particles studied in high energy physics for decades
but have not been observed yet [3]. Proposed in 1937 by Ettore Majorana, they are associated
with real solutions of the Dirac equation [1], and as a consequence, they are their antiparticles,
without electric charge. Particles like electrons are described by their energy, momentum, and spin.
In a solid an electron can occupy a energy level, and an unoccupied level is called
a hole. Majorana fermions can emerge as a quantum superposition of an electron and a hole
that move freely, with each one having the same
direction, or spin. This Majorana fermion spin can interact with the spin of atomic nuclei in the material,
so it ought to be seen using nuclear magnetic resonance techniques, they predict [3].
From a high energy physics perspective, Majorana fermions are essentially a half of an ordinary
Dirac fermion. Due to the particle-hole redundancy, a single fermionic state is associated with each pair
of $\pm E$ energy levels, being the presence or absence of a fermion in this state, defines a two-level
system with energy splitting $E$. The existence of the Majorana fermions can be helpful to explain why the
universe has a final asymmetry between matter and antimatter, once Majorana neutrinos obey all of the
Sakharov requirements [4]. However, in a solid state perspective, Majorana fermions could be
used to encode quantum information [5, 6] and also to tune the heat and charge
transport [7], if they indeed emerge spatially separated as a zero mode states [8].
Many-body electronic states can be described by the formalism of second quantization. In this formalism,
electrons are represented by creation and annihilation operators. Each electron of the system can be seen
as a superposition of two Majorana fermions as
$$\displaystyle\gamma_{j1}=c_{j}^{\dagger}+c_{j},\quad\gamma_{j2}=i(c_{j}^{%
\dagger}-c_{j}),$$
(1)
where $c_{j}^{\dagger}$ and $c_{j}$ are respectively the creation and annihilation operators
for electrons, with quantum numbers denoted by index $j$. Additonally, $\gamma$ annihilates or
creates a Majorana fermion. Because $\gamma^{\dagger}=\gamma$, to create or to destroy such particle
has the same effect on the system. Furthermore, from inverting the transformation, we observe that a Majorana
fermion consists of a superposition of electron and hole degrees of freedom. Systems with superconductor order
can exhibit such kind of collective behavior, where their quasi-particles are indeed a product of such
superposition [9]. Therefore, Majorana fermions are expected to emerge in superconductor materials.
However, in most physical systems, the two Majorana fermions comprising the electron are interlaced in the space
making no sense to describe them as isolated particles. To find it spatially separated the system must present
topological properties, e.g., the Kitaev chain [10], topological insulator with induced superconductivity
on the surface [11], topological superconductors [12], and others. In this way, it is an important
issue to obtain and understand the effective potential of emergent Majorana fermions in different contexts, which
could be used, for example, to obtain thermodynamic properties, study the dynamic of these quasiparticles in the
presence of interaction as well as another physical properties of condensed matter systems in which such
particle-like structures appear.
In this work, we obtain the effective potential for emergent Majorana fermions on both, the one-dimensional
Kitaev Model (KM) and a TI surface with induced superconductivity. The analytical calculations have been
performed by using the functional integral techniques in the complex time representation, extracting the natural
logarithm of the partition function [13]. In the case of KM, the calculation of the effective potential
allows us to determine the dependence on the temperature of the superconductor parameter (SP). Additionally, we
show the existence of a phase transition from the trivial to the topological phase by analyzing the mixture of
electrons and holes into the system. These obtained results motivate us to use similar techniques to find the
effective potential for a TI surface with induced superconductivity (See Fig. 1) [3, 12].
We have then shown that this effective potential depends on the type of superconductor gap. For $s$-wave,
the obtained result is in agreement with high-T${}_{c}$ superconductivity theory for cuprates, which assumes
that superconductivity appears in the $CuO$ planes. This result is very interesting because it gives some
insights on the physical mechanism behind the superconductivity in high-T${}_{c}$. For $p$-wave, the obtained
effective potential at zero temperature shows the existence of a continuous quantum phase transition separating
the trivial and the superconductor states as a function of the electron-electron interaction. For non zero
temperature, the gap equation also exhibits the qualitative behavior of the SP as a function of the temperature.
This work is divided as follows: In section II, we obtain and discuss the results of the
effective potential of a KM. Section III brings the discussions on the effective potential of Majorana
fermions appearing in IT with induced supercondiuctivity. Finally, in Section IV,
we present our conclusions and perspectives on the obtained results.
II The Kitaev Model
The Kitaev chain is the simplest model exhibiting Majorana fermions. It consists of a 1D system of
spinless fermions that interacts with the nearest neighbor. It is an exactly soluble model and provides
a useful place to study Majorana fermions in 1D space [3]. This model allows considering a
1D superconductivity order with spinless fermions. The Hamiltonian of KM can be written as
$$\displaystyle\mathcal{H}=\sum_{j}\left[-t\left(c_{j}^{\dagger}c_{j+1}+c_{j+1}^%
{\dagger}c_{j}\right)-\mu\left(c_{j}^{\dagger}c_{j}-\frac{1}{2}\right)\right.$$
$$\displaystyle\left.+\Delta\left(c_{j}^{\dagger}c_{j+1}^{\dagger}+c_{j+1}c_{j}%
\right)\right]-\frac{\Delta^{2}}{g},$$
(2)
where $t$ and $\Delta$ are respectively the hopping and SP, $\mu$ is
the electronic chemical potential, and $g$ is the electron-electron interaction. Let us assume
that $\Delta$ is real and consider a system with open boundary conditions. Using the inverse
representation of Eq. (1), given by
$$c_{j}=\frac{1}{2}\left(\gamma_{j1}+i\gamma_{j2}\right),\quad\quad c_{j}^{%
\dagger}=\frac{1}{2}\left(\gamma_{j1}-i\gamma_{j2}\right),$$
(3)
the Hamiltonian of KM can be rewritten in the Majorana basis as
$$\displaystyle\mathcal{H}=\frac{i}{2}\sum_{j}\left[-\mu\gamma_{j,1}\gamma_{j,2}%
+\left(t+\Delta\right)\gamma_{j,2}\gamma_{j+1,1}\right.$$
$$\displaystyle\left.+\left(-t+\Delta\right)\gamma_{j,1}\gamma_{j+1,2}\right]-%
\frac{\Delta^{2}}{g}\,.$$
(4)
The partition function for Majorana fermions can be then obtained by
performing a functional integral in time complex representation [13], that is,
$$\mathcal{Z}=\int\left[id\gamma^{\dagger}\right]\left[d\gamma\right]\exp\left[-%
\int_{0}^{\beta}d\tau\sum_{j}\left(\gamma^{\dagger}\partial_{\tau}\gamma+%
\mathcal{H}\right)\right],$$
where $\mathcal{H}$ is a $2\times 2$ matrix
that depends on the Nambu fields
$\gamma^{\dagger}=\left(\gamma_{1}^{\dagger}\,\,\,\gamma_{2}^{\dagger}\right)$ and
$\gamma=\left(\gamma_{1}\,\,\,\gamma_{2}\right)^{T}$
in the real space and, $\beta=1/T$. It is worth notice that those fields are independent and they can be
integrated separately. To calculate this integral, we can proceed with a Fourier transformation
of the Majorana operator in both, space and time, in order to obtain a partition function in the momentum space.
That is,
$$\mathcal{Z}=\int\left[id\gamma^{\dagger}\right]\left[d\gamma\right]\exp\left[-%
\sum_{n,p}\beta\left(i\gamma_{n,p}^{\dagger}\mathcal{A}\gamma_{n,p}+\frac{%
\Delta^{2}}{g}\right)\right];$$
(5)
where,
$$\displaystyle\gamma_{n,p}^{\dagger}=\left(\gamma_{1,n,p}^{\dagger}\,\,\gamma_{%
2,n,p}^{\dagger}\right);\quad\mathcal{A}=\left(\begin{array}[]{cc}\omega_{n}&D%
^{*}\\
-D&\omega_{n}\\
\end{array}\right);$$
$$\displaystyle D=\left[\mu+2t\cos\left(p\right)+2i\Delta\sin\left(p\right)%
\right]/4,$$
(6)
$\omega_{n}$ are the Matsubara frequencies for fermions [13].
Eq. (5) is not merely the electronic partition function of the Hamiltonian (II)
with a transformation (3) implemented. Indeed, it is the Majorana partition function obtained
from Hamiltonian (II). A simple change of basis in the electronic partition function does
not provide the correct answer to the Majorana one. This model presents a topological phase when $|2t|>|\mu|$
and otherwise, it is trivial. In the topological phase, the solutions show separated Majorana zero modes.
Since in the momenta space, the Majorana operator works as a Grassmann variable, the integral given in Eq.
(5) is reduced to a Gaussian integral, whose effective potential can be promptly
calculated. Therefore, after performing the sum on the Matsubara frequencies, we have that the effective
potential is given by
$$\displaystyle\text{V}_{\text{eff}}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{\beta}\ln\mathcal{Z}$$
(7)
$$\displaystyle=$$
$$\displaystyle\frac{\Delta^{2}}{g}-\frac{1}{\beta}\int dp\left[\beta|{D}|+2\ln%
\left(1+e^{-\beta|{D}|}\right)\right]\,.$$
It must be highlighted that the sign of the chemical potential distinguishes electrons (negative chemical potential)
and holes (positive chemical potential). Nevertheless, unlike calculations performed for free fermions [13],
the contributions of electrons and holes to the effective potential must not be separated, as a consequence of the
system’s topology. This fact will be evidenced when we are performing the calculation of the number of particles.
The characteristic behavior of the SP as a function of temperature can be obtained using the minimum of the $V_{\rm eff}$.
Additionally, $\Delta$ must not change if we are considering electronic or Majorana effective potential. In this
way, by assuming that $t=\Delta$ and $\mu=0$, we obtain
$$T^{\prime}=\frac{\Delta^{\prime}}{\text{arctanh}\left(\Delta^{\prime}\right)},$$
(8)
where we have defined $T\equiv T^{\prime}\Delta_{0}/4$ and $\Delta\equiv\Delta^{\prime}\Delta_{0}$, with
$\Delta_{0}=\frac{1}{4}\int dp\,\sin^{2}p$. $\Delta_{0}$ can be interpreted as a finite value
that does not contribute qualitatively to the obtained results and it can be determined by assuming the
existence of an energy cutoff, $\Lambda$, considered as the limit of the first Brillouin zone,
$\Lambda=\pi/a$, where $a$ is the lattice spacing. In this context, we can notice that
Eq. (8) agrees qualitatively with experimental data for the behavior of the SP in such
a way that $\Delta$ has a finite and positive value when $T=0$, decreasing to zero as $T$
increases, as shown in Fig. 2.
The chemical potential, can be then determined in the limit of infinite volume, with the number
of electrons given by
$$\displaystyle N$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\beta}\frac{\partial}{\partial\mu}\mathcal{Z}$$
(9)
$$\displaystyle=$$
$$\displaystyle\frac{1}{16}\int dp\frac{\mu+2t\cos(p)}{|{D}|}\tanh\left(\frac{%
\beta|{D}|}{2}\right).$$
The integrand in Eq. (9) is the density of electrons in the momentum space, in such a way that a
positive density corresponds to electrons while a negative one corresponds to holes. If $|{2t}|<|{\mu}|$
(trivial phase), the electronic density is always positive or negative, meaning the system has only electrons
or holes depending on if the chemical potential is positive or negative. On the other hand, if
$|{2t}|>|{\mu}|$ (topological phase), the electronic density can be positive and negative, showing a
mixture of electrons and holes in the system. This fact evidences the existence of a topologial phase and
spattially separated Majorana zero modes because both electrons and holes are constituents of the system.
In Fig. 3, we show the behavior of the number of electrons and holes for $\Delta=1$ and different
values of $\mu$ and $t$, for $T=0$. In the topological phase, occurring when
$\mu=0$ and $t=1$, the density of particles changes from positive to negative values. On the
other hand, for the trivial phase, appearing when $\mu=2$, $t=1$, and when $\mu=-2$, $t=-1$,
the density of particles is always positive or negative, respectively referring to electrons (blue-dashed line)
and holes (red-dashed line). The topology of the considered system is independent on the SP,
although we have assumed an exact value for $\Delta$, its magnitude is not relevant for the phenomenon,
but only for the amplitude of the particle number. The same effect is observed for non-zero temperature.
III Induced Superconductivity in TI Surface
In a previous work, Fu and Kane [11] have considered an s-wave superconductor covering a TI, showing
that the superconductivity can be induced in the surface of a 3D TI by proximity effects. In this context, aiming
to perform a more general analysis, we consider a superconductor with parameter $\Delta(r,r^{\prime})$ inducing
superconductivity in the surface states of 3D TI. The interface between the superconductor and the TI can
be described by the Hamiltonian
$$\displaystyle H$$
$$\displaystyle=$$
$$\displaystyle\int dr~{}dr^{\prime}\left\{c_{\uparrow r}^{\dagger}\Delta(r,r^{%
\prime})c_{\downarrow r^{\prime}}^{\dagger}-c_{\uparrow r}\Delta^{*}(r,r^{%
\prime})c_{\downarrow r^{\prime}}\right.$$
$$\displaystyle\left.\hskip-5.690551pt-\frac{{|\Delta(r,r^{\prime})|}^{2}}{g}+%
\frac{\delta(r-r^{\prime})}{2}\left[c_{\downarrow r^{\prime}}^{\dagger}p_{+}c_%
{\uparrow r}+c_{\uparrow r}p_{+}c_{\downarrow r^{\prime}}^{\dagger}\right.\right.$$
$$\displaystyle\left.\left.\hskip-5.690551pt+c_{\uparrow r}^{\dagger}p_{-}c_{%
\downarrow r^{\prime}}+c_{\downarrow r^{\prime}}p_{-}c_{\uparrow r}^{\dagger}-%
2\sum_{\sigma=\uparrow r,\downarrow r^{\prime}}\mu\left(c_{\sigma}^{\dagger}c_%
{\sigma}-\frac{1}{2}\right)\right]\right\},$$
where $p_{\pm}=p_{x}\pm ip_{y}$ depends on the momentum operator, $c_{\sigma}^{\dagger}$ and $c_{\sigma}$
are the creation and annihilation operators for electrons, and $g$ is the electron coupling constant.
The presence of a vortex in the SP,
which depends on the nature of the superconductor material, can yield zero mode states with the spinor
structure having components that obey the Majorana fermions conditions. This fact implies the existence of
Majorana bound states in the surface of the TI. Using Eq. (LABEL:Heletron) we can determine the effective potential
for emergent Majorana fermions using the partition function. For such purpose, the Hamiltonian has to be rewritten
as a function of Majorana operators in the momentum space, using the inverse of relation (1).
The partition function, as a functional integral in time complex representation, has the form
$$\displaystyle\mathcal{Z}$$
$$\displaystyle=$$
$$\displaystyle\left[\prod_{n}\prod_{p,p^{\prime}}\prod_{\alpha}\int id\gamma_{%
\alpha,n}^{\dagger}d\gamma_{\alpha,n}\right]$$
$$\displaystyle\exp\left\{\sum_{p,p^{\prime}}\left[\sum_{n}i\gamma_{\alpha,n}^{%
\dagger}\mathcal{D}\gamma_{\alpha,n}-\frac{{|\Delta(p,-p^{\prime})|}^{2}}{g}%
\right]\right\}\,,$$
where the Majorana Nambu field is defined as
$$\displaystyle\gamma=\left(\gamma_{1\uparrow}(p)\quad\gamma_{2\uparrow}(p)\quad%
\gamma_{1\downarrow}(p^{\prime})\quad\gamma_{2\downarrow}(p^{\prime})\right)^{%
T}\,,$$
(12)
and the matrix $\mathcal{D}$ is given by
$$\displaystyle\mathcal{D}=\frac{\beta\delta_{p,p^{\prime}}}{4}\left(\begin{%
array}[]{cc}\mathcal{W}+\bar{\mu}&\mathcal{P}_{-}-\bar{\Delta}\\
\mathcal{P}_{+}+\bar{\Delta}^{*}&\mathcal{W}+\bar{\mu}\end{array}\right)\,,$$
being
$$\displaystyle\mathcal{W}=2\left(\begin{array}[]{cc}\omega_{n}&0\\
0&\omega_{n}\end{array}\right),\bar{\mu}=\left(\begin{array}[]{cc}0&-\mu\\
\mu&0\end{array}\right),$$
$$\displaystyle\mathcal{P}_{\pm}=\frac{1}{2}\left(\begin{array}[]{cc}-i(p_{\pm}+%
p_{\pm}^{\prime})&(p_{\pm}+p_{\pm}^{\prime})\\
-(p_{\pm}+p_{\pm}^{\prime})&-i(p_{\pm}+p_{\pm}^{\prime})\end{array}\right),$$
$$\displaystyle\bar{\Delta}=\frac{1}{\delta_{p,p^{\prime}}}\left(\begin{array}[]%
{cc}i\Delta(p,-p^{\prime})&\Delta(p,-p^{\prime})\\
-\Delta(p,-p^{\prime})&i\Delta(p,-p^{\prime})\end{array}\right)\,.$$
(13)
$\Delta(p,-p^{\prime})$ is the Fourier transform of $\Delta(r,r^{\prime})$, and $\mu$ is the chemical potential
for Majorana fermions. The SP can assume different wave configurations, as $s$-wave and $p$-wave for example.
In both cases, we can write this parameter as
$\Delta(p,-p^{\prime})=i|{\Delta(p)}|\delta_{p,p^{\prime}}=i|{\Delta}|f(p)%
\delta_{p,p^{\prime}}$, where $f(p)$
characterizes the symmetry and $|{\Delta}|$ is a constant. Additionally, the integration in
Eq. LABEL:particaofinal) can also be solved using a Gaussian integration. After performing the summation
over the Matsubara frequencies, the effective potential is (assuming $\mu=0$ for simplicity)
$$\displaystyle\text{V}_{\text{eff}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{p}\left\{\frac{|{\Delta(p)}|^{2}}{g}-\sqrt{|{p}|^{2}+|{%
\Delta(p)}^{2}|}\right.$$
(14)
$$\displaystyle\left.-\frac{2}{\beta}\ln\left[1+e^{-\beta\sqrt{|{p}|^{2}+|{%
\Delta(p)}^{2}|}}\right]\right\}.$$
At first place, we will considerer the $s-$wave case. Therefore, by assuming that $\Delta(r,r^{\prime})=|{\Delta}|\delta(r-r^{\prime})$, which results in $f(p)=-i$. The effective potential in the continuum momentum space then becomes
$$\displaystyle\text{V}_{\text{eff}}$$
$$\displaystyle=$$
$$\displaystyle-2\pi\int_{0}^{\Lambda}dp~{}p\left\{\frac{2}{\beta}\ln\left[1+e^{%
+\beta k}\right]-k\right\}$$
(15)
$$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\frac{\pi%
\Lambda^{2}|{\Delta(p)}|^{2}}{g}\,,$$
where $k=\sqrt{p^{2}+|{\Delta(p)}|^{2}}=\sqrt{p^{2}+|{\Delta}|^{2}}$.
To evaluate this integral, we use polar coordinates, whereby the energy cutoff $\Lambda$ depends on
the lattice parameter of the TI, we use a approximate circular first Brillouin zone
(good aproximation for several TIs) and we can perform the integration using the cutoff as the
value of the momentum in the limit of the Brillouin zone [14, 15].
Following the same procedure used in Section II, we can analyze the minimum of the potential in
order to obtain the behavior of the SP as a function of the temperature, which leads to the following
gap equation
$$\displaystyle\frac{\partial}{\partial\Delta}\text{V}_{\text{eff}}=-\int_{0}^{%
\Lambda^{2}}dx~{}\frac{2\pi|{\Delta}|}{\sqrt{|{\Delta}|^{2}+x}}$$
$$\displaystyle\tanh\left(\frac{\beta}{2}\sqrt{|{\Delta}|^{2}+x}\right)$$
(16)
$$\displaystyle+\frac{2\pi\Lambda^{2}|{\Delta}|}{g}\,.$$
Apart from constants, the equation (16) agrees with previously obtained results describing the
behavior of $\Delta$ in quasi-two-dimensional Dirac electronic systems, as a function of the temperature
[16]. Indeed, this result was expected once the present work is considering an electronic system,
although described by the effective potential of Majorana fermions with $\Delta$ being the same parameter
in both effective potentials, for Majorana or electrons. Additionally, the obtained result corresponds to
the same one obtained by the spin fermion model, candidate to explain how high-T${}_{c}$ superconductivity
emerges in $CuO$ planes of cuprates systems [17, 18, 19]. This fact may
suggest the existence of a natural emergence of Majorana fermions in the $CuO$ planes in cuprates,
which is an opened issue under investigation.
From considering a $p$-wave superconductor, we have that $\Delta(p)=|{\Delta}|\left(p_{x}+ip_{y}\right)$
in the effective potential (14). After perform the Matsubara summation, we obtain the same result
given in Eq. (15), but with the new $p$-wave parameter, $\Delta(p)$,
instead the $s$-wave parameter.
In the equilibrium of the system, occuring when it is in the minimum of the equation (15)
(derivative equal zero), one can observe the existence of a critical value of the electron coupling constant
$g_{c}$. Indeed, if $g<g_{c}$, there is no induced superconductivity on the TI surface. Thus, it can be observed
that there is a critical $g$ value allowing an induced superconductivity in the surface states of the
topological insulators by proximity effect. This condition leads to the SP at zero temperature, $\Delta_{0}$, as
$$\displaystyle|\Delta_{0}|=\left\{\begin{array}[]{ccc}0&,\,~{}\text{if}&g\leq 4%
\Lambda/3\\
\sqrt{\left(\frac{3g}{4\Lambda}\right)^{2}-1}&,\,~{}\text{if}&g>4\Lambda/3.\\
\end{array}\right.$$
(17)
Then, a continuous quantum phase transition occurs in the critical value $g_{c}=4\Lambda/3$,
which separates the normal from the superconductor phase. Additionally, the second derivative
shows that it is indeed a minimum of the system.
Aiming to obtain the magnitude of $g_{c}$, we must multiply $p_{\pm}$ in Eq. (LABEL:Heletron)
by $\hbar v_{f}$, where $v_{f}$ corresponds to the Fermi velocity of the surface carriers of the topological insulator.
In this way, we obtain that $g_{c}=4\hbar v_{f}\Lambda/3$, evidencing that $g_{c}$ depends only on the TI parameters,
since $\Lambda=1/a$ is related with the first Brillouin zone ($a$ is the lattice parameter for the TI).
This is a very interesting result, once it implies that if $g_{sc}<g_{c}$, where $g_{sc}$ is the electron-electron
interaction in the superconductor material, superconductivity will not be induced in the TI surface by proximity effects.
As an example, we can consider the parameters of Bi${}_{2}$Se${{}_{3}}$, $\hbar v_{f}=2.87$ eVÅ , and the largest
lattice parameter $a=30.5$ Å [15, 20]. In this way, we obtain $g_{c}=0.4$ eV.
Moreover, we can calculate $g_{sc}$ using [21]
$$\displaystyle g_{sc}\approx-\frac{1}{N(0)\ln\left(\frac{T_{c}}{\Theta_{D}}%
\right)},$$
(18)
where $T_{c}$ is the critical temperature for superconductors transitions, $\Theta_{D}$ is the
Debye temperature, and $N(0)$ is the density of states of the superconductor. If we consider, for example,
the Pb parameters, that is, $T_{c}=7.19$ K, $\Theta_{D}=105$ K and $N(0)=0.49$
states/eV [22, 21], we obtain $g_{sc}\approx 0.76$ eV. Since $g_{sc}>g_{c}$ it will
be expected that Pb can induce superconductivity in Bi${}_{2}$Se${{}_{3}}$. Indeed, this result is corroborated
by experimetnal results in which induced superconductivity has been observed in Bi${}_{2}$Se${{}_{3}}$
in the presence of Pb [23].
At finite temperature, the minimum of the effective potential for $p$-wave produces a gap equation as follow
$$\frac{3g}{2\Lambda^{3}\sqrt{1+{\text{abs}{\Delta}}^{2}}}\int_{0}^{\Lambda}dp~{%
}p~{}\tanh\left(\frac{\beta p}{2}\sqrt{1+\text{abs}{\Delta}^{2}}\right)=1.$$
(19)
The integration of this equation in momentum space results in an SP dependence on the temperature.
By assuming $\Lambda=1$, Fig. 4 presents the characteristic behavior of $\Delta$,
which decreases when the temperature increases. Furthermore, it can be noticed that an increase of $g$ yields a
decrease in the critical temperature below which the system becomes a superconductor. Besides that, the value of the
SP in zero temperature corresponds to $|{\Delta_{0}}|$ calculated previously.
IV Conclusions
We used the functional integral techniques in order to obtain the effective potential of Majorana fermions for two different systems which may present Majorana fermions modes spatially separated. Those effective potentials describe, in both cases, the expected behavior of the SP as a function of the temperature.
In the case of the Kitaev chain, we showed that the SP has a positive and finite value when
$T=0$, decreasing when $T$ increases. We have also obtained the density of particle in momentum space as a function of $p$, showing the differences between topological and trivial phases concerning the presence of holes and electrons in the system.
From considering a topological insulator surface with induced $s$-wave superconductivity, we have obtained a gap equation similar to that obtained for superconductivity in quasi-two-dimensional Dirac electronic systems. Our results suggest the natural emergence of Majorana fermions in $CuO$ planes. For $p$-wave and zero temperature, it was showed the existence of a continuous quantum phase transition separating the normal and the superconductor states. The obtained critical electron-electron interaction depends on the parameters of the TI. It is shown that superconductivity will be induced on the surface of IT only if their electron-electron interaction is higher than a critical value of $g_{c}$.
For non zero temperature, we have obtained that the gap equation exhibits the expected behavior of the $\Delta$ as a function of the temperature. As prospects for future investigations, we are considering the possibility of the emergence of Majorana
fermions in the CuO planes in cuprates, an important issue to be responsed if we think in applications such as quantum
computation based on these elusive particles.
Acknowledgements.
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The authors also thank CNPq (Grant Nos. 401132/2016-1 and 309484/2018-9) and FAPEMIG for financial support.
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The origin of the Langevin equation and the calculation of the mean squared displacement: Let’s set the record straight
K. Razi Naqvi
[email protected]
Department of Physics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
(December 4, 2020)
Abstract
Ornstein and his coauthors, who constructed a dynamical theory of Brownian motion, taking the equation $mdv/dt=-\zeta v+X$ as their starting point, usually named the equation after Einstein alone or after both Einstein and Langevin; furthermore, Ornstein, who was the first to extract from this equation the correct expression for $\overline{\Delta^{2}}$, the mean-squared distance covered by a Brownian particle, credited de Haas-Lorentz, rather than Langevin, for finding the stationary limit of $\overline{\Delta^{2}}$. A glance at Einstein’s 1907 paper, titled “Theoretical remarks on Brownian motion”, should suffice to convince one that it is not unfair to attribute the conception of the above equation, now universally known as the Langevin equation, to Einstein. Langevin’s avowed aim in his 1908 article was to recover, through a route that was ‘infinitely more simple’, Einstein’s 1905 expression for the diffusion coefficient, but a careful reading of Langevin’s paper shows that—depending on how one interprets his description of the statistical behavior of the random force $X$ appearing in the above equation—his analysis is at best incomplete, and at worst a mere tautology. Since textbook accounts are based on the interpretation that renders the proof fallacious, alternative derivations, which are adaptations of those given by de Haas-Lorentz and Ornstein, are presented here. Some neglected aspects of the contents of Ornstein’s early papers on Brownian motion are also brought to light.
I Introduction
It has become customary${}^{1}$ to mark the birth of stochastic differential equations with the publication of the 1908 piece${}^{2,3}$ by Langevin—because he was the first to inscribe such an equation. It is equally common to regard Ornstein as the principal architect of the modern theory of Brownian motion—because he showed, in a series of seminal papers,${}^{4-9}$ how one may extract physically significant information from what is now called Langevin’s equation. Yet it does not appear to be widely known that, in nearly all his papers, Ornstein named the underlying stochastic equation after Einstein alone or after both Einstein and Langevin.
Though the question “Did Ornstein have any grounds for associating the archetypal stochastic equation with Einstein?” will be answered in this article, it will not occupy center-stage. My principal purpose is rather to draw attention to an unsatisfactory feature of Langevin’s ostensibly simple derivation of an expression for the mean-squared displacement. One of his propositions (that relating to the vanishing of the average value of the product of the particle position $x$ and the random force $X$), on which the entire argument hinges, is offered without any proof; the singular ease with which the final result is arrived at provokes unease, rather than approval, when one notices that the average values of $vX$ and $xv$, two closely related products involving the velocity $v$, cannot vanish, but they must do so according to the reasoning employed by Langevin.
I argue here that Langevin’s logic is seriously flawed, and its continued presentation to beginners no better than the dispensation of a nostrum to an unsuspecting patient. A demolition plan, such as that proposed here, stands a better chance of adoption if it is accompanied by a constructive suggestion; accordingly, I also present an alternative derivation, inspired by the edifice constructed by Ornstein and co-authors; though not as short as the original Langevin version, it is logically sound, and yields much greater insight. I also draw attention to an alternative treatment, due originally to de Haas-Lorentz,${}^{10}$ who may be called a forerunner of Ornstein, and to some unaccountably neglected aspects of Ornstein’s work.
II Langevin, his forerunners, contemporaries and aftercomers
Using different approaches, Einstein${}^{11}$ and Smoluchowski${}^{12}$ had arrived, shortly before Langevin turned his attention to the issue, at discordant expressions for the diffusion coefficient $D$ of a spherical particle. Langevin lodged two claims at the start of his analysis: “I have been able to determine, first of all, that a correct application of the method of Mr. Smoluchowski leads one to recover the formula of Mr. Einstein precisely, and, furthermore, that it is easy to give a demonstration that is infinitely more simple by means of a method that is entirely different”. [Emphasis in the original]
It will be well to pause at this point and recall what Einstein wrote in his 1907 paper. I recapitulate, in the next two paragraphs, his statements after some condensation and a modicum of paraphrasing, but without changing his notation.
II.1 Einstein’s 1907 paper
If we knew nothing of the molecular theory of heat, Einstein wrote,${}^{13}$ we should expect the following to happen. Suppose that we impart to a particle suspended in a liquid a certain velocity applied to it from without; then the particle will come to a halt as a result of the friction of the liquid. If we ignore the inertia of the latter and assume that the resistance experienced by a particle moving with velocity $v$ is $6\pi kPv$, where $k$ is the viscosity of the liquid and $P$ the radius of the particle, we obtain the equation
$$m{dv\over dt}=-6\pi kPv.$$
(1)
From this it follows that the velocity will fall to a tenth of its original value in a time $\theta=\ln 10\,m/(6\pi kP)$; for a spherical particle of radius 2.5 $\mu$m suspended in water, $\theta$ comes out to be 330 ns.
If we take into consideration the molecular theory of heat, Einstein reminded the reader, we have to modify this conception. We must continue to assume, as before, that the particle will nearly completely lose its initial velocity in the very short time $\theta$ through friction. But, at the same time, we must assume that the particle gets new impulses during this time by some process that is the inverse of viscosity, so that it retains a speed which on an average is equal to that implied by the Maxwell-Boltzmann distribution. But since we must imagine that direction and magnitude of these impulses are (approximately) independent of the initial velocity of the particle, we must conclude that the velocity of the particle will be very greatly altered in the extraordinary short time $\theta$, and, indeed, in a totally random manner.
It is clear that Einstein conceived, already in 1907, a modified version of Eq. (1), the modification entailing the addition, on the right-hand side, of a random force. We shall see (in the next section) that his assumptions about the randomness of the additional force were echoed by Langevin when the latter put pen to paper. ‘Let us with Einstein consider a colloidal particle,’ wrote Ornstein et al.${}^{9}$ in 1927 (without citing a specific Einstein article); they presented in their own phrase the above argument from Einstein’s 1907 note, and wrote down the amended equation immediately afterwards. Thus, it seems safe to assume that Ornstein was familiar with the note, and it was this knowledge that motivated him to credit Einstein with the germinal idea for converting Eq. (1) into a stochastic differential equation.
II.2 Langevin’s 1908 paper
Langevin accepted the validity of the equipartition principle by writing
$$m\overline{v^{2}}={RT\over N},$$
(2)
where a bar denotes ‘an average extended over a large number of identical particles’, and the other symbols have their usual meanings. Using $\mu$ and $a$ instead of Einstein’s $k$ and $P$, respectively, he wrote the equation of motion in the form
$$m{d^{2}x\over dt^{2}}=-6\pi\mu a{dx\over dt}+X,$$
(3)
and added immediately afterwards: “About the complementary force $X$, we know that it is indifferently positive and negative and that its magnitude is such that it maintains the agitation of the particle, which the viscous resistance would stop without it”. It is to be noted that he says ‘we know’ while Einstein wrote ‘we must assume’.
Langevin multiplied Eq. (3) by $x$, and arrived, after some elementary manipulations at the following equation:
$${m\over 2}{d^{2}x^{2}\over dt^{2}}-mv^{2}=-3\pi\mu a{dx^{2}\over dt}+Xx.$$
(4)
Next, he took an ensemble average, argued that ‘the average value of the term $Xx$ is evidently null by reason of the irregularity of the complementary forces $X$’, replaced the average $m\overline{v^{2}}$ by its equipartition value, introduced the symbol
$$z=\overline{dx^{2}\over dt}={d\,\overline{x^{2}}\over dt},$$
(5)
and arrived finally at his key equation:
$${m\over 2}{dz\over dt}+3\pi\mu az={RT\over N}.$$
(6)
He was seeking after an expression for $z$, whose stationary limit was to be equated to $2D$; this limit, to be denoted by $z_{\infty}$, emerges as soon as one sets $dz_{\infty}/dt=0$. The result agrees with Langevin, who took a slightly longer route and found
$$D={RT\over 6\pi\mu aN},$$
(7)
thereby duplicating Einstein’s result.
II.3 An assessment of Langevin’s paper
It is worth remarking first that Langevin’s use of symbols and his further manipulations are reminiscent of the steps choreographed by Clausius in a paper where he derived the virial theorem.${}^{14}$ Clausius used the symbol $X$ for the deterministic force when he wrote Newton’s equation of motion for a particle (moving in a non-resistive medium), $m(d^{2}x/dt^{2})=X$, and made use of the following identity:
$$2x{d^{2}x\over dt^{2}}+2\left({dx\over dt}\right)^{2}={d^{2}x^{2}\over dt^{2}}.$$
(8)
Neither the conception of Eq. (3) nor its transformation into a differential equation for $x^{2}$ can be called truly original. Be that as it may, it was Langevin who actually wrote down Eq. (3) and—regardless of whose brainchild it was—he eventually became its eponym.
My main concern is with Langevin’s justification for setting $\overline{xX}=0$. Many later authors have simply taken his word for the vanishing of this average. The reception accorded to his analysis may be summed up by quoting from a recent review:${}^{15}$ “Langevin’s derivation, however, was spectacularly simple and direct compared to the others (this is probably why it is Langevin’s derivation rather than Einstein’s that is usually found in modern textbooks)”. Among those who have endorsed Langevin’s logic, one may mention Feynman,${}^{16}$ Reif,${}^{17}$, Heer${}^{18}$ and the authors${}^{19}$ of a recent treatise devoted to the Langevin equation.
I quote Feynman (who uses the symbol $F$ for $X$):
Now what about $x$ times the force? If the particle happens to have gone a certain distance $x$, then, since the irregular force is completely irregular and does not know where the particle started from, the next impulse can be in any direction relative to $x$. If $x$ is positive, there is no reason why the average force should also be in that direction. It is just as likely to be one way as the other. The bombardment forces are not driving it in a definite direction. So the average value of $x$ times $F$ is zero. [Italics in the original]
This line of reasoning would lead one to infer, as has indeed been done, that $\overline{vX}$ and $\overline{xv}$ also vanish. Let us deal with $\overline{vX}$ first. J. D. van der Waals Jr.${}^{20}$ multiplied Eq. (3) by $v$, took an ensemble average, and set $\overline{vX}=0$ to find
$${m\over 2}{d\,\overline{v^{2}}\over dt}=-6\pi\eta a\overline{v^{2}}.$$
(9)
Van der Waals knew that Eq. (9) was unacceptable, since it implied that $\overline{v^{2}}$ must decay to zero, contravening the equipartition principle. There are two ways to resolve the difficulty: one may either scrap the supposition $\overline{vX}=0$, or discard Eq. (3) itself. Unfortunately, van der Waals made the latter choice, and spent some effort in formulating his own theory of Brownian movement, using a different stochastic equation. Though van der Waals did not succeed in winning support for his point of view, he contributed indirectly by engaging Ornstein in a fierce debate. In the course of rebutting van der Waals, Ornstein and Burger (O&B) had to concede that Eq. (3) is an approximation, albeit an excellent one.${}^{7}$ The limitations of Langevin’s equation, which are easily perceived (see below), ought to be emphasized even when it is introduced in an elementary course.
To notice the untenability of the assumption $\overline{xv}=0$, made by Heer,${}^{18}$ one only has to note that since
$${d\,\overline{x^{2}}\over dt}=2\,\overline{xv}\quad\hbox{(identically)},$$
(10)
the vanishing of $\overline{xv}$ implies the vanishing of $D$ itself.
II.4 The treachery of physical intuition
It seems devilishly difficult to divine, on the basis of physical intuition alone, the statistical independence of two random variables in a dynamical problem.${}^{21}$ Let no one forget that Maxwell assumed, in his first attempt to deduce the law of distribution of molecular velocities, that the distribution of the velocity component along a particular cartesian axis was independent of the values along the other two axes.${}^{22}$ According to Chapman and Cowling,${}^{22}$ “it would be natural to suppose that these components are not independent”. Seven years later, Maxwell repudiated this assumption, and made a second attempt,${}^{23}$ which too proved to be unsatisfactory when scrutinized by Boltzmann. After attributing the proof presented in their book to Lorentz (who improved Boltzmann’s exposition) Chapman and Cowling comment: “This proof also is open to some objection, because of the assumption $\ldots$ that there is no correlation between the velocity and the position of a molecule”.
Let us return now to Langevin, who accepted that $z$ eventually attains a stationary value. After averaging Eq. (4), he should have seen that only one conclusion was justified: the value of $\overline{xX}$ is (or becomes, at long times) a constant that cannot be smaller than $-RT/N$. Had he allowed for the possibility $\overline{xX}\neq 0$, he would have found, instead of Eq. (6),
$${m\over 2}{dz\over dt}+3\pi\mu az={RT\over N}+\overline{xX},$$
(11)
from which one can only deduce that
$$D={z_{\infty}\over 2}={1\over 6\pi\mu a}\left({RT\over N}+\overline{xX}\right).$$
(12)
Since Langevin’s aim was to validate Eq. (7), the assumption $\overline{xX}=0$ reduces his demonstration to a tautology.
The foregoing arguments amount to little more than an elaboration of a footnote in which O&B, who used the symbol $\Delta$ for the mean-squared displacement, stated: “From the formula (1) [Eq. (3) here] we can deduce the relation $\overline{\Delta^{2}}=bt$ if we introduce suppositions, it is however impossible to find the value of $b$, without penetrating into the mechanism of the Brownian motion”.${}^{7}$
II.5 Statistical properties of the random acceleration
I find it hard to evade the conclusion that Langevin was tempted into making an error that has dodged detection on account of its seductive simplicity. Still, I will now adopt the exegesis, offered by O&B as well as Manoliu and Kittel,${}^{24}$ that Langevin’s pithy phrase ‘indifferently positive and negative’ is to be interpreted as follows: it implies not only that $\overline{X}=0$ but also a supplementary relation, concerning the autocorrelation of $X$, from which one can deduce the vanishing of the cross-correlation $\overline{xX}$. But these authors also noted the necessity of proving the vanishing of $\overline{xX}$ from the second relation. It is imperative to do so because, as Manoliu and Kittel point out,${}^{24}$ physical intuition might lead one to conclude that $\overline{xX}\neq 0$; after all, an encyclopaedia entry,${}^{25}$ cited by them, does warn the reader: “The mean random force vanishes, but the mean product of the random force into the displacement [in three dimensions] is equal to $-3kT$; the frequent assertion that this product vanishes is mistaken”. Since proving the relation $\overline{xX}\neq 0$ entails, even if one takes the short cut devised here, a good deal of preparatory work, the route taken by Langevin, which has been praised for its brevity and simplicity, turns into a wearisome trek (from the student’s point of view), making Einstein’s treatment look like a jaunt—unless the teacher is prepared to replace the assertion ‘it is self-evident that $\overline{xX}=0$’ by the assurance ‘it can be shown that $\overline{xX}=0$’.
Enough has now been said to convince the reader that Langevin’s derivation of the mean squared displacement must be rectified, or replaced by a cogent alternative. All the ingredients for accomplishing the latter task were assembled by de Haas-Lorentz, but she stopped short, as did Langevin before her, of going beyond the stationary limit. It was Ornstein who, actually worked out, using his own approach, the complete expression in 1917. The prospects of adding something novel to this topic are infinitesimal indeed, but a repackaging of the material scattered in Ornstein’s many publications seems worthwhile. I wish to present a slightly different and shorter account than appears in the 1930 article${}^{26}$ of Uhlenbeck and Ornstein (U&O). As a preliminary, however, I wish to recall some statements made by Ornstein in his first paper.
III Ornstein’s 1917 paper
Ornstein began his first paper on Brownian motion by citing the works of Smoluchowski and Burger; referring to the thesis of de Haas-Lorentz, later published as a monograph,${}^{10}$ he remarked:
It is worth observing that the way different averages depend on time may be calculated from the results obtained by Mrs. de Haas-Lorentz by a slightly more careful transition of the limit than was necessary for the object she had put herself (viz. the determination of the stationary condition). First, I want to determine these averages by a new method …
The relation used by de Haas-Lorentz and mentioned by Ornstein has the following form (though they use different symbols):
$$m{dv\over dt}=-\zeta v+X,$$
(13)
which has some advantages over the form chosen by Langevin: The use, on the right-hand side, of the symbol $\zeta$ rather than $6\pi\eta a$ saves some writing, besides extending the applicability of the equation to non-spherical particles; the introduction of the particle velocity $v$ also simplifies the resulting formulas, but the main reason for the change is to emphasize that velocity, rather than the position, is the quantity of greater interest here. Upon dividing Eq. (13) by $m$, and setting
$$\alpha\equiv{\zeta\over m}\quad\hbox{and}\quad A\equiv{X\over m}.$$
(14)
one gets the still neater form
$${dv\over dt}=-\alpha v+A.$$
(15)
Ornstein carried out a formal integration of Eq. (15) to get
$$v=v_{0}e^{-\alpha t}+e^{-\alpha t}\int_{0}^{t}\!e^{\alpha t_{1}}A(t_{1})dt_{1}%
,\quad v_{0}\equiv v(0).$$
(16)
The fact that $A(t_{1})$ on the right-hand side is a random function did not worry him unduly, since he knew that he would attribute physical significance only to averaged quantities. It is important to recall at this stage that, in the works published during 1917–1930, Ornstein calculated an average over those molecules which start, at time $t=0$, with a definite velocity (say $v_{0}$); I will refer to an average of this type as a $v_{0}$-average, and distinguish it from an overall average (the quantity considered by Langevin) by appending the symbol $v_{0}$ to the bar signifying a particular average. A $v_{0}$-average can itself be averaged over all initial velocities (distributed according to the Maxwell-Boltzmann law). If one’s aim is to obtain only the overall average, it often proves more convenient to bypass the $v_{0}$-averaging. For the sake of completeness, let us take note of a mean value that will be needed shortly. If we take a $v_{0}$-average of Eq. (15), we arrive, after setting $\overline{A\,}^{v_{0}}=0$, at the equation
$${d\,\overline{v\,}^{v_{0}}\over dt}=-\alpha\,\overline{v\,}^{v_{0}},$$
(17)
whose solution is
$$\overline{v\,}^{v_{0}}=v_{0}e^{-\alpha t}.$$
(18)
This is, of course, the same result as would be obtained by taking a $v_{0}$-average of Eq. (16).
At this point, I would like to part company—at least temporarily—with Ornstein and turn to the problem of calculating the quantity $\overline{xA}$.
IV Emending the Langevin analysis
Since we are dealing with a system that is in thermal equilibrium, we can immediately state
$$\overline{v}=0,\quad\quad\overline{v^{2}}={RT\over Nm}=\beta,\hbox{ say.}$$
(19)
Our task is to deduce the statistical properties of $A$, using only Eqs. (15) and (19). If we take an ensemble average of Eq. (16), we get, after setting $\overline{v_{0}}=0$ on the right-hand side,
$$\overline{v}=\int_{0}^{t}\!e^{\alpha t_{1}}\overline{A(t_{1})}dt_{1}=0,$$
(20)
from which follows the relation $\overline{A}=0$.
Averaging the equation found by multiplying Eq. (15) by $2v$ gives
$${d\,\overline{v^{2}}\over dt}=-2\alpha\overline{v^{2}}+2\overline{vA}=0,$$
(21)
which shows, after it is combined with Eq. (19), that
$$\overline{vA}=\alpha\beta.$$
(22)
On the other hand, multiplication of Eq. (16) by $A(t)$ and averaging gives
$$\overline{Av}=e^{-\alpha t}\int_{0}^{t}\!e^{\alpha t_{1}}\overline{A(t)A(t_{1}%
)}dt_{1}.$$
(23)
On equating the right-hand sides of the last two equations one finds
$$\alpha\beta=e^{-\alpha t}\int_{0}^{t}\!e^{\alpha t_{1}}\overline{A(t)A(t_{1})}%
dt_{1},$$
(24)
a relation that can only hold if $\overline{A(t)A(t_{1})}$ possesses the two properties expressed below; in the first place
$$\overline{A(t)A(t_{1})}=\cases{0,&if $t_{1}\neq t$\cr\cr\infty&if $t_{1}=t$,\cr}$$
(25a)25a( 25a )
and secondly
$$\int_{0}^{t}\!\overline{A(t)A(t_{1})}dt_{1}=\alpha\beta.$$
(25b)25b( 25b )
We have now completed the preliminaries for finding $\overline{Ax}$. Integration of Eq. (15) from 0 to $t$ provides a relation connecting $x$, $v$, and $A$:
$$v(t)-v_{0}=-\alpha\bigl{[}x(t)-x(0)\bigr{]}+\int_{0}^{t}\!A(t_{1})dt_{1}$$
(26)
We now multiply Eq. (26) by $A(t)$ and take an average to find, after some obvious rearrangement,
$$\overline{Ax}={1\over\alpha}\left[\int_{0}^{t}\!A(t)A(t_{1})dt_{1}-\overline{%
vA}\right]=0.$$
(27)
The second equality follows from the fact that each term within the square brackets equals $\alpha\beta$. After this demonstration, we can justifiably retrace the steps that took Langevin from Eq. (4) to Eq. (6), and integrate our version of Eq. (6) to get
$${d\,\overline{xv}\over dt}=-\alpha\overline{\,xv\,}+\beta,$$
(28)
which can itself integrated to obtain
$$\overline{xv}={\beta\over\alpha}\left(1-e^{-\alpha t}\right).$$
(29)
Since $d\overline{\,x^{2}\,}/dt=2\overline{\,xv\,}$, we finally get, with $x_{0}\equiv x(0)$,
$$\overline{x^{2}\,}=x_{0}^{2}+{2\beta\over\alpha}\left[t-{1\over\alpha}\left(1-%
e^{-\alpha t}\right)\right],$$
(30)
a relation that Langevin could have derived but chose not to do so. He was reluctant, as was de Haas-Lorentz a few years later, to go beyond the expression derived by Einstein. Needless to say, this result was first obtained by Ornstein${}^{5}$ by averaging his $v_{0}$-average $\overline{\,x^{2}\,}^{v_{0}}$ over a Maxwellian distribution of initial velocities.
IV.1 A closer look at correlations
We see from Eq. (29) that the correlation of $x$ and $v$, assumed to be zero initially, rises exponentially (with a time constant $1/\alpha$) to its stationary value. On the other hand, the average $\overline{vA}$ turned out to be a constant; this is simply because we have assumed the corresponding time constant to be zero. It is tempting to tinker with the autocorrelation of $A$ and assign it a small but non-zero decay time by setting
$$\overline{A(t_{1})A(t_{2})}=C\lambda{e^{-\lambda|t_{2}-t_{1}|}},\quad C=\hbox{%
constant}.$$
(31)
Should we change our mind, we can always recover the results previously obtained by letting $\lambda\to\infty$ and assigning $C$ an appropriate value. Substituting the right-hand side of the last equation into Eq. (23) and integrating we get
$$\overline{\,vA\,}={C\lambda\over\lambda+\alpha}\left[1-e^{-(\lambda+\alpha t)}%
\right].$$
(32)
Though this result looks appealing, it should be rejected, when dealing with a system that is in thermal equilibrium, because a time-dependent value of $\overline{\,vA\,}$ violates the equipartition principle: according to Eq. (21), $\overline{\,vA\,}$ must be a constant, and this constancy is incompatible with any choice that makes the autocorrelation $\overline{\,A(t_{1})A(t_{2})\,}$ decay at a finite rate. Ornstein and van Wijk (O&vW), who dealt with overall averages rather than $v_{0}$-averages, emphasized that the statistical properties of $A$ are determined as soon as one writes down Eqs. (15) and assumes that the system is in thermal equilibrium.${}^{5}$ Manoliu and Kittel overlooked this restriction when they replaced the white spectrum of $A$ by a Lorentzian;${}^{24}$ it will be pointed out presently that the spectrum of $A$ can be made colored (nonwhite) only by modifying the equation of motion itself. The introduction of Eq. (31) did not lead to an immediate generalization of our treatment, but it can still serve as a crutch for those students who are unable to jump from Eq. (24) to Eq. (25), or feel uncomfortable with an alternative (given below) involving the Dirac delta function; in the next section, it will serve as a part of a scheme devised for generalizing Eq. (15).
IV.2 A critique of the Langevin equation itself
Let us set $m=1$ so that we may speak of the force rather than acceleration, and let $w(t)=-\alpha v+A$ denote the total force acting on the particle. O&B agreed with van der Waals that for $t=0$, $\overline{w(t)\,}^{v_{0}}=0$. Besides it is evident, they remarked, that for $t$ infinite the average value of $w(t)$ undergoes no influence from $v_{0}$ and therefore must be zero. They noted that the course of $\overline{w(t)\,}^{v_{0}}=0$ must be such that it starts from zero, attains a maximum (or minimum) and then falls to zero again. On the other hand, Eqs. (15) and (18) show that
$$\overline{w\,}^{v_{0}}=-\alpha\overline{v\,}^{v_{0}}+\overline{A\,}^{v_{0}}=-%
\alpha e^{-\alpha t}v_{0}.$$
(33)
Fig. 1. Plots showing the time dependence of the average force acting on a Brownian particle that started with a velocity $v_{0}>0$. The solid curve depicts the idealized behavior predicted by Eq. (15), whereas the dotted curve is obtained by plotting the right-hand side of Eq. (34) with $\lambda=10\alpha$ and $C=12$.
Assuming that $v_{0}>0$, they drew two curves, one of which, labelled by them as ‘Einstein’, is a plot of the right-hand side of Eq. (33), and the other (which they call the true curve) is a schematic plot of the behaviour described above. I have taken the liberty of converting their qualitative, freehand figure to a quantitative plot, by using the following expression
$$\overline{w\,}^{v_{0}}=C{\bigl{(}e^{-\lambda t}-e^{-\alpha t}\bigr{)}\over%
\lambda-\alpha},\quad(C=\hbox{constant}).$$
(34)
The reasons for choosing the above form will be explained shortly; at present, it is enough to note that the resulting curve resembles that drawn by O&B, and to proceed by quoting their comments on the shapes of the curves:
For $t=0$ the line, which represents this course deviates from the true curve. The important agreement existing between Einstein’s theory and the experiment now makes us presume, that the true $w(t)$-$t$ curve and the curve according to Einstein only deviate from each other for short times after the departure of the particle with the velocity $v_{0}$ so that the maximum in the true curve lies close to $t=0$, and that from this maximum onward it descends pretty well exponentially according to Einstein’s curve. It goes without saying that these are only assumptions, which a calculation of the true $\overline{w(t)\,}^{v_{0}}$ curve must prove from the molecular theory. We are however of the opinion that it is worthwhile to point to this possible interpretation of Einstein’s master-stroke in the theory of the Brownian motion.
It is remarkable that O&B recognized the limitations of Eq. (15) as far back as 1918, and even pointed out the nature of the refinment that was needed. A generalization of Langevin’s equation in the form
$${dv\over dt}=-\int_{-\infty}^{t}\alpha(t-t^{\prime})v(t^{\prime})+A(t).$$
(35)
came almost fifty years later.${}^{27}$ It can be shown that $\alpha(t-t^{\prime})$ is proportional to the autocorrelation of $A$, which explains the choice made in Eq. (34): if we take a $v_{0}$-average of Eq. (35), and use a zero-order approximation $\overline{v\,}^{v_{0}}=v_{0}\exp(-\alpha t)$ on the right-hand side, and apply Eq. (31), the integral would equal (apart from a multiplicative constant) the right-hand side of Eq. (34).
V A random walk in velocity space
Both Langevin and Ornstein began with two claims: each stated that though the desired result(s) could be found by a re-working of an existing analysis, each would rather present a derivation of his own. Ornstein mentioned that the time dependent averages can be calculated from the results obtained by de Haas-Lorentz,${}^{10}$ who credited the gist of the method used by her to a 1910 article (not concerned with Brownian motion) by Einstein and Hopf.${}^{28}$ She integrated Eq. (15) from $t=(N-1)\tau$ to $t=(N-1)\tau$, thereby converting it into a difference equation:
$$v_{N}-v_{N-1}=-\alpha v_{N-1}\tau+I_{N-1},$$
(36a)
where
$$I_{N-1}=\int_{(N-1)\tau}^{N\tau}A(t_{1})dt_{1}$$
(36b)
The time interval $\tau$ is so short that $v$ does not change substantially during this interval, but the fluctuations in $A$ are supposed to be extremely rapid even on this short time scale. With this background, we can already state two statistical properties of the random term on the right-hand side of Eq. (36a):
$$\overline{I_{k}\,}^{v_{0}}=0$$
(37)
$$\overline{I_{j}I_{k}}^{v_{0}}=\delta_{ij}\Pi_{1},\quad\delta_{ij}=\cases{1,&if%
$j=k$\cr\cr 0,&otherwise.\cr}$$
(38)
We have arrived at Eq. (36a) by discretizing Eq. (15), but an alternative point of view, adopted by Gunther and Weaver,${}^{29}$ is worth a mention: One can regard Eq. (36a) as the basic equation in its own right, $I_{0}$, $I_{1}$, $\cdots$ being a sequence of mutually independent random variables; a great virtue of Eq. (36a) is its amenability to Monte Carlo simulation.${}^{29,30}$ Regardless of one’s interpretation, Eq. (36a) can be immediately understood as describing a random walk in velocity space, and its solution requires only summation of geometric series, as demonstrated first by de Haas-Lorentz; her contribution has been underestimated, partly because she herself called it the method of Einstein and Hopf, but mainly because she did not extract the full time-dependence from her expressions.
It follows from iteration of Eq. (36a) that $v_{N}$ can be arranged as a sum. Since this section is supposed to provide a discrete variant of Ornstein’s approach, we will express the sum so as to highlight the correspondence between Eq. (16) and its discrete counterpart shown below:
$$v_{N}=v_{0}\rho^{N}+\rho^{N}\sum_{k=1}^{N}I_{k-1}\rho^{-k},$$
(39)
where
$$\rho\equiv(1-\alpha\tau).$$
(40)
Let us now take a $v_{0}$-average of Eq. (39), and arrive, after using Eq. (37),
at the relation
$$\overline{v_{N}\,}^{v_{0}}=v_{0}\rho^{N}$$
(41)
We would expect, on letting $N\to\infty$ and $\tau\to 0$, to recover the results found by Ornstein if we make the identifications $N\tau=t$, so that $v_{N}=v(t)$, $x_{N}=x(t)$, and so on. Since we are going to let $\tau\to 0$, it is permissible to write
$$\rho=e^{-\alpha\tau},\quad\hbox{or}\quad\rho^{N}=e^{-\alpha t}.$$
(42)
We have now established that, as $\tau\to 0$, one gets
$$\overline{v\,}^{v_{0}}=v_{0}e^{-\alpha t},$$
(43)
a result that coincides with that stated in Eq. (18).
Upon squaring-and-averaging, Eq. (39), and making use of Eqs. (37) and (38), we are left with
$$\displaystyle\overline{v_{N}^{2}}^{v_{0}}$$
$$\displaystyle=$$
$$\displaystyle v_{0}^{2}\rho^{2N}+\rho^{2N}\Pi_{1}\sum_{k=0}^{N-1}\rho^{-2k}$$
(44)
$$\displaystyle=$$
$$\displaystyle v_{0}^{2}\rho^{2N}+\Pi_{1}\left[{\rho^{2N}-1\over 1-\rho^{-2}}%
\right].$$
As we let $\tau\to 0$, the first term on the right-hand side goes into $v_{0}^{2}e^{-2\alpha t}$, whereas the factor within the square brackets will approach
$${1\over 2\alpha\tau}(1-e^{-2\alpha t}).$$
Since $\overline{v^{2}}^{v_{0}}$ must tend to $\beta$ as $t\to\infty$, we choose the limiting value
$$\lim_{\tau\to 0}{\Pi_{1}\over\tau}=2\alpha\beta,$$
(45)
and arrive thereby at the result
$$\overline{v^{2}}^{v_{0}}=v_{0}^{2}e^{-2\alpha t}+\beta(1-e^{-2\alpha t}),$$
(46)
derived first by Ornstein.
There is no need to go further since Gunther and Weaver${}^{29}$ have shown how to obtain $\overline{v^{2}\,}^{v_{0}}$ as well as $\overline{x^{2}\,}^{v_{0}}$, but two comments are still in order. First, they were apparently unaware that the bulk of the calculation had already been done by de Haas-Lorentz; secondly, they arrive at a formula which may be written (when allowance is made for the difference in notation and a typographical error in their paper) as follows:
$$\displaystyle\overline{(x-x_{0})^{2}}^{v_{0}}$$
$$\displaystyle=$$
$$\displaystyle{v_{0}^{2}\over\alpha^{2}}(1-e^{-\alpha t})^{2}$$
(47)
$$\displaystyle+{2\beta\over\alpha}\Bigl{[}t-{3-4e^{-\alpha t}+e^{-2\alpha t}%
\over 2\alpha}\Bigr{]}.$$
At this stage, they only point out that if one sets $v_{0}^{2}=\beta$, the expression for the mean-squared displacement given in Eq. (47) reduces to that ‘obtained using the classical Langevin equation’, namely that given in Eq. (30). Of course, both expressions pertain to the Langevin equation, and their difference can be traced to different averaging procedures.
Continuing in the same vein, one can recover all the mean values calculated by U&O; a verification of this statement will not be offered in these pages.
VI The delta function
Ornstein’s handling of integrals involving the autocorrelation $\overline{A(t_{2})A(t_{1})\,}^{v_{0}}$ reveals an uncanny prescience of an impulse function (some ten years before Dirac${}^{31}$ introduced the delta function). Readers who do not have easy access to Ornstein’s 1917 article may examine the two 1927 articles, where the original calculations are repeated. In all these papers, Ornstein encounters the autocorrelation when he calculates $\overline{v^{2}\,}^{v_{0}}$; by squaring both sides of Eq. (16) and performing a $v_{0}$-average he finds
$$\displaystyle\overline{v^{2}\,}^{v_{0}}$$
$$\displaystyle=$$
$$\displaystyle v_{0}^{2}e^{-2\alpha t}+$$
(48)
$$\displaystyle\quad\int_{0}^{t}\!dt_{1}\,e^{\alpha t_{1}}\!\int_{0}^{t}\,e^{%
\alpha t_{2}}\overline{A(t_{1})A(t_{2})\,}^{v_{0}}dt_{2}.$$
At this point he argues that $\overline{A(t_{1})A(t_{2})\,}^{v_{0}}$ must be different from zero, since $\overline{v^{2}\,}^{v_{0}}$ for $t=\infty$ is positive, but it is clear that the correlation can be appreciable only when the difference between $t_{1}$ and $t_{2}$ is small. He introduces $\tau=t_{2}-t_{1}$ as a new variable and proceeds as follows:
$$\displaystyle\int_{0}^{t}\!dt_{1}\,e^{\alpha t_{1}}\!\int_{0}^{t}\,e^{\alpha t%
_{2}}\overline{A(t_{1})A(t_{2})\,}^{v_{0}}dt_{2}=$$
$$\displaystyle\quad\quad\quad\quad\int_{0}^{t}\!dt_{1}\,e^{2\alpha t_{1}}\!\int%
\overline{A(t_{1})A(t_{1}+\tau)\,}^{v_{0}}d\tau,$$
(49)
adding that the replacement of $t_{1}+t_{2}$ in the exponent by $2t_{1}$ is justified because only the region around $\tau=0$ contributes to the second integral. Finally, he states that it is easily seen that the second integral is independent of $t_{1}$ and $t_{2}$, and is a characteristic constant for the problem at hand, and may be denoted by $a_{1}$. To a modern reader, it should be evident that Ornstein identified the autocorrelation $\overline{A(t_{1})A(t_{2})\,}^{v_{0}}$ with $a_{1}\delta(t_{1}-t_{2})$. In the 1917 article, he also evaluated $\overline{v(t)A(t)\,}^{v_{0}}$ through the same reasoning and found that its value to be $a_{1}/2$. It will be instructive to repeat this calculation using delta calculus. We have
$$\displaystyle\overline{vA\,}^{v_{0}}$$
$$\displaystyle=$$
$$\displaystyle e^{\alpha t}\int_{0}^{t}\,e^{\alpha t_{1}}\overline{v(t)A(t_{1})%
\,}^{v_{0}}dt_{1}$$
(50)
$$\displaystyle=$$
$$\displaystyle e^{\alpha t}\int_{0}^{t}\,e^{\alpha t_{1}}a_{1}\delta(t-t_{1})dt%
_{1}={a_{1}\over 2}$$
since
$$\int_{0}^{\epsilon}\delta(x)dx={1\over 2}\int_{-\epsilon}^{\epsilon}\delta(x)%
dx={1\over 2}$$
(51)
Ornstein never used the name delta function but he did use, in two joint publications with Uhlenbeck, the symbol $\delta$. In their 1930 article,${}^{26}$ U&O mention that
$$\left({1\over 4\pi Dt}\right)^{1/2}e^{-(x-x_{0})^{2}/4Dt}$$
is that solution of the diffusion equation which “for $t=0$, becomes $\delta(x-x_{0})$, when $\delta(x)$ means the function defined by the properties
$$\displaystyle\delta(x)$$
$$\displaystyle=$$
$$\displaystyle 0\quad\hbox{for}\quad x\neq 0$$
$$\displaystyle\int_{-\infty}^{\infty}\delta(x)dx$$
$$\displaystyle=$$
$$\displaystyle 1\hbox{"}.$$
Seven years later, Ornstein and Uhlenbeck${}^{32}$ used the symbol $\delta(E-\varepsilon)$ for expressing the condition “that at $t=0$ all particles have the energy $E$,” and added “$\delta(E-\varepsilon)$ is the well-known singular peak function”. However neither U&O nor O&vW chose to take advantage of the delta calculus, and preferred to carry out their manipulations using the pre-Dirac manoeuvres pioneered by Ornstein; in these two papers, the authors change both variables by setting $u=(t_{1}+t_{2})$ and $w=(t_{1}-t_{2})$. I will not speculate about their preference, but will now raise, and later answer, a different question: Why has Ornstein not been credited with the invention of the delta calculus?
The answer is not far to seek. The second section of the 1927 paper of Dirac is devoted to notation, where he states:
One cannot go far in the development of the theory of $\ldots$ without needing a notation for that function of $\ldots$ $x$ that is equal to zero except when $x$ is small, and whose integral through a range that contains the point $x=0$ is equal to unity. We shall use the symbol $\delta(x)$ to denote this function, i.e., $\delta(x)$ is defined by
$$\delta(x)=0\quad\hbox{when}\quad x\neq 0,$$
and
$$\int_{-\infty}^{\infty}\!dx\;\delta(x)=1.$$
Strictly, of course, $\delta(x)$ is not a proper function of $x$, but can be regarded only as a limit of a certain sequence of functions. All the same one can use $\delta(x)$ as though it were a proper function for practically all the purposes of $\ldots$ without getting incorrect results. One can also use the differential coefficients of $\delta(x)$, namely $\delta^{\prime}(x)$, $\delta^{\prime\prime}(x)\ldots$, which are even more discontinuous and less “proper” than $\delta(x)$ itself.
A few elementary properties of these functions will now be given so as not to interrupt the argument later. We can obviously take $\delta(x)=\delta(-x)$, $\delta^{\prime}(x)=-\delta^{\prime}(-x)$, etc. $\ldots$. If $f(x)$ is any regular function $\ldots$, we have
$$\int_{-\infty}^{\infty}\!f(x)\,\delta(a-x)dx=f(a),$$
so that the operation of multiplying by $\delta(a-x)$ and integrating with respect to $x$ is equivalent to the operation of substituting $a$ for $x$.
What we now call Dirac’s delta function might well have been named after Ornstein, if the latter had interrupted his argument to introduce, with appropriate fanfare, a new function, spelling out its principal properties once and for all, and giving it a smart name.
VII The naming of the process and the process of naming
With the exception of two papers, where Ornstein was the only author, and a third, where he had more than one coauthor, each of his other papers in the period 1917–33 was a joint effort with a single coauthor: Burger, Uhlenbeck, van Wijk and Zernike. Though Uhlenbeck is the first author of the 1930 article, the random process investigated in the article is now called Ornstein-Uhlenbeck process. The person responsible for this reversal of order is, as far as I can see, Doob,${}^{33}$ who must have been under the impression that Ornstein was the first author (as can be verified by examining his list of references); since he refers, throughout his paper, to ‘Ornstein and Uhlenbeck’ it stands to reason that he introduced the name ‘O.U. process’; subsequent authors have chosen to replace the initials by complete names without reversing their order. Since singling out Uhlenbeck among all the coauthors named above seems (to me) unfair, and including them all is clearly impractical, the choice ‘Ornstein process’ would have been more equitable, but the name ‘Ornstein-Uhlenbeck process is here to stay.
Ornstein died in 1941.${}^{34}$ Four years later, Ming Chen Wang and Uhlenbeck published their celebrated review article.${}^{35}$ By now, they were unequivocal in calling Eq. (15) the Langevin equation; they also used the appellation ‘the Dirac delta function’ and used it for specifying the temporal behavior of the autocorrelation of $A$ by stating the relation
$$\overline{A(t_{1})A(t_{2})}=2\alpha\beta\delta(t_{1}-t_{2}).$$
VIII Acknowledgment
I am very grateful to O. L. J. Gijzeman for his friendly help in acquiring copies of some works by Ornstein and van der Waals, and a copy of ref. 4.
1.
See, for example, E. Nelson, “Dynamical Theories of Brownian Motion, (Princeton University Press, 2001), 2nd. edition, p. 45. Posted at the web at: http://www.math.princeton.edu/$\sim$nelson/books/
bmotion.pdf
2.
P. Langevin, “Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530-533 (1908)
3.
D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la theorie du mouvement brownien, C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079-1081 (1997). I have followed the translation presented in this article—apart from replacing “M.” (the French abbreviation for Monsieur) by “Mr.” (the corresponding abbreviation in English for Mister).
4.
L. S. Ornstein, A Survey of his Work from 1908 to 1933, (Utrecht, 1933). This volume contains a complete list of his works (and that of his close associates) published up to 1932 and a few in 1933; each entry contains not only the title of the article but also a brief comment on the contents. For articles published in the Proceedings of the Royal Academy of Amsterdam, references are given to the original as well as the English translation, and the year for the translated work is stated to be the same as the original, and I have followed this practice (which differs from that followed by Ornstein himself). Articles on Brownian motion are listed in Section a (a26, a27, a30, a32, a33, a39, a40, a43); among the contributions to Brownian motion which appeared after the publication of this book, only ref. 5 is of interest in the present context.
5.
L. S. Ornstein and W. R. van Wijk, “On the derivation of distribution functions in problems of Brownian motion,” Physica 1, 235–254 (1933).
6.
L. S. Ornstein, “On the Brownian motion,” Proc. Royal Acad. Amsterdam 21, 96–108 (1917).
7.
L. S. Ornstein and H. C. Burger, “On the theory of the Brownian motion”, Proc. Royal Acad. Amst. 21, 922–931 (1919).
8.
L. S. Ornstein, “Zur Theorie der Brownschen Bewegung für Systeme, worin mehrere Temperaturen vorkommen,” Z. Phys. 41, 848–856 (1927).
9.
L. S. Ornstein, H. C. Burger, J. Taylor and W. Clarkson, “The Brownian movement of a galvanometer coil and the influence of temperature of the outer circuit,” Proc. Royal Soc. A. 115, 391–406 (1927).
10.
G. L. de Haas-Lorentz, Die Brownsche Bewegung und einige verwandte Erscheinungen (Viewig, Braunschweig, 1913).
11.
A. Einstein, Investigations on the theory of the Brownian movement (Dover, New York, 1956).
12.
M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung der Suspsnsionen,” Ann. Phys. 21, 756–780 (1906).
13.
A. Einstein, “Theoretische bemerkungen über die Brownsche Bewegung,” Z. Elektrochem. 13, 41–42 (1907).
14.
R. Calusius, “On a mechanical theorem applicable to heat,” Phil. Mag. 40, 122–127 (1870).
15.
M. D. Haw, “Colloidal suspensions, Brownian motion, molecular reality: a short history,” J. Phys. Condens. Matter 14, 7769–7779 (2002).
16.
R. P. Feynman, Lectures on Physics (Addison-Wesley, Reading, Massachusetts, 1963), Vol. 1, Chap. 41.
17.
F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965) p. 565.
18.
C. V. Heer, Lectures on Physics (Academic, New York, 1972) p. 417.
19.
W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron, The Langevin Equation (World Scientific, Singapore, 2004) p. 14.
20.
J. D. van der Waals Jr., “On the Theory of the Brownian movement, ”Proc. Royal Acad. Amsterdam 20, 1254–1271 (1918). The author of this article was the son of J. D. van der Waals, well-known for his equation of state and the winner of the 1910 Nobel Prize in Physics.
21.
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases (C.U.P., Cambridge, 1970) p. 73.
22.
J. C. Maxwell, “Illustrations of the dynamical theory of gases,” Phil. Mag. 19, 19–32 (1860).
23.
J. C. Maxwell, “On the dynamical theory of gases,” Phil. Trans. Royal Soc. 157, 49–88 (1867).
24.
A. Manoliu and C. Kittel, “Correlation in the Langevin theory of Brownian motion,” Am. J. Phys. 47, 678 (1979).
25.
J. Thewlis (Ed.), Encyclopaedic Dictionary of Physics (Pergamon, Oxford, 1961) Vol. 1, pp. 511–512.
26.
G. E. Uhlenbeck and L. S. Ornstein, “On the theory of the Brownian motion,” Phys. Rev. 36, 823–841 (1930).
27.
R. Kubo, “Brownian motion and nonequilibrium statistical mechanics,” Science 233, 330–334 (1986).
28.
A. Einstein and L. Hopf, “Statistische Untersuchung der Bewegung eines Resonators in einem Strahlungsfeld,” Ann. Phys. 33, 1105–1115 (1910).
29.
L. Gunther and D. L.Weaver, “Monte Carlo simulation of Brownian motion with viscous drag,” Am. J. Phys. 46, 543–545 (1978).
30.
J. D. Doll and D. L. Freeman, “Randomly exact methods,”Science 234, 1356–1360 (1986).
31.
P. A. M. Dirac, “The physical interpretation of the quantum dynamics,” Proc. Royal Soc. A, 113, 621–641 (1927).
32.
L. S. Ornstein and G. E. Uhlenbeck, “Some kinetic problems regarding the motion of neutrons through paraffine,” Physica 4, 478–486 (1937).
33.
J. L. Doob, “The Brownian movement and stochastic equations,” Annals of Mathematics, 43, 351–369 (1942).
34.
R. C. Mason, “Leonard Salomon Ornstein,” Science 102, 638–639 (1945).
35.
Ming Chen Wang and G. E. Uhlenbeck, “On the theory of the Brownian motion II,” Rev. Mod. Phys. 17, 323–342 (1945). |
Second order intuitionistic propositional logic
of the real line is decidable111Partially supported by MNiSW Grant N N206 355836
Konrad Zdanowski
University of Cardinal Stefan Wyszyński, Warsaw
()
Abstract
It is known that the set of tautologies of second order intuitionistic
propositional logic, $\mathrm{IPC}2$, is undecidable.
Here, we prove that the sets of formulas of $\mathrm{IPC}2$ which are
true in the algebra of open subsets of reals or rationals
are decidable.
1 Basic definitions
We investigate the second order intuitionistic propositional logic, denoted as $\mathrm{IPC}2$
(for a detailed treatment of this logic we refer to the book [SU]).
The set of formulas
of this logic is the same as in the classical case. We have standard propositional
connectives, universal and existential quantifiers and the set of propositions. Firstly, we present
the set of axioms and rules in the Gentzen style. Then, we define various semantics for $\mathrm{IPC}2$
and describe their status with respect to completeness and decidability of the tautology problem.
Below, $\Gamma$ is a multiset of formulas of $\mathrm{IPC}2$, $\psi$,
$\varphi$ and $\rho$ are formulas of $\mathrm{IPC}2$ and $p$ is a proposition.
The letters I and E in names of rules stand for the “introduction”
and “elimination”, respectively.
1.
Axioms:
$$\Gamma,\psi\vdash\psi.$$
2.
Rules for conjunction:
$$\displaystyle\frac{\Gamma\vdash\psi\ \ ||\ \ \Gamma\vdash\varphi}{\Gamma\vdash%
\psi\land\varphi}\,\textrm{($\land$I)},$$
$$\displaystyle\frac{\Gamma\vdash\varphi\land\psi}{\Gamma\vdash\varphi},\ \ \ %
\frac{\Gamma\vdash\varphi\land\psi}{\Gamma\vdash\psi}\,\textrm{($\land$E)}.$$
3.
Rules for disjunction:
$$\displaystyle\frac{\Gamma\vdash\varphi}{\Gamma\vdash\varphi\lor\psi},\ \ \ %
\frac{\Gamma\vdash\psi}{\Gamma\vdash\varphi\lor\psi}\,\textrm{($\lor$I)},$$
$$\displaystyle\frac{\Gamma,\varphi\vdash\rho\ \ ||\ \ \Gamma,\psi\vdash\rho\ \ %
||\ \ \Gamma\vdash\varphi\lor\psi}{\Gamma\vdash\rho}\,\textrm{($\lor$E).}$$
4.
Rules for implication:
$$\displaystyle\frac{\Gamma,\varphi\vdash\psi}{\Gamma\vdash\varphi\Rightarrow%
\psi}\,\textrm{($\Rightarrow$I)},$$
$$\displaystyle\frac{\Gamma\vdash\varphi\Rightarrow\psi\ \ ||\ \ \Gamma\vdash%
\varphi}{\Gamma\vdash\psi}\,\textrm{($\Rightarrow$E).}$$
5.
The rule ex falso quodlibet:
$$\displaystyle\frac{\Gamma\vdash\bot}{\Gamma\vdash\varphi}\,\textrm{($\bot$E)}.$$
6.
Rules for quantifiers:
$$\displaystyle\frac{\Gamma\vdash\varphi}{\Gamma\vdash\forall p\,\varphi}\,%
\textrm{($\forall$I)},$$
$$\displaystyle\frac{\Gamma\vdash\forall p\,\varphi}{\Gamma\vdash\varphi[p:=\psi%
]}\,\textrm{($\forall$E)},$$
$$\displaystyle\frac{\Gamma\vdash\varphi[p:=\psi]}{\Gamma\vdash\exists p\,%
\varphi}\,\textrm{($\exists$I)},$$
$$\displaystyle\frac{\Gamma\vdash\exists p\,\varphi\ \ ||\ \ \Gamma,\varphi%
\vdash\psi}{\Gamma\vdash\psi}\,\textrm{($\exists$E)}.$$
In rules ($\forall$I) and ($\exists$E) we have a restriction that the variable
$p$ should not occur as a free variable of $\Gamma$ or $\psi$
We denote the above calculus with $\mathrm{IPC}2$ as well.
Later, we consider sets of tautologies of $\mathrm{IPC}2$
for various kinds of algebraic semantics.
However, these sets will contain the set of theorems of the above
calculus with one exception of $\mathrm{IPC}2^{-}$ (defined later).
By $\mathrm{IPC}$ we denote intuitionistic propositional logic (without quantification)
and the corresponding calculus defined by removing from the above one the rules
for quantifiers.
In the above formulation we have no special rules and even no symbol for negation.
This is so, because we do not treat negation as a primitive but rather we define
$\neg\varphi$ as $\varphi\Rightarrow\bot$. We define also $\varphi\Leftrightarrow\psi$ as
$(\varphi\Rightarrow\psi)\land(\psi\Rightarrow\varphi)$. It is known that in the given calculus one can
define from $\forall$ and $\Rightarrow$ all other connectives and quantifiers. So, the following
formulas are provable,
$$\displaystyle\bot$$
$$\displaystyle\Leftrightarrow\forall p\,p$$
$$\displaystyle\varphi\lor\psi$$
$$\displaystyle\Leftrightarrow\forall p((\varphi\Rightarrow p)\Rightarrow((\psi%
\Rightarrow p)\Rightarrow p)),$$
$$\displaystyle\varphi\land\psi$$
$$\displaystyle\Leftrightarrow\forall p((\varphi\Rightarrow(\psi\Rightarrow p))%
\Rightarrow p),$$
$$\displaystyle\exists q\varphi(q)$$
$$\displaystyle\Leftrightarrow\forall p(\forall q(\varphi(q)\Rightarrow p)%
\Rightarrow p).$$
In the proof of our main theorem we use this fact implicitly by restricting our
translation to formulas with $\forall$ and $\Rightarrow$, only. Let us mention that
we do need $\forall$ quantifier to define other connectives, see e.g. [SU] or [SU10].
It was proved by Löb (see [L76]) that the above calculus has an undecidable provability problem.
Even the $\forall$–free fragment of this logic is undecidable (see [SU10]).
Before Löb’s article, Gabbay in [G74] considered
$\mathrm{IPC}2$ extended by a scheme called the axiom of constant domains ($\mathrm{CD}$),
$$\forall p(\varphi\lor\psi(p))\Rightarrow(\varphi\lor\forall p\psi(p)),$$
where $p$ is not free in $\varphi$. In the context of first order
intuitionistic logic this scheme was introduced by Grzegorczyk in [G64] and it is also
called Grzegorczyk scheme (one should not confuse this with Grzegorczyk axiom
in modal logic). Gabbay showed undecidability of ($\mathrm{IPC}2+\mathrm{CD}$), see [G74],
but his proof was later corrected by Sobolev in [S77]. Gabbay claimed that his result
generalizes to the case without $\mathrm{CD}$. According to Gabbay, the generalization
could be obtained by the finite axiomatizability of $\mathrm{CD}$ over $\mathrm{IPC}2$.
Nevertheless, it seems that there is no obvious method to define such an axiomatization.
Sobolev also considered logics without full
comprehension axioms which correspond in our setting to rules $\forall E$ and $\exists I$.
In the restricted versions of both rules we demand that the formula $\psi$ is atomic.
Let us call
this logic $\mathrm{IPC}2^{-}$. Sobolev showed then that any logic between $\mathrm{IPC}2^{-}$ and ($\mathrm{IPC}2+\mathrm{CD}$)
is undecidable.
Now, we will discuss various semantics for $\mathrm{IPC}2$. There are two kinds
of popular semantics for intuitionistic logics, one constructed using Kripke models
and the other one based on Heyting algebras. In the Kripkean approach semantics
is given by Kripke frames $(C,\leq,\{D_{c}\colon c\in C\})$, where
each $D_{c}$ is a subset of $\mathcal{P}(C)$ and all $X\in D_{c}$ are upward closed
with respect to $\leq$. Moreover, for each $c\leq c^{\prime}\in C$ we have $D_{c}\subseteq D_{c^{\prime}}$.
The set $C$ is the set of possible worlds and, for each $c\in C$, the set $D_{c}$ is the
range of second order quantification in the world $c$. Then, the value of a given formula $\varphi$
as the set of possible worlds at which $\varphi$ is true
may be given by a usual inductive definition. In order to satisfy unrestricted versions
of rules $\forall E$ and $\exists I$ one needs to require that for any formula $\varphi$, any $c\in C$
and for any valuation $v$ of propositions into $D_{c}$, if $X_{\varphi,v}$
is the set of worlds greater or equal $c$ at which
$\varphi$ is satisfied with $v$, then $X_{\varphi,v}\in D_{c}$, for each $c\in C$.
This class of models gives a sound and complete semantics for $\mathrm{IPC}2$. If we drop
the last condition concerning rules $\forall E$ and $\exists I$ then we get
a sound and complete semantics for $\mathrm{IPC}2^{-}$.
The logics with $\mathrm{CD}$ are complete w.r.t. “constant domains semantics” where all $D_{c}$’s are
equal, see [G74]. Then, the range for quantifiers is the same in all possible worlds.
Let us stress that $D_{c}$’s may not contain all upward closed subsets of $C$.
Adding the condition that $D_{c}$’s contain all upward closed subsets of $C$ gives us
the, so called, principal semantics.
The set of tautologies of principal Kripkean semantics is recursively isomorphic
to the classical second order logic as proved by
Kremer, see [K97a].
Now, we turn to algebraic semantics. In the classical case the sound and complete semantics
for second order propositional logic is given by boolean algebras. In the intuitionistic case the sound
and complete semantics for $\mathrm{IPC}$ is given by Heyting algebras.
A Heyting algebra $(H,\leq,\cap,\cup,\rightarrow,0,1)$ is a distributive lattice with top and bottom elements augmented with the pseudo-complement operation $\rightarrow$ which interprets implication. It is required that the following is well defined for $a,b\in H$,
$$a\rightarrow b=\max\{c\in H\colon c\cap a\leq b\}.$$
In the case of $\mathrm{IPC}$, Heyting algebras give the sound and
complete semantics, e.g., by constructing a Heyting algebra from a Kripke model where all upward closed sets from the Kripke model form the universe of the algebra.
In the algebraic semantics we have two ways in which we can interpret quantification.
In the, so called, principal semantics quantifiers range over all elements of a given algebra
and their meaning is given by infinite joins and meets, which have to exist. In the non-principal
semantics quantifiers range over a distinguished subset of an algebra.
In the second order case the relation between Kripkean and algebraic semantics is not
as straightforward as in the quantifier free case. An obvious way to translate a Kripke frame $(C,\leq,\dots)$
into a special kind of Heyting algebra, a topology, would be to define
a topology of all upward closed subsets of $C$.
However, not all upward closed subsets of a Kripke model
are within its domain of quantification. Moreover, for different possible worlds $c\in C$
we may have different domains $D_{c}$.
Only if we consider principal Kripkean semantics then for a given
frame $(C,\leq,\dots)$ we may define a topology
of all upward closed subsets of $C$ (see, e.g., Exercise 2.8 in [SU]).
For such topology the satisfaction relation is preserved
from the one of the Kripke frame.
Lately, Philip Kramer in a personal communication
expressed his strong confidence that the complexity of the set of tautologies
for principal algebraic semantics is as hard as in the case of principal Kripkean
semantics. Kramer made his statement in his article [K97b] (p. 296)
claiming that a nontrivial extension of methods from [K97a]
would be needed. In a non-principal case a recent article by Kramer, [K13], establishes
its completeness w.r.t. $\mathrm{IPC}2$.
A special case of Heyting algebras is given by topologies. Let us describe a satisfiability relation for $\mathrm{IPC}2$
and topological principal semantics.
Let $\mathcal{T}=(T,\mathcal{O}(T))$ be an arbitrary topology, where $\mathcal{O}(T)$ is the set of open subsets of $T$,
and let $v\colon\mathrm{PROP}\longrightarrow\mathcal{O}(T)$ be a valuation
from the set of propositions. Then, we may define a value of a given formula of $\mathrm{IPC}2$ in $\mathcal{T}$ under
$v$, denoted as $\llbracket\varphi\rrbracket^{\mathcal{T}}_{v}$, by recursion on the complexity of the formula:
1.
$\llbracket\bot\rrbracket^{\mathcal{T}}_{v}=0$,
2.
$\llbracket p\rrbracket^{\mathcal{T}}_{v}=v(p)$,
3.
$\llbracket\psi\wedge\gamma\rrbracket^{\mathcal{T}}_{v}=\llbracket\psi%
\rrbracket^{\mathcal{T}}_{v}\cap\llbracket\gamma\rrbracket^{\mathcal{T}}_{v}$,
4.
$\llbracket\psi\vee\gamma\rrbracket^{\mathcal{T}}_{v}=\llbracket\psi\rrbracket^%
{\mathcal{T}}_{v}\cup\llbracket\gamma\rrbracket^{\mathcal{T}}_{v}$,
5.
$\llbracket\psi\Rightarrow\gamma\rrbracket^{\mathcal{T}}_{v}=\rm{int}((T%
\setminus\llbracket\psi\rrbracket^{\mathcal{T}}_{v})\cup\llbracket\gamma%
\rrbracket^{\mathcal{T}}_{v})$,
6.
$\llbracket\exists p\psi\rrbracket^{\mathcal{T}}_{v}=\bigcup_{a\in\mathcal{O}(T%
)}\llbracket\psi\rrbracket^{\mathcal{T}}_{v(p\mapsto a)})$,
7.
$\llbracket\forall p\psi\rrbracket^{\mathcal{T}}_{v}=\rm{int}(\bigcap_{a\in%
\mathcal{O}(T)}\llbracket\psi\rrbracket^{\mathcal{T}}_{v(p\mapsto a)})$.
We say that $\varphi$ is true in $\mathcal{T}$ under $v$ if $\llbracket\varphi\rrbracket^{\mathcal{T}}_{v}=T$.
It is known that the topologies of open subsets of ${{\mathbb{R}}}$ or $\mathbb{Q}$ form a sound and complete
semantics for $\mathrm{IPC}$. It can be easily shown that it is not the case for $\mathrm{IPC}2$.
Indeed, for each two $r,r^{\prime}\in{{\mathbb{R}}}$ there is a homeomorphism of ${{\mathbb{R}}}$
into itself mapping $r$ to $r^{\prime}$. It follows that if we have a sentence $\psi$
of $\mathrm{IPC}2$ then either
$\llbracket\psi\rrbracket^{{{\mathbb{R}}}}_{v}={{\mathbb{R}}}$ or $\llbracket\psi\rrbracket^{{{\mathbb{R}}}}_{v}=\emptyset$
(note that the value of $\llbracket\psi\rrbracket^{{{\mathbb{R}}}}_{v}$ does not depend on $v$, for a sentence $\psi$).
Therefore, for each sentence $\psi$ of $\mathrm{IPC}2$, $\psi\lor\neg\psi$ is true in ${{\mathbb{R}}}$.
Of course, some sentences of that form are not provable in $\mathrm{IPC}2$.
One can check that if we take an arbitrary quantifier free formula $\varphi(p_{1},\dots,p_{n})$ which is a classical
tautology and is not an intuitionistic tautology then for $\psi:=\forall p_{1}\dots\forall p_{n}\varphi$, the formula
$\psi\lor\neg\psi$ is not provable in $\mathrm{IPC}2$.
The very same argument works also for $\mathbb{Q}$.
On the other hand the sentence $\neg\forall p(p\lor\neg p)$ is true in ${{\mathbb{R}}}$ (and in $\mathbb{Q}$) though it
is not valid intuitionistically and moreover it is a classical contrtautology.
It follows that the $\mathrm{IPC}2$ theory of ${{\mathbb{R}}}$ or $\mathbb{Q}$ is not a subset of
classical tautologies.
Despite the above facts, the topologies of the real and rational lines are natural semantics
for intuitionistic logics. Firstly, it is natural to ask about second order propositional
theory of these models which are kind of standard models for $\mathrm{IPC}$.
Secondly, one can see $\mathrm{IPC}2$ over ${{\mathbb{R}}}$ or $\mathbb{Q}$ as a language capable of expressing some
properties of these topologies. Thus, we may ask about the decidability of topological theories
of ${{\mathbb{R}}}$ or $\mathbb{Q}$ expressible in $\mathrm{IPC}2$.
We show here, that $\mathrm{IPC}2$ tautologies of principal semantics of reals or rationals
are easier than in the general case, namely decidable.
The method used in the proof is an interpretation of these theories into the monadic theory
of infinite binary tree, proved to be decidable by Rabin’s result (for details
on this subject we refer to [GWT02]).
Let $T^{\omega}=\left\{0,1\right\}^{*}$ be the set of finite binary sequences.
The infinite binary tree is a structure $\mathcal{T}^{\omega}=(T^{\omega},s_{0},s_{1},\leq)$, where
$s_{0}(u)=u0$ and $s_{1}(u)=u1$ and $u\leq v$ when $u$ is an initial segment of $v$.
A path in $\mathcal{T}^{\omega}$ is an infinite set $P\subseteq T^{\omega}$ such that $P$ is closed
on initial segments and is linearly ordered by $\leq$. The empty sequence is denoted by $\varepsilon$.
The monadic second order logic is an extension of first order logic by second order quantifiers ranging
over subsets of a given universe. Rabin’s theorem states that the monadic second order theory
of $\mathcal{T}^{\omega}$ is decidable. We will denote this theory by $\mathrm{S2S}$.
It should be noted that the complexity of $\mathrm{S2S}$ is non-elementary.
It became a standard method to show decidability of various problems by reducing them
to $\mathrm{S2S}$. In the next section, we exhibit a reduction for theories
of $\mathrm{IPC}2$ of reals and rationals.
2 Decidability on ${{\mathbb{R}}}$ and $\mathbb{Q}$
2.1 Interpretation in S2S
We give interpretations of $\mathrm{IPC}2$ theories of $\mathcal{O}({{\mathbb{R}}})$ and $\mathcal{O}(\mathbb{Q})$ in $\mathrm{S2S}$.
A similar though a bit simpler
interpretation was used in [Z04] showing decidability of $\mathrm{IPC}2$ (and S4 with
propositional quantification) on trees of height and arity $\leq\omega$ (in the principal
semantics).
Theorem 1
The $\mathrm{IPC}2$ theories of the open subsets of reals and the open subsets
of rational numbers are interpretable in $\mathrm{S2S}$.
Proof.
Let us recall that we write $s_{0}(x)$ and $s_{1}(x)$ to denote
respectively the left and the right successors, and $x\leq y$ to denote
that $x$ is on the path from the root of the tree to $y$.
Firstly, we give an interpretation of the $\mathrm{IPC}2$ theory of reals.
Instead of thinking about ${{\mathbb{R}}}$ we take an open interval $(0,1)$ which has the same topological
properties. In particular any topological operation is taken in $(0,1)$, e.g., the closure of
$(0,1/3)$ is $(0,1/3]$.
Each real number $r\in(0,1)$ may be seen as its binary representation $0.a_{0}a_{1}a_{2}\dots$,
where $a_{i}\in\left\{0,1\right\}$ and $r=\Sigma_{i=0}^{\infty}a_{i}2^{-i-1}$.
Such representations can be interpreted as infinite paths in $T^{\omega}$.
A binary sequence $0.a_{0}a_{1}a_{2}\dots$ is therefore a path
$\left\{\varepsilon,s_{a_{0}}(\varepsilon),s_{a_{1}}s_{a_{0}}(\varepsilon),%
\dots\right\}$.
In what follows we will use both representations: infinite $\left\{0,1\right\}$–sequences
$a_{0}a_{1}a_{2}\dots$ (without the leading “$0$.”)
and paths in $T^{\omega}$.
In order to have the unique representation of each real we exclude
sequences which have only finitely many zeros.
We can define on such infinite paths in $T^{\omega}$
the topology inherited from $(0,1)$.
What we need is to assure that formulas of $\mathrm{IPC}2$
can be effectively translated into formulas of $\mathrm{S2S}$ such that
they will be equivalent modulo the translation of open sets of both topologies.
An infinite binary path represents a real from $(0,1)$ if and only if it is not
of the form $0^{\omega}$ or $u1^{\omega}$ for some $u\in\left\{0,1\right\}^{*}$.
The set of such paths is obviously definable in $\mathrm{S2S}$.
A formula $\textrm{Path}(X)$ stating that $X$ is an infinite path is just
$$(X(x)\land X(y)\Rightarrow(x\leq y\lor y\leq x))\land$$
$$\forall x(X(x)\Rightarrow(X(s_{0}(x))\lor X(s_{1}(x))))\land X(\varepsilon).$$
Then a formula $U(X)$ defining the set of paths which represent some real
can be written as:
$$\textrm{Path}(X)\land\forall x\big{(}X(x)\Rightarrow\exists z\geq x\;X(s_{0}(z%
))\big{)}\land\exists x\,X(s_{1}(x)).$$
The first conjunct of the above formula states that $X$ is an infinite path, the second one
states that there are infinitely many $0$’s in $X$ and the third conjunct states that there is at least one $1$ in $X$.
For a set $X$ such that $\textrm{Path}(X)$, let $r(X)$ be a real represented by $X$.
We need to represent not only real numbers but also
subsets of $(0,1)$. For a subset $S\subseteq T^{\omega}$, by
$R(S)$ we denote a set of reals such that their corresponding paths are contained
in $S$,
$$R(S)=\left\{r(X)\colon U(X)\land X\subseteq S\right\}.$$
We will represent open subsets of $(0,1)$ by their closed complements. For a set $S\subseteq T^{\omega}$,
$R(S)$ is closed in $(0,1)$ if the following formula, $\mathrm{Closed}(S)$, holds:
$$\displaystyle\forall X\subseteq S\{[\textrm{Path}(X)\land\exists y(X(s_{0}(y))%
\land\forall z\geq s_{0}(y)\,\neg X(s_{0}(z))))]\Rightarrow$$
$$\displaystyle\exists Y\subseteq S\exists y[\textrm{Path}(Y)\land Y(y)\land X(y)\land$$
$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ X(s_{0}(y))\land\forall z\geq s_{0}(y%
)(X(z)\Rightarrow X(s_{1}(z)))\land$$
$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ Y(s_{1}(y))\land\forall z\geq s_{1}(y%
)(Y(z)\Rightarrow Y(s_{0}(z)))]\}.$$
The formula above states that if a path $X$ of the form $u01^{\omega}$ is a subset of $S$ then there
is a path $Y\subseteq S$ of the form $u10^{\omega}$. The condition is necessary because we do not allow paths of the form $u1^{\omega}$ to represent reals (and satisfy the predicate $U(X)$).
Thus, if we have such a path
$X\subseteq S$, then we require that $S$ contains also a path $Y$ such that $r(X)=r(Y)$ and $U(Y)$.
Otherwise, it could happen that for some sequence of sets
$\{X_{i}\subseteq S\colon i\in\omega\land U(X_{i})\}$ such that $\lim_{i\rightarrow\infty}r(X_{i})=r(X)$
(in fact all $r(X_{i})$ may be less than $r(X)$) there is no $Y\subseteq S$ such that $U(Y)$ and $r(Y)=r(X)$.
We use two facts about sets satisfying formula $\mathrm{Closed}(S)$.
Claim 2
1.
For any $C$ closed in $(0,1)$ there exists $S\subseteq T^{\omega}$ such that
$\mathrm{Closed}(S)$ and $R(S)=C$.
2.
For any $S\subseteq T^{\omega}$ such that $\mathrm{Closed}(S)$, $R(S)$ is closed in $(0,1)$.
Proof.
To show $1$ it is enough to take $S=\bigcup\{X\subseteq T^{\omega}\colon U(X)\land r(X)\in C\}$.
Then, $\mathrm{Closed}(S)$ holds. Indeed, let a path $u01^{\omega}$ be a subset
of $S$, then there is a sequence of reals $r_{i}\in C$, for $i\in\omega$ such that $r_{i}=r(u01^{i}v_{i})$,
for infinite binary words $v_{i}$ where all paths $u01^{i}v_{i}$
are subsets of $C$.
The sequence $r_{i}$ converges to a real $r(u10^{\omega})$
and since $C$ is closed $r(u01^{\omega})\in C$ and so $u10^{\omega}$ is a subset of $S$.
Obviously, $C\subseteq R(S)$. To prove the converse let us assume that $r=r(X)\in R(S)$
for some $X\subseteq R(S)$ such that $U(X)$. Let us assume towards a contradiction that $r\not\in C$.
We have two cases to consider. The first one when $X$ is of the form $u10^{\omega}$ and
the second, complementary case. We consider the latter. Then, for each $i\in\omega$
there is $r_{i}\in C$ and $X_{i}\subseteq S$ such that $U(X_{i})$, $r_{i}=r(X_{i})$ and $X$ has a common initial segment
with $X_{i}$ of length $i$. This is so because any element of $S$ belongs to a path representing
a real from $C$. Now, $r=\lim_{i\rightarrow\infty}r_{i}$ and, since $C$ is closed, $r\in C$, a contradiction.
As for the case of $X=u10^{\omega}$ we repeat the same reasoning either with $X$ or
with a path $u01^{\omega}$. In both cases we get the same contradiction $r(X)=r(u01^{\omega})\in C$.
To show $2$ let $r_{i}\in R(S)$ be a sequence of reals converging to some $r\in(0,1)$.
Let $P_{i}\subseteq S$ be such that $U(P_{i})$ and $r_{i}=r(P_{i})$ and let $P\subseteq T^{\omega}$
be such that $U(P)$ and $r=r(U)$.
If $P_{i}$ are of the form $u01^{n_{i}}v_{i}$, for some strictly increasing sequence $n_{i}$, then $P$
is a path $u10^{\omega}$ and,
by $\mathrm{Closed}(S)$, $P\subseteq S$. It follows that $r\in R(S)$.
Otherwise, $P$ is a path with infinitely many $1$’s and $r=\sum_{i\in\omega}2^{-n_{i}}$,
for some strictly increasing sequence $n_{i}$. Now, if $|r-r_{i}|<2^{-n_{i}-2}$ then $P$ and $P_{i}$
have a common initial segment of length $n_{i}$.
We obtain that
$P\subseteq\bigcup_{i\in\omega}P_{i}\subseteq S$ and, therefore, $r\in R(S)$.
$\Box$
The above claim shows that sets of the form $\mathrm{Closed}(S)$ are a good representation
of closed subsets of $(0,1)$. We can write an $\mathrm{S2S}$ formula $\mathrm{clBelong}(X,S)$ expressing that a real $r(X)$ belongs to a closed set $R(S)$.
It has the form
$$U(X)\land\mathrm{Closed}(S)\land X\subseteq S.$$
Similarly, we can express that a closed set $R(S)$ is included in a set $R(T)$ with
$$\forall X(\mathrm{clBelong}(X,S)\Rightarrow\mathrm{clBelong}(X,T)).$$
Let us state a useful lemma about definability in $\mathrm{S2S}$.
Lemma 3
For each $\mathrm{S2S}$ formula $\varphi(X)$ with $X$ a free second order variable and possibly with some first
and second order parameters there exists a formula $\min_{\varphi}(X)$ such that
•
if there exists a unique minimal closed set $C\subseteq(0,1)$ such that
$\varphi(X)$ is true for any $X$ with $C=R(X)$,
then $\min_{\varphi}(X)$ is true only about sets $X$ satisfying $C=R(X)$,
•
$\min_{\varphi}(X)$ if false for any set $X$, otherwise.
Proof.
We write a formula $\min_{\varphi}(X)$ as
$$\varphi(X)\land\mathrm{Closed}(X)\land$$
$$\forall Y((\mathrm{Closed}(Y)\land\varphi(Y))\Rightarrow\forall Z((U(Z)\land%
\mathrm{clBelong}(Z,X))\Rightarrow\mathrm{clBelong}(Z,Y))).$$
$\Box$
Now,we define an inductive translation of an $\mathrm{IPC}2$ formula $\varphi(p_{1},\dots,p_{n})$
into an $\mathrm{S2S}$ formula $\varphi^{*}(T,T_{1},\dots,T_{n})$. We represent open sets by its closed complements.
We require the following property:
for all open subsets $R,R_{1},\dots,R_{n}$ of $(0,1)$ and
all $X,X_{1},\dots,X_{n}\subseteq T^{\omega}$ such that $\mathrm{Closed}(X)$, $R=(0,1)\setminus R(X)$
and $\mathrm{Closed}(X_{i})$, $R_{i}=(0,1)\setminus R(X_{i})$, for $i\leq n$, we have the equivalence,
$$[\varphi]^{(0,1)}_{\left\{p_{i}\mapsto R_{i}\right\}}=R\textrm{\ \ if and only%
if\ \ }$$
$$(\left\{0,1\right\}^{*},s_{0},s_{1},\leq)\models\varphi^{*}[X,X_{1},\dots,X_{n%
}].$$
If $\varphi=\bot$, then $\varphi^{*}=\forall xT(x)$ (note that if $X=T^{\omega}$
then $R(X)=(0,1)$ and we want the complement of $X$ to be the empty set).
If $\varphi=p_{i}$, then
$\varphi^{*}=\forall Y(U(Y)\Rightarrow(\mathrm{clBelong}(Y,T)\Leftrightarrow%
\mathrm{clBelong}(Y,T_{i})))$.
For $\varphi=(\psi_{1}\Rightarrow\psi_{2})$, we have
$$\displaystyle[\varphi]^{(0,1)}_{v}$$
$$\displaystyle=$$
$$\displaystyle\rm{int}\left(((0,1)\setminus[\psi_{1}]^{(0,1)}_{v})\cup[\psi_{2}%
]^{(0,1)}_{v}\right)$$
$$\displaystyle=$$
$$\displaystyle\max\{O\subseteq(0,1)\colon O\textrm{ is open }\land O\subseteq((%
0,1)\setminus[\psi_{1}]^{(0,1)}_{v})\cup[\psi_{2}]^{(0,1)}_{v}\}$$
$$\displaystyle=$$
$$\displaystyle(0,1)\setminus\min\{C\subseteq(0,1)\colon C\textrm{ is closed }\land$$
$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ([%
\psi_{1}]^{(0,1)}_{v}\cap((0,1)\setminus[\psi_{2}]^{(0,1)}_{v}))\subseteq C\}.$$
By properties of the topology the above maximum and minimum exist. We need to write
a formula $\psi^{*}(T,T_{1},\dots,T_{n})$ such that with parameters $X_{1},\dots,X_{n}$ substituted for
$T_{1},\dots,T_{n}$, respectively, it will be true only about the unique $T$ with
$$R(T)=C_{0}=\min\{C\subseteq(0,1)\colon C\textrm{ is closed }\land([\psi_{1}]^{%
(0,1)}_{v}\cap((0,1)\setminus[\psi_{2}]^{(0,1)}_{v}))\subseteq C\}.$$
Let $\widehat{\varphi}(T,T_{1},\dots,T_{n})$ be a formula
$$\mathrm{Closed}(T)\land\psi^{*}_{1}(T^{*}_{1},T_{1},\dots,T_{n})\land\psi^{*}_%
{2}(T^{*}_{2},T_{1},\dots,T_{n})\land$$
$$\forall X((U(X)\land\neg\mathrm{clBelong}(X,T^{*}_{1})\land\mathrm{clBelong}(X%
,T^{*}_{2}))\Rightarrow\mathrm{clBelong}(X,T)).$$
The formula above expresses the definitional property of $C_{0}$ in $\mathrm{S2S}$ and the topology of $T^{\omega}$
inherited from $(0,1)$. Now, as $\varphi^{*}(T,T_{1},\dots,T_{n})$ we take
the formula $\min_{\widehat{\varphi}(T,\dots)}(T,T_{1},\dots,T_{n})$ from Lemma 3 where the minimum is taken over $T$. The formula $\varphi^{*}(T,\dots)$ is true only about the set $C_{0}$ what
proves the inductive thesis for $\varphi$.
If $\varphi=\forall p_{n}\psi(p_{n})$ then
$$\displaystyle[\varphi]^{(0,1)}_{v}$$
$$\displaystyle=$$
$$\displaystyle\rm{int}(\bigcap_{O\textrm{ is open }}[\psi]^{(0,1)}_{v(p\mapsto O%
)})$$
$$\displaystyle=$$
$$\displaystyle\max\{S\subseteq(0,1)\colon S\textrm{ is open and for all open $O%
\subseteq(0,1)$, }S\subseteq[\psi]^{(0,1)}_{v(p\mapsto O)}\}$$
$$\displaystyle=$$
$$\displaystyle(0,1)\setminus\min\{C\subseteq(0,1)\colon C\textrm{ is closed and }$$
$$\displaystyle\textrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %
for all open $O\subseteq(0,1)$, }(0,1)\setminus[\psi]^{(0,1)}_{v(p\mapsto O)}%
\subseteq C\}.$$
Since any topology is a complete Heyting algebras, the above sets are well defined.
The last expression can be translated to an $\mathrm{S2S}$ formula. Let $\widehat{\varphi}(T)$ be the following
formula
$$\displaystyle\mathrm{Closed}(T)\land$$
$$\displaystyle\forall W\forall T_{n}[(\mathrm{Closed}(W)\land\mathrm{Closed}(T_%
{n})\land\psi^{*}(W,T_{1},\dots,T_{n}))\Rightarrow$$
$$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall Y((U(Y)\land\mathrm%
{clBelong}(Y,W))\Rightarrow\mathrm{clBelong}(Y,T))].$$
Now, using Lemma 3, we can write $\varphi^{*}(T)$ as
$\min_{\widehat{\varphi}(T)}(T,T_{1},\dots,T_{n-1})$.
The above translation gives us decidability of $\mathrm{IPC}2$ on $(0,1)$ since for any
$\mathrm{IPC}2$ sentence $\varphi$,
$$\varphi$$ is true in $$(0,1)$$ if and only if
$$\forall T\forall X((\mathrm{Closed}(T)\land\varphi^{*}(T)\land U(X))%
\Rightarrow\neg\mathrm{clBelong}(X,T))$$
is true in $$\mathcal{T}^{\omega}$$.
A similar procedure gives also decidability of the $\mathrm{IPC}2$ theory of open subsets of rationals.
One needs to use the fact that the topology of dyadic rationals from $(0,1)$
is isomorphic to the topology of $\mathbb{Q}$.
Then, the set of paths which correspond to
these rationals is easily definable as paths of the form $u10^{\omega}$,
for some $u\in\left\{0,1\right\}^{*}$. Now, let $U_{\mathbb{Q}}(X)$ be a formula which defines
these paths. In order to obtain an interpretation of the $\mathrm{IPC}2$ theory of open subsets of rationals
one should restrict the universe of the given above interpretation to infinite paths
satisfying $U_{\mathbb{Q}}$. Syntactically, one should replace each occurrence of $U(X)$
with $U_{\mathbb{Q}}(X)$.
$\Box$
The above reduction gives a non elementary upper bound on the complexity of $\mathrm{IPC}2$
on reals or rationals. We conjecture that the complexity of these theories is in fact elementary.
Acknowledgments. I would like to thank Paweł Urzyczyn for urging me to write
this paper.
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Spontaneous traveling waves in oscillatory systems with cross diffusion
V. N. Biktashev
Department of Mathematical Sciences,
University of Liverpool, Liverpool L69 7ZL, UK
M. A. Tsyganov
Institute of Theoretical and Experimental Biophysics,
Pushchino, Moscow Region, 142290, Russia
(December 7, 2020)
Abstract
We identify a new type of pattern formation in spatially distributed
active systems. We simulate one-dimensional two-component systems
with predator-prey local interaction and pursuit-evasion taxis
between the components. In a sufficiently large domain, spatially
uniform oscillations in such systems are unstable with respect to
small perturbations. This instability, through a transient regime
appearing as spontanous focal sources, leads to establishment of
periodic traveling waves. The traveling waves regime is
established even if boundary conditions do not favor such solutions.
The stable wavelength are within a range bounded both from above and
from below, and this range does not coincide with instability bands
of the spatially uniform oscillations.
pacs: 87.10.+e, 02.90.+p
I Introduction
Dissipative structures, i.e. patterns in spatially extended
systems away from equilibrium have been intensively studied for many
decades. A very comprehensive review can be found in
Cross and Hohenberg (1993); results obtained since then would
probably require an even more extensive review. A very popular
class of mathematical models is the reaction-diffusion systems with
diagonal diffusion matrices. There have been numerous indications
that non-diagonal elements in diffusion matrices, i.e. cross-diffusion, can lead to new nontrivial effects not observed in
classical reaction-diffusion systems, e.g. quasi-solitons in
systems with excitable reaction part
Tsyganov et al. (2003, 2004); Biktashev et al. (2004); Tsyganov and Biktashev (2004); Biktashev and Tsyganov (2005). However oscillatory systems are more prevalent
than excitable and nontrivial effects of cross-diffusion in oscillatory systems have not been
studied yet. Here we consider an example where the
reaction part of the system is dissipative while the diffusion part
is not. We describe spontaneously generated periodic waves, and
identify the features of these waves that indicate that we are
dealing here with a phenomenon not seen before.
A general formulation of a reaction-diffusion system with nonlinear diffusion is
$$\frac{\partial\mathbf{u}}{\partial t}=\mathbf{f}(\mathbf{u})+\nabla(\mathbf{D}%
(\mathbf{u})\nabla\mathbf{u}),\qquad\mathbf{u},\mathbf{f}\in\mathbb{R}^{N},\;%
\mathbf{D}\in\mathbb{R}^{N\times N}.$$
(1)
Both the reaction term $\mathbf{f}(\mathbf{u})$ and the diffusion term $\nabla(\mathbf{D}(\mathbf{u})\nabla\mathbf{u})$
in the right-hand side represent dissipative processes, which for
diffusion implies that matrix $\mathbf{D}\in\mathbb{R}^{N\times N}$
is positive (semi-)definite, typically diagonal
with non-negative elements. A huge amount of results have been
obtained about pattern formation described by such models. However,
many physical situations lead to non-diagonal elements in $\mathbf{D}$, i.e. cross-diffusion (see e.g. discussions in Tsyganov et al. (2007); Vanag and Epstein (2009)).
Some such situations may be adequately
described by $\mathbf{D}$ whose eigenvalues have zero real part, e.g. when the
self-diffusion of components is negligible. In such cases reaction
part is dissipative and the “diffusion” part is not. Physical
consequences of such ambivalence are little understood yet.
Cross diffusion has been seen to produce interesting phenomena, such
as fronts, pulses and stationary periodic structures (see e.g. del Castillo Negrete et al. (2002); Chung and
Peacock-López (2007)
among many other works), however phenomenologically similar regimes
are known in reaction-diffusion systems, too.
In a recent series of works we have described unusual phenomena, such
as quasi-solitons and their variations, in excitable systems in which
linear or nonlinear cross-diffusion was added to or replaced
self-diffusion (see e.g. Tsyganov et al. (2003, 2004); Biktashev et al. (2004); Tsyganov and Biktashev (2004); Biktashev and Tsyganov (2005)). The ability of a medium
to conduct solitary waves is stipulated by its excitable kinetics
described by the reaction term $\mathbf{f}(\mathbf{u})$, whereas specifics of their
interaction are also due to the cross-diffusion terms. However,
excitability is a relatively exotic, albeit very important, type of
behaviour compared to oscillations. For instance, in population
dynamics, plausible excitable predator-prey models have been proposed
Truscott and Brindley (1994) but we are not aware of reliable observations
of natural systems described by such models. On the other hand,
oscillatory behaviour in predator-prey systems is textbook material
Murray (2002); Britton (2003) and there are plentiful observational
data on traveling waves in cyclic populations
Sherratt and Smith (2008).
Solitary waves in oscillatory systems are not feasible, and it is not
clear what new features cross-diffusion may impose.
The purpose of this communication is to describe new phenomena we have
observed in oscillatory systems with “pursuit-evasion” nonlinear
cross-diffusion interaction between the components.
II The models
We consider two predator-prey models with cross-diffusion terms of
“pursuit-evasion” mutual taxis,
$$\displaystyle\frac{\partial u}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle f(u,v)+D_{u}\frac{\partial^{2}u}{\partial x^{2}}+h_{-}\frac{%
\partial}{\partial x}\left(u\frac{\partial v}{\partial x}\right),$$
$$\displaystyle\frac{\partial v}{\partial t}$$
$$\displaystyle=$$
$$\displaystyle g(u,v)+D_{v}\frac{\partial^{2}v}{\partial x^{2}}-h_{+}\frac{%
\partial}{\partial x}\left(v\frac{\partial u}{\partial x}\right),$$
(2)
for $(x,t)\in[0,L]\times[0,t_{\max}]$
for two reaction kinetics, the Truscott-Brindley (TB) model
Truscott and Brindley (1994)
$$\displaystyle f(u,v)$$
$$\displaystyle=$$
$$\displaystyle\beta u(1-u)-vu^{2}/(u^{2}+\nu^{2}),$$
$$\displaystyle g(u,v)$$
$$\displaystyle=$$
$$\displaystyle\gamma vu^{2}/(u^{2}+\nu^{2})-wv,$$
(3)
where $\beta=0.43$, $\nu=0.053$, $\gamma=0.1$ and $w=0.055$ unless
stated otherwise, and the
Rosenzweig-MacArthur (MA) model
Rosenzweig and
MacArthur (1963); Britton (2003); Sherratt and Smith (2008)
$$\displaystyle f(u,v)$$
$$\displaystyle=$$
$$\displaystyle\beta u(1-u)-vu/(u+\nu),$$
$$\displaystyle g(u,v)$$
$$\displaystyle=$$
$$\displaystyle\gamma vu/(u+\nu)-wv,$$
(4)
where $\beta=1$, $\nu=0.3$, $\gamma=0.15$ and $w=0.03$ unless stated
otherwise.
Here $u$ represents
prey, $v$ predators, the term with $h_{+}$ describes pursuit of prey by
predators and the term with $h_{-}$ describes evasion of predators by prey.
The simulation were done on an interval $x\in[0,L]$ with
periodic or Neumann boundary conditions for both
components, using forward Euler stepping in time,
center differences for the diffusion terms and upwind difference for
the taxis terms, see Tsyganov et al. (2004) for details and justification.
Except where stated otherwise, we used discretization
steps $\Delta x=0.1$ and $\Delta t=4\cdot 10^{-4}$.
III Numerical observations
Figs. 1 and 2 illustrate the phenomenon of the
spontaneous onset of periodic waves. Starting from arbitrary spatially-uniform
intial conditions at $t=0$, after a transient allowed to establish
uniform oscillations, perturbations were introduced and subsequent
evolution observed. The perturbation was introduced at half of the
grid points chosen randomly, where at $t=300$ the values of $u$ were
replaced by randomly chosen numbers in the interval between $0.15$ and
$0.45$. Fig. 1 shows space-time density plots and fig. 2 illustrate
selected profiles of the emerging wavetrains.
In the TB model with periodic boundary conditions (fig. 1(a)),
after a “random” transient lasting two or three bulk oscillation
periods, patterns start to emerge: waves start “from nowhere” and
annihilate upon collision with other such waves. After a few periods
of such collisions, the waves propagating leftwards win over and a
periodic wavetrain establishes which then persists. Different seeds
in the random number generator produce solutions differing in details
but always leading to periodic trains, leftward and rightward
propagating with equal probability (compare
density plots and wave profiles in fig. 1 and fig. 2,
which corresponded to different simulations with the same parameter sets).
Impenetrable boundaries do not allow periodic wavetrain solutions;
however the tendency to establish periodic wavetrains is observed even
then. In fig. 1(b) rightward propagating waves win over. Their impact
with the right boundary $x=L$ is with partial reflection, when the
reflected wave is weak and soon decays; note that this behaviour is
typical for collision of solitary excitation waves in such systems
Tsyganov et al. (2004). The left boundary has a quenching effect, but at a
distance from it waves emerge spontaneously. This distance varies
irregularly, indicating that spontaneous generation of waves is
associated with an instability, thus sensitive dependence on initial
conditions and probably chaotic dynamics. This irregular pattern
persists for a long time.
This behaviour is in a contrast with a system with the same
kinetics but pure diffusional spatial terms: in fig. 1(c), similar
initial random perturbations lead very quickly to re-establishment of
spatially uniform oscillations.
The parameters used in fig. 1(a) are close to the boundary of
the oscillatory regime in the TB model (achieved e.g. at
$w\approx 0.053$ with other parameters fixed). When parameters are
further into the oscillatory region, spontaneous generation of
periodic wavetrains is still observed, although the transient period
of spontaneous wavelet generations and collisions lasts longer, see
fig. 1(d).
We have also found that prevalence of the “evasion” taxis ($h_{-}$ coefficient) helps
generation of periodic trains, but $h_{+}=0$ is not necessary, and such
generation can be observed with the “pursuit” taxis present as well,
see fig. 1(e).
Spontaneous generation of periodic trains is observed in the RM model
as well, see fig. 1(f).
The spontaneously emerging periodic wavetrains typically had
wavelengths in a limited range. To check whether this is dictated by
initial conditions or is due to limitations of the system, we
performed simulations in a circle, i.e. an interval with periodic
boundary conditions, of a slowly changing length $L$. We started from
an established propagating wave in a circle. Then we changed the
length $L$ of the circle in small steps, allowing sufficient time
between the steps for the waves to adjust. During the simulation we
monitored the number of waves $n$, determined via the number of points
where $u$ crossed the level $u=u_{*}=0.2$, and the periods $T$ defined as
intervals between $u$ crossing the level $u=u_{*}$ in the positive
direction. Results of one such simulation are shown in fig. 3.
The number of waves $n$ in the interval did not remain constant
(fig. 3(a)), but spontaneously adjusted so as to keep the
average wavelength within certain limits: between approximately 2.5 and 8 in the
simulation shown. This number was not a unique function of the
interval length: changing $L$ upwards and downwards produced different
dependencies $n(L)$, i.e. we have hysteresis. Simulations at slower
rate of change of $L$ slightly changed the $n(L)$ dependencies but the
hysteresis stayed. Near the transition points where $n$ changed the
value, the propagation of the waves was non-stationary, and was always
for $L$ just below the transitional value, whether it was decreasing
(fig. 3(b)) or increasing (fig. 3(c)). Increasing $L$
had a noticeably more destabilizing effect than decreasing.
The nature of the non-stationary solutions is illustrated by the
density plots shown in fig. 3(d). Starting from an $n=1$
solution, an increase of $L$ above the value of $L\approx 5$ leads to
an instability of the steady propagating wave solution. This is a
soft, Eckhaus-type
instability and leads to a mild modulation of the wave, producing
a seemingly two-periodic motion. The amplitude of the modulations
grows as $L$ increases, until at $L=7.6$ a qualitative transformation
occurs. A gap between the wave and its own copy around the circle
grows so big that at a certain moment it is sufficient to allow
spontaneous generation of another wave, leading to an $n=2$
solution. This solution is steady, i.e. propagates without
modulations, until $L$ grows so big it in turn becomes unstable etc.
IV Preliminary theoretical considerations
Substantial theoretical analysis of the phenomenon of the spontaneous
traveling periodic waves is beyond the scope of this communication.
Here we consider one naive approach and then some known pattern
formation mechanisms, which a priori might look relevant to this
phenomenon, only to eliminate them, as not providing a satisfactory
explanation. We will refer to the
historical review by
Cross and Hohenberg (1993, p. 870) (CH for brevity), and to
a recent symmetry based classification of instabilities and
bifurcations of periodic dissipative waves and structures given by
Rademacher and Scheel (2007, p. 2680) (RS for
brevity).
It is not captured by lambda-omega approach
The simple class of two-component reaction-diffusion systems
introduced by Kopell and Howard (1973) and called
“lambda-omega systems”, and closely related to the complex Ginzburg-Landau
equation, allows exact solutions in the form of periodic waves. It has
offered qualitative insight in many nonlinear wave phenomena,
including periodic waves in cyclic
populations Sherratt and Smith (2008). However, it does not seem to be
helpful in our present case. The modification of the lambda-omega
system, corresponding to the choice of signs of taxis terms in (2)
is
$$\frac{\partial z}{\partial t}=(\Lambda(|z|)+i\Omega(|z|))z-i\nabla^{2}z$$
(5)
where $z$ is a complex dynamic variable representing $u+iv$,
and the purely imaginary diffusivity here corresponds to the absence of
self-diffusion, $D_{u}=D_{v}=0$. Then the periodic traveling wave ansatz
$z=a\,\exp[i(\omega t-kx)]$, $a,\omega,k\in\mathbb{R}$, gives the finite system
$$\Lambda(a)=0,\qquad\omega=\Omega(a)-k^{2},$$
i.e. all waves have the same amplitude which is a root of $\Lambda()$,
and exist for all wavelengths $k$ rather than in a finite interval.
Stability analysis and consideration of nonzero self-diffusion do
not help either.
It does not emerge via Turing mechanism.
The instability of
spatially-uniform solutions in favour of non-oscillatory,
spatially-periodic solutions with periods in a finite range is, of
course, a defining feature of the Turing patterns, called just so by
RS and classified as type $I_{s}$ in CH nomenclature. Cross-diffusion
can provide an alternative to the original Turing’s short range
inhibition - long range activation condition. Indeed Turing-type
instabilities and spontaneously occurring, self-supporting
time-stationary spatially periodic patterns have been observed in
locally multistable systems with cross-diffusion
del Castillo Negrete et al. (2002). Our present observations are
different in that here we are dealing with time-oscillating phenomena
not just space-oscillating.
It does not emerge via Turing-Hopf mechanism.
A Hopf bifurcation of the spatially uniform equilibrium at a
nonzero wavelength, is called “Hopf”, “oscillatory Turing” and
“Turing Hopf” instability by RS, classifed as type $I_{o}$ in CH
nomenclature, and also known as short-wave instability or
finite-wavelength instability. It can lead to stable periodic
propagating waves, in lasers, fluid convection and reaction-diffusion
models
Swift and Hohenberg (1977); Haken (1983); Livshits (1983); Lega et al. (1994).
In reaction-diffusion context, such waves have been observed
experimentally and in simulations in populations and BZ reaction
Mendelson and Lega (1998); Vanag and Epstein (2002). However, the standard
way such instability occurs in systems (1) implies existence of an
equilibrium that is stable with respect to spatially uniform
perturbations, which we do not have here, and it only can occur if
$N\geq 3$ whereas we have only two components, $u$ and $v$.
Specifically, for $\mathbf{u}(x,t)=\mathbf{u}_{r}+\mathbf{v}\,e^{\lambda t+ikx}$ where
$\mathbf{u}_{r}=(u_{r},v_{r})$ is the spatially uniform equilibrium and $|\mathbf{v}|\ll 1$,
we have the characteristic equation
$$\det\left(\mathbf{F}_{r}-\mathbf{D}_{r}k^{2}-\lambda\mathbf{I}\right)=0,$$
where
$\mathbf{F}_{r}=\mathbf{F}(\mathbf{u}_{r})=\left(\partial\mathbf{f}/\partial%
\mathbf{u}\right)_{\mathbf{u}=\mathbf{u}_{r}}=\left[\begin{array}[]{cc}f_{11}&%
f_{12}\\
f_{21}&f_{22}\end{array}\right]$
is the Jacobian matrix of the reaction terms and $\mathbf{D}_{r}=\mathbf{D}(\mathbf{u}_{r})$ is the
diffusion matrix, both evaluated at the equilibrium. Considering for
simplicity the cases of fig. 1(a,b,d,f) where $\mathbf{D}_{r}=\left[\begin{array}[]{cc}0&h_{-}u_{r}\\
0&0\end{array}\right]$, we have
$$\lambda=\frac{1}{2}\left(f_{11}+f_{22}\pm\sqrt{(f_{11}-f_{22})^{2}+4f_{12}f_{2%
1}-f_{21}h_{-}u_{r}k^{2}}\right)$$
which for
$k^{2}>\max\left(((f_{11}-f_{22})^{2}+4f_{12}f_{21})/(h_{-}u_{r})\,,\,0\right)$
gives oscillatory behaviour of perturbations, but then
$\mathrm{Re}(\lambda)=(f_{11}+f_{22})/2=\mathrm{const}$ whereas it has to have a
maximum at a positive $k^{2}$ for this mechanism to be relevant.
It does not emerge via Turing-Hopf instability of spatially uniform oscillations
The next possible candidate is the instability of spatially uniform
oscillations with respect to perturbations which nonzero freqency and
nonzero wavenumber. This case is not considered in the CH
nomenclature, and is called “Hopf” instability of spatially homogeneous
oscillations, with the same variants as in the previous case, by RS.
This instability looks plausible as spatially homogeneous (spatially uniform) oscillations in our
systems are indeed possible and even stable in small spatial
domains, so we have investigated this possibility with particular
care. As limit cycles in the point systems of (3) and (4) can
not be described analytically, the investigation of stability has to
be done numerically. We have considered solutions of the form
$\mathbf{u}(x,t)=\mathbf{u}_{o}(t)+\mathrm{Re}\left(\mathbf{v}(t)e^{ikx}\right)$ with $|\mathbf{v}|\ll 1$, which
gives a coupled system of ordinary differential equations
$$\displaystyle\displaystyle{\frac{\mathrm{d}\mathbf{u}_{o}}{\mathrm{d}t}}$$
$$\displaystyle=\mathbf{f}(\mathbf{u}_{o}),$$
(6a)
$$\displaystyle\displaystyle{\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}}$$
$$\displaystyle=\left(\mathbf{F}(\mathbf{u}_{o})-\mathbf{D}(\mathbf{u}_{o})k^{2}%
\right)\mathbf{v},$$
(6b)
with parameter $k$. We solved system (6) forward in time
with initial conditions for bulk oscillations $\mathbf{u}_{o}$ in the basin of
attraction of the limit cycle, and arbitrary nonzero initial conditions
for the perturbation $\mathbf{v}$. Then we estimated the Lyapunov exponent for the
$\mathbf{v}$-subsystem, $\lambda(k)=\lim_{t\to\infty}t^{-1}\ln(||\mathbf{v}(t)||)$.
The estimation was done by finding maxima of the first
component of $\mathbf{v}(t)$ and linearly fitting their logarithms against
$t$, for an interval of large enough values of $t$. For selected
values of $k$ we used two linear independent sets of initial
conditions for $\mathbf{v}$, to eliminate the theoretical possibility of
accidentally chosing initial conditions that did not lead to the maximal
exponent.
The resulting graph $\lambda(k)$ for the TB model at the same
parameters as in
fig. 1(a) and 3 is shown on fig. 4(a). For
comparison, we also show histograms of the empirical wavenumbers
observed in simulations shown in fig. 3, calculated as $k=2\pi n/L$, separately for the growing and decreasing $L$.
Fig. 4(b) shows similar graphs made for the RM model
at the same parameters as in fig. 1(f). It is clear that,
although there are finite bands of wavenumbers producing growing
perturbations, the
actually selected wavenumbers are not the same as those of
the fastest growing perturbations, and for the TB model they even partly fall in the interval of
decaying perturbations.
Moreover, the growing perturbations of the spatially uniform oscillations in
fact do not represent propagating periodic waves, but standing
waves. This is illustrated in fig. 4(c) where we show a density
plot of a simulation of the full model, similar to fig. 1(a)
but with different initial conditions. Here we chose initial
conditions as spatially uniform oscillations plus a very small perturbation
sinusoidal in space. Note that for the limit of infinitely small
perturbation amplitudes this exactly corresponds to system
(6).
We conclude that although the cross-diffusion driven instability does indeed
take place in the considered examples, the waves that emerge are in
fact quite different from the spontaneous periodic traveling waves.
Spontaneous sources as a precursor of spontaneous periodic waves
The periodic standing waves emerging via the cross-diffusion driven
instability described above, are in turn unstable themselves.
Fig. 4(d) shows a continuation of simulation of
fig. 4(c). The standing waves are observed for a long time, as
they are stable within the space of functions with spatial period
$2\pi/k=L/6$, and the numerical initial conditions are almost exactly
periodic with that period, up to small errors resulting from finite
precision arithmetics. The small symmetry-breaking numerical errors
allow for an instability of the periodic standing waves to develop,
during which some of the standing waves occur later than others. When
this instability sufficiently develops, there is a sudden, “hard”
transition to propagating waves. The spatial period of the propagating
waves is twice longer than the spatial period of preceding standing
waves. We stress that the traveling waves do not appear via anything
like “bifurcation” from standing waves, at least in the examples we
considered.
Notice that the long transient solution shown in fig. 4(c,d) is
a periodic standing wave by its symmetry, but it also looks like a
periodic set of focal sources, synchronously sending out solitary
waves which then annihilate each other. As can be seen in
fig. 1, apart from the symmetry, this sort of transient before
the onset of periodic waves is typical, and only its duration varies
in different simulations. That is, the special initial conditions in
fig. 4(c,d) only affect the symmetry and the duration of the
transient, but not its qualitative character. A similar route to
traveling waves via unstable periodic set of “focal sources”
standing waves is obseved in the RM model.
V Conclusion
The considered examples demonstrate an unusual type of behaviour. The
systems are oscillatory, but the spatially uniform oscillations are
unstable. The systems can also demonstrate standing periodic waves,
which are also unstable. These instabilities lead to periodic
propagating waves, which seems to be the only stable regime. This
regime emerges spontaneously even when boundary conditions disallow
propagating waves. The periods of the waves can be in a certain
interval with strict boundaries, both upper and lower. Nearer the
upper end of the interval, i.e. at longer wavelengths, the periodic
waves do not propagate steadily but are modulated. Transition from
steady to modulated propagation is soft and has empirical features of
a supercritical Hopf bifurcation (of a relative equilibrium), i.e. possibly an Eckhaus mechanism.
The defining features described above are sufficiently generic, and
the phenomenon of spontaneous periodic traveling waves does not
disappear as the parameters are varied, nor it is restricted just to
one model. This behaviour does not fall into existing classification
of pattern formation scenarios. The detailed mechanisms
of spontaneous generation and maintenance of periodic traveling waves
require further investigation. However, it is clear that
cross-diffusion is an essential factor, since its replacement with,
or adding of significant amount of, self diffusion
eliminates the effect. Cross-diffusion phenomena are known in a in a
variety of physical situations. For example, spontaneous periodic
waves have been observed in a Burridge-Knopoff mathematical model of
earthquakes Cartwright et al. (1997, 1999). That
model belongs to the class (1), with only one nonzero element
of matrix $\mathbf{D}$, as in our simulations shown in fig. 1(a) and
(g) but constant, and
excitable FitzHugh-Nagumo local kinetics. It is not known whether
the spontaneous waves in the Burridge-Knopoff model have a finite
interval of allowed wavenumbers, as illustrated by fig. 3 for
our case, however other described features of those waves are
similar to those described here and are likely to have a
similar nature. Further investigation of the mechanism of generation
of such waves is a subject for further study which is of
broad physical interest as a new pattern forming mechanism in
dissipative spatially distributed systems.
Returning to the application that originally motivated this study,
attempts to explain waves observed in cyclic biological populations,
using reaction-diffusion models, had to involve spatially-nonuniform
external factors, e.g. sites of increased mortality due to
environmental conditions
Sherratt et al. (2002); Sherratt and Smith (2008). Such factors are needed
to disallow uniform oscillations. Our present results imply that such
factors may not be necessary if cross-diffusion interaction are taken
into account as the uniform oscillations may be unstable and waves
form spontaneously.
Acknowledgments
We are grateful to O. Piro for helpful advice.
The study was supported in part by RFBR grant 07-04-00363 (Russia) and by a
grant from the Research Centre for Mathematics and Modelling of
Liverpool University (UK).
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Geometric explanation of the rich-club phenomenon in complex networks
Máté Csigi
MTA-BME Information Systems Research Group, Budapest
University of Technology and Economics, H-1117 Budapest, Magyar
tudósok krt. 2, Hungary
Attila Kőrösi
MTA-BME Information Systems Research Group, Budapest
University of Technology and Economics, H-1117 Budapest, Magyar
tudósok krt. 2, Hungary
József Bíró
MTA-BME Information Systems Research Group, Budapest
University of Technology and Economics, H-1117 Budapest, Magyar
tudósok krt. 2, Hungary
Zalán Heszberger
MTA-BME Information Systems Research Group, Budapest
University of Technology and Economics, H-1117 Budapest, Magyar
tudósok krt. 2, Hungary
Yury Malkov
Federal state budgetary institution of science, Institute of Applied Physics of the Russian
Academy of Sciences, 46 Ul’yanov Street, 603950 Nizhny Novgorod, Russia
András Gulyás
Abstract
The rich club organization (the presence of highly connected hub
core in a network) influences many structural and functional
characteristics of networks including topology, the efficiency of
paths and distribution of load. Despite its major role, the
literature contains only a very limited set of models capable of
generating networks with realistic rich club structure. One
possible reason is that the rich club organization is a divisive
property among complex networks which exhibit great diversity, in
contrast to other metrics (e.g. diameter, clustering or degree
distribution) which seem to behave very similarly across many
networks. Here we propose a simple yet powerful geometry-based
growing model which can generate realistic complex networks with
high rich club diversity by controlling a single geometric
parameter. The growing model is validated against the Internet,
protein-protein interaction, airport and power grid networks.
keywords: complex networks, rich club coefficient, metric space, geometry
Introduction
The rich club organization plays a central role in the structure and
function of networks
[1, 2, 3, 4, 5, 6, 7, 8]. Some
networks (e.g. the human brain [7], airport networks,
social networks [1] and the Internet
[8]) have a strong rich club meaning that their hubs
are densely connected to each other. Others (e.g. protein-protein
interaction networks [1], the power grid
[9]) behave quite the contrary as the subgraphs
made out of their hubs are very sparse. This high variation across
networks is illustrated in Figure 1, which shows the
normalized rich club coefficient $\rho(k)$ [1]
as the function of degree $k$ for the airport network, the Internet
and the protein-protein interaction network.
The explanation and reproduction of this great rich club diversity is
highly non trivial. The state-of-the-art models targeting the rich
club organization are based on heavy randomization techniques
[10, 11, 12, 13],
which shuffle network connections until a given organization structure
is artificially imitated. Although these randomization-based models
are fairly usable, they do not give deeper insight into the mechanisms
causing this diversity during the evolution of the networks.
Consequently, growing models capable of incorporating various
rich club networks in a simple and intuitive manner would be useful
towards deeper understanding the underlying evolutionary reasons of
this diversity.
Here we propose a simple geometry-based growing model which can
explain the emergence of the rich club variability in real networks by
adjusting a single spatial parameter. Our model is built upon the
real-world observation that in some networks the establishment of very
long connections is not feasible. For example in power grid networks,
the electric current cannot be transferred efficiently (i.e. without
huge losses of energy) over large distances without intermediate
transformations at middle stations
[14, 15]. Similarly, optical
networks apply signal re-generators for the transmission of light
signals over large distances to be able to sustain the signal-to-noise
ratio [16]. Also in certain social networks,
middlemen as intermediate nodes may play crucial role in enhancing
cooperation between the individuals or groups [17].
Such networks seem to implement an “artificial” threshold above
which no direct connections are allowed. Other networks do not have
such inherent thresholds and the length of the edges is only limited
by the “natural” geometric boundary of the network. For example in
airport networks we can find very long links, because transferring
passengers over large distances is not an issue with the current
aviation technologies.
In this paper we confine these observations into a simple geometric
growing model, in which we introduce a length threshold for creating
edges. We show that such a growing model can naturally reproduce and
account for the experienced diversity in the rich-club organization of
networks, while keeping other network statistics (diameter, degree
distribution and clustering) intact. The applied geometric
representation of networks is an active and quickly advancing research
direction in network science[18]. There are
numerous studies describing networks as random geometric graphs,
performing some functions
[19, 20] (e.g. navigation,
information transmission) or structural properties (e.g. small-world,
clustering, modularity)
[21, 22] of networks in a
geometric context, and disclosing some fundamental relations between
topology and hidden metric spaces[23]. A the proper
choice of geometry (e.g. Euclidean, Bolyai-Lobachevskian hyperbolic
geometry or other metric space) can also promote the interpretation of
numerous network processes [24, 25, 26].
Results
In our model $N$ nodes are randomly generated one after another on an
Euclidean 2D $R$-disk with uniformly distributed coordinates. When adding
a new node, it selects the $m$ closest nodes already residing on the
disk. The distances between the new node and the old ones are
calculated by the Euclidean distances normalized by a function of the
old node degrees (as in the Growing Homophilic model[22]).
If this “effective” distance between the new node
and a selected one is smaller than the threshold $T$ then they are
directly connected, otherwise a so-called "bridge" node to the
midpoint of the two nodes is established and connects to both nodes.
The formal description of the network generation process is performed
in panel (a) of Figure 2, while panel
(b) shows a small network generated with the model.
Time evolution of the model, as new nodes are inserted into the network
at different stages is shown in Figure 2. For the sake of
simplicity, in this illustration the distance normalization by node
degrees is omitted. At the beginning of the
generation process, many bridge nodes are inserted as the distance
between the nodes is typically larger than $T$ (see panel (c)
in Figure 2). As the network grows, the average node density and
degrees increases, so the typical normalized distance between the
nodes will fall below $T$ and no more bridge nodes are added (panel
(d) in Figure 2). From this stage the model
falls back to the growing homophilic model analyzed in
[22]. Setting $T$ to a very large value
(e.g. $T>2R$) completely recovers the model in
[22] because bridge nodes are never inserted to
the graph. We show, that by varying $T$, the model generates complex
networks with diverse rich-club organization, while having scale-free
degree distribution, small diameter and large clustering. In the
remaining of the paper we will use the settings summarized in
Table 1 in our analytical and simulation results.
Number of bridge nodes
First, we show that the total number of bridge nodes quickly converges
to a relatively small value compared to the reasonable network size
($N$) during the generation of the graph, and this value is
independent of the graph size. To support this observation we give a
recursive estimation of the expected number of new bridge nodes
generated at each step of the model, and based on this recursion a
mathematical expression is given to the limit of the expected total
number of bridge nodes (see methods for more details Methods).
By analyzing the recursion one can show that the expected number of bridge nodes at step $N$ denoted by $b_{N}$ can approximately be expressed in the form
$$b_{N}\approx\exp(-f_{1}N+f_{2}\log N+f_{3})\ .$$
(1)
where the functions $f_{1}$, $f_{2}$ and $f_{3}$ may depend on $R,T,m$ but
are independent from $N$. From this it immediately follows, that for the total number of bridge nodes $B_{N}$
$$B_{N}\approx\int_{x=1}^{N}\exp(-f_{1}x+f_{2}\log x+f_{3}){\rm d}x\rightarrow%
\exp(f_{3}){E}_{-f_{2}}(f_{1})\ {\rm as}\ N\rightarrow\infty$$
(2)
where ${E}$ is the exponential integral function. The vanishing term
during the convergence in $B_{N}$ is $N^{1+f_{2}}E_{-f_{2}}(f_{1}N)$ and
also approximately exponential.
In Figure 4 the expected total number of bridge
nodes ($B_{N}$) calculated by recursion (6) is plotted
in each iteration together with the simulation result for the same
parameters. The two plots readily justify that $B_{N}$ has a
characteristic flat after certain iterations, which means that $B_{N}$
converges to a finite fixed value during the graph generation
process. This also illustrates that for sufficiently large network the
total number of bridge nodes is negligible comparing to the network
size. Furthermore, according to statistical tests the overall
distribution of the nodes on the $R$-disk is apparently not affected
by the bridge nodes, and still can be treated as uniform.
Diameter, clustering and degree distribution
The diameter of all three generated networks (see
Table 1) is around $9$-$10$, similar to the real
networks (Table 2). Figure 4
shows that the diameter of the $T=12$ networks is an approximate
logarithmic function of the network size, which confirms the
small-world property. Also the generated networks have high clustering
coefficients with values very close to that of real networks.
Finally, Table 2 confirms that the clustering
coefficient is insensitive to the threshold parameter.
Now we show that the generated networks has scale-free degree
distribution independently of $T$.
Theorem 1
The networks produced by the model have scale-free degree
distribution with $\gamma=3$ when $N\to\infty$.
Proof: Suppose we compute the effective distance as
${{d}_{\text{eff}}}=\frac{{{d}_{Euc}}}{\sqrt{k}}$. At each insertion
step the algorithm connects a new element to exactly $m$ neighbors
that globally minimize the normalized distance. To infer the degree
distribution of the neighbor elements, we temporary fix the distance
to the $m+1$-th nearest neighbor $d_{\text{eff}}^{m+1}$ and randomly
shuffle positions of the $m$ neighbor nodes under the condition that
they all remain the $m$ nearest neighbors with respect to the new
element (i.e. having effective distance to the new element less than
$d_{\text{eff}}^{m+1}$).
For every possible value of the neighbor degree $k$, possible element
positions are bounded in the initial Euclidean space by a radius
${{r}_{\text{Euc}}}=d_{\text{eff}}^{m+1}\sqrt{k}$. Since the nodes are
distributed uniformly in the Euclidean space, the probability of
having an element with degree k proportional to the
$r_{\text{Euc}}$-ball volume. Thus under fixed $d_{\text{eff}}^{m+1}$
the overall probability of connecting to an element with degree $k$ is
proportional to $(k)$.
The probability inferred for a fixed value of $d_{\text{eff}}^{m+1}$
does not depend on either the value of $d_{\text{eff}}^{m+1}$, or the
positions of the nodes that are not the closest neighbors of the
inserted elements, so that is true for every possible positions of the
elements in the Euclidean space and overall probability of connection
to a node is proportional to its degree $(k)$. This means that new
nodes connect to the old ones with probability proportional to $k$,
which is equivalent to the Barabasi-Albert model
[27], proved to produce scale-free networks
with $\gamma=3$.
The Figure 6 shows the degree distributions of three
networks generated with our model with various values of $T$. The plot
readily confirms that the degree distributions are indeed scale-free
with $\gamma=3$ independently of $T$.
Rich-club coefficient
Although the insertion of bridge nodes keeps degree distribution,
clustering and diameter intact, the simulation results plotted in
Figure 6 clearly show that the graphs generated
by the model differ greatly in their rich-club organization depending
on $T$. Setting $T$ to the diameter of the $R$-disk ($T=100$, red
triangles in Figure 6), the model does not limit
the lengths of the edges artificially, so the only limiting factor is
the natural geometry of the disk itself. In this case we obtain a
network with a strong rich-club, similarly to the airport
network. Conversely, adjusting $T$ to $12$, the model will create only
edges having $d_{\text{eff}}<12$. This is a strong “artificial”
limitation for the edge lengths imposed by the generation process. As
a result, the model yields a network with no rich-club ($T=12$, blue
squares in Figure 6), likewise the PPI network.
We note the appealing similarity between
Figure 6 and Figure 1, showing
the rich-club diversity in real networks.
Discussion
An intriguing question could be whether our model captures something
fundamental from the growth processes of real networks, or exhibit
similar rich-club diversity simply by chance. For answering this
question we have performed the CCDF’s (Complementary Cumulative
Distribution Function) of the normalized edge length distribution in a
rich-club (airports with flights in the US) and a non rich-club
network (the North American Power Grid) together with the networks
generated with our model in
Figure 7. Panel (a) shows
continuously significant (on all length scale) decrease of edge length
distributions before the final “natural” cutoff for the airport and
the $T=100$ networks caused by the geometry of the continent and the
$R$-disk respectively. On panel (b) however we can observe a
clearly visible plateau before the cutoff of the edge lengths in the
power grid network. This means that edge lengths are much denser near
the cutoff, which in this case is rather “artificial” and caused by
the growth process of the network and not the underlying geometry. Our
model produces a very similar edge length distribution for the setting
$T=12$. These results hint that networks having no rich-clubs use a
very similar limiting for the length of the connections as our model
do. As a consequence, this length-limiting phenomenon can also account
for the emergence of the observed diverse rich-club organization in
real networks.
These two examples also underline that our method is parsimonious in a
sense that the rich club organization can be tuned by only a single
geometric threshold parameter in a growing homophilic model. We think
the results presented in this paper are strong indications that the
rich club diversity can be placed at all on a growing/evolutionary
perspective, and provide deeper insight into the mechanisms
resulting certain rich club behavior during the growth of networks.
Methods
Data acquisition
The topology of the AS level Internet has been downloaded from CAIDA
(Center for Applied Internet Data Analysis, www.caida.org). We
have downloaded the airport network from the OpenFlights database
(www.openflights.org). We used the DIP [28]
database as a source for the protein-protein interaction network of
the S. cerevisiae. Finally, the map of the north american power grid
has been downloaded from [29].
Recursive estimation of the number of bridge nodes
Let $A(r,T,R)$ be the area of the intersection of an $r-$centered disk
with radius $T$ and the $R-$disk, and let $p(r,T,R)$ be the fraction
of $A(r,T,R)$ and the area of the $R-$disk, i.e.
$p(r,T,R)=\frac{A(r,T,R)}{R^{2}\pi}$. Further, let us assume that
there are already $j$ nodes in the network. The $(j+1)^{\rm th}$
randomly generated node will connect to the $m$ nearest neighbors. For
calculating the necessary bridge nodes in this step, the task is to
determine what are the nodes among the $m$ nearest neighbors which are
farther than $T$. To ease the computation, the degrees of the
neighbors are substituted by their expectation values (denoted by
$\bar{k}_{j}$ and to be determined later) subject to the whole network at
this stage. Since the effective distance is computed as the Euclidean
distance divided by $\sqrt{\bar{k}_{j}}$, it is approximately equivalent
to investigate the expected number of points among the $m$ nearest
ones being outside of the $(j+1)^{\rm th}$ node $T\sqrt{\bar{k}_{j}}$ -
radius vicinity. This will be equal to the expected number of newly
inserted bridge nodes at this step. Denote the radial coordinate of
the $(j+1)^{\rm th}$ node by $r$ and assume that the previously
generated random points and established bridge nodes are still evenly
distributed on the $R-$disk. With this, the probability that
$i,0\leq i\leq j$ nodes among the $j$ ones are closer to the
$(j+1)^{\rm th}$ node than $T\sqrt{\bar{k}_{j}}$ is
$$\binom{j}{i}p(r,T\sqrt{\bar{k}_{j}},R)^{i}(1-p(r,T\sqrt{\bar{k}_{j}},R))^{j-i}$$
and hence the expected number of necessary bridge nodes at this step is
$$\sum_{i=0}^{\min(m-1,j-1)}\min(m-i,j-i)\binom{j}{i}p(r,T\sqrt{\bar{k}_{j}},R)^%
{i}(1-p(r,T\sqrt{\bar{k}_{j}},R))^{j-i}:=\beta_{j}(r,T\sqrt{\bar{k}_{j}},R,m)$$
Note, that this is still a conditional expectation value which is to
be de-conditioned by the density of the radial coordinate $r$. Towards
the de-conditioning, first the function $A(r,T,R)$ is to be
determined. Clearly, $A(r,T,R)=T^{2}\pi$ if $r\leq R-T$, i.e. there
is no intersection of the two disks. Otherwise, if $r\geq R-T$ then
by using straightforward geometrical calculations
$$A(r,T,R)=\alpha R^{2}+\gamma T^{2}-2\sqrt{s(s-R)(s-T)(s-r)}$$
(3)
where
$$\alpha=\arccos\frac{R^{2}+r^{2}-T^{2}}{2rR}\ ,\ \gamma=\arccos\frac{r^{2}+T^{2%
}-R^{2}}{2rT}\ ,\ s=\frac{R+T+r}{2}\ .$$
(4)
Now, the de-conditioning is possible with
$p(r,T,R)=A(r,T,R)/(R^{2}\pi)$ and the density of the radial coordinate
$r$, which is $\frac{2r}{R^{2}}$. Further, let $j(N)=N+b_{1}+b_{2}+\ldots+b_{N}$ where
$N$ is the randomly generated points and $b_{l}\ l=1,\ldots,N$ is the
expected number of bridge nodes established after the $l^{\rm th}$
random node. For completing the recursive estimation, the expected
degree should also be expressed upon the $l^{\rm th}$ random node
generation. This is
$$\bar{k}_{l}=\frac{2(lm-\frac{m(m+1)}{2}+2\sum_{i=1}^{l}b_{i})}{l+\sum_{i=1}^{l%
}b_{i}}\ ,\ {\rm for}\ l>m\ ,\ {\rm else}\ \bar{k}_{l}=\frac{2(\frac{l(l-1)}{2%
}+2\sum_{i=1}^{l}b_{i})}{l+\sum_{i=1}^{l}b_{i}}\ .$$
(5)
For $l=1$ let $b_{1}=0$, $\bar{k}_{1}=0$ and let $B_{N}=\sum_{i=1}^{N}b_{i}$.
The main recursion can now be expressed as
$$\!\!\!\!\!\!\!\!\!\!\!\!b_{N+1}=\int_{r=0}^{R}\beta_{j(N)}(r,T\sqrt{\bar{k}_{j%
(N)}},R,m)\frac{2r}{R^{2}}{\rm d}r\ .$$
(6)
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Author contributions statement
M.Cs. and A.G. have developed the model and contributed to the experiments. A.K., J.B., Z.H. and Y.M. contributed to the analysis and the numerical results. All authors reviewed the manuscript.
Additional information
Competing financial interests
The authors declare no competing financial interests. |
Option Pricing and Hedging with Small Transaction Costs111We are grateful to Aleš Černý, Christoph Czichowsky, Paolo Guasoni, Marcel Nutz, and Mete Soner for fruitful discussions. We also thank Ren Liu for his careful reading of the manuscript. Part of this work was completed while the second author was visiting Columbia University. He thanks Ioannis Karatzas and the university for their hospitality.
Jan Kallsen
Christian-Albrechts-Universität zu Kiel, Westring 383, D-24098 Kiel, Germany, email [email protected].
Johannes Muhle-Karbe
ETH Zürich, Departement für Mathematik, Rämistrasse 101, CH-8092, Zürich, Switzerland, and Swiss Finance Institute, email [email protected]. Partially supported by the National Centre of Competence in Research Financial Valuation and Risk Management (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management), of the Swiss National Science Foundation (SNF).
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Abstract
An investor with constant absolute risk aversion trades a risky asset with general Itô-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.
Mathematics Subject Classification: (2010) 91G20, 91G10, 91G80.
JEL Classification: G13, G11.
Keywords: transaction costs, indifference pricing and hedging, exponential utility, asymptotics
1 Introduction
The pricing and hedging of derivative securities is a central theme of Mathematical Finance. In complete markets, the risk incurred by selling any claim can be offset completely by dynamic trading in the underlying. Then, there is only one price compatible with the absence of arbitrage, namely the initial value of the replicating portfolio. This line of reasoning is torn to pieces by the presence of even the small bid-ask spreads present in the most liquid financial markets. Transaction costs make (super-)replication prohibitively expensive [27], thereby calling for approaches that explicitly balance the gains and costs of trading.
An economically appealing choice is the utility-indifference approach put forward by Hodges and Neuberger [14] as well as Davis, Panas, and Zariphopoulou [9].222Cf. Leland [19] for an alternative approach, where trading only takes place at exogenous discrete times. For an investor with given preference structure, the idea is to determine a “fair” price by matching the maximal expected utilities that can be attained with and without the claim. Both [14] and [9] focus on investors with constant absolute risk aversion for tractability. Nevertheless, the numerical computation of the solution turns out to be quite challenging, involving multidimensional nonlinear free boundary problems already for plain vanilla call options written on a single risky asset with constant investment opportunities.
In reality, transaction costs are small, and continue to decline across financial markets. Therefore, asymptotic expansions for small spreads have been proposed to “reveal the salient features of the problem while remaining a good approximation to the full but more complicated model” [29]. For small costs, a formal asymptotic analysis of the model of Davis et al. [9] has been carried out by Whalley and Wilmott [29].333See Bichuch [3] for a rigorous proof. A different limiting regime, where absolute risk version becomes large as the spread tends to zero, is studied by Barles and Soner [1]. In both cases, the computation of indifference prices boils down to the solution of certain inhomogeneous Black-Scholes equations, whereas the corresponding hedging strategies are determined explicitly at the leading order.
For small costs, the present study provides formal asymptotics for essentially general continuous asset price dynamics and arbitrary contingent claims. As in the extant literature, we also focus on investors with constant absolute risk aversion, for which the cash additivity of the corresponding exponential utility functions allows to handle the option position by a change of measure.444In the absence of an option position, asymptotics for the optimal policy and the associated welfare can also be obtained for general utilities and with intermediate consumption, see Soner and Touzi [28] as well as the forthcoming companion paper of the present study [17]. Both with and without an option position, the leading-order optimal trading strategies consist of keeping the number of risky shares in a time and state dependent no-trade region around their frictionless counterparts. The width of the latter is determined by the following tradeoff: Large fluctuations of the frictionless optimizer call for a wide buffer in order to reduce trading costs. Conversely, wildly fluctuating asset prices cause the investor’s positions to deviate substantially from the frictionless target, thereby necessitating closer tracking. Accordingly, the ratio of local fluctuations – measured in terms of the local quadratic variation both for the frictionless optimizer and the risky asset – is the crucial statistic determining the optimal trading boundaries in the presence of small transaction costs. The corresponding welfare loss – and in turn the indifference price adjustments – turn out to be given by the squared width of the no-trade region, suitably averaged with respect to both time and states.
The pricing implications of small transaction costs depend on the interplay between the frictionless pure investment and hedging strategies. If no trading takes place in the absence of an option position, then the costs incurred by hedging the claims necessitate a higher premium. This changes, however, if trades prescribed by the hedge partially offset the rebalancing of the investor’s pure investment position. Then, maybe surprisingly, a smaller compensation may in fact be sufficient for the risk incurred by selling the claims if transaction costs are taken into consideration.
The remainder of the article is organized as follows. The main results – explicit formulas for the leading-order optimal policy and welfare – are presented in Section 2. Subsequently, we discuss how they can be adapted to deal with utility-indifference pricing and hedging. The derivations of all results are collected in Appendix A. They are based on formal perturbation arguments, applied to the martingale optimality conditions for a frictionless “shadow price”, which admits the same leading-order optimal strategy and utility as the original market with transaction costs.
2 Optimal Investment
Consider a market with two assets, a riskless one with price normalized to one and a risky one trading with proportional transaction costs. This means that one has to pay a higher ask price $(1+\varepsilon)S_{t}$ when purchasing the risky asset but only receives a lower bid price $(1-\varepsilon)S_{t}$ when selling it. Here, $\varepsilon>0$ is the relative bid-ask spread and the mid price $S_{t}$ follows a general, not necessarily Markovian, Itô process:
$$dS_{t}=b_{t}^{S}dt+\sqrt{c^{S}_{t}}dW_{t},$$
(2.1)
for a standard Brownian motion $W$. In this setting, an investor with exponential utility function $U(x)=-e^{-px}$, i.e., with constant absolute risk aversion $p>0$, trades to maximize the certainty equivalent $-\frac{1}{p}\log E[e^{-pX^{\phi}_{T}}]$ over all terminal wealths $X^{\phi}_{T}$ at time $T$ corresponding to self-financing trading strategies $\phi$.555The costs of setting up and liquidating the portfolio are only incurred once and are therefore of order $O(\varepsilon)$. Hence, they do not impact the investor’s welfare at the leading order $\varepsilon^{2/3}$, and we disregard them throughout. The optimal number of shares in the absence of frictions ($\varepsilon=0$) is denoted by $\varphi_{t}$; we assume it to be a sufficiently regular Itô process with local quadratic variation $d\langle\varphi\rangle_{t}/dt$, which is satisfied in most applications.
The dynamics (2.1) are formulated in discounted terms. If the safe asset earns a constant interest rate $r>0$, one can reduce to this case by discounting, replacing risk aversion $p$ by $e^{rT}p$.
2.1 Optimal Policy
For small transaction costs $\varepsilon$, an approximately optimal666That is, this strategy matches the optimal certainty equivalent, at the leading order for small costs. strategy $\varphi^{\varepsilon}_{t}$ is to engage in the minimal amount of trading necessary to keep the number of risky shares within the following buy and sell boundaries around the frictionless optimizer $\varphi_{t}$:
$$\Delta\varphi^{\pm}_{t}=\pm\left(\frac{3}{2p}\frac{d\langle\varphi\rangle_{t}}%
{d\langle S\rangle_{t}}\varepsilon S_{t}\right)^{1/3}.$$
(2.2)
The random and time varying no-trade region $[\varphi_{t}+\Delta\varphi_{t}^{-},\varphi_{t}+\Delta\varphi_{t}^{+}]$ is symmetric around the frictionless optimizer $\varphi_{t}$, and its half-width is given by the cubic root of three parts:
i)
the constant $3/2p$ that only depends on the investor’s risk aversion but not on the underlying probabilistic model.
ii)
the observable (absolute) half-width $\varepsilon S_{t}$ of the bid-ask spread.
iii)
the fluctuations of the frictionless optimizer, measured in terms of its local quadratic variation $d\langle\varphi\rangle_{t}$, normalized by the market’s local fluctuations $d\langle S\rangle_{t}$. Tracking more wildly fluctuating strategies requires a wider buffer to reduce trading costs. Conversely, large fluctuations in the asset prices cause large fluctuations of the investor’s risky position, thereby necessitating closer tracking to reduce losses due to displacement from the frictionless target position.
Unless the planning horizon is postponed to infinity [6, 8, 26], transaction costs introduce horizon effects even with a constant investment opportunity set [20]. Nevertheless, the local dynamics of the frictionless optimizer $\varphi_{t}$ alone always act as a sufficient statistic for the asymptotically optimal trading boundaries (2.2) – the investor does not hedge against the presence of a small constant friction. These optimal trading boundaries are “myopic” in the sense that they are of the same form as for the local utility maximizers considered by Martin [23]. By definition, these also behave myopically in the absence of frictions, unlike the exponential investors considered here, whose optimal policies generally include an intertemporal hedging term reflecting future investment opportunities. Somewhat surprisingly, the generally different frictionless strategies enter the leading order trading boundaries in the same way through their local fluctuations.
More general preferences and intermediate consumption are studied by Soner and Touzi [28] and in a forthcoming companion paper of the present study [17].
2.2 Welfare
The utility associated to the above policy can also be quantified, thereby allowing to assess the welfare impact of transaction costs. To this end, let $\mathrm{CE}$ and $\mathrm{CE}^{\varepsilon}$ denote the certainty equivalents without and with transaction costs $\varepsilon$, respectively, i.e., the cash amounts that yield the same utility as trading optimally in the market. Then, for small transaction costs $\varepsilon$:
$$\mathrm{CE}^{\varepsilon}\sim\mathrm{CE}-\frac{p}{2}E_{Q}\left[\int_{0}^{T}(%
\Delta\varphi^{+}_{t})^{2}d\langle S\rangle_{t}\right],$$
(2.3)
and this leading-order optimal performance is achieved by the policy from Section 2.1. Hence, the leading-order loss due to transaction costs is of order $O(\varepsilon^{2/3})$ as in the Black-Scholes model [26]. It is given by an average of the squared half-width of the optimal no-trade region. The latter has to be computed with respect to a clock that runs at the speed $d\langle S\rangle_{t}=c^{S}_{t}dt$ of the market’s local variance, i.e., losses due to transaction costs accrue more rapidly in times of frequent price moves. Moreover, this average has to be determined under the marginal pricing measure $Q$ associated to the frictionless utility maximization problem, i.e., the equivalent martingale measure that minimizes the entropy with respect to the physical probability. The leading-order effect of transaction costs can therefore be interpreted as the price of a path-dependent contingent claim, computed under the investor’s marginal pricing measure.
As first pointed out by Rogers [25] (also compare Goodman and Ostrov [12]), the utility loss due to transaction costs is composed of two parts. On the one hand, there is the displacement loss due to following the strategy $\varphi^{\varepsilon}$ instead of the frictionless maximizer $\varphi$. In addition, there is the loss due to the costs directly incurred by trading. Rogers observed that, for small transaction costs, the leading orders of these two losses coincide for the optimal policy. For investors with constant absolute risk aversion, we complement this by the insight that two thirds of the leading-order welfare loss are incurred due to trading costs, whereas the remaining one third is caused by displacement. Surprisingly, this holds irrespective of the model for the risky asset and the investor’s risk aversion.
3 Indifference Pricing and Hedging
Due to the cash additivity of the exponential utility function, the above results can be adapted to optimal investment in the presence of a random endowment, thereby leading to asymptotic formulas for utility-based prices and hedging strategies.
Indeed, suppose that at time $t=0$, the investor sells a claim $H$ maturing at time $T$, for a premium $\pi(H)$. Then, her investment problem becomes
$$\textstyle{\sup_{\phi}}E\left[-e^{-p(X^{\phi}_{T}+\pi(H)-H)}\right]=e^{-p\pi(H%
)}E[e^{pH}]\textstyle{\sup_{\phi}}E_{P^{H}}\left[-e^{-pX^{\phi}_{T}}\right],$$
where $P^{H}$ is the equivalent probability with density $dP^{H}/dP=e^{pH}/E[e^{pH}]$. Hence, we are back in the above setting of pure investment; only the dynamics of the risky asset $S$ change when passing from the physical probability $P$ to $P^{H}$, and the frictionless optimizer changes accordingly. Then, with small transaction costs $\varepsilon$, the optimal policy in the presence of the random endowment $-H$ corresponds to the minimal amount of trading to keep the number of risky assets within the following buy and sell boundaries around the frictionless optimizer $\varphi_{t}^{H}$:777Note that the square-bracket processes of continuous processes are invariant under equivalent measure changes, i.e., it does not matter whether they are computed under $P$ or $P^{H}$.
$$\Delta\varphi^{H,\pm}_{t}=\left(\frac{3}{2p}\frac{d\langle\varphi^{H}\rangle_{%
t}}{d\langle S\rangle_{t}}\varepsilon S_{t}\right)^{1/3}.$$
(3.1)
Hence, at the leading order, the optimal investment strategy in the presence of a random endowment again prescribes the minimal amount of trading to remain within a buffer around its frictionless counterpart, whose width can be calculated from the local variations of the latter. Again by appealing to the results for the pure investment problem, the corresponding certainty equivalent is found to be
$$\displaystyle\mathrm{CE}^{\varepsilon,H}\sim\mathrm{CE}^{H}-\frac{p}{2}E_{Q^{H%
}}\left[\int_{0}^{T}(\Delta\varphi_{t}^{H,+})^{2}d\langle S\rangle_{t}\right],$$
(3.2)
where $\mathrm{CE}^{H}$ and $Q^{H}$ denote the frictionless certainty equivalent and minimal entropy martingale measure in the presence of the claim $H$, respectively. With this result at hand, the corresponding utility indifference price $\pi^{\varepsilon}(H)$ of Hodges and Neuberger [14] can be computed by matching (3.2) with the investor’s certainty equivalent (2.3) in the absence of the claim. At the leading order, it turns out that the frictionless indifference price $\pi^{0}(H)$ – which makes the frictionless certainty equivalents $\mathrm{CE}^{H},\mathrm{CE}$ with and without the claim coincide – has to be corrected by the difference between the effects of transaction costs with and without the claim:
$$\pi^{\varepsilon}(H)\sim\pi^{0}(H)+\frac{p}{2}\left(E_{Q^{H}}\left[\int_{0}^{T%
}(\Delta\varphi^{H,+}_{t})^{2}d\langle S\rangle_{t}\right]-E_{Q}\left[\int_{0}%
^{T}(\Delta\varphi^{+}_{t})^{2}d\langle S\rangle_{t}\right]\right).$$
(3.3)
As a consequence, the indifference price can be either higher or lower than its frictionless counterpart, depending on whether higher or lower trading costs are incurred due to the presence of the claim.
3.1 Complete Markets
Even in the absence of frictions, utility-based prices and hedging strategies are typically hard to compute unless the market is complete, and we first focus on this special case in the sequel. Then, there is a unique equivalent martingale measure $Q$, and frictionless indifference prices coincide with expectations under the latter:
$$\pi^{0}(H)=E_{Q}[H].$$
Moreover, any claim $H$ can then be hedged perfectly by a replicating strategy $\Delta_{t}^{H}$:888As this notation indicates, this is just the usual delta hedge in a Markovian setting, i.e., the derivative of the option price with respect to the underlying.
$$H=E_{Q}[H]+\int_{0}^{T}\Delta^{H}_{t}dS_{t}.$$
Consequently, the random endowment can simply be removed from the optimal investment problem:
$$\textstyle{\sup_{\phi}}E\left[-e^{-p(x+\int_{0}^{T}\phi_{t}dS_{t}+\pi(H)-H)}%
\right]=e^{-p(\pi(H)-E_{Q}[H])}\textstyle{\sup_{\phi}}E\left[-e^{-p(x+\int_{0}%
^{T}(\phi_{t}-\Delta^{H}_{t})dS_{t})}\right].$$
As a result, the optimal strategy for investors with constant absolute risk aversion is to hedge away the random endowment and, in addition to that, invest as in the pure investment problem:
$$\varphi^{H}_{t}=\varphi_{t}+\Delta^{H}_{t}.$$
(3.4)
The (monetary) boundaries of the corresponding leading-order optimal no-trade region for small transaction costs $\varepsilon$ can in turn be determined by inserting (3.4) into (3.1):
$$\Delta\varphi^{\pm,H}_{t}S_{t}=\pm\left(\frac{3}{2p}\frac{d\langle\varphi+%
\Delta^{H}\rangle_{t}}{d\langle S\rangle_{t}}S_{t}^{4}\right)^{1/3}\varepsilon%
^{1/3}.$$
(3.5)
To shed more light on this first-order correction due to the presence of small transaction costs, write999For $d\varphi_{t}=b^{\varphi}_{t}dt+\sqrt{c^{\varphi}_{t}}dW_{t}$, this is obtained by setting $\Gamma^{\varphi}_{t}=\sqrt{c^{\varphi}_{t}/c^{S}_{t}}$ and $a^{\varphi}_{t}=b^{\varphi}_{t}-b^{S}_{t}\sqrt{c^{\varphi}_{t}/c^{S}_{t}}$; the argument for $\Delta^{H}$ is analogous.
$$d\varphi_{t}=\Gamma^{\varphi}_{t}dS_{t}+a^{\varphi}_{t}dt,\quad d\Delta^{H}_{t%
}=\Gamma^{H}_{t}dS_{t}+a^{H}_{t}dt.$$
(3.6)
The gammas $\Gamma_{t}^{\varphi}$ and $\Gamma_{t}^{H}$ describe the sensitivities (of the diffusive parts) of the strategies $\varphi_{t}$ and $\Delta_{t}^{H}$ with respect to price moves.101010In a Markovian setting, Itô’s formula shows that these processes indeed coincide with the usual notion of an option’s “gamma”, i.e., the second derivative of the option price with respect to the underlying. With this notation,
$$\frac{d\langle\varphi+\Delta^{H}\rangle_{t}}{d\langle S\rangle_{t}}S_{t}^{4}=(%
\Gamma_{t}^{\varphi}S_{t}^{2}+\Gamma_{t}^{H}S_{t}^{2})^{2}.$$
(3.7)
Hence, the width of the no-trade region (3.5) is determined by the cash-gamma of the investor’s portfolio, that is, the sensitivity of the frictionless optimal risky position to changes in the risky asset. If shocks to the risky asset cause the frictionless position to move a lot, then the investor should keep a wider buffer around it to save transaction costs. For the Black-Scholes model, where the frictionless optimal risky position in the pure investment problem is constant, (3.5) reduces to the formula derived by Whalley and Wilmott [29, Section 4].
Inserting (3.5) into (3.3), the corresponding utility indifference price for small transaction costs is found to be:
$$\displaystyle\pi^{\varepsilon}(H)$$
$$\displaystyle\sim E_{Q}[H]+\left(\frac{9p}{32}\right)^{1/3}\varepsilon^{2/3}E_%
{Q}\left[\int_{0}^{T}\left[\left(\frac{d\langle\varphi+\Delta^{H}\rangle_{t}}{%
d\langle S\rangle_{t}}\right)^{2/3}-\left(\frac{d\langle\varphi\rangle_{t}}{d%
\langle S\rangle_{t}}\right)^{2/3}\right]S_{t}^{2/3}d\langle S\rangle_{t}\right]$$
$$\displaystyle=E_{Q}[H]+\left(\frac{9p}{32}\right)^{1/3}\varepsilon^{2/3}E_{Q}%
\left[\int_{0}^{T}\left[|\Gamma^{\varphi}_{t}S_{t}^{2}+\Gamma^{H}_{t}S_{t}^{2}%
|^{4/3}-|\Gamma^{\varphi}_{t}S_{t}^{2}|^{4/3}\right]\frac{d\langle S\rangle_{t%
}}{S_{t}^{2}}\right].$$
(3.8)
The correction compared to the frictionless model is therefore – up to a constant – determined by the $Q$-expected time-average of the difference between (suitable powers of) the future cash-gamma of the investor’s optimal position with and without the option, scaled by the infinitesimal variance of the relative returns. For the Black-Scholes model, one readily verifies that (3.1) can be rewritten in terms of the inhomogeneous Black-Scholes equation of Whalley and Wilmott [29, Section 3.3].
Representation (3.1) implies that the investor should charge a higher price than in the frictionless case if delta-hedging the option increases the sensitivity of her position with respect to price changes of the risky asset. Conversely, a lower premium is sufficient if delta hedging the claim reduces this sensitivity. The impact of trade size and risk aversion depends on the relative importance of investment and hedging. Since the underlying frictionless market is complete, the perfect hedge $\Delta_{t}^{H}$ and in turn its gamma $\Gamma_{t}^{H}$ are independent of the investor’s risk aversion, but scale linearly with the number $n$ of claims sold. Conversely, the optimal investment strategy $\varphi_{t}$ and its gamma $\Gamma_{t}^{\varphi}$ are independent of the option position sold, but scale linearly with the inverse of risk aversion. Consequently, the comparative statics of utility-based prices and hedges with transaction costs – which depend on both quantities – are ambiguous in general. However, they can be analyzed in more detail if either the option position or the pure investment dominates.
Marginal Investment
First, we focus on the case where the investor’s primary focus lies on the pricing and risk management of her option position. This regime applies if the cash-gamma $\Gamma_{t}^{\varphi}$ of the pure investment under consideration is negligible compared to its counterpart $\Gamma_{t}^{nH}$ for the option position $nH$. In particular, this occurs if the risky asset is assumed to be a martingale with vanishing risk premium, as in Hodges and Neuberger [14], so that no investment is optimal without the option position, or in the asymptotic regime of Barles and Soner [1], where the option position increases as the spread becomes small.
If the contribution of the pure investment strategy is negligible, Formula (3.1) for the indifference price per claim reads as:
$$\frac{\pi^{\varepsilon}(nH)}{n}\sim E_{Q}[H]+\left(\frac{9pn\varepsilon^{2}}{3%
2}\right)^{1/3}E_{Q}\left[\int_{0}^{T}|\Gamma^{H}_{t}S_{t}^{2}|^{4/3}\frac{d%
\langle S\rangle_{t}}{S_{t}^{2}}\right].$$
(3.9)
For a marginal pure investment, small transaction costs therefore always lead to a positive price correction compared to the frictionless case. The interpretation is that the transaction costs incurred by carrying out the approximate hedge necessitate a higher premium. The size of this effect depends on the relative magnitudes of trade size, risk aversion, and transaction costs. Trade size $n$ and risk aversion $p$ both enter the leading order correction through their cubic roots, and therefore in an interchangeable manner.111111Barles and Soner [1] consider the case where the product $pn\varepsilon^{2}$ converges to a finite limit. In the Black-Scholes model, they characterize the limiting price per claim as the solution to an inhomogeneous Black-Scholes equation. However, this limit does not coincide with the right-hand side of (3.9), whose derivation assumes only transaction costs to be small while risk aversion is fixed.
If the contribution of the pure investment strategy is negligible, the optimal trading strategy in the presence of the claims can be directly interpreted as a utility-based hedge. In view of (3.5), the latter corresponds to keeping the number of risky shares within a no-trade region around the frictionless delta-hedge $\Delta^{nH}_{t}=n\Delta^{H}_{t}$; the maximal monetary deviations allowed are:
$$\Delta\varphi^{nH,\pm}S_{t}=\pm\frac{n^{2/3}\varepsilon^{1/3}}{p^{1/3}}\left(%
\frac{3}{2}(\Gamma^{H}_{t}S_{t}^{2})^{2}\right)^{1/3}.$$
(3.10)
Higher risk aversion induces closer tracking of the frictionless target here, leading to more trading and in turn higher prices (cf. Formula (3.9)).
Semi-Static Delta-Gamma Hedging
“Delta-gamma hedging” is often advocated in order to reduce the impact of transaction costs, cf., e.g., [4, p. 129]. The above results allow to relate this idea to the semi-static hedging of a claim $H$, by dynamic trading in the underlying risky asset and a static position $n^{\prime}$ in some other claim $H^{\prime}$ set up at time zero. In the frictionless case, the choice of $n^{\prime}$ does not matter, since any such position can be offset by delta-hedging with the underlying. With transaction costs, this no longer remains true. Suppose the risky asset is a martingale ($P=Q$) and $H$, $H^{\prime}$ are traded at their frictionless prices $E_{Q}[H],E_{Q}[H^{\prime}]$ with some transaction costs of order $O(\varepsilon)$. Then, (3.9) applied to the claim $H-n^{\prime}H^{\prime}$ shows that the leading-order frictional certainty equivalent of selling one unit of $H$ – and hedging it with a static position of $n^{\prime}$ units of $H^{\prime}$ and optimal dynamic trading in the underlying – is given by:
$$-\left(\frac{9p}{32}\right)^{1/3}\varepsilon^{2/3}E_{Q}\left[\int_{0}^{T}\left%
|\Gamma^{H}_{t}S_{t}^{2}-n^{\prime}\Gamma^{H^{\prime}}_{t}S_{t}^{2}\right|^{4/%
3}\frac{d\langle S\rangle_{t}}{S_{t}^{2}}\right].$$
Maximizing this certainty equivalent in $n^{\prime}$ therefore amounts to minimizing a suitable average of the future (cash) gamma of the total option position $H-n^{\prime}H^{\prime}$.
If other options are only used once for hedging, the total position should thus not be made gamma-neutral at any one point in time. The exception is when both option gammas are approximately constant over the horizon under consideration; then, it is optimal to make the total option position gamma-neutral. This situation occurs if the static option position is only held briefly, and before maturity of either claim. Hence, it seems reasonable to conjecture that one should trade to remain close to a delta-gamma neutral position if hedging dynamically with both the underlying and an option.121212Compare [11] for related results in a pure investment problem with a stock and an option.
Marginal Option Position
Now, let us turn to the converse situation of a marginal option position, i.e., the sale of a small number $n$ of claims $H$. For incomplete markets without frictions, the limiting price for $n\to 0$, called marginal utility-based price, is a linear pricing rule, namely the expectation under the frictionless minimal entropy martingale measure, independent of both trade size and risk aversion. The leading-order correction for small $n$ is linear both in the trade size $n$ and in the investor’s risk aversion $p$ [22, 2, 18]. Hence, both quantities are interchangeable also in this setting: doubling risk aversion has the same effect as doubling the trade size.
Let us now derive corresponding results for the incompleteness caused by imposing small transaction costs in an otherwise complete market. Then, Taylor expanding (3.1) for small $n$ yields:131313Here and in the sequel, we report the leading-order terms for small transaction costs $\varepsilon$ and small trade size $n$.
$$\displaystyle\frac{\pi^{\varepsilon}(nH)}{n}\sim E_{Q}[H]$$
$$\displaystyle+\left(\frac{9p}{32}\right)^{1/3}\varepsilon^{2/3}E_{Q}\left[\int%
_{0}^{T}\frac{4}{3}|\Gamma^{\varphi}_{t}S_{t}^{2}|^{4/3}\frac{\Gamma^{H}_{t}}{%
\Gamma^{\varphi}_{t}}\frac{d\langle S\rangle_{t}}{S_{t}^{2}}\right]$$
$$\displaystyle+n\left(\frac{9p}{32}\right)^{1/3}\varepsilon^{2/3}E_{Q}\left[%
\int_{0}^{T}\frac{2}{9}|\Gamma^{\varphi}_{t}S_{t}^{2}|^{4/3}\left(\frac{\Gamma%
^{H}_{t}}{\Gamma^{\varphi}_{t}}\right)^{2}\frac{d\langle S\rangle_{t}}{S_{t}^{%
2}}\right].$$
Recall that the optimal frictionless strategy $\varphi$ and its gamma $\Gamma^{\varphi}$ are independent of the trade size $n$ but scale linearly with the inverse of risk aversion $p$. Hence, the frictionless scalings are robust with respect to small transaction costs: The limiting price per claim for $n\to 0$ is also linear in the claim, and independent of trade size and risk aversion. Moreover, these quantities both enter linearly, and therefore in an interchangeable manner, in the leading order correction term for small trade sizes.
For a small option position, the sign of the prize correction compared to the complete frictionless market depends on the interplay between the pure investment strategy and the hedge for the claim. The pure investment strategy is typically negatively correlated with price shocks, $\Gamma_{t}^{\varphi}<0$ (e.g., in the Black-Scholes model). Then, the difference between the frictionless price and the limiting price with transaction costs is determined by the sign of the option’s gamma. If the latter is positive as for European call or put options, then the marginal utility-based price taking into account transaction costs is smaller than its frictionless counterpart. This is because utility-based investment strategies are typically of contrarian type, i.e., decreasing when the risky price rises, whereas the delta-hedge of a European call or put is increasing with the value of its underlying. Consequently, hedging the claim allows the investor to save transaction costs, so that she is willing to sell the claim for a smaller premium. This rationale, however, is only applicable if the fluctuations of the hedging position are small enough to be absorbed by the investor’s other investments. In particular, the price adjustment is always positive for a marginal pure investment.
Let us also consider how the investor’s trading strategy changes in the presence of a small number of claims. Taylor expanding (3.5) for small $n$ shows that it is optimal to refrain from trading as long as the risky position remains within a bandwidth of
$$\Delta\varphi^{nH,\pm}_{t}S_{t}\sim\pm\Delta\varphi^{\pm}_{t}S_{t}\left(1+%
\frac{2}{3}n\frac{\Gamma^{H}_{t}}{\Gamma^{\varphi}_{t}}\right)$$
around the frictionless optimal position $(\varphi_{t}+n\Delta^{H}_{t})S_{t}$. The interpretation for the ratio of gammas is the same as for the corresponding utility-based prices above: If the trades prescribed by the delta hedge partially offset moves of the pure investment strategy, then the resulting reduced sensitivity to price shocks allows the investor to use a smaller no-trade region than in the absence of the claims. As for risk aversion, note that whereas the investor’s pure investment and the corresponding trading boundaries $\Delta\varphi_{t}^{\pm}$ in the absence of the claim scale with her risk tolerance $1/p$, the adjustment due to the presence of the claims does not. For small option positions, it is linear in trade size but independent of risk aversion as is true for the frictionless hedge.
3.2 Incomplete Markets
In incomplete markets, simple formulas for indifference prices and hedging strategies can typically only be obtained in the limit for a small number of claims, even in the absence of frictions. If the trade size $n$ is small, [22, 2, 18] show that the optimal strategy $\varphi_{t}$ for the pure investment problem should be complemented be $n\xi_{t}$, where $\xi_{t}$ is the mean-variance optimal hedge for the claim, determined under the marginal pricing measure $Q$, i.e.,
$$\xi_{t}=\frac{d\langle V,S\rangle_{t}}{d\langle S\rangle_{t}},$$
where $V$ denotes the $Q$-martingale generated by the payoff $H$. As a consequence, the leading-order adjustment of the portfolio due to the presence of the claim is linear in trade size and independent of risk aversion, as in the complete case discussed above. The corresponding indifference price per claim converges to the expectation under the marginal pricing measure, which is again independent of trade size and risk aversion. The leading-order adjustment for larger trade sizes is given by the $pn/2$-fold of the minimal $Q$-expected squared hedging error, i.e.,
$$\frac{\pi^{0}(nH)}{n}\sim E_{Q}[H]+\frac{pn}{2}E_{Q}\left[\left(H-E_{Q}[H]-%
\int_{0}^{T}\xi_{t}dS_{t}\right)^{2}\right].$$
(3.11)
As a result, it is linear both in trade size and risk aversion. In this setting of a small option position held in a potentially incomplete frictionless market, we now discuss the implications of small transaction costs.
Negligible Risk Premium
Let us first consider the case where the risky asset is a martingale under the physical probability. Then, no trading is optimal for the pure investment problem, $\varphi_{t}=0$, and the minimal entropy martingale measure coincides with the physical probability, $Q=P$. As a consequence, the monetary trading boundaries (3.1) around the frictionless strategy $n\xi_{t}$ are given by:
$$\Delta\varphi^{nH,\pm}_{t}S_{t}\sim\pm\frac{n^{2/3}\varepsilon^{1/3}}{p^{1/3}}%
\left(\frac{3}{2}\frac{d\langle\xi\rangle_{t}}{d\langle S\rangle_{t}}S_{t}^{4}%
\right)^{1/3}.$$
In view of (3.7), this is the same formula as in the complete case (3.10), with the perfect hedge replaced by the mean-variance optimal one. In particular, the scalings in trade size and risk aversion are robust to incompleteness in the frictionless market, as long as the option position is small.
To determine the corresponding leading order price correction, insert the above trading boundaries into (3.3) and note that $Q=P$, $\Delta\varphi_{t}^{+}=0$, as well as $dQ^{nH}/dP=1+O(n)$. As a consequence:
$$\frac{\pi^{\varepsilon}(nH)}{n}\sim E[H]+\frac{pn}{2}E\left[\left(H-E[H]-\int_%
{0}^{T}\xi_{t}dS_{t}\right)^{2}\right]+\left(\frac{9pn\varepsilon^{2}}{32}%
\right)^{1/3}E\left[\int_{0}^{T}\left(\frac{d\langle\xi\rangle_{t}}{d\langle S%
\rangle_{t}}S_{t}^{4}\right)^{2/3}\frac{d\langle S\rangle_{t}}{S_{t}^{2}}%
\right].$$
The second term is the correction due to transaction costs, which once more parallels the complete case (3.9), with the mean-variance optimal hedge again replacing the replicating strategy. The first term is the correction (3.11) due to the incompleteness of the frictionless market, which is proportional to the minimal squared hedging error and hence vanishes in the complete case. At the leading order, the two price corrections therefore separate; their relative sizes are determined by the magnitude of risk aversion $p$ times trade size $n$, compared to the spread $\varepsilon$.
Nontrivial Risk Premium
If the pure investment strategy $\varphi_{t}$ is not negligible, the trading boundaries (3.1) around the frictionless strategy $\varphi_{t}+n\xi_{t}$ are given by
$$\Delta\varphi^{nH,\pm}_{t}\sim\Delta\varphi_{t}^{\pm}\left(1+\frac{2n}{3}\frac%
{d\langle\varphi,\xi\rangle_{t}}{d\langle\varphi\rangle_{t}}\right).$$
In the small claim limit, the interpretations from the complete case are therefore robust as well: If shocks to the frictionless investment and hedging strategies are negatively correlated, one should keep a smaller buffer with the claim, and conversely for the case of a positive correlation.
Concerning the pricing implications of small transaction costs added to incomplete frictionless markets, the situation is somewhat more involved. The reason is that the presence of the claim changes the impact of the transaction costs in two different ways. On the one hand, it affects the trading strategy that is used: the investor passes from staying within $\Delta\varphi^{\pm}_{t}$ around the pure investment strategy $\varphi_{t}$, to keeping within $\Delta\varphi_{t}^{nH,\pm}$ around $\varphi_{t}^{nH}=\varphi_{t}+n\xi_{t}+O(n^{2})$. On the other hand, even after hedging the claim, the latter still induces some unspanned risk in incomplete markets, and therefore affects the investor’s marginal evaluation rule. That is, the marginal pricing measure changes from $Q$, with density proportional to the marginal utility $U^{\prime}(\int_{0}^{T}\varphi_{t}dS_{t})$ associated to the pure investment strategy, to $Q^{nH}$, with density proportional to the marginal utility augmented by the $n$ claims, $U^{\prime}(\int_{0}^{T}\varphi_{t}^{H}dS_{t}-nH)$ (compare [24, Theorem 1.1]). By Formula (3.3) and Taylor expansion for small $n$, the leading-order price impact of small transaction costs is then found to be given by:
$$\frac{p}{2}\left(E_{Q}\left[\int_{0}^{T}\frac{4n}{3}(\Delta\varphi^{+}_{t})^{2%
}\frac{d\langle\varphi,\xi\rangle_{t}}{d\langle\varphi\rangle_{t}}d\langle S%
\rangle_{t}\right]+E\left[\left(\frac{dQ^{nH}}{dP}-\frac{dQ}{dP}\right)\int_{0%
}^{T}(\Delta\varphi^{+}_{t})^{2}d\langle S\rangle_{t}\right]\right).$$
The first term is due to changing the trading strategy; it is already visible for complete frictionless markets. The second term reflects the change of the marginal pricing measure due to the presence of the claim, which does not take place in complete frictionless markets with a unique equivalent martingale measure. As for the hedging strategy above, the sign of the first term depends on the correlation of shocks to investment and hedging strategies. To examine the sign of the second term, notice that $\varphi_{t}^{H}=\varphi_{t}+n\xi_{t}+O(n^{2})$, Taylor expansion, and the $Q$-martingale property of the wealth process $\int_{0}^{T}\xi_{t}dS_{t}$ yield
$$\frac{dQ^{nH}}{dP}=\frac{e^{-p(\int_{0}^{T}(\varphi_{t}+n\xi_{t})dS_{t}+O(n^{2%
})-nH)}}{E[e^{-p(\int_{0}^{T}(\varphi_{t}+n\xi_{t})dS_{t}+O(n^{2})-nH)}]}=%
\frac{dQ}{dP}\left(1+np\left(H-E_{Q}[H]-\int_{0}^{T}\xi_{t}dS_{t}\right)\right%
)+O(n^{2}).$$
As a result, the second term in the price impact of small transaction costs is given by the covariance between the shortfall of the frictionless utility-based hedge and the cumulated transaction costs effect, measured by the average squared width of the no-trade region:
$$E\left[\left(\frac{dQ^{nH}}{dP}-\frac{dQ}{dP}\right)\int_{0}^{T}(\Delta\varphi%
^{+}_{t})^{2}d\langle S\rangle_{t}\right]\sim npE_{Q}\left[\left(H-E_{Q}[H]-%
\int_{0}^{T}\xi_{t}dS_{t}\right)\int_{0}^{T}(\Delta\varphi^{+}_{t})^{2}d%
\langle S\rangle_{t}\right].$$
Hence, incompleteness of the frictionless market increases the impact of transaction costs, if these tend to accrue more rapidly when the imperfect utility-based hedge also does badly, i.e., when the different sources of incompleteness tend to cluster. In contrast, the premium for the option is decreased if the two risks are negatively correlated and thereby diversify the investor’s portfolio. The same correlation adjustment also occurs in frictionless markets, when passing from the marginal pricing measure for the pure investment problem to its counterpart in the presence of a small option position. In this sense, the marginal pricing implications of transaction costs are therefore the same as for selling a path dependent option with payoff $\int_{0}^{T}(\Delta\varphi^{+}_{t})^{2}d\langle S\rangle_{t}$. It is important to emphasize, however, that this is not the case at all for the corresponding hedge.
Appendix A Derivation of the Main Results
In the following, the main results are derived by applying formal perturbation arguments to the martingale optimality conditions for a frictionless shadow price. The latter is a “least favorable” frictionless market extension in the sense that it evolves in the bid-ask spread, thereby leading to potentially more favorable trading prices, but admits an optimal policy that only entails the purchase resp. sale of risky shares when the shadow price coincides with the ask resp. bid price.
The observation that such a shadow price should always exist can be traced back to Jouini and Kallal [15] as well as Cvitanić and Karatzas [7] (also cf. Loewenstein [21]). Starting with Kallsen and Muhle-Karbe [16], this concept has recently also been used for the computation and verification of optimal policies in simple settings. Since shadow prices are not known a priori, they have to be determined simultaneously with the optimal policy. Here, we show how to do so for general continuous asset prices, approximately for small costs. In contrast to most previous asymptotic results, we do not first solve the problem for arbitrary costs $\varepsilon>0$ and then expand the solution around $\varepsilon=0$. Instead, we directly tackle the much simpler approximate problem for $\varepsilon\sim 0$, in the same spirit as in the approach of Soner and Touzi [28].
Throughout, mathematical formalism is treated liberally. For example, we do not state and verify technical conditions warranting the uniform integrability of local martingales, interchange of integration and differentiation, and the uniformity of estimates. Rigorous proofs have been worked out in the present setting for the Black-Scholes model [13, 3], and by Soner and Touzi [28] for an infinite-horizon consumption problem in a Markovian setup.
A.1 Notation
Throughout, we write $\phi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S$ for the stochastic integral $\int_{0}^{\cdot}\phi_{t}dS_{t}$. The identity process is denoted by $I_{t}=t$ and for any Itô process $X$ we write $b^{X}$ and $\sigma^{X}$ for its local drift and diffusion coefficients, respectively, in the sense that $dX_{t}=b^{X}_{t}dt+\sigma^{X}_{t}dW_{t}$ for a standard Brownian motion $W$. Finally, for Itô processes $X$ and $Y$, we denote by $c^{X,Y}_{t}=d\langle X,Y\rangle_{t}/dt$ their local quadratic covariation; if $X=Y$ we abbreviate to $d\langle X\rangle_{t}/dt=c_{t}^{X,X}=c_{t}^{X}$.
A.2 Martingale Optimality Conditions
In this section, we formally derive conditions ensuring that a family $(\varphi^{\varepsilon})_{\varepsilon>0}$ of frictional strategies is approximately optimal as the spread $\varepsilon$ becomes small. For the convenience of the reader, we first briefly recapitulate their exact counterparts in the frictionless case.
Frictionless Optimality Conditions
In the absence of transaction costs ($\varepsilon=0$), the following duality result is well known (cf., e.g., [10]): The wealth process $x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S$ corresponding to a trading strategy $\varphi$ is optimal, if (and essentially only if) there exists a process $Z$ satisfying the following optimality conditions:
i)
$Z$ is a martingale.
ii)
$ZS$ is a martingale.
iii)
$Z_{T}=U^{\prime}(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_%
{T})$.
The first two conditions imply that $Z$ is – up to normalization – the density of an equivalent martingale measure $Q$ for $S$. The third identifies it as the solution to a dual minimization problem, linked to the primal maximizer by the usual first-order condition.
Let us briefly recall why conditions i)-iii) imply the optimality of $\varphi$. To this end, let $\psi$ be any competing strategy. Then, the concavity of the utility function $U$ and condition iii) imply
$$\displaystyle E[U(x+\psi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T%
})]$$
$$\displaystyle\leq E[U(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{%
{}}S_{T})]+E[U^{\prime}(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}%
}{{}}S_{T})(\psi-\varphi)\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{%
T}]$$
$$\displaystyle=E[U(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S%
_{T})]+E[Z_{0}]E_{Q}[(\psi-\varphi)\stackrel{{\scriptstyle\mbox{\tiny$\bullet$%
}}}{{}}S_{T}].$$
Since $S$ and in turn the wealth process $(\psi-\varphi)\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S$ is a $Q$-martingale by conditions i) and ii), the second expectation vanishes and the optimality of $\varphi$ follows.
Approximate Optimality Conditions with Transaction Costs
Now, let us derive approximate versions of conditions i), ii), iii) in the presence of small transaction costs $\varepsilon$, ensuring the approximate optimality of a family $(\varphi^{\varepsilon})_{\varepsilon>0}$ of strategies, at the leading order $O(\varepsilon^{2/3})$ as $\varepsilon$ becomes small. The exact optimal strategies converge to their frictionless counterpart. Hence, it suffices to consider families $(\psi^{\varepsilon})_{\varepsilon>0}$ of strategies converging to the frictionless optimizer $\varphi$, i.e., $\psi^{\varepsilon}=\varphi+o(1)$.
Let $(\varphi^{\varepsilon})_{\varepsilon>0}$ be a candidate family of strategies whose optimality we want to verify. As above, for any family of competitors $(\psi^{\varepsilon})_{\varepsilon>0}$, the concavity of $U$ implies:
$$E\left[U(X^{\psi^{\varepsilon}}_{T})\right]\leq E\left[U(X^{\varphi^{%
\varepsilon}}_{T})\right]+E\left[U^{\prime}(X^{\varphi^{\varepsilon}}_{T})(X^{%
\psi^{\varepsilon}}_{T}-X^{\varphi^{\varepsilon}}_{T})\right],$$
(A.1)
where $X_{T}^{\psi^{\varepsilon}},X_{T}^{\varphi^{\varepsilon}}$ denote the payoffs generated by trading the strategies with transaction costs. Now, suppose we can find shadow prices $S^{\varepsilon}$ evolving in the bid-ask spreads $(1\pm\varepsilon)S$, matching the trading prices $(1\pm\varepsilon)S$ in the original market with transaction costs whenever the respective strategies $\varphi^{\varepsilon}$ trade. Then, the frictional wealth process associated to $\varphi^{\varepsilon}$ evidently coincides with its frictionless counterpart for $S^{\varepsilon}$, i.e., $X^{\varphi^{\varepsilon}}_{T}=x+\varphi^{\varepsilon}\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}_{T}$. For any other strategy, trading in terms of $S^{\varepsilon}$ rather than with the original bid-ask spread can only increase wealth, since trades are carried out at potentially more favorable prices: $X^{\psi^{\varepsilon}}_{T}\leq x+\psi^{\varepsilon}\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}_{T}$. Together with (A.1), this implies:
$$E\left[U(X^{\psi^{\varepsilon}}_{T})\right]\leq E\left[U(X^{\varphi^{%
\varepsilon}}_{T})\right]+E\left[U^{\prime}(X^{\varphi^{\varepsilon}}_{T})(%
\psi^{\varepsilon}-\varphi^{\varepsilon})\stackrel{{\scriptstyle\mbox{\tiny$%
\bullet$}}}{{}}S^{\varepsilon}_{T}\right].$$
(A.2)
Now, suppose we can find a process $Z^{\varepsilon}$ satisfying the following approximate versions of i), ii), iii) above:
$\mbox{i}^{\varepsilon})$
$Z^{\varepsilon}$ is approximately a martingale, in that its drift rate $b^{Z^{\varepsilon}}$ is of order $O(\varepsilon^{2/3})$.
$\mbox{ii}^{\varepsilon})$
$Z^{\varepsilon}S^{\varepsilon}$ is approximately a martingale, in that its drift rate $b^{Z^{\varepsilon}S^{\varepsilon}}$ is of order $O(\varepsilon^{2/3})$.
$\mbox{iii}^{\varepsilon})$
$Z^{\varepsilon}_{T}=U^{\prime}(x+\varphi^{\varepsilon}\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}_{T})+O(\varepsilon^{2/3})$.
Then, since $\psi^{\varepsilon}-\varphi^{\varepsilon}=o(1)$, Condition $\mbox{iii}^{\varepsilon})$ implies that (A.2) can be rewritten as
$$E\left[U(X^{\psi^{\varepsilon}}_{T})\right]\leq E\left[U(X^{\varphi^{%
\varepsilon}}_{T})\right]+E\left[Z^{\varepsilon}_{T}((\psi^{\varepsilon}-%
\varphi^{\varepsilon})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{%
\varepsilon}_{T})\right]+o(\varepsilon^{2/3}).$$
Applying integration by parts twice yields
$$\displaystyle Z^{\varepsilon}((\psi^{\varepsilon}-\varphi^{\varepsilon})%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon})$$
$$\displaystyle=Z^{\varepsilon}(\psi^{\varepsilon}-\varphi^{\varepsilon})%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}+((\psi^{%
\varepsilon}-\varphi^{\varepsilon})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$%
}}}{{}}S^{\varepsilon})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}Z^{%
\varepsilon}+(\psi^{\varepsilon}-\varphi^{\varepsilon})\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}\langle Z^{\varepsilon},S^{\varepsilon}\rangle$$
$$\displaystyle=\left((\psi^{\varepsilon}-\varphi^{\varepsilon})\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}-(\psi^{\varepsilon}-%
\varphi^{\varepsilon})S^{\varepsilon}\right)\stackrel{{\scriptstyle\mbox{\tiny%
$\bullet$}}}{{}}Z^{\varepsilon}+(\psi^{\varepsilon}-\varphi^{\varepsilon})%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}(Z^{\varepsilon}S^{%
\varepsilon}),$$
and in turn
$$E\left[U(X^{\psi^{\varepsilon}}_{T})\right]\leq E\left[U(X^{\varphi^{%
\varepsilon}}_{T})\right]+E\left[\left((\psi^{\varepsilon}-\varphi^{%
\varepsilon})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}%
-(\psi^{\varepsilon}-\varphi^{\varepsilon})S^{\varepsilon}\right)\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}Z_{T}^{\varepsilon}+(\psi^{\varepsilon}%
-\varphi^{\varepsilon})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}(Z^{%
\varepsilon}S^{\varepsilon})_{T}\right]+o(\varepsilon^{2/3}).$$
The second expectation is given by the integrated expected drift rate of its argument,
$$\left((\psi^{\varepsilon}-\varphi^{\varepsilon})\stackrel{{\scriptstyle\mbox{%
\tiny$\bullet$}}}{{}}S^{\varepsilon}-(\psi^{\varepsilon}-\varphi^{\varepsilon}%
)S^{\varepsilon}\right)b^{Z^{\varepsilon}}+(\psi^{\varepsilon}-\varphi^{%
\varepsilon})b^{Z^{\varepsilon}S^{\varepsilon}},$$
which is of order $o(\varepsilon^{2/3})$, by conditions $\mbox{i}^{\varepsilon})$ and $\mbox{ii}^{\varepsilon})$ above and because $\psi^{\varepsilon}-\varphi^{\varepsilon}=o(1)$. Hence,
$$E\left[U(X^{\psi^{\varepsilon}}_{T})\right]\leq E\left[U(X^{\varphi^{%
\varepsilon}}_{T})\right]+o(\varepsilon^{2/3}),$$
and the expected utilities of the candidate family $(\varphi^{\varepsilon})_{\varepsilon>0}$ therefore dominate those of the competitors $(\psi^{\varepsilon})_{\varepsilon>0}$ at the leading order $O(\varepsilon^{2/3})$. In summary, the strategies $(\varphi^{\varepsilon})_{\varepsilon>0}$ are indeed approximately optimal if we can find a shadow price $S^{\varepsilon}$ and an approximate martingale density $Z^{\varepsilon}$ satisfying the approximate optimality conditions $\mbox{i}^{\varepsilon})$, $\mbox{ii}^{\varepsilon})$, $\mbox{iii}^{\varepsilon})$.
A.3 Derivation of a Candidate Policy
We now look for strategies $\varphi^{\varepsilon}$, shadow prices $S^{\varepsilon}$, and approximate martingale densities $Z^{\varepsilon}$ satisfying the approximate optimality conditions $\mbox{i}^{\varepsilon})-\mbox{iii}^{\varepsilon})$. Write
$$\varphi^{\varepsilon}=\varphi+\Delta\varphi,\quad S^{\varepsilon}=S+\Delta S.$$
Motivated by previous asymptotic results [29, 13, 3, 28], we assume that the deviations of the optimal strategy with transaction costs from the frictionless optimizer are asymptotically proportional to the cubic root of the spread:
$$\Delta\varphi=O(\varepsilon^{1/3}).$$
(A.3)
Since the shadow price $S^{\varepsilon}$ has to lie in the bid-ask spread $(1\pm\varepsilon)S$, we must have
$$\Delta S=O(\varepsilon).$$
(A.4)
In addition, we assume that $\Delta S$ is an Itô process with drift and diffusion coefficients satisfying:
$$b^{\Delta S}=O(\varepsilon^{1/3}),\quad\sigma^{\Delta S}=O(\varepsilon^{2/3}).$$
(A.5)
All of these assumptions will turn out to be consistent with the results of our calculations below, see Section A.4. Now, notice that
$$\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{%
\varepsilon}=\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S+\Delta%
\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S+O(\varepsilon^{2/3}),$$
because (A.3) and (A.5) give $\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S=O(%
\varepsilon^{2/3})$, and integration by parts in conjunction with (A.4) and (A.5) shows $\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S=O(%
\varepsilon^{2/3})$. Therefore,
$$U^{\prime}(x+\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}%
}}{{}}S^{\varepsilon}_{T})=pe^{-p(x+\varphi^{\varepsilon}\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{\varepsilon}_{T})}=pe^{-p(x+\varphi%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})}(1-p\Delta\varphi%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})+O(\varepsilon^{2/3}).$$
The factor $pe^{-p(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})}$ coincides with the terminal value of the frictionless martingale density $Z$ (cf. the frictionless optimality condition iii) above). The process
$$Z^{\varepsilon}=Z(1-p\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}%
}}{{}}S)$$
therefore is a martingale (because $Z$ is the density of a martingale measure for $S$), satisfying Condition $\mbox{i}^{\varepsilon})$, for which $\mbox{iii}^{\varepsilon})$ holds as well. It remains to determine $\Delta S$ and $\Delta\varphi$ for which $\mbox{ii}^{\varepsilon})$ holds, too. Integration by parts yields
$$Z^{\varepsilon}S^{\varepsilon}-Z^{\varepsilon}_{0}S^{\varepsilon}_{0}=S^{%
\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}Z^{\varepsilon}+%
Z^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{%
\varepsilon}+\langle Z^{\varepsilon},S^{\varepsilon}\rangle.$$
Since the martingale $Z^{\varepsilon}$ has zero drift, it follows that the drift rate of $Z^{\varepsilon}S^{\varepsilon}$ is given by
$$b^{Z^{\varepsilon}S^{\varepsilon}}=Z^{\varepsilon}(b^{S}+b^{\Delta S})+c^{Z^{%
\varepsilon},S+\Delta S}.$$
(A.6)
As $b^{\Delta S}=O(\varepsilon^{1/3})$ by assumption, and $Z^{\varepsilon}=Z(1-p\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}%
}}{{}}S)$, it follows that
$$Z^{\varepsilon}(b^{S}+b^{\Delta S})=Z\left(b^{S}-p(\Delta\varphi\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S)b^{S}+b^{\Delta S}\right)+O(%
\varepsilon^{2/3}).$$
(A.7)
Moreover, writing the frictionless martingale density as a stochastic exponential $Z=\scr{E}(N)=1+Z\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}N$, it follows from (A.5) and integration by parts that
$$\displaystyle\langle Z^{\varepsilon},S+\Delta S\rangle=$$
$$\displaystyle\langle Z(1-p\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$%
\bullet$}}}{{}}S),S\rangle+O(\varepsilon^{2/3})$$
$$\displaystyle=$$
$$\displaystyle Z\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\left(\langle
N%
,S\rangle-p(\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S)%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle N,S\rangle-p\Delta%
\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle S,S\rangle%
\right)+O(\varepsilon^{2/3}),$$
so that
$$c^{Z^{\varepsilon},S+\Delta S}=Z\left(c^{N,S}-p(\Delta\varphi\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S)c^{N,S}-p\Delta\varphi c^{S}\right)+O%
(\varepsilon^{2/3}).$$
(A.8)
Then, inserting (A.7) and (A.8) into (A.6) and using that $b^{S}+c^{N,S}=0$ by Girsanov’s theorem because $ZS=\scr{E}(N)S$ is a martingale by the frictionless optimality condition ii), gives
$$\displaystyle b^{Z^{\varepsilon}S^{\varepsilon}}$$
$$\displaystyle=Z\left(b^{S}+c^{N,S}-p(\Delta\varphi\stackrel{{\scriptstyle\mbox%
{\tiny$\bullet$}}}{{}}S)(b^{S}+c^{N,S})+b^{\Delta S}-p\Delta\varphi c^{S}%
\right)+O(\varepsilon^{2/3})$$
$$\displaystyle=Z(b^{\Delta S}-p\Delta\varphi c^{S})+O(\varepsilon^{2/3}).$$
To make the drift of $Z^{\varepsilon}S^{\varepsilon}$ vanish – up to terms of order $O(\varepsilon^{2/3})$ – in accordance with $\mbox{ii}^{\varepsilon})$, it is therefore necessary that
$$b^{\Delta S}=p\Delta\varphi c^{S}+O(\varepsilon^{2/3}).$$
(A.9)
This drift condition naturally leads to an ansatz of the form $\Delta S=f(\Delta\varphi)$. Then, since the shadow price $S^{\varepsilon}=S+\Delta S$ has to move from the ask price $(1+\varepsilon)S$ to the bid price $(1-\varepsilon)S$ as $\Delta\varphi$ varies between some buy boundary $\Delta\varphi^{-}$ and some sell boundary $\Delta\varphi^{+}$, the function $f$ has to satisfy the boundary conditions
$$f(\Delta\varphi^{-})=\varepsilon S,\quad f(\Delta\varphi^{+})=-\varepsilon S.$$
(A.10)
Moreover, even though the process $\Delta\varphi$ is reflected to remain between the trading boundaries, these singular terms should vanish in the dynamics of $\Delta S$, so that the shadow price $S^{\varepsilon}=S+\Delta S$ does not allow for arbitrage. By Itô’s formula, this implies that the derivative of $f$ should vanish at the boundaries:
$$f^{\prime}(\Delta\varphi^{-})=0,\quad f^{\prime}(\Delta\varphi^{+})=0.$$
(A.11)
The simplest family of functions capable of matching these boundary conditions is given by the symmetric cubic polynomials
$$f(x)=\alpha x^{3}-\gamma x.$$
With this ansatz, (A.11) gives
$$\Delta\varphi^{\pm}=\pm\sqrt{\frac{\gamma}{3\alpha}},$$
and (A.10) implies
$$\gamma=\left(\frac{1}{2}\varepsilon S\right)^{2/3}3\alpha^{1/3}.$$
Moreover, Itô’s formula applied to $f(x)=\alpha x^{3}-\gamma x$ yields
$$\Delta S-\Delta S_{0}=(3\alpha\Delta\varphi^{2}-\gamma)\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}\Delta\varphi+3\alpha\Delta\varphi\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle\Delta\varphi,\Delta\varphi\rangle.$$
Now, notice that the optimal trading strategy $\varphi^{\varepsilon}=\varphi+\Delta\varphi$ with transaction costs is necessarily of finite variation. Assuming it is also continuous then implies $\langle\Delta\varphi\rangle=\langle\varphi\rangle$. Moreover, since $\varphi^{\varepsilon}$ is constant except at the trading boundaries (where $\Delta\varphi=\Delta\varphi^{\pm}$ and in turn $3\alpha\Delta\varphi^{2}-\gamma=0$), we also have
$$(3\alpha\Delta\varphi^{2}-\gamma)\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}%
}{{}}\Delta\varphi=-(3\alpha\Delta\varphi^{2}-\gamma)\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}\varphi.$$
Thus,
$$\Delta S-\Delta S_{0}=-(3\alpha\Delta\varphi^{2}-\gamma)\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\varphi+3\alpha\Delta\varphi\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle\varphi\rangle,$$
and the drift coefficient of $\Delta S$ is given by
$$b^{\Delta S}=3\alpha\Delta\varphi c^{\varphi}+O(\varepsilon^{2/3}).$$
Comparing this to the leading-order term in (A.9), we obtain
$$\alpha=\frac{p}{3}\frac{c^{S}}{c^{\varphi}},$$
and in turn
$$\gamma=\left(\frac{3p^{1/2}}{2}\sqrt{\frac{c^{S}}{c^{\varphi}}}S\right)^{2/3}%
\varepsilon^{2/3}$$
as well as
$$\Delta\varphi^{\pm}=\pm\sqrt{\frac{\gamma}{3\alpha}}=\pm\left(\frac{3}{2p}%
\frac{c^{\varphi}}{c^{S}}\varepsilon S\right)^{1/3}.$$
At the first order, this determines the optimal strategy $\varphi^{\varepsilon}$ with transaction costs as the minimal amount of trading necessary to remain in the randomly changing interval $[\varphi+\Delta\varphi^{-},\varphi+\Delta\varphi^{+}]$ around the frictionless optimizer $\varphi$.
A.4 Approximate Optimality
The above considerations assumed that the coefficients $\alpha,\gamma$ are constant, but then lead to stochastic processes $\alpha_{t},\gamma_{t}$, which seems contradictory at first glance. However, we can verify a fortiori that this choice does indeed satisfy $\mbox{i}^{\varepsilon})-\mbox{iii}^{\varepsilon})$. To see this set, for $\alpha,\gamma$ as above,
$$\Delta S_{t}=\alpha_{t}\Delta\varphi_{t}^{3}-\gamma_{t}\Delta\varphi_{t}$$
and let the strategy $\varphi^{\varepsilon}=\varphi+\Delta\varphi$ correspond to the minimal amount of trading necessary to remain within the boundaries $\Delta\varphi^{\pm}$ around the frictionless optimizer $\varphi$. Then by definition, the process $S^{\varepsilon}:=S+\Delta S$ takes values in the bid-ask spread $[(1-\varepsilon)S,(1+\varepsilon)S]$ and coincides with the bid resp. ask price whenever $\varphi^{\varepsilon}$ reaches the selling boundary $\varphi+\Delta\varphi^{+}$ resp. the buying boundary $\varphi-\Delta\varphi^{+}$ as required for a shadow price. Concerning the dynamics of $\Delta S$, notice that integration by parts (now taking into account the stochasticity of $\alpha$ and $\gamma$) and Itô’s formula give
$$\displaystyle\Delta S-\Delta S_{0}=$$
$$\displaystyle\alpha\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}(\Delta%
\varphi^{3})+\Delta\varphi^{3}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{%
}}\alpha+\langle\alpha,\Delta\varphi^{3}\rangle-\gamma\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}\Delta\varphi-\Delta\varphi\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}\gamma-\langle\gamma,\Delta\varphi\rangle$$
$$\displaystyle=$$
$$\displaystyle(3\alpha\Delta\varphi^{2}-\gamma)\stackrel{{\scriptstyle\mbox{%
\tiny$\bullet$}}}{{}}\Delta\varphi+(3\alpha\Delta\varphi)\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle\Delta\varphi\rangle$$
$$\displaystyle\quad+\Delta\varphi^{3}\stackrel{{\scriptstyle\mbox{\tiny$\bullet%
$}}}{{}}\alpha-\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
\gamma+(3\Delta\varphi^{2})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
\langle\alpha,\Delta\varphi\rangle-\langle\gamma,\Delta\varphi\rangle$$
$$\displaystyle=$$
$$\displaystyle-(3\alpha\Delta\varphi^{2}-\gamma)\stackrel{{\scriptstyle\mbox{%
\tiny$\bullet$}}}{{}}\varphi+(3\alpha\Delta\varphi)\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}\langle\varphi\rangle$$
$$\displaystyle\quad+\Delta\varphi^{3}\stackrel{{\scriptstyle\mbox{\tiny$\bullet%
$}}}{{}}\alpha-\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
\gamma-(3\Delta\varphi^{2})\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
\langle\alpha,\varphi\rangle+\langle\gamma,\varphi\rangle.$$
(A.12)
Here we have used for the last equality that $\varphi^{\varepsilon}=\varphi+\Delta\varphi$ only moves on the set $\Delta\varphi=\Delta\varphi^{\pm}$ where $3\alpha\Delta\varphi^{2}-\gamma=0$, and that $\Delta\varphi=-\varphi+\varphi^{\varepsilon}$ only differs from $-\varphi$ by a finite variation term. If the risky asset $S$, the frictionless optimizer $\varphi$, as well as their local quadratic variation processes $c^{S},c^{\varphi}$ (and in turn the processes $\alpha$ and $\gamma$) follow sufficiently regular Itô processes, this representation shows that this property is passed on to $\Delta S$. Moreover, since $\Delta\varphi=O(\varepsilon^{1/3})$, $\gamma=O(\varepsilon^{2/3})$, and $\alpha=O(1)$ (and the same asymptotics are valid for the drift and diffusion coefficients of $\alpha$ and $\gamma$), its diffusion coefficient is indeed of order $O(\varepsilon^{2/3})$ and its drift rate is of order $O(\varepsilon^{1/3})$. More specifically, the latter is given by $b^{\Delta S}=3\alpha\Delta\varphi c^{\varphi}+O(\varepsilon^{2/3})$; hence, by definition of $\alpha$, the drift condition (A.9) and in turn the approximate optimality condition $\mbox{ii}^{\varepsilon})$ is indeed satisfied for the shadow price $S^{\varepsilon}$ and the strategy $\varphi^{\varepsilon}$. Consequently, the latter is approximately optimal for small spreads.
A.5 Computation of the Leading-Order Utility Loss
Let us now compute – at the leading order $O(\varepsilon^{2/3})$ – the expected utility that can be obtained by applying the strategy $\varphi^{\varepsilon}$. Since the latter is approximately optimal, this will then also determine the leading-order impact of transaction costs on the certainty equivalent of trading optimally in the market.
To do this, the analysis of the previous section needs to be refined. Including a second term in the Taylor expansion of the utility function, and taking into account $\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S^{%
\varepsilon}=\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S+\Delta%
\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S+\varphi^{%
\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S$, where $\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S$ is of order $O(\varepsilon^{2/3})$,141414This follows using integration by parts to write $\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S%
=\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S+%
\varphi\Delta S-\varphi_{0}\Delta S_{0}-\Delta S\stackrel{{\scriptstyle\mbox{%
\tiny$\bullet$}}}{{}}\varphi-\langle\varphi,\Delta S\rangle$, and recalling that the drift and diffusion coefficients of $\Delta S$ are of order $O(\varepsilon^{1/3})$ and $O(\varepsilon^{2/3})$, respectively, whereas $\Delta S$ and $\Delta\varphi$ are of order $O(\varepsilon)$ resp. $O(\varepsilon^{1/3})$. gives:
$$\displaystyle E[U(x+\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$%
\bullet$}}}{{}}S^{\varepsilon}_{T})]=E[U(x+\varphi\stackrel{{\scriptstyle\mbox%
{\tiny$\bullet$}}}{{}}S_{T})]$$
$$\displaystyle+E[U^{\prime}(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet%
$}}}{{}}S_{T})(\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
S_{T}+\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
\Delta S_{T})]$$
$$\displaystyle+\frac{1}{2}E[U^{\prime\prime}(x+\varphi\stackrel{{\scriptstyle%
\mbox{\tiny$\bullet$}}}{{}}S_{T})(\Delta\varphi\stackrel{{\scriptstyle\mbox{%
\tiny$\bullet$}}}{{}}S_{T})^{2}]+O(\varepsilon).$$
For the exponential utility function $U(x)=-e^{-px}$, the marginal utility $U^{\prime}(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})$ needs to be normalized by $E[U^{\prime}(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})%
]=-pE[U(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})]$ to obtain the density of an equivalent martingale measure $Q$ for $S$. Since, moreover, the absolute risk-aversion $-U^{\prime\prime}/U^{\prime}=p$ is constant, it follows that
$$\displaystyle E[U(x+\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$%
\bullet$}}}{{}}S^{\varepsilon}_{T})]=E[U(x+\varphi\stackrel{{\scriptstyle\mbox%
{\tiny$\bullet$}}}{{}}S_{T})]\left(1-pE_{Q}[\varphi^{\varepsilon}\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S_{T}]+\frac{p^{2}}{2}E_{Q}[(%
\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})^{2}]%
\right)+O(\varepsilon),$$
where we have used that the expectation of the $Q$-martingale $\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S$ vanishes. The second correction term $\frac{p^{2}}{2}E_{Q}[(\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$%
}}}{{}}S_{T})^{2}]$ represents the leading-order relative utility loss due to displacement, incurred by trading $\varphi^{\varepsilon}$ instead of the frictionless optimizer $\varphi$ at the frictionless price $S$. The first correction term $-pE_{Q}[\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}%
}\Delta S_{T}]$ measures the utility loss incurred directly due to transaction costs, when trades are carried out at the shadow price $S^{\varepsilon}$ rather than at the mid price $S$.
Let us first focus on the displacement loss $\frac{p^{2}}{2}E_{Q}[(\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$%
}}}{{}}S_{T})^{2}]$. Integration by parts gives
$$(\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T})^{2}=2(%
\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S)\Delta\varphi%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S_{T}+\Delta\varphi^{2}%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle S\rangle_{T}.$$
As the first term is a $Q$-martingale, it follows that the leading-order displacement loss is given by
$$\frac{p^{2}}{2}E_{Q}[(\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$%
}}}{{}}S_{T})^{2}]=\frac{p^{2}}{2}E_{Q}[\Delta\varphi^{2}\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle S\rangle_{T}].$$
Now, consider the direct transaction cost loss $-pE_{Q}[\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}%
}\Delta S_{T}]$. Integration by parts and $\Delta S=O(\varepsilon)$ yield
$$\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S=-\langle%
\varphi,\Delta S\rangle+O(\varepsilon).$$
First taking into account the dynamics of $\Delta S$ (cf. (A.12)), and then inserting the definitions of $\alpha$, $\gamma$ and the trading boundaries $\Delta\varphi^{+}$, as well as $c^{S}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}I=\langle S\rangle$, it follows that
$$\displaystyle-pE_{Q}[\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}%
\Delta S_{T}]$$
$$\displaystyle=-pE_{Q}[(3\alpha\Delta\varphi^{2}-\gamma)c^{\varphi}\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}I_{T}]+O(\varepsilon)=-p^{2}E_{Q}[(%
\Delta\varphi^{2}-(\Delta\varphi^{+})^{2})\stackrel{{\scriptstyle\mbox{\tiny$%
\bullet$}}}{{}}\langle S\rangle_{T}]+O(\varepsilon).$$
The remaining term $-pE_{Q}[\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S%
_{T}]$ can again by computed by integrating the drift rate of the argument of the expectation (here, $b^{\Delta S,Q}$ denotes the drift of $\Delta S$ under the measure $Q$):
$$\displaystyle-pE_{Q}[\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}%
}}{{}}\Delta S_{T}]$$
$$\displaystyle=-pE_{Q}[\Delta\varphi b^{\Delta S,Q}\stackrel{{\scriptstyle\mbox%
{\tiny$\bullet$}}}{{}}I_{T}]$$
$$\displaystyle=-pE_{Q}[\Delta\varphi(b^{\Delta S}+c^{N,\Delta S})\stackrel{{%
\scriptstyle\mbox{\tiny$\bullet$}}}{{}}I_{T}]=-pE_{Q}[\Delta\varphi b^{\Delta S%
}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}I_{T}]+O(\varepsilon).$$
Here, the second equality follows from Girsanov’s theorem, and the third one holds since $\Delta\varphi=O(\varepsilon^{1/3})$ and the diffusion coefficient of $\Delta S$ is of order $O(\varepsilon^{2/3})$ by (A.12). Combined with the drift condition (A.9) and $c^{S}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}I=\langle S\rangle$, this yields
$$-pE_{Q}[\Delta\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\Delta S%
_{T}]=-p^{2}E_{Q}[\Delta\varphi^{2}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$%
}}}{{}}\langle S\rangle_{T}]+O(\varepsilon).$$
As a consequence, the total relative utility loss directly caused by transaction costs is given by
$$-pE_{Q}[\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}%
}\Delta S_{T}]=p^{2}E_{Q}[((\Delta\varphi^{+})^{2}-2\Delta\varphi^{2})%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle S\rangle_{T}]+O(%
\varepsilon).$$
To further simplify the formulas for both parts of the utility loss, replace – at the leading order $O(\varepsilon^{2/3})$ – the terms $\Delta\varphi^{2}$ by their expectation $\frac{1}{3}(\Delta\varphi^{+})^{2}$ under the uniform distribution on $[\Delta\varphi^{-},\Delta\varphi^{+}]$ (compare [25, 12]), which is justified below. Then, the displacement loss is determined as $\frac{p^{2}}{6}E_{Q}[(\Delta\varphi^{+})^{2}\stackrel{{\scriptstyle\mbox{\tiny%
$\bullet$}}}{{}}\langle S,S\rangle_{T}]+o(\varepsilon^{2/3})$, and the transaction cost loss is found to be given by twice that value. Hence, the total utility loss due to transaction costs is given by
$$E[U(x+\varphi^{\varepsilon}\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}S%
^{\varepsilon}_{T})]=E[U(x+\varphi\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}%
}}{{}}S_{T})]\left(1+\frac{p^{2}}{2}E_{Q}\left[(\Delta\varphi^{+})^{2}%
\stackrel{{\scriptstyle\mbox{\tiny$\bullet$}}}{{}}\langle S\rangle_{T}\right]%
\right)+o(\varepsilon^{2/3}),$$
and the claimed formula for the certainty equivalent follows by taking logarithms and Taylor expansion.
To complete the argument, it remains to verify that we can indeed pass to the uniform distribution for $\Delta\varphi$ at the leading order. To this end, define $D=\Delta\varphi\sigma^{S}$, which is an Itô process reflected to stay between the boundaries $D^{\pm}=\Delta\varphi^{\pm}\sigma^{S}$. In the interior of $[D^{-},D^{+}]$, the strategy $\varphi^{\varepsilon}$ is constant, so that $\Delta\varphi=-\varphi$. Hence, the drift rate $b^{D}$ and the diffusion coefficient $\sigma^{D}$ of $D$ are both of order $O(1)$. Now, fix a mesh $0=t^{\varepsilon}_{0}<\ldots<t^{\varepsilon}_{N^{\varepsilon}}=T$ with mesh size of order $O(\varepsilon^{1/3})$, and write
$$\int_{0}^{T}\Delta\varphi_{u}^{2}d\langle S\rangle_{u}=\int_{0}^{T}D_{u}^{2}du%
=\sum_{i=1}^{N^{\varepsilon}}\int_{t^{\varepsilon}_{i-1}}^{t^{\varepsilon}_{i}%
}D_{u}^{2}du.$$
(A.13)
Rescale $D$ by dividing by $\varepsilon^{1/3}$ and integrating over $v=u/\varepsilon^{2/3}$ instead of $u$, obtaining
$$\int_{t^{\varepsilon}_{i-1}}^{t^{\varepsilon}_{i}}D_{u}^{2}du=\varepsilon^{4/3%
}\int_{t^{\varepsilon}_{i-1}/\varepsilon^{2/3}}^{t^{\varepsilon}_{i}/%
\varepsilon^{2/3}}\left(\frac{D_{\varepsilon^{2/3}v}}{\varepsilon^{1/3}}\right%
)^{2}dv.$$
(A.14)
The drift and diffusion coefficient of the rescaled integrand $(\varepsilon^{-1/3}D_{\varepsilon^{2/3}v})_{v\geq 0}$ are given by $b^{D}_{\varepsilon^{2/3}v}\varepsilon^{1/3}=O(\varepsilon^{1/3})$ and $\sigma^{D}_{\varepsilon^{2/3}v}=O(1)$, respectively. Hence, at the leading order, it equals driftless Brownian motion with constant volatility $\sigma^{D}_{t^{\varepsilon}_{i}}$ on $t^{\varepsilon}_{i-1}/\varepsilon^{2/3}\leq v\leq t^{\varepsilon}_{i}/%
\varepsilon^{2/3}$. Reflected Brownian motion on the interval $[D^{-}_{\varepsilon^{2/3}v}/\varepsilon^{1/3},D^{+}_{\varepsilon^{2/3}v}/%
\varepsilon^{1/3}]$ (whose boundaries are constant and equal to $D^{\pm}_{t^{\varepsilon}_{i-1}}$, at the leading order) has a uniform stationary distribution with second moment $(D_{t^{\varepsilon}_{i-1}}^{+})^{2}/3\varepsilon^{2/3}$. Consequently, the ergodic theorem [5, II.35 and II.36] implies
$$\int_{t^{\varepsilon}_{i-1}/\varepsilon^{2/3}}^{t^{\varepsilon}_{i}/%
\varepsilon^{2/3}}\left(\frac{D_{\varepsilon^{2/3}v}}{\varepsilon^{1/3}}\right%
)^{2}dv=\frac{t^{\varepsilon}_{i}-t^{\varepsilon}_{i-1}}{\varepsilon^{2/3}}%
\left(\frac{(D^{+}_{t^{\varepsilon}_{i-1}})^{2}}{3\varepsilon^{2/3}}+o(1)%
\right).$$
Combining this with (A.13) and (A.14) and letting the mesh size go to zero, we obtain the assertion:
$$\int_{0}^{T}\Delta\varphi_{u}^{2}d\langle S,S\rangle_{u}=\int_{0}^{T}D_{u}^{2}%
du=\frac{1}{3}\int_{0}^{T}(D_{u}^{+})^{2}du+o(\varepsilon^{2/3}).$$
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A study of Cousin complexes through the dualizing complexes
Mohammad T. Dibaei
Institute for Studies in Theoretical Physics and
Mathematics, P.O.Box 19395-5746, Tehran, Iran.
Abstract.
For the Cousin complex of certain modules, we
investigate finiteness of cohomology modules, local duality
property and injectivity of its terms. The existence of canonical
modules of Noetherian non-local rings and the Cousin complexes
of them with respect to the height filtration are discussed .
Key words and phrases:Cousin complexes, dualizing complexes, Gorenstein modules.
This research is supported in part by MIM Grant P82–104.
Address: Mosaheb Institute of Mathematics, Teacher Training University,
599 Taleghani Avenue, 19165 Tehran, Iran.
E–mail address: [email protected] .
2000 Mathematics Subject Classification: 13D25;
13H10; 13D45
Introduction
This Paper is the continuation of [DT1] and [DT2]. We have seen in
[DT2] that if $M$ is a finitely generated module over a local ring $A$ which
possesses the fundamental dualizing complex $I^{\bullet}$,
then, under certain conditions on $M$, ${\operatorname{Hom}}_{A}(M,I^{\bullet}$)
represents the Cousin complex of the module
$H^{{\dim}A-{\dim}_{A}(M)}({\operatorname{Hom}}_{A}(M,I^{\bullet}$)),
the $({\dim}{A}-{{\dim}_{A}}(M))$-th cohomology module of the complex
${\operatorname{Hom}}_{A}($M$,I^{\bullet}$), with respect to an appropriate
filtration of ${\operatorname{Spec}}(A)$; and that we can reconstruct the Cousin
complex of the module $M$ by means of the fundamental dualizing
complex (see the proof of [DT2, Lemma 3.1]). In
section 2, we pursue our expectation that the Cousin complexes of such modules will inherit some properties of
the dualizing complex of the ring itself. We will show that if
$({A},{\mathfrak{m}})$ is a Noetherian local ring ( not necessarily
possessing a dualizing complex) such that all of its formal
fibres are Cohen-Macaulay rings, $M$ is a finitely generated $A$-module
which satisfies the condition $({S}_{2})$ of Serre and
${\operatorname{Min}}_{\widehat{A}}(\widehat{M})={\operatorname{Ass}}_{\widehat%
{A}}(\widehat{M})$,
then all the cohomology modules
of $C_{A}(M)$, the Cousin complex of $M$ with respect to the
$M$-height filtration, are finitely generated $A$-modules (a result proved also,
under different hypotheses, by T. Kawasaki in [K, Theorem 1.1]), and
also they satisfy a local duality property which is analogous to
that of the Grothendieck local duality. Here, $\widehat{M}$ denotes the
completion of $M$ with respect to the ${\mathfrak{m}}$-adic topology. We
present, in section 3, a number of applications which come out of these results
and those of [DT1] and [DT2] .
In the remainder of the paper we study the Cousin complex
of certain modules over Noetherian non-local ring $R$. In section 4
we recall the notion of canonical modules for such a ring $R$ and
prove the existence of them when $R$ possesses a dualizing
complex and satisfies $({S}_{2})$. As a result we present a partial
generalization of [BH, Proposition 3.3.18]. In section 5, we generalize [DT1, Corollary
3.4] for non-local case
and give a characterization for the Cousin
complex of a canonical module w.r.t. the height filtration to be
a dualizing complex.
Finally, we give an explicit description for all indecomposables
injective modules which improves [DT1, Corollary 3.3 ] .
1. Preliminaries
Throughout $A$ is a Noetherian local ring of dimension $d$ with
the maximal ideal ${\mathfrak{m}}$, and $M$ is a finitely generated
$A$–module of ${\dim}_{A}(M)=s$. A finitely generated $A$-module
$K_{M}$ (if it exists) is called the $canonical$ $module$ of $M$ if
$K_{M}\otimes_{A}\widehat{A}\cong{\operatorname{Hom}}_{A}({{H}}^{s}_{\mathfrak{%
m}}(M),{E}({A}/{\mathfrak{m}}))$,
where ${{H}}^{s}_{\mathfrak{m}}(M)$ is the s–th local cohomology module
of $M$ w.r.t. ${\mathfrak{m}}$, and ${E}({A}/{\mathfrak{p}})$ is the injective
envelope of the $A$–module ${A}/{\mathfrak{p}}$ with
${\mathfrak{p}}\in{\operatorname{Spec}}({A})$. The canonical module of $M$ (if exists)
is unique up to isomorphism (see [HK, Lemma 5.8] ).
1.1. Some remarks
If $A$ possesses a dualizing complex, then
it possesses the fundamental dualizing complex
${I^{\bullet}:0\longrightarrow I^{0}\overset{\delta^{0}}{\longrightarrow}{I}^{1%
}\overset{\delta^{1}}{\longrightarrow}\cdots\overset{\delta^{d-1}}{%
\longrightarrow}{I}^{d}\longrightarrow 0}$,
which we call “ the dualizing complex ” (see [H]), with
the
following properties :(i) for each $i\geq 0$, $H^{i}(I^{\bullet})$, the $i$-th cohomology
module of $I^{\bullet}$, is finitely generated.(ii) $I^{i}=\underset{{\mathfrak{p}}\in{{\operatorname{Spec}}(A)},{\dim}(A/{%
\mathfrak{p}})=d-i}{\bigoplus}E(A/{\mathfrak{p}})$, $i=0,1,...,d$.If $A$ possesses the dualizing complex $I^{\bullet}$, then the
module $K_{M}:=H^{d-s}({\operatorname{Hom}}_{A}(M,I^{\bullet}))$ is the
canonical module of $M$. If $K_{M}$ is the canonical module
of $M$, it is easy to see that $\widehat{(K_{M})}\cong{K_{\widehat{M}}}$
is the canonical module of $\widehat{M}$, as
$\widehat{A}$-module.For the module $M$, we set ${\operatorname{Min}}_{A}(M)$ to denote the set of
all minimal elements of ${\operatorname{Supp}}_{A}(M)$, and
${\operatorname{Assh}}_{A}(M)=\{{\mathfrak{p}}\in{\operatorname{Supp}}_{A}(M):{%
\dim}_{A}(A/{\mathfrak{p}})={\dim}_{A}(M)\}$.
Also $M$ is said to satisfy $(S_{n})$ if
${0pt}_{{A}_{\mathfrak{p}}}({M}_{\mathfrak{p}})\geq{\min}\{n,{\operatorname{ht}%
}_{M}({\mathfrak{p}})\}$ for all ${\mathfrak{p}}\in{\operatorname{Supp}}_{A}(M)$.
A filtration of ${\operatorname{Spec}}(A)$ is a descending sequence
$\mathcal{F}=(F_{i})_{i\geq 0}$ of subsets of ${\operatorname{Spec}}(A)$, so
that,
$F_{0}\supseteq F_{1}\supseteq\cdots\supseteq F_{i}\supseteq\cdots,$
with the property
that, for each $i\geq 0$, each member of $\partial F_{i}=F_{i}-F_{i+1}$ is a minimal member of $F_{i}$, with respect to
inclusion . We say that $\mathcal{F}$ admits $M$ if
${\operatorname{Supp}}_{A}(M)\subseteq F_{0}$. Suppose that $\mathcal{F}$ is a
filtration of ${\operatorname{Spec}}(A)$ that admits $M$. The Cousin complex
$C(\mathcal{F},M)$ for $M$ with respect to $\mathcal{F}$ has the
form
$0\overset{d^{-2}}{\longrightarrow}M\overset{d^{-1}}{\longrightarrow}M^{0}%
\overset{d^{0}}{\longrightarrow}M^{1}\longrightarrow\cdots\longrightarrow M^{n%
-1}\overset{d^{n-1}}{\longrightarrow}M^{n}\longrightarrow\cdots$
with $M^{n}=\bigoplus_{{\mathfrak{p}}\in{\partial{F}_{n}}}({\operatorname{Coker}}d^{%
n-2})_{\mathfrak{p}}$ for all $n\geq 0$, and with differentiation
$d^{n}$, as recalled in [T].
Set ${\mathcal{H}}_{M}=(H_{i})_{i\geq 0}$ to be the
$M$-height filtration of ${\operatorname{Spec}}(A)$, i.e. $H_{i}=\{{\mathfrak{p}}\in{\operatorname{Supp}}_{A}(M):{\operatorname{ht}}_{M}(%
{\mathfrak{p}})\geq i\}$. We denote the Cousin
complex of $M$ with respect to ${\mathcal{H}}_{M}$ by $C_{A}(M)$.
Set $\bar{A}=A/0:_{A}M$. Then $M$ has a natural
structure as $\bar{A}$-module. It is straightforward to see that
each term of the complex $C_{A}(M)$ has a natural $\bar{A}$-module
structure and each differentiation of $C_{A}(M)$ is an
$\bar{A}$–homomorphism. Moreover, it is straightforward to see
that :
1.2. Lemma
If M is a finitely generated A-module and
$\bar{A}:=A/0:_{A}M$, then there exists an isomorphism of
complexes $C_{A}(M)\cong C_{\bar{A}}(M)$.
The following lemma will be used later.
1.3. Lemma
[P, Theorem 3.5] Suppose that all formal
fibres of A are Cohen-Macaulay. If M is a finitely generated
A-module, then there is a morphism of complexes $u^{\bullet}:C_{A}(M)\bigotimes_{A}\widehat{A}\longrightarrow C_{\widehat{A}}(%
\widehat{M})$
which is a monomorphism. Moreover the quotient complex
$Q^{\bullet}$, in the exact sequence
$0\longrightarrow C_{A}(M)\bigotimes_{A}\widehat{A}\overset{u^{\bullet}}{%
\longrightarrow}C_{\widehat{A}}(\widehat{M})\longrightarrow Q^{\bullet}\longrightarrow
0$,
is an exact complex, so that, for each $i\geq 0$,
there exists an $\widehat{A}$–isomorphism $H^{i}(C_{A}(M))\bigotimes_{A}\widehat{A}\cong H^{i}(C_{\widehat{A}}(\widehat{M%
}))$.
1.4. Convention
For a complex $C^{\bullet}:0\longrightarrow C^{-1}\overset{\theta^{-1}}{\longrightarrow}C^{0}%
\overset{\theta^{0}}{\longrightarrow}C^{1}\overset{\theta^{1}}{\longrightarrow}\cdots$ of $A$ –module and $A$–homomorphisms, we
denote
$C^{\prime}:0\longrightarrow C^{0}\overset{\theta^{0}}{\longrightarrow}C^{1}%
\overset{\theta^{1}}{\longrightarrow}\cdots$ and $({C^{\prime}})^{\ast}:0\longrightarrow H^{0}(C^{\prime})\longrightarrow C^{0}%
\longrightarrow C^{1}\longrightarrow\cdots$.
2. Some properties of Cousin complexes
In this section we
establish some properties of certain complexes by means of
dualizing complexes. First we show that these Cousin complexes
have finitely generated cohomologies.
2.1. Theorem
Let A be a ring with Cohen-Macaulay
formal fibres. Assume that M satisfies $(S_{2})$ and
${\operatorname{Min}}_{\widehat{A}}(\widehat{M})={\operatorname{Assh}}_{%
\widehat{A}}(\widehat{M})$.
Then
${C_{A}(M)^{\prime}}$ has finitely generated cohomology modules.
Proof. Since $M$ satisfies $(S_{2})$, the Cousin
complex $C_{A}(M)$ is exact at $M$ and $M^{0}$ (see [SSc, Example
4.4]). Thus $H^{0}(C_{A}(M)^{\prime})=M$. So it is enough to prove that
$H^{i}(C_{A}(M)^{\prime})$ is finitely generated for all $i>0$. Note that,
for $i>0$, we have $H^{i}(C_{A}(M)^{\prime})=H^{i}(C_{A}(M))$. By
1.3, we have $H^{i}(C_{A}(M))\bigotimes_{A}\widehat{A}\cong H^{i}(C_{\widehat{A}}(\widehat{M%
}))$. Therefore
$C_{\widehat{A}}(\widehat{M})$ is also exact at $\widehat{M}$ and
$(\widehat{M})^{0}$ ; so that $\widehat{M}$ satisfies $(S_{2})$ as
$\widehat{A}$–module. Since ${\operatorname{Min}}_{\widehat{A}}(\widehat{M})={\operatorname{Assh}}_{%
\widehat{A}}(\widehat{M})$, by [DT2, Theorem 3.2], all
cohomology modules $H^{i}(C_{\widehat{A}}(\widehat{M}))$ are
finitely generated $\widehat{A}$–module. Now, by [M, Exercise
7.3], the claim follows. $\Box$
2.2. Corollary
Assume
that the ring $A$ satisfies $(S_{2})$ and all formal fibres of
$A$ are Cohen-Macaulay. Then $C_{A}(A)$, the Cousin complex of
$A$, has finitely generated cohomology modules.
Proof. By [M, Theorem 23.9], $\widehat{A}$
satisfies
$(S_{2})$ and
thus ${\operatorname{Min}}(\widehat{A})={\operatorname{Assh}}(\widehat{A})$ (see[DT1, Remark
1.3]). $\Box$
For a ring $A$ and a property $P$, the $P$
locus of $A$ is defined to be the set $P(A)=\{{\mathfrak{p}}\in{\operatorname{Spec}}(A):P$ holds for $A_{\mathfrak{p}}\}$. We show that the
$(S_{n})$ locus of any $(S_{2})$ local ring with Cohen-Macaulay
formal fibres is an open subset of ${\operatorname{Spec}}(A)$ for all $n\geq 2$
.
2.3. Corollary
If A satisfies $(S_{2})$ and all formal
fibres of $A\longrightarrow\widehat{A}$ are Cohen-Macaulay
, then for each $n\geq 0$, $S_{n}(A)$ is an open subset of
${\operatorname{Spec}}(A)$, in the Zariski topology. In particular, CM(A)
is an open subset of ${\operatorname{Spec}}(A)$.
Proof. It follows that $\widehat{A}$ is
$(S_{2})$. We assume that $n\geq 3$. Set $U_{i}={\operatorname{Spec}}(A)-{\operatorname{Supp}}_{A}(H^{i}(C_{A}(A))),1%
\leq i\leq n-2$. Each
$U_{i}$ is an open subset of ${\operatorname{Spec}}A$, because
${\operatorname{Supp}}_{A}(H^{i}(C_{A}(A)))=V(0:_{A}H^{i}(C_{A}(A)))$ by 2.2.
Set $W=\cap_{i=1}^{n-2}U_{i}$. We show that $S_{n}(A)=W$. Let
${\mathfrak{p}}\in S_{n}(A)$; so that $A_{\mathfrak{p}}$ is $(S_{n})$. Thus, by
[SSc, Example 4.4], $H^{n}(C_{A_{\mathfrak{p}}}(A_{\mathfrak{p}}))=0$ for $1\leq i\leq n-2$. Therefore, by [S1, Theorem 3.5], we have that
${\mathfrak{p}}\in U_{n}$ for all $i$, $1\leq i\leq n-2$; that is ${\mathfrak{p}}\in W$. In a
similar way, we have $W\subseteq S_{n}(A)$.$\square$
Next, we state a local duality property for
the Cousin complexes of certain modules.
2.4. Theorem
(Local duality for certain Cousin
complexes). Assume that all formal fibres of A are
Cohen-Macaulay, M satisfies $(S_{2})$, and that
${\operatorname{Min}}_{\widehat{A}}(\widehat{M})={\operatorname{Assh}}_{%
\widehat{A}}(\widehat{M})$.
Then, for each $i\geq 0$, $D_{A}H^{i}(C_{A}(M)^{\prime})\cong H_{\widehat{{\mathfrak{m}}}}^{s-i}(K_{%
\widehat{M}})$, where $D_{A}:={\operatorname{Hom}}_{A}(-,E(A/{\mathfrak{m}}))$. Moreover, if M admits a canonical
module, then the completion signs on the right hand side of the
above isomorphism can be removed.
Proof. Set
$\bar{A}=A/0:_{A}M$ and
$\bar{\widehat{A}}=\widehat{A}/0:_{\widehat{A}}\widehat{M}$. It
is straightforward to see that $\bar{\widehat{A}}$ and
$\widehat{\bar{A}}$ are isomorphic rings. Let $J^{\bullet}$ be the
dualizing complex for $\widehat{A}$ and assume that
$I^{\bullet}={\operatorname{Hom}}_{\widehat{A}}(\bar{\widehat{A}},J^{\bullet})$
such that $I^{0}={\operatorname{Hom}}_{\widehat{A}}(\bar{\widehat{A}},J^{d-s})$. Hence $I^{\bullet}$ is the dualizing complex for
$\bar{\widehat{A}}$. As seen in the proof of 2.1, $\widehat{M}$
satisfies $(S_{2})$ as $\widehat{A}$–module. It is easy to see
that $\widehat{M}$ also satisfies $(S_{2})$ as
$\bar{\widehat{A}}$-module. Since
${\dim}_{\bar{\widehat{A}}}(\widehat{M})={\dim}(\bar{\widehat{A}})$,
we have, by the proof of [DT2, Lemma 3.1], the isomorphism of
complexes
$$C_{\bar{\widehat{A}}}(\widehat{M})^{\prime}\cong{\operatorname{Hom}}_{\bar{%
\widehat{A}}}(K_{\widehat{M}},I^{\bullet}).$$
Therefore, by 2.1 and [B-ZS, Corollary 2.5], we have
(1)
$$D_{\bar{\widehat{A}}}H^{i}(C_{\bar{\widehat{A}}}(\widehat{M})^{\prime})\cong H%
_{\bar{\widehat{{\mathfrak{m}}}}}^{s-i}(K_{\widehat{M}})$$
for all $i\geq 0$.
On the other hand each formal fibre of $\bar{A}$ is also a formal
fibre of $A$ and $\bar{\widehat{A}}\cong\widehat{\bar{A}}$. Hence,
from 1.3, we have
(2)
$$H^{i}(C_{\widehat{\bar{A}}}(\widehat{M})^{\prime})\cong H^{i}(C_{\bar{A}}(M)^{%
\prime})\otimes_{\bar{A}}\widehat{\bar{A}},$$
for all $i>0$. From (1) and (2), we obtain
(3)
$${\operatorname{Hom}}_{\widehat{\bar{A}}}(H^{i}(C_{\bar{A}}(M)^{\prime})\otimes%
_{\bar{A}}\widehat{\bar{A}},E_{\widehat{\bar{A}}}(\widehat{\bar{A}}/\widehat{%
\bar{{\mathfrak{m}}}}))\cong H_{\widehat{\bar{{\mathfrak{m}}}}}^{s-i}(K_{%
\widehat{M}}).$$
The left hand side of (3) is
isomorphic to
$${\operatorname{Hom}}_{\bar{A}}(H^{i}(C_{\bar{A}}(M)^{\prime}),{\operatorname{%
Hom}}_{\widehat{\bar{A}}}(\widehat{\bar{A}},E_{\widehat{\bar{A}}}(\widehat{%
\bar{A}}/\widehat{\bar{{\mathfrak{m}}}})))$$
which, in turn, is isomorphic to
${\operatorname{Hom}}_{\bar{A}}(H^{i}(C_{\bar{A}}(M)^{\prime}),E_{\widehat{\bar%
{A}}}(\widehat{\bar{A}}/\widehat{\bar{{\mathfrak{m}}}}))$.
Thus, we have from (3), the isomorphism
(4)
$${\operatorname{Hom}}_{\bar{A}}(H^{i}(C_{\bar{A}}(M)^{\prime}),E_{\bar{A}}({%
\bar{A}}/{\bar{{\mathfrak{m}}}}))\cong H_{\bar{\widehat{{\mathfrak{m}}}}}^{s-i%
}(K_{\widehat{M}}).$$
Assume that $N$ is an $A$, $\bar{A}$–bimodule such that, for $a\in A$ and $x\in N$, $ax=\bar{a}x$, where $-:A\longrightarrow\bar{A}$ is the natural map. Then we have
$$\begin{array}[]{llll}{\operatorname{Hom}}_{\bar{A}}(N,E_{\bar{A}}(\bar{A}/\bar%
{{\mathfrak{m}}}))&\cong{\operatorname{Hom}}_{\bar{A}}(N,{\operatorname{Hom}}_%
{A}(\bar{A},E(A/{\mathfrak{m}})))\\
&\cong{\operatorname{Hom}}_{A}(N\otimes_{\bar{A}}\bar{A},E(A/{\mathfrak{m}}))%
\\
&\cong{\operatorname{Hom}}_{A}(N,E(A/{\mathfrak{m}})).\end{array}$$
By Independence Theorem for the local cohomologies, we
have $H_{\bar{\widehat{{\mathfrak{m}}}}}^{s-i}(K_{\widehat{M}})\cong H_{\widehat{{%
\mathfrak{m}}}}^{s-i}(K_{\widehat{M}})$.
Put all these together, we obtain, from (4) and 1.2, that
${\operatorname{Hom}}_{A}(H^{i}(C_{A}(M)^{\prime}),E(A/{\mathfrak{m}}))\cong H_%
{\widehat{{\mathfrak{m}}}}^{s-i}(K_{\widehat{M}}),i=0,1,...,$
as $A$ and $\widehat{A}$–modules.
If
$M$ admits a canonical module $K_{M}$, we then have
$\widehat{(K_{M})}\cong K_{\widehat{M}}$, and by the Artinianness of
$H_{{\mathfrak{m}}}^{s-i}(K_{M})$, we get the final claim. $\Box$
3. Applications
First we show that over a local ring with
Cohen–Macaulay formal fibres, certain $f$–modules are also
generalized Cohen–Macaulay modules. Recall that $M$ is called
generalized Cohen–Macaulay (abbr. g.CM) if there exists
$r\geq 1$ such that, for each system of parameters $x_{1},...,x_{s}$ for $M$ and for all $i=1,...,s$,
${\mathfrak{m}}^{r}[((x_{1},...,x_{i-1})M:x_{i})/(x_{1},...,x_{i-1})M]=0.$
Note that, by [ScTC, (3.2) and (3.3)], $M$ is a g.CM
module if and only if $H_{{\mathfrak{m}}}^{i}(M)$ is of finite length for
all $i=0,1,...,s-1$ .
An $A$–module $M$ is called an
$f$–module if for each system of parameters $x_{1},...,x_{s}$
for $M$
${\operatorname{Supp}}_{A}[((x_{1},...,x_{i-1})M:x_{i})/(x_{1},...,x_{i-1})M]%
\subseteq{\{{\mathfrak{m}}\}}$
for all $i=1,...,s$. It is clear that if $M$ is g.CM
module then it is an $f$–module.
3.1. Theorem
(Compare [ScTC, (3.8)]).
Assume that all formal fibres of A are Cohen–Macaulay. Let M
be an A–module such that
${\operatorname{Min}}_{\widehat{A}}(\widehat{M})={\operatorname{Assh}}_{%
\widehat{A}}(\widehat{M})$.
If M is an f–module with ${0pt}_{A}(M)\geq 2$, then
M is a
g.CM module.
Proof. By a straightforward argument and using the
equivalent definition of $f$–module [T, Lemma 1.2 (ii)], it can
be shown that $M$ is $(S_{2})$ and that
${\operatorname{Min}}_{A}(M)={\operatorname{Assh}}_{A}(M)$.
Now, by [T, Lemma 1.2 (iv)], the $M$–height filtration of
Spec($A$) is the same as the $M$–dimension filtration
$\mathcal{D}$ of Spec($A$), where $\mathcal{D}=(D_{i})_{i\geq 0},D_{i}=\{{\mathfrak{p}}\in{\operatorname{Supp}}_{%
A}(M):{\dim}(A/{\mathfrak{p}})\leq s-i\}$. Thus,
by [DT1, Lemma 3.1], there exists an isomorphism
$C_{A}(M)=C(\mathcal{D},M)\cong C(\mathcal{U},M)$
(over ${\operatorname{Id}}_{M}$), where $C(\mathcal{U},M)$ is the complex of
modules of generalized fractions on $M$ with respect to the chain
of triangular subsets $\mathcal{U}=(U_{i})_{i\geq 1}$ on $A$,
defined by
$U_{i}=\{(x_{1},\cdots,x_{i})\in A^{i}:$
there exists $j$ with $0\leq j\leq i$ such that $x_{1},\cdots,x_{j}$
is an s.s.o.p. for $M$ and $x_{j+1}=\cdots=x_{i}=1\}$
(See [DT1] for details). By [SZ, Corollary 2.3 and
Theorem 2.4],
$H^{i-1}(C_{A}(M))\cong H_{{\mathfrak{m}}}^{i}(M),i=1,\cdots,s-1$.
Therefore, by
Theorem 2.1, $H_{{\mathfrak{m}}}^{i}(M)$ is of finite length for all
$i=0,1,\cdots,s-1.\Box$
3.2. Corollary
Assume
that all formal fibres of A are Cohen-Macaulay. If A is an f-ring
with ${0pt}(A)\geq 2$, then A is a g.CM ring.
Proof. As we have seen in the proof of 3.1, $A$
is $(S_{2})$. By [M, Theorem 23.9], $\widehat{A}$ satisfies $(S_{2})$.
Thus ${\operatorname{Min}}(\widehat{A})={\operatorname{Assh}}(\widehat{A})$ (see [AG, Lemma 1.1]). Now
the result follows from Theorem 3.1. $\Box$
Our next
application studies the injectivity of the terms of the Cousin
complex $C_{A}(M)$.
In [S2],
a finitely generated $A$-module $M$ is defined to be a Gorenstein $A$-module
whenever its Cousin complex provides a minimal injective
resolution. It is also proved that if $A$ admits a canonical
module $\Omega$, then any Gorenstein $A$-module is isomorphic to
the direct sum of a finite number of copies of $\Omega$ [S3, Theorem
2.1].
It is known that if $A$ does not have a canonical
module and has a Gorenstein module, then it has a unique
indecomposable Gorenstein module $G$ and every Gorenstein
$A$-module is isomorphic to a direct sum of a finite number of
copies $G$ (see[FFGR and S2]). Here we extend this result and show
that for any finitely generated module $M$, over a complete $(S_{2})$
local ring $A$ which satisfies $(S_{2})$, if $0:_{A}M=0$ and
$C_{A}(M)^{\prime}$ is an injective complex, then $M$ is isomorphic to a
direct sum of copies of a uniquely determined indecomposable
one.
3.3. Theorem
Let A satisfy $(S_{2})$ and
suppose that it possesses a dualizing complex. Assume that M
satisfies $(S_{2})$ and $0:_{A}M=0$. The following
statements are equivalent:
(i) $C_{A}(M)^{\prime}$ is an injective
complex;
(ii) $M$ is isomorphic to a direct sum
of a finite number of copies of the canonical module $K$ of the ring
$A$.
Proof. (i)$\Rightarrow$ (ii). We do not need
$A$ to satisfy $(S_{2})$ in this part. The proof is a
straightforward adaptation of the argument in [S3, Theorem
2.1(v)]. Let $K$ denote the canonical module of $A$.
Let
$I^{\bullet}:0\longrightarrow I^{0}\longrightarrow I^{1}\longrightarrow\cdots%
\longrightarrow I^{d}\longrightarrow 0$
be the dualizing complex for $A$ so that $K=H^{0}(I^{\bullet})$. By
the proof of [DT2, Lemma 3.1], $C_{A}(M)\cong{\operatorname{Hom}}_{A}(K_{M},I^{\bullet})^{\ast}$, where
$K_{M}={\operatorname{Hom}}_{A}(M,K)$. Hence all cohomology modules
of $C_{A}(M)$ are finitely generated (see [S4, Lemma 3.4(ii)]).
By [S5, Theorem], ${\operatorname{Hom}}_{A}(K_{M},I^{d})\cong H_{{\mathfrak{m}}}^{d}(M)$. As $H_{{\mathfrak{m}}}^{d}(M)$ is an Artinian
injective $A$-module, we may write $H_{\mathfrak{m}}^{d}(M)\cong\oplus_{i=1}^{n}E(A/{\mathfrak{m}})$, say. Using the Matlis functor
${\operatorname{Hom}}_{A}(-,E(A/{\mathfrak{m}}))$ and that $I^{d}=E(A/{\mathfrak{m}})$, we
obtain $K_{M}\otimes_{A}\widehat{A}\cong(\oplus_{i=1}^{n}A)\otimes_{A}\widehat{A}$. This implies, by[HK,
Lemma 5.8], that $K_{M}\cong\oplus_{i=1}^{n}A$. Hence we have
$H_{{\mathfrak{m}}}^{d}(K_{M})\cong\oplus_{i=1}^{n}H_{{\mathfrak{m}}}^{d}(A)$.
On the other hand, by Grothendieck’s Local Duality
Theorem [B-ZS, Corollary 2.5] and the fact that $M$ satisfies
$(S_{2})$ so $C_{A}(M)$ is exact at point $-1,0$ (see [SSc,
Example 4.4]), we obtain
$H_{{\mathfrak{m}}}^{d}(K_{M})\cong{\operatorname{Hom}}_{A}(H^{0}(C_{A}(M)^{%
\prime}),E(A/{\mathfrak{m}}))\cong{\operatorname{Hom}}_{A}(M,E(A/{\mathfrak{m}%
}))$.
By applying the Matlis functor again, we get
$M\otimes_{A}\widehat{A}\cong{\operatorname{Hom}}_{A}(H_{{\mathfrak{m}}}^{d}(K_%
{M}),E(A/{\mathfrak{m}}))\cong{\operatorname{Hom}}_{A}(\oplus_{i=1}^{n}H_{{%
\mathfrak{m}}}^{d}(A),E(A/{\mathfrak{m}}))\cong(\oplus_{i=1}^{n}K)\otimes_{A}%
\widehat{A}$. Now,
by [HK, Lemma 5.8], $M\cong\oplus_{i=1}^{n}K$.
(ii)$\Rightarrow$(i). We have
${\operatorname{Supp}}_{A}(M)={\operatorname{Supp}}_{A}(K)={\operatorname{Spec}%
}(A)$ (see [A, (1.8)] and
[AG, Lemma 1.1]). It is routine to check that $C_{A}(M)\cong\oplus_{i=1}^{n}C_{A}(K)$. As ${\operatorname{Min}}A={\operatorname{Assh}}A$ and the
dimension filtration and the height filtration of ${\operatorname{Spec}}(A)$
are the same (see [A, (1.9)]), the claim follows by [DT1,
Corollary 3.4].$\Box$
3.4. Corollary
Assume that $\widehat{A}$ satisfies
$(S_{2})$. Then the following statements are equivalent :
(i) $C_{\widehat{A}}(\widehat{A})^{\prime}$ is an injective complex of
$\widehat{A}$-modules;
(ii) A is the canonical
module of A.
Moreover, if A satisfies one of the
above equivalent conditions, then A is Gorenstein if and only if
$\widehat{A}$ satisfies $(S_{n})$ for some $n\geq(1/2){\dim}A+1$.
Proof. (i)$\Rightarrow$ (ii). Set $\Omega$ for
the canonical module of $\widehat{A}$. By 3.3, $\widehat{A}\cong\Omega^{n}$ for some $n$. Thus $H_{\widehat{{\mathfrak{m}}}}^{d}(\widehat{A})\cong\oplus_{i=1}^{n}H_{\widehat{%
{\mathfrak{m}}}}^{d}(\Omega)\cong\oplus_{i=1}^{n}E(\widehat{A}/\widehat{{%
\mathfrak{m}}})$ and, by applying
${\operatorname{Hom}}_{\widehat{A}}(-,E(\widehat{A}/\widehat{{\mathfrak{m}}}))$, we get $\Omega\cong\widehat{A}^{n}$. Thus $\widehat{A}^{n^{2}}=\widehat{A}$, which
implies $n=1$ and so $A$ is the canonical module of $A$.
(ii)$\Rightarrow$(i). As $\widehat{A}$ is the canonical
module of $\widehat{A}$ and $\widehat{A}$ satisfies $(S_{2})$,
$C_{\widehat{A}}(\widehat{A})^{\prime}$ is the dualizing complex of $\widehat{A}$
[DT1, Corollary 3.4].
For the last part, we may
assume that $A$ is complete. By [SSc, Example 4.4], $C_{A}(A)$
is exact at points $-1,0,1,...,n-2$, from which it follows, by
Theorem 2.4, that $H_{{\mathfrak{m}}}^{{\dim}A-i}(A)=0$ for
$0<i\leq n-2$. On the other hand, as $A$ satisfies $(S_{n})$,
$H_{{\mathfrak{m}}}^{i}(A)=0$ for all $i<$min$\{d,n\}$. As
${\dim}A-(n-2)\leq n$, it follows that $H_{{\mathfrak{m}}}^{i}(A)=0$ for
all $i<d$, which imply the exactness of $C_{A}(A)$. The other
side is trivial.$\Box$
4. Canonical modules of non–local rings
Recall that, for a Noetherian (not necessarily local) ring $R$, the canonical module
of R (if it exists) is a finite $R$– module $K$ such that
$K_{{\mathfrak{m}}}$, the localization of $K$ at any maximal ideal
${\mathfrak{m}}$ of $R$, is the canonical module of $R_{{\mathfrak{m}}}$. In
order to generalize our results to the non–local case one might
ask whether a canonical module exists even when $R$ possesses
a dualizing complex. We will show that, if $R$ satisfies
$(S_{2})$ and all formal fibres of $R_{{\mathfrak{m}}}$, for any maximal ideal
${\mathfrak{m}}$ of $R$, are
Cohen-Macaulay, then existence of a canonical module for $R$ is
equivalent to the statement that $R$ possesses a dualizing complex.
Throughout, $R$ is a Noetherian ring of finite dimension
which is not necessarily local.
Assume that $R$ possesses a dualizing complex
$I^{\bullet}$ and $t({\mathfrak{p}};I^{\bullet}),{\mathfrak{p}}\in{\operatorname{Spec}}R$, denotes the unique integer $i$
for which ${\mathfrak{p}}$ occurs in $I^{i}$ (see [H, page 23]).
4.1. Proposition
Assume that R satisfies
$(S_{2})$ and that it possesses a dualizing complex
$I^{\bullet}$. If ${\mathfrak{p}},{\mathfrak{q}}\in{\operatorname{Min}}(R)$ such that
${\mathfrak{p}}\subseteq{\mathfrak{r}}$ and ${\mathfrak{q}}\subseteq{\mathfrak{r}}$ for some
${\mathfrak{r}}\in{\operatorname{Spec}}(R)$, then $t({\mathfrak{p}};I^{\bullet})=t({\mathfrak{q}};I^{\bullet})$.
Proof. We may assume that $R$ is
a local ring and that its maximal ideal is ${\mathfrak{r}}$. As $R$
satisfies $(S_{2})$ and possesses a dualizing complex, then
${\operatorname{Min}}(R)={\operatorname{Assh}}(R)$ [A; 1.1]. Therefore $t({\mathfrak{p}};I^{\bullet})=t({\mathfrak{q}};I^{\bullet}).\Box$
4.2. Notation
Assume that $R$ satisfies $(S_{2})$ and
that
$I^{\bullet}:0\longrightarrow I^{0}\overset{\delta^{0}}{\longrightarrow}I^{1}%
\overset{\delta^{1}}{\longrightarrow}\cdots\overset{\delta^{l-1}}{%
\longrightarrow}I^{l}\longrightarrow 0$
is a
dualizing complex for $R$. It follows that ${\operatorname{Ass}}_{R}(I^{0})\subseteq{\operatorname{Min}}R$. Assume that
${\operatorname{Ass}}_{R}(I^{0})\neq{\operatorname{Min}}(R)$. Let $r$ be the greatest
integer such that $X:={\operatorname{Min}}(R)\bigcap{\operatorname{Ass}}_{R}(I^{r})\neq\emptyset$. Set, for each
$i\geq 0$,
$X_{i}=\{{\mathfrak{p}}\in{\operatorname{Ass}}_{R}(I^{i}):{\mathfrak{p}}$ contains some element of
$X\}$;
$X_{i}^{\prime}={\operatorname{Ass}}_{R}(I^{i})\setminus X_{i}$;
$I_{1}^{i}=\oplus_{{\mathfrak{p}}\in X_{i}}E(A/{\mathfrak{p}}),I_{2}^{i}=\oplus%
_{{\mathfrak{p}}\in X_{i}^{\prime}}E(A/{\mathfrak{p}})$,
so
that we have $I^{i}=I_{1}^{i}\oplus I_{2}^{i}$.
4.3. Proposition
With the notations as in 4.2,
${\operatorname{Hom}}_{R}(I_{1}^{i},I_{2}^{i+1})=0={\operatorname{Hom}}_{R}(I_{%
2}^{i},I_{1}^{i+1})$.
Proof. If ${\operatorname{Hom}}_{R}(I_{1}^{i},I_{2}^{i+1})\neq 0$, then ${\operatorname{Hom}}_{R}(I_{1}^{i},E(R/{\mathfrak{p}}))\neq 0$ for some ${\mathfrak{p}}\in X_{i+1}^{\prime}$. Assume that
$f:I_{1}^{i}\longrightarrow E(R/{\mathfrak{p}})$ is an
$R$–homomorphism and that $x\in I_{1}^{i}$. Let
$x=x_{1}+\cdots+x_{s}$, where $x_{j}\in E(R/{\mathfrak{p}}_{j}),1\leq j\leq s$. By definition of $X_{i+1}^{\prime}$, we have
${\mathfrak{p}}_{1}\bigcap\cdots\bigcap{\mathfrak{p}}_{s}\nsubseteqq{\mathfrak{%
p}}$.
Take $t\in{\mathfrak{p}}_{1}\bigcap\cdots\bigcap{\mathfrak{p}}_{s}\setminus{%
\mathfrak{p}}$.
Hence $t^{m}x=0$ for some positive integer $m$. On the other
hand the map $E(R/{\mathfrak{p}})\overset{t^{m}}{\longrightarrow}E(R/{\mathfrak{p}})$ is an isomorphism. Thus $t^{m}f(x)=f(t^{m}x)=0$
implies that $f(x)=0$.
To show that ${\operatorname{Hom}}_{R}(I_{2}^{i},I_{1}^{i+1})=0$,
we may assume, on the contrary, that
${\operatorname{Hom}}_{R}(I_{2}^{i},E(R/{\mathfrak{p}}))\neq 0$ for some ${\mathfrak{p}}\in X_{i+1}$. So we may assume that
${\mathfrak{p}}^{\prime}\subseteq{\mathfrak{p}}$, for some ${\mathfrak{p}}^{\prime}\in X_{i}^{\prime}$. By
localizing at ${\mathfrak{p}}$, we get ${\operatorname{Min}}(R_{{\mathfrak{p}}})={\operatorname{Assh}}(R_{{\mathfrak{p%
}}})$, because $R_{{\mathfrak{p}}}$ satisfies $(S_{2})$
and $R_{{\mathfrak{p}}}$ possesses a dualizing complex. As
${\mathfrak{p}}^{\prime}R_{{\mathfrak{p}}}$ contains a minimal element
${\mathfrak{q}}R_{{\mathfrak{p}}}\in{\operatorname{Assh}}(R_{{\mathfrak{p}}})$, we get, by 4.1,
$t({\mathfrak{q}};I^{\bullet})=r$ and that ${\mathfrak{q}}\in X$. This contradicts with the
definition of $X_{i}^{\prime}.\Box$
4.4. Theorem
Assume that R satisfies $(S_{2})$ and
that it possesses a dualizing complex. Then R possesses a
dualizing complex
$J^{\bullet}:0\longrightarrow J^{0}\longrightarrow J^{1}\longrightarrow\cdots%
\longrightarrow J^{d}\longrightarrow 0,d={\dim}R,$
such that ${\operatorname{Ass}}_{R}(J^{0})={\operatorname{Min}}(R)$. In
particular $R$ admits a canonical module.
Proof. The proof is influenced by [H, Lemma 3.1]. Suppose that
$I^{\bullet}:0\longrightarrow I^{0}\overset{\delta^{0}}{\longrightarrow}I^{1}%
\overset{\delta^{1}}{\longrightarrow}\cdots\overset{\delta^{l-1}}{%
\longrightarrow}I^{l}\longrightarrow 0$
is the dualizing complex for $R$. Assume further that
${\operatorname{Ass}}(I^{0})\neq{\operatorname{Min}}(R)$ and that $r$ is the greatest integer
with $X:={\operatorname{Min}}(R)\bigcap{\operatorname{Ass}}_{R}(I^{r})\neq\emptyset$. We set
$X_{i},X^{\prime}_{i},I_{1}^{i}$ and $I_{2}^{i}$ as in 4.2. Note that
$I_{1}^{i}=0$ and $I_{2}^{i}=I^{i}$ for $0\leq i<r$
(see [S6, Lemma 3.3]).
We construct a dualizing complex
$J^{\bullet}:0\longrightarrow J^{0}\overset{\eta_{0}}{\longrightarrow}J^{1}%
\overset{\eta_{1}}{\longrightarrow}\cdots$
as follows.
Set $J^{i}=I_{1}^{i+r}\oplus I_{2}^{i}$ for all $i\geq 0$, and
define $\eta^{i}:J^{i}\longrightarrow J^{i+1}$ by
$\eta^{i}(x+y)=\delta_{1}^{i+r}(x)+\delta_{2}^{i}(y)$ for $x\in I_{1}^{i+r},y\in I_{2}^{i}$, where
$\delta_{1}^{j}:=\delta^{j}\mid_{I_{1}^{j}}$ and $\delta_{2}^{j}:=\delta^{j}\mid_{I_{2}^{j}},j\geq 0$. It follows
from Proposition 4.3 that $J^{\bullet}$ is a complex. To show that
$H^{i}(J^{\bullet})$ is a finitely generated $R$–module for all
$i\geq 0$, we note that, by a straightforward argument, there are
two natural isomorphisms
$H^{i}(I^{\bullet})\cong({\operatorname{Ker}}\delta_{1}^{i}/{\operatorname{Im}}%
\delta_{1}^{i-1})\oplus({\operatorname{Ker}}\delta_{2}^{i}/{\operatorname{Im}}%
\delta_{2}^{i-1})$,
$H^{i}(J^{\bullet})\cong({\operatorname{Ker}}\delta_{1}^{i+r}/{\operatorname{Im%
}}\delta_{1}^{i+r-1})\oplus({\operatorname{Ker}}\delta_{2}^{i}/{\operatorname{%
Im}}\delta_{2}^{i-1}),i\geq 0.$
Therefore $J^{\bullet}$ is a dualizing complex for $A$. Now we have $J^{0}=I_{1}^{r}\oplus I^{0}$ and thus
${\operatorname{Ass}}_{R}(I^{0})\subsetneqq{\operatorname{Ass}}_{R}(J^{0})$. So after a finite
number of steps we are finished.
Finally, let
$J^{\bullet}$ be a dualizing complex with
${\operatorname{Min}}(R)={\operatorname{Ass}}_{R}(J^{0})$. For each ${\mathfrak{m}}\in{\operatorname{Max}}(R)$, the
complex
$0\longrightarrow(J^{0})_{{\mathfrak{m}}}\longrightarrow(J^{1})_{{\mathfrak{m}}%
}\longrightarrow\cdots\longrightarrow(J^{t({\mathfrak{m}};J^{\bullet}}))_{{%
\mathfrak{m}}}\longrightarrow 0$
is the dualizing complex for
$R_{{\mathfrak{m}}}$, so that, by Grothendieck’s Local Duality Theorem
[B-ZS, Corollary 2.5], $(H^{0}(J^{\bullet}))_{{\mathfrak{m}}}$ is the
canonical module of
$R_{{\mathfrak{m}}}$. Thus $H^{0}(J^{\bullet})$ is a canonical module of $R$. $\Box$
As an application, we can give a partial generalization of
[BH, Proposition 3.3.18].
4.5. Theorem
Assume that R
satisfies $(S_{2})$ and possesses a dualizing complex
$I^{\bullet}:0\longrightarrow I^{0}\longrightarrow I^{1}\longrightarrow\cdots%
\longrightarrow I^{d}\longrightarrow 0$,
with $d={\dim}R,I^{i}=\underset{{\operatorname{ht}}{\mathfrak{p}}=i}{\oplus}E(R/{%
\mathfrak{p}}),i=0,1,\cdots$
and that $K_{R}$ is a canonical module of $R$.
(a) The following conditions
are equivalent:
(i) $K_{R}$ has a rank;
(ii) ${\operatorname{rank}}K_{R}=1$;
(iii) $R$ is
generically Gorenstein ( that is $R_{{\mathfrak{p}}}$ is a Gorenstein ring
for all mini-
mal prime ideals ${\mathfrak{p}}$ of $R$).
(b) If $K_{R}$ satisfies $(S_{3})$ and the equivalent
conditions of (a) hold, then $K_{R}$ can be identified with an
ideal of height 1 or equals $R$. In the first case $R/K_{R}$, is
an $(S_{2})$ ring with the canonical module $R/K_{R}$, the ring
itself.
Proof. The proof is parallel to that of [BH,
Proposition
3.3.18] and we present it for the convenience of the reader.
(a). (i)$\Rightarrow$(ii)$\Rightarrow$(iii). Set $Q$ to be
the ring of total fractions of $R$ and let $K_{R}\otimes_{R}Q$ be
a free $Q$–module of rank $r$, say. Let ${\mathfrak{p}}\in{\operatorname{Min}}(R)$. As
${\operatorname{Min}}(R)={\operatorname{Ass}}(R)={\operatorname{Ass}}_{R}(K_{R})$ (see [DT1, 1.3]), we
know that $K_{R_{{\mathfrak{p}}}}\cong(K_{R})_{{\mathfrak{p}}}$ and, by [BH,
Proposition 1.4.3], the $R_{{\mathfrak{p}}}$–module $(K_{R})_{{\mathfrak{p}}}$ is
free of rank $r$. As the dualizing complex of $R_{{\mathfrak{p}}}$ is
$(I^{\bullet})_{{\mathfrak{p}}}:0\longrightarrow I_{{\mathfrak{p}}}^{0}\longrightarrow
0$, we get $(R_{{\mathfrak{p}}})^{r}\cong(K_{R})_{{\mathfrak{p}}}\cong E(R/{\mathfrak{p}})$ from which
it follows that $r=1$, and thus $R_{{\mathfrak{p}}}$ is Gorenstein.
(iii)$\Rightarrow$(i). Note that
${\operatorname{Min}}(R)={\operatorname{Ass}}(R)$, and thus [BH, Proposition 1.4.3] implies
that $K_{R}$ has rank 1.
As ${\operatorname{Ass}}R={\operatorname{Ass}}(K_{R}),K$
is torsion free. Thus [BH, 1.4.18] implies that $K_{R}$ is
isomorphic to a sub–module of a free $R$–module of rank 1, and
it may be identified with an ideal of $R$ which we again denote by
$K_{R}$.
If ${\dim}R=0$, we get $K_{R}\cong R$, so we
may assume ${\dim}R>0$, and also $K_{R}$ is a proper ideal of $R$.
By [BH, Proposition 1.4.3], $K_{R}$ has a free sub–module
${\mathfrak{a}}$, which is also an ideal of $R$ of rank 1. Assuming ${\mathfrak{a}}=xR$ with $x$ is a base for ${\mathfrak{a}},x$ is $R$–regular and
$K_{R}$–regular. Let ${\mathfrak{p}}$ be a prime ideal containing
$K_{R}$. Applying the functor ${\operatorname{Hom}}_{R_{{\mathfrak{p}}}}(-,(I^{\bullet})_{{\mathfrak{p}}})$ on the exact sequence $0\longrightarrow K_{R}R_{{\mathfrak{p}}}\longrightarrow R_{{\mathfrak{p}}}%
\longrightarrow R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}}\longrightarrow 0$, we get the exact
sequence
$$\begin{array}[]{lll}0&\longrightarrow H^{0}({\operatorname{Hom}}_{R_{{%
\mathfrak{p}}}}(R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}},(I^{\bullet})_{{%
\mathfrak{p}}}))\longrightarrow H^{0}((I^{\bullet})_{{\mathfrak{p}}})%
\longrightarrow H^{0}({\operatorname{Hom}}_{R_{{\mathfrak{p}}}}(K_{R}R_{{%
\mathfrak{p}}},(I^{\bullet})_{{\mathfrak{p}}}))\\
&\longrightarrow H^{1}({\operatorname{Hom}}_{R_{{\mathfrak{p}}}}(R_{{\mathfrak%
{p}}}/K_{R}R_{{\mathfrak{p}}},(I^{\bullet})_{{\mathfrak{p}}}))\longrightarrow H%
^{1}((I^{\bullet})_{{\mathfrak{p}}})\longrightarrow\cdots.\end{array}$$
Note that $H^{1}(I^{\bullet})_{{\mathfrak{p}}}=0$ as $K_{R_{{\mathfrak{p}}}}$
satisfies $(S_{3})$ (see [DT2, Proposition 2.5]).
On the
other hand, we have
$H^{0}({\operatorname{Hom}}_{R_{{\mathfrak{p}}}}(R_{{\mathfrak{p}}}/K_{R}R_{{%
\mathfrak{p}}},(I^{\bullet})_{{\mathfrak{p}}}))\cong 0:_{K_{R}R_{{\mathfrak{p}%
}}}K_{R}R_{{\mathfrak{p}}}\subseteq 0:_{R_{{\mathfrak{p}}}}K_{R}R_{{\mathfrak{%
p}}}=0$ because
$K_{R}R_{{\mathfrak{p}}}$ is the canonical module of $R_{{\mathfrak{p}}}$ and
$R_{{\mathfrak{p}}}$ satisfies $(S_{2})$. Therefore we get the exact
sequence
$o\longrightarrow K_{R}R_{{\mathfrak{p}}}\longrightarrow R_{{\mathfrak{p}}}%
\longrightarrow H^{1}({\operatorname{Hom}}_{R_{{\mathfrak{p}}}}(R_{{\mathfrak{%
p}}}/K_{R}R_{{\mathfrak{p}}},(I^{\bullet})_{{\mathfrak{p}}}))\longrightarrow 0$
which
implies that $H^{1}({\operatorname{Hom}}_{R_{{\mathfrak{p}}}}(R_{{\mathfrak{p}}}/K_{R}R_{{%
\mathfrak{p}}},(I^{\bullet})_{{\mathfrak{p}}}))\cong(R/K_{R})_{{\mathfrak{p}}}$. It follows that
${\dim}(R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}})={\operatorname{ht}}{%
\mathfrak{p}}-1$. Using the
Grothendieck local duality shows that $R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}}$
is the canonical module of $R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}}$. As $K_{R}R_{{\mathfrak{p}}}$ contains an $R_{{\mathfrak{p}}}$–regular element, we have
${\operatorname{ht}}_{R_{{\mathfrak{p}}}}(K_{R}R_{{\mathfrak{p}}})\geq 1$. Since
${\dim}(R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}})={\operatorname{ht}}{%
\mathfrak{p}}-1$, we get
${\operatorname{ht}}(K_{R})=1$.
For the final part, we may assume that ${\dim}R>3$. As
$R$ is $(S_{2})$ and $K_{R}$ is $(S_{3})$, from the exact sequence
$H_{{\mathfrak{p}}R_{{\mathfrak{p}}}}^{i}(R_{{\mathfrak{p}}})\longrightarrow H_%
{{\mathfrak{p}}R_{{\mathfrak{p}}}}^{i}(R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p%
}}})\longrightarrow H_{{\mathfrak{p}}R_{{\mathfrak{p}}}}^{i+1}(K_{R}R_{{%
\mathfrak{p}}})$, we get $H_{{\mathfrak{p}}R_{{\mathfrak{p}}}}^{i}(R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak%
{p}}})=0$ for $i=0,1$.
This shows that $R_{{\mathfrak{p}}}/K_{R}R_{{\mathfrak{p}}}$ satisfies
$(S_{2}).\Box$
We can also generalize[DT1, Corollary 3.4].
4.6. Theorem
Assume that $R$ satisfies $(S_{2})$, and
that ${\dim}R<\infty$. The following statements are equivalent.
(i) R possesses a dualizing complex;
(ii) R admits a canonical module K, and $C(\mathcal{H},K)^{\prime}$, the induced complex of the Cousin complex of K with respect
to the height filtration $\mathcal{H}=(H_{i})_{i\geq 0}$ with
$H_{i}=\{{\mathfrak{p}}\in{\operatorname{Spec}}(R):{\operatorname{ht}}({%
\mathfrak{p}})\geq i\}$, is a
dualizing complex for R;
(iii)R admits a canonical module K and $H^{i}(C(\mathcal{H},K)^{\prime})$ is finitely a generated R–module for all $i\geq 1$.
Proof. (i)$\Rightarrow$(ii). By 4.4, there exists
a
dualizing complex
$I^{\bullet}:0\longrightarrow I^{0}\overset{\delta^{0}}{\longrightarrow}I^{1}%
\overset{\delta^{1}}{\longrightarrow}\cdots$
for $R$ such that
${\operatorname{Ass}}_{R}(I^{0})={\operatorname{Min}}(R)$. Set $K={\operatorname{Ker}}\delta^{0}$. As seen
in 4.4, $K$ is a canonical module of $R$ and
${\operatorname{Ass}}_{R}(K)={\operatorname{Min}}(R)$. Now, by [DT1, Theorem 2.4(iv)],
$C(\mathcal{H},K)^{\prime}$ is a dualizing complex for $R$.
(ii)$\Rightarrow$(iii) is clear.
(iii)$\Rightarrow$(i).
For each ${\mathfrak{m}}\in{\operatorname{Max}}(R)$, we have, by [S1, Theorem 3.5],
$C(\mathcal{H},K)_{{\mathfrak{m}}}\cong C(\mathcal{H_{{\mathfrak{m}}}},K_{{%
\mathfrak{m}}})$, where $\mathcal{H}_{{\mathfrak{m}}}$ is the height filtration
of $R_{{\mathfrak{m}}}$. Therefore, by [DT1, Corollary 3.4], $(C(\mathcal{H},K)^{\prime})_{{\mathfrak{m}}}$ is a dualizing complex for $R_{{\mathfrak{m}}}$. Since $R_{{\mathfrak{p}}}$
satisfies $(S_{2})$ for all ${\mathfrak{p}}\in{\operatorname{Spec}}(R)$, by the same
argument as in the proof of [DT1, Corollary 3.4], each term of
$C(\mathcal{H},K)^{\prime}$ is an injective module. Thus, by [S4,
Theorem 4.2], $C(\mathcal{H},K)^{\prime}$ is a dualizing complex for
$R$. $\Box$
5. Indecomposable injective modules structure
In this
section, by using of a particular dualizing complex for an
$(S_{2})$ Noetherian ring $R$ of finite dimension, we give an explicit description for the
structure of all indecomposable injective modules. In [DT1,
Corollary 3.3], it is shown that for each ${\mathfrak{p}}\in{\operatorname{Spec}}(R)$,
there exists a finitely generated $R$–module $T$, depending on
${\mathfrak{p}}$, such that $E(R/{\mathfrak{p}})$ is a module of generalized
fractions of $T$. Here we will show that $T$ can be replaced by a canonical
module of $R$ and that it does not depend on ${\mathfrak{p}}$.
Our approach involves the concept of a chain of
triangular subsets on $R$ explained in [O, page 420]. Such a
chain $\mathcal{U}=(U_{i})_{i\geq 1}$ determines a complex
$C(\mathcal{U},M)$ of modules of generalized fractions on an
$R$–module $M$, that is
$C(\mathcal{U},M):0\longrightarrow M\overset{e^{0}}{\longrightarrow}U_{1}^{-1}M%
\overset{e^{1}}{\longrightarrow}\cdots\overset{e^{i-1}}{\longrightarrow}U_{i}^%
{-i}M\overset{e^{i}}{\longrightarrow}U_{i+1}^{-i-1}M\overset{e^{i+1}}{%
\longrightarrow}\cdots$
in which
$e^{0}(m)=\frac{m}{(1)}$ for all $m\in M$ and
$e^{i}(\frac{m}{(u_{1},\cdots,u_{i})})=\frac{m}{(u_{1},\cdots,u_{i},1)}$ for all $i\geq 1$, $m\in M$,
and $(u_{1},\cdots,u_{i})\in U_{i}$. Note that in the complex
$C(\mathcal{U},M)$, $U_{i+1}^{-i-1}M$ is regarded as the $i$–th term, so that
$H^{i}(C(\mathcal{U},M))={\operatorname{Ker}}e^{i+1}/{\operatorname{Im}}e^{i}$, $i\geq 0$,
and $H^{-1}(C(\mathcal{U},M))={\operatorname{Ker}}e^{0}$.
Assume that $R$ satisfies $(S_{2})$ and possesses a
dualizing complex, so that $R$ possesses a dualizing complex
$I^{\bullet}:0\longrightarrow I^{0}\overset{\delta^{0}}{\longrightarrow}I^{1}%
\overset{\delta^{1}}{\longrightarrow}\cdots\overset{\delta^{d-1}}{%
\longrightarrow}I^{d}\longrightarrow 0,d={\dim}R$,
such that ${\operatorname{Ass}}_{R}(I^{0})={\operatorname{Min}}(R)$. Set
$K={\operatorname{Ker}}\delta^{0}$, and consider the induced extended complex
$I^{\ast}:0\longrightarrow K\hookrightarrow I^{0}\overset{\delta^{0}}{%
\longrightarrow}I^{1}\overset{\delta^{1}}{\longrightarrow}\cdots%
\longrightarrow I^{d}\longrightarrow 0,$ .
For each ${\mathfrak{p}}\in{\operatorname{Ass}}_{R}(I^{0})$, the complex
$0\longrightarrow(I^{0})_{{\mathfrak{p}}}\longrightarrow 0$ is the
dualizing complex for $R_{{\mathfrak{p}}}$, so that $K_{{\mathfrak{p}}}\cong E(R/{\mathfrak{p}})$. Hence ${\operatorname{Ass}}_{R}(K)={\operatorname{Min}}(R)$. Thus, by [DT1,
Proposition 3.2], there is a unique isomorphism of complexes (over
${\operatorname{Id}}_{K}$) from $I^{\ast}$ to $C(\mathcal{V},K)$, the complex
of modules of generalized fractions on $K$ with respect to the
chain of triangular subsets $\mathcal{V}=(V_{i})_{i\geq 1}$ on
$R$, defined by
$V_{i}=\{(v_{1},\cdots,v_{i})\in R^{i}:{\operatorname{ht}}_{R}((v_{1},\cdots,v_%
{j}))\geq j$ for all $j$ with $1\leq j\leq i\}$.
Now, we restate [DT1, Corollary 3.3] in a more
appropriate form.
5.1. Corollary
Assume that $R$ satisfies $(S_{2})$
and that it possesses a dualizing complex, so that $R$ admits a
canonical module $K$, say. Then, for each ${\mathfrak{p}}\in{\operatorname{Spec}}(R)$,
$E(R/{\mathfrak{p}})\cong(V_{{\operatorname{ht}}{\mathfrak{p}}}\times(R%
\setminus{\mathfrak{p}}))^{-{\operatorname{ht}}{\mathfrak{p}}-1}K$
where $V_{r}$
is the triangular subset of $R^{r}$ defined in the paragraph
just before the corollary.
Acknowledgment. I thank M. Tousi for his comment on
2.3. I also thank the referee for the invaluable comments on the
manuscript.
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The free group on $n$ generators modulo $n+u$ random relations as $n$ goes to infinity
Yuan Liu and Melanie Matchett Wood
Department of Mathematics
University of Wisconsin-Madison
480 Lincoln Drive
Madison, WI 53705 USA
and
American Institute of Mathematics
600 East Brokaw Road
San Jose, CA 95112 USA
[email protected]
Abstract.
We show that, as $n$ goes to infinity, the free group on $n$ generators, modulo $n+u$ random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen-Lenstra heuristics. For each $n$, these random groups belong to the few relator model in the Gromov model of random groups.
1. Introduction
For an integer $u$ and positive integers $n$, we study the random group given by the free group $F_{n}$ on $n$ generators modulo $n+u$ random relations. In particular we find that these random groups have a nice limiting behavior as $n\rightarrow\infty$ and we explicitly describe the limiting random group.
There are two ways to take relations in a “uniform” way: 1)complete $F_{n}$ to the profinite complete group $\hat{F}_{n}$ on $n$ generators and take relations with respect for Haar measure, or 2)take relations from $F_{n}$ uniformly among words up to length $\ell$ and then let $\ell\rightarrow\infty$. In Proposition 14.1, we show that the random groups obtained from the second method weakly converge, as $\ell\rightarrow\infty$, to the random groups obtained from the first method.
For a positive integer $n$, let $\hat{F}_{n}$ be the profinite free group on $n$ generators. For an integer $u$, we define the random group $X_{u,n}$ by taking the quotient of $\hat{F}_{n}$ by (the closed, normal subgroup generated by) $n+u$ independent random generators, taken from Haar measure on $\hat{F}_{n}$. We need to define a topology to make precise the convergence of $X_{u,n}$ as $n\rightarrow\infty$.
Let $S$ be a set of (isomorphism classes of) finite groups. Let $\bar{S}$ be the smallest set of groups containing $S$ that is closed under taking quotients, subgroups, and finite direct products. For a profinite group $G$, we write $G^{\bar{S}}$ for its pro-$\bar{S}$ completion.
We consider the set $\mathcal{P}$ of isomorphism classes of profinite groups $G$ such that $G^{\bar{S}}$ is finite for all finite sets $S$ of finite groups. All finitely generated profinite groups are in $\mathcal{P}$ and all groups in
$\mathcal{P}$ are small in the sense of [FJ08, Section 16.10].
We define a topology on $\mathcal{P}$ in which the basic opens are, for each finite set $S$ of finite groups and finite group $H$, the sets $U_{S,H}:=\{G\mid G^{\bar{S}}\simeq H\}$.
Theorem 1.1.
Let $u$ be an integer. Then there is a probability measure $\mu_{u}$ on $\mathcal{P}$ for the $\sigma$-algebra of Borel sets such that
as $n\rightarrow\infty$, the distributions of $X_{u,n}$ weakly converge to $\mu_{u}$.
We give these $\mu_{u}$ explicitly in fact. See Equation (3.2) for a formula for $\mu_{u}$ on each basic open, and see Section 12 for several other interesting examples of the values of these measures. In fact, we prove in Theorem 11.4 a stronger form of convergence than weak convergence, which, in particular, tell us the measure of any finite group. In particular, we have
(1.2)
$$\mu_{u}(\textrm{trivial group})=\prod_{\begin{subarray}{c}G\textrm{ finite %
simple}\\
\textrm{abelian group}\end{subarray}}\prod_{i=u+1}^{\infty}(1-|G|^{-i})\prod_{%
\begin{subarray}{c}G\textrm{ finite simple}\\
\textrm{non-abelian group}\end{subarray}}e^{-|\operatorname{Aut}(G)|^{-1}|G|^{%
-u}},$$
which is $\approx.4357$ when $u=1$.
The abelian group version of this problem has been well-studied, as the limiting groups when $u=0,1$ are the random groups of the Cohen-Lenstra heuristics. The first factor above, as a product over primes, is very familiar from the random groups of the Cohen-Lenstra heuristics, but here it naturally appears as part of a product over all finite simple groups.
Cohen and Lenstra [CL84] defined certain random abelian groups that they predicted gave the distribution of class groups of random quadratic fields. Friedman and Washington [FW89] later realized that these random abelian groups arose as the limits of cokernels of random matrices, which is just a rewording of the abelianization of our construction above. These random abelian groups are universal, in the sense that, as $n\rightarrow\infty$, taking ${\mathbb{Z}}^{n}$ modulo almost any collection of $n+u$ independent relations will give these same random abelian groups, even if the relations are taken from strange and lopsided distributions [Woo15].
One motivation for our work is to develop a non-abelian version of the random abelian groups of Cohen and Lenstra, in order to eventually be able to model non-abelian versions of class groups of random number fields. Boston, Bush, and Hajir [BBH16] have defined random pro-$p$ groups that they conjecture model the pro-$p$ generalizations of class groups of random imaginary quadratic fields. In their definition, they were able to use special properties of $p$-groups to give a definition that avoids the limit as $n\rightarrow\infty$ that we study above (or rather, reduces the question of the limit as $n\rightarrow\infty$ to the abelian case, which was already understood).
There is a large body of work on the Gromov, or density, model of random groups (see [Oll05] for an excellent introduction). In this model, one takes $F_{n}$ modulo $r(\ell)$ random relations uniform among words of length $\ell$, and studies the behavior as $\ell\rightarrow\infty$.
When $r(\ell)$ grows like $(2n-1)^{d\ell}$ this is called the density $d$ model.
There has been a great amount of work to understand, as $\ell\rightarrow\infty$, what properties hold asymptotically almost surely for these groups (e.g see [Oll05, Oll10] for an overview and [CW15, KK13, Mac16, OW11] for some more recent examples). Our $X_{u,n}$ are limits as $\ell\rightarrow\infty$ of density $0$ models of these random groups. However, the emphasis of our work is different from much of the previous work on the Gromov model of random groups. That work has often emphasized a random group with given generators, and we consider only the isomorphism class of the group and focus on the convergence to a limiting random variable. Also, from the point of view of our topology, any Gromov random group with $r(\ell)\rightarrow\infty$ weakly converges to the trivial group (see Proposition 14.2). Our topology is aimed at understanding finite quotients of groups, and is rather different than the topology due to Chabauty [Cha50] and Grigorchuk [Gri84] on the space of marked groups that emphasizes the geometry of the Cayley graphs but isn’t well behaved on isomorphism classes of groups.
The closest previous work to ours is that of Dunfield and Thurston [DT06]. They studied $F_{n}$ modulo $r$ random relations (with both methods described above of taking relations) in order to contrast those random groups with random $3$-manifold groups. Their main consideration was the probability that these random groups (for fixed $n$ and $r$) had a quotient map to a fixed finite group. They do observe [DT06, Theorem 3.10] that for a fixed non-abelian finite simple group $G$, the distribution of the number of quotient maps to $G$ has a Poisson limiting behavior as $n\rightarrow\infty$; this is the first glimpse of the nice limiting behavior as $n\rightarrow\infty$ that we study in this paper.
Jarden and Lubotzky [JL06] studied the normal subgroup of $\hat{F}_{n}$ generated by a fixed number of random elements, in particular proving that when it is infinite index that it is almost always the free profinite group on countably many generators. Our work here complements theirs, as they have determined the structure of the random normal subgroup and we determine the structure of the quotient by this random normal subgroup.
The bulk of this paper is devoted to showing the existence of the measure $\mu_{u}$ of Theorem 1.1. Let $\mu_{u,n}$ be the distribution of our random group $X_{u,n}$. Since the basic opens in our topology are also closed, it is clear that if $\mu_{u}$ exists, then for any basic open $U$ we have $\mu_{u}(U)=\lim_{n\rightarrow\infty}\mu_{u,n}(U).$
The argument for the existence of $\mu_{u}$ breaks into two major parts. The first part is to show the limit
$\lim_{n\rightarrow\infty}\mu_{u,n}(U)$ exists. The second part is to show that these measures on basic opens define a countably additive measure. After giving some notation and basic definitions in Section 2, we will give the values of $\mu_{u}$ on basic opens in Section 3 for easy reference. Then in Section 4 we set up the strategy for proving
$\lim_{n\rightarrow\infty}\mu_{u,n}(U)$ exists, which is entirely group theoretical. This argument will take us through Section 8. It is easy to express
$\mu_{u,n}$ in group theory terms involving $\hat{F}_{n}$. However, such expressions do not allow one to take a limit as $n\rightarrow\infty$, and so the main challenge is to extract $\hat{F}_{n}$ from the description of the probabilities so that they only involve the number $n$ and group theoretical quantities that do not depend on $n$. This requires several steps. In Section 5, we express the probabilities in terms of multiplicities of certain groups appearing in $\hat{F}_{n}$. In Section 6, we bound what possible groups can have positive multiplicities. In Section 7, we relate the multiplicities to a count of certain surjections, and finally in Section 8 we count these surjections in another way that eliminates $\hat{F}_{n}$ from our description of the probabilities.
The next challenge is to show the countable additivity of the $\mu_{u}$ that we have then defined on basic opens. It follows from Fatou’s lemma that for a finite set $S$ of finite groups,
$$\sum_{H\text{ is finite}}\lim_{n\rightarrow\infty}\mu_{u,n}(U_{S,H})\leq 1.$$
However, a priori, this inequality may be strict. In the limit as $n$ goes to infinity there could be escape of mass.
To show that this does not occur, we require bounds on the $\mu_{u,n}(U_{S,H})$ that are sufficiently uniform in $n$. The difficultly is that our group theoretical expressions do not easily lend themselves to the kind of bounds useful for an analytic argument. We obtain the necessary uniformity by considering a notion of chief factor pairs, which generalizes the notion of a chief factor of a group to also include the conjugation action on the chief factor. We are able to bound the size of the outer action of conjugation on chief factors for a given $S$ in Section 6, which then, combined with an induction on $S$, gives us the uniformity necessary to show in Section 9 that the above inequality is actually an equality. That is the heart of the proof of countable additivity, which we show in Section 9.
Once we have established the existence of the measure $\mu_{u}$ with the desired measure on basic opens, Theorem 1.1 follows immediately in Section 10. In Section 11, we give the measures of sets of the form $\{X\in\mathcal{P}\mid X^{\bar{S}}\simeq H^{\bar{S}}\}$ for arbitrary sets $S$ of finite groups, and see that $\mu_{u}$ and $\lim_{n\rightarrow\infty}\mu_{u,n}$ agree there, giving a stronger convergence than the weak convergence of Theorem 1.1. There, one inequality is automatic, and we then we argue that we either have the necessary uniformity to get equality, or that the larger probability is $0$, which also gives equality.
The result of Section 11 then allows us to compute measures of many different Borel sets, and in Section 12, we give many examples including the trivial group, infinite groups, and distributions of the abelianization and pro-nilpotent quotient. In Section 13, we see that the measures $\mu_{u}$ give positive measure to any basic open where groups can be generated by $u$ more relations than generators. Finally, in Section 14, we compare the profinite model used in this paper to the discrete group model described above.
This is the beginning of investigation into these random groups, and there are many further questions we would like to understand. Are these measures universal in the sense of [Woo15], i.e. would we still get $\mu_{u}$ as $n\rightarrow\infty$ even if we took our relations from a different measure? Are these measures determined by their moments, which in [HB94, Lemma 17], [FK06, Section 4.2], [EVW09, Lemma 8.1], [Woo17, Theorem 8.3], and [BW17, Theorem 1.4] has been an important tool to identify analogous random groups? What is the measure of the set of all infinite groups when $u\geq 0$ (see Example 12.8, and note by [JL06] this implies the normal subgroup generated by the relations is free on countably many generators with probability $1$)? What is the measure of the set of finitely generated groups, and of finitely presented groups? Do the $\mu_{u,n}$ converge strongly to $\mu_{u}$? Besides their inherent interest, many of these questions have implications for the possible connections to number theory described above.
2. Notation and basic group theoretical definitions
2.1. Notation
Whenever we take a quotient by relations, we always mean by the closed, normal subgroup generated by those relations. For elements $x_{1},\dots$ of a group $G$, we write $[x_{1},\dots]_{G}$ for the closed normal subgroup of $G$ generated by $x_{1},\dots$.
We write $G\simeq H$ to mean that $G$ and $H$ are isomorphic. For profinite groups, we always mean isomorphic as profinite groups. For two groups $G$ and $H$, we write $G=H$ when there is an obvious map from one of $G$ or $H$ to the other (e.g. when $H$ is defined as a quotient or subgroup of $G$) and that map is an isomorphism.
For a group $G$, we write $G^{j}$ for the direct product of $j$ copies of $G$. If $H$ is a subgroup of $G$, then we denote the centralizer of $H$ by $\mathbf{C}_{G}(H)$.
When we say a set of finite groups, we always mean a set of isomorphism classes of finite groups.
2.2. $F$-groups
If $F$ is a group, an $F$-group is a group $G$ with an action of $F$.
A morphism of $F$-groups is a group homomorphism that respects the $F$-action. An $F$-subgroup is a subgroup $G$ such that $f(G)=G$ for all $f\in F$, and an $F$-quotient is a group quotient homomorphism that respects the $F$-action.
An irreducible $F$-group is an $F$-group with no normal
$F$-subgroups except the trivial subgroup and the group itself.
We write $\operatorname{Hom}_{F}(G_{1},G_{2})$ for the $F$-group morphisms from $G_{1}$ to $G_{2}$ and $h_{F}(G):=|\operatorname{Hom}_{F}(G,G)|$. We write $\operatorname{Sur}_{F}(G_{1},G_{2})$ for the $F$-group surjections from $G_{1}$ to $G_{2}$, and $\operatorname{Aut}_{F}(G)$ for the $F$-group automorphisms of $G$.
For a sequence $x_{k}$ in an $F$-group $G$, let $[x_{1},\dots]_{F}$ be the closed normal $F$-subgroup of $G$ generated by the $x_{k}$.
2.3. $H$-extensions
For a group $H$, an $H$-extension is a group $E$ with a surjective morphism $\pi:E\rightarrow H.$
If $\pi:E\rightarrow H$ and $\pi^{\prime}:E^{\prime}\rightarrow H$ are $H$-extensions, a morphism from $(E,\pi)$ to $(E^{\prime},\pi^{\prime})$ is a group homomorphism $f:E\rightarrow E^{\prime}$ such that $\pi=\pi^{\prime}\circ f$. If $\pi:E\rightarrow H$ and $\pi^{\prime}:E^{\prime}\rightarrow H$ are $H$-extensions, we write $\operatorname{Sur}_{H}(\pi,\pi^{\prime})$ for the set of surjective morphisms from $(G,\pi)$ to $(G^{\prime},\pi^{\prime})$.
For an $H$-extension $E$, we write $\operatorname{Aut}_{H}(E,\pi)$ for the automorphisms of $(E,\pi)$ as an $H$-extension.
If $(E,\pi)$ is an $H$-extension, a sub-$H$-extension is a subgroup $E^{\prime}$ of $E$ with $\pi|_{E^{\prime}}$, such that $\pi|_{E^{\prime}}$ is surjective.
Note that when $\ker\pi$ is abelian, it is an $H$-group under conjugation in $E$.
2.4. Pro-$\bar{S}$ completions and level $S$ groups
Given a set $S$ of finite groups, we let $\bar{S}$ denote the smallest set of groups containing $S$ that is closed under taking quotients, subgroups and finite direct products. (This is called the variety of groups generated by $S$.) Given a profinite group $G$, we write $G^{\bar{S}}$ for its pro-$\bar{S}$ completion, which is defined as
$$G^{\bar{S}}=\varprojlim_{M}G/M,$$
where the inverse limit is taken over all closed normal subgroups $M$ of $G$ such that $G/M\in\bar{S}$.
Definition.
For a set $S$ of finite groups, we say that a profinite group $G$ is level $S$ if $G\in\bar{S}$. Also, for a positive integer $\ell$, let $S_{\ell}$ be the set consisting of all groups whose order is less than or equal to $\ell$. Then we say $G$ is level $\ell$ if $G\in\bar{S}_{\ell}$. Note that for $G\in\mathcal{P}$ we have that $G$ is level $S$ if and only if $G=G^{\bar{S}}$.
3. Definition of $\mu_{u}$
For finite set $S$ of finite groups and finite group $H$, let $U_{S,H}:=\{X\in\mathcal{P}\,|\,X^{\bar{S}}\simeq H\}.$
For integers $n\geq 1$ and $u>-n$, we have a measure $\mu_{u,n}$ on the $\sigma$-algebra of Borel sets of $\mathcal{P}$ such that $\mu_{u,n}(A)=\operatorname{Prob}(X_{u,n}\in A)$.
We will define a measure $\mu_{u}$, for each integer $u$, at first as a measure on the algebra $\mathcal{A}$ of sets generated by the $U_{S,H}$. For $A\in\mathcal{A}$, we define
(3.1)
$$\mu_{u}(A):=\lim_{n\rightarrow\infty}\mu_{u,n}(A).$$
We will below establish that 1) this limit exists when $A=U_{S,H}$ (see Theorem 8.1, and Equation (3.2) just below, in which we give the value of the limit), and hence for any $A\in\mathcal{A}$ since the limit is compatible with finite sums and subtraction from $1$; and 2) $\mu_{u}$ is countably additive on $\mathcal{A}$ (see Theorem 9.1). These two results represent the bulk of the work of the paper. Then by Carathéodory’s extension theorem, it follows that $\mu_{u}$ extends uniquely to a probability measure on $\mathcal{P}$.
3.1. Value of $\mu_{u}$ on basic open sets
Given a finite group $H$, let $\mathcal{A}_{H}$ be the set of isomorphism classes of non-trivial finite abelian irreducible $H$-groups. Let $\mathcal{N}$ be the set of isomorphism classes of finite groups that are isomorphic to $G^{j}$ for some finite simple non-abelian group $G$ and a positive integer $j$.
Let $S$ be a set of finite groups, and $H$ a finite level $S$ group.
For $G\in\mathcal{A}_{H}$, we define the quantity
$$\lambda(S,H,G):=(h_{H}(G)-1)\sum_{\begin{subarray}{c}\textrm{isom. classes of %
$H$-extensions $(E,\pi)$}\\
\textrm{such that $\ker\pi$ isom. $G$ as $H$-groups,}\\
\textrm{and $E$ is level $S$}\end{subarray}}\frac{1}{|\operatorname{Aut}_{H}(E%
,\pi)|}.$$
We will see in Remark 8.13 that for $G\in\mathcal{A}_{H}$, the number $\lambda(S,H,G)$ is an integer power of $h_{H}(G)$.
If $G\in\mathcal{N}$, we define
$$\lambda(S,H,G):=\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-%
extensions $(E,\pi)$}\\
\textrm{such that $\ker\pi$ isom. $G^{j}$ as groups, }\\
\textrm{$\ker\pi$ irred. $E$-group,}\\
\textrm{and $E$ is level $S$}\end{subarray}}\frac{1}{|\operatorname{Aut}_{H}(E%
,\pi)|}.$$
The definitions are not quite parallel in the abelian and non-abelian cases, but this is unavoidable given the different behavior of abelian and non-abelian simple groups.
It will follow from Theorem 8.1 below that for a finite set $S$ of finite groups and a finite level $S$ group $H$, we have
(3.2)
$$\displaystyle\mu_{u}(U_{S,H})=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=0}^{\infty}(1-\lambda(S,H,G)\frac{%
h_{H}(G)^{-i-1}}{|G|^{{u}}})$$
$$\displaystyle\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{subarray}}e^{-|G|^{%
-u}\lambda(S,H,G)}.$$
Theorem 11.4 gives the analogous result for an infinite set $S$.
We will see in Section 6 that for finite $S$ only finitely many elements of $\mathcal{A}_{H}$ and $\mathcal{N}$ contribute non-trivially to this product.
4. Setup and organization of the proofs
The proof of Equation (3.2) will be established from Section 5 to Section 8, which are dominated by group theoretical methods. Here we outline the proof for the reader’s convenience.
Suppose $n$ is a positive integer, $S$ is a finite set of finite groups, and $H$ is a finite level $S$ group. Then $(\hat{F}_{n})^{\bar{S}}$ is a finite group [Neu67, Cor. 15.72] and $(X_{u,n})^{\bar{S}}$ has the same distribution as the quotient of $(\hat{F}_{n})^{\bar{S}}$ by $n+u$ independent, uniform random relations $r_{1},\cdots r_{n+u}$ from $(\hat{F}_{n})^{\bar{S}}$. By the definition of $\mu_{u}$, we have that
$$\mu_{u}(U_{S,H})=\lim_{n\to\infty}\operatorname{Prob}((\hat{F}_{n})^{\bar{S}}/%
[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{n})^{\bar{S}}}\simeq H).$$
We consider a normal subgroup $N$ of $(\hat{F}_{n})^{\bar{S}}$ with an isomorphism $(\hat{F}_{n})^{\bar{S}}/N\simeq H$. Let $M$ be the intersection of all maximal proper $(\hat{F}_{n})^{\bar{S}}$-normal subgroups of $N$. We denote $F=(\hat{F}_{n})^{\bar{S}}/M$ and $R=N/M$. Then for independent, uniform random elements $r_{1},\cdots,r_{n+u}$ of $(\hat{F}_{n})^{\bar{S}}$, we have that $[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{n})^{\bar{S}}}=N$ if and only if $R$ is the normal subgroup generated by the images of $r_{1},\cdots,r_{n+u}$ in $F$. Indeed, the “only if” direction is clear; and if $[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{n})^{\bar{S}}}/M=R$, then $[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{n})^{\bar{S}}}=N$ since $[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{n})^{\bar{S}}}$ being contained in a proper maximal $(\hat{F}_{n})^{\bar{S}}$-normal subgroup of $N$ would imply that its image is contained in a proper maximal $F$-normal subgroup of $R$.
Any two surjections from $(\hat{F}_{n})^{\bar{S}}$ to $H$ are isomorphic as $H$-extensions [Lub01, Proposition 2.2]. Thus, the short exact sequence
(4.1)
$$1\rightarrow R\rightarrow F\rightarrow H\rightarrow 1$$
does not depend (up to isomorphisms of $F$ as an $H$-extension) on the choice of the normal subgroup $N$.
Definition.
Given a finite set $S$ of finite groups, a positive integer $n$ and a finite level $S$ group $H$, the short exact sequence defined in Equation (4.1) is called the fundamental short exact sequence associated to $S$, $n$ and $H$.
By the above arguments, $\operatorname{Prob}((\hat{F}_{n})^{\bar{S}}/[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{%
n})^{\bar{S}}}\simeq H)$ equals the number of normal subgroups $N$ of $(\hat{F}_{n})^{\bar{S}}$ with $(\hat{F}_{n})^{\bar{S}}/N\simeq H$ times the probability that independent, uniform random elements $x_{1},\cdots,x_{n+u}\in F$ normally generate $R$. Note that the number of such normal subgroups $N$ is $|\operatorname{Sur}((\hat{F}_{n})^{\bar{S}},H)|/|\operatorname{Aut}(H)|$, and there is a one-to-one correspondence between $\operatorname{Sur}((\hat{F}_{n})^{\bar{S}},H)$ and $\operatorname{Sur}(\hat{F}_{n},H)$. It follows that
(4.2)
$$\operatorname{Prob}((\hat{F}_{n})^{\bar{S}}/[r_{1},\cdots,r_{n+u}]_{(\hat{F}_{%
n})^{\bar{S}}}\simeq H)=\frac{|\operatorname{Sur}(\hat{F}_{n},H)|}{|%
\operatorname{Aut}(H)|}\operatorname{Prob}([x_{1},\cdots,x_{n+u}]_{F}=R).$$
It therefore suffices to compute $\operatorname{Prob}([x_{1},\cdots,x_{n+u}]_{F}=R)$.
Note that $R$ is an $F$-group under the conjugation action. It will follow from Lemma 5.11 that $R$ is a direct product of irreducible $F$-groups.
Theorem 5.1 will prove the formula for $\operatorname{Prob}([x_{1},\cdots,x_{n+u}]_{F}=R)$ for $F$ and $R$ where $R$ is a direct product of irreducible $F$-groups, in terms of the multiplicities of the various irreducible $F$-group factors of $R$. In Section 6, we will give some criteria for which irreducible $F$-groups can appear in $R$. Then in Section 7, we will relate the multiplicities of irreducible factors in $R$ to the number of normal subgroups of $R$ with specified quotients. In Section 8, we will count these normal subgroups of $R$ in another way in order to finally give an explicit formula for $\mu_{u,n}(U_{S,H})$. This formula will be explicit enough that we can easily take the limit as $n\rightarrow\infty$, giving
Equation (3.2).
5. Generating probabilities for products of irreducible $F$-groups
Throughout this section, we let $n\geq 1$ and $u>-n$ be integers, $F$ a group, and $R$ a finite product of finite irreducible $F$-groups. (We don’t require $R$ to be a subgroup of $F$.) The goal of this section is to prove the following theorem which gives the probability that the normal $F$-subgroup generated by $n+u$ random elements of $R$ is the whole group.
Theorem 5.1.
Let $F$ be a group and $G_{i}$ be finite irreducible $F$-groups for $i=1,\dots,k$ such that for $i\neq j$, we have that $G_{i}$ and $G_{j}$ are not isomorphic $F$-groups, and let $m_{i}$ be non-negative integers.
Let $R=\prod_{i=1}^{k}G_{i}^{m_{i}}$.
Then
$$\operatorname{Prob}([x_{1},\dots,x_{n+u}]_{F}=R)=\prod_{\begin{subarray}{c}1%
\leq i\leq k\\
G_{i}\textrm{ abelian}\end{subarray}}\prod_{j=0}^{m_{i}-1}(1-h_{F}(G_{i})^{j}|%
G_{i}|^{-n-u})\prod_{\begin{subarray}{c}1\leq i\leq k\\
G_{i}\textrm{ non-abelian}\end{subarray}}(1-|G_{i}|^{-n-u})^{m_{i}}$$
where the $x_{i}$ are independent, uniform random elements of $R$.
Remark 5.2.
Given a finite abelian irreducible $F$-group $G$, if we let $\mathfrak{m}$ be maximal such that $G^{\mathfrak{m}}$ can be generated by one element as an $F$-group, then we have
$h_{F}(G)^{\mathfrak{m}}=|G|$. This follows from Theorem 5.1 because if we take $m_{i}=\mathfrak{m}$, the probability that one element generates $G^{m_{i}}$ is positive, but if we take $m_{i}=\mathfrak{m}+1$ the probability is $0$.
We will build up to Theorem 5.1 through several lemmas.
First, we determine the structure of normal $F$-subgroups of products of irreducible $F$-groups.
Lemma 5.3.
If $G_{i}$ are irreducible $F$-groups and $N$ is an $F$-subgroup of $\prod_{i=1}^{m}G_{i}$ that projects to $1$ or $G_{i}$ in each factor, then there exists a subset $J\subset\{1,\dots,m\}$ such that the projection of $N$ to $\prod_{i\in J}G_{i}$ is an isomorphism.
Proof.
We prove this by induction on $m$. Let $\pi_{m}$ be the projection map from $\prod_{i=1}^{m}G_{i}$ to $G_{m}$, and $\pi$ the projection map from $N$ to $\prod_{i=1}^{m-1}G_{i}$. Since $\pi_{m}(N)$ is $1$ or $G_{m}$, and $\pi_{m}(\ker\pi)$ is a normal $F$-subgroup of
$\pi_{m}(N)$, we have $\pi_{m}(\ker\pi)$ is $1$ or $G_{m}$. If $\pi_{m}(\ker\pi)=1$, then since $\ker\pi\cap\ker\pi_{m}=1$, we have $\ker\pi=1$ and $N$ is isomorphic to $\pi(N)$.
If $\pi_{m}(\ker\pi)=G_{m}$, then $N$ is isomorphic to $\pi(N)\times G_{m}$.
In either case, we apply the inductive hypothesis to $\pi(N)$ and conclude the lemma.
∎
Lemma 5.4.
Let $G_{1}$ and $G_{2}$ be irreducible $F$-groups. Then any homomorphism of $F$-groups $\phi:G_{1}\rightarrow G_{2}$ with normal image is either trivial or an isomorphism.
Proof.
If it is not trivial, then $\ker(\phi)$ is a normal $F$-subgroup and so
must be trivial, and $\operatorname{im}(\phi)$ is a normal $F$-subgroup and
must be $G_{2}$, so it is a bijection.
∎
Lemma 5.5.
Let $G_{i}$ be irreducible $F$-groups for $i=1,\dots,k$ such that for $i\neq j$, we have that $G_{i}$ and $G_{j}$ are not isomorphic as $F$-groups.
Let $N$ be a normal $F$-subgroup of $\prod_{i=1}^{k}G_{i}^{m_{i}}$, then $N=\prod_{i=1}^{k}N_{i},$
where $N_{i}$ is a normal $F$-subgroup of $G_{i}^{m_{i}}$.
Proof.
Since $N$ is a normal $F$-subgroup of $\prod_{i=1}^{k}G_{i}^{m_{i}}$, its projection to each factor $G_{i}$ is normal $F$-subgroup of $G_{i}$, hence it’s either 1 or $G_{i}$. By Lemma 5.3, we can write $N$ abstractly as $\prod_{i=1}^{k}G_{i}^{n_{i}}$ and $N_{i}=G_{i}^{n_{i}}$.
From Lemma 5.4, we see that for $i\neq j$ the projection $N_{i}\to G_{j}^{m_{j}}$ is trivial, and it follows that $N_{i}$ is the subgroup of elements of $N$ that are trivial in the projections to $G_{j}^{m_{j}}$ for all $j\neq i$. Finally, if $n\in N_{i}$, then we can see that any $\prod_{i=1}^{k}G_{i}^{m_{i}}$ conjugate of $n$ is trivial in the projections to $G_{j}^{m_{j}}$ for all $j\neq i$ and in is $N$. Hence $N_{i}$ is a normal $F$-subgroup of $G_{i}^{m_{i}}$.
∎
The followings are two corollaries of Lemma 5.5.
Corollary 5.6.
Let $G_{i}$ be irreducible $F$-groups for $i=1,\dots,k$ such that for $i\neq j$, we have that $G_{i}$ and $G_{j}$ are not isomorphic as $F$-groups.
Let $N$ be a normal $F$-subgroup of $\prod_{i=1}^{k}G_{i}^{m_{i}}$. Then $N=\prod_{i=1}^{k}G_{i}^{m_{i}}$ if and only if
$\pi_{i}(N)=G_{i}^{m_{i}}$ for each projection $\pi_{i}:N\rightarrow G_{i}^{m_{i}}$.
Corollary 5.7.
Let $G_{i}$ be finite irreducible $F$-groups for $i=1,\dots,k$ such that for $i\neq j$, we have that $G_{i}$ and $G_{j}$ are not isomorphic as $F$-groups, and let $m_{i}$ be non-negative integers.
Let $R=\prod_{i=1}^{k}G_{i}^{m_{i}}$.
Then
$$\operatorname{Prob}([x_{1},\dots,x_{n+u}]_{F}=R)=\prod_{i=1}^{m}\operatorname{%
Prob}([y_{i,1},\dots,y_{i,n+u}]_{F}=G_{i}^{m_{i}}),$$
where the $x_{k}$ are independent, uniform random elements of $G$, and the $y_{i,k}$
are independent, uniform random elements of $G_{i}^{m_{i}}.$
The next lemma will help us determine when $[y_{i,1},\dots,y_{i,n+u}]_{F}=G_{i}^{m_{i}}$.
Lemma 5.8.
Let $G$ be an irreducible $F$-group. If $G$ is non-abelian, then a normal $F$-subgroup $N$ of $G^{m}$
is all of $G^{m}$ if and only if it is non-trivial in each of the $m$ projections to $G$.
If $G$ is abelian, then a normal $F$-subgroup $N$ of $G^{m}$
is all of $G^{m}$ if and only if the projection onto the product of the first $m-1$ factors is surjective and the projection of $N$
onto the $m$th factor does not factor through the projection onto the product of the first $m-1$ factors.
Proof.
The only if direction is clear.
We let $\pi$ be the projection of $N$ onto the first $m-1$ factors of $G^{m}$ and $\pi^{m}$ the projection onto the last factor.
For the other direction, for non-abelian $G$ we induct and so we have by the inductive hypothesis $\pi(N)=G^{m-1}$.
For $G$ abelian we have $\pi(N)=G^{m-1}$ as a hypothesis.
We consider $\pi_{m}(\ker\pi)$, which must be $1$ or $G$. If $\pi_{m}(\ker\pi)$ is $G$, then we see $N=G^{m}$, as it includes element with every possible first $m-1$ coordinates, and then an element with trivial first $m-1$ coordinates and every possible $m$th coordinate.
Now we show that we cannot have $\pi_{m}(\ker\pi)=1.$
Suppose for the sake of contradiction that $\pi_{m}(\ker\pi)=1.$ Then
since $\ker\pi_{m}\cap\ker\pi=1$, we have $\ker\pi=1,$ and $\pi$ is an isomorphism on $N$, and in particular $\pi_{m}$ factors through $\pi$. So given our hypotheses, this can only happen when $G$ is non-abelian.
We write elements $(a,b)\in G^{m-1}\times G$. Since $\pi_{m}(N)$ is non-trivial, but must be $G$ be the irreducibility of $G$. For every $b\in G$, we have some $a\in G^{m-1}$ such that $(a,b)\in N$.
However, since $N$ is normal, that means $(a,gbg^{-1})\in N$ for every $g\in G$. Since $\pi_{m}$ factors through $\pi$, we have that $b=gbg^{-1}$ for every $b,g\in G$, which is a contradiction, since above we saw we can only be in this case if $G$ is non-abelian.
∎
Lemma 5.8 lets us compute the probabilities appearing in the right-hand side of Corollary 5.7 in the following two corollaries.
Corollary 5.9.
If $G$ is a finite non-abelian irreducible $F$-group, and $y_{k}$ for $k=1,\dots,n+u$
are independent, uniform random elements of $G^{m}$, then
$$\operatorname{Prob}([y_{1},\dots,y_{n+u}]_{F}=G^{m})=(1-|G|^{-n-u})^{m}.$$
Corollary 5.10.
If $G$ is a finite abelian irreducible $F$-group, and $y_{k}$ for $k=1,\dots,n+u$
are independent, uniform random elements of $G^{m}$, then
$$\operatorname{Prob}([y_{1},\dots,y_{n+u}]_{F}=G^{m})=\prod_{k=0}^{m-1}(1-h_{F}%
(G)^{k}|G|^{-n-u}).$$
Proof.
Let $\pi_{k}$ be the projection of $G^{m}$ onto the $k$th factor, and $\Pi_{k}$ the projection of $G^{m}$ to the first $k$ factors.
We have
$$\displaystyle\operatorname{Prob}([y_{1},\dots,y_{n+u}]_{F}=G^{m})$$
$$\displaystyle=$$
$$\displaystyle\prod_{k=0}^{m-1}\operatorname{Prob}(\Pi_{k+1}([y_{1},\dots,y_{n+%
u}]_{F})=G^{k+1}\,|\,\Pi_{k}([y_{1},\dots,y_{n+u}]_{F})=G^{k}).$$
We condition on the values of $\Pi_{k}(y_{i})$, and we still have, with this conditioning, that the $\pi_{k+1}(y_{i})$ are uniform, independent random in $G$. By Lemma 5.8, given $\Pi_{k}([y_{1},\dots,y_{n+u}]_{F})=G^{k}$, we will have
$\Pi_{k+1}([y_{1},\dots,y_{n+u}]_{F})=G^{k+1},$ exactly if the map
$\pi_{k+1}|_{[y_{1},\dots,y_{n+u}]_{F}}$ does not factor through
$\Pi_{k}|_{[y_{1},\dots,y_{n+u}]_{F}}$.
We have a total of $|G|^{n+u}$ choices for the $(n+u)$-tuple $(\pi_{k+1}(y_{1}),\cdots,\pi_{k+1}(y_{n+u}))$. Call
choice for $(\pi_{k+1}(y_{1}),\dots,\pi_{k+1}(y_{n+u}))$ bad
if $\pi_{k+1}|_{[y_{1},\dots,y_{n+u}]_{F}}$ factors through $\Pi_{k}|_{[y_{1},\dots,y_{n+u}]_{F}}$.
Since $\Pi_{k}([y_{1},\dots,y_{n+u}]_{F})=G^{k}$, there are $|\operatorname{Hom}_{F}(G^{k},G)|$ choices for
maps from $G^{k}$ to $G$, each of which gives a bad choice for $(\pi_{k+1}(y_{1}),\dots,\pi_{k+1}(y_{n+u}))$ (and all bad choices arise this way).
For two maps in $\operatorname{Hom}_{F}(G^{k},G)$ to give the same bad choice, they would have to agree on $\Pi_{k}(y_{i})$ for all $i$, and since
$\Pi_{k}([y_{1},\dots,y_{n+u}]_{F})=G^{k}$, this would imply the two maps in $\operatorname{Hom}_{F}(G^{k},G)$ would be the same.
Thus there are $|\operatorname{Hom}_{F}(G^{k},G)|$ bad choices in $|G|^{n+u}$ for the $\pi_{k+1}(y_{i})$, and as
$|\operatorname{Hom}_{F}(G^{k},G)|=h_{F}(G)^{k}$, the corollary follows.
∎
Theorem 5.1 now
follows from Corollaries 5.7, 5.9, and 5.10. Also, we can now prove the following lemma which is key for our general approach in Section 4.
Lemma 5.11.
Let $G$ be a finite group, and let $N$ be a normal subgroup of $G$.
Let $M$ be the intersection of all maximal proper, $G$-normal subgroups of $N$.
Then $N/M$ is a $G/M$-group under the action of conjugation.
We have that $N/M$ is isomorphic, as an $G/M$-group, to a direct product of irreducible $G/M$-groups. Moreover, among these irreducible $G/M$-groups, the abelian ones all have the action of $G/M$ factor through $G/N$, so are also irreducible $G/N$-groups.
Proof.
We consider $N$ as a $G$-group under conjugation.
A subgroup of $N$ is a normal subgroup of $G$ if and only if it is a $G$-subgroup of $N$.
Taking the quotient modulo $M$ gives us a containment respecting bijection between the $G$-subgroups of $N$ containing $M$ and the $G/M$-subgroups of $N/M$. Since all maximal proper $G$-subgroups of $N$ contain $M$, the quotient map gives us a bijection between
the maximal proper $G$-subgroups of $N$ and the maximal proper $G/M$-subgroups of $N/M$, and in particular the quotient $M/M=1$ is the quotient of all the maximal proper $G/M$-subgroups of $N/M$.
Let $M_{i}$ be the maximal proper $G/M$-subgroups of $N/M$.
Each $(N/M)/M_{i}$ is an irreducible $G/M$-group. We have that $N/M$ is a subgroup of $\prod_{i}(N/M)/M_{i}$ that surjects into each factor, $N/M$ is isomorphic to a direct product of irreducible $G/M$-groups by Lemma 5.3.
On an abelian irreducible $G/M$-group factor, conjugation by any element in $N/M$ gives the trivial group action, so we have the last statement of the lemma.
∎
6. Determining factors appearing in $R$
Throughout this section, we assume $S$ is a set of finite groups, $n$ is a positive integer, and $H$ is a finite level $S$ group. If $S$ is finite, then we let
$$1\rightarrow R\rightarrow F\rightarrow H\rightarrow 1$$
be the fundamental short exact sequence associated to $S$, $n$ and $H$ (see Section 4). In this section, we will bound which irreducible $F$-groups are possible factors in $R$.
A finite irreducible $F$-group is characteristically simple and thus, as a group, a direct product $\Gamma^{m}$ of isomorphic simple groups.
First, when the group $H$ is fixed, Lemma 6.1 will bound the possible power $m$ for factors in $R$. For fixed $S$, Corollary 6.12 will then bound the possible simple group $\Gamma$ for factors in $R$. We take a slightly longer than necessary route to Corollary 6.12 because along the way we will develop the technology to prove Corollary 6.13,
which will later be critical in Section 9 for our proof of countable additivity of $\mu_{u}$.
Lemma 6.1.
Let $(E,\pi)$ be an $H$-extension such that $G=\ker\pi$ is a finite irreducible $E$-group. Then $G$ is isomorphic to $\Gamma^{m}$ for some finite simple group $\Gamma$ and $m\leq|H|$.
Proof.
Let $G\simeq\Gamma^{m}$, where $\Gamma$ is a finite simple group. If $\Gamma={\mathbb{Z}}/p{\mathbb{Z}}$, then $G\subset\mathbf{C}_{E}(G)$ and we have that the map $H\rightarrow\operatorname{Aut}(G)=\operatorname{GL}_{m}({\mathbb{Z}}/p{\mathbb%
{Z}})$ defined by conjugation action is an irreducible representation of $H$. Since for any non-zero vector $v\in{\mathbb{F}}_{p}^{m}$, the vectors $hv$, for $h\in H$, span a subrepresentation of $H$, we have the dimension $m$ is at most $|H|$.
If $\Gamma$ is non-abelian, then consider an embedding $\iota:\Gamma\hookrightarrow\Gamma^{m}$ such that the image is a normal subgroup. There is an element $a=(a_{1},\cdots,a_{m})\in\iota(\Gamma)$ such that $a_{i}$ is not the identity element for some $i$.
Let $b\in\Gamma^{m}$ have $j$th coordinate $1$ for $j\neq i$ and $i$th coordinate $\gamma\in\Gamma$. Then since $\iota(\Gamma)$ is normal, we have that the commutator $[a,b]\in\iota(\Gamma)$. The element $[a,b]$ is trivial in all but the $i$th coordinate, where it is $[a_{i},\gamma]$. So the intersection of $\iota(\Gamma)$ and the $i$th factor (which is a normal subgroup of $\Gamma$) contains
$[a_{i},\gamma]$ for some non-trivial $a_{i}\in\Gamma$ and all $\gamma\in\Gamma$. Since $\Gamma$ is a non-abelian simple group, this means
the intersection of $\iota(\Gamma)$ and the $i$th factor is non-trivial, and hence all of the $i$th factor. So $\iota(\Gamma)$ is exactly the $i$th factor of $\Gamma^{m}$. We have thus showed that a normal subgroup of $\Gamma^{m}$ that is isomorphic to $\Gamma$ must be one of the $m$ factors. So we have a well-defined map $\operatorname{Aut}(\Gamma^{m})\to S_{m}$ (the symmetric group on $m$ elements), and note that $\operatorname{Inn}(\Gamma^{m})$ is in the kernel of this map. If $\Gamma^{m}$ is an irreducible $E$-group, the action of $H$ on the factors must be transitive, which proves $m\leq|H|$.
∎
Recall that a chief series of a finite group $G$ is a chain of normal subgroups
(6.2)
$$1=G_{0}\lhd G_{1}\lhd\cdots\lhd G_{r}=G$$
such that for each $0\leq i\leq r-1$, $G_{i}$ is normal in $G$ and the quotient group $G_{i+1}/G_{i}$ is a minimal normal subgroup of $G/G_{i}$. If $M$ is a minimal normal subgroup of $G$, then define $\rho_{M}$ to be the homomorphism
$$\displaystyle\rho_{M}:G$$
$$\displaystyle\to$$
$$\displaystyle\operatorname{Aut}(M)$$
$$\displaystyle g$$
$$\displaystyle\mapsto$$
$$\displaystyle(x\mapsto gxg^{-1})_{x\in M}.$$
The kernel of $\rho_{M}$ is the centralizer $\mathbf{C}_{G}(M)$ of $M$ in $G$. So $\rho_{M}$ gives an isomorphism from $G/\mathbf{C}_{G}(M)$ to the subgroup $\rho_{M}(G)$ of $\operatorname{Aut}(M)$. In fact, since $M$ is a minimal normal subgroup of $G$, it is a direct product of isomorphic simple groups. If $M$ is a direct product of isomorphic abelian simple groups, i.e. an elementary abelian $p$-group, then $\rho_{M}(M)=\operatorname{Inn}(M)=1$; otherwise, $\rho_{M}(M)=\operatorname{Inn}(M)\simeq M$. Thus, $\operatorname{Inn}(M)$ is always a normal subgroup of $\rho_{M}(G)$ and $\rho_{M}(G)/\operatorname{Inn}(M)\simeq G/(M\cdot\mathbf{C}_{G}(M))$.
Definition.
A chief factor pair is a pair of finite groups $(M,A)$ such that $M$ is an irreducible $A$-group and the $A$-action on $M$ is faithful (hence $A$ is naturally a subgroup of $\operatorname{Aut}(M)$). In particular, the chief series (6.2) gives a sequence of chief factor pairs $(G_{i+1}/G_{i},\rho_{G_{i+1}/G_{i}}(G/G_{i}))$, and we call them chief factor pairs of the series (6.2).
Definition.
Two chief factor pairs $(M_{1},A_{1})$ and $(M_{2},A_{2})$ are isomorphic if there exists an isomorphism $\alpha:M_{1}\to M_{2}$ such that the induced isomorphism $\alpha^{*}:\operatorname{Aut}(M_{1})\to\operatorname{Aut}(M_{2})$ maps $A_{1}$ to $A_{2}$.
The following is an analog of the Jordan-Hölder Theorem.
Lemma 6.3.
Let $G$ be a finite group. Suppose there are two chief series of $G$:
(6.4)
$$\displaystyle 1=G_{0}\lhd G_{1}\lhd\cdots\lhd G_{r}=G$$
(6.5)
and
$$\displaystyle 1=I_{0}\lhd I_{1}\lhd\cdots\lhd I_{s}=G.$$
Then
(1)
$r=s$;
(2)
the list of isomorphism classes of chief factor pairs $\Big{\{}\Big{(}G_{i+1}/G_{i},\rho_{G_{i+1}/G_{i}}(G/G_{i})\Big{)}_{i=0}^{r-1}%
\Big{\}}$ is a rearrangement of the list $\Big{\{}\Big{(}I_{i+1}/I_{i},\rho_{I_{i+1}/I_{i}}(G/I_{i})\Big{)}_{i=0}^{s-1}%
\Big{\}}$.
Proof.
We prove this by induction on $|G|$. The case that $|G|=1$ is trivial. Assume the lemma is true for all groups of order less than $k$ and $G$ is a group of order $k$. If $G_{1}=I_{1}$ then
$$\displaystyle 1\lhd G_{2}/G_{1}\lhd\cdots\lhd G_{r}/G_{1}=G/G_{1}$$
and
$$\displaystyle 1\lhd I_{2}/I_{1}\lhd\cdots I_{s}/I_{1}=G/I_{1}$$
are two chief series of $G/G_{1}$. So the lemma is proved for $G$ by the induction hypothesis.
Assume $G_{1}\neq I_{1}$. Since they are minimal normal subgroups, $G_{1}\cap I_{1}=1$ and $G_{1}I_{1}=G_{1}\times I_{1}$. Define $J_{2}$ to be the product $G_{1}I_{1}$. Then $J_{2}/G_{1}\simeq I_{1}$ is a minimal normal subgroup of $G/G_{1}$ and we can construct a chief series of $G$ passing through $G_{1}$ and $J_{2}$
(6.6)
$$1\lhd G_{1}\lhd J_{2}\lhd J_{3}\lhd\cdots\lhd J_{t}=G.$$
Comparing chief series (6.4) and (6.6), it follows that $r=t$ and
$$\displaystyle\left\{\Big{(}G_{i+1}/G_{i},\rho_{G_{i+1}/G_{i}}(G/G_{i})\right)_%
{i=0}^{r-1}\Big{\}}$$
$$\displaystyle\sim$$
$$\displaystyle\Big{\{}\Big{(}G_{1},\rho_{G_{1}}(G)\Big{)},\Big{(}J_{2}/G_{1},%
\rho_{J_{2}/G_{1}}(G/G_{1})\Big{)},$$
$$\displaystyle\Big{(}J_{i+1}/J_{i},\rho_{J_{i+1}/J_{i}}(G/J_{i})\Big{)}_{i=2}^{%
t-1}\Big{\}}$$
where the symbol $\sim$ means “is a rearrangement of”. Let $\pi$ be the quotient map $G\to G/G_{1}$. As $G_{1}\unlhd\mathbf{C}_{G}(I_{1})$, if an element in $G$ centralizes $I_{1}$, then its image under $\pi$ centralizes $\pi(I_{1})=J_{2}/G_{1}$. It follows that $\pi(\mathbf{C}_{G}(I_{1}))\subseteq\mathbf{C}_{G/G_{1}}(J_{2}/G_{1})$. Conversely, if $a$ is an element in $G$ such that $\pi(a)\in\mathbf{C}_{G/G_{1}}(J_{2}/G_{1})$, then for every $h\in I_{1}$, we have $\pi(aha^{-1})=\pi(h)$, which indicates that there exists $g\in G_{1}$ such that $aha^{-1}=hg$. But $I_{1}\unlhd G$, so $aha^{-1}\in I_{1}$. It follows from $I_{1}\cap G_{1}=1$ that $g=1$, and hence $a\in\mathbf{C}_{G}(I_{1})$, which proves $\pi(\mathbf{C}_{G}(I_{1}))=\mathbf{C}_{G/G_{1}}(J_{2}/G_{1})$. Thus we have
$$\displaystyle\faktor{G}{\mathbf{C}_{G}(I_{1})}$$
$$\displaystyle\cong$$
$$\displaystyle\faktor{G/G_{1}}{\mathbf{C}_{G}(I_{1})/G_{1}}$$
$$\displaystyle\cong$$
$$\displaystyle\faktor{G/G_{1}}{\mathbf{C}_{G/G_{1}}(J_{2}/G_{1})}.$$
Therefore the chief factor pairs $\Big{(}I_{1},\rho_{I_{1}}(G)\Big{)}$ and $\Big{(}J_{2}/G_{1},\rho_{J_{2}/G_{1}}(G/G_{1})\Big{)}$ are isomorphic. So the list (6) is
(6.8)
$$\sim\Big{\{}\Big{(}G_{1},\rho_{G_{1}}(G)\Big{)},\Big{(}I_{1},\rho_{I_{1}}(G)%
\Big{)},\Big{(}J_{i+1}/J_{i},\rho_{J_{i+1}/J_{i}}(G/J_{i})\Big{)}_{i=2}^{t-1}%
\Big{\}}.$$
Similarly, by comparing the following chief series of $G$
(6.9)
$$1\lhd I_{1}\lhd J_{2}\lhd J_{3}\lhd\cdots\lhd J_{t}=G.$$
with (6.5), we finish the proof of the lemma.
∎
Definition.
If $G$ is a finite group, then define $\mathcal{CF}(G)$ to be the set consisting of all isomorphism classes of chief factor pairs of a chief series of $G$ ($\mathcal{CF}(G)$ does not depend on the choice of chief factor series by Lemma 6.3). If $T$ is a set of finite groups, then
$$\mathcal{CF}(T):=\bigcup\limits_{G\in T}\mathcal{CF}(G).$$
The following lemma shows that every factor in $R$ comes from $\mathcal{CF}(\bar{S})$.
Lemma 6.10.
Let $S$ be a finite set of finite groups, and $R$, $F$ and $H$ as defined at the beginning of this section. If $G$ is an irreducible $F$-subgroup of $R$, then $(G,\rho_{G}(F))\in\mathcal{CF}(\bar{S})$ and $\rho_{G}(F)/\operatorname{Inn}(G)$ is isomorphic to a quotient of $H$.
Proof.
Since $F$ is a finite level $S$ group and $G$ is a minimal normal subgroup of $F$, we have $(G,F/\mathbf{C}_{F}(G))\in\mathcal{CF}(\bar{S})$. Further, $R$ is a direct product of irreducible $F$-groups, so $R$ is contained in $\mathbf{C}_{F}(G)\cdot G$ and it follows that $(F/\mathbf{C}_{F}(G))/\operatorname{Inn}(G)=F/(\mathbf{C}_{F}(G)\cdot G)$ is a quotient of $H$.
∎
In the rest of this section, we will bound the size of chief factor pairs.
Lemma 6.11.
If $S$ is a set of finite groups that is closed under taking subgroups and quotients, then $\mathcal{CF}(\bar{S})=\mathcal{CF}(S)$.
Proof.
Since $\bar{S}$ is the closure of $S$ under taking finite direct products, quotients and subgroups, it suffices to show that none of these three actions creates new chief factor pairs not belonging to $\mathcal{CF}(S)$.
First, taking direct products and quotients does not create new chief factor pairs. If $G$ and $J$ are finite groups with chief series $1\lhd G_{1}\lhd\cdots\lhd G_{r}=G$ and $1\lhd J_{1}\lhd\cdots\lhd J_{s}=J$. Then the following chief series of $G\times J$
$$1\lhd G_{1}\times 1\lhd\cdots G\times 1\lhd G\times J_{1}\lhd\cdots\lhd G\times
J$$
implies that $\mathcal{CF}(G\times J)=\mathcal{CF}(G)\cup\mathcal{CF}(J)$. If $N$ is a normal subgroup of $G$, then $\mathcal{CF}(G/N)\subseteq\mathcal{CF}(G)$ since we can always find a chief series of $G$ passing through $G/N$.
Finally, assume $J$ is a subgroup of $G$ for $G\in\bar{S}$ such that $\mathcal{CF}(G)\subseteq\mathcal{CF}(S)$. We want to prove $\mathcal{CF}(J)\subseteq\mathcal{CF}(S)$. Let $1\lhd G_{1}\lhd\cdots\lhd G_{r}=G$ be a chief series of $G$. We can construct a chief series of $J$ that passes through $G_{i}\cap J$ for every $i=1,\cdots,r$. The chief factor pairs achieved from the elements between $G_{i}\cap J$ and $G_{i+1}\cap J$ are achieved from the group $J/(G_{i}\cap J)\simeq(J\cdot G_{i})/G_{i}$, which is a subgroup of $G/G_{i}$. Thus it’s enough to consider the positions between $1$ and $G_{1}\cap J$. Since $(G_{1},\rho_{G_{1}}(G))\in\mathcal{CF}(G)\subseteq\mathcal{CF}(S)$, there is a group $G^{\prime}\in S$ and a minimal subgroup $G^{\prime}_{1}$ of $G^{\prime}$ such that the chief factors $(G_{1},\rho_{G_{1}}(G))$ and $(G^{\prime}_{1},\rho_{G^{\prime}_{1}}(G^{\prime}))$ are isomorphic, i.e. $\exists$ $\alpha:G_{1}\overset{\sim}{\to}G^{\prime}_{1}$ such that $\alpha^{*}:\operatorname{Aut}(G_{1})\overset{\sim}{\to}\operatorname{Aut}(G^{%
\prime}_{1})$ maps $\rho_{G_{1}}(G)$ to $\rho_{G^{\prime}_{1}}(G^{\prime})$. Define $A:=\rho_{G_{1}}(J)=(J\cdot\mathbf{C}_{G}(G_{1}))/\mathbf{C}_{G}(G_{1})$ that is a subgroup of $\rho_{G_{1}}(G)$. Note that the action of $A$ on $G_{1}$ actually stabilizes $G_{1}\cap J$. Let $J^{\prime}:=\rho_{G^{\prime}_{1}}^{-1}(\alpha^{*}(A))$ and $J^{\prime}_{1}=\alpha(G_{1}\cap J)$. So $J^{\prime}$ is a subgroup of $G^{\prime}$ satisfying the following short exact sequence
$$1\to\mathbf{C}_{G^{\prime}}(G^{\prime}_{1})\to J^{\prime}\to\alpha^{*}(A)\to 1.$$
and $J^{\prime}_{1}$ is a subgroup of $G^{\prime}_{1}\cap J^{\prime}$.
Since $\mathbf{C}_{G^{\prime}}(G_{1}^{\prime})\leq\mathbf{C}_{J^{\prime}}(G^{\prime}_%
{1}\cap J^{\prime})$, the action of $J^{\prime}$ via conjugation on $G^{\prime}_{1}\cap J^{\prime}$ factors through $\alpha^{*}(A)$. Also, since the $\alpha^{*}(A)$ action on $G^{\prime}_{1}$ stabilizes $J^{\prime}_{1}$, we have that $J^{\prime}_{1}$ is a normal subgroup of $J^{\prime}$. Because $G_{1}\cap J$ with the action of $A$ is isomorphic to $J^{\prime}_{1}$ with the action of $\alpha^{*}(A)$, every chief factor pair of $G$ achieved from positions between 1 and $G_{1}\cap J$ is also a chief factor pair of $J^{\prime}$ achieved via a series passing through $J^{\prime}_{1}$. Finally, $J^{\prime}$ as a subgroup of $G^{\prime}$ belongs to $S$, so $\mathcal{CF}(J)\subseteq\mathcal{CF}(S)$ and we prove the lemma.
∎
Corollary 6.12.
If $S$ is a set of finite groups, and $\Gamma\in\bar{S}$ is a simple group, then $\Gamma$ is in the closure of $S$ under taking subgroups and quotients.
Proof.
If $\Gamma\in\bar{S}$ is a simple group, then $(\Gamma,\operatorname{Inn}(\Gamma))\in\mathcal{CF}(\bar{S})$. By Lemma 6.11, $\Gamma$ is in the closure of $S$ under taking subgroups and quotients.
∎
Corollary 6.13.
Let $S$ be a finite set of finite groups. Then $\mathcal{CF}(\bar{S})$ is a finite set. Moreover, if $\ell$ is the upper bound of the orders of groups in $S$, then for any pair $(M,A)\in\mathcal{CF}(\bar{S})$, the quotient $A/\operatorname{Inn}(M)$ is of level $\ell-1$.
Proof.
Without lost of generality, let’s assume $S$ is closed under taking subgroups and quotients. By Lemma 6.11, $\mathcal{CF}(\bar{S})=\mathcal{CF}(S)$ is finite, and for any chief factor pair $(M,A)\in\mathcal{CF}(\bar{S})$, there is a group $G\in S$ such that $(M,A)\in\mathcal{CF}(G)$. If $M$ is abelian, then $|M||A|\leq|G|\leq\ell$; otherwise, $M$ is non-abelian and $|A|=|M||A/\operatorname{Inn}(M)|\leq|G|\leq\ell$. In either case, we have $|A/\operatorname{Inn}(M)|\leq\ell-1$.
∎
7. Counting maximal quotients of irreducible $F$-groups
In order to apply Theorem 5.1 to a group that we know, abstractly, to be a product of irreducible $F$-groups, we need to know the multiplicities of the various irreducible $F$-groups in the product. In this section, we relate those multiplicities to a count of surjections.
Theorem 7.1.
Let $G_{i}$ be finite irreducible $F$-groups for $i=1,\dots,k$ such that $G_{i}$ and $G_{j}$ are not isomorphic for $i\neq j$. Then if $G_{j}$ is abelian
$$\#\operatorname{Sur}_{F}\left(\prod_{i=1}^{k}G_{i}^{m_{i}},G_{j}\right)=h_{F}(%
G_{j})^{m_{j}}-1$$
and if $G_{j}$ is non-abelian
$$\#\operatorname{Sur}_{F}\left(\prod_{i=1}^{k}G_{i}^{m_{i}},G_{j}\right)=m_{j}|%
\operatorname{Aut}_{F}(G_{j})|.$$
This theorem follows immediately from Lemma 5.4 and the following lemmas.
Lemma 7.2.
Let $G_{i}$ be finite irreducible $F$-groups for $i=1,\dots,k$ such that for $i\neq j$, we have that $G_{i}$ and $G_{j}$ are not isomorphic. The restriction map
$$\operatorname{Sur}_{F}\left(\prod_{i=1}^{k}G_{i}^{m_{i}},G_{j}\right)%
\rightarrow\operatorname{Hom}_{F}\left(G_{j}^{m_{j}},G_{j}\right)$$
is a bijection to $\operatorname{Sur}_{F}\left(G_{j}^{m_{j}},G_{j}\right)\subset\operatorname{Hom%
}_{F}\left(G_{j}^{m_{j}},G_{j}\right)$.
Proof.
Note that in a surjection, each $G_{i}$ must go to a normal subgroup of $G_{j}$, and so by Lemma 5.4 the restriction to every $G_{i}$ factor for $i\neq j$ is trivial. So that proves the above restriction map is injective.
The restriction map is surjective to $\operatorname{Sur}_{F}\left(G_{j}^{m_{j}},G_{j}\right)$ since $G_{j}^{m_{j}}$ is a quotient of $\prod_{i=1}^{k}G_{i}^{m_{i}}$.
∎
Lemma 7.3.
Let $G$ be a finite irreducible $F$-group and $m$ a positive integer.
We have
$$\operatorname{Hom}_{F}(G^{m},G)\subset\operatorname{Hom}_{F}(G,G)^{m}$$
by restriction to each factor.
If $G$ is abelian, then this inclusion is an equality. If $G$ is non-abelian, then we have that $\operatorname{Hom}_{F}(G^{m},G)$ is the subset of the
$m$-tuples $\operatorname{Hom}_{F}(G,G)^{m}$ where at most $1$ coordinate is a non-trivial morphism in $\operatorname{Hom}_{F}(G,G)$.
The only homomorphism that is not surjective among those above is the trivial morphism.
Proof.
If $G$ is abelian, then for $\phi_{i}\in\operatorname{Hom}_{F}(G,G)$, we have a morphism $\phi:G^{m}\rightarrow G$ such that $\phi(a_{1},\dots,a_{m})=\prod_{i=1}^{m}\phi_{i}(a_{i})$.
Note for $\phi\in\operatorname{Hom}_{F}(G^{m},G)$, with restrictions $\phi_{i}$ to the factors, we have that $a\in\phi_{i}(G)$ and $b\in\phi_{j}(G)$ commute for $i\neq j$. Since $\phi_{i}(G)$ is $1$ or $G$, if $G$ is non-abelian we see that at most one $\phi_{i}$ can be non-trivial. Moreover, clearly the $m$-tuples $\operatorname{Hom}_{F}(G,G)^{m}$ where at most $1$ coordinate is a non-trivial morphism in $\operatorname{Hom}_{F}(G,G)$ give elements of $\operatorname{Hom}_{F}(G^{m},G)$.
For an $F$-morphism $G\rightarrow G$, if it is non-trivial, it must be injective (since its kernel is a normal $F$-subgroup), and thus surjective.
∎
8. Determination of $\mu_{u,n}$ on basic open sets
The goal of this section is to prove Theorem 8.1, in which we will give $\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)$ for every finite set $S$ and finite level $S$ group $H$, i.e. determine the measures of the basic open sets in the distributions coming from our random groups.
Throughout this section, we assume $n\geq 1$, $S$ is a finite set of finite groups, $H$ is a finite level $S$ group and
$$1\rightarrow R\rightarrow F\rightarrow H\rightarrow 1$$
is the fundamental short exact sequence associated to $S$, $n$ and $H$.
For any abelian irreducible $H$-group $G$, we define $m(S,n,H,G)$ to be the multiplicity of $G$ in $R$ as an $H$-group under conjugation (see Lemma 5.11). Let $G$ be a non-abelian finite group. Let $G_{i}$ be the irreducible $F$-group structures one can put on $G$. Then we define $m(S,n,H,G)$ to be the sum (over $i$) of the multiplicity of the $G_{i}$ in $R$ as an $F$-group under conjugation.
Equation (4.2) and Theorem 5.1 allow us to express $\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)$ in terms of the multiplicities
$m(S,n,H,G)$. The work of this section will be to find
explicit formulas for these $m(S,n,H,G)$ (given in Corollaries 8.8 and 8.10).
Theorem 8.1.
Let $S$ be a finite set of finite groups
and $H$ a finite level $S$ group. Let $n\geq 1$ and $u>-n$ be integers.
Then
$$\displaystyle\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{|\operatorname{Sur}(\hat{F}_{n},H)|}{|\operatorname{Aut}(H)%
||H|^{n+u}}\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\end{subarray}}\prod_{%
k=0}^{m(S,n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{n+u}}})\prod_{\begin{subarray}%
{c}G\in\mathcal{N}\end{subarray}}(1-|G|^{-{n-u}})^{m(S,n,H,G)},$$
and we have
$$\displaystyle\lim_{n\rightarrow\infty}\operatorname{Prob}((X_{u,n})^{\bar{S}}%
\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{\infty}(1-\lambda(S,H,G)\frac{%
h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{%
subarray}}e^{-|G|^{-u}\lambda(S,H,G)}.$$
Further, if $G\in\mathcal{A}_{H}\cup\mathcal{N}$ is isomorphic as a group to $\Gamma^{j}$ for some simple group $\Gamma$, and either 1) $\Gamma$ is not in the closure of $S$ under taking subgroups and quotients, or 2) $j>|H|$, then
$m(S,n,H,G)=\lambda(S,H,G)=0$.
Remark 8.4.
The products over $\mathcal{A}_{H}$ and
$\mathcal{N}$ appearing in Theorem 8.1 are actually finite products (except for trivial terms), because of the last statement in the theorem.
Remark 8.5.
We will show in Section 11 that statement of Theorem 8.1 also works for an arbitrary set $S$ of finite groups.
First, we need to define the Möbius function on a poset of $H$-extensions. Given a finite group $H$, there is a poset $\mathcal{E}_{H}$ of $H$-extensions (not isomorphism classes of $H$-extensions) where $(E,\pi)\leq(E^{\prime},\pi^{\prime})$ if $(E,\pi)$ is a sub-$H$-extension of $(E^{\prime},\pi^{\prime})$. (This relation is defined for literal sub-$H$-extensions and not $H$-extensions just isomorphic to a subextension.)
We let $\nu(D,E)$ be the Möbius function of this poset (we drop the maps to $H$ in the notation but they are implicit) so that for two $H$-extensions $D$ and $E$ we have
$$\displaystyle\nu(E,E)$$
$$\displaystyle=1$$
$$\displaystyle\nu(D,E)$$
$$\displaystyle=-\sum_{\begin{subarray}{c}D^{\prime}\in\mathcal{E}_{H}\\
D<D^{\prime}\leq E\end{subarray}}\nu(D^{\prime},E)\quad\quad\textrm{ if $D\neq
E$}$$
so that in particular
$$\displaystyle\nu(D,E)$$
$$\displaystyle=0\textrm{ if $D$ is not a sub-$H$-extension of $E$}$$
$$\displaystyle\sum_{\begin{subarray}{c}D^{\prime}\in\mathcal{E}_{H}\\
D\leq D^{\prime}\leq E\end{subarray}}\nu(D^{\prime},E)$$
$$\displaystyle=\begin{cases}1&\textrm{if $D=E$}\\
0&\textrm{otherwise}.\end{cases}$$
Theorem 7.1 relates our key multiplicities $m(S,n,H,G)$ to the number of $F$-surjections from $R$ to $G$. An $F$-surjection $R\rightarrow G$ has a kernel $K$, and we have a surjection from our $H$-extension $(\hat{F}_{n})^{\bar{S}}\rightarrow H$ to the $H$-extension $F/K\rightarrow H$. The next proposition will count such surjections of $H$-extensions.
Proposition 8.6.
Let $n\geq 1$ be an integer, $S$ a finite set of finite groups, and $H$ a finite level $S$ group.
Let $(\hat{F}_{n})^{\bar{S}}\stackrel{{\scriptstyle\rho}}{{\rightarrow}}H$ be an $H$-extension structure on $(\hat{F}_{n})^{\bar{S}}$.
Let $E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}H$ be a finite $H$ extension. We have
$$|\operatorname{Sur}(\rho,\pi)|=\begin{cases}\sum_{\begin{subarray}{c}D\in%
\mathcal{E}_{H},D\leq E\end{subarray}}\nu(D,E)\left(\frac{|D|}{|H|}\right)^{n}%
&\textrm{ if $E$ is level $S$}\\
0&\textrm{ otherwise}.\end{cases}$$
Proof.
If $(\hat{F}_{n})^{\bar{S}}\rightarrow E$ is a surjection, then $E$ is level $S$. If $E$ is level $S$ and $(D,\psi)\leq(E,\pi)$, surjections $(\hat{F}_{n})^{\bar{S}}\rightarrow D$ exactly correspond to surjections $\hat{F}_{n}\rightarrow D$, i.e. choices of image for each generator $x_{1},\dots,x_{n}$ of $\hat{F}_{n}$ such that their images generate $D$.
For each generator $x_{i}$ of $\hat{F}_{n}$, we have a fixed coset of $\ker(\pi)$ in $D$ it can land in to actually obtain a surjection compatible with the maps to $H$.
We have that the number of homomorphisms $\hat{F}_{n}\rightarrow D$ where the generators go to the appropriate cosets is $(|D|/|H|)^{n}$.
Let $E^{\prime}$ be a subgroup of $D$ that could be generated by some $y_{1},\dots,y_{n}$ with each $y_{i}$ in the required cosets of $\ker(\pi)$. Since $\rho$ is a surjection, it follows that $\pi(E^{\prime})=H$.
So we have
$$\left(\frac{|D|}{|H|}\right)^{n}=\sum_{\begin{subarray}{c}(E^{\prime},\phi)\in%
\mathcal{E}_{H}\\
(E^{\prime},\phi)\leq(D,\psi)\end{subarray}}|\operatorname{Sur}_{H}(\rho,\phi)|.$$
Using Möbius inversion, we obtain the result. We can sum the above as follows. Given a finite $H$-extension $(E,\pi)$ of level $S$, we have
$$\displaystyle\sum_{(D,\psi)\leq(E,\pi)}\nu(D,E)\left(\frac{|D|}{|H|}\right)^{n}$$
$$\displaystyle=\sum_{(D,\psi)\leq(E,\pi)}\nu(D,E)\sum_{\begin{subarray}{c}(E^{%
\prime},\phi)\leq(D,\psi)\end{subarray}}|\operatorname{Sur}_{H}(\rho,\phi)|$$
$$\displaystyle=\sum_{(E^{\prime},\phi)\leq(E,\pi)}|\operatorname{Sur}_{H}(\rho,%
\phi)|\sum_{(E^{\prime},\phi)\leq(D,\psi)\leq(E,\pi)}\nu(D,E)$$
$$\displaystyle=|\operatorname{Sur}_{H}(\rho,\pi)|,$$
as desired.
∎
Now we will build on Proposition 8.6 to find $|\operatorname{Sur}_{F}(R,G)|$, after which we can then use Theorem 7.1 to find the $m(S,n,H,G)$. We will first do the case of abelian $G$, and then non-abelian $G$.
Proposition 8.7 (Counting surjections from $R$ to abelian $G$).
Let $H$, $F$, and $R$ be defined as at the beginning of this section.
Let $G$ be an abelian irreducible $F$-group.
Then
$$\displaystyle|\operatorname{Sur}_{F}(R,G)|=|\operatorname{Aut}_{F}(G)|\sum_{%
\begin{subarray}{c}\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ as an $H$-group}\\
\textrm{$E$ is level $S$}\end{subarray}}\frac{\sum_{\begin{subarray}{c}D\in%
\mathcal{E}_{H},D\leq E\end{subarray}}\nu(D,E)\left(\frac{|D|}{|H|}\right)^{n}%
}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
if the action of $F$ on $G$ factors through $F\rightarrow H$ (i.e. elements of $R$ act trivially on $G$) and $|\operatorname{Sur}_{F}(R,G)|=0$ otherwise.
Proof.
We have that $|\operatorname{Sur}_{F}(R,G)|$ is $|\operatorname{Aut}_{F}(G)|$ times
the number of $F$-subgroups $M$ of $R$ such
that $R/M$ under $F$-conjugation is isomorphic to $G$ as an $F$-group.
If $M$ is an $F$-subgroup of $R$ such that $R/M$ is abelian, then the action of $F$ via conjugation on $R/M$ factors through $H$ (because conjugation by elements from $R$ is trivial in $R/M$ as $R/M$ is abelian).
So suppose that the action of $F$ on $G$ factors through $H$.
We have
the number of $F$-subgroups $M$ of $R$ such
that $R/M$ is isomorphic to $G$ as an $F$-group is
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of}\\
\textrm{$H$-extensions $(E,\pi)$}\end{subarray}}\#\left\{F\textrm{-subgroups $%
M$ of $R$}\,\Bigg{|}\,\begin{aligned} &\displaystyle(F/M\rightarrow H)\simeq(E%
,\pi)\textrm{ as $H$-exts,}\\
&\displaystyle\textrm{$R/M\simeq G$ as $F$-groups }\end{aligned}\right\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of}\\
\textrm{$H$-extensions $(E,\pi)$}\\
\ker\pi\simeq G\textrm{ as groups}\end{subarray}}\#\left\{F\textrm{-subgroups %
$M$ of $R$}\,\Bigg{|}\,\begin{aligned} &\displaystyle(F/M\rightarrow H)\simeq(%
E,\pi)\textrm{ as $H$-exts,}\\
&\displaystyle\textrm{$R/M\simeq G$ as $F$-groups }\end{aligned}\right\}.$$
Given that $R/M$ is abelian (which is guaranteed by the group isomorphism $\ker\pi\simeq G$ and the $H$-extension isomorphism $(F/M\rightarrow H)\simeq(E,\pi)$), since the action of $F$ on $R/M$ factors through $H$, we have that
$R/M$ is isomorphic to $G$ as an $F$-group if and only if it is isomorphic to $G$ as an $H$-group.
Given $(F/M\rightarrow H)\simeq(E,\pi)$, this is the same as requiring $\ker\pi\simeq G$ as an $H$-group.
Thus the above sum is equal to
$$\displaystyle\sum_{\begin{subarray}{c}\text{isom. classes of $H$-extension }(E%
,\pi)\\
\ker\pi\text{ isom. $G$ as an $H$-group}\end{subarray}}\#\left\{F\text{-%
subgroups $M$ of $R$}\mid(F/M\rightarrow H)\simeq(E,\pi)\text{ as $H$-exts}\right\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}\text{isom. classes of $H$-extension }(E%
,\pi)\\
\ker\pi\text{ isom. $G$ as an $H$-group}\end{subarray}}\frac{\#\left\{(M,\phi)%
\,\Bigg{|}\,\begin{aligned} &\displaystyle M\text{ an $F$-subgroup of }R\\
&\displaystyle\phi:(F/M\rightarrow H)\simeq(E,\pi)\end{aligned}\right\}}{|%
\operatorname{Aut}_{H}(E,\pi)|}.$$
Note that the data $(M,\phi)$ above is exactly the same as the data of a surjection of $H$-extensions from $F\rightarrow H$ to $E\rightarrow H$.
Now let $(E,\pi)$ be an $H$-extension with $\ker\pi$ (via conjugation) an abelian irreducible $H$-group. Consider a surjection of $H$-extensions from $(\hat{F}_{n})^{\bar{S}}\rightarrow H$ to $E\rightarrow H$, in which the map $(\hat{F}_{n})^{\bar{S}}\rightarrow E$ has kernel $K$. Let $N$ denote the kernel of $(\hat{F}_{n})^{\bar{S}}\to H$. Then $N/K$ is an irreducible $(\hat{F}_{n})^{\bar{S}}$-group, and so $K$ is a maximal proper $(\hat{F}_{n})^{\bar{S}}$-subgroup of $N$. So the map $(\hat{F}_{n})^{\bar{S}}\rightarrow E$ factors through $F$. On the other hand, any surjection of $H$-extensions from $F\rightarrow H$ to $E\rightarrow H$ clearly extends to a surjection of $H$-extensions from $(\hat{F}_{n})^{\bar{S}}\rightarrow H$ to $E\rightarrow H$. Thus the above sum is equal to
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-extensions %
$(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ as an $H$-group}\end{subarray}}\frac{|%
\operatorname{Sur}_{H}((\hat{F}_{n})^{\bar{S}}\rightarrow H,\pi)|}{|%
\operatorname{Aut}_{H}(E,\pi)|}.$$
The result now follows from applying Proposition 8.6 above, after dividing out by the number of choices of isomorphism to $(E,\pi)$.
∎
We now can determine the multiplicities of the abelian irreducible $F$-groups in $R$ by combining Theorem 7.1 and Proposition 8.7.
Corollary 8.8 (Multiplicities of abelian $G$ in $R$).
Let $H$, $F$ and $R$ be as above. Let $G$ be an abelian irreducible $H$-group.
Then
$$\frac{h_{H}(G)^{m(S,n,H,G)}-1}{h_{H}(G)-1}=\sum_{\begin{subarray}{c}\textrm{%
isom. classes of $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ as an $H$-group}\\
\textrm{$E$ is level $S$}\end{subarray}}\frac{\sum_{\begin{subarray}{c}D\in%
\mathcal{E}_{H},D\leq E\end{subarray}}\nu(D,E)\left(\frac{|D|}{|H|}\right)^{n}%
}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
Next, we will apply a similar plan to obtain the multiplicities of the non-abelian $G$, but there is an important difference from the abelian case.
When $\ker(E\stackrel{{\scriptstyle\pi}}{{\rightarrow}}H)$ is non-abelian, a surjection of $H$-extensions $F\rightarrow E$ still gives an $F$-group structure on $\ker\pi$ by conjugation in $E$, but, unlike in the case when $\ker\pi$ is abelian, that $F$-group structure is not necessarily determined by the isomorphism type of the $H$-extension $(E,\pi).$
So in this case it is most convenient to add together, for each possible underlying group $G$ of a non-abelian irreducible $F$-group, all surjections $F\rightarrow E$ over all $G$ extensions $E$ of $H$.
Proposition 8.9 (Counting surjections from $R$ to non-abelian $G$).
Let $H$, $F$ and $R$ be as above.
Let $G$ be a finite non-abelian group.
Let $G_{i}$ be the pairwise non-isomorphic irreducible $F$-group structures on $G$ for $1\leq i\leq k$ ($k$ may be $0$).
Then
$$\displaystyle\sum_{i=1}^{k}\frac{|\operatorname{Sur}_{F}(R,G_{i})|}{|%
\operatorname{Aut}_{F}(G_{i})|}=$$
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-extensions %
$(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\\
\textrm{$E$ is level $S$}\end{subarray}}\frac{\sum_{\begin{subarray}{c}D\in%
\mathcal{E}_{H},D\leq E\end{subarray}}\nu(D,E)\left(\frac{|D|}{|H|}\right)^{n}%
}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
Proof.
We note that ${|\operatorname{Sur}_{F}(R,G_{i})|}/{|\operatorname{Aut}_{F}(G_{i})|}$ is the number of $F$-subgroups of $R$ whose corresponding quotient is isomorphic to $G_{i}$ as an $F$-group.
We have
$$\displaystyle\sum_{i=1}^{k}\frac{|\operatorname{Sur}_{F}(R,G_{i})|}{|%
\operatorname{Aut}_{F}(G_{i})|}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{k}\sum_{\begin{subarray}{c}\textrm{isom. classes of}%
\\
\textrm{ $H$-extensions $(E,\pi)$}\end{subarray}}\#\left\{\textrm{$F$-%
subgroups $M$ of $R$}\,\Bigg{|}\,\begin{aligned} &\displaystyle(F/M\rightarrow
H%
)\simeq(E,\pi)\text{ as $H$-exts}\\
&\displaystyle R/M\simeq G_{i}\text{ as $F$-groups }\end{aligned}\right\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{k}\sum_{\begin{subarray}{c}\textrm{isom. classes of}%
\\
\textrm{ $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\end{subarray}}\#\left\{\textrm{$F$-subgroups $M$%
of $R$}\,\Bigg{|}\,\begin{aligned} &\displaystyle(F/M\rightarrow H)\simeq(E,%
\pi)\text{ as $H$-exts}\\
&\displaystyle R/M\simeq G_{i}\text{ as $F$-groups }\end{aligned}\right\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of}\\
\textrm{ $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\end{subarray}}\sum_{i=1}^{k}\#\left\{\textrm{$F$%
-subgroups $M$ of $R$}\,\Bigg{|}\,\begin{aligned} &\displaystyle(F/M%
\rightarrow H)\simeq(E,\pi)\text{ as $H$-exts}\\
&\displaystyle R/M\simeq G_{i}\text{ as $F$-groups }\end{aligned}\right\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of}\\
\textrm{ $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\end{subarray}}\sum_{i=1}^{k}\#\left\{%
\textrm{$F$-subgroups $M$ of $R$}\,\Bigg{|}\,\begin{aligned} &\displaystyle(F/%
M\rightarrow H)\simeq(E,\pi)\text{ as $H$-exts}\\
&\displaystyle R/M\simeq G_{i}\text{ as $F$-groups }\end{aligned}\right\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-extensions %
$(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\end{subarray}}\#\{\textrm{$F$-subgroups $M%
$ of $R\,|\,(F/M\rightarrow H)\simeq(E,\pi)$ as $H$-exts }\}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-extensions %
$(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\end{subarray}}\frac{\#\left\{(M,\phi)\,%
\Bigg{|}\,\begin{aligned} &\displaystyle\textrm{$M$ an $F$-subgroup of $R$}\\
&\displaystyle\textrm{$\phi:(F/M\rightarrow H)\simeq(E,\pi)$ as $H$-exts}\end{%
aligned}\right\}}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
The second equality follows because $(F/M\rightarrow H)\simeq(E,\pi)$ and $R/M\simeq G_{i}$ imply that $\ker\pi\simeq G$.
The fourth equality follows because $(F/M\rightarrow H)\simeq(E,\pi)$ and $R/M$ and irreducible $F$-group implies that $\ker\pi$ is an irreducible $E$-group. The final equality follows because $\ker\pi\simeq G$ and $\ker\pi$ and irreducible $E$-group and $(F/M\rightarrow H)\simeq(E,\pi)$ implies that $R/M$ is isomorphic to some $G_{i}$ as an $F$-group.
Note that the data $(M,\phi)$ in the final equation above is exactly the same as the data of a surjection of $H$-extensions from $F\rightarrow H$ to $E\rightarrow H$.
Now let $(E,\pi)$ be an $H$-extension with $\ker\pi$ (via conjugation) an irreducible $E$-group. Consider a surjection of $H$-extensions from $(\hat{F}_{n})^{\bar{S}}\rightarrow H$ to $E\rightarrow H$, in which the map $(\hat{F}_{n})^{\bar{S}}\rightarrow E$ has kernel $K$. Again, we let $N$ denote the kernel of $(\hat{F}_{n})^{\bar{S}}\rightarrow H$. Then $N/K$ is an irreducible $(\hat{F}_{n})^{\bar{S}}$-group, and so $K$ is a maximal proper $(\hat{F}_{n})^{\bar{S}}$-subgroup of $N$. So the map $(\hat{F}_{n})^{\bar{S}}\rightarrow E$ factors through $F$. On the other hand, any surjection of $H$-extensions from $F\rightarrow H$ to $E\rightarrow H$ clearly extends to a surjection of $H$-extensions from $(\hat{F}_{n})^{\bar{S}}\rightarrow H$ to $E\rightarrow H$.
Thus, the above sum is equal to
$$\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\end{subarray}}\frac{|\operatorname{Sur}((%
\hat{F}_{n})^{\bar{S}}\rightarrow H,\pi)|}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
Then by applying Proposition 8.6 we obtain the result.
∎
We now can determine the multiplicities of the non-abelian irreducible $F$-groups in $R$ by combining Theorem 7.1 and Proposition 8.9.
Corollary 8.10 (Multiplicities of non-abelian $G$ in $R$).
Let $H$, $F$ and $R$ be as above.
Let $G$ be an non-abelian finite group.
Then
$$m(S,n,H,G)=\sum_{\begin{subarray}{c}\textrm{isom. classes of $H$-extensions $(%
E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\\
\textrm{$E$ level $S$}\end{subarray}}\frac{\sum_{\begin{subarray}{c}D\in%
\mathcal{E}_{H},D\leq E\end{subarray}}\nu(D,E)\left(\frac{|D|}{|H|}\right)^{n}%
}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
Finally, before we prove Theorem 8.1, we need the following lemma, whose proof is straightforward.
Lemma 8.11.
Suppose $x_{1},x_{2},\dots$ is a sequence of real numbers with limit $x$, and $y_{1},\dots$ is a sequence of real numbers with limit $\infty$. Let $a>1$ be a real number.
Then $f(x)=\prod_{i=1}^{\infty}(1-xa^{-i})$ is continuous in $x$ and
$$\lim_{n\rightarrow\infty}\prod_{i=1}^{y_{n}}(1-x_{n}a^{-i})=\prod_{i=1}^{%
\infty}(1-xa^{-i}).$$
Proof of Theorem 8.1.
Equation (4.2) and Theorem 5.1, combined with the multiplicities given by Corollaries 8.8 and 8.10, establish Equation (8.1) of Theorem 8.1.
By the definition of $\lambda(S,H,G)$ defined in Section 3.1 and Corollaries 8.8 and 8.10, we see that $\lambda(S,H,G)$ is related to $m(S,n,H,G)$ as follows:
(8.12)
$$\displaystyle\lim_{n\rightarrow\infty}\frac{h_{H}(G)^{m(S,n,H,G)}}{|G|^{n}}$$
$$\displaystyle=\lambda(S,H,G)$$
for $$G\in\mathcal{A}_{H}$$
$$\displaystyle\lim_{n\rightarrow\infty}\frac{m(S,n,H,G)}{|G|^{n}}$$
$$\displaystyle=\lambda(S,H,G)$$
$$\displaystyle\textrm{for $G\in\mathcal{N}$}.$$
Remark 8.13.
Note that since by Remark 5.2, for $G\in\mathcal{A}_{H}$, we have that $|G|$ is a power of $h_{H}(G)$, it follows that $\lambda(S,H,G)$ is an integral power of
$h_{H}(G)$, and that the limit above stabilizes for sufficiently large $n$.
We next establish the final statement of Theorem 8.1. Since
any irreducible $F$-group factor of $R$ is in $\bar{S}$, Corollary 6.12 shows that it is a power of a simple group in the closure of $S$ under taking subgroups and quotients.
Lemma 6.1 shows that the power is bounded by $|H|,$ showing the final statement of Theorem 8.1.
To establish Equation (8.1), it will suffice to take the limit of a factor in Equation (8.1)
corresponding to a single $G$ (since there are only finitely many $G$ with non-trivial factors, independent of $n$, by Lemma 6.10 and Corollary 6.13).
The factor in Equation (8.1) for a $G\in\mathcal{A}_{H}$ is
$$\prod_{k=0}^{m(S,n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{n+u}}})=\prod_{i=1}^{m(%
S,n,H,G)}(1-\frac{h_{H}(G)^{m(S,n,H,G)}h_{H}(G)^{-i}}{|G|^{{n+u}}}).$$
If there are no extensions $E$ in the sum in Corollary 8.8, then $m(S,n,H,G)$ and $\lambda(S,H,G)$ are $0$. Otherwise $\lambda(S,H,G)>0$, and thus it follows from Equation (8.12) that $m(S,n,H,G)\rightarrow\infty$ as $n\rightarrow\infty$.
So using Lemma 8.11 and Equation (8.12), we obtain the limit in Equation (8.1) for a single factor $G\in\mathcal{A}_{H}$. In a similar but simpler fashion, from Equation (8.12), we obtain the limit in Equation (8.1) for a single factor $G\in\mathcal{N}$. This completes the proof of Theorem 8.1.
∎
9. Countably additivity of $\mu_{u}$
The goal of this section is to prove Theorem 9.1 which states that $\mu_{u}$ defined in Equation (3.1) is countably additive on the algebra $\mathcal{A}$.
It then follows from Carathéodory’s extension theorem that $\mu_{u}$ can be uniquely extended to a measure on the Borel sets of $\mathcal{P}$.
The heart the proof of Theorem 9.1 is Theorem 9.2. We will first prove Theorem 9.2 in Section 9.1, and then prove Theorem 9.1 in Section 9.2.
Theorem 9.1.
Let $u$ be an integer. Then $\mu_{u}$ is countably additive on the algebra $\mathcal{A}$ generated by the $U_{S,H}$ for $S$ a finite set of finite groups and $H$ a finite group.
Theorem 9.2.
Let $\ell$ be a positive integer. Recall that $S_{\ell}$ is defined to be the set consisting of all groups of order less than or equal to $\ell$. For a non-negative integer $u$, we have
$$\sum_{H\text{ is finite and level }\ell}\mu_{u}(U_{S_{\ell},H})=1.$$
9.1. Proof of Theorem 9.2
Assume $\ell$ is a positive integer, $H$ is a finite level $\ell$ group and $\tilde{H}=H^{\bar{S}_{\ell-1}}$ .
In Lemmas 9.3 and 9.4, we will first give upper bounds for the number of irreducible factors $G$ with non-zero $m(S_{\ell},n,H,G)$ for some $n$ that are isomorphic to a given underlying group $M$.
Lemma 9.3.
Suppose $M$ is a direct product of isomorphic abelian simple groups. Then
$$\#\left\{G\in\mathcal{A}_{H}\,\Bigg{|}\,\begin{aligned} &\displaystyle G\simeq
M%
\text{ and }\\
&\displaystyle m(S_{\ell},n,H,G)\neq 0\text{ for some }n\end{aligned}\right\}%
\leq\sum_{(M,A)\in\mathcal{CF}(\bar{S}_{\ell})}|\operatorname{Sur}(\widetilde{%
H},A)|,$$
where the notation on the right-hand side above means that the sum is taken over all chief factor pairs in $\mathcal{CF}(\bar{S}_{\ell})$ whose first components are isomorphic to $M$ as groups.
Proof.
We give an injection
$$\left\{G\in\mathcal{A}_{H},\phi\,\Bigg{|}\,\begin{aligned} &\displaystyle\phi:%
G\simeq M\text{ and }\\
&\displaystyle m(S_{\ell},n,H,G)\neq 0\text{ for some }n\end{aligned}\right\}%
\rightarrow\{(M,A)\in\mathcal{CF}(\bar{S}_{\ell}),\pi\,|\,\pi\in\operatorname{%
Sur}(H,A)\}.$$
Consider $G\in\mathcal{A}_{H}$ and $\phi:G\simeq M$ such that $m(S_{\ell},n,H,G)\neq 0$ for some $n$. Assume
$$1\rightarrow R\rightarrow F\rightarrow H\rightarrow 1$$
is the fundamental short exact sequence associated to $S_{\ell},$ $n,$ and $H$. Then $G$ appears as a factor in $R$ and $(G,\rho_{G}(F))\in\mathcal{CF}(\bar{S}_{\ell})$. Using $\phi:G\simeq M$, we have that the quotient $\rho_{G}(F)$ of $H$ acts on $M$, and so $(M,\rho_{G}(F))\in\mathcal{CF}(\bar{S}_{\ell}).$
We let $\pi$ be the quotient map from $H$ to $\rho_{G}(F)$. Given $(M,A)\in\mathcal{CF}(\bar{S}_{\ell})$ and $\pi\in\operatorname{Sur}(H,A)$, we can use $\pi$ to give $M$ the structure of an irreducible $H$-group, and let $\phi$ be the identity. This recovers $G$ and $\phi$, though possibly without $m(S_{\ell},n,H,G)\neq 0$.
By Corollary 6.13, if $(M,A)\in\mathcal{CF}(\bar{S}_{\ell})$, then $A/\operatorname{Inn}(M)\simeq A$ is a group of level $\ell-1$. Then by the definition of pro-$\bar{S}$ completion, $\operatorname{Sur}(H,A)$ is one-to-one corresponding to $\operatorname{Sur}(\widetilde{H},A)$ and we finish the proof.
∎
Similarly, for non-abelian irreducible factors, we have the following lemma.
Lemma 9.4.
Suppose $M$ is a direct product of isomorphic non-abelian simple groups. Then
$$\displaystyle\#\left\{\text{isom. classes of $H$-extensions }(E,\pi)\,\Bigg{|}%
\,\begin{aligned} &\displaystyle\ker\pi\simeq M\text{ is irred. $E$-group}\\
&\displaystyle E\text{ is level }\ell\end{aligned}\right\}$$
$$\displaystyle\leq$$
$$\displaystyle\sum_{(M,A)\in\mathcal{CF}(\bar{S}_{\ell})}|\operatorname{Sur}(%
\widetilde{H},A/\operatorname{Inn}(M))|.$$
Proof.
We give an injection
$$\displaystyle\left\{\text{isom. classes of $H$-extensions }(E,\pi)\,\Bigg{|}\,%
\begin{aligned} &\displaystyle\ker\pi\simeq M\text{ is irred. $E$-group}\\
&\displaystyle E\text{ is level }\ell\end{aligned}\right\}$$
$$\displaystyle\to$$
$$\displaystyle\{(M,A)\in\mathcal{CF}(\bar{S}_{\ell}),\phi\mid\phi\in%
\operatorname{Sur}(H,A/\operatorname{Inn}(M))\}.$$
Consider an isomorphism class of $H$-extension $(E,\pi)$ such that $\ker\pi\simeq M$ is an irreducible $E$-group and $E$ is level $\ell$. Then $(\ker\pi,\rho_{\ker\pi}(E))\in\mathcal{CF}(\bar{S}_{\ell})$, and $\rho_{\ker\pi}$ induces a surjection $\phi:H\to\rho_{\ker\pi}(E)/\operatorname{Inn}(M)$ since $\rho_{\ker\pi}$ is an isomorphism when restricted on $\ker\pi$ that maps $\ker\pi$ to $\operatorname{Inn}(M)$. Suppose $(M,A)\in\mathcal{CF}(\bar{S}_{\ell})$ and $\phi\in\operatorname{Sur}(H,A/\operatorname{Inn}(M))$. If two $H$-extensions $(E_{1},\pi_{1})$ and $(E_{2},\pi_{2})$ both map to $(M,A)$ and $\phi$, then from the diagram below we see that $E_{1}$ and $E_{2}$ are both the fiber product of $\phi$ and $A\to A/\operatorname{Inn}(M)$, so $(E_{1},\pi_{1})$ and $(E_{2},\pi_{2})$ are isomorphic as $H$-extensions. Therefore, the map defined at the begin of the proof is an injection. Then the lemma follows as $\operatorname{Sur}(H,A)$ is one-to-one corresponding to $\operatorname{Sur}(\widetilde{H},A)$.
{tikzcd}
E_2
\arrow[bend left]drrπ_2
\arrow[swap, bend right]ddrρ_kerπ_2
\arrow[dashed]drι& &
& E_1 \arrowrπ_1
\arrowdρ_kerπ_1&
H \arrowdϕ
& A \arrowr &
A/Inn(M)
∎
Let $P_{u,n}(U_{S_{\ell},H})$ denote the product in Equation (8.1), i.e.
$$P_{u,n}(U_{S_{\ell},H})=\prod_{G\in\mathcal{A}_{H}}\prod_{k=0}^{m(S_{\ell},n,H%
,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{n+u}})\prod_{G\in\mathcal{N}}(1-|G|^{-n-u})^%
{m(S_{\ell},n,H,G)}.$$
Lemma 9.5.
Suppose $\ell>1$ ,$n\geq 1$ and $u>-n$ are integers and $\widetilde{H}$ is a finite level $\ell-1$ group. Then there exists a non-zero constant $c(u,\ell,\widetilde{H})$ depending on $u,\ell$ and $\widetilde{H}$ such that, for every finite level $\ell$ group $H$ with $H^{\bar{S}_{\ell-1}}=\widetilde{H}$,
either $P_{u,n}(U_{S_{\ell},H})\geq c(u,\ell,\widetilde{H})$ or $P_{u,n}(U_{S_{\ell},H})=0$ .
Proof.
For each $G\in\mathcal{A}_{H}$, $G$ is a direct product of isomorphic abelian simple groups, i.e. $G$ is a direct product of ${\mathbb{Z}}/p{\mathbb{Z}}$ for some prime $p$. By Remark 5.2, $h_{H}(G)$ is a power of $p$. Note that both of the trivial map (every element maps to 1) and the identity map of $G$ respect the $H$-action, therefore $h_{H}(G)>1$ and the product $\prod_{k=0}^{m(S_{\ell},n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{n+u}})$ is either 0 or greater than $\prod_{k=1}^{\infty}(1-p^{-k})\geq\prod_{k=1}^{\infty}(1-2^{-k})$. If $P_{u,n}(U_{S_{\ell},H})\neq 0$, then
$$\displaystyle\prod_{G\in\mathcal{A}_{H}}\prod_{k=0}^{m(S_{\ell},n,H,G)-1}(1-%
\frac{h_{H}(G)^{k}}{|G|^{n+u}})$$
$$\displaystyle=$$
$$\displaystyle\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\text{ and }\\
m(S_{\ell},n,H,G)\neq 0\end{subarray}}\prod_{k=0}^{m(S_{\ell},n,H,G)-1}(1-%
\frac{h_{H}(G)^{k}}{|G|^{n+u}})$$
$$\displaystyle\geq$$
$$\displaystyle\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\text{ and }\\
m(S_{\ell},n,H,G)\neq 0\text{ for some }n\end{subarray}}\prod_{k=1}^{\infty}(1%
-2^{-k})$$
$$\displaystyle=$$
$$\displaystyle\left[\prod_{k=1}^{\infty}(1-2^{-k})\right]^{\#\{G\in\mathcal{A}_%
{H}\mid\,m(S_{\ell},n,H,G)\neq 0\text{ for some }n\}}$$
$$\displaystyle\geq$$
$$\displaystyle\left[\prod_{k=1}^{\infty}(1-2^{-k})\right]^{\sum\limits_{\begin{%
subarray}{c}(M,A)\in\mathcal{CF}(\bar{S}_{\ell})\\
M\text{ is abelian}\end{subarray}}|\operatorname{Sur}(\widetilde{H},A)|},$$
where the last inequality follows from Lemma 9.3.
Therefore, if $P_{u,n}(U_{S_{\ell},H})$ is non-zero, then its abelian part has a lower bound depending only on $\ell$ and $\widetilde{H}$. Similarly, for the non-abelian part, we consider
$$\displaystyle\prod_{G\in\mathcal{N}}(1-|G|^{-n-u})^{m(S_{\ell},n,H,G)}$$
$$\displaystyle\geq$$
$$\displaystyle\prod_{G\in\mathcal{N}}\left[(1-\frac{1}{2})^{2}\right]^{\frac{m(%
S_{\ell},n,H,G)}{|G|^{n+u}}}$$
$$\displaystyle=$$
$$\displaystyle\left[(1-\frac{1}{2})^{2}\right]^{\sum\limits_{G\in\mathcal{N}}%
\frac{m(S_{\ell},n,H,G)}{|G|^{n+u}}}.$$
By Corollary 8.10, we have
$$\displaystyle\sum_{G\in\mathcal{N}}\frac{m(S_{\ell},n,H,G)}{|G|^{n+u}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{G\in\mathcal{N}}|G|^{-u}\left(\sum_{\begin{subarray}{c}%
\text{isom. classes of }H\text{-extensions }(E,\pi)\\
\ker\pi\text{ isom. $G$ }\\
\ker\pi\text{ irred. $E$-group}\\
E\text{ is level }\ell\end{subarray}}\frac{|G|^{-n}\sum_{D\in\mathcal{E}_{H},D%
\leq E}\nu(D,E)\frac{|D|}{|H|}^{n}}{|\operatorname{Aut}_{H}(E,\pi)|}\right)$$
$$\displaystyle\leq$$
$$\displaystyle\sum_{G\in\mathcal{N}}|G|^{-u}\#\left\{\text{isom. classes of $H$%
-extensions }(E,\pi)\,\Bigg{|}\,\begin{aligned} &\displaystyle\ker\pi\simeq G%
\text{ is irred. $E$-group}\\
&\displaystyle E\text{ is level }\ell\end{aligned}\right\}$$
$$\displaystyle\leq$$
$$\displaystyle\sum_{G\in\mathcal{N}}|G|^{-u}\left(\sum_{(G,A)\in\mathcal{CF}(%
\bar{S}_{\ell})}|\operatorname{Sur}(\widetilde{H},A/\operatorname{Inn}(G))|\right)$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}(G,A)\in\mathcal{CF}(\bar{S}_{\ell})\\
G\text{ non-abelian}\end{subarray}}|G|^{-u}|\operatorname{Sur}(\widetilde{H},A%
/\operatorname{Inn}(G))|.$$
The first inequality above follows from the fact that $|\operatorname{Sur}(\rho,\pi)|$ in Proposition 8.6 is less than or equal to $|G|^{n}$. The second inequality follows by Lemma 9.4.
It shows that the non-abelian part also has a lower bound depending on $u,\ell$ and $\widetilde{H}$. By Corollary 6.13, $\mathcal{CF}(\bar{S}_{\ell})$ is a finite set, so these lower bounds for abelian part and non-abelian parts are both non-zero. Then we proved the theorem.
∎
Now, we establish the inductive step that is crucial in the proof of Theorem 9.2.
Lemma 9.6.
Let $\ell>1$, $n\geq 1$, $u>-n$ be integers, and $\widetilde{H}$ be a finite level $\ell-1$ group. Then
$$\lim_{n\to\infty}\sum_{\begin{subarray}{c}H\text{ is finite level }\ell\\
\text{s.t. }\widetilde{H}=H^{\bar{S}_{\ell-1}}\end{subarray}}\mu_{u,n}(U_{S_{%
\ell},H})=\sum_{\begin{subarray}{c}H\text{ is finite level }\ell\\
\text{s.t. }\widetilde{H}=H^{\bar{S}_{\ell-1}}\end{subarray}}\mu_{u}(U_{S_{%
\ell},H}).$$
Proof.
Assume $H$ is finite and level $\ell$ such that $\widetilde{H}=H^{\bar{S}_{\ell-1}}$. Then either $\mu_{u,n}(U_{S_{\ell},H})=0$ or
$$\displaystyle\mu_{u,n}(U_{S_{\ell},H})$$
$$\displaystyle=$$
$$\displaystyle\frac{|\operatorname{Sur}(\hat{F}_{n},H)|}{|\operatorname{Aut}(H)%
||H|^{n+u}}P_{u,n}(U_{S_{\ell},H})$$
$$\displaystyle\geq$$
$$\displaystyle\frac{1}{2}c(u,\ell,\widetilde{H})\frac{1}{|\operatorname{Aut}(H)%
||H|^{u}}$$
for $n>i(H)$, where $i(H)$ is the smallest integer such that $|\operatorname{Sur}(\hat{F}_{n},H)|/|H|^{n}\geq\frac{1}{2}$ for all $n>i(H)$ (note that $i(H)$ is finite since $\lim_{n\to\infty}|\operatorname{Sur}(\hat{F}_{n},H)|/|H|^{n}=1$).
Let’s call $H$ achievable if it is finite level $\ell$ and there exists $n$ such that $\mu_{u,n}(U_{S_{\ell},H})\neq 0$ (we will give an equivalent definition in Section 13).
The function $\mu_{u,n}(U_{S_{\ell},H})$ of $H$ is dominated by the function of $H$ that is $\frac{1}{|\operatorname{Aut}(H)||H|^{u}}$ when $H$ is achievable and $0$ otherwise.
We have
$$\displaystyle\sum_{\begin{subarray}{c}H\text{ is achievable}\\
\text{ s.t. }\widetilde{H}\simeq H^{\bar{S}_{\ell-1}}\end{subarray}}\frac{1}{|%
\operatorname{Aut}(H)||H|^{u}}$$
$$\displaystyle=$$
$$\displaystyle\lim_{n\to\infty}\sum_{\begin{subarray}{c}H\text{ is achievable}%
\\
\text{ s.t. }\widetilde{H}\simeq H^{\bar{S}_{\ell-1}}\\
\text{and }i(H)<n\end{subarray}}\frac{1}{|\operatorname{Aut}(H)||H|^{u}}$$
$$\displaystyle\leq$$
$$\displaystyle\lim_{n\to\infty}\frac{2}{c(u,\ell,\widetilde{H})}\sum_{\begin{%
subarray}{c}H\text{ is achievable}\\
\text{ s.t. }\widetilde{H}\simeq H^{\bar{S}_{\ell-1}}\\
\text{and }i(H)<n\end{subarray}}\mu_{u,n}(U_{S_{\ell},H})$$
$$\displaystyle\leq$$
$$\displaystyle\frac{2}{c(u,\ell,\widetilde{H})}.$$
Thus by Lebesgue’s Dominated Convergence Theorem, we have
$$\displaystyle\lim_{n\to\infty}\sum_{\begin{subarray}{c}H\text{ is finite level%
}\ell\\
\text{ s.t. }\widetilde{H}\simeq H^{\bar{S}_{\ell-1}}\end{subarray}}\mu_{u,n}(%
U_{S_{\ell},H})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}H\text{ is finite level }\ell\\
\text{ s.t. }\widetilde{H}\simeq H^{\bar{S}_{\ell-1}}\end{subarray}}\lim_{n\to%
\infty}\mu_{u,n}(U_{S_{\ell},H}),$$
which completes the lemma.
∎
Proof of Theorem 9.2.
We proceed by induction on $\ell$. When $\ell=1$, note that the trivial group is the only group that is finite level 1 and it’s obvious that $\mu_{u}(U_{S_{1},1})=1$. Assume the theorem is true for $\ell-1$, i.e.
$$\sum\limits_{\widetilde{H}\text{ is finite level }\ell-1}\mu_{u}(U_{S_{\ell-1}%
,\widetilde{H}})=1.$$
We see that for any finite level $\ell-1$ group $\widetilde{H}$
$$\displaystyle\mu_{u}(U_{S_{\ell-1},\widetilde{H}})$$
$$\displaystyle=$$
$$\displaystyle\lim_{n\to\infty}\mu_{u,n}(U_{S_{\ell-1},\widetilde{H}})$$
$$\displaystyle=$$
$$\displaystyle\lim_{n\to\infty}\sum_{\begin{subarray}{c}H\text{ is finite level%
}\ell\\
\text{s.t. }\widetilde{H}=H^{\bar{S}_{\ell-1}}\end{subarray}}\mu_{u,n}(U_{S_{%
\ell},H})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\begin{subarray}{c}H\text{ is finite level }\ell\\
\text{s.t. }\widetilde{H}=H^{\bar{S}_{\ell-1}}\end{subarray}}\mu_{u}(U_{S_{%
\ell},H}),$$
where the second equality above follows from the definition of $\mu_{u,n}$ on basic open sets and the last step follows from Lemma 9.6.
Therefore, we finish the proof by
$$\displaystyle\sum_{H\text{ is finite level }\ell}\mu_{u}(U_{S_{\ell},H})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\widetilde{H}\text{ is finite level }\ell-1}\sum_{\begin{%
subarray}{c}H\text{ is finite level }\ell\\
\text{s.t. }\widetilde{H}=H^{\bar{S}_{\ell-1}}\end{subarray}}\mu_{u}(U_{S_{%
\ell},H})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\widetilde{H}\text{ is finite level }\ell-1}\mu_{u}(U_{S_{%
\ell-1},\widetilde{H}})$$
$$\displaystyle=$$
$$\displaystyle 1.$$
∎
9.2. Proof of Theorem 9.1
We will use the following corollary of Theorem 9.2.
Corollary 9.7.
Let $\ell$ be a positive integer, and $B=\cup_{j=1}^{\infty}U_{S_{\ell},H_{j}}$ for some finite groups $H_{j}$ such that
$U_{S_{\ell},H_{j}}\neq U_{S_{\ell},H_{j^{\prime}}}$ for $j\neq j^{\prime}$. Suppose that $B\in\mathcal{A}$, the algebra of sets generated by the basic open sets $U_{S,G}$ for a finite set $S$ of finite groups and a finite level $S$ group $G$.
Let $u$ be an integer.
Then $\mu_{u}(B)=\sum_{j=1}^{\infty}\mu_{u}(U_{S_{\ell},H_{j}})$.
Proof.
Let $G_{j}$ be the level $\ell$ finite groups not among the $H_{j}$. Then for every positive integer $M$, we have
$$\sum_{j=1}^{M}\mu_{u}(U_{S_{\ell},H_{j}})\leq\mu_{u}(B)\leq 1-\sum_{j=1}^{M}%
\mu_{u}(U_{S_{\ell},G_{j}}).$$
Taking limits as $M\rightarrow\infty$ gives
$$\sum_{j=1}^{\infty}\mu_{u}(U_{S_{\ell},H_{j}})\leq\mu_{u}(B)\leq 1-\sum_{j=1}^%
{\infty}\mu_{u}(U_{S_{\ell},G_{j}})=\sum_{j=1}^{\infty}\mu_{u}(U_{S_{\ell},H_{%
j}}),$$
where the last equality is by Theorem 9.2.
∎
Proof of Theorem 9.1.
Since $\mu_{u}(A)$ is defined as a limit of measures $\mu_{u,n}(A)$, it is immediate that $\mu_{u}$ is finitely additive because finite sums can be exchanged with the limit. If we have disjoint sets $A_{n}\in\mathcal{A}$ with $A=\cup_{n\geq 1}A_{n}\in\mathcal{A}$, by taking $B_{n}=A\setminus\cup_{j=1}^{n}A_{j}$, it suffices to show that for $B_{1}\supset B_{2}\supset\dots$ (with $B_{n}\in\mathcal{A}$) with $\cap_{n\geq 1}B_{n}=\emptyset$ we have $\lim_{n\rightarrow\infty}\mu_{u}(B_{n})=0$.
We can assume, without loss of generality, that
for each $\ell\geq 1$, we have $B_{\ell}=\cup_{j}U_{S_{\ell},G_{\ell,j}}$ (i.e. $B_{\ell}$ is defined at level $\ell$).
(Note that when all groups in $S$ have order at most $m$ that $U_{S,H}$ is a union of sets of the form $U_{S_{m},G}$ for varying $G$. We can always insert redundant $B_{i}$’s if the level required to define the $B_{\ell}$ increase quickly.)
We will show by contradiction that $\lim_{\ell\rightarrow\infty}\mu_{u}(B_{\ell})=0$.
Suppose, instead that there is an $\epsilon>0$ such that for all $\ell$, we have $\mu_{u}(B_{\ell})\geq\epsilon$.
It follows from Corollary 9.7 that for each $\ell$ we have a subset $K_{\ell}\subset B_{\ell}$ such that $\mu_{u}(B_{\ell}\setminus K_{\ell})<\epsilon/2^{\ell+1}$ and $K_{\ell}$ is a finite union of $U_{S_{\ell},G_{\ell,j}}$.
Next, let $C_{\ell}=\cap_{j=1}^{\ell}K_{j}$. Then
$\mu_{u}(B_{\ell}\setminus C_{\ell})<\epsilon/2$, since
$$\displaystyle\mu_{u}(B_{\ell}\setminus C_{\ell})=$$
$$\displaystyle\mu_{u}(B_{\ell}\setminus K_{\ell})+\mu_{u}(K_{\ell}\setminus K_{%
\ell}\cap K_{\ell-1})+\cdots+\mu_{u}(K_{\ell}\cap\cdots\cap K_{2}\setminus K_{%
\ell}\cap\cdots\cap K_{1})$$
$$\displaystyle<$$
$$\displaystyle\epsilon/2^{\ell+1}+\mu_{u}(B_{\ell-1}\setminus K_{\ell-1})+%
\cdots+\mu_{u}(B_{1}\setminus K_{1})$$
$$\displaystyle<$$
$$\displaystyle\epsilon/2^{\ell+1}+\epsilon/2^{\ell}+\cdots+\epsilon/2^{2}.$$
So $\mu_{u}(C_{\ell})\geq\epsilon/2$ for each $\ell$ and in particular it is non-empty. Note $C_{\ell+1}\subset C_{\ell}$ for all $\ell$. Pick $x_{\ell}\in C_{\ell}$ for all $\ell$.
Note $C_{\ell}$ is defined at level $\ell$ and a finite union of the basic open sets $U_{S_{\ell},G_{\ell,j}}$.
Pick an $H_{1}$ so that infinitely many of the $x_{\ell}$ are in $U_{S_{1},H_{1}}$ (this is possible since all $x_{\ell}$ are in $C_{1}$ and there are only finitely many $U_{S_{1},H}$ that make up $C_{1}$), and then disregard the $x_{\ell}$ that are not in $U_{S_{1},H_{1}}$. In particular note $U_{S_{1},H_{1}}\subset C_{1}$. Then pick $H_{2}$ so that infinitely many of the remaining $x_{\ell}$ are in $U_{S_{2},H_{2}}$, and disregard the $x_{\ell}$ that are not. Since all of the remaining $x_{\ell}$ are in $U_{S_{1},H_{1}}$, we have $U_{S_{2},H_{2}}\subset U_{S_{1},H_{1}}$. Also note $U_{S_{2},H_{2}}\subset C_{2}$. We continue this process and then consider the profinite group $H$ that is the inverse limit of the $H_{i}$’s. Since $H\in U_{S_{\ell},H_{\ell}}\subset C_{\ell}\subset B_{\ell}$ for all $\ell$, we have a point $H\in\cap_{\ell\geq 1}B_{\ell}$ which is a contradiction.
∎
10. Proof of Theorem 1.1
The last section established the existence of the probability measure $\mu_{u}$ on Borel sets of $\mathcal{P}$.
Now we are able to give the proof of Theorem 1.1, the weak convergence of the $\mu_{u,n}$ to $\mu_{u}$.
Proof of Theorem 1.1.
Note that the weak convergence $\mu_{u,n}\Rightarrow\mu_{u}$ is equivalent to that
$$\liminf_{n\to\infty}\mu_{u,n}(U)\geq\mu_{u}(U)$$
for all open sets $U$. In the topological space $\mathcal{P}$, every open set is a countable disjoint union of basic open sets. Assume $U=\cup_{i\geq 1}U_{i}$ is an open set, where $U_{i}$ are disjoint basic open sets. By Fatou’s lemma, we have
$$\mu_{u}(U)=\sum_{i\geq 1}\mu_{u}(U_{i})=\sum_{i\geq 1}\lim_{n\to\infty}\mu_{u,%
n}(U_{i})\leq\liminf_{n\to\infty}\sum_{i\geq 1}\mu_{u,n}(U_{i})=\liminf_{n\to%
\infty}\mu_{u,n}(U).$$
∎
11. For arbitrary set $S$
In this section, we let $S$ be an arbitrary (not necessarily finite) set of finite groups and consider the value of $\mu_{u}$ on the specific type of Borel sets
$$V_{S,H}:=\{X\in\mathcal{P}\mid X^{\bar{S}}\simeq H\}$$
for a finite level $S$ group $H$. We will first prove an analogue of Theorem 8.1 for an arbitrary set $S$ (see Theorem 11.4), the proof of which shows that Equation (8.1) gives the value $\mu_{u}(V_{S,H})$.
Note that $V_{S,H}$ is not a basic open set, but is the intersection of a sequence of basic open sets. Since we will approximate $S$ by increasing finite subsets, we need the following lemma.
Lemma 11.1.
Consider two sets $T\subset T^{\prime}$ of finite groups. For any positive integer $n$, finite group $H$ of level $T$, and $G\in\mathcal{A}_{H}\cup\mathcal{N}$, we have
$m(T,n,H,G)\leq m(T^{\prime},n,H,G).$ Also if $T_{1}\subset T_{2}\subset\cdots$ are finite sets of finite groups, then $m(T_{m},n,H,G)$ eventually stabilizes as $m\rightarrow\infty$.
Proof.
Consider the case when $G$ is abelian.
Let $\rho:\hat{F}^{\bar{T}_{m}}\rightarrow H$ be a surjection.
Corollary 8.8 and Proposition 8.6 give
$$\frac{h_{H}(G)^{m(T_{m},n,H,G)}-1}{h_{H}(G)-1}=\sum_{\begin{subarray}{c}%
\textrm{isom. classes of $H$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ as an $H$-group}\\
\textrm{$E$ is level $T_{m}$}\end{subarray}}\frac{|\operatorname{Sur}(\rho,\pi%
)|}{|\operatorname{Aut}_{H}(E,\pi)|}.$$
The right-hand side is clearly non-decreasing in $m$. There are only finitely many isomorphism classes of $H$-extensions whose kernel is isomorphic to $G$, which proves the stabilization. The case of non-abelian $G$ is similar.
∎
Definition.
Let $S$ be a set of finite groups, $n$ a positive integer, and $H$ a finite level $S$ group. Let $T_{1}\subset T_{2}\subset\cdots$ be finite sets of finite groups such that $\cup_{m\geq 1}T_{m}=S$. For any $G\in\mathcal{A}_{H}\cup\mathcal{N}$, we define
$m(S,n,H,G)=\lim_{m\to\infty}m(T_{m},n,H,G).$
Remark 11.2.
It’s clear that $m(S,n,H,G)$ does not depend on the choice of the increasing sequence $T_{i}$, and $m(S,n,H,G)$ is always a non-negative integer.
It is actually easier to determine $\mu_{u}(V_{S,H})$, as we will in the next lemma, than to
find $\lim_{n\rightarrow\infty}\mu_{u,n}(V_{S,H}),$ which we will do in Theorem 11.4.
Lemma 11.3.
Let $S$ be a set of finite groups.
Let $T_{1}\subset T_{2}\subset\cdots$ be finite sets of finite groups such that $\cup_{m\geq 1}T_{m}=S$.
Let $H$ be a finite group of level $S$. Let $u$ be an integer.
Then
$$\displaystyle\mu_{u}(V_{S,H})$$
$$\displaystyle=$$
$$\displaystyle\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}\operatorname{%
Prob}((X_{u,n})^{\bar{T}_{m}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{\infty}(1-\lambda(S,H,G)\frac{%
h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{%
subarray}}e^{-|G|^{-u}\lambda(S,H,G)}.$$
Proof.
First of all, since $\mu_{u}$ is a measure, we have
$$\mu_{u}(V_{S,H})=\mu_{u}(\cap_{m\geq 1}U_{T_{m},H^{\bar{T}_{m}}})=\lim_{m\to%
\infty}\mu_{u}(U_{T_{m},H^{\bar{T}_{m}}})=\lim_{m\to\infty}\lim_{n\to\infty}%
\operatorname{Prob}((X_{u,n})^{\bar{T}_{m}}\simeq H).$$
By definition, we have that $\lambda(T,H,G)$ is non-decreasing in $T$, i.e. if $T\subset T^{\prime}$ then
$\lambda(T,H,G)\leq\lambda(T^{\prime},H,G)$. Further, again by definition, we have
$$\lambda(S,H,G)=\lim_{m\rightarrow\infty}\lambda(T_{m},H,G).$$
When $m$ is sufficiently large such that $H$ is level $T_{m}$, we have
$$\displaystyle\lim_{n\rightarrow\infty}\operatorname{Prob}((X_{u,n})^{\bar{T}_{%
m}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{\infty}(1-\lambda(T_{m},H,G)%
\frac{h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{%
subarray}}e^{-|G|^{-u}\lambda(T_{m},H,G)}$$
by Equation (8.1) since $T_{m}$ is finite. For each $G\in\mathcal{A}_{H}$, the factor
$$\prod_{i=1}^{\infty}(1-\lambda(T_{m},H,G)\frac{h_{H}(G)^{-i}}{|G|^{{u}}})$$
is a limit of terms
$$\prod_{i=1}^{m(T_{m},n,H,G)}(1-\frac{h_{H}(G)^{m(T_{m},n,H,G)}h_{H}(G)^{-i}}{|%
G|^{{n+u}}}),$$
by Lemma 8.11,
each of which is a probability and in the interval $[0,1]$.
Since the factors
$$\prod_{i=1}^{\infty}(1-\lambda(T_{m},H,G)\frac{h_{H}(G)^{-i}}{|G|^{{u}}})\quad%
\quad\textrm{and}\quad\quad e^{-|G|^{-u}\lambda(T_{m},H,G)}$$
are all in $[0,1]$ and are non-increasing in $m$, we have the second equality in the following
$$\displaystyle\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}\operatorname{%
Prob}((X_{u,n})^{\bar{T}_{m}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{|\operatorname{Aut}(H)||H|^{u}}%
\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{%
\infty}(1-\lambda(T_{m},H,G)\frac{h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{%
subarray}{c}G\in\mathcal{N}\end{subarray}}e^{-|G|^{-u}\lambda(T_{m},H,G)}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\lim_{m\rightarrow\infty}\prod_{i=1}^{%
\infty}(1-\lambda(T_{m},H,G)\frac{h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{%
subarray}{c}G\in\mathcal{N}\end{subarray}}\lim_{m\rightarrow\infty}e^{-|G|^{-u%
}\lambda(T_{m},H,G)}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{\infty}(1-\lambda(S,H,G)\frac{%
h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{%
subarray}}e^{-|G|^{-u}\lambda(S,H,G)}.$$
The last equality uses the continuity from Lemma 8.11.
∎
Theorem 11.4.
The statement in Theorem 8.1 also works for an arbitrary set $S$ of finite groups.
Proof.
Let $T_{m}$ be the subset of $S$ of all groups of order at most $m$ in $S$.
Since $H$ is level $S$, for large enough $m$ we have that $H^{\bar{T}_{m}}=H$, and from now on we only consider $m$ this large.
We can show that $G^{\bar{S}}\simeq H$ if and only if for every $m\geq 1$ we have $G^{\bar{T}_{m}}\simeq H^{\bar{T}_{m}}.$
Since $G^{\bar{T}_{m}}$ is a quotient of $G^{\bar{S}}$, the “only if” direction is clear. If we take the inverse limit of the sets $\operatorname{Isom}(G^{\bar{T}_{m}},H^{\bar{T}_{m}})$, with the natural maps, we have an inverse limit of non-empty finite sets, which is non-empty.
An element of this inverse limit gives us an isomorphism $G^{\bar{S}}\simeq H^{\bar{S}}$.
From this, and the basic properties of a measure, and Equation (8.1) for finite $S$, we have that
$$\displaystyle\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\lim_{m\rightarrow\infty}\frac{|\operatorname{Sur}(\hat{F}_{n},H)%
|}{|\operatorname{Aut}(H)||H|^{n+u}}\prod_{\begin{subarray}{c}G\in\mathcal{A}_%
{H}\end{subarray}}\prod_{k=0}^{m(T_{m},n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{n%
+u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{subarray}}(1-|G|^{-{n-u}})%
^{m(T_{m},n,H,G)}.$$
From Lemma 11.1, we have that
$$\prod_{k=0}^{m(T_{m},n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{n+u}}})\quad\quad%
\textrm{and}\quad\quad(1-|G|^{-{n-u}})^{m(T_{m},n,H,G)}$$
are non-increasing in $m$, and as they are probabilities they are in the interval $[0,1]$. Thus it follow from basic analysis that
$$\displaystyle\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{|\operatorname{Sur}(\hat{F}_{n},H)|}{|\operatorname{Aut}(H)%
||H|^{n+u}}\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\end{subarray}}\lim_{m%
\rightarrow\infty}\prod_{k=0}^{m(T_{m},n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{n%
+u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{subarray}}\lim_{m%
\rightarrow\infty}(1-|G|^{-{n-u}})^{m(T_{m},n,H,G)}$$
By definition of $m(S,n,H,G)$, we have that $\lim_{m\rightarrow\infty}m(T_{m},n,H,G)=m(S,n,H,G)$ (and the latter is finite).
Thus, we have
$$\displaystyle\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{|\operatorname{Sur}(\hat{F}_{n},H)|}{|\operatorname{Aut}(H)%
||H|^{n+u}}\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\end{subarray}}\prod_{%
k=0}^{m(S,n,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{n+u}}})\prod_{\begin{subarray}%
{c}G\in\mathcal{N}\end{subarray}}(1-|G|^{-{n-u}})^{m(S,n,H,G)},$$
which is Equation (8.1) for arbitrary $S$.
Next, towards Equation (8.1) for arbitrary $S$, we will show that the order of the limits in Lemma 11.3 could be exchanged.
For every $m$, we have $\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)\leq\operatorname{Prob}((X_{u,%
n})^{\bar{T}_{m}}\simeq H)$ and so
(11.5)
$$\displaystyle\limsup_{n\rightarrow\infty}\operatorname{Prob}((X_{u,n})^{\bar{S%
}}\simeq H)$$
$$\displaystyle\leq$$
$$\displaystyle\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}\operatorname{%
Prob}((X_{u,n})^{\bar{T}_{m}}\simeq H)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{\infty}(1-\lambda(S,H,G)\frac{%
h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{%
subarray}}e^{-|G|^{-u}\lambda(S,H,G)}.$$
From here we consider two cases.
Case 1 will be the following:
$$\sum_{G\in\mathcal{A}_{H}}\frac{\lambda(S,H,G)}{h_{H}(G)|G|^{u}}+\sum_{G\in%
\mathcal{N}}\frac{\lambda(S,H,G)}{|G|^{u}}\quad\textrm{diverges}.$$
In case $1$, the product in Equation (11.5) is $0$, and we have proven $\lim_{n\rightarrow\infty}\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)=0$, establishing Equation (8.1).
Case 2 will be the following:
$$\sum_{G\in\mathcal{A}_{H}}\frac{\lambda(S,H,G)}{h_{H}(G)|G|^{u}}+\sum_{G\in%
\mathcal{N}}\frac{\lambda(S,H,G)}{|G|^{u}}\quad\textrm{converges}.$$
We define a minimal non-trivial $H$-extension $(E,\pi)$ to be an $H$-extension whose only quotient $H$-extensions are itself and the trivial one. These are exactly the $H$-extensions with $\ker\pi$ an irreducible $E$-group (under conjugation). Also, these are exactly the $H$-extensions $(E,\pi)$ such that $\ker\pi$ is an abelian irreducible $H$-group or $\ker\pi$ is a power of a non-abelian simple group and an irreducible $E$-group.
Since $|\operatorname{Aut}(E)|\geq|\operatorname{Aut}_{H}(E,\pi)|$ and $h_{H}(G)\geq 2$, we have
$$\sum_{G\in\mathcal{A}_{H}}\frac{\lambda(S,H,G)}{h_{H}(G)|G|^{u}}+\sum_{G\in%
\mathcal{N}}\frac{\lambda(S,H,G)}{|G|^{u}}\geq\frac{1}{2}\sum_{\begin{subarray%
}{c}(E,\pi)\textrm{ min. non-triv. $H$-extension }\\
\textrm{$E$ level $S$}\end{subarray}}|\operatorname{Aut}(E)|^{-1}|G|^{-u}.$$
Since we are in case 2, the sum on the right converges, and
$$\displaystyle\lim_{m\rightarrow\infty}\sum_{\begin{subarray}{c}(E,\pi)\textrm{%
min. non-triv. $H$-extension }\\
\textrm{$E$ level $S$, but not level $T_{m}$}\end{subarray}}|\operatorname{Aut%
}(E)|^{-1}|E|^{-u}$$
$$\displaystyle=$$
$$\displaystyle|H|^{-u}\lim_{m\rightarrow\infty}\sum_{\begin{subarray}{c}(E,\pi)%
\textrm{ min. non-triv. $H$-extension }\\
\textrm{$E$ level $S$, but not level $T_{m}$}\end{subarray}}|\operatorname{Aut%
}(E)|^{-1}|G|^{-u}$$
$$\displaystyle=$$
$$\displaystyle 0.$$
If $(X_{u,n})^{\bar{S}}\not\simeq H,$ but $(X_{u,n})^{\bar{T}_{m}}\simeq H$ for some $m$, then $X_{u,n}$ has a surjection to $H$ and thus $X_{u,n}$ has a surjection to some minimal non-trivial $H$-extension $(E,\pi)$ of level $S$ but not level $T_{m}$.
Note that
$$\displaystyle{\mathbb{P}}(X_{u,n}\textrm{ has a surjection to $E$})$$
$$\displaystyle\leq{\mathbb{E}}(\textrm{quotients of $X_{u,n}$ isom. to $E$})$$
$$\displaystyle=|\operatorname{Aut}(E)|^{-1}{\mathbb{E}}(|\operatorname{Sur}(X_{%
u,n},E)|)$$
$$\displaystyle=|\operatorname{Aut}(E)|^{-1}\frac{|\operatorname{Sur}(\hat{F}_{n%
},E)|}{|E|^{n+u}}$$
$$\displaystyle\leq|\operatorname{Aut}(E)|^{-1}|E|^{-u}.$$
Thus,
$$\displaystyle\operatorname{Prob}((X_{u,n})^{\bar{S}}\simeq H)\geq\operatorname%
{Prob}((X_{u,n})^{\bar{T}_{m}}\simeq H)-\sum_{\begin{subarray}{c}(E,\pi)%
\textrm{ min. non-triv. $H$-extension }\\
\textrm{$E$ level $S$, but not level $T_{m}$}\end{subarray}}|\operatorname{Aut%
}(E)|^{-1}|E|^{-u}$$
and
$$\displaystyle\liminf_{n\rightarrow\infty}\operatorname{Prob}((X_{u,n})^{\bar{S%
}}\simeq H)$$
$$\displaystyle\geq$$
$$\displaystyle\lim_{n\rightarrow\infty}\operatorname{Prob}((X_{u,n})^{\bar{T}_{%
m}}\simeq H)-\sum_{\begin{subarray}{c}(E,\pi)\textrm{ min. non-triv. $H$-%
extension }\\
\textrm{$E$ level $S$, but not level $T_{m}$}\end{subarray}}|\operatorname{Aut%
}(E)|^{-1}|E|^{-u}$$
Now we take a $\lim_{m\rightarrow\infty}$ of both sides and conclude Equation (8.1) for arbitrary $S$.
Finally, note that if $m(S,n,H,G)\neq 0$, then $m(T_{m},n,H,G)\neq 0$ for some $m$, and so the last statement of Theorem 8.1 for infinite $S$ follows from the same statement for finite $S$.
∎
Though this doesn’t follow from weak convergence (see Proposition 14.2 and Remark 14.3, for example), we see here that $\mu_{u}$ and $\lim_{n\rightarrow\infty}\mu_{u,n}$ agree on the $V_{S,H}$.
Corollary 11.6.
Let $S$ be a set of finite groups and $H$ a finite level $S$ group. Then we have
$$\lim_{n\to\infty}\mu_{u,n}(V_{S,H})=\mu_{u}(V_{S,H}).$$
Proof.
In the proof of Theorem 11.4, we showed that
$$\lim_{n\to\infty}\lim_{m\to\infty}\operatorname{Prob}((X_{u,n})^{\bar{T}_{m}}%
\simeq H)=\lim_{m\to\infty}\lim_{n\to\infty}\operatorname{Prob}((X_{u,n})^{%
\bar{T}_{m}}\simeq H).$$
By Lemma 11.3, the right-hand side in the above equation is $\mu_{u}(V_{S,H})$. Also, since $\mu_{u,n}$ are measures on $\mathcal{P}$, we have
$\lim_{m\to\infty}\operatorname{Prob}((X_{u,n})^{\bar{T}_{m}}\simeq H)=\mu_{u,n%
}(V_{S,H})$.
∎
12. Examples of the values of $\mu_{u}$
In this section, we will apply Theorem 11.4 to compute $\mu_{u}(A)$ for some interesting Borel sets $A$.
Example 12.1 (Trivial group).
Let $S$ contain every finite group. Then the trivial group is the only element in $V_{S,1}$. By Lemma 6.1, if $(E,\pi)$ is an extension of the trivial group such that $\ker\pi$ is irreducible $E$-group, then $E$ is a finite simple group. Then it follows from the definition of $\lambda(S,H,G)$ that
$$\lambda(S,1,G)=\begin{cases}1&G\text{ is an abelian simple group}\\
|\operatorname{Aut}(G)|^{-1}&G\text{ is a non-abelian simple group}\\
0&otherwise\end{cases}$$
By Theorem 11.4, we have
$$\mu_{u}(\text{trivial group})=\prod_{p\text{ prime}}\prod_{i=u+1}^{\infty}(1-p%
^{-i})\prod_{\begin{subarray}{c}G\text{ finite simple}\\
\text{non-abelian group}\end{subarray}}e^{-|G|^{-u}|\operatorname{Aut}(G)|^{-1%
}}.$$
The above product over prime integers is zero if and only if $u\leq 0$. When $u\geq 1$, by the classification of finite simple groups, the number of finite simple groups of given order is at most 2. Note that $|\operatorname{Aut}(G)|\geq|\operatorname{Inn}(G)|=|G|$ for every non-abelian simple group $G$. We have
$$\prod_{\begin{subarray}{c}G\text{ finite simple}\\
\text{non-abelian group}\end{subarray}}e^{-|G|^{-u}|\operatorname{Aut}(G)|^{-1%
}}\geq\exp(-\sum_{\begin{subarray}{c}G\text{ finite simple}\\
\text{non-abelian group}\end{subarray}}|G|^{-u-1})>0,$$
which shows that $\mu_{u}(\text{trivial group})>0$ if and only if $u\geq 1$. By using the classification of finite simple groups, we are able to give the following approximations
$$\mu_{u}(\text{trivial group})\approx\begin{cases}0.4357&\text{when }u=1\\
0.7168&\text{when }u=2\\
0.8616&\text{when }u=3.\end{cases}$$
We observe that the product over non-abelian factors is very close to 1 and cannot be seen in this many digits.
Example 12.2 (Any infinite group).
Again let $S$ contain all finite groups. Let $H$ be an infinite profinite group in $\mathcal{P}$, and $H_{\ell}$ denote the pro-$\bar{S}_{\ell}$ completion of $H$. Since $U_{S_{\ell},H_{\ell}}$ is a sequence of basic opens that is decreasing in $\ell$ and $\cap_{\ell}U_{S_{\ell},H_{\ell}}=\{H\}$, we obtain
$$\displaystyle\mu_{u}(\{H\})$$
$$\displaystyle=$$
$$\displaystyle\lim_{\ell\to\infty}\mu_{u}(U_{S_{\ell},H_{\ell}})$$
$$\displaystyle\leq$$
$$\displaystyle\lim_{\ell\to\infty}\frac{1}{|\operatorname{Aut}(H_{\ell})||H_{%
\ell}|^{u}}.$$
Note that $H$ is the inverse limit of $H_{\ell}$, so $\lim_{\ell\to\infty}|H_{\ell}|=\infty$. It follows that $\mu_{u}(\{H\})=0$ when $u\geq 1$. When $u=0$, since $\{H\}$ is contained in the Borel set $A:=V_{\{\text{all abelian groups}\},H^{ab}}$ and $\mu_{0}(A)=0$ (see Example 12.4), we have $\mu_{0}(\{H\})=0$.
Example 12.3 (Pro-$p$ abelianization).
Let $p$ be a prime integer and $S$ the set consisting of all finite abelian $p$-groups. Then $\bar{S}=S$ and $({\mathbb{Z}}/p{\mathbb{Z}},1)$ is the only element in $\mathcal{CF}(\bar{S})$. Let $H$ be a finite abelian $p$-group of generator rank $d$. Then for any $G\in\mathcal{A}_{H}\cup\mathcal{N}$, the factor in Theorem 8.1 associated to $G$ is $1$, unless $G={\mathbb{Z}}/p{\mathbb{Z}}$ with the trivial $H$-action.
We consider the Borel set $V_{S,H}$ that is the set of all profinite groups whose maximal abelian pro-$p$ quotient is $H$. For any integer $n\geq d$ , there is a normal subgroup $N$ of $(\hat{F}_{n})^{\bar{S}}=({\mathbb{Z}}_{p})^{n}$ such that the corresponding quotient is $H$. Since $H$ is finite, $N$ is isomorphic to $({\mathbb{Z}}_{p})^{n}$ with the trivial $(\hat{F}_{n})^{\bar{S}}$-action, which shows that $m(S,n,H,{\mathbb{Z}}/p{\mathbb{Z}})=n$ and $\lambda(S,H,{\mathbb{Z}}/p{\mathbb{Z}})=1$. By Lemma 11.3, we have
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{i=1}^{\infty}(%
1-p^{-i-u}).$$
When $u<0$, this probability is 0, which is as expected since we can never get a finite quotient of $({\mathbb{Z}}_{p})^{n}$ with fewer than $n$ relators. When $u\geq 0$, we get a finite group with probability 1. When $u=0$ or $1$, these are the measures used in the Cohen-Lenstra heuristics for class groups of quadratic number fields.
More generally, let’s consider an infinite abelian pro-$p$ group $H$ in $\mathcal{P}$. Since $H\in\mathcal{P}$, the pro-$\overline{\{{\mathbb{Z}}/2{\mathbb{Z}}\}}$ completion of $H$ is finite, so $H$ is finitely generated, i.e. $H=H_{1}\times({\mathbb{Z}}_{p})^{r}$ for a finite abelian $p$-group $H_{1}$ and a positive integer $r$. Let $T_{j}:=\{{\mathbb{Z}}/p^{j}{\mathbb{Z}}\}$. So $\bar{T}_{j}$ is an increasing sequence and $\cup\bar{T}_{j}=S$. Assume $n\geq d$ and $j$ is greater than the exponent of $H_{1}$. Then we have
$$(\hat{F}_{n})^{\bar{T}_{j}}=({\mathbb{Z}}/p^{j}{\mathbb{Z}})^{n}\text{ and }H^%
{\bar{T}_{j}}=H_{1}\times({\mathbb{Z}}/p^{j}{\mathbb{Z}})^{r}.$$
So $m(T_{j},n,H,{\mathbb{Z}}/p{\mathbb{Z}})=n-r$, $\lambda(T_{j},H,{\mathbb{Z}}/p{\mathbb{Z}})=p^{-r}$ and
$$\displaystyle\mu_{u}(V_{T_{j},H})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H^{\bar{T}_{j}})||H_{1}|^{u}p^{jru}}%
\prod_{i=1+u+r}^{\infty}(1-p^{-i})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{|\operatorname{Aut}(H_{1})||H_{1}|^{2r+u}p^{jr(r+u)}}%
\prod_{i=1}^{r}(1-p^{-i})^{-1}\prod_{i=1+u+r}^{\infty}(1-p^{-i}),$$
since
$$|\operatorname{Aut}(H^{\bar{T}_{j}})|=|\operatorname{Aut}(H_{1})||H_{1}|^{2r}p%
^{jr^{2}}\prod_{i=1}^{r}(1-p^{-i}).$$
It follows that $\mu_{u}(V_{S,H})=\lim_{j\to\infty}(V_{T_{j},H})>0$ if and only if $u+r=0$, in which case
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H_{1})||H_{1}|^{-u}}\prod_{i=1-u%
}^{\infty}(1-p^{-i}).$$
So we see that when $u<0$, $\mu_{u}(V_{S,H})>0$ if and only if the (torsion-free) rank of $H$ is $-u$ and we get the groups in such form with probability 1.
Example 12.4 (Abelianization).
Similar to the example above, when $S$ is the set of all finite abelian groups and $H$ is a finite abelian group, we have
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{p\textrm{ %
prime}}\prod_{i=1}^{\infty}(1-p^{-i-u}),$$
which is $0$ if $u\leq 0$ and is positive if $u\geq 1$. If $H=H_{1}\times(\hat{{\mathbb{Z}}})^{r}$, then
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H_{1})||H_{1}|^{-u}}\prod_{p%
\textrm{ prime}}\prod_{i=1-u}^{\infty}(1-p^{-i})>0$$
if $u=-r<0$ and $\mu_{u}(V_{S,H})=0$ otherwise.
In order to consider the pro-$p$ quotients of our random groups, we will first need to recall the definitions of some $p$-group invariants.
Let $H$ be a finite $p$-group of generator rank $d$. The relation rank of $H$ is defined to be $r(H):=\dim_{{\mathbb{F}}_{p}}\operatorname{H}^{2}(H,{\mathbb{Z}}/p{\mathbb{Z}})$. On the other hand, there is a normal subgroup $N$ of $\hat{F}_{d}$ such that $\hat{F}_{d}/N\simeq H$. Define $N^{*}:=[N,\hat{F}_{d}]\cdot N^{p}$. Then $N^{*}$ is the minimal $\hat{F}_{d}$-normal subgroup of $N$ such that $N/N^{*}$ is a finite elementary abelian $p$-group with trivial $\hat{F}_{d}/N^{*}$-action. Then $\hat{F}_{d}/N^{*}$ is called the $p$-covering group of $H$, and $N/N^{*}$ is called the $p$-multiplicator of $H$, and $\dim_{{\mathbb{F}}_{p}}(N/N^{*})$ is called the $p$-multiplicator rank of $H$. One can show that $\operatorname{H}^{2}(H,{\mathbb{Z}}/p{\mathbb{Z}})\simeq N/N^{*}$, so $r(H)$ equals to the $p$-multiplicator rank of $H$.
Lemma 12.5.
Let $H$ be a finite $p$-group of generator rank $d$, $S$ the set of all finite $p$-groups, and $G$ the $H$-group that is isomorphic to ${\mathbb{Z}}/p{\mathbb{Z}}$ with trivial $H$-action. Then $m(S,d,H,G)=r(H)$ and $m(S,n,H,G)=r(H)+n-d$ for every $n\geq d$.
Proof.
Since the intersection of every normal subgroup and the center of a finite $p$-group is nontrivial, every finite $p$-group acts trivially on all of its minimal normal subgroups, which implies $\mathcal{CF}(\bar{S})=\{({\mathbb{Z}}/p{\mathbb{Z}},1)\}$. Recall that $m(S,n,H,G)$ is defined to be $\lim_{i\to\infty}m(T_{i},n,H,G)$, where $T_{i}$ is an increasing sequence of finite sets of groups such that $\cup T_{i}=S$. When $i$ is sufficiently large such that $T_{i}$ contains the $p$-covering group of $H$, the map $\rho:(\hat{F}_{d})^{\bar{T}_{i}}\to H$ factors through the $p$-covering group of $H$. Let $1\to R\to F\to H\to 1$ be the fundamental short exact sequence associated to $T_{i},d,H$. It’s not hard to check that $R$ is also the maximal quotient of $\ker\rho$ that is an elementary abelian $p$-group with the trivial $F$-action. Therefore, $R$ is the $p$-multiplicator of $H$ and $m(S,d,H,G)=m(T_{i},d,H,G)=r(H)$.
Assume $n\geq d$. We can find a surjection $\rho_{1}:\hat{F}_{n+1}\to H$ and generators $x_{1},\cdots,x_{n+1}$ of $\hat{F}_{n+1}$ such that $\rho_{1}(x_{n+1})=1$. Let $\rho_{2}$ be the restriction of $\rho_{1}$ on the subgroup generated by $x_{1},\cdots,x_{n}$. Then $\rho_{2}:\hat{F}_{n}\to H$ is a surjection. Let $1\to R_{1}\to F_{1}\overset{\pi_{1}}{\to}H\to 1$ and $1\to R_{2}\to F_{2}\overset{\pi_{2}}{\to}H\to 1$ be the fundamental short exact sequences associated to $T_{i},n+1,H$ and $T_{i},n,H$ that arise from $\rho_{1}$ and $\rho_{2}$ respectively. These constructions allow us to get a surjection $\pi:F_{1}\to F_{2}$ with $\pi_{1}=\pi_{2}\circ\pi$, and a generator set $y_{1},\cdots,y_{n+1}$ of $F_{1}$ such that $\pi(y_{n+1})=1$. Since $y_{n+1}\in\ker\pi_{1}$ and $F_{1}$ acts trivially on $R_{1}$, the subgroup generated by $y_{n+1}$, which is isomorphic to ${\mathbb{Z}}/p{\mathbb{Z}}$, is normal in $F_{1}$. It implies that $R_{1}\simeq R_{2}\times{\mathbb{Z}}/p{\mathbb{Z}}$ and $m(T_{i},n+1,H,G)=m(T_{i},n,H,G)+1$. By induction on $n$, we finish the proof of the lemma.
∎
Example 12.6 (Pro-$p$ quotient).
Let $H$ be a finite $p$-group of generator rank $d$, and $S$ the set of all finite $p$-groups, and $G$ the $H$-group that is isomorphic to ${\mathbb{Z}}/p{\mathbb{Z}}$ with trivial $H$-action. Since $\mathcal{CF}(\bar{S})=({\mathbb{Z}}/p{\mathbb{Z}},1)$, we have
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{i=1}^{\infty}(%
1-\frac{\lambda(S,H,G)}{p^{i+u}}).$$
By Equation (8.12) and Lemma 12.5, $\lambda(S,H,G)=p^{r(H)-d}$. So
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{i=1+u-r(H)+d}^%
{\infty}(1-p^{-i}),$$
and $\mu_{u}(V_{S,H})>0$ if and only if $u\geq r(H)-d$.
Given $u$, if $X_{n,u}^{\bar{S}}$ has generator rank $d$ with
$d^{2}/4\geq d+u$, we have that the $X_{n,u}^{\bar{S}}$ is necessarily infinite by the Golod-Shafarevich inequality. We can see from the pro-$p$ abelianization that we get groups $X_{n,u}^{\bar{S}}$ with each generator rank $d\geq\min(0,-u)$ with positive probability.
All groups in $\mathcal{P}$ have their pro-$p$ quotient finitely generated.
Example 12.7 (Pro-nilpotent quotient).
When $S$ is the set of all finite nilpotent groups and $H$ is a finite nilpotent group with Sylow $p$-subgroup $H_{p}$ of generator rank $d_{p}$, we have
$$\mu_{u}(V_{S,H})=\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{p\textrm{ %
prime}}\prod_{i=1+u-r(H_{p})+d_{p}}^{\infty}(1-p^{-i}).$$
Let $W$ be the set of profinite groups $G$ such that there are only finitely many primes $p$ such that the maximal pro-$p$ quotient of $G$ has generator rank $\geq max(2,-u+1)$.
By the Borel-Cantelli lemma, we can see that $\mu_{u}(W)=1$, and thus $\mu_{u}$ assigns probability $1$ to the set of groups who pro-nilpotent quotient is finitely generated.
Example 12.8 (All infinite groups).
When $u\leq 0$, we have $\mu_{u}(\{\textrm{infinite groups}\})=1$ (which can already be seen on the abelianization).
When $u>0$, we have $0<\mu_{u}(\{\textrm{infinite groups}\})<1$, since $\mu_{u}(\{\textrm{trivial group}\})>0$ and there is positive probability of infinite pro-p quotient.
This was seen for the $\mu_{u,n}$ in [JL06].
13. Which groups appear?
In this section, we consider the question of when $\mu_{u}$ is $0$ on our basic opens $U_{S,H}$.
In order for a basic open $U_{S,H}$ to have positive probability for $\mu_{u,n}$, the group $H$ needs to be able to be generated as a pro-$\bar{S}$ group with $n$ generators and $n+u$ relations. We will see in Proposition 13.2 that the same criterion holds for
$\mu_{u,n}$.
We start with a lemma about the number of generators and relations required to present a pro-$\bar{S}$ group.
Lemma 13.1.
Let $S$ be a finite set of finite groups and $u$ an integer. Let $H$ be a finite pro-$\bar{S}$ group that can be generated by $d$ generators. If $H$ can be presented as a pro-$\bar{S}$ group by $m$ generators and $m+u$ relations, then $H$ can be presented as a pro-$\bar{S}$ group by $d$ generators and $d+u$ relations.
This is the same as the situation when $S$ is the set of all profinite groups and $H$ is a finite group (see [Lub01, Theorem 0.1]), but contrasts to the more general situation of presenting $H$ as a finite group, where the analog is a long-standing open question (see [Gru76, Lecture 1: Question 3]).
Proof.
Suppose for the sake of contradiction that we have a counterexample, and consider one with $m$ minimal. We have that
$$\displaystyle\mu_{u,m}(U_{S,H})$$
$$\displaystyle=$$
$$\displaystyle\frac{|\operatorname{Sur}(\hat{F}_{m},H)|}{|\operatorname{Aut}(H)%
||H|^{m+u}}\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\end{subarray}}\prod_{%
k=0}^{m(S,m,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{m+u}}})\prod_{\begin{subarray}%
{c}G\in\mathcal{N}\end{subarray}}(1-|G|^{-{m-u}})^{m(S,m,H,G)}$$
$$\displaystyle>$$
$$\displaystyle 0.$$
In particular, since $|G|$ is a power of $h_{H}(G)$ this implies that for $G\in\mathcal{A}_{H}$, we have
$$h_{H}(G)^{m(S,m,H,G)-1}|G|^{-m-u}\leq h_{H}(G)^{-1}.$$
However, since we have a minimal counterexample, we have that $m>d$ and
$$\displaystyle\mu_{u,m-1}(U_{S,H})$$
$$\displaystyle=$$
$$\displaystyle\frac{|\operatorname{Sur}(\hat{F}_{m-1},H)|}{|\operatorname{Aut}(%
H)||H|^{m-1+u}}\prod_{\begin{subarray}{c}G\in\mathcal{A}_{H}\end{subarray}}%
\prod_{k=0}^{m(S,m-1,H,G)-1}(1-\frac{h_{H}(G)^{k}}{|G|^{{m-1+u}}})\prod_{%
\begin{subarray}{c}G\in\mathcal{N}\end{subarray}}(1-|G|^{-m+1-u})^{m(S,m-1,H,G)}$$
$$\displaystyle=$$
$$\displaystyle 0.$$
By the final statement of Theorem 8.1, we have that one of the factors is $0$.
Since $m>d$, we have $|\operatorname{Sur}(\hat{F}_{m-1},H)|\neq 0$. If $H$ is the trivial group, the lemma is clear. Thus we can assume $d\geq 1$ and $m\geq 2$, and so for $G\in\mathcal{N}$ we have $(1-|G|^{-m+1-u})^{m(S,m-1,H,G)}>0$.
Thus, for some $G\in\mathcal{A}_{H}$, we have
$$h_{H}(G)^{m(S,m-1,H,G)-1}|G|^{-m+1-u}\geq 1.$$
If $\rho_{n}:(\hat{F}_{n})^{\bar{S}}\rightarrow H$ is a surjection, we have
$$\displaystyle(h_{H}(G)^{m(S,n,H,G)}-1)|G|^{-n}$$
$$\displaystyle=(h_{H}(G)-1)\sum_{\begin{subarray}{c}\textrm{isom. classes of $H%
$-extensions $(E,\pi)$}\\
\textrm{$\ker\pi$ isom. $G$ }\\
\textrm{$\ker\pi$ irred. $E$-group}\\
\textrm{$E$ is level $S$}\end{subarray}}\frac{|\operatorname{Sur}(\rho_{n},\pi%
)|}{|\operatorname{Aut}_{H}(E,\pi)||G|^{n}}.$$
Any surjection from $\rho_{n}$ to $\pi$ can be extended to a surjection from $\rho_{n+1}$ to $\pi$ in $|G|$ ways.
So, $(h_{H}(G)^{m(S,n,H,G)}-1)|G|^{-n}$ is non-decreasing in $n$.
So we have
$$\frac{h_{H}(G)^{m(S,m,H,G)}-1}{|G|^{m}}\geq\frac{h_{H}(G)^{m(S,m-1,H,G)}-1}{|G%
|^{m-1}},$$
and then we have
$$\displaystyle|G|^{u}h_{H}(G)$$
$$\displaystyle\leq h_{H}(G)^{m(S,m-1,H,G)}|G|^{-m+1}$$
$$\displaystyle\leq h_{H}(G)^{m(S,m,H,G)}|G|^{-m}+|G|^{-m+1}-|G|^{-m}$$
$$\displaystyle\leq|G|^{u}+|G|^{-m+1}-|G|^{-m}.$$
Since $m>d$ and $H$ can be generated by $d$ generators, the number of relations $m+u$ has to be positive. From above, we have $|G|^{m+u}(h_{H}(G)-1)\leq(|G|-1)$. Then this is a contradiction, since $|G|\geq 2$ and $h_{H}(G)\geq 2$.
∎
Lemma 13.1 leads to the following definition.
Definition.
Let $S$ be a finite set of finite groups and $u$ be an integer. We call a finite group $H$ with generator rank $d$ achievable (with $S$ and $u$ implicit) if it can be generated as a pro-$\bar{S}$ group with $d$ generators and $d+u$ relations.
Proposition 13.2.
Let $S$ be a finite set of finite groups and $u$ be an integer. Then for a finite group $H$ we have
that
$\mu_{u}(U_{S,H})>0$ if $H$ is achievable and $\mu_{u}(U_{S,H})=0$ otherwise.
So given $u$, our measure $\mu_{u}$ is supported on those groups in $\mathcal{P}$ who pro-$\bar{S}$ completion is achievable (for $u,S$) for every finite set $S$ of finite groups.
Note that given $S$, any finite pro-$\bar{S}$ group $H$ is achievable for $u$ sufficiently large.
Example 13.3.
From [GKKL07, Theorem A], we have that every finite simple group can be presented as a profinite group with $2$ generators and $18$ relations.
Thus if $u\geq 16$ and $S$ is a finite set of finite groups with $H\in\bar{S}$ a simple group, then $H$ is achievable.
Example 13.4.
If $S$ is the set of all groups of order $32$ and $u\leq 0$, we can see that $H={\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ is not achievable. To obtain $H$ as a quotient of $F_{2}$, it is easy to compute we need at least 3 relations (for both generators to be order $2$ and for them to commute with each other).
Remark 13.5.
Proposition 13.2 need not hold for infinite $S$. For example, if $S$ is the set of all finite abelian groups, then any finite abelian group $H$ can be presented as an abelian group with $n$ generators and $n$ relations, but $\mu_{0}(V_{S,H})=0$. (See Example 12.4.) This is because the product over $G\in\mathcal{A}_{H}$ is contains factors $(1-p^{-1})$ for each prime $p$ and thus is $0$ even though no individual factor is $0$. Further, if $S$ is the set of all groups and $H\in\mathcal{P}$, we have $\mu_{0}(V_{S,H})\leq\mu_{0}(V_{\textrm{\{abelian groups\}},H^{ab}})=0$. Some of those groups $H$ can be profinitely presented with $n$ generators and $n$ relations.
It is an interesting open question to understand in general for which infinite $S$ and finite $H$ does the product in Equation 8.1 give $\mu_{u}(V_{S,H})=0$ even when none of the factors in the product is $0$.
Proof of Proposition 13.2.
By Lemma 13.1, if $H$ is not achievable, then $\mu_{u,n}(U_{S,H})=0$ for all $n$ and $\mu_{u}(U_{S,H})=0$.
Suppose that $\mu_{u}(U_{S,H})=0$.
Then using Theorem 8.1 and Remark 8.4, we must have that one of the factors in
$$\displaystyle\frac{1}{|\operatorname{Aut}(H)||H|^{u}}\prod_{\begin{subarray}{c%
}G\in\mathcal{A}_{H}\end{subarray}}\prod_{i=1}^{\infty}(1-\lambda(S,H,G)\frac{%
h_{H}(G)^{-i}}{|G|^{{u}}})\prod_{\begin{subarray}{c}G\in\mathcal{N}\end{%
subarray}}e^{-|G|^{-u}\lambda(S,H,G)}.$$
is $0$, i.e. for some $G\in\mathcal{A}_{H}$ we have $\lambda(S,H,G)h_{H}(G)^{-1}|G|^{-u}\geq 1$. Recall by Remark 5.2 that $|G|$ is a power of $h_{H}(G)$ and thus so is $\lambda(S,H,G)$. In fact, for sufficiently large $n$, we have $\lambda(S,H,G)=h_{H}(G)^{m(S,n,H,G)}|G|^{-n}$.
Thus, for sufficiently large $n$, we have $h_{H}(G)^{m(S,n,H,G)-1}\geq|G|^{n+u}$ and $\mu_{u,n}(U_{S,H})=0$.
However if we can present $H$ as a pro-$\bar{S}$ group with $d$ generators and $d+u-k$ relations with $k\geq 0$, we can add $m$ generators for any $m$ and $m+k$ relations to trivialize those generators, to present $H$ with $d+m$ generators and $d+m+u$ relations for all $m\geq 0$, which implies $\mu_{u,n}(U_{S,H})>0$ for $n$ sufficiently large.
∎
14. Comparision to non-profinite groups
Let $Y_{u,n,\ell}$ be $F_{n}$ modulo $n+u$ random relations uniform from words of length at most $\ell$. In this section, we will compare this model to our $X_{u,n}$. To put the groups on the same footing, we take the profinite completions $\hat{Y}_{u,n,\ell}$ of the $Y_{u,n,\ell}$. Alternatively, we could enlarge our measure space to include non-profinite groups, with the same definition of basic opens. Since our topology would not separate groups with the same profinite completion, we might as well consider only the profinite completions. (Note by [OW11] and [Ago13], at density $<1/6$, these groups are asymptotically almost surely residually finite and thus inject into their profinite completions.)
The following is almost the same as [DT06, Lemma 4.4], but we include it here for completeness.
Proposition 14.1.
Given integers $n,u$, we have that the distributions $\nu_{u,n,\ell}$ of the $\hat{Y}_{u,n,\ell}$ weakly converge to $X_{u,n}$.
Proof.
As in the proof of Theorem 1.1, it suffices to show that for each finite group $H$ and finite set $S$ of groups that
$$\lim_{\ell\rightarrow\infty}\nu_{u,n,\ell}(U_{S,H})=\mu_{u,n}(U_{S,H}).$$
Thus we are asked to compare the quotient of the finite group $(\hat{F}_{n})^{\bar{S}}$ by the image of random uniform words of length at most $\ell$ versus by uniform random relators. However as $\ell\rightarrow\infty$ the image of a random uniform words of length at most $\ell$ converges to the uniform distribution on $(\hat{F}_{n})^{\bar{S}}$, by the fundamental theorem on irreducible, aperiodic finite state Markov chains [Dur10, Theorem 6.6.4].
∎
Next we see that taking a number of relations that is going to infinity always gives groups weakly converging to the trivial group in our topology. This includes all positive density Gromov random groups as well as plenty of density $0$ random groups.
Proposition 14.2.
Let $u(\ell)$ be an integer valued function of the positive integers that goes to $\infty$ as $\ell\rightarrow\infty$.
Then $\nu_{u(\ell),n,\ell}$ weakly converge to the probability measure supported on the trivial group as $\ell\rightarrow\infty$.
Proof.
Fix a finite set $S$ of finite groups and a finite group $H$.
Fix an integer $v$. For $u(\ell)\geq v$, we have that
$$\operatorname{Prob}(\hat{Y}_{u(\ell),n,\ell}\textrm{ has a surjection to $H$})%
\leq\operatorname{Prob}(\hat{Y}_{v,n,\ell}\textrm{ has a surjection to $H$}).$$
Since the set of groups with a surjection to $H$ is open and closed by Proposition 14.1, we have that
$$\lim_{\ell\rightarrow\infty}\operatorname{Prob}(\hat{Y}_{v,n,\ell}\textrm{ has%
a surjection to $H$})=\operatorname{Prob}(X_{v,n}\textrm{ has a surjection to%
$H$}).$$
It is easy to see using the approach of our paper that
$${\mathbb{E}}(|\operatorname{Sur}(X_{v,n},H|)=\frac{|\operatorname{Sur}(F_{n},H%
)|}{|H|^{n+v}}\leq|H|^{-v}.$$
Thus
$$\limsup_{\ell\rightarrow\infty}\operatorname{Prob}(\hat{Y}_{u(\ell),n,\ell}%
\textrm{ has a surjection to $H$})\leq|H|^{-v}$$
for every $v$, and so $\lim_{\ell\rightarrow\infty}\operatorname{Prob}(\hat{Y}_{u(\ell),n,\ell}%
\textrm{ has a surjection to $H$})=0$.
Thus, for every $U_{S,H}$ with $H$ non-trivial, we have that
$$\lim_{\ell\rightarrow\infty}\nu_{u(\ell),n,\ell}(U_{S,H})=0.$$
For $u(\ell)\geq v$, we have that
$$\operatorname{Prob}(\hat{Y}_{u(\ell),n,\ell}^{\bar{S}}\textrm{ trivial})\geq%
\operatorname{Prob}(\hat{Y}_{v,n,\ell}^{\bar{S}}\textrm{ trivial}).$$
By Proposition 14.1, we have that
$$\lim_{\ell\rightarrow\infty}(\operatorname{Prob}(\hat{Y}_{v,n,\ell}^{\bar{S}}%
\textrm{ trivial}))=\operatorname{Prob}(X_{v,n}^{\bar{S}}\textrm{ trivial}).$$
So
$$\liminf_{\ell\rightarrow\infty}\operatorname{Prob}(\hat{Y}_{u(\ell),n,\ell}^{%
\bar{S}}\textrm{ trivial})\geq\limsup_{v\rightarrow\infty}\operatorname{Prob}(%
X_{v,n}^{\bar{S}}\textrm{ trivial}).$$
From Equation (1.2), we have that $\limsup_{v\rightarrow\infty}\operatorname{Prob}(X_{v,n}^{\bar{S}}\textrm{ %
trivial})=1$.
(We can control the size of the product in Equation (1.2), for example, by using the fact that there are at most 2 finite simple groups of any particular order.)
Thus, for every $U_{S,1}$, we have that
$$\lim_{\ell\rightarrow\infty}\nu_{u(\ell),n,\ell}(U_{S,1})=1.$$
∎
Remark 14.3.
Proposition 14.2 might seem surprising at first. The groups $Y_{u(\ell),n,\ell}$ are plenty interesting as $\ell\rightarrow\infty$. In particularly they are asymptotically almost surely infinite at density $<1/2$ [Gro93],
and residually finite at density $<1/6$ [OW11, Ago13], and so have many finite quotients. The above shows that those quotients are escaping off to infinity, however. Just as a very interesting sequence of numbers might go to $0$, an interesting sequence of random groups can converge to the trivial group. A better analogy might be that a sequence of integers with interesting asymptotic growth that goes to $0$ $p$-adically. This shows that, at low densities, the weak convergence of $\nu_{u(\ell),n,\ell}$ in $\ell$ is not as strong
as the convergence of the $\mu_{u,n}$ in $n$ that we see in Corollary 11.6. In particular
$$\lim_{\ell\rightarrow\infty}\nu_{u(\ell),n,\ell}(\textrm{trivial group})=0\neq
1.$$
Acknowledgements
We thank Nigel Boston, Persi Diaconis, Tullia Dymarz, Benson Farb, Turbo Ho, Peter Sarnak, Mark Shusterman, and Tianyi Zheng for helpful conversations.
This work was done with the support of an American Institute of Mathematics Five-Year Fellowship, a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, a Vilas Early Career Investigator Award, and National Science Foundation grants DMS-1652116 and DMS-1301690.
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Spectral distribution Method for neutrinoless double beta decay:
Results for ${}^{82}$Se and ${}^{76}$Ge
V.K.B. Kota
Physical Research Laboratory, Ahmedabad 380009, India
Department of Physics, Laurentian Unversity, Sudbury, ON P3E 2C6,
Canada
\runningheads
Spectral distribution Method for neutrinoless double beta
decayV.K.B. Kota, R.U. Haq
{start}
1, \coauthorR.U. Haq2
1
2
{Abstract}
Statistical spectral distribution method based on shell model and random matrix
theory is developed for calculating neutrinoless double beta decay nuclear
transition matrix elements. First results obtained for ${}^{82}$Se and ${}^{76}$Ge
using the spectral method are close to the available shell model results.
1 Introduction
Neutrinoless double beta decay ($0\nu\beta\beta$ or NDBD) which involves
emission of two electrons without the accompanying neutrinos and which violates
lepton number conservation has been an important and challenging problem both
for the experimentalists and theoreticians. Recent neutrino oscillation
experiments have demonstrated that neutrinos have mass
[1, 2, 3]. The observation of $0\nu\beta\beta$ decay is
expected to provide information regarding the absolute neutrino mass which is,
as yet, not known. As a result, experimental programs to observe this decay
have been initiated at different laboratories across the globe and already are
in advanced stages of development. The most recent results for $0\nu\beta\beta$ decay of ${}^{136}$Xe have been reported by EXO-200 collaboration
[4] and KamLand-Zen collaboration [5]. They give a lower limit of
$3.4\times 10^{25}$ yr for the half-life. Further, phase I results from GERDA
experiment [6] for ${}^{76}$Ge gives a lower limit of $3.0\times 10^{25}$ yr for the half-life. Nuclear transition matrix elements (NTME) are
the essential ingredient for extracting the neutrino mass from the half lives
[7]. There has been considerable effort to obtain NTME for various
candidate nuclei and they have been calculated theoretically using a variety of
nuclear models: (i) large scale shell model; (ii) quasi-particle random phase
approximation and its variants; (iii) proton-neutron interacting boson model;
(iv) particle number and angular momentum projection including configuration
mixing within the generating coordinate method framework; (v) projected
Hartree-Fock-Bogoliubov method with pairing plus quadrupole-quadrupole
interaction. A detailed comparative study of the results from these various
methods is discussed in [8, 9]. In addition, more recently the so
called deformed shell model based on Hartree-Fock single particle states has
been used for the candidate nuclei in the A=60-90 region [10]. It is
important to note that the predictions of various models for NTME vary typically
from 2 to 6 [8].
The statistical spectral distribution method (SDM) developed by French and
collaborators for nuclear structure is well documented [11] and the
operation of embedded Gaussian orthogonal ensemble of random matrices (EGOE) in
nuclear shell model spaces forms the basis for SDM [12]. With this, it
is natural that one should develop and apply SDM for calculating the NTME for
$0\nu\beta\beta$ and compare the results with those obtained using shell model
and other models. This is addressed in the present paper with first results for
${}^{82}$Se and ${}^{76}$Ge. The essential point is that NTME can be viewed as a
transition strength (square of the matrix element connecting a given initial
state to a final state by a transition operator) generated by the NDBD operator
that is two-body in nature. Therefore, SDM for transition strengths as given in
[13, 14, 15, 11] can be used as the starting point for further
developments and applications. Let us add that SDM is sometimes called moment
method.
We now give a preview. Section 2 gives a brief discussion of the relation
between neutrino mass and NTME and then deals with the structure of the NDBD
transition operator. Section 3 deals with the details of SDM for transition
strengths as applicable for NDBD. In Section 4, we present SDM results for
${}^{82}$Se and ${}^{76}$Ge NDBD NTME. For these two nuclei, experiments SuperNEMO
and GERDA+MAJORANA respectively are under development to measure NDBD half
lives. Finally, conclusions and future outlook are given in Section 5.
2 Neutrinoless double beta decay and NTME
In $0\nu\beta\beta$, the half-life for the 0${}^{+}_{i}$ ground state (gs) of a
initial even-even nucleus decay to the 0${}^{+}_{f}$ gs of the final even-even nucleus
is given by [7]
$$\left[T_{1/2}^{0\nu}(0^{+}_{i}\to 0^{+}_{f})\right]^{-1}=G^{0\nu}\left|M^{0\nu%
}(0^{+})\right|^{2}\left(\displaystyle\frac{\left\langle m_{\nu}\right\rangle}%
{m_{e}}\right)^{2}\;,$$
(1)
where $\left\langle m_{\nu}\right\rangle$ is the effective neutrino mass (a combination of
neutrino mass eigenvalues and also involving the neutrino mixing matrix). The
$G^{0\nu}$ is a phase space integral (kinematical factor); tabulations for
$G^{0\nu}$ are available. The $M^{0\nu}$ represents NTME of the NDBD transition
operator and it is a sum of a Gamow-Teller like ($M_{GT}$), Fermi like ($M_{F}$)
and tensor ($M_{T}$) two-body operators. Since it is well known that the tensor
part contributes only up to 10% of the matrix elements, we will neglect the
tensor part. Then, from the closure approximation which is well justified for
NDBD, we have
$$\begin{array}[]{rcl}M^{0\nu}(0^{+})&=&M^{0\nu}_{GT}(0^{+})-\displaystyle\frac{%
g_{V}^{2}}{g_{A}^{2}}M^{0\nu}_{F}(0^{+})=\left\langle 0^{+}_{f}\mid\mid{\cal O%
}(2:0\nu)\mid\mid 0^{+}_{i}\right\rangle\;,\\
{\cal O}(2:0\nu)&=&\displaystyle\sum_{a,b}{\cal H}(r_{ab},\overline{E})\tau_{a%
}^{+}\tau_{b}^{+}\left(\sigma_{a}\cdot\sigma_{b}-\displaystyle\frac{g_{V}^{2}}%
{g_{A}^{2}}\right)\;.\end{array}$$
(2)
As seen from Eq. (2), NDBD half-lives are generated by the two-body
transition operator ${\cal O}(2:0\nu)$; note that $a,b$ label nucleons. The $g_{A}$
and $g_{V}$ are the weak axial-vector and vector coupling constants. The
${\cal H}(r_{ab},\overline{E})$ in Eq. (2) is called the ‘neutrino
potential’. Here, $\overline{E}$ is the average energy of the virtual
intermediate states used in the closure approximation. The form given by Eq.
(2) is justified only if the exchange of the light majorana
neutrino is indeed the mechanism responsible for the NDBD. With the phase space
factors fairly well known, all one needs are NTME $|M^{0\nu}(0^{+})|=\left|\left\langle 0^{+}_{f}\mid\mid{\cal O}(2:0\nu)\mid\mid
0%
^{+}_{i}\right\rangle\right|$. Then, measuring the
half-lives makes it possible to deduce neutrino mass using Eq. (1).
The neutrino potential is of the form ${\cal H}(r_{ab},\overline{E})=[R/r_{ab}]\,\Phi(r_{ab},\overline{E})$ where $R$ in fm units is the nuclear
radius and similarly $r_{ab}$ is in fm units. A simpler form for the function
$\Phi$, involving sine and cosine integrals, as given in [7] and
employed in [10], is used in the present work. It is useful to note that
$\Phi(r_{ab},\overline{E})\sim\exp({-\frac{3}{2}\frac{\overline{E}}{\hbar c}r_{%
ab}})$. The effects of short-range correlations in the wavefunctions are
usually taken into account by multiplying the wavefunction by the Jastrow
function $[1-\gamma_{3}e^{-\gamma_{1}r_{ab}^{2}}(1-\gamma_{2}r_{ab}^{2})]$.
There are other approaches [16] for taking into account the short range
correlations but they are not considered here. Now, keeping the wavefunctions
unaltered, the Jastrow function can be incorporated into ${\cal H}(r_{ab},\overline{E})$ giving an effective ${\cal H}_{eff}(r_{ab},\overline{E})$,
$${\cal H}(r_{ab},\overline{E})\to{\cal H}_{eff}(r_{ab},\overline{E})={\cal H}(r%
_{ab},\overline{E})[1-\gamma_{3}\;e^{-\gamma_{1}\;r_{ab}^{2}}(1-\gamma_{2}\;r_%
{ab}^{2})]^{2}\;.$$
(3)
The choice of the values for the parameters $\gamma_{1}$, $\gamma_{2}$ and
$\gamma_{3}$ is given in Section 4.
Let us say that for the nuclei under consideration, protons are in the single
particle (sp) orbits $j^{p}$ and similarly neutrons in $j^{n}$. Using the usual
assumption that the radial part of the sp states are those of the harmonic
oscillator, the proton sp states are completely specified by
(${\bf n}^{p},\ell^{p},j^{p}$) with ${\bf n}^{p}$ denoting oscillator radial quantum number so
that for a oscillator shell ${\cal N}^{p}$, $2{\bf n}^{p}+\ell^{p}={\cal N}^{p}$. Similarly, the
neutron sp states are (${\bf n}^{n},\ell^{n},j^{n}$). In terms of the creation
($a^{\dagger}$) and annihilation ($a$) operators, normalized two-particle
(antisymmetrized) creation operator $A^{J}_{\mu}(j_{1}j_{2})=(1+\delta_{j_{1}j_{2}})^{-1/2}(a^{\dagger}_{j_{1}}a^{%
\dagger}_{j_{2}})^{J}_{\mu}$ and then
$A^{J}_{\mu}\left|0\right\rangle=\left|(j_{1}j_{2})J\mu\right\rangle$ represents a normalized
two-particle state. At this stage, it is important to emphasize that we are
considering only $0^{+}$ to $0^{+}$ transitions in $0\nu\beta\beta$ and therefore
only the $J$ scalar part of ${\cal O}(2:0\nu)$ will contribute to $M^{0\nu}$. With
this, the NDBD transition operator can be written as,
$${\cal O}(2:0\nu)=\displaystyle\sum_{j_{1}^{p}\geq j_{2}^{p};j_{3}^{n}\geq j_{4%
}^{n};J}{\cal O}_{j_{1}^{p}\,j_{2}^{p};j_{3}^{n}\,j_{4}^{n}}^{J}(0\nu)%
\displaystyle\sum_{\mu}A^{J}_{\mu}(j_{1}^{p}j_{2}^{p})\left\{A^{J}_{\mu}(j_{3}%
^{n}j_{4}^{n})\right\}^{\dagger}\;.$$
(4)
Here, ${\cal O}_{j_{1}^{p}\,j_{2}^{p};j_{3}^{n}\,j_{4}^{n}}^{J}(0\nu)=\left\langle(j_%
{1}^{p}\,j_{2}^{p})JM\mid{\cal O}(2:0\nu)\mid(j_{3}^{n}\,j_{4}^{n})JM\right%
\rangle_{a}$ are two-body matrix
elements (TBME) and ‘$a$’denotes antisymmetrized two-particle
wavefunctions; $J$ is even for $j_{1}=j_{2}$ or $j_{3}=j_{4}$. The TBME are obtained
by using the standard approach based on Brody-Moshinisky brackets and Talmi
integrals.
3 Spectral distribution method for NDBD
3.1 State densities and Gaussian form
Let us consider shell model sp orbits $j^{p}_{1},j^{p}_{2},\ldots,j^{p}_{r}$ with $m_{p}$
protons distributed in them. Similarly, $m_{n}$ neutrons are distributed in
$j^{n}_{1},j^{n}_{2},\ldots,j^{n}_{s}$ orbits. Then, the proton configurations are
$\widetilde{m_{p}}=[m_{p}^{1},m_{p}^{2},\ldots,m_{p}^{r}]$ where $m_{p}^{i}$ is number of protons in the
orbit $j_{i}^{p}$ with $\sum_{i=1}^{r}\,m_{p}^{i}=m_{p}$. Similarly, the neutron
configurations are $\widetilde{m_{n}}=[m_{n}^{1},m_{n}^{2},\ldots,m_{n}^{s}]$ where $m_{n}^{i}$ is number
of neutrons in the orbit $j_{i}^{n}$ with $\sum_{i=1}^{s}\,m_{n}^{i}=m_{n}$. With these,
$(\widetilde{m_{p}},\widetilde{m_{n}})$’s denote proton-neutron configurations. The nuclear effective
Hamiltonian is one plus two-body, $H=h(1)+V(2)$ and we assume that the one-body
part $h(1)$ includes the mean-field producing part of the two-body interaction.
Thus, $V(2)$ is the irreducible two-body part of $H$ [11]. From now on, for
simplicity we shall denote $h=h(1)$ and $V=V(2)$. The state density $I^{H}(E)$,
with $\left\langle\left\langle--\right\rangle\right\rangle$ denoting trace, can be written as a sum of the
partial densities defined over $(\widetilde{m_{p}},\widetilde{m_{n}})$,
$$\begin{array}[]{l}I^{(m_{p},m_{n})}(E)=\left\langle\left\langle\delta(H-E)%
\right\rangle\right\rangle^{(m_{p},m_{n})}=\displaystyle\sum_{(\widetilde{m_{p%
}},\widetilde{m_{n}})}\,\left\langle\left\langle\delta(H-E)\right\rangle\right%
\rangle^{(\widetilde{m_{p}},\widetilde{m_{n}})}\\
\\
=\displaystyle\sum_{(\widetilde{m_{p}},\widetilde{m_{n}})}\,I^{(\widetilde{m_{%
p}},\widetilde{m_{n}})}(E)=\displaystyle\sum_{(\widetilde{m_{p}},\widetilde{m_%
{n}})}\,d(\widetilde{m_{p}},\widetilde{m_{n}})\,\rho^{(\widetilde{m_{p}},%
\widetilde{m_{n}})}(E)\;.\end{array}$$
(5)
Here, $d(\widetilde{m_{p}},\widetilde{m_{n}})$ is the dimension of the configuration $(\widetilde{m_{p}},\widetilde{m_{n}})$ and
$\rho^{(\widetilde{m_{p}},\widetilde{m_{n}})}(E)$ is normalized to unity. For strong enough two-body
interactions (this is valid for nuclear interactions [11]), the operation
of embedded GOE of one plus two-body interactions [EGOE(1+2)] will lead to
Gaussian form for the partial densities $\rho^{(\widetilde{m_{p}},\widetilde{m_{n}})}(E)$ and therefore,
$$I^{(m_{p},m_{n})}(E)=\displaystyle\sum_{(\widetilde{m_{p}},\widetilde{m_{n}})}%
\;I^{(\widetilde{m_{p}},\widetilde{m_{n}})}_{\cal G}(E)\;.$$
(6)
In Eq. (6), ${\cal G}$ denotes Gaussian. The Gaussian partial densities
are defined by the centroids $E_{c}(\widetilde{m_{p}},\widetilde{m_{n}})=\left\langle H\right\rangle^{(%
\widetilde{m_{p}},\widetilde{m_{n}})}$ and
variances $\sigma^{2}(\widetilde{m_{p}},\widetilde{m_{n}})=\left\langle H^{2}\right%
\rangle^{(\widetilde{m_{p}},\widetilde{m_{n}})}-[E_{c}(\widetilde{m_{p}},%
\widetilde{m_{n}})]^{2}$. Expressions for these follow easily from trace propagation
methods [17, 11]. In practical applications to nuclei, Eq. (6)
has to be applied in fixed-$J$ spaces [18, 19] or an approximate $J$
projection has to be carried out [11, 20, 21]. We will return to this
question in Section 3.4.
3.2 Transition strength densities and bivariate Gaussian form
Given a transition operator ${\cal O}$, the spectral distribution method for
transition strengths starts with the transition strength density
$I^{H}_{\cal O}(E_{i},E_{f})$,
$$I^{H}_{\cal O}(E_{i},E_{f})=I(E_{f})|\left\langle E_{f}\mid{\cal O}\mid E_{i}%
\right\rangle|^{2}I(E_{i})$$
(7)
where $E$’s are eigenvalues of $H$. A plausible way to proceed now [13]
is to first construct the transition strength density with $H=h=\sum_{r}\epsilon_{r}n_{r}$; $n_{r}$ is the number operator for the orbit $r$ and $\epsilon_{r}$
are the sp energies (spe). As the configurations $(\widetilde{m_{p}},\widetilde{m_{n}})$ are eigenstates
of $h$, it is straightforward to construct $I^{h}_{{\cal O}}$ [11].
Next the interaction $V$, the two-body part of $H$, is switched on. Then, the
role of $V$ is to locally spread $I^{h}_{{\cal O}}$ and therefore the strength
density will be a bivariate convolution of $I^{h}_{\cal O}$ and $\rho^{V}_{{\cal O}}$;
the spreading function (normalized to unity) $\rho^{V}_{{\cal O}}$ is a
bivariate distribution. For strong enough interactions, operation of EGOE(1+2)
generates bivariate Gaussian form for $\rho^{V}_{{\cal O}}$ and this result has been
established for NDBD type operators in [15]. Applying this, with some
additional approximations as discussed ahead, will give
$$\begin{array}[]{l}\left|\left\langle E_{f}\mid{\cal O}\mid E_{i}\right\rangle%
\right|^{2}=\displaystyle\sum_{{\widetilde{m}}_{i},{\widetilde{m}}_{f}}%
\displaystyle\frac{I^{{\widetilde{m}}_{i}}_{{\cal G}}(E_{i})I^{{\widetilde{m}}%
_{f}}_{{\cal G}}(E_{f})}{I^{m_{i}}(E_{i})I^{m_{f}}(E_{f})}\left|\left\langle{%
\widetilde{m}}_{f}\mid{\cal O}\mid{\widetilde{m}}_{i}\right\rangle\right|^{2}%
\\
\times\displaystyle\frac{\rho^{V}_{{\cal O}:biv-{\cal G}}(E_{i},E_{f},{\cal E}%
_{{\cal O}:V}({\widetilde{m}}_{i}),{\cal E}_{{\cal O}:V}({\widetilde{m}}_{f}),%
\sigma_{{\cal O}:V}({\widetilde{m}}_{i}),\sigma_{{\cal O}:V}({\widetilde{m}}_{%
f}),\zeta_{{\cal O}:V}({\widetilde{m}}_{i},{\widetilde{m}}_{f}))}{\rho_{{\cal G%
}}^{{\widetilde{m}}_{i}}(E_{i})\rho_{{\cal G}}^{{\widetilde{m}}_{f}}(E_{f})}\;%
;\\
\left|\left\langle{\widetilde{m}}_{f}\mid{\cal O}\mid{\widetilde{m}}_{i}\right%
\rangle\right|^{2}=\left[d({\widetilde{m}}_{i})d({\widetilde{m}}_{f})\right]^{%
-1}\;\displaystyle\sum_{\alpha,\beta}\left|\left\langle{\widetilde{m}}_{f},%
\alpha\mid{\cal O}\mid{\widetilde{m}}_{i},\beta\right\rangle\right|^{2}\;.\end%
{array}$$
(8)
This is the basic equation that allows one to use SDM for the calculation of
NTME $M^{0\nu}$. In order to apply this, we need the marginal centroids
${\cal E}_{{\cal O}:V}({\widetilde{m}}_{i})$ and ${\cal E}_{{\cal O}:V}({\widetilde{m}}_{f})$, marginal variances
$\sigma^{2}_{{\cal O}:V}({\widetilde{m}}_{i})$ and $\sigma^{2}_{{\cal O}:V}({\widetilde{m}}_{f})$ and the correlation
coefficient $\zeta_{{\cal O}:V}({\widetilde{m}}_{i},{\widetilde{m}}_{f})$ defining $\rho^{V}_{{\cal O}:biv-{\cal G}}$.
Also, we need $\left|\left\langle{\widetilde{m}}_{f}\mid{\cal O}\mid{\widetilde{m}}_{i}\right%
\rangle\right|^{2}$. Note that ${\widetilde{m}}=(\widetilde{m_{p}},\widetilde{m_{n}})$ in actual applications and further, the angular momentum quantum
numbers for the parent and daughter nuclei involved in $0\nu\beta\beta$ decay
need to be considered. We will turn to these now.
3.3 SDM for NTME for $0\nu\beta\beta$ decay
Firstly, the marginal centroids and variances in Eq. (8) are
approximated, following random matrix theory [15], to the corresponding
state density centroids and variances (see for example [14, 13]) giving
${\cal E}_{{\cal O}:V}((\widetilde{m_{p}},\widetilde{m_{n}})_{r})\approx E_{c}(%
(\widetilde{m_{p}},\widetilde{m_{n}})_{r})=\left\langle H\right\rangle^{(%
\widetilde{m_{p}},\widetilde{m_{n}})_{r}}$ and $\sigma^{2}_{{\cal O}:V}((\widetilde{m_{p}},\widetilde{m_{n}})_{r}))\approx%
\sigma^{2}((\widetilde{m_{p}},\widetilde{m_{n}})_{r}))=\left\langle V^{2}%
\right\rangle^{(\widetilde{m_{p}},\widetilde{m_{n}})_{r}}$; $r=i,f$. For the correlation
coefficient $\zeta$, there is not yet any valid form involving configurations.
Therefore, the only plausible way forward currently is to estimate $\zeta$ as a
function of $(m_{p},m_{n})$ using random matrix theory given in [15, 13].
Then, the definition of $\zeta$ is
$$\begin{array}[]{l}\zeta_{{\cal O}:V}(m_{p},m_{n})=\\
\displaystyle\frac{\left\langle{\cal O}(2:0\nu)^{\dagger}\,V\,{\cal O}(2:0\nu)%
\,V\right\rangle^{(m_{p},m_{n})}}{\displaystyle\sqrt{\left\langle{\cal O}(2:0%
\nu)^{\dagger}\,V^{2}\,{\cal O}(2:0\nu)\right\rangle^{(m_{p},m_{n})}\;\left%
\langle{\cal O}(2:0\nu)^{\dagger}\,{\cal O}(2:0\nu)\,V^{2}\right\rangle^{(m_{p%
},m_{n})}}}\;.\end{array}$$
(9)
Results in Sections 5 and 6 of [15], obtained using EGOE representation
for both the ${\cal O}$ and $V$ operators, will allow one to obtain $\zeta(m_{p},m_{n})$.
For nuclei of interest, using the numerical results in Table 2 of [15],
it is seen that $\zeta\sim 0.6-0.8$. These values are used in the $M^{0\nu}$
calculations reported in Section 4 ahead.
In order to apply Eq. (8), in addition to the marginal
centroids, variances and $\zeta$, we also need an expression for $\left|\left\langle{\widetilde{m}}_{f}\mid{\cal O}\mid{\widetilde{m}}_{i}\right%
\rangle\right|^{2}$, the configuration mean square matrix
element of the transition operator. Applying the propagation theory given in
[17] will give,
$$\begin{array}[]{l}\left|\left\langle(\widetilde{m_{p}},\widetilde{m_{n}})_{f}%
\mid{\cal O}(2:0\nu)\mid(\widetilde{m_{p}},\widetilde{m_{n}})_{i}\right\rangle%
\right|^{2}=\left\{d[(\widetilde{m_{p}},\widetilde{m_{n}})_{f}]\right\}^{-1}\\
\times\;\displaystyle\sum_{\alpha,\beta,\gamma,\delta}\;\displaystyle\frac{m^{%
i}_{n}(\alpha)\,[m^{i}_{n}(\beta)-\delta_{\alpha\beta}]\,[N_{p}(\gamma)-m^{i}_%
{p}(\gamma)]\,[N_{p}(\delta)-m^{i}_{p}(\delta)-\delta_{\gamma\delta}]}{N_{n}(%
\alpha)\,[N_{n}(\beta)-\delta_{\alpha\beta}]\,N_{p}(\gamma)\,[N_{p}(\delta)-%
\delta_{\gamma\delta}]}\\
\times\displaystyle\sum_{J_{0}}\;\left[{\cal O}^{J_{0}}_{\gamma^{p}\delta^{p}%
\alpha^{n}\beta^{n}}(0\nu)\right]^{2}(2J_{0}+1)\;;\\
(\widetilde{m_{p}},\widetilde{m_{n}})_{f}=(\widetilde{m_{p}},\widetilde{m_{n}}%
)_{i}\times\left(1^{+}_{\gamma_{p}}1^{+}_{\delta_{p}}1_{\alpha_{n}}1_{\beta_{n%
}}\right)\;.\end{array}$$
(10)
Note that in Eq. (10), the final configuration is defined by
removing one neutron from orbit $\alpha$ and another from $\beta$ and then
adding one proton in orbit $\gamma$ and another in orbit $\delta$. Also,
$N_{p}(\alpha)$ is the degeneracy of the proton orbit $\alpha$ and similarly
$N_{n}(\gamma)$ for the neutron orbit $\gamma$.
3.4 Angular momentum decomposition of transition strengths
For NDBD NTME calculations, to complete the transition strength theory given by
Eqs. (8) - (10), we need $J$ projection as the
quantity of interest is
$$\left|\left\langle E_{f}J_{f}=0\mid{\cal O}\mid E_{i}J_{i}=0\right\rangle%
\right|^{2}$$
where $E_{i}$ and $E_{f}$ are the ground state energies of the parent and daughter
nuclei respectively and similarly $J_{i}$ and $J_{f}$. Firstly, note that
$$\begin{array}[]{l}\left|\left\langle E_{f}J_{f}=0\mid{\cal O}(2:0\nu)\mid E_{i%
}J_{i}=0\right\rangle\right|^{2}\\
=\displaystyle\frac{\left\langle\left\langle[{\cal O}(2:0\nu)]^{\dagger}X(H,J^%
{2},E_{f},J_{f}){\cal O}(2:0\nu)Y(H,J^{2},E_{i},J_{i})\right\rangle\right%
\rangle^{(m_{p}^{i}m_{n}^{i})}}{\left\langle\left\langle X(H,J^{2},E_{f},J_{f}%
)\right\rangle\right\rangle^{(m_{p}^{f}m_{n}^{f})}\left\langle\left\langle Y(H%
,J^{2},E_{i},J_{i})\right\rangle\right\rangle^{(m_{p}^{i}m_{n}^{i})}}\\
=\displaystyle\frac{I^{(m_{p}^{i}m_{n}^{i}),(m_{p}^{f}m_{n}^{f})}_{{\cal O}(2:%
0\nu)}(E_{i},E_{f},J_{i},J_{f})}{I^{(m_{p}^{i}m_{n}^{i})}(E_{i},J_{i})I^{(m_{p%
}^{f}m_{n}^{f})}(E_{f},J_{f})}\;;\\
X(H,J^{2},E_{f},J_{f})=\delta(H-E_{f})\delta(J^{2}-J_{f}(J_{f}+1))\;,\\
Y(H,J^{2},E_{i},J_{i})=\delta(H-E_{i})\delta(J^{2}-J_{i}(J_{i}+1))\;.\end{array}$$
(11)
The four variate density $I_{{\cal O}}(E_{i},E_{f},J_{i},J_{f})=I_{{\cal O}}(E_{i},E_{f})\rho_{{\cal O}}%
(J_{i},J_{f}:E_{i},E_{f})$ where $\rho$ is a conditional density. Now, using
the fact that $J_{f}$ ($J_{i}$) is uniquely determined by $J_{i}$ ($J_{f}$) for the
${\cal O}(2:0\nu)$ operator and the $J$-factoring used in [13] will give the
approximation
$$I_{{\cal O}(2:0\nu)}(E_{i},E_{f},J_{i},J_{f})\sim I_{{\cal O}(2:0\nu)}(E_{i},E%
_{f})\sqrt{C_{J_{i}}(E_{i})C_{J_{f}}(E_{f})}\;.$$
(12)
The function $C_{J}(E)$ involves spin cut-off factor as given below. In addition,
we have the well established result [17, 21, 13, 11] $I(E,J)=I(E)C_{J}(E)$. Using these will give,
$$\begin{array}[]{l}\left|\left\langle E_{f}J_{f}=0\mid{\cal O}(2:0\nu)\mid E_{i%
}J_{i}=0\right\rangle\right|^{2}=\displaystyle\frac{\left|\left\langle E_{f}%
\mid{\cal O}(2:0\nu)\mid E_{i}\right\rangle\right|^{2}}{\displaystyle\sqrt{C_{%
J_{i}=0}(E_{i})C_{J_{f}=0}(E_{f})}}\;\;;\\
\\
C_{J_{r}}(E_{r})=\displaystyle\frac{(2J_{r}+1)}{\displaystyle\sqrt{8\pi}\;%
\sigma^{3}_{J}(E_{r})}\exp-(2J_{r}+1)^{2}/8\sigma_{J}^{2}(E_{r})\stackrel{{%
\scriptstyle J_{r}=0}}{{\longrightarrow}}\displaystyle\frac{1}{\displaystyle%
\sqrt{8\pi}\sigma^{3}_{J}(E_{r})}\end{array}$$
(13)
where $r=i,f$. Note that $\sigma_{J}^{2}(E)=\left\langle J_{Z}^{2}\right\rangle^{E}$ is the energy
dependent spin cut-off factor. In the approximation $C_{J_{i}=0}(E_{i})\sim[\sqrt{8\pi}\sigma^{3}_{J}(E_{i})]^{-1}$ (similarly for $C_{J_{f}=0}(E_{f})$), we have
used the fact that in general $\sigma_{J}(E)>>1$. The spin cut-off factor can be
calculated using SDM [11, 20, 21]. Carrying this out for the nuclei of
interest in the present study, it is seen that $\sigma_{J}(E)\sim 3-4$ with $E$
varying up to 5 MeV excitation. Similarly, for lower $2p-1f$ shell nuclei
studied in [20], $\sigma_{J}(E)\sim 4-6$. Because of the uncertainties
in using spin decomposition via Eq. (13), in the present work
$M^{0\nu}$ is calculated by varying $\sigma_{J}$ from 3 to 6. In principle it is
possible to avoid the use of spin cut-off factors (see Section 5).
4 SDM results for ${}^{82}$Se and ${}^{76}$Ge $0\nu\beta\beta$ NTME
In the first application of SDM given in Section 3, we have chosen ${}^{82}$Se as
large shell model results, obtained using an easily available and well
established effective interaction, for the NTME for the $0\nu\beta\beta$ decay
to ${}^{82}$Kr are available in [22]. In addition, SuperNEMO experiment
will be measuring ${}^{82}$Se $0\nu\beta\beta$ decay half-life [23].
In the shell model calculations, ${}^{56}$Ni is the core and the valence protons
and neutrons in ${}^{82}$Se and ${}^{82}$Kr occupy the $f_{5/2}pg_{9/2}$ orbits
${}^{1}p_{3/2}$, ${}^{0}f_{5/2}$, ${}^{1}p_{1/2}$ and ${}^{0}g_{9/2}$. The effective
interaction used is JUN45. The spe and TBME defining JUN45 are given in
[24]. In the SDM application, same shell model space, spe and TBME are
employed. Firstly, all the proton-neutron configurations are generated for both
${}^{82}$Se and ${}^{82}$Kr. Number of positive parity configurations is 316 for
${}^{82}$Se and 1354 for ${}^{82}$Kr.
Using the formula in [17] and the JUN45 interaction, the centroids and
variances defining the Gaussian partial densities in Eq. (6) are
calculated. These will also give the marginal centroids and variances in Eq.
(8). The average width ($\overline{\sigma}$) for ${}^{82}$Se
configurations is 3.34 MeV with a 9% fluctuation. Similarly, for ${}^{82}$Kr,
$\overline{\sigma}=4.7$ MeV with a 5% fluctuation. Proceeding further, the
TBME ${\cal O}_{j_{1}^{p}\,j_{2}^{p};j_{3}^{n}\,j_{4}^{n}}^{J}(0\nu)$ defining the $0\nu\beta\beta$ transition operator are calculated and they are 259 in number for the
chosen set of sp orbits. The choices made for the various parameters in the
transition operator are (i) $R=1.2A^{1/3}$ fm; (ii) $b=1.003A^{1/6}$ fm; (iii)
$\overline{E}=1.12A^{1/2}$ MeV; (iv) $g_{A}/g_{V}=1$ (quenched); (v)
$\gamma_{1}=1.1\,fm^{-2}$, $\gamma_{2}=0.68\,fm^{-2}$ and $\gamma_{3}=1$ (these are
Miller-Spencer Jastrow correlation parameters). Then, applying Eq.
(10), the configuration mean square matrix elements of the
transition operator are obtained for all the configurations. With all these, in
order to apply Eqs. (8) and (13), we need the
ground states of ${}^{82}$Se and ${}^{82}$Kr and also the values of the $\zeta$ and
$\sigma_{J}$ parameters.
Using the so called Ratcliff procedure [11, 25], the ground states are
determined in SDM. For this one needs a reference level with energy ($E_{R}$)
and angular momentum and parity $J^{\pi}$ value ($J_{R}^{\pi}$) and also the total
number of states up to and including the reference level ($N_{R}$). The constraint
in choosing the reference level is that the $J^{\pi}$ values for all levels up to
the reference level should be known with certainty. Satisfying this, we have,
from the most recent data [26], for ${}^{82}$Se the values $E_{R}=1.735$MeV,
$J_{R}^{\pi}=4^{+}$ and $N_{R}=21$. Similarly, for ${}^{82}$Kr we have $E_{R}=2.172$MeV,
$J_{R}^{\pi}=0^{+}$ and $N_{R}=34$. The ground states are found to be $\sim 3\sigma$
below the lowest configuration centroid.
After obtaining the ground states, the ground to ground NTME are calculated
using Eq. (8) with $J$-decomposition via Eq.
(13). For the correlation coefficient $\zeta$ the values $0.6$,
$0.65$, $0.7$ and $0.8$ are used as stated in Section 3.3. Similarly, assuming
$\sigma_{J}(E_{i}(gs))=\sigma_{J}(E_{f}(gs))=\sigma_{J}$, the values chosen for
$\sigma_{J}$ are $3$, $4$, $5$ and $6$ as stated in Section 3.4. With increasing
$\zeta$ and $\sigma_{J}$ values, it is easy to see that the NTME $M^{0\nu}$ will
increase. The values of NTME for $\zeta=0.6$ and $\sigma_{J}=3$, 4, 5 and 6 are 1,
1.54, 2.15 and 2.83 respectively. Similarly, for $\zeta=0.65$, $0.7$ and $0.8$
they are $(1.18,1.82,2.54,3.34)$, $(1.38,2.12,2.97,3.9)$ and
$(1.78,2.74,3.82,5.03)$ respectively. With these and using $\sigma_{J}\sim 3-4$
and $\zeta\sim 0.7-0.8$ will give $M^{0\nu}\sim 2-3$ in SDM while the shell
model value given in [22] by Horoi et al., using JUN45 interaction and
same Jastrow parameters, is 2.59. It is important to note that the shell model
results include a more detailed transition operator and other modifications. In
addition, with a different interaction Poves et al. [27] obtained the
shell model value to be $\sim 2.18$. As already stated in the introduction,
with other nuclear models $M^{0\nu}\sim 3-6$. Thus, it is plausible to conclude
that SDM is useful for calculating NTME for $0\nu\beta\beta$. For further
confirmation of this, in a second example ${}^{76}$Ge is considered and
GERDA+MAJORANA experiments will measure the ${}^{76}$Ge $0\nu\beta\beta$ decay
half life in future [28].
For ${}^{76}$Ge to ${}^{76}$Se NDBD NTME, same shell model inputs are used as above
and similarly the parameters in the transition operator. Number of positive
parity proton-neutron configurations is 958 for ${}^{76}$Ge and 2604 for
${}^{76}$Se. The $\overline{\sigma}$ for ${}^{76}$Ge configurations is 4.4 MeV with
a 6% fluctuation. Similarly, for ${}^{76}$Se, $\overline{\sigma}=5.51$ MeV with
a 4% fluctuation. For the ground state determination we have [26], $(E_{R},J_{R}^{\pi},N_{R})=(2.02\mbox{MeV},\,4^{+},\,37)$ for ${}^{76}$Ge and $(E_{R},J_{R}^{\pi},N_{R})=(1.79\mbox{MeV},\,2^{+},\,33)$ for ${}^{76}$Se. The ground states
here are also $\sim 3\sigma$ below the lowest configuration centroid. With all
these, the NTME are calculated and their values for $\zeta=0.65$ and
$\sigma_{J}=3$, 4 , 5 and 6 are 1.02, 1.56, 2.19 and 2.87 respectively. Similarly,
for $\zeta=0.7$ and $0.8$ they are $(1.29,1.98,2.77,3.63)$ and $(1.96,3.01,4.21,5.54)$ respectively. Shell model result from Horoi et al. [29],
obtained using JUN45 interaction and same Jastrow parameters, is 2.72 while it
is 2.3 from Poves et al. [27] shell model calculations. Clearly, the
SDM values with $\zeta\sim 0.7-0.8$ and $\sigma_{j}\sim 4$ are close to the
shell model results.
5 Conclusions and future outlook
In the present paper SDM for calculating NTME for NDBD is described with all the
relevant equations. As first examples, results for ${}^{82}$Se and ${}^{76}$Ge are
presented and the SDM results are seen to be close to the shell model values.
It is clearly important that the SDM formulation given in Section 3 should be
tested. This is possible by constructing complete shell model Hamiltonian
matrix, in the configuration-$J$ basis, for the parent and daughter nuclei
(with $J^{\pi}$ values fixed) and the transition matrix generated by the action
of the transition operator on each of the parent states taking to the daughter
states. Although this might seem complicated for realistic nuclei, a pseudo NDBD
nucleus such as ${}^{24}$Mg could used for the test.
Another direction for a better SDM calculation is to evaluate all the
configuration centroids and variances with fixed-$J$ using for example, the
large scale computer codes developed recently by Sen’kov et al. [18];
note that we need $E_{c}(({\widetilde{m}}_{p},{\widetilde{m}}_{n}),J=0)$ and $\sigma(({\widetilde{m}}_{p},{\widetilde{m}}_{n}),J=0)$.
However, the methods used by Sen’kov et al. need to be extended to derive a
formula (or a viable method for computing) $\left|\left\langle({\widetilde{m}}_{p},{\widetilde{m}}_{n})_{f}J_{f}=0\mid{%
\cal O}\mid({\widetilde{m}}_{p},{\widetilde{m}}_{n})_{i}J_{i}=0\right\rangle%
\right|^{2}$. With these, it is possible in the near
future to apply the theory described in Section 3 without using Eq.
(13) for the calculation of NTME.
Most important is to improve SDM theory with a better treatment of $\zeta$
including its definition with configuration partitioning, although $\zeta$ via
Eq. (9) and its extensions could be made tractable. In future,
this need to be addressed.
It is useful to add that Eq. (10) easily gives the total
transition strength sum, the sum of the strengths from all states of the parent
nucleus to all the states of the daughter nucleus and this depends only on the
sp space considered. For the ${}^{82}$Se the total strength sum is 31239 and for
${}^{76}$Ge it is 54178. Thus, $M^{0\nu}(0^{+})$ is a very small fraction of the
total strength generated by the NDBD transition operator. Starting from Eq.
(8), it is possible to obtain the total strength (NEWSR)
originating from the ground state of the parent nucleus and also the linear and
quadratic energy weighted strength sums. These may prove to be useful in putting
constraints on the nuclear models being used for NDBD studies. This will be
addressed in future.
Finally, using SDM [11, 30] it is possible to study orbit occupancies and
GT distributions in various NDBD nuclei. These results can be compared with
available experimental data and will provide tests for the goodness of SDM for
NDBD. Results of these studies as well as the $M^{0\nu}$ results for heavier
${}^{124}$Sn, ${}^{130}$Te and ${}^{136}$Xe nuclei are in the process.
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Komondor: a Wireless Network Simulator for Next-Generation High-Density WLANs
††thanks: This work has been partially supported by a Gift from CISCO University Research Program (CG#890107) & Silicon Valley Community Foundation, by the Spanish Ministry of Economy and Competitiveness under the Maria de Maeztu Units of Excellence Programme (MDM-2015-0502), and by the Catalan Government under grant SGR-2017-1188. The work by S. Barrachina-Muñoz is supported by an FI grant from the Generalitat de Catalunya.
1st Sergio Barrachina-Muñoz
Wireless Networking (WN)
Universitat Pompeu Fabra
Barcelona, Spain
[email protected]
2nd Francesc Wilhelmi
Wireless Networking (WN)
Universitat Pompeu Fabra
Barcelona, Spain
[email protected]
3rd Ioannis Selinis
Inst. for Com. Systems 5GIC
University of Surrey
Guildford, United Kingdom
[email protected]
4th Boris Bellalta
Wireless Networking (WN)
Universitat Pompeu Fabra
Barcelona, Spain
[email protected]
Abstract
Komondor is a wireless network simulator for next-generation wireless local area networks (WLANs).
The simulator has been conceived as an accessible (ready-to-use) open source tool for research on wireless networks and academia.
An important advantage of Komondor over other well-known wireless simulators lies in its high event processing rate, which is furnished by the simplification of the core operation. This allows outperforming the execution time of other simulators like ns-3, thus supporting large-scale scenarios with a huge number of nodes.
In this paper, we provide insights into the Komondor simulator and overview its main features, development stages and use cases. The operation of Komondor is validated in a variety of scenarios against different tools: the ns-3 simulator and two analytical tools based on Continuous Time Markov Networks (CTMNs) and the Bianchi’s DCF model. Results show that Komondor captures the IEEE 802.11 operation very similarly to ns-3. Finally, we discuss the potential of Komondor for simulating complex environments – even with machine learning support – in next-generation WLANs by easily developing new user-defined modules of code.
Wireless network simulator, high-density, WLAN, IEEE 802.11ax, dynamic channel bonding, machine learning
I Introduction
The Institute of Electrical and Electronics Engineers (IEEE) 802.11 Wireless Local Area Networks (WLANs) are evolving fast to satisfy the new strict requirements in terms of data rate and user density. In particular, various IEEE 802.11 amendments have been introduced in the past few years or are under active development to accommodate the need for higher capacity, exponential growth in number of devices, and novel use-cases. [1]. An example of next-generation high-density deployment is depicted in Fig. 1 where multiple WLANs are allocated with different channels and dynamic channel bonding (DCB) policies.
Of a particular interest is the IEEE 802.11ax (11ax) amendment [2], that is under active development and which was introduced to address the demands and challenges that WLANs will face in the congested 2.4/5 GHz bands [3]. Other important amendments for next-generation wireless networks are the IEEE 802.11ay [4] and EXtreme Throughput (XT) 802.11 [5], which aim to exploit the 60 GHz and $\leq 6$ GHz frequency bands, respectively. Amendments like the aforementioned ones lay the foundation of next generation WLANs by including new features such as multiple-antenna techniques like Downlink/Uplink Multi-User Multiple-Input-Multiple-Output (DL/UL MU-MIMO), spatial reuse techniques like BSS coloring, and efficient use of channel resources like DL/UL Orthogonal Frequency Division Multiple Access (OFDMA). Therefore, it becomes necessary to provide reliable simulation tools able to assess the performance and behavior of next-generation WLANs in multiple scenarios/cases, specially in high-density deployments.
In this paper, we present Komondor,111All of the source code of Komondor, under the GNU General Public License v3.0., is open, and potential contributors are encouraged to participate. The repository can be found at https://github.com/wn-upf/Komondor an open source, event-driven simulator based on the CompC++ COST library [6]. Komondor is focused on fulfilling the need for assessing the novel features introduced in recent and future amendments, which may be endowed with applications driven by machine learning (ML). The motivation for developing and building the presented wireless network simulator is motivated by:
i)
The lack of analytical models for capturing next-generation techniques in spatially distributed and/or high-density deployments.
ii)
The lack of next-generation WLAN-oriented simulators.
iii)
The complexity of extending simulators comprising an exhaustive implementation of the physical (PHY) layer.
iv)
The large (or intractable) execution time required by other simulators to simulate high-density deployments.
v)
The need for conveniently incorporating ML-based agents in the simulation tool.
In short, Komondor is designed to efficiently implement new functionalities by relying on flexible and simplified PHY layer dependencies, to be faster than most off-the-shelf simulators, and to provide a gentle learning curve to new users.
The rest of the paper is organized as follows. Section II overviews the related work with respect to the available simulators. Section III describes the Komondor’s architecture and lists its already available functionalities. Then, Section IV presents the validation results of the basic operation of Komondor against the well-known network simulator ns-3 [7] and two analytical models. Section V provides insights on potential use cases of Komondor. Section VI concludes the paper.
II Wireless network simulators
Wireless network simulators can be categorized into continuous-time and discrete-event. On the one hand, continuous-time simulators continuously keep track of the system dynamics by dividing the simulation time into very small periods of time. On the other hand, in discrete-event simulators, events are used to characterize changes in the system. Accordingly, for the latter, events are ordered in time and normally allow running faster simulations than continuous-time simulators. In addition, discrete-event simulators allow tracing events with higher precision.
From the family of discrete-event driven network simulators, only a few ones are publicly available. OMNET++ [8] is a component-based C++ simulation library that is not open-source and is used for modeling communication networks and distributed multiprocessor systems. OPNET [9] is another commercial simulator that allows the integration of external components. NetSim [10] was conceived to provide an accurate simulation model oriented to the world wide web. To that purpose, the simulator was written in Java, which compromises simulation time with programming flexibility. When it comes to open source network simulators, a MATLAB-based link-level simulator was presented in [11] for supporting the IEEE 802.11g/n/ac/ah/af technologies. The ns-2 simulator [12], is another network simulator known for its accuracy and the integration with the network animator. Finally, the ns-3, which was introduced in 2006 to replace the ns-2, presents significant advantages over the ns-2 due to its detailed simulation features, becoming very popular among the research community [13]. Table I highlights in a nutshell the most important characteristics of the overviewed network simulators and Komondor.
Among the family of overviewed discrete-event simulators, we highlight the ns-3 open-source simulator due to its popularity and use it as a baseline for comparing against Komondor. Despite the plethora of features that are supported in ns-3, it has some inherent limitations, such as the high complexity for developing new features/models as an extension of the simulator core. In particular, compatibility with the already existing/supported models is required and must be carefully ensured. For example, beamforming for previous mature amendments (i.e. IEEE 802.11n/ac) is not available yet, owing to the effort required to integrate it. Moreover, the integration of new features strongly depends on the willingness of the community to contribute to the development.
With respect to the IEEE 802.11ax operation – rates and support for information elements are being developed – the implementation is mostly based on the Draft 1.0 [14]. Such a draft dates from 2016 and does not include most of the core IEEE 802.11ax functionalities. At the time of submitting this paper, only the Single-User Protocol Data Unit (SU PPDU) and MIMO with up to four antennas are supported in ns-3, whereas OFDMA and MU-MIMO are not supported in the official distribution [15].
Apart from the official resources, we find few ns-3 works publicly available that support IEEE 802.11ax features, which may (or may not) be integrated in future releases. For example, we highlight the works with regard to the OFDMA that have been carried out by Getachew Redieteab et al. (based on the IEEE 802.11ax specification framework document [16]) and Cisco [17]. However, none of these works completely follow the latest developments in the IEEE 802.11ax standard and have not been validated through extensive simulations and testbed results, as had previously occurred with the OFDM [18]. In addition to OFDMA, the spatial reuse operation (i.e., BSS Color [19]) is under active development, whereas extensions of the capture effect have been applied to ns-3 to follow the IEEE 802.11ax guidelines and studied in [20] and in a testbed [21].
III Komondor Design Principles
III-A Architecture
Komondor aims to realistically capture the operation of WLANs. Henceforth, it reproduces actual transmissions on a per-packet basis. To that purpose, Komondor is based on the COST library, which allows building interactions between components (e.g., wireless nodes, buffers, packets) through synchronous and asynchronous events. While the former are messages explicitly exchanged between components through input/output ports, the latter are based on timers.
In practice, components perform a set of operations until a significant event occurs. For instance, a node that is decreasing its backoff may freeze it when an overlapping node occupies the channel. The beginning and end of such a transmission are examples of significant events, whereas decreasing the backoff counter is not. Nevertheless, events may be triggered by different timers. In the previous example, a node’s transmission begins once the backoff timer terminates (i.e., the backoff timer triggers the beginning of the transmission), while the end of the transmission is triggered by the packet transmission timer. Fig. 2 shows the schematic of a COST component, which is composed by inports, outports, and a set of timers.
III-B IEEE 802.11 Features
Komondor entails a long-term project in which several contributors are involved. That is, the simulator is continuously evolving to include novel techniques and generally improve performance. The current version of Komondor (v2.0) includes the following fully tested IEEE 802.11ax features:
•
Distributed coordination function (DCF): the Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) captures the basic Wi-Fi operation for accessing the channel. Moreover, Contention Window (CW) adaptation is considered.
•
Buffering and packet aggregation: several traffic generator models are implemented in Komondor such as deterministic, Poisson or full-buffer. Besides, multiple media access control protocol data unit (MPDU) can be aggregated into the same PLCP protocol data unit (PPDU) in order to reduce the generated communication overheads.
•
Dynamic channel bonding (DCB): multiple channel widths can be selected during transmissions by implementing DCB policies in order to maximize the spectrum efficiency. Some of these policies were already evaluated in [22, 23].
•
Modulation coding scheme (MCS) selection: the MCS is negotiated between any transmitter-receiver pair according to the Signal-to-Interference-and-Noise Ratio (SINR), thus supporting multiple transmission rates.
•
Ready-to-send/Clear-to-send (RTS/CTS) and Network Allocation Vector (NAV): virtual carrier sensing is implemented in order to minimize the number of collisions by hidden-nodes.
Future development stages are under progress including other features such as OFDMA, MU-MIMO transmissions, beamforming, spatial reuse, and ML-based configuration.
III-C Execution Flowchart
Komondor is composed of several modules that allow performing simulations with a high degree of freedom. Fig. 3 summarizes the operational mode of Komondor from a user’s point of view. A more detailed user’s guide providing a quick-start and guided execution examples is available in [24].
III-C1 Input and Setup/Start
as for the execution console command for starting Komondor simulations, arguments are designed in a simple and efficient way. Examples of console arguments are the file names of the inputs, the activation flags of the logs, the simulation time and the random seed. In addition, input files (in CSV format) are used to define the environment and have been conceived in a way that the user can easily modify important simulation parameters such as the traffic load, the path-loss model, or the data packet size.
Once the environment is generated and nodes are initialized, traffic is exchanged between nodes until the simulation time runs out.
III-C2 Stop and Output
when the simulation finishes, the closing is handled and statistics are gathered. Then, extensive and detailed performance statistics are per default provided by Komondor (e.g., throughput, delay, spectrum utilization, or collisions). Moreover, the user can efficiently include as much as metrics as desired.
III-D States and events
The Komondor’s core operation is based on states, which represent the status (or situation) in which a node can be involved. A state diagram summarizing both states and transitions is shown in Fig. 4. Roughly, a given node starts in the SENSING state, where multiple events can occur (e.g., a new packet is buffered or a new transmission is detected). Then, according to the noticed event, the node transits to the corresponding state.
III-D1 States
we depict below each state and how a node must behave in front of new events.
•
SENSING: a node senses the channel with two main purposes. First, to follow the CSMA/CA operation to gain access to the channel (in case there is backlogged data in the buffer/s). Second, to wait for incoming transmissions, so that either carrier sensing or receiving procedures are held. In case of being immersed in a backoff procedure, a node detecting a “new transmission” event would sense the power received in its primary channel, and assess whether to freeze the backoff countdown. Similarly, whenever an “end transmission” event occurs, the channel is sensed in order to determine whether the backoff counter can be resumed or remain paused.
•
TRANSMIT: a transmitter node is currently transmitting a packet. No matter what events may occur, during the packet transmission, the node blocks its receiver capabilities and remains in the same state until the transmission is finished.
•
RECEIVE: when a node is receiving and decoding an incoming packet, it will behave in front of a new event according to its implication in the channel of interest. Of special importance are those new transmission events triggered by other nodes that have gained access to the medium. Specifically, if a new transmission generates enough interference, the ongoing reception will be discarded, thus leading to a packet loss.
•
WAIT states: these states allow modeling the situations where a node that transmitted a packet is expecting for the corresponding response. Namely, after transmitting RTS, CTS or DATA packets, the transmitter will wait for the corresponding CTS, DATA or ACK/BACK packets, respectively. If the response packet is not received before the corresponding timeout is triggered, the transmitter assumes that either the transmitted packet or the response packet is lost and resets to SENSING state. Wait states are particularly useful to detect packet losses when anomalies in the network (e.g., hidden terminal problem) occur.
•
NAV: when a node enters in NAV state due to a successful reception of an RTS or CTS addressed to a different destination, it sets a NAV timer and keeps listening to its primary channel. If a new RTS or CTS is successfully received during the NAV, the timer is updated, provided that the new NAV time is larger than the current remaining time.
III-D2 Events
each time a node performs an action that can affect the system (e.g., it starts transmitting a frame), an event is announced. Events in Komondor are lined up on the time axis and handled by the core entity. Events management is similar in ns-3. However, the latter exhibits a significant limitation, since events that are scheduled at the exact same time can be executed in any order. Such a development feature may lead to unpredictable results and is incompatible with real-world situations in which events can occur simultaneously. Some inconsistencies may occur in case that the execution order affects to multiple simultaneous events (e.g., two packets arriving at the exact same time). To solve this, Komondor, which is also a discrete-event simulator, employs temporal variables to compare the exact timestamps at which two or more events were generated. As a result, Komondor is able to successfully simulate the behavior of simultaneous events while keeping the logic of the states.
III-E Developing new modules
Komondor has been conceived to be easily modified and extended. In particular, several modules have been provided to represent different simulation capabilities (e.g., propagation, channel access or traffic generation). Accordingly, Komondor can be potentially extended to support the operation of other IEEE 802.11 amendments such as 11n, 11ac, 11ad or 11ay. In addition, ML-based modules can also be introduced. A complete manual can be found at the repository [25].
IV Validation
In this Section we validate the operation of Komondor and show its potential for dealing with high-density scenarios. For the former, we define a set of illustrative scenarios and compare the results with the ones obtained with ns-3.222Details on the ns-3 implementation used in the simulations presented throughout this paper can be found at https://github.com/wn-upf/Komondor/tree/master/Documentation/Validation/ns-3. For instance, this implementation includes the 11ax residential scenario propagation loss [26] and has a PLCP training duration updated according to the 11ax amendment [2]. In addition to ns-3, a mutual validation is performed with the Continuous Time Markov Networks (CTMNs) modeling introduced in [27], and which is extended for spatially distributed networks in the Spatial-Flexible Continuous Time Markov Network (SFCTMN) framework [22]. As for high-density scenarios, we make use of the Bianchi’s DCF analytical model [28] to validate the results in fully-overlapping deployments, where all the nodes are within the carrier sense of the others. The results shown in the following subsections were obtained according to the parameters defined in Table II of the Appendix A. Note that full-buffer traffic is assumed in all the scenarios throughout this work for comparative purposes.
IV-A Analyzing toy Scenarios
Komondor has been conceived as a friendly and ready-to-use wireless network simulator that can be used by researches and teachers to study fundamental networking issues. In particular, scenarios and environment configurations can be conveniently modified through structured input files. The scenarios proposed in this Section are a clear example of toy scenarios where different networking concepts such as flow starvation or additive interference take place. Furthermore, a given user can easily analyze WLAN scenarios through the implemented logs generation system and statistics reporting. Accordingly, particular phenomena in the PHY and medium access control (MAC) layers can be tracked (e.g., channel contention, packet collisions, physical carrier sensing, energy detection, or buffer dynamics).
IV-B Basic Operation
We first aim to validate the basic IEEE 802.11 operation of the DCF implemented in Komondor when RTS/CTS is applied. For that, we consider a single Access Point (AP) scenario (we name it Scenario 1) with one and two stations (STAs), where full-buffer downlink traffic is held. The two-STAs case allows us to assess the proper behavior of Komondor in presence of multiple STAs. To validate this scenario, we compare the Komondor results with the ones provided by ns-3 and the SFCTMN framework. Fig. 5 shows the simulation results obtained from each tool, for packet aggregation ($N_{\text{agg}}=64$) and no-aggregation ($N_{\text{agg}}=1$). We note that the average throughput obtained by each simulation tool is almost identical, either for packet aggregation or not. In addition, having multiple STAs leads to the same result as for a single one since the destination STA is picked at random in every transmission.
IV-C Complex inter-WLAN interactions
In order to validate the behavior of Komondor in front of more complex inter-WLAN interactions, we now focus on the three-WLANs scenarios shown in Fig. 6. We name them Scenario 2a-2d. The interactions occurring in such scenarios are illustrated through CTMNs, where states333Note that CTMN states are not related by any means to Komondor states. represent the WLANs that are currently transmitting. Note that each of these scenarios reflects different situations that are of particular interest since they generalize different well-known phenomena in wireless networks:
•
Fully overlapping (Fig. 5(a)): all the nodes cause contention to all the others when transmitting. For that, the distance between consecutive APs and between AP and STA of the same WLAN is set to $d_{\text{AP,AP}}=d_{\text{AP,STA}}=2$ m, respectively.
•
Flow starvation (Fig. 5(c)): contention is triggered in a pair-wise manner, so that $\text{WLAN}_{\text{A}}$ and $\text{WLAN}_{\text{C}}$ do not interfere each other. For that, the distance is set to $d_{\text{AP,AP}}=4$ m and $d_{\text{AP,STA}}=2$ m. Note that this case could be also extended to show a hidden node effect if $\text{AP}_{\text{A}}$ or $\text{AP}_{\text{C}}$ were intended to transmit to a STA located at the location of $\text{AP}_{\text{B}}$.
•
Potential overlap (Fig. 5(e)):
contention only occurs at $\text{WLAN}_{\text{B}}$ when both $\text{WLAN}_{\text{A}}$ and $\text{WLAN}_{\text{C}}$ transmit concurrently. Otherwise, the channel is sensed as free. Note that, in this case, packets are successfully transmitted in $\text{WLAN}_{\text{B}}$ whenever it access the channel. The distances are $d_{\text{AP,AP}}=5$ m and $d_{\text{AP,STA}}=2$ m for $\text{WLAN}_{\text{A}}$ and $\text{WLAN}_{\text{C}}$, and $d_{\text{AP,STA}}=3$ m for $\text{WLAN}_{\text{B}}$.
•
No overlapping (Fig. 5(g)): none of the nodes causes contention to any other when transmitting. That is, every WLAN operates like in isolation. The distances in this case are $d_{\text{AP,AP}}=10$ m and $d_{\text{AP,STA}}=2$ m.
The average throughput experienced by each WLAN in each scenario is shown in Fig. 7. As previously done, we compare the performance of Komondor with ns-3 and SFCTMN. Note that results gathered by both Komondor and ns-3 are very similar in all the cases. Concerning the differences on the average throughput values estimated by both simulators and SFCTMN, we observe two phenomena with respect to backoff collisions in topologies of Scenario 2a and 2c. First, in 2a, the throughput is slightly higher when the capture effect condition is ensured. This is due to the fact that concurrent transmissions (or backoff collisions) are permitted and captured in the simulators. Second, the most notable difference is given in 2c, which is caused by the assumption of continuous time backoffs in the CTMN. These are clear examples of the limitations of the analytical tool.
IV-D High-density and simulator performance
Finally, we assess the performance of Komondor when dealing with high-density scenarios. Notice that being able to simulate scenarios with a large number of nodes is a key feature due to the ever-increasing trend towards short-range and dense deployments. In this situation, we show the results of different fully-overlapping scenarios, ranging from 1 to 50 WLANs, each consisting in of one AP and one STA. The validation is performed against the Bianchi’s analytical model and ns-3. The MCS for all the WLANs is set to 256-QAM. Fig. 8 shows the results in terms of throughput (average and aggregate) and collision probability obtained for fully overlapping networks of different sizes. For comparison purposes, the simulation time used in each scenario has been set to 100 seconds, for both Komondor and ns-3. As shown, Komondor maintains its accuracy with respect to Bianchi’s model, even when dealing with a lot of nodes. Regarding ns-3, slight differences are noticed in the collisions probability due to the error rate model, where collisions are based on the dropped RTS frames and the use of the Extended Interframe Space (EIFS). Moreover, differences in the throughput increase with the number of nodes, as previously addressed in [29].
To conclude this section, we provide insights into the execution complexity of Komondor. Fig. 9 shows the execution time and the number of generated events in Komondor for each number of WLANs.444Note that the execution time is strongly dependent of the computer used and its status at the moment of performing the simulation. In our case, we used an Intel Core i5-4300U CPU @ 1.9 GHz x 4 and 7.7 GiB memory. Regarding ns-3’s execution complexity, significantly higher results were obtained. For instance, for 16 WLANs (i.e., 32 nodes), the simulation time was around 227 seconds, in contrast to the 11 seconds (95% time reduction) required by Komondor. In addition, the number of generated events in ns-3 was around 23 times higher than in Komondor.
V Komondor and potential use cases
Apart from small deployments consisting of few WLANs under single-channel operation [30], more complex scenarios capturing DCB or high-density scenarios have been already validated and analyzed by using Komondor. In this section, we briefly discuss further potential uses such as the implementation of next-generation WLAN techniques or the inclusion of learning agents to perform efficient spectrum access and spatial reuse.
V-A Potential usage
Complex wireless environments can be already extensively simulated by Komondor as a result of its reduced computational complexity in comparison to other well-known simulators such as ns-3.
A prominent example of a complex scenario mixing both high-density deployments and DCB is discussed in [22], where authors assessed the performance of different DCB policies versus node density (see Fig. 1). In [23], a similar deployment is analyzed while considering different traffic loads.
A set of scenarios including DCB is shown in Fig. 10, which were validated in Komondor’s validation report v0.1.555Komondor’s validation report v0.1: https://github.com/wn-upf/Komondor/blob/master/Documentation/Other/validation_report_v01.pdf. New features like MIMO, beamforming and MU communications through OFDMA and/or MU-MIMO are currently under development.
V-B Machine learning agents
In addition to simulating advanced techniques proposed by the latest IEEE 802.11 amendments, Komondor permits including intelligent agents. In particular, agents are embedded to APs (see Fig. 10(a)) to perform the following operations (see Fig. 10(b)): i) monitor WLAN’s performance, ii) run an implemented learning method, and iii) suggest new configurations to be applied by the WLAN, according to generated knowledge.
The application of intelligent agents has been previously studied in [31, 30], where decentralized learning is employed to both Transmit Power Control (TPC) and Carrier Sense Threshold (CST) adjustment.
VI Conclusions
In this work, we presented Komondor, a wireless network simulator that stems from the need of providing a reliable and low-complexity simulation tool able to capture the operation of novel WLAN mechanisms like DCB or spatial reuse. The operation of Komondor has been validated against the ns-3 simulator and analytical tools such as CTMNs and Bianchi’s DCF model. In this regard, we have shown its effectiveness when dealing with high-density scenarios, thereby outperforming ns-3 with respect to the simulation time. Finally, we have discussed the potential of Komondor regarding complex scenarios and ML integration. In particular, a preliminary ML-based architecture is already implemented, so that intelligent agents can rule self-configuring operations at different communication levels. The Komondor project is expected to move forward and include novel mechanisms such as OFDMA, MU-MIMO, and the spatial reuse operation, naming a few among others.
Appendix A Simulation parameters
Table II describes the PHY and MAC parameters used for simulations. The duration of the RTS, CTS and data frame is computed as follows:
$$\displaystyle T_{\text{RTS}}=T_{\text{PHY-leg}}+\left\lceil\frac{L_{SF}+L_{%
\text{RTS}}}{L_{s,l}}\right\rceil\sigma_{\text{leg}}\text{,}$$
$$\displaystyle T_{\text{CTS}}=T_{\text{PHY-leg}}+\left\lceil\frac{L_{SF}+L_{%
\text{CTS}}}{L_{s,l}}\right\rceil\sigma_{\text{leg}}\text{,}$$
$$\displaystyle T_{\text{D}}=T_{\text{HE-SU}}+\left\lceil\frac{L_{\text{SF}}+L_{%
\text{MH}}+N_{\text{agg}}L_{\text{D}}}{L_{s,l}}\right\rceil\sigma\text{.}$$
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LES of an Inclined Jet into a
Supersonic Turbulent Crossflow
Antonino Ferrante${}^{1}$, Georgios Matheou${}^{2}$, Paul E. Dimotakis${}^{2}$,
Mike Stephens${}^{3}$, Paul Adams${}^{3}$, Richard Walters${}^{3}$
${}^{1}$ Aeronautics & Astronautics,
University of Washington, Seattle, WA 98195, USA
${}^{2}$ Graduate Aeronautical Laboratories,
California Institute of Technology, Pasadena, CA 91125, USA
${}^{3}$ Data Analysis and Assessment Center, U.S. Army
Engineer Research and Development Center, MS 39180, USA
Abstract
This short article describes flow parameters, numerical method, and animations of the fluid dynamics video “LES of an Inclined Jet into a Supersonic Turbulent Crossflow” (high-resolution and
low-resolution video).
We performed large-eddy simulation with the sub-grid scale (LES-SGS) stretched-vortex model of momentum and scalar transport to study the gas-dynamics interactions of a helium inclined round jet into a supersonic ($M=3.6$) turbulent (Re${}_{\theta}$$\ =13\times 10^{3}$) air flow over a flat surface.
The video shows the temporal development of Mach-number and magnitude of density-gradient in the mid-span plane, and isosurface of helium mass-fraction and $\lambda_{2}$ (vortical structures).
The identified vortical structures are sheets, tilted tubes, and discontinuous rings. The vortical structures are shown to be well correlated in space and time with helium mass-fraction isosurface ($Y_{\rm He}=0.25$).
1 Flow parameters
Figure 1 shows the flow schematic. Helium is injected through an inclined round jet into a supersonic turbulent air flow over a flat surface.
In the present investigation, the jet axis forms a 30${}^{\circ}$ angle with the streamwise direction of the air flow.
The flow parameters of air and helium are reported in Table 1.
The jet diameter, $d$, is 3.23$\times 10^{-3}$ m, and the boundary layer thickness, $\delta$, of the air flow is 2$\times 10^{-2}$ m as in the experimental study of Maddalena, Campioli & Schetz (2006). The air free-stream Mach number is 3.6, the jet Mach number is 1.0, and the jet to free-stream momentum ratio, $\overline{q}=(\rho U^{2})_{\rm j}/(\rho U^{2})_{\rm e}$, is 1.75.
The Reynolds number of the air flow based on the momentum thickness is ${\sl Re}_{\theta}=U_{\rm e}\theta/\nu_{\rm w}=13\times 10^{3}$ (${\sl Re}_{\delta}=U_{\rm e}\delta/\nu_{\rm w}=113\times 10^{3}$), where $U_{\rm e}$ is the free-stream air velocity and $\nu_{\rm w}$ is the kinematic viscosity of air computed at the wall for adiabatic wall conditions.
2 Numerical method
Large-eddy simulation was performed using the sub-grid scale (LES-SGS) stretched-vortex model of momentum and scalar transport developed by Pullin and co-workers Misra & Pullin (1997); Voelkl et al. (2000); Pullin (2000); Kosović et al. (2002).
A hybrid numerical approach Hill & Pullin (2004); Pantano et al. (2007) with low numerical dissipation that uses tuned centered finite differences (TCD) in smooth flow regions, and a weighted essentially non-oscillatory (WENO) scheme Liu et al. (1994); Jiang & Shu (1996) around discontinuities and ghost-fluid boundaries is employed.
The level-set approach with the ghost-fluid method Fedkiw et al. (1999) is used to treat the complex boundary (Fig. 1) where no-slip and adiabatic boundary conditions are applied.
The framework for block-structured Adaptive Mesh Refinement in Object-oriented C++ (AMROC) by Deiterding (2003) was adopted.
The computational domain is a parallelepiped with sides $-0.025\leavevmode\nobreak\ \mbox{m}\leq x\leq 0.0774$ m, $-0.006\leavevmode\nobreak\ \mbox{m}\leq y\leq 0.0452$ m, and $-0.0256\leavevmode\nobreak\ \mbox{m}\leq z\leq 0.0256$ m (i.e., $-7.74\leq x/d\leq 24$, $-1.86\leq y/d\leq 14$, and $-7.92\leq z/d\leq 7.92$), in the streamwise, wall-normal and spanwise directions, respectively. The flat-wall boundary is the $y=0$ plane and the center of the jet-exit plane is at the axes origin (Fig. 1).
The basic mesh is $256\times 128\times 128=4.2\times 10^{6}$ grid points with a base mesh spacing of $\Delta x_{\rm b}=4\times 10^{-4}$ m. The dynamic mesh refinement (AMR=2) adds one level of finer mesh on the coarse mesh, increasing the effective total number of mesh points to approximately $16\times 10^{6}$ with a fine-mesh spacing of $\Delta x_{\rm f}=2\times 10^{-4}$ m. The use of AMR=2 saves about half grid points with respect to the fully refined mesh of $33\times 10^{6}$ grid points.
The Kolmogorov length-scale, $\eta$, of the shear layer created between the high-speed helium jet and low-speed air stream is of about 1.5 $\mu$m, i.e., $\eta$ is of $O(100)$ times smaller than the fine-mesh spacing. The SGS-TKE is less than 20% of the total TKE in most of the flow field . Thus, the resolution criterion for a sufficiently resolved LES-SGS Pope (2004) is satisfied.
2.1 Inflow conditions
The transition and spatial development of the helium jet were found to be strongly dependent on the inflow conditions of the crossflow Ferrante et al. (2009-1511). These results indicate that correct turbulent inflow conditions are necessary to predict the main flow characteristics, dispersion and mixing of a gaseous jet in a supersonic turbulent crossflow.
A methodology for the generation of synthetic turbulent inflow conditions for LES of spatially developing, supersonic, turbulent wall-bounded flows has been developed by Ferrante et al. (2010).
A brief description is here given. Inflow conditions are computed as the sum of zero-pressure gradient, compressible turbulent-boundary-layer (ZPG-CTBL) mean profiles and turbulence fluctuations. First, the friction velocity, $u_{\tau}$, and velocity profile von Kármán (1930) of the incompressible ZPG-TBL are computed by prescribing the Reynolds number based on boundary-layer thickness, $Re_{\delta}$ Ferrante & Elghobashi (2004).
The resulting velocity profile is then transformed into the velocity profile of the ZPG-CTBL at $M_{\rm e}=3.6$ according to van Driest van Driest (1951); Smits & Dussauge (2006); Knight (2006-498).
Last, the temperature profile of the ZPG-CTBL is computed using the Walz formula Walz (1969) assuming an adiabatic wall.
Inflow turbulence fluctuations are generated by modifying the methodology of Ferrante & Elghobashi (2004) to supersonic flows at high-Reynolds number.
A model spectrum Pope (2000)of turbulence kinetic energy, $E(k)$, and the Reynolds stresses, $\left\langle u_{i}u_{j}\right\rangle^{+}$, are prescribed at the inflow plane.
The Reynolds stresses at $M_{\rm e}=3.6$ are obtained by scaling the incompressible Reynolds stresses of a ZPG-TBL at Re${}_{\theta}$=2900 obtained via DNS Ferrante & Elghobashi (2005) with the theoretical local profile $\rho/\rho_{\rm w}$.
Such scaling is justified by the DNS results of Guarini et al. (2000).
3 Animations
The fluid dynamics video “LES of an Inclined Jet into a Supersonic Turbulent Crossflow” (high-resolution and
low-resolution video) shows five animations (A1 to A5)111Each animation lasts about 23 s and shows 30 frames/s for a total of 696 frames. The temporal development of the flow is shown for about $10^{-3}$ s. Thus, the flow is shown 23,000 times slower than at its actual speed.
:
1.
Mach-number, $M$, contours in the mid-span plane;
2.
contours of density-gradient magnitude, $|\nabla\rho|$, in the mid-span plane;
3.
isosurface of helium mass-fraction, $Y_{\rm He}=0.25$;
4.
isosurface of $\lambda_{2}$ (vortical structures)222The vortical structures are educed using the $\lambda_{2}$-method Jeong & Hussain (1995), where $\lambda_{2}$ is defined as the second largest eigenvalue of the tensor $(S_{ik}S_{kj}+\Omega_{ik}\Omega_{kj})$, where $S_{ij}\equiv(\partial_{j}U_{i}+\partial_{i}U_{j})/2$ is the strain rate tensor, and $\Omega_{ij}\equiv(\partial_{j}U_{i}-\partial_{i}U_{j})/2$ is the rotation rate tensor.;
5.
overlapped isosurface of helium mass-fraction $Y_{\rm He}=0.25$ (yellow), and vortical structures (blue).
The Mach-number contours (A1) show that boundary-layer turbulence/bow-shock interaction results in shock-wave unsteadiness that affects the roll-up of the shear-layer (formed between air stream and helium jet) and, consequently, modulates in time the size and shape of the barrel-shock (black region near the jet exit).
The contours of the density-gradient magnitude (A2) show the bow-shock and the shear-layer formed in between the air stream after the bow-shock and the expanded helium jet. Both contours show large-scale structures advected downstream. Jet unsteadiness, lateral and wall-normal helium dispersion, and three-dimensional structure of the helium jet are shown in the animation of $Y_{\rm He}=0.25$ isosurface (A3).
The vortical structures (isosurface of $\lambda_{2}$ shown in A4) are sheets near the jet exit where the shear formed between the air-stream and the helium jet is large. Downstream the jet exit, the vortical structures are mostly tilted tubes which sometimes look like discontinuous rings.
In A5, the isosurface of $Y_{\rm He}=0.25$ (yellow) mostly envelopes the isosurface of $\lambda_{2}$ (blue). The two isosurfaces, showing helium-jet puffs and vortical structures, are well correlated in space and time. The vortical structures look like muscles that move the isosurface of helium mass-fraction, contributing to helium dispersion and helium mass-fraction convoluted isosurface.
Acknowledgments
This work was supported by AFOSR Grants FA9550-04-1-0020 and FA9550-04-1-0389, and by Caltech funds. The authors would like to thank Dr. C. Pantano for support with the AMROC software, the Center of Advanced Computing Research (CACR) at Caltech for computing time.
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Prophet Inequalities with Linear Correlations
and Augmentations
Nicole Immorlica
([email protected])
Microsoft Research.
Sahil Singla
([email protected])
Princeton University and
Institute for Advanced Study. Supported in part by the
Schmidt Foundation.
Bo Waggoner
([email protected])
University of Colorado.
( January 16, 2021)
Abstract
In a classical online decision problem, a decision-maker who is trying
to maximize her value inspects a sequence of arriving items to learn
their values (drawn from known distributions), and decides when to
stop the process by taking the current item. The goal is to prove a
“prophet inequality”: that she can do approximately as well as a
prophet with foreknowledge of all the values. In this work, we
investigate this problem when the values are allowed to be correlated.
Since non-trivial guarantees are impossible for arbitrary correlations,
we consider a natural “linear” correlation structure introduced by Bateni et al. [BDHS15] as a generalization of the common-base value model of Chawla et al. [CMS15].
A key challenge is that
threshold-based algorithms, which are commonly used for prophet
inequalities, no longer guarantee good performance for linear correlations.
We relate this
roadblock to another “augmentations” challenge that might be of
independent interest: many existing prophet inequality algorithms are not robust to
slight increase in the values of the arriving items.
We leverage this intuition to prove bounds
(matching up to constant factors) that decay gracefully with the
amount of correlation of the arriving items.
We extend these results to the case of selecting multiple items
by designing a new $(1+o(1))$ approximation ratio algorithm that is robust
to augmentations.
\setitemize
itemsep=2pt,topsep=0pt,parsep=0pt
\setstretch1.1
Contents
1 Introduction
1.1 Linear Correlations Model
1.2 Techniques and the Augmented Prophets Problem
1.3 Related Work
2 Model and Fixed-Threshold Algorithms
2.1 Model and Notation
2.2 Fixed-Threshold Algorithms
3 Our Approach of Handling Correlations via Augmentations
4 Selecting a Single Item
4.1 Bounded Column Sparsity
4.2 Bounded Row Sparsity
4.3 Lower Bounds
5 Selecting Multiple Items
5.1 Bounded Column Sparsity
5.2 Bounded Row Sparsity
5.3 Multiple-Items Augmentation Lemma
A Unweighted Linear Correlations
A.1 Notation and Proof Overview
A.2 Proof
B Negatively Correlated Values
C Bounded $s_{\text{col}}$ and Small Cardinality Constraint
D Missing Proofs
D.1 Missing Proofs from Section 2
D.2 Missing Proofs from Section 5.3
1 Introduction
In classic optimal-stopping problems, a decision-maker wishes to select one of a set $[n]=\{1,\dots,n\}$ of options whose values are distributed according to a known joint distribution. Option $i$ materializes at time $i$, revealing its value $X_{i}$. The decision-maker must then either select option $i$, receiving a value of $X_{i}$, or permanently reject it and continue. Her goal is to choose an option whose value, in expectation over her selection algorithm and the randomness in the problem instance, obtains at least a $1/\alpha$-fraction of the expected maximum value for some approximation ratio $\alpha\geq 1$. Such an approximation is referred to as a “prophet inequality” as it compares the decision-maker’s performance to that of a prophet who knows the realizations of all values in advance and can always stop at the maximum. Examples of optimal-stopping problems include hiring, in which an employer interviews a sequence of candidates and must make a hiring decision on the spot; or house-buying in a sellers’ market, in which a buyer must make an offer at the open house.
These optimal-stopping problems became more popular in the last 15 years particularly because of their applications in mechanism design. E.g., an $\alpha$-prophet inequality often implies a posted-pricing mechanism that gets a $1/\alpha$-fraction of the maximum welfare for a sequence of bidders arriving online (see related work in Section 1.3).
The approximability of optimal stopping problems depends heavily on the correlations of the option values. In the case where all the values are independent, a tight ${2}$ approximation ratio was shown by Krengel and Sucheston [KS78]. In 1984, Samuel-Cahn [SC84] presented a simple median-of-maximum “threshold-based” rule with the same performance: compute $\tau$ as the median of the distribution of the maximum value, and stop at the first $X_{i}$ exceeding $\tau$. Other threshold rules are also known to obtain ${2}$ approximation, e.g., Kleinberg and Weinberg [KW12] showed this for $\tau=\tfrac{1}{2}\mathop{\mathbb{E}}\displaylimits\left[\max_{i}X_{i}\right]$.
When the values are negatively correlated, the problem intuitively becomes even easier than the independent case: observing and rejecting a low $X_{i}$ increases the chances of seeing a high $X_{i^{\prime}}$ in the future (and vice versa accepting a large $X_{i}$ decreases the chances of having missed a high $X_{i^{\prime}}$ in the future).
A simple implication of threshold-based algorithms is a ${2}$ approximation ratio for negatively correlated values (see Appendix B).
Rinott and Samuel-Cahn [RSC${}^{+}$87, RSC91] indeed show that the value of the optimal stopping algorithm for negative correlations exceeds that of the independent case, holding the marginals fixed.
On the other hand, with general and positive correlation structures, no algorithm can guarantee better than $\Omega({n})$ approximation ratio (as is known from Hill and Kertz [HK92], and will also be a special case of our lower bounds). Therefore, the question is how to impose a structure on the correlations that both models interesting applications and allows for better bounds.
1.1 Linear Correlations Model
We consider a linear correlations model in which there exists a set of $m$ independent variables $Y_{1},\dots,Y_{m}$, with each option value $X_{i}$ being a positive linear combination of some subset:
$$\textbf{X}=A\cdot\textbf{Y}$$
where $A$ is a nonnegative matrix. The algorithm is given the matrix $A$ and the distributions of all the $Y_{j}$s, but when $X_{i}$ arrives, it only finds $X_{i}$ and not any of the realizations of the $Y_{j}$s.
This model was introduced in an auctions context by Bateni et al. [BDHS15], where it was inspired by the common-base value model of Chawla et al. [CMS15].
It has two natural parameters capturing the degree of correlation of an instance. If each row of $A$ has at most $s_{\text{row}}$ nonzero entries (row sparsity $s_{\text{row}}$), this implies that each $X_{i}$ only depends on at most $s_{\text{row}}$ different $Y_{j}$s. On the other hand, if each column of $A$ has at most $s_{\text{col}}$ nonzero entries (column sparsity $s_{\text{col}}$), this implies that each $Y_{j}$ is only relevant to at most $s_{\text{col}}$ different $X_{i}$s.
General applications.
Linear correlations occur in applications when each option $i$ is defined by the degree $A_{ij}$ to which it exhibits each feature $j\in[m]$. The value $\sum_{j}A_{ij}Y_{j}$ of the option is then determined by the values $Y_{j}$ of its features, which is unknown to the decisionmaker. Of particular interest in our setting are applications with many features, such as the hiring and house-buying examples often used to motivate prophet problems. Other relevant applications include selecting hotel rooms, restaurants, and movies. Here we elaborate on the hiring and house-buying examples, noting how they naturally exhibit column or row sparsity.111Note however that our bounds are expressed in terms of the minimum of row and column sparsity of an instance, and hence apply even to instances with high row/column sparsity.
In a hiring application, the features of a candidate might include where he received his education, his major, his work experience in each relevant industry, aspects of his personality, etc. When the employer interviews a candidate, she learns how much she likes him, but not how to attribute her value for the candidate to particular features (every school/major/industry is a different feature). If the candidate pool is diverse, so that candidates come from many different schools/majors/industries, we might expect the instance to have a low column sparsity. Similarly, in house-buying, the features of a house might include the commute time, the square footage, and various bells-and-whistles like the existence of a patio, a hot-tub, a roof-deck, the number of parking spaces, if any. Again, the value of a house is a linear combination of the value of its features, but when seeing a house, the buyer may only be able to access and articulate an overall valuation. If each house has a limited number of bells-and-whistles, we expect the instance to have low row-sparsity.
Mechanism design applications: Welfare for interdependent values.
Prophet inequalities can directly imply social welfare and revenue guarantees for sequential posted-price mechanisms [CHMS10, KW12].
In the simplest model, a single item is for sale to a sequence of arriving bidders with values $X_{1},\dots,X_{n}$, drawn from distributions known to the seller.
A threshold-$\tau$ stopping rule immediately translates to a posted price $\tau$.
The item is purchased by the first bidder whose value satisfies $X_{i}\geq\tau$.
In particular, social welfare is the value of the bidder who purchases the item, so a prophet inequality directly translates to a social welfare guarantee.222Revenue guarantees, at least in the classical independent-$X_{i}$ model, can be obtained using a threshold in virtual value space.
While we show that threshold-based policies fail for linearly correlated values, we obtain positive results with inclusion-threshold policies.
These correspond to offering a fixed posted price to a predetermined subset of buyers, while the others are automatically rejected.
For linearly correlated bidder values, our positive results immediately imply social welfare guarantees using such inclusion-posted-price mechanisms.
Here, linear correlations capture some component of common values in bidder preferences.
Namely, there are different features $Y_{1},\dots,Y_{m}$ of the object, with bidder $i$ placing weight $A_{ij}$ on feature $j$.
In the mechanism-design setting, it is particularly natural to make our assumption that the decisionmaker (here, the seller) is not able to access $Y_{ij}$ when value $X_{i}$ arrives.
Results.
We start from the observation that threshold-based algorithms cannot guarantee good approximation ratios as soon as any correlations are introduced.
Therefore, we define inclusion-threshold algorithms that probabilistically include a subset of the arrivals for consideration and take the first arrival in this subset to exceed a threshold.
We first design an inclusion-threshold algorithm to obtain an $O(s_{\text{col}})$ approximation ratio, i.e., a guarantee that degrades gracefully as the amount of correlation increases, from the known $O(1)$ bound for the independent case to the known $\Theta(n)$ worst-case bound.
Then, we design a more complex inclusion-threshold algorithm to obtain an $O({s_{\text{row}}})$ approximation ratio, i.e., another gracefully degrading guarantee.
Together, these prove an $O\big{(}{\min\left\{{s_{\text{row}}},{s_{\text{col}}}\right\}}\big{)}$ approximation guarantee for the linear correlations model.
We then design a lower bound instance and prove that this is tight up to constants, i.e., no algorithm can guarantee better than an $\Omega\big{(}{\min\left\{{s_{\text{row}}},{s_{\text{col}}}\right\}}\big{)}$ approximation.
Theorem 1.1 (Informal Theorem 4.1).
For the linearly correlated prophet problem, there exists an $O({\min\{{s_{\text{col}}},{s_{\text{row}}}\}})$ approximation ratio algorithm.
Finally, we extend these results to the case of selecting a subset of up to $r$ of the arriving options with a goal of maximizing their expected sum (also known as an $r$-uniform matroid constraint). It is known that for independent distributions, $1+o(1)$ approximation ratio prophet inequality algorithms are possible for the case of large $r$ [HKS07, Ala11]. We show a similar result for linearly correlated instances with bounded column sparsity $s_{\text{col}}$.
Theorem 1.2 (Informal Theorem 5.1).
For the linearly correlated prophet problem where we select $r$ options, there exists a $1+o(1)$ approximation ratio algorithm when $r\gg s_{\text{col}}$.
The case of bounded row sparsity, however, turns out to be harder: regardless of $r$, no algorithm can guarantee better than an $\Omega({s_{\text{row}}})$ approximation ratio for unbounded $s_{\text{col}}$, as in the $r=1$ case.
1.2 Techniques and the Augmented Prophets Problem
A crucial technique for our results is to introduce and solve the augmented prophets problem.
The idea is to suppose we have an instance with independent random variables $Z_{1},\dots,Z_{n}$ and an algorithm, say a threshold rule, guaranteeing some approximation ratio.
Now suppose we “augment” the instance by sending instead the values $X_{1}:=Z_{1}+W_{1},~{}\dots,~{}X_{n}:=Z_{n}+W_{n}$ where the $W_{i}$s are nonnegative.
Does the algorithm (which does not know $W_{i}$s) continue to guarantee its original approximation ratio (measured against the maximum $Z_{i}$)?
One would hope so, as each arriving option has only increased, while the benchmark has not.
However, this turns out not to be true for the median-of-maximum threshold rule.
E.g., if $Z_{i}\sim\text{Bernoulli}(p)$ i.i.d. for $p\ll\tfrac{1}{n}$, the median is zero, and augmenting the first arrival to a miniscule positive value causes the (strict) median threshold rule to always take it, resulting in an arbitrarily poor approximation.
Luckily, we show that the half-of-expected-maximum threshold algorithm retains its approximation guarantees, even when the $W_{i}$s are chosen by an adversary depending on the past $X_{i^{\prime}}$s, i.e., $i^{\prime}<i$.
Armed with this “augmentation lemma”, we use subsampling to obtain inclusion sets of arrivals $\{X_{i}\}$ with significant independent portions $\{Z_{i}\}$.
In bounded $s_{\text{col}}$ case, direct subsampling of arrivals succeeds.
The case where $s_{\text{row}}$ is bounded but $Y_{j}$ can appear in any number of arrivals is more challenging.
We show it suffices to obtain a contention-resolution style subsampling of the arrivals such that each $Y_{j}$ is well-represented, but only with its maximum coefficient $A_{ij}$.
We then use a graph-theoretic argument to construct such a scheme.
The augmented prophets problem is also our key technique for the linearly correlated prophets problem with an $r$-uniform matroid constraint. In this case, however, we notice that none of the existing $1+o(1)$ algorithms are robust to augmentations.
Hence we design a new $1+o(1)$ algorithm and prove its robustness using a much more sophisticated analysis involving a sequence of thresholds and “buckets” with different cardinality constraints.
By combining this augmentation result with random partitioning of the input, we obtain the $1+o(1)$ approximation for the $r$-uniform matroid problem with fixed $s_{\text{col}}$.
Theorem 1.3 (Informal Lemma 3.2 and 3.4).
For the augmented prophets problem, there exists a ${2}$ approximation ratio algorithm when selecting a single option and a $1+o(1)$ approximation ratio when selecting $r\gg 1$ options.
One can also view the augmented prophets problem as capturing correlations induced by a mischievous wish-granting genie who awards bonuses $W_{i}\geq 0$ at each step, but tries to choose them so as to worsen the algorithm’s performance. We think this problem is of independent interest and will find further applications in designing robust prophet inequality algorithms.
1.3 Related Work
The last decade has seen significant interest in prophet inequalities motivated by their applications in mechanism design.
Many works focus on multiple-choice prophet inequality problems.
This includes prophet inequalities for uniform matroids in [HKS07, Ala11], for general matroids in [CHMS10, Yan11, KW12, FSZ16, LS18], for matchings and combinatorial auctions in [AHL12, FGL15, DFKL17, EHKS18], and for arbitrary packing constraints in [Rub16, RS17]. There has also been a lot of work on variants of prophet inequalities: the prophet secretary problem where the values arrive in a random order [EHLM17, AEE${}^{+}$17, CFH${}^{+}$17, EHKS18, ACK18, LS18, CSZ19], and the limited information setting where we only have sample-access to distributions [AKW, CDFS19, RWW19]. All these work assume mutually independent values, whereas capturing correlations and designing robust algorithms is the main challenge in our work.
Rinott and Samuel-Cahn [RSC${}^{+}$87, RSC91, RSC92] study correlated prophet inequalities. However, their techniques are not applicable to our work because their positive results hold only for negatively correlated values. Furthermore, their benchmark is the expected maximum of independent values having the same marginal distributions. This benchmark could be a factor $n$ larger than the expected maximum for positively correlated values.
Our approach via the augmented prophets problem is also related to the line of work on designing robust stochastic optimization algorithms. Since algorithms that assume known input distributions tend to over-fit, here the goal is to design algorithms that are robust to adversarial noise (see [Dia18, Moi18, DKK${}^{+}$16, LRV16, CSV17, DKK${}^{+}$18, EKM18, LMPL18, BGSZ20] and references therein). Our single-item and multiple-items augmentation algorithms can be seen as robust prophet inequality algorithms that retain their guarantees even when the input distributions are augmented by an adversary. Another relevant reference is that of Dütting and Kesselheim [DK19], which gives prophet inequalities assuming only probability distributions that are $\epsilon$-close (in some metric) to the true distributions. Their technical results, however, are not useful here because augmented distributions can be very far from the original distributions.
2 Model and Fixed-Threshold Algorithms
2.1 Model and Notation
In the linear correlations model, there are $n$ random variables $X_{1},X_{2},\ldots,X_{n}$ that linearly depend on $m$ independent nonnegative random variables (sometimes called features) $Y_{1},Y_{2},\ldots,Y_{m}$ as
$$\textbf{X}=A\cdot\textbf{Y},$$
where matrix $A$ only contains non-negative entries. Let $s_{\text{row}}$ denote the row-sparsity of $A$ (maximum number of nonzero entries in any row) and $s_{\text{col}}$ denote the column-sparsity of $A$ (maximum number of nonzero entries in any column).
An online algorithm is initially given $A$ and the distributions of $Y_{1},\dots,Y_{m}$.
Then, it observes the realizations of $X_{1},\dots,X_{n}$ one at a time.
After observing $X_{i}$, the algorithm decides either to stop and take the reward $X_{i}$, ending the process, or to reject $X_{i}$ and continue to $X_{i+1}$.
Given such an algorithm ALG, we abuse notation by writing ALG for the reward of the algorithm, a random variable.
The algorithm has an approximation ratio of $\alpha$ for some $\alpha(n,s_{\text{row}},s_{\text{col}})$ if for all $n,s_{\text{row}},s_{\text{col}}$ and all instances of the problem with these parameters,
$$\mathbb{E}[\text{ALG}]\geq\frac{1}{\alpha(n,s_{\text{row}},s_{\text{col}})}%
\cdot\mathbb{E}[\max_{i}X_{i}].$$
Such a guarantee is often called a prophet inequality because it compares the algorithm to a “prophet” that can predict the realizations of all $X_{i}$ in advance and take $\max_{i}\{X_{i}\}$ every time.
We use the notation $(\cdot)^{+}$ to mean $\max\{\cdot,0\}$.
Our examples frequently use random variables that are either zero or some fixed positive value.
We say the variable is active if it takes its positive value.
We sometimes say that $X_{i}$ “includes” $Y_{j}$ if $A_{ij}>0$.
2.2 Fixed-Threshold Algorithms
A fixed-threshold algorithm selects a single threshold $\tau$ and takes the first arrival $X_{i}$ that exceeds $\tau$.
We refer to such an algorithm as $\text{ALG}_{\tau}$.
Fixed-threshold algorithms have been very successful in prophet inequality design.
However, our first result shows their severe limitation for even mildly correlated prophet inequalities.
Lemma 2.1.
In the linear correlations model, even for $s_{\text{row}}=s_{\text{col}}=2$ there exist instances where every fixed threshold $\tau$ algorithm $\text{ALG}_{\tau}$ has an approximation ratio at least $\Omega({n})$.
The full proof is deferred to Appendix D, but the instance is important and described next.
Intuitively, the problem is that cases where an arrival $X_{i}$ just crosses the threshold may be correlated with some later $X_{i^{\prime}}$ being very large.
Taking $X_{i}$ prevents the algorithm from ever obtaining the gains from $X_{i^{\prime}}$.
Our proof uses the following “tower” variables, which will also be useful later.
Definition 2.2 (Tower $Y$ variables).
Given $\epsilon>0$, define the tower Y variables as $Y_{i}=\frac{1}{\epsilon^{i}}$ with probability $\epsilon^{i}$ and $Y_{i}=0$ otherwise for each $i\in\{1,\dots,m\}$.
Example 2.3 ($s_{\text{row}}=s_{\text{col}}=2$ tower instance).
Take the tower Y variables.
Let $A$ be an $n\times n$ matrix with entry $A_{i,i}=1$ for all $i$ and $A_{i,i+1}=\epsilon$ for $i\in[1,n-1]$. All other entries are $0$. Visually,
$$\displaystyle X_{1}=Y_{1}+\epsilon Y_{2},\qquad X_{2}=Y_{2}+\epsilon Y_{3},%
\qquad\ldots,\qquad X_{n}=Y_{n}.$$
We have $s_{\text{col}}=s_{\text{row}}=2$. The point is that if $X_{i}$ is nonzero, then almost certainly $X_{i}=\frac{1}{\epsilon^{i}}$.
But in this case, a threshold algorithm cannot distinguish between the more likely case that $Y_{i}=\frac{1}{\epsilon^{i}}$, in which case it should stop and take $X_{i}$, and the unlikely case that $Y_{i}=0$ and $Y_{i+1}=\frac{1}{\epsilon^{i+1}}$, in which case it should wait and take $X_{i+1}$.
Indeed, these coefficients of matrix $A$ play an important role, and in Appendix A we show that when entries of $A$ are restricted to being only $0$ or $1$, there exists a constant-factor approximation fixed-threshold algorithm.
This raises the question whether for a general matrix $A$ any policy can achieve a better approximation, let alone a simple policy.
Our positive results will show that relatively simple inclusion-threshold algorithms can achieve tight prophet inequalities.
Definition 2.4.
An inclusion-threshold algorithm selects a subset $S\subseteq\{1,\dots,n\}$ and threshold $\tau$, possibly both at random, and selects the first $X_{i}$ such that $i\in S$ and $X_{i}\geq\tau$.
In other words, it commits to a subset $S$ of arrivals and applies a threshold policy to those $X_{i}$, ignoring the others.
3 Our Approach of Handling Correlations via Augmentations
In analysis of prophet inequalities, the problem is to upper-bound the expected maximum of the variables $X_{i}$ as compared to one’s algorithm.
An important and common approach is to use the fact that for any threshold $\tau$,
$$\displaystyle\mathbb{E}[\max_{i}X_{i}]\quad\leq\quad\tau+\mathbb{E}[\max_{i}%
\left(X_{i}-\tau\right)^{+}]\quad\leq\quad\tau+\sum_{i}\mathbb{E}[\left(X_{i}-%
\tau\right)^{+}].$$
(1)
When all the arrivals $X_{i}$ are independent, it is known that one can always select $\tau$ such that the left and right sides of (1) differ by at most a constant factor, i.e., $\frac{e}{e-1}\approx 1.6$ (this is related to the correlation gap [ADSY12]).
In fact, the prototypical prophets analysis shows that setting some threshold $\tau$ allows $\text{ALG}_{\tau}$ to approximate the right hand side up to a constant factor.
However, when $\{X_{i}\}$ are correlated, this could be a very loose upper bound. E.g., consider $X_{1}=\cdots=X_{n}=Y_{1}\sim\text{Bernoulli}(p)$.
The left side equals $p$ while the right side equals $\tau+np(1-\tau)=np+\tau(1-np)\geq np$ for $p<\frac{1}{n}$.
So the right side can be a factor $n$ larger than the left, and we cannot hope to approximate the right side with any algorithm.
One approach to correlated prophets could be a direct analysis of the right-hand side of (1) in cases of limited correlation.
Here, we take a different approach.
The first key idea is to use inclusion-threshold algorithms.
To see why, consider a first attempt: discard certain $X_{i}$ such that we are only left with a subset that are all independent of each other.
Now a standard prophet algorithm that only considers these $X_{i}$ would obtain a constant factor of the maximum in this subset.
One could then hope to argue that this subset’s maximum approximates the original maximum up to a factor depending on the amount of correlation.
Indeed, one can show that this approach succeeds on the tower instance in Example 2.3 with $s_{\text{row}}=s_{\text{col}}=2$, e.g., by including every other $X_{i}$.
But in general this approach cannot give tight bounds. This is because
each $X_{i}$ contains $s_{\text{row}}$ variables, each of which can appear in up to $s_{\text{col}}-1$ other $X_{i^{\prime}}$, so including $X_{i}$ requires eliminating $\approx s_{\text{row}}s_{\text{col}}$ other variables.
Our goal is to achieve approximations to within $\min\{s_{\text{row}},s_{\text{col}}\}$ factors even if $\max\{s_{\text{row}},s_{\text{col}}\}=n$.
So in addition to including only a subset of $X_{i}$, we will use a second key idea: decompose each variable as $X_{i}=Z_{i}+W_{i}$, where the $Z_{i}$s satisfy independence requirements and $W_{i}$s are viewed as “bonus” augmentations.
We will show that $\mathbb{E}[\max_{i}Z_{i}]$ is an approximation to $\mathbb{E}[\max_{i}X_{i}]$.
Then we will compete with $\mathbb{E}[\max_{i}Z_{i}]$.
However, the augmentations add an additional challenge, requiring us to solve the following problem.
Definition 3.1 (Single-Item Augmented Prophets Problem).
The algorithm is given the distributions of a set of independent nonnegative random variables $Z_{1},\dots,Z_{n}$.
Then, it observes one at a time the realizations of $X_{i}=Z_{i}+W_{i}$ for $i\in[1,n]$, where each $W_{i}$ is nonnegative and satisfies that $Z_{i}$ is independent of $X_{1},\dots,X_{i-1}$ for each $i$.
The algorithm chooses at each step whether to continue or stop and obtain value $X_{i}$.
It must compete with $\mathbb{E}[\max_{i}Z_{i}]$.
One can view the augmented prophets problem as capturing correlations induced by a mischievous genie who awards bonuses $W_{i}\geq 0$ at each step so as to worsen the algorithm’s performance.
We note that the genie cannot base her choices on the future, i.e., $W_{i}$ is a random variable that may be correlated with $Z_{i^{\prime}}$ if $i^{\prime}\leq i$ but not if $i^{\prime}>i$.
Intuitively, it might seem like algorithms for prophets problems should continue to perform well, as the genie can only increase the rewards at each step.
However, this is not true for the classical median stopping rule, i.e, for $\tau=$ the median of $\max_{i}Z_{i}$ (see an example in Section 1.2).
Luckily, a different threshold rule is robust:
Lemma 3.2 (Single-item Augmentation lemma).
For the augmented prophets problem, a fixed threshold algorithm with $\tau=\frac{1}{2}\mathbb{E}[\max_{i}Z_{i}]$ guarantees
$\frac{\mathbb{E}[\text{ALG}_{\tau}]}{\mathbb{E}[\max_{i}Z_{i}]}\geq\frac{1}{2}$.
Proof.
We “augment” a standard prophet inequality proof.
Let $P=\Pr[\max_{i}X_{i}\geq\tau]$. Now,
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]$$
$$\displaystyle=P\cdot\tau+\sum_{i}\Pr[X_{i^{\prime}}<\tau~{}(\forall i^{\prime}%
<i)]\cdot\mathbb{E}\big{[}(X_{i}-\tau)^{+}\mid X_{i^{\prime}}<\tau~{}(\forall i%
^{\prime}<i)\big{]}$$
$$\displaystyle\geq P\cdot\tau+\sum_{i}(1-P)\cdot\mathbb{E}\big{[}(X_{i}-\tau)^{%
+}\mid X_{i^{\prime}}<\tau~{}(\forall i^{\prime}<i)\big{]}$$
because $\text{ALG}_{\tau}$ selects no element with probability $1-P$.
Nonnegativity of $W_{i}$ implies $X_{i}\geq Z_{i}$, so
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]$$
$$\displaystyle\geq P\cdot\tau+\sum_{i}(1-P)\cdot\mathbb{E}\big{[}(Z_{i}-\tau)^{%
+}\mid X_{i^{\prime}}<\tau~{}(\forall i^{\prime}<i)\big{]}$$
$$\displaystyle=P\cdot\tau+(1-P)\cdot\mathbb{E}\Big{[}\sum_{i}(Z_{i}-\tau)^{+}%
\Big{]}$$
because $Z_{i}$ is independent of the event $\{X_{i^{\prime}}<\tau~{}(\forall i^{\prime}<i)\}$. Since $\sum_{i}(Z_{i}-\tau)^{+}\geq\max_{i}(Z_{i}-\tau)^{+}$,
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]$$
$$\displaystyle\geq P\cdot\tau+(1-P)\cdot\mathbb{E}\big{[}\max_{i}(Z_{i}-\tau)^{%
+}\big{]}$$
$$\displaystyle\geq P\cdot\tau+(1-P)\cdot\mathbb{E}\big{[}\max_{i}Z_{i}-\tau\big%
{]}\quad=\quad P\cdot\tau+(1-P)\tau\quad=\quad\tau.$$
This proves that $\mathbb{E}[\text{ALG}_{\tau}]\geq\frac{1}{2}\mathbb{E}[\max_{i}Z_{i}]$, as claimed.
∎
Multiple Items.
The key idea in proving our $1+o(1)$ approximation ratio result for selecting multiple items for bounded $s_{\text{col}}$ is to extend the augmentation lemma to cardinality constraints.
Definition 3.3 (Multiple-Items Augmented Prophets Problem).
In the augmented prophets problem with cardinality constraint $r$, the algorithm is given the distributions of a set of independent nonnegative random variables $Z_{1},\dots,Z_{n}$.
Then, it observes one at a time the realizations of $X_{i}=Z_{i}+W_{i}$ for $i\in[1,n]$, where each $W_{i}$ is nonnegative and satisfies that each $Z_{i}$ is independent of $X_{1},\dots,X_{i-1}$.
The algorithm chooses at each step to reject or accept $X_{i}$ subject to taking at most $r$ variables total.
It must compete with $\mathbb{E}\left[\sum_{i=1}^{r}Z^{(i)}\right]$.
Since none of the prior $1+o(1)$ approximation ratio algorithms for multiple-items is robust to augmentations, in Section 5.3 we design a new algorithm to prove the following multiple-items augmentation lemma.
Lemma 3.4.
(Multiple-Items Augmentation Lemma).
There is an algorithm for the augmented prophets problem with cardinality constraint $r$ achieving a $\Big{(}1+O\big{(}\frac{(\log r)^{3/2}}{r^{1/4}}\big{)}\Big{)}$ approximation ratio.
Next, we utilize the single-item augmentation lemma, along with careful decompositions of $\{X_{i}\}$, to separately attack the single-item problem for the cases of bounded $s_{\text{row}}$ and $s_{\text{col}}$.
4 Selecting a Single Item
In this section we prove our main theorem.
Later, we will also address cases where the algorithm can take multiple items.
Theorem 4.1.
There exists an inclusion-threshold algorithm for the linearly correlated prophet problem with approximation ratio $O({\min\{{s_{\text{col}}},{s_{\text{row}}}\}})$.
We will first show in Proposition 4.2 that an inclusion-threshold algorithm guarantees $O({s_{\text{col}}})$; then in Proposition 4.4 that an inclusion-threshold algorithm achieves $O({s_{\text{row}}})$. The algorithm that runs one of these according to which of $s_{\text{col}},s_{\text{row}}$ is smaller is an inclusion-threshold algorithm achieving the claimed performance.
We will see that bounded column sparsity is the easier case, requiring a simpler algorithm and analysis.
For the case of bounded row sparsity, we will need much more careful reasoning about dependencies and correlations between $Y_{j}$.
This difficulty will also manifest quantitatively when we move to the cardinality-constraint setting in Section 5, where better bounds will be achievable only in the case of bounded column sparsity.
4.1 Bounded Column Sparsity
Recall that column sparsity $s_{\text{col}}$ is the maximum number of times a given feature $Y_{j}$ may appear with nonzero coefficient.
We now give a relatively straightforward algorithm for achieving $\Omega\left(\frac{1}{s_{\text{col}}}\right)$ fraction of $\mathbb{E}[\max_{i}X_{i}]$.
The idea is similar to the “first attempt” described in Section 3, using the single-item augmentation lemma (Lemma 3.2) to overcome the challenges discussed there.
Proposition 4.2.
There exists an inclusion-threshold algorithm for the linearly correlated prophet problem with approximation ratio ${2e\cdot s_{\text{col}}}$.
Proof.
Choose $S\subseteq[n]$ by including each $i\in[n]$ independently with probability $\frac{1}{s_{\text{col}}}$.
This gives the inclusion subset; now we define the threshold $\tau$.
Assign each $Y_{j}$ to the first surviving $X_{i}$ that includes it, i.e., for each $i\in S$, construct a set $T_{i}:=\big{\{}j:A_{ij}>0\text{ and }A_{i^{\prime}j}=0~{}(\forall i^{\prime}%
\in S\text{ where }i^{\prime}<i)\big{\}}$.
Let $Z_{i}=\sum_{j\in T_{i}}A_{ij}Y_{j}$ and set $\tau=\frac{1}{2}\mathbb{E}[\max\{Z_{1},\ldots,Z_{n}\}]$.
If $i\not\in S$, then $T_{i}=\emptyset$ and $Z_{i}=0$.
By definition of an inclusion-threshold algorithm (Definition 2.4), we automatically reject any $X_{i}$ such that $i\not\in S$, and we select the first arriving $X_{i}$ such that $i\in S$ and $X_{i}\geq\tau$.
Now, by construction, we can write $X_{i}=Z_{i}+W_{i}$ where each $Z_{i}$ contains only variables $Y_{j}$ not appearing in any prior $X_{i^{\prime}}$ for $i^{\prime}\in S$ and $i^{\prime}<i$.
So $Z_{i}$ is independent of the preceding $X_{i^{\prime}}$ under consideration.
Hence by the single-item augmentation lemma (Lemma 3.2),
$\mathbb{E}[\text{ALG}]\geq\frac{1}{2}\mathbb{E}[\max_{i}Z_{i}].$
Next, we show that $\mathbb{E}[\max_{i}Z_{i}]$ is comparable to $\mathbb{E}[\max_{i}X_{i}]$.
Claim 4.3.
$\mathbb{E}[\max\{Z_{1},\ldots,Z_{n}\}]\geq\frac{1}{e\cdot s_{\text{col}}}%
\mathbb{E}[\max\{X_{1},\ldots,X_{n}\}]$, where
the expectation is over the construction of $S$ as well as $Y_{1},\dots,Y_{n}$.
Proof of Claim 4.3.
We prove that for every fixed realization of $Y_{1},\dots,Y_{n}$, the inequality holds in expectation over $S$.
Let $X_{i^{*}}=\max_{i}X_{i}$.
For each $Y_{j}$ with $A_{i^{*}j}>0$, we claim $\Pr[j\in T_{i}]\geq\frac{1}{e\cdot s_{\text{col}}}$ because $X_{i}$ survives with probability $\frac{1}{s_{\text{col}}}$ and independently, the other at most $s_{\text{col}}-1$ variables $X_{i^{\prime}}$ with $A_{i^{\prime}j}>0$ all fail to survive with probability at least333We often use the inequality $\left(1-\frac{1}{N}\right)^{N-1}=\left(\frac{N-1}{N}\right)^{N-1}\geq\frac{1}{e}$, which follows from $\left(\frac{N}{N-1}\right)^{N-1}=\left(1+\frac{1}{N-1}\right)^{N-1}\leq e$. $\left(1-\frac{1}{s_{\text{col}}}\right)^{s_{\text{col}}-1}\geq\frac{1}{e}$.
(If $s_{\text{col}}=1$, then this probability is $1$.)
So with probability only over the construction of $S$,
$$\displaystyle\mathbb{E}_{S}[Z_{i^{*}}]\quad=\quad\sum_{j}\Pr[j\in T_{i^{*}}]A_%
{i^{*}j}Y_{j}\quad\geq\quad\frac{1}{e\cdot s_{\text{col}}}\sum_{j}A_{i^{*}j}Y_%
{j}\quad=\quad\frac{1}{e\cdot s_{\text{col}}}X_{i^{*}}.$$
So we have $\mathbb{E}_{S}[Z_{i^{*}}]\geq\frac{1}{e\cdot s_{\text{col}}}\max_{i}X_{i}$, so $\mathbb{E}_{S}[\max_{i}Z_{i}]\geq\frac{1}{e\cdot s_{\text{col}}}\max_{i}X_{i}$.
This holds for each fixed realization of $Y_{1},\dots,Y_{n}$, so it holds in expectation.
∎
Claim 4.3 completes the proof of Proposition 4.2, as we have
$$\mathbb{E}[\text{ALG}]\quad\geq\quad\frac{1}{2}\mathbb{E}[\max_{i}Z_{i}]\quad%
\geq\quad\frac{1}{2e\cdot s_{\text{col}}}\mathbb{E}[\max_{i}X_{i}].\qed$$
4.2 Bounded Row Sparsity
Recall that row sparsity $s_{\text{row}}$ implies that each $X_{i}$ only depends on at most $s_{\text{row}}$ different features $Y_{j}$; however, a given $Y_{j}$ may appear in arbitrarily many $X_{i}$s.
In this section, for notational convenience, we assume without loss of generality that $\max_{i}\{A_{ij}\}=1$ for all $j$.
(If this is not the case, one can renormalize each column and redefine a scaled version of $Y_{j}$.)
We prove the following:
Proposition 4.4.
There exists an inclusion-threshold algorithm for the linearly correlated prophets problem achieving approximation ratio ${2e^{3}\cdot s_{\text{row}}}$.
This case requires more care.
There does not seem to be an analogous approach to randomly excluding $X_{i}$, as for bounded column sparsity.
Moreover, an important observation is that the $X_{i}$ cannot be treated identically in a manner oblivious to the structure of $A$.
For every “important” row that ought to be included, there can be many unimportant rows.
Indeed, we can take any instance and prepend it with arbitrarily many variables of the form $X_{i}=Y_{1}$ without changing the row sparsity $s_{\text{row}}$.
An oblivious inclusion-threshold algorithm would essentially keep only variables from this prefix, ignoring the actual instance.
Before the formal proof of Proposition 4.4, we develop a tool to address this challenge.
The key idea is to design an inclusion scheme that, for any instance structure, allows each $Y_{j}$ to be both represented and “independent” with a reasonably high probability.
Here independence refers to not sharing an $X_{i}$ with any other included $Y_{j^{\prime}}$.
Inspired by contention-resolution schemes, which have a similar flavor, we define a representative construction of a subset of the $Y_{j}$ and corresponding $X_{i}$ with $A_{ij}=1$, where $X_{i}$ is matched to $Y_{j(i)}$.
Definition 4.5.
Consider a randomized selection of $S\subseteq\{1,\dots,n\}$ and $T\subseteq\{1,\dots,m\}$ of equal size with a perfect matching $j(i)$ satisfying $A_{ij(i)}=1$.
Call this construction $\alpha$-representative or $\alpha$-rep. if
(i)
for all $j\in\{1,\dots,m\}$, we have $\Pr[j\in T]\geq\alpha$, and
(ii)
for all $i,i^{\prime}\in S$, $i\not=i^{\prime}$, we have $A_{i^{\prime}j(i)}=0$.
Note that we cannot hope for better than an $O(\frac{1}{s_{\text{row}}})$-rep. construction, as any inclusion of some $Y_{j}$ can rule out $s_{\text{row}}-1$ other features.
This raises the question of whether one can achieve $\Omega(\frac{1}{s_{\text{row}}})$-rep.
Lemma 4.6.
For any linearly correlated instance there exists a $\frac{1}{e^{2}\cdot s_{\text{row}}}$ rep. construction.
Proof.
For each $Y_{j}$, define its primary $i(j)$ by picking any $i$ such that $A_{ij}=1$ (by our renormalization assumption, there is at least one).
Consider a directed graph $G$ where the nodes are $\{1,\dots,m\}$ representing the independent variables $Y_{j}$.
There is a directed edge $(j,j^{\prime})$ if $A_{i(j)j^{\prime}}>0$, i.e., $j$ points to all other $j^{\prime}$ who are included in its primary variable $X_{i(j)}$.
We note that both edges $(j,j^{\prime})$ and $(j^{\prime},j)$ might be present.
The key property is that each vertex in $G$ has out degree $\leq s_{\text{row}}-1$, as $Y_{j}$ has only one primary $X_{i(j)}$ and at most $s_{\text{row}}-1$ other variables $j^{\prime}$ have $A_{i(j)j^{\prime}}>0$.
Because average in-degree equals average out-degree, this implies there exists a vertex with in-degree at most $s_{\text{row}}-1$.
Applying this argument recursively, we get the following claim.
Claim 4.7.
There exists an order $\pi$ of the vertices of $G$ such that for every $j$, the induced subgraph on $\pi(1),\ldots,\pi(j)$ satisfies that the in-degree of $\pi(j)$ is at most $s_{\text{row}}-1$.
Proof.
As shown, there exists some $j$ with in-degree at most $s_{\text{row}}-1$.
Set $\pi(m)=j$.
Now delete $j$ from the graph, including all edges to and from $j$.
In this graph again all out-degrees are at most $s_{\text{row}}-1$, so we can recursively construct $\pi(m-1),\dots,\pi(1)$.
∎
Consider all the $Y_{j}$ variables in the order $\pi$ given by Claim 4.7.
Initialize $S,T=\emptyset$.
On considering $j$, if $j$ does not have an edge with any vertex $j^{\prime}\in T$ (neither incoming nor outgoing), then independently with probability $\frac{1}{s_{\text{row}}}$, add $j$ to $T$ and add $j(i)$ (its primary variable) to $S$.
With the remaining probability, do nothing and continue. We show that this randomized construction of $S,T$ satisfies the two properties of a $\frac{1}{e^{2}\cdot s_{\text{row}}}$ rep. construction.
For Property (i), note each $j$ has at most $2(s_{\text{row}}-1)$ total edges (both incoming and outgoing) to nodes $j^{\prime}$ appearing prior to $j$ in the permutation: $j$ has at most $s_{\text{row}}-1$ outgoing edges in total, and by construction of $\pi$, has at most $s_{\text{row}}-1$ incoming edges from nodes prior to $j$ in $\pi$.
So when we reach $j$ in the permutation, we consider it with probability at least the probability that all these $2(s_{\text{row}}-1)$ neighbors have been rejected, which is at least $\left(1-\frac{1}{s_{\text{row}}}\right)^{2(s_{\text{row}}-1)}\geq\frac{1}{e^{2}}$; and then we include it with probability $\frac{1}{s_{\text{row}}}$.
This shows that each $j$ is included with probability at least $\frac{1}{e^{2}\cdot s_{\text{row}}}$.
Next, for Property (ii), consider any $i,i^{\prime}\in S$ with respective partners $j,j^{\prime}\in T$.
We must show $A_{ij^{\prime}}=0$.
Note that by construction, $i=i(j)$ and $i^{\prime}=i(j^{\prime})$, i.e., they are the primary variables for $j,j^{\prime}$.
Either $j$ was selected into $T$ before or after $j^{\prime}$.
In either case, the second variable could only be selected if edge $(j,j^{\prime})$ did not exist in the graph, which implies $A_{ij^{\prime}}=0$.
This completes the proof of Lemma 4.6.
∎
Given our representative construction, we are ready to complete the algorithm and proof.
Proof of Proposition 4.4.
The algorithm is an inclusion-threshold algorithm.
Its inclusion set $S$ is obtained by calling our representative construction of Lemma 4.6, which also produces a choice $j(i)$ for each $i\in S$.
Define $Z_{i}=Y_{j(i)}$ for each $i\in S$ and $Z_{i}=0$ if $i\not\in S$.
Set $\tau=\frac{1}{2}\mathbb{E}[\max_{i}Z_{i}]$.
By the second property of representative constructions, $Z_{i}$ is independent of $X_{i^{\prime}}$ for all $i^{\prime}\in S,i^{\prime}\neq i$.
Therefore, by the Augmentation Lemma (Lemma 3.2),
$$\displaystyle\mathbb{E}[\text{ALG}]\geq\frac{1}{2}\mathbb{E}[\max_{i}Z_{i}].$$
(2)
Combining this with the following Lemma 4.8 will prove Proposition 4.4.
Lemma 4.8.
$\mathbb{E}[\max_{i}Z_{i}]\geq\frac{1}{e^{3}\cdot s_{\text{row}}}\mathbb{E}[%
\max_{i}X_{i}]$.
Before proving Lemma 4.8, we will need one more idea.
Let $R_{i}=\{j:A_{ij}>0\}$, the variables included in $X_{i}$.
Notice that we may have $|R_{i}\cap T|\geq 2$, i.e., multiple variables $Y_{j}$ are members of $X_{i}$ and appear in the construction $T$.
This can occur when $X_{i}$ is not primary for any of them.444An illuminating instance is: $X_{i}=Y_{i}$ for all $i\leq s_{\text{row}}$ and $X_{s_{\text{row}}+1}=0.99(Y_{1}+\dots+Y_{s_{\text{row}}})$.
It will help to lower-bound the probability that $Y_{j}$ is the unique member of $R_{i}\cap T$.
Claim 4.9.
Under the construction of Lemma 4.6, for each $j\in R_{i}$, $\Pr[R_{i}\cap T=\{j\}]\geq\frac{1}{e^{3}\cdot s_{\text{row}}}$.
Proof.
We have $\Pr[j\in T]\geq\frac{1}{e^{2}\cdot s_{\text{row}}}$ by the representative construction.
Meanwhile, conditioned on any other decisions, each $j^{\prime}$ is included in $T$ with probability at most $\frac{1}{s_{\text{row}}}$, because it is considered with probability at most $1$ and included independently with probability $\frac{1}{s_{\text{row}}}$ conditioned on being considered.
So $\Pr[R_{i}\cap T=\{j\}|j\in T]\geq\big{(}1-\frac{1}{s_{\text{row}}}\big{)}^{s_{%
\text{row}}-1}\geq\frac{1}{e}$.
∎
Proof of Lemma 4.8.
Fix the realizations of all $Y_{j}$.
Let $i^{*}=\operatorname{argmax}_{i}X_{i}$.
First, notice that
$$\displaystyle\mathbb{E}[\max_{i}Z_{i}]$$
$$\displaystyle=\quad\mathbb{E}[\max_{j\in T}\big{\{}Y_{j}\big{\}}]\quad\geq%
\quad\mathbb{E}\big{[}\max_{j\in T\cap R_{i^{*}}}\big{\{}A_{i^{*}j}\cdot Y_{j}%
\big{\}}\big{]}.$$
Now the expected maximum of elements in $T\cap R_{i^{*}}$ is at least the sum over the elements of each’s contribution to the max, which is at least the chance it is the unique survivor times its value. This allows us to relate a max to a sum, and it relies on the fact that the representative construction’s randomness is independent of the realizations of the $\{Y_{j}\}$. Thus,
$$\displaystyle\mathbb{E}[\max_{i}Z_{i}]~{}\geq~{}\mathbb{E}\big{[}\max_{j\in T%
\cap R_{i^{*}}}\big{\{}A_{i^{*}j}\cdot Y_{j}\big{\}}\big{]}~{}\geq~{}\sum_{j}%
\Pr\big{[}T\cap R_{i^{*}}=\{j\}\big{]}\cdot A_{i^{*}j}Y_{j}~{}\geq~{}\sum_{j}%
\frac{1}{e^{3}\cdot s_{\text{row}}}A_{i^{*}j}Y_{j},$$
where the last inequality uses Claim 4.9. Since $\sum_{j}A_{i^{*}j}Y_{j}=X_{i^{*}}$, we have
$\mathbb{E}[\max_{i}Z_{i}]\geq\frac{1}{e^{3}\cdot s_{\text{row}}}X_{i^{*}}$ .
Taking expectations over $Y_{1},\dots,Y_{n}$ completes the proof.
∎
Finally, combining (2) with Lemma 4.8 completes the proof of Proposition 4.4.
∎
4.3 Lower Bounds
We now give a matching hardness result, showing that no algorithm can do better than our results in the previous section up to constant factors.
Example 4.10 (General tower instance).
Take the tower Y variables (recall $Y_{i}=\frac{1}{\epsilon^{i}}$ with probability $\epsilon^{i}$, and $0$ otherwise).
Given input integer $c$, set $n=s_{\text{row}}=s_{\text{col}}=c$.
Let $A$ be an $n\times n$ matrix with entry $A_{i,j}=0$ for $j<i$ and $A_{i,j}=\epsilon^{j-i}$ for $j\geq i$.
Visually,
$$\displaystyle X_{1}=Y_{1}+\epsilon Y_{2}+~{}\cdots~{}\cdots~{}+\epsilon^{n}Y_{%
n},\qquad X_{2}=Y_{2}+\epsilon Y_{3}+\cdots+\epsilon^{n-1}Y_{n},\qquad\ldots,%
\qquad X_{n}=Y_{n}.$$
The difficulty here amplifies that of Example 2.3.
If any of $Y_{i},Y_{i+1},\dots,Y_{n}$ are active, then this will cause $X_{i}$ to be nonzero.
Assuming only one of these variables is active (by far the most likely case), it is impossible for the algorithm to tell whether to stop or continue.
It will turn out that this instance is hard even if the algorithm is given additional power.
Definition 4.11.
In the fractional variant of the prophet problem, at each arrival $i$, the algorithm may choose to take a fraction $p_{i}$ of the current arrival $X_{i}$ subject to always taking at most one unit in total, i.e., $\sum_{i=1}^{n}p_{i}\leq 1$ with probability $1$.
Its reward is $\sum_{i=1}^{n}p_{i}X_{i}$.
One strategy available in the fractional prophet problem is to spend the entire budget on a single arrival, which is an algorithm for the standard prophets problem.
So a lower bound for the fractional problem immediately implies a lower bound for the prophets problem.
Theorem 4.12.
In the linearly correlated prophet problem, even if fractional, no online algorithm can guarantee a smaller approximation ratio than ${\min\{{s_{\text{col}}},{s_{\text{row}}}\}}$.
Proof.
We consider a family of instances of Example 4.10 where $n=s_{\text{col}}=s_{\text{row}}$.
Since for every $i$ we have $X_{i}\geq Y_{i}$, we get that for sufficiently small $\epsilon$,
$$\displaystyle\mathbb{E}[\max_{i}\{X_{i}\}]\quad\geq\quad\mathbb{E}[\max_{j}\{Y%
_{j}\}]\quad=\quad\sum_{i=1}^{n}\epsilon^{i}\prod_{j>i}(1-\epsilon^{j})\cdot%
\frac{1}{\epsilon^{i}}\quad\geq\quad n(1-\epsilon)^{n}\quad\geq\quad n(1-n%
\epsilon).$$
On the other hand, we show every online algorithm which may even fractionally select elements has value at most $1/(1-\epsilon)^{2}$. This implies the approximation ratio can be ${n}(1-n\epsilon)(1-\epsilon)^{2}$, which tends to $n$ as $\epsilon\rightarrow 0$.
Lemma 4.13.
Suppose at arrival $i$ we have $p=\sum_{i^{\prime}<i}p_{i^{\prime}}$ and $X_{i}=1/\epsilon^{i}$.
Conditioned on this event, any online algorithm obtains expected value from elements $i,\dots,n$ at most $({1-p})/{\epsilon^{i}}$.
Before proving Lemma 4.13, we use it to prove that every algorithm has $O(1)$ expected value. Notice that if $X_{i}>1/\epsilon^{i}$ then the online algorithm should never accept any fraction of the element $X_{i}$ as $X_{i+1}$ is guaranteed to be larger. Hence by Lemma 4.13, the optimal algorithm ALG takes the smallest $i$ for which $X_{i}=1/\epsilon^{i}$, which means
$$\displaystyle\mathbb{E}[\text{ALG}]$$
$$\displaystyle\leq\sum_{i\geq 1}\Pr\big{[}\big{(}X_{j}>1/\epsilon^{j}\text{ for%
all }j<i\big{)}~{}\&~{}\big{(}X_{i}=1/\epsilon^{i}\big{)}\big{]}\cdot\frac{1}%
{\epsilon^{i}}$$
$$\displaystyle=\sum_{i\geq 1}\Pr[Y_{i-1}=1/\epsilon^{i-1}]\cdot\big{(}\Pr[X_{i}%
=1/\epsilon^{i}]\big{)}\cdot\frac{1}{\epsilon^{i}}$$
$$\displaystyle\leq\sum_{i\geq 1}\Pr[Y_{i-1}=1/\epsilon^{i-1}]\cdot\big{(}\sum_{%
j\geq i}\Pr[Y_{j}=1/\epsilon^{j}]\big{)}\cdot\frac{1}{\epsilon^{i}}$$
$$\displaystyle=\sum_{i\geq 1}\epsilon^{i-1}\sum_{j\geq i}\epsilon^{j}\cdot\frac%
{1}{\epsilon^{i}}\quad\leq\quad\sum_{i\geq 1}\frac{\epsilon^{i-1}}{1-\epsilon}%
\quad\leq\quad\frac{1}{(1-\epsilon)^{2}}.\qed$$
Now we prove the missing lemma.
Proof of Lemma 4.13.
We prove this lemma by reverse induction on $i$. It is immediately true for $i=n$, as spending the entire remaining budget $1-p$ on acquiring $X_{i}=\frac{1}{\epsilon^{i}}$ is optimal.
To prove the inductive step, notice the optimal online algorithm can be written as a convex combination of the following two algorithms: one that spends the entire remaining budget of $(1-p)$ on $X_{i}$ and another one that spends no budget on $X_{i}$ and plays optimally afterwards. We argue that the first algorithm is better, which means its expected value is $({1-p})/{\epsilon^{i}}$.
Observe that the second algorithm obtains nonnegative reward only if one of $Y_{j}$s for $j>i$ is active. In this case $X_{i+1}=1/\epsilon^{i+1}$ (note it cannot be larger because $X_{i}=1/\epsilon^{i}$), and hence by induction hypothesis the optimal online algorithm gets value $X_{i+1}=(1-p)/\epsilon^{i+1}$. Thus, the expected value of the algorithm is
$$\Pr\big{[}\exists j>i\text{ s.t. }Y_{j}=1/\epsilon^{j}\mid X_{i}=1/\epsilon^{i%
}\big{]}\cdot(1-p)/\epsilon^{i+1}.$$
We show $\Pr\big{[}\exists j>i\text{ s.t. }Y_{j}=1/\epsilon^{j}\mid X_{i}=1/\epsilon^{i%
}\big{]}\leq\epsilon$, which implies the first algorithm is always better. To see this, notice
$$\displaystyle\Pr\big{[}\exists j>i\text{ s.t. }Y_{j}=1/\epsilon^{j}\mid X_{i}=%
1/\epsilon^{i}\big{]}$$
$$\displaystyle=\frac{\Pr\big{[}\big{(}\exists j>i\text{ s.t. }Y_{j}=1/\epsilon^%
{j}\big{)}~{}\&~{}\big{(}X_{i}=1/\epsilon^{i}]\big{)}\big{]}}{\Pr[X_{i}=1/%
\epsilon^{i}]}$$
$$\displaystyle=\frac{\sum_{j>i}\Pr\big{[}\big{(}Y_{j}=1/\epsilon^{j}\big{)}~{}%
\&~{}\big{(}X_{i}=1/\epsilon^{i}]\big{)}\big{]}}{\sum_{j\geq i}\Pr\big{[}\big{%
(}Y_{j}=1/\epsilon^{j}\big{)}~{}\&~{}\big{(}X_{i}=1/\epsilon^{i}]\big{)}\big{]}}$$
$$\displaystyle=\frac{\sum_{j>i}\epsilon^{j}/(1-\epsilon^{j})\cdot\prod_{j^{%
\prime}\geq i}(1-\epsilon^{j^{\prime}})}{\sum_{j\geq i}\epsilon^{j}/(1-%
\epsilon^{j})\cdot\prod_{j^{\prime}\geq i}(1-\epsilon^{j^{\prime}})}\quad=%
\quad\frac{\sum_{j>i}\epsilon^{j}/(1-\epsilon^{j})}{\sum_{j\geq i}\epsilon^{j}%
/(1-\epsilon^{j})}.$$
Now using $1+x\leq\frac{1}{1-x}\leq 1+2x$ for $0\leq x<0.5$, we get
$$\displaystyle\Pr\big{[}\exists j>i\text{ s.t. }Y_{j}=1/\epsilon^{j}\mid X_{i}=%
1/\epsilon^{i}\big{]}$$
$$\displaystyle\leq\frac{\sum_{j\geq i+1}\epsilon^{j}\cdot(1+2\epsilon^{j})}{%
\sum_{j\geq i}\epsilon^{j}\cdot(1+\epsilon^{j})}$$
$$\displaystyle=\frac{\epsilon^{i+1}-\epsilon^{n+1}+(2\epsilon^{2i+2}-2\epsilon^%
{2n+2})/(1+\epsilon)}{\epsilon^{i}-\epsilon^{n+1}+(\epsilon^{2i}-\epsilon^{2n+%
2})/(1+\epsilon)}\quad\leq\quad\epsilon,$$
where the last inequality uses $\epsilon<1/2$ and $\epsilon^{n}<2$.
∎
5 Selecting Multiple Items
In this section, we show that our approach via augmentations extends to a variant of the problem in which one may take up to $r\in\mathbb{N}$ of the arriving variables $X_{1},\dots,X_{n}$.
Let $Q\subseteq\{1,\dots,n\}$ be the indices chosen by ALG with $|Q|\leq r$. Let $X^{(i)}$ denote the $i$th-largest realized variable (we later use notation $Z^{(i)}$ for the $i$th-largest among $\{Z_{1},\dots,Z_{n}\}$ as well).
We have
$$\text{ALG}=\sum_{i\in Q}X_{i}\quad\text{ while }\quad\textsc{OPT}=\sum_{i=1}^{%
r}X^{(i)}.$$
We refer to this problem as a cardinality constraint of $r$.
It is also referred to as selecting an independent set of a rank-$r$ matroid in the special case of $r$-uniform matroids.
In this setting, there will be significant differences between row and column sparsity assumptions.
We will show that for bounded column sparsity $s_{\text{col}}$, one can design $(1+o(1))$-approximation algorithms for cardinalities $r\to\infty$, while this does not hold for bounded row sparsity $s_{\text{row}}$.
5.1 Bounded Column Sparsity
As $r\to\infty$, we will show for bounded column sparsity an approximation ratio approaching $1$.
Theorem 5.1.
For a fixed $s_{\text{col}}$, the linearly correlated prophets problem with cardinality constraint $r$ admits a $\Big{(}1+O\big{(}(\frac{s_{\text{col}}}{r})^{1/3}\log r\big{)}\Big{)}$-approximation.
The key idea is to prove an augmentation lemma for selecting multiple items (restated below).
See 3.4
In Section 5.3 we prove this augmentation lemma, but before we use it to prove Theorem 5.1.
The idea of the reduction is that by randomly partitioning the variables into $s_{\text{col}}/\epsilon^{\prime}$ “groups” gives us multiple independent Augmented Prophets problems.
We think of each group as a subproblem of selecting $\epsilon^{\prime}r/s_{\text{col}}$ elements and use the Augmentation Lemma to approximately solve it.
Proof of Theorem 5.1.
Formally, let there be $c=s_{\text{col}}/\epsilon^{\prime}$ sets, which we call groups, $B_{1},\dots,B_{c}$.
For each $X_{i}$, place it in a group $j\in\{1,\dots,c\}$ chosen uniformly at random. For each $X_{i}$, let
$$X_{i}^{\prime}=\sum{a_{ij}Y_{j}\cdot\mathbf{1}[\text{$Y_{j}$ only appears once%
in the group containing $X_{i}$]}}$$
denote the sum of $X_{i}$’s components $Y_{j}$ that do not appear with any other variable in the group containing $X_{i}$. Let $\textsc{OPT}_{j}^{\prime}$ denote the sum of the largest $\epsilon^{\prime}r/s_{\text{col}}$ elements $X_{i}^{\prime}$ in group $B_{j}$.
Claim 5.2.
$\mathbb{E}\big{[}\sum_{j}\textsc{OPT}_{j}^{\prime}\big{]}\geq(1-\epsilon^{%
\prime})\cdot\mathbb{E}[\textsc{OPT}].$
Proof.
Consider a fixed $X_{i}$. Condition on $X_{i}$ landing in a group. Notice that each of its $Y_{j}$s have at least $1-\epsilon^{\prime}$ chance of appearing only with $X_{i}$ in this group, and hence it contributes to $\textsc{OPT}_{j}^{\prime}$.
∎
Since each group $B_{j}$ forms a separate instance of the Augmented Prophets problems, we can apply the Augmentation Lemma 3.4 on each of them. Let $\text{ALG}_{j}$ denote the algorithm’s performance in group $j$. By selecting $\epsilon^{\prime}$ which is less than
$\frac{\log r}{\sqrt{\epsilon^{\prime}r/s_{\text{col}}}}$, we get
$$\sum_{j}\mathbb{E}[\text{ALG}_{j}]\quad\geq\quad\sum_{j}(1-O(\epsilon^{\prime}%
))\cdot\mathbb{E}[\textsc{OPT}^{\prime}_{j}]\quad\geq\quad(1-O(\epsilon^{%
\prime}))\cdot\mathbb{E}[\textsc{OPT}],$$
where the last inequality uses Claim 5.2.
∎
5.2 Bounded Row Sparsity
For cardinality constraints, the symmetry between bounds for row and column sparsity breaks: One cannot guarantee better than a $\Theta({s_{\text{row}}})$ approximation in general even as $r\to\infty$.
In fact, this follows by reducing to our previous hardness result for fractional prophets.
Theorem 5.3.
No algorithm for linearly correlated prophets with cardinality constraint can guarantee better than an $\Omega({s_{\text{row}}})$-approximation, even as $r\to\infty$.
Finally, we show that the $\Omega(\frac{1}{s_{\text{row}}})$ upper bound for single item can be extended to the cardinality constraint setting.
The intuition is straightforward, as we can simply instantiate $r$ parallel versions of our previous single item algorithm and assign arriving variables to each at random.
The analysis needs to show that no more than a constant factor is lost due to cases where members of OPT are sent to the same bucket.
Theorem 5.4.
For all $r$, there is an $O({s_{\text{row}}})$-approximation for the linearly correlated prophet problem with cardinality constraint $r$.
We now present formal proofs of the last two theorems.
Proof of Theorem 5.3.
For any $r,s_{\text{row}}$, we construct the general tower instance of Example 4.10 with parameter $s_{\text{row}}$, but we simply make $r$ copies of each variable $X_{i}$ (not independent, but exact copies).
The row sparsity is unchanged.
Now we will make the problem easier in two steps.
(1) Allow variables to arrive in batches of size $r$.
The algorithm may select any subset of the variables (until fulfilling its cardinality constraint), then reject the rest and receive the next batch.
This is a strictly easier problem, so an algorithm’s performance can only improve.
Of course this will not help on this instance, since each batch of $r$ are all identical, and it will turn out to be optimal to either take them all or none.
(2) Now instead we send the original tower instance (with no duplication), and we give the algorithm a cardinality constraint of $1$, but we allow it to pick a fractional amount of each variable $X_{i}$.
In other words, we return exactly to the setting of Theorem 4.12.
In each case, the algorithm sees the same information before making each decision, i.e., the value of the current $X_{i}$.
In the fractional problem, the algorithm can pick any fraction $p_{i}$ of $X_{i}$, so long as the total amount picked is at most one.
In problem (1) above, the algorithm can pick any fraction $\frac{c}{r}$ of the variables that equal the original $X_{i}$, as long as it has does not exceed a total of $\frac{r}{r}$.
The fractional problem allows more choice and the benchmarks (normalized) are the same, since the maximum of the duplicated instance will take all $r$ copies of the largest $X_{i}$.
So the fractional problem is only easier.
We now invoke Theorem 4.12, which gives an lower bound of $\Omega({s_{\text{row}}})$-approximation on the fractional problem.
(Note that $s_{\text{row}}$ did not change during the above reduction, although $s_{\text{col}}$ did.)
∎
Proof of Theorem 5.4.
Given an instance, we create $r$ buckets $B_{1},\dots,B_{r}$ and place each variable $X_{i}$ in a bucket $B_{j}$ uniformly at random.
We then run our algorithm for the linearly correlated prophet problem with bounded $s_{\text{row}}$ in each bucket $j$ (call it $\text{ALG}_{j}$), selecting one item.
Given a fixed assignment of variables to buckets, in each bucket $B_{j}$ we have by the algorithm’s guarantee that, with expectation over realizations of $\{X_{i}:i\in B_{j}\}$, by Proposition 4.4
$$\mathbb{E}_{\textbf{X}}[\text{ALG}_{j}]\geq\Omega\left(\frac{1}{s_{\text{row}}%
}\right)\cdot\mathbb{E}_{\textbf{X}}[\max_{X_{i}\in B_{j}}X_{i}].$$
Taking expectations over both buckets and variables, we have
$$\displaystyle\mathbb{E}[\text{ALG}]~{}=~{}\sum_{j=1}^{r}\mathbb{E}_{B}\mathbb{%
E}_{\textbf{X}}\big{[}\text{ALG}_{j}\big{]}~{}\geq~{}\Omega\left(\frac{1}{s_{%
\text{row}}}\right)\sum_{j=1}^{r}\mathbb{E}_{B}\mathbb{E}_{\textbf{X}}\big{[}%
\max_{X_{i}\in B_{j}}X_{i}\big{]}~{}=~{}r\cdot\Omega\left(\frac{1}{s_{\text{%
row}}}\right)\mathbb{E}_{\textbf{X}}\mathbb{E}_{B}\big{[}\max_{X_{i}\in B_{1}}%
X_{i}\big{]},$$
where the last equality is by symmetry of the buckets.
Now since $\textsc{OPT}=X^{(1)}+\cdots+X^{(r)}$ where $X^{(i)}$ is the $i$th largest variable, and since for fixed variable realizations $\Pr[X^{(i)}=\max_{X_{i^{\prime}}\in B_{1}}X_{i^{\prime}}]=\frac{1}{r}\prod_{i^%
{\prime}=1}^{i-1}(1-\frac{1}{r})\geq\frac{1}{e\cdot r}$, we get
$$\displaystyle\mathbb{E}[\text{ALG}]$$
$$\displaystyle\geq r\cdot\Omega\left(\frac{1}{s_{\text{row}}}\right)\mathbb{E}_%
{\textbf{X}}\Big{[}\sum_{i=1}^{r}\Pr[X^{(i)}=\max_{X_{i^{\prime}}\in B_{1}}X_{%
i^{\prime}}]\cdot X^{(i)}\Big{]}$$
$$\displaystyle\geq r\cdot\Omega\left(\frac{1}{s_{\text{row}}}\right)\mathbb{E}_%
{\textbf{X}}\Big{[}\sum_{i=1}^{r}\frac{1}{e\cdot r}X^{(i)}\Big{]}\quad=\quad%
\Omega\left(\frac{1}{s_{\text{row}}}\right).\qed$$
5.3 Multiple-Items Augmentation Lemma
In Section 3 we showed a $2$-approximation single-item augmentation lemma using the half of expected-maximum as a threshold. In this section, we prove a $1+o(1)$ approximation multiple-items augmentation lemma (Lemma 3.4), assuming the cardinality constraint $r$ is sufficiently large.
We first give a surrogate benchmark $\textsc{OPT}^{\prime}$ that competes with OPT.
We then define the algorithm and show that it competes with $\textsc{OPT}^{\prime}$.
Surrogate benchmark.
The analysis hinges on theshold $\tau_{0}:=\frac{\mathbb{E}[\textsc{OPT}]}{\epsilon}$, where we call variables $Z_{i}$ and $X_{i}$ that fall above the threshold heavy and below light.
For light variables, we exclude the scenario where they are very small, below some threshold $\tau_{c}\leq\epsilon\frac{\mathbb{E}[\textsc{OPT}]}{r}$.
We will generally use the prime symbol $\prime$ to denote a version of a variable that is zeroed out if it’s too heavy (or too light).
Let
$$\displaystyle Z_{i}^{\prime}=Z_{i}\cdot\mathbf{1}[\tau_{c}\leq Z_{i}<\tau_{0}]%
\quad\text{and}\quad Z^{(i)\prime}=Z^{(i)}\cdot\mathbf{1}[\tau_{c}\leq Z_{i}<%
\tau_{0}].$$
Let
$$\displaystyle\textsc{OPT}^{\prime}$$
$$\displaystyle=\textsc{OPT}_{1}^{\prime}+\textsc{OPT}_{2}^{\prime}\qquad\text{where}$$
$$\displaystyle\textsc{OPT}_{1}^{\prime}$$
$$\displaystyle=\max_{i}~{}Z_{i}\cdot\mathbf{1}[Z_{i}\geq\tau_{0}]\quad\text{and%
}\quad\textsc{OPT}_{2}^{\prime}={\textstyle\sum_{j=1}^{r}Z^{(j)\prime}}.$$
Note that $\textsc{OPT}^{\prime}$ only considers one heavy variable and throws all other heavy variables away.
Nevertheless, in Appendix D we will show the following claim that it competes with OPT.
Claim 5.5.
$\mathbb{E}[\textsc{OPT}^{\prime}]\geq(1-2\epsilon)\mathbb{E}[\textsc{OPT}]$.
Algorithm overview.
For intuition, the problem with setting any particular fixed threshold is that variables with insignificant values of $Z_{i}$ can be boosted by the adversary to some $X_{i}$ just above the threshold.
The algorithm would use up its $r$ slots and be unable to take the later, larger arrivals that contribute to OPT.
Therefore, we will define a sequence of thresholds, each a factor of $1-\epsilon$ apart.
The algorithm will have a certain number of slots $\tilde{r}_{j}$ for the “bucket” $j$ of arrivals between any two thresholds.
For this fixed bucket, such a boosting strategy by the adversary can only cost the algorithm a factor of $1-\epsilon$.
Roughly, this strategy will cover the case where OPT is concentrated.
To allow for cases where most of OPT comes from very rare, very large variables, we will also reserve a slot for such variables and analyze it separately.
Algorithm definition.
We define a sequence of thresholds.
Recall that $\textsc{OPT}=\sum_{j=1}^{r}Z^{(j)}$ where $Z^{(j)}$ is the $j$th-largest of $Z_{1},\dots,Z_{n}$.
Let $c=\left\lceil\frac{1}{\epsilon}\ln\frac{r}{\epsilon^{2}}\right\rceil$ and define thresholds
$$\displaystyle\tau_{j}$$
$$\displaystyle=(1-\epsilon)^{j}\frac{\mathbb{E}[\textsc{OPT}]}{\epsilon}$$
$$\displaystyle(j=0,\dots,c)$$
Observation 5.6.
The largest threshold $\tau_{0}=\frac{\mathbb{E}[\textsc{OPT}]}{\epsilon}$ and the smallest satisfies $\tau_{c}\leq\frac{\epsilon\cdot\mathbb{E}[\textsc{OPT}]}{r}$.
Proof.
$\tau_{0}$ is immediate, and we have $(1-\epsilon)^{c}\leq e^{-\epsilon c}\leq\exp\left(-\epsilon\frac{1}{\epsilon}%
\ln\frac{r}{\epsilon^{2}}\right)=\frac{\epsilon^{2}}{r}$.
∎
Now we define the size of each bucket.
Recall that we abuse notation by writing $Z_{i}\in\textsc{OPT}$ if $Z_{i}$ is one of the $r$ variables included in the OPT solution.
Let
$$\displaystyle r_{j}$$
$$\displaystyle=\mathbb{E}\left|\left\{i:Z_{i}\in\textsc{OPT},\tau_{j}\leq Z_{i}%
\leq\tau_{j-1}\right\}\right|$$
$$\displaystyle(j=1,\dots,c)$$
$$\displaystyle\beta$$
$$\displaystyle=3\sqrt{r\ln(c/\epsilon)}$$
$$\displaystyle\tilde{r}_{j}$$
$$\displaystyle=r_{j}+\beta$$
$$\displaystyle(j=1,\dots,c)$$
$$\displaystyle\tilde{r}_{0}$$
$$\displaystyle=1.$$
We first define an algorithm $\text{ALG}^{\prime}$ that does not quite achieve the cardinality constraint $r$.
We will then modify it to obtain ALG with only a small loss in performance.
$\text{ALG}^{\prime}$ initializes $b_{j}=0$ for $j=0,\dots,c$ and proceeds as follows when a variable $X_{i}$ arrives.
1.
If $X_{i}<\tau_{c}$, we discard $X_{i}$ and continue.
2.
Otherwise, let $j=\min\{j^{\prime}:X_{i}\geq\tau_{j^{\prime}}\}$.
3.
If $b_{j}<\tilde{r}_{j}$, we take $X_{i}$ and increment $b_{j}$.
Otherwise (bucket $j$ is full), increment $j$ and repeat this step.
If $j>c$, stop and discard $X_{i}$.
In other words, we attempt to assign $X_{i}$ to its original bucket, but if that is full, we allow it to fall into buckets reserved for smaller variables (higher indices $j$).
Now, the final algorithm ALG is defined as follows: run $\text{ALG}^{\prime}$, but each time $\text{ALG}^{\prime}$ takes an arrival $X_{i}$, discard it independently with probability $\epsilon$.
If a variable is not discarded, but the cardinality constraint $r$ is reached, then discard it anyways.
Analysis.
We show in Appendix D that ALG approximates $\text{ALG}^{\prime}$.
Lemma 5.7.
For all $\epsilon\geq\frac{9\left(\ln r\right)^{3/2}}{r^{1/4}}$, we have $\mathbb{E}[\text{ALG}]\geq(1-2\epsilon)\mathbb{E}[\text{ALG}^{\prime}]$.
Now, we analyze $\text{ALG}^{\prime}$.
Let us define the contributions of $\text{ALG}^{\prime}$ and $\textsc{OPT}^{\prime}$ bucket-by-bucket.
The top bucket of $\text{ALG}^{\prime}$ is split into cases where the corresponding $Z_{i}$ is heavy or light.
The following are random sets:
$$\displaystyle O_{j}$$
$$\displaystyle=\{i\in\textsc{OPT}^{\prime}:\tau_{j}\leq Z_{i}<\tau_{j-1}\}$$
$$\displaystyle(j=1,\dots,c)$$
$$\displaystyle O_{0}$$
$$\displaystyle=\{i\in\textsc{OPT}^{\prime}:\tau_{0}\leq Z_{i}\}$$
$$\displaystyle B_{j}$$
$$\displaystyle=\{i\in\text{ALG}:\tau_{j}\leq X_{i}<\tau_{j-1}\}$$
$$\displaystyle(j=1,\dots,c)$$
$$\displaystyle B_{0}^{\text{light}}$$
$$\displaystyle=\{i\in\text{ALG}^{\prime}:Z_{i}<\tau_{0}\leq X_{i}\}$$
$$\displaystyle B_{0}^{\text{heavy}}$$
$$\displaystyle=\{i\in\text{ALG}^{\prime}:\tau_{0}\leq Z_{i}\}.$$
We use the notation $i\in\textsc{OPT}^{\prime}$ to denote that $i$ contributes to $\textsc{OPT}^{\prime}$, i.e. either $Z_{i}$ is the largest among $\{Z_{j}\}$ or $Z_{i}^{\prime}$ is among the $r$ largest of $\{Z_{j}^{\prime}\}$.
Similarly, we write $i\in\text{ALG}^{\prime}$ to mean that the algorithm takes $X_{i}$.
Now, we break down $\text{ALG}^{\prime}$ as follows.
$$\displaystyle\text{ALG}^{\prime}$$
$$\displaystyle=\text{ALG}_{1}^{\prime}+\text{ALG}_{2}^{\prime}\qquad\text{where}$$
$$\displaystyle\text{ALG}_{1}^{\prime}=\sum_{i\in B_{0}^{\text{heavy}}}X_{i}$$
$$\displaystyle\quad\text{and}\quad\text{ALG}_{2}^{\prime}=\sum_{i\in B_{0}^{%
\text{light}}}X_{i}+\sum_{j=1}^{c}\sum_{i\in B_{j}}X_{i}.$$
In other words, $\text{ALG}_{1}^{\prime}$ tracks the contribution of the special “heavy” bucket, but only in the case where the underlying variable $Z_{i}$ is heavy.
$\text{ALG}_{2}^{\prime}$ tracks the remaining case and all other buckets.
Notice these definitions are only for the purpose of analysis, as $Z_{i}$ is not observable to the algorithm.
Finally, we define
$$P:=\Pr[\max_{i}X_{i}\geq\tau_{0}].$$
A key point will be that if $P$ is large, then the algorithm will often get some variable larger than $\tau_{0}$, which is good enough to compete with OPT.
Lemma 5.8.
$\mathbb{E}[\text{ALG}^{\prime}]\geq\frac{P}{\epsilon}\mathbb{E}[\textsc{OPT}]$.
Proof.
With probability $P$, some $X_{i}$ exceeds $\tau_{0}=\frac{\mathbb{E}[\textsc{OPT}]}{\epsilon}$.
Since Bucket $0$ is reserved for such arrivals with budget $b_{0}=1$, the algorithm gets such $X_{i}$ if this occurs, so its expectation is $\geq P\tau_{0}$.
∎
Thus, if $P\geq\epsilon$, we are already done.
The rest of the analysis will leverage cases where $P$ is small.
Lemma 5.9.
$\mathbb{E}[\text{ALG}_{1}^{\prime}]\geq(1-P)\mathbb{E}[\textsc{OPT}_{1}^{%
\prime}]$.
Proof.
$$\displaystyle\mathbb{E}[\text{ALG}_{1}^{\prime}]$$
$$\displaystyle=\mathbb{E}\Big{[}\sum_{i}X_{i}\cdot\mathbf{1}[i\in B_{0}^{\text{%
heavy}}]\Big{]}\quad\geq\quad\mathbb{E}\Big{[}\sum_{i}Z_{i}\cdot\mathbf{1}[i%
\in B_{0}^{\text{heavy}}]\Big{]}$$
$$\displaystyle=\sum_{i}\Pr[\text{$b_{0}=0$ when $i$ arrives}]\cdot\mathbb{E}%
\left[Z_{i}\cdot\mathbf{1}[Z_{i}\geq\tau_{0}]\right]\qquad\text{(using %
independence)}$$
$$\displaystyle\geq(1-P)\sum_{i}\mathbb{E}\left[Z_{i}\cdot\mathbf{1}[Z_{i}\geq%
\tau_{0}]\right]$$
$$\displaystyle\geq(1-P)\mathbb{E}[\max_{i}\{Z_{i}\cdot\mathbf{1}[Z_{i}\geq\tau_%
{0}]\}]~{}~{}=~{}~{}(1-P)\cdot\mathbb{E}[\textsc{OPT}_{1}^{\prime}].\qed$$
The final piece of the argument is to show that the “buckets” strategy works, i.e., it cannot be disrupted by augmentations.
The idea is that we have reserved an accurate number of slots in each bucket for the case where there is no augmentation.
An augmented variable $X_{i}$ can take away a bucket slot from some $Z_{i^{\prime}}$, with $Z_{i^{\prime}}\gg Z_{i}$, but then it will contribute about as much to $\text{ALG}^{\prime}$ as $Z_{i^{\prime}}$ did to $\textsc{OPT}_{2}^{\prime}$.
In this case, we should be concerned that $\textsc{OPT}_{2}^{\prime}$ gets both $Z_{i}$ and $Z_{i^{\prime}}$ while $\text{ALG}^{\prime}$ only gets $X_{i}$, with $X_{i^{\prime}}$ disappearing thanks to the bucket being full.
However, the algorithm allows such an $X_{i^{\prime}}$ to “trickle down” into a lower-tier bucket, in particular, the slot that is not being used by $X_{i}$.
And if this slot is full as well, then in any case $\text{ALG}^{\prime}$ is competing with $\textsc{OPT}_{2}^{\prime}$.
Lemma 5.10.
For $\epsilon\geq\frac{9\left(\ln r\right)^{3/2}}{r^{1/4}}$, we have $\mathbb{E}[\text{ALG}_{2}^{\prime}]\geq(1-\epsilon)^{2}\mathbb{E}[\textsc{OPT}%
_{2}^{\prime}]$.
Proof.
First, let $C$ denote the event that none of the $\textsc{OPT}_{2}^{\prime}$ buckets are filled to the $\tilde{r}_{j}$ capacities, i.e. $C$ is the event that $|O_{j}|\leq\tilde{r}_{j}$ for all $j=1,\dots,c$.
Claim 5.11.
$\Pr[C]\geq 1-\epsilon$.
Proof.
Recall that $r_{j}=\mathbb{E}|O_{j}|$.
Because $|O_{j}|$ is a sum of independent Bernoulli random variables, we have by a standard Bernstein bound that
$$\Pr[|O_{j}|-r_{j}\geq\sqrt{2r_{j}t}+t]\leq e^{-t}.$$
Setting $t=\ln\frac{c}{\epsilon}$, we get a bound of $\frac{\epsilon}{c}$; a union bound over the buckets will complete the proof.
We just need to show that $\sqrt{2r_{j}\ln\frac{c}{\epsilon}}+\ln\frac{c}{\epsilon}\leq\beta$, as then the probability of exceeding $r_{j}+\beta$ is only smaller.
As in the proof of Claim D.1, for this choice of $\epsilon$, we have $\ln\frac{c}{\epsilon}\leq\ln(r)$, and $\ln(r)\leq r$, so
$$\textstyle\sqrt{2r_{j}\ln\frac{c}{\epsilon}}+\ln\frac{c}{\epsilon}\leq\sqrt{2r%
\ln\frac{c}{\epsilon}}+\sqrt{r\ln\frac{c}{\epsilon}}\leq\beta.\qed$$
Next we argue that at each tier of thresholds, $\text{ALG}^{\prime}$ is getting just as many variables as $\textsc{OPT}_{2}^{\prime}$, even if their identities are different.
First, a helpful property:
Claim 5.12.
Conditioned on $C$, suppose $|B_{j}|<\tilde{r}_{j}$ for some $j\geq 1$.
Then $|B_{0}^{\text{light}}|+\sum_{j^{\prime}=1}^{j}|B_{j^{\prime}}|\geq\sum_{j^{%
\prime}=1}^{j}|O_{j^{\prime}}|$.
Proof.
Consider any arrival $i$ that contributes to the right side.
We claim it is also counted on the left.
We know $X_{i}\geq Z_{i}\geq\tau_{j}$, because the variable contributes to the right side.
So the algorithm will attempt to place $X_{i}$ in some assigned bucket $j^{\prime}\leq j$.
If it does not succeed because the bucket is full, it will proceed to $j^{\prime}+1,\dots,$ and possibly eventually $j$.
Because $|B_{j}|<\tilde{r}_{j}$, we know there is space for $i$ in bucket $j$, so the algorithm definitely takes $i$ in bucket $j$ or earlier.
By definition, $Z_{i}<\tau_{0}$, so $i$ cannot be a member of $B_{0}^{\text{heavy}}$.
Therefore, it is counted by the left side.
∎
Now we can show the key fact.
Claim 5.13.
Conditioned on $C$, we have for all $j=1,\dots,c$ that
$$\textstyle\left|B_{0}^{\text{light}}\right|+\sum_{j^{\prime}=1}^{j}|B_{j^{%
\prime}}|\geq\sum_{j^{\prime}=1}^{j}|O_{j^{\prime}}|.$$
Proof.
By induction on $j$.
For $j=1$, we must show $|B_{0}^{\text{light}}|+|B_{1}|\geq|O_{1}|$.
Recall that an arrival $i$ is a member of $O_{1}$ if $\tau_{1}\leq Z_{i}<\tau_{0}$.
There are two cases.
If $|B_{1}|=\tilde{r}_{1}$, i.e. the bucket is full, then the case is proven as we have assumed event $C$, which implies $|O_{1}|\leq\tilde{r}_{1}$.
Otherwise, the case follows by Claim 5.12.
Now consider $j>1$.
If $|B_{j}|<\tilde{r}_{j}$, i.e. the bucket is not full, then the case follows by Claim 5.12.
Otherwise, i.e. bucket $j$ is full, then we have $|B_{j}|\geq|O_{j}|$ because of property $C$.
Combining this with the induction hypothesis proves that $|B_{0}^{\text{light}}|+\sum_{j^{\prime}=1}^{j}|B_{j^{\prime}}|\geq\sum_{j^{%
\prime}=1}^{j}|O_{j^{\prime}}|$.
∎
We are now ready to complete the proof of Lemma 5.10.
From Claim 5.13, given event $C$, we can make a one-to-one mapping from contributions $Z_{i}^{\prime}$ of $\textsc{OPT}_{2}^{\prime}$ to contributions $X_{j}$ of $\text{ALG}_{2}^{\prime}$, such that $X_{j}$ is in the same bucket or a higher bucket than $Z_{i}^{\prime}$.
(I.e. map all elements of $O_{1}$ to elements of $B_{1}$ or $B_{0}^{\text{light}}$; map all elements of $O_{2}$ to remaining elements of these or to elements of $B_{2}$; and so on.)
For each such pair, we have $X_{j}\geq(1-\epsilon)Z_{i}^{\prime}$ because, at worst, both are in the same bucket.555For example, we may have $Z_{i}^{\prime}\approx\tau_{4}$ while $X_{j}=\tau_{3}=(1-\epsilon)\tau_{4}$.
In total, this implies that, conditioned on $C$, we always have $\text{ALG}_{2}^{\prime}\geq(1-\epsilon)\textsc{OPT}_{2}^{\prime}$; and $C$ occurs with probability at least $1-\epsilon$.
∎
Corollary 5.14.
For $\epsilon\geq\frac{9(\ln r)^{3/2}}{r^{1/4}}$, we have $\mathbb{E}[\text{ALG}^{\prime}]\geq(1-\epsilon)^{2}\cdot\mathbb{E}[\textsc{OPT%
}^{\prime}]$.
Proof.
If $P\geq\epsilon$, then by Lemma 5.8, we have $\mathbb{E}[\text{ALG}^{\prime}]\geq\mathbb{E}[\textsc{OPT}]\geq\mathbb{E}[%
\textsc{OPT}^{\prime}]$, proving the claim.
Otherwise, by Lemma 5.9, $\mathbb{E}[\text{ALG}_{1}^{\prime}]\geq(1-P)\mathbb{E}[\textsc{OPT}_{1}^{%
\prime}]\geq(1-\epsilon)\mathbb{E}[\textsc{OPT}_{1}^{\prime}]$; and by Lemma 5.10, $\mathbb{E}[\text{ALG}_{2}^{\prime}]\geq(1-\epsilon)^{2}\mathbb{E}[\textsc{OPT}%
_{2}^{\prime}]$.
So in the case $P<\epsilon$, we have
$$\displaystyle\mathbb{E}[\text{ALG}^{\prime}]\quad=\quad\mathbb{E}[\text{ALG}_{%
1}^{\prime}]+\mathbb{E}[\text{ALG}_{2}^{\prime}]$$
$$\displaystyle\geq\quad(1-\epsilon)\mathbb{E}[\textsc{OPT}_{1}^{\prime}]+(1-%
\epsilon)^{2}\mathbb{E}[\textsc{OPT}_{2}^{\prime}]$$
$$\displaystyle\geq\quad(1-\epsilon)^{2}\mathbb{E}[\textsc{OPT}^{\prime}].\qed$$
Proof of Augmentation Lemma 3.4.
For $\epsilon\leq\frac{1}{2}$, $\epsilon\geq\frac{9(\ln r)^{3/2}}{r^{1/4}}$, we have:
$$\displaystyle\mathbb{E}[\text{ALG}]\stackrel{{\scriptstyle\text{Lemma~{}\ref{%
lemma:alg-approx-algprime}}}}{{\geq}}(1-2\epsilon)\mathbb{E}[\text{ALG}^{%
\prime}]\stackrel{{\scriptstyle\text{Corollary \ref{cor:algprime-competes-%
optprime}}}}{{\geq}}(1-2\epsilon)^{3}\mathbb{E}[\textsc{OPT}^{\prime}]%
\stackrel{{\scriptstyle\text{Claim \ref{claim:onlyOneLarge}}}}{{\geq}}(1-2%
\epsilon)^{4}\mathbb{E}[\textsc{OPT}].$$
So we obtain a $\big{(}1+O(\epsilon)\big{)}$-approximation.
∎
Appendix A Unweighted Linear Correlations
In this section we consider a linear correlations model where the nonnegative matrix $A$ from $\textbf{X}=A\cdot\textbf{Y}$ is unweighted, i.e., each of its entry is either $0$ or $1$. Alternately, for $i\in[n]$ there are known sets $S_{1},S_{2},\ldots,S_{n}\subseteq[m]$ such that $X_{i}=\sum_{j\in S_{i}}Y_{j}$.
Our lower bound tower instance from §4.3 no longer holds as it crucially expolits that matrix $A$ has entries that decrease exponentially in $\epsilon$. Can we do better than a $\Theta\left({\min\{s_{\text{col}},s_{\text{row}}\}}\right)$ approximation ratio?
One might wonder if there exists an alternate hardness instance that only has $0-1$ entries in $A$. We show that this is not the case. In fact, there exist simple threshold-based constant approximation algorithms.
Theorem A.1.
The unweighted linear correlations problem has a fixed threshold constant-factor approximation algorithm.
The main intuition in the proof of Theorem A.1 is that for the unweighted problem each independent $Y_{j}$ has limited “influence” on the $X_{i}$s. This is because either $Y_{j}$ appears with coefficient $0$, in which case it has no influence on the value $X_{i}$, or it appears with coefficient $1$, in which case it has the same influence in the value of every such $X_{i}$. A threshold algorithm is therefore difficult to fool because unlike the tower instance, it is not possible to have a scenario where a $Y_{j}$ is very large but our algorithm selects it within an $X_{i}$ where it appears with a small coefficient $\epsilon$.
For readers familiar with the revenue-maximization result of Babaioff et al. [BILW14], i.e., the best of selling items individually and selling all the items together in a single bundle is a constant factor approximation to optimal revenue, our result has a similar flavor, although the technical details are quite different. We decompose our problem instance into a “core” and a “tail” part. The tail consists of cases where any $Y_{j}$ exceeds a boundary $\tau$; the core, the rest. For the tail case we show that approximating $\mathbb{E}[\max_{j}\{Y_{j}\}]$ (the best individual item) suffices, and in the core case we can approximate the best bundle $X_{i}$.
This argument will show that there is one fixed threshold $\tau_{\text{core}}$ such that the algorithm taking the first arrival above $\tau_{\text{core}}$ achieves a constant approximation to the optimal core contribution to $\max\{X_{i}\}$; and similarly for $\tau_{\text{tail}}$ and the tail part.
There remains a corner case, where we show a fixed threshold equal to the boundary $\tau$ gives a constant factor.
Thus, for any given instance, one can select the appropriate choice among $\tau,\tau_{\text{core}},\tau_{\text{tail}}$, giving a fixed-threshold constant-factor approximation algorithm.
A.1 Notation and Proof Overview
We first choose a real number $\tau$ representing a boundary.
Let $p_{j}=\Pr[Y_{j}>\tau]$. We set $\tau$ such that $\prod_{j}(1-p_{j})=1/2$, i.e., with half probability all $Y_{j}$ are below $\tau$.
We let the set $A$ be all “heavy” $Y_{j}$ variables: $A:=\{j:Y_{j}>\tau\}$.
Recall that for each $X_{i}$, the set of active indices is $S_{i}$, so we have $X_{i}=\sum_{j\in S_{i}}Y_{j}$.
We first upper-bound OPT by contributions from the core event that $A=\emptyset$ (all $Y_{j}$ are small) and from the remaining tail event.
Claim A.2.
$$\mathbb{E}[\textsc{OPT}]\leq\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]+\sum_{j}%
p_{j}\cdot\mathbb{E}[Y_{j}\mid Y_{j}>\tau].$$
Proof.
For any outcome of $Y_{j}$s, we separately count those in $A$ (larger than $\tau$) and the rest, and relax the objective to always take the large ones:
$$\displaystyle\max_{i}X_{i}\quad=\quad\max_{i}\{\sum_{j\in S_{i}}Y_{j}\}\quad%
\leq\quad\max_{i}\{\sum_{j\in S_{i}\cap\overline{A}}Y_{j}\}+\sum_{j\in A}Y_{j}.$$
Now taking expectations on both sides,
$$\displaystyle\mathbb{E}[\textsc{OPT}]\quad=\quad\mathbb{E}[\max_{i}X_{i}]$$
$$\displaystyle\leq\quad\mathbb{E}[\max_{i}\{\sum_{j\in S_{i}\cap\overline{A}}Y_%
{j}\}]+\mathbb{E}[\sum_{j\in A}Y_{j}]$$
$$\displaystyle\leq\quad\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]+\sum_{j}p_{j}%
\cdot\mathbb{E}[Y_{j}\mid Y_{j}>\tau].$$
To justify that $\mathbb{E}[\max_{i}\{\sum_{j\in S_{i}\cap\overline{A}}Y_{j}\}]\leq\mathbb{E}[%
\max_{i}X_{i}\mid A=\emptyset]$, use a coupling argument: First draw all $Y_{j}$ from their initial distributions and consider the value of $\max_{i}\{\sum_{j\in S_{i}\cap\overline{A}}Y_{j}\}$. Now take any variables $Y_{j}>\tau$, and redraw them until they fall below $\tau$.
The value of the inner sum can only increase, but now we are exactly obtaining $\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]$.
∎
Given that Claim A.2 upper-bounds OPT by the sum of two terms, our proof goes in two steps.
First, we approximate the tail contributions $\sum_{j}p_{j}\cdot\mathbb{E}[Y_{j}\mid Y_{j}>\tau]$.
In Lemma A.3 we use the Augmentation Lemma 3.2 to
give a simple fixed threshold-$\tau_{\text{tail}}$ algorithm with expected value $\Omega(\max\{Y_{j}\})$. This is a bit surprising because the lower bound in §4.3 actually proves such a result is not possible for weighted linear correlations. In Claim A.4, we show that $\Omega(\max\{Y_{j}\})$ suffices to capture the tail term.
To capture the core contributions $\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]$, in Claim A.6 we argue that for all instances where $Y_{j}$s are bounded by $\tau$, we can use concentration of XOS functions to argue that $\Pr[\max_{i}X_{i}>1/2\cdot\mathbb{E}[\max_{i}X_{i}]]$ is at least a constant, and hence a simple fixed threshold-$\tau_{\text{core}}$ algorithm suffices.
This second step also holds for prophets with weighted linear correlations.
There is also a corner case where $\tau$ is too large to apply concentration.
But in this case, setting a threshold $\tau$ will directly achieve a constant factor.
A.2 Proof
Lemma A.3.
For the prophet inequality problem with unweighted linear correlations, there exists a fixed threshold algorithm with expected value $\Omega(\max\{Y_{j}\})$.
Proof.
Define $Z_{i}$ to be the sum of $Y_{j}$s that appear in $X_{i}$ and have not appeared in any $X_{i^{\prime}}$ for $i^{\prime}<i$. Since every $Y_{j}$ appears in some $Z_{i}$, we know $\max\{Z_{i}\}\geq\max\{Y_{j}\}$. The Augmentation Lemma 3.2 now completes the proof.
∎
Now we argue that $\mathbb{E}[\max_{j}Y_{j}]$ takes care of the second term in Claim A.2.
Claim A.4.
$$\mathbb{E}[\max_{j}Y_{j}]\geq\frac{1}{2}\cdot\sum_{j}p_{j}\cdot\mathbb{E}[Y_{j%
}\mid Y_{j}>\tau].$$
Proof.
$$\displaystyle\mathbb{E}[\max_{j}Y_{j}]$$
$$\displaystyle\geq\sum_{j}\Pr[A=\{j\}]\cdot\mathbb{E}[\max_{j}Y_{j}\mid A=\{j\}]$$
$$\displaystyle\geq\sum_{j}\frac{p_{j}}{2}\cdot\mathbb{E}[Y_{j}\mid A=\{j\}],$$
where the second inequality uses $\Pr[A=\{j\}]=\Pr[Y_{j}>\tau]\Pr[Y_{j^{\prime}}\leq\tau(\forall j^{\prime}\neq j%
)]\geq(p_{j})\left(\frac{1}{2}\right)$ and that $\max_{j}Y_{j}$ given that $A=\{j\}$ is the same as $Y_{j}$.
∎
Corollary A.5.
For any instance with unweighted linear correlations, there exists $\tau_{\text{tail}}$ such that the algorithm setting a fixed threshold of $\tau_{\text{tail}}$ obtains $\mathbb{E}[\text{ALG}]\geq\Omega\left(\sum_{j}p_{j}\cdot\mathbb{E}[Y_{j}\mid Y%
_{j}>\tau]\right)$.
We now turn to the core portion of contributions to OPT.
Claim A.6.
Let $V=\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]$.
If the boundary satisfies $\tau\leq V/10$, there exists $\tau_{\text{core}}$ such that the algorithm setting a fixed threshold of $\tau_{\text{core}}$ obtains $\mathbb{E}[\text{ALG}]\geq\Omega(V)$.
Proof.
Let $Y_{j}^{\prime}$ be a copy of $Y_{j}$ conditioned on falling into the range $[0,\tau]$.
Let $X_{i}^{\prime}=\sum_{j\in S_{i}}Y_{j}^{\prime}$ and let $W=\max_{i}X_{i}^{\prime}$.
We have $\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]=\mathbb{E}[W]=V$.
Now, note that $W$ is an XOS function of the independent $Y_{i}^{\prime}$ variables (meaning is a maximum of weighted combinations).
Thus we can apply the concentration of XOS functions (more generally, for self-bounding functions, see e.g. [Von10]) to get
$$\Pr\big{[}W<(1-\delta)\cdot\mathbb{E}[W]\big{]}\leq\exp\Big{(}-\delta^{2}\cdot%
\frac{\mathbb{E}[W]}{2\tau}\Big{)}.$$
In particular, for $\delta=1/2$ we get
$$\Pr\Big{[}W<\frac{1}{2}\mathbb{E}[W]\Big{]}\quad\leq\quad\exp\Big{(}-\frac{%
\mathbb{E}[W]}{8\tau}\Big{)}\quad\leq\gamma,$$
for some constant $\gamma<1$, using that $\tau\leq\mathbb{E}[W]/10$.
Hence the expected value of an algorithm that sets a threshold of $\frac{1}{2}\mathbb{E}[W]$ is at least
$$\Pr\Big{[}W\geq\frac{1}{2}\mathbb{E}[W]\Big{]}\cdot\frac{1}{2}\mathbb{E}[W]%
\geq\Omega\Big{(}\mathbb{E}[W]\Big{)}.\qed$$
Now we have all the tools to prove our main theorem.
Proof of Theorem A.1.
By Claim A.2, one of the follwing is at least a ${2}$-approximation to the prophet: $V:=\mathbb{E}[\max_{i}X_{i}\mid A=\emptyset]$, and $\mathbb{E}[\sum_{j}p_{j}\cdot\mathbb{E}[Y_{j}\mid Y_{j}>\tau]$.
Suppose it is the latter.
Then by Claim A.4, setting a fixed threshold of $\tau_{\text{tail}}$ gives a constant-factor approximation.
So suppose we have $V\geq\mathbb{E}[\textsc{OPT}]/2$.
If $\tau>V/10$, then we can set a fixed threshold of $\tau$: with probability at least $\frac{1}{2}$, some $X_{i}\geq\tau$ (because some $Y_{j}\geq\tau$), so we obtain performance at least $\frac{1}{2}\tau\geq\frac{1}{40}\mathbb{E}[\textsc{OPT}]$.
Finally, if $V\geq\mathbb{E}[\textsc{OPT}]/2$ and $\tau\leq V/10$, then by Claim A.6, setting a fixed threshold of $\tau_{\text{core}}$ gives a constant-factor approximation.
∎
Appendix B Negatively Correlated Values
In this section we show how a ${2}$ approximation ratio for negatively correlated values is a simple implication of prior work on threshold algorithms.
Formally, one common definition of negative correlation is negative association, which is said to hold for $\{X_{j}\}$ if for all monotone increasing functions $f,g$ and disjoint subsets $S,S^{\prime}\subseteq\{1,\dots,n\}$, for all $a,b\in\mathbb{R}$ we have
$$\Pr\Big{[}f(X_{j}:j\in S)\geq a\Big{]}\leq\Pr\Big{[}f(X_{j}:j\in S)\geq a\mid g%
(X_{j}:j\in S^{\prime})\leq b\Big{]}.$$
Let $\tau=\frac{1}{2}\mathbb{E}[\max_{i}X_{i}]$ and let $P=\Pr[\max_{i}X_{i}\geq\tau]$.
We can write down precisely the usual prophet proof with just one line requiring additional justification.
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]$$
$$\displaystyle=\textstyle P\cdot\tau+\sum_{i=1}^{n}\Pr[X_{i^{\prime}}<\tau(%
\forall i^{\prime}<i)]\cdot\mathbb{E}\left[(X_{i}-\tau)^{+}\mid X_{i^{\prime}}%
<\tau(\forall i^{\prime}<i)\right]$$
$$\displaystyle\textstyle\geq P\cdot\tau+(1-P)\cdot\sum_{i=1}^{n}\mathbb{E}\left%
[(X_{i}-\tau)^{+}\mid X_{i^{\prime}}<\tau(\forall i^{\prime}<i)\right]$$
$$\displaystyle\textstyle\geq P\cdot\tau+(1-P)\cdot\sum_{i=1}^{n}\mathbb{E}\left%
[(X_{i}-\tau)^{+}\right],$$
where the last inequality uses negative association as $(X_{i}-\tau)^{+}$ is a monotone function as is $\max_{i^{\prime}<i}X_{i^{\prime}}$. We can also use weaker notions of negative correlation for the last inequality, e.g., NLODS as in Section 2 of [RSC92].
Now repeating the old prophet inequality analysis,
$$\sum_{i=1}^{n}\mathbb{E}\Big{[}(X_{i}-\tau)^{+}\Big{]}\quad\geq\quad\mathbb{E}%
\Big{[}\sum_{i=1}^{n}(X_{i}-\tau)^{+}\Big{]}\quad\geq\quad\mathbb{E}\Big{[}%
\max_{i}X_{i}-\tau\Big{]}\quad=\quad\tau.$$
Thus,
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]\quad\geq\quad P\cdot\tau+(1-P)\cdot%
\tau\quad=\quad\tau.$$
Appendix C Bounded $s_{\text{col}}$ and Small Cardinality Constraint
For the $r$-uniform matroid problem with bounded $s_{\text{col}}$, we have shown in Section 5.1 a $1+o(1)$ approximation for large $r$ tending to infinity.
Here, we complement that result with a gracefully-improving approximation ratio for all $r$ that smoothly interpolates between $O\left({s_{\text{col}}}\right)$ for $r=1$ (the classic result) and $O(1)$ for $r\geq s_{\text{col}}$.
The approach is an extension of our algorithm for bounded column sparsity in the $r=1$ case.
Theorem C.1.
For any $s_{\text{col}},r$, the linearly correlated prophets problem with column sparsity $s_{\text{col}}$ and cardinality constraint $r$ admits an approximation ratio of ${2e^{2}}\cdot\max\left\{1,\frac{s_{\text{col}}}{r}\right\}$.
In other words, as $r=2,3,\ldots,s_{\text{col}}$, the guarantee improves to a constant factor times $\frac{2}{s_{\text{col}}},\frac{3}{s_{\text{col}}},\ldots,1$.
Proof.
Let there be $r$ sets (“buckets”) $B_{1},\dots,B_{r}$.
If $r<s_{\text{col}}$, let there also be a “discard pile” $B_{0}$.
Let $c=\max\{r,s_{\text{col}}\}$.
For each $X_{i}$, place it in a bucket $j\in\{1,\dots,r\}$ each chosen with probability $\frac{1}{c}$.
If $r<s_{\text{col}}$, then with the remaining probability of $\frac{s_{\text{col}}-r}{s_{\text{col}}}$, place $X_{i}$ in the discard bucket $B_{0}$.
For each bucket $j=1,\dots,r$, give the bucket a cardinality constraint of $1$ item and run the following algorithm (based on the $r=1$ case).
Let $S_{j}=\{i:X_{i}\in B_{j}\}$ and note that they are disjoint for different $j$.
When a variable $X_{i}$ arrives, send it to the algorithm for its bucket, or if it is in $B_{0}$, discard $X_{i}$ and continue.
In bucket $j$, we run an inclusion-threshold algorithm with $S_{j}$ and with $\tau_{j}$ to be determined next.
Assign each $Y_{j^{\prime}}$ to the first $X_{i}\in B_{j}$ that includes it, i.e., let $T_{i}=\{j^{\prime}:A_{ij^{\prime}}>0\text{ and }A_{i^{\prime}j^{\prime}=0}(%
\forall i^{\prime}<i,i^{\prime}\in S_{j})\}$.
Let $Z_{i}=\sum_{j^{\prime}\in T_{i}}A_{ij^{\prime}}Y_{j^{\prime}}$, and let $\tau_{j}=\frac{1}{2}\mathbb{E}[\max_{i\in S_{j}}Z_{i}]$.
Claim C.2.
For each bucket $j\in\{1,\dots,r\}$, the expected value selected by the algorithm is at least $\frac{1}{2e}\cdot\mathbb{E}[\max_{i\in S_{j}}X_{i}]$.
Proof.
By construction each $Z_{i}$ is independent of all previous $X_{i}$ in the same bucket, so by the Augmentation Lemma 3.2, bucket $j$ obtains expected reward at least $\frac{1}{2}\mathbb{E}[\max_{i\in S_{j}}Z_{i}]$ with randomness over the variables.
For each $Y_{j^{\prime}}$ with $A_{ij^{\prime}}>0$, we claim $\Pr[j\in T_{i}]\geq\frac{1}{e}$ because there are at most $s_{\text{col}}-1$ other variables $X_{i^{\prime}}$ that include $Y_{j^{\prime}}$, and each misses bucket $j$ with probability at least $1-\frac{1}{s_{\text{col}}}$, so they all miss bucket $j$ with probability at least $(1-\frac{1}{s_{\text{col}}})^{s_{\text{col}}-1}\geq\frac{1}{e}$.
In this case, we must have $j\in T_{i}$.
So for each fixed $X_{i}$, we have with probability only over the bucket assignments and construction of $S_{j}$,
$$\displaystyle\mathbb{E}[Z_{i}]\quad=\quad\sum_{j^{\prime}}\Pr[j^{\prime}\in T_%
{i}]\cdot A_{ij^{\prime}}Y_{j^{\prime}}\quad\geq\quad\frac{1}{e}\sum_{j^{%
\prime}}A_{ij^{\prime}}Y_{j^{\prime}}\quad=\quad\frac{1}{e}X_{i}.$$
Combining these facts, each bucket $j$ obtains expected reward at least $\frac{1}{2e}\mathbb{E}[\max_{i\in S_{j}}X_{i}]$.
∎
Write $\mathbb{E}_{B}$ for an expectation taken over the bucketing and $\mathbb{E}_{\textbf{X}}$ for expectation over the realizations of the variables.
The above gives that for every set of variable realizations, $\mathbb{E}_{B}[\text{ALG}]\geq\frac{1}{2e}\sum_{j=1}^{r}\mathbb{E}_{B}[\max_{i%
\in S_{j}}X_{i}]$.
By linearity of expectation and symmetry of the buckets, we have
$$\displaystyle\mathbb{E}_{\textbf{X},B}[\text{ALG}]$$
$$\displaystyle\geq\frac{r}{2e}\mathbb{E}_{\textbf{X},B}\big{[}\max_{i\in S_{1}}%
X_{i}\big{]}.$$
(3)
Claim C.3.
$\mathbb{E}_{\textbf{X},B}\big{[}\max_{i\in S_{1}}X_{i}\big{]}\geq\frac{1}{e%
\cdot c}\cdot\mathbb{E}_{\textbf{X}}[\textsc{OPT}]$, where $c=\max\{r,s_{\text{col}}\}$.
Proof.
Let the random variable $X^{(i)}$ equal the $i$th-largest realized variable, i.e., in particular
OPT
$$\displaystyle=\sum_{i=1}^{r}X^{(i)}.$$
Recall that any fixed variable falls into bucket $B_{1}$ with probability $\frac{1}{c}$.
Fixing realizations of $X^{(1)},\dots,X^{(r)}$, we have with probability taken only over the buckets,
$$\displaystyle\mathbb{E}_{B}\big{[}\max_{i\in S_{1}}X_{i}\big{]}$$
$$\displaystyle\geq\sum_{i=1}^{r}\Pr\left[X^{(i)}\in B_{1}\text{ and }X^{(i^{%
\prime})}\not\in B_{1}(\forall i^{\prime}<i)\right]X^{(i)}$$
$$\displaystyle=\sum_{i=1}^{r}\frac{1}{c}\left(1-\frac{1}{c}\right)^{i-1}X^{(i)}%
\quad\geq\quad\frac{1}{e\cdot c}\sum_{i=1}^{r}X^{(i)}\quad=\quad\frac{1}{e%
\cdot c}\textsc{OPT}.$$
Now taking an expectation on both sides over the realizations of X proves the claim.
∎
Finally, combine Claim C.3 with Inequality (3) to get
$$\displaystyle\mathbb{E}[\text{ALG}]\quad\geq\quad\frac{r}{2e}\mathbb{E}_{%
\textbf{X}}\mathbb{E}_{B}\big{[}\max_{i\in S_{1}}X_{i}\big{]}\quad\geq\quad%
\frac{r}{2e^{2}c}\mathbb{E}_{\textbf{X}}[\textsc{OPT}],$$
which proves Theorem C.1.
∎
Appendix D Missing Proofs
D.1 Missing Proofs from Section 2
See 2.1
Proof of Lemma 2.1.
We consider the $s_{\text{row}}=s_{\text{col}}=2$ tower instance (Example 2.3), with sufficiently small $\epsilon$ chosen later.
We claim that $\mathbb{E}[\max_{i}X_{i}]=\Omega(n)$ while any fixed threshold-$\tau$ algorithm has $\mathbb{E}[\text{ALG}_{\tau}]\leq 3$.
First, we bound $\mathbb{E}[\text{ALG}_{\tau}]$.
Let $p_{j}=\Pr[\text{$\text{ALG}_{\tau}$ takes $X_{j}$ and $Y_{j}$ is active}]$.
In this case the algorithm’s reward includes $Y_{j}=\frac{1}{\epsilon^{j}}$ with coefficient $1$.
Let $p_{j}^{\prime}=\Pr[\text{$\text{ALG}_{\tau}$ takes $X_{j-1}$ and $Y_{j}$ is %
active}]$.
In this case its reward includes $Y_{j}=\frac{1}{\epsilon^{j}}$ with coefficient $\epsilon$.
We note that $p_{j},p_{j}^{\prime}\leq\Pr[\text{$Y_{j}$ is active}]=\epsilon^{j}$.
By summing over $Y_{1},\dots,Y_{n}$, we have
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]\quad=\quad\sum_{j=1}^{n}\Big{(}p_{j%
}\frac{1}{\epsilon^{j}}+(p_{j}^{\prime})(\epsilon)\frac{1}{\epsilon^{j}}\Big{)%
}\quad\leq\quad\sum_{j=1}^{n}\Big{(}p_{j}\frac{1}{\epsilon^{j}}+\epsilon\Big{)%
}\quad=\quad n\epsilon+\sum_{j=1}^{n}p_{j}\frac{1}{\epsilon^{j}}.$$
We argue that $p_{j}=0$ for all but at most two terms.
Let $j^{*}$ satisfy ${\epsilon^{-(j^{*}+1)}}<\tau\leq{\epsilon^{-j^{*}}}$, or $j^{*}=1$ if $\tau\leq\frac{1}{\epsilon}$.
We claim $p_{j}=0$ if $j\leq j^{*}-2$: assuming $\epsilon<\frac{1}{2}$, we have
$$X_{j}\quad\leq\quad\frac{2}{\epsilon^{j}}\quad\leq\quad\frac{1}{\epsilon^{j+1}%
}\quad<\quad\tau,$$
so $X_{j}$ is never taken.
We also claim $p_{j}=0$ if $j\geq j^{*}+1$: If $Y_{j}$ is active, then
$$X_{j-1}\quad\geq\quad\epsilon\frac{1}{\epsilon^{j}}\quad\geq\quad\frac{1}{%
\epsilon^{j-1}}\quad\geq\quad\tau,$$
so $X_{j-1}$ is taken and $X_{j}$ is not.
So we have $p_{j}=0$ unless $j\in\{j^{*}-1,j^{*}\}$, in which case $p_{j}\leq\epsilon^{j}$.
So
$$\displaystyle\mathbb{E}[\text{ALG}_{\tau}]\quad\leq\quad n\epsilon+\sum_{j=1}^%
{n}p_{j}\frac{1}{\epsilon^{j}}\quad\leq\quad n\epsilon+2\quad\leq\quad 3$$
for $\epsilon\leq\frac{1}{n}$.
For the benchmark, since $\max_{i}\{X_{i}\}\geq\max_{j}\{Y_{j}\}$, it suffices to show $\mathbb{E}[\max_{j}Y_{j}]=\Omega(n)$. Now,
$$\displaystyle\mathbb{E}[\max_{j}\{Y_{j}\}]$$
$$\displaystyle=\textstyle\sum_{i=1}^{n}\Pr[Y_{j}=0~{}(\forall j>i)]\cdot\Pr[Y_{%
i}\neq 0]\cdot\Big{(}\frac{1}{\epsilon^{i}}\Big{)}$$
$$\displaystyle\textstyle\geq\Pr[Y_{j}=0~{}(\forall j)]\cdot\sum_{i=1}^{n}\Pr[Y_%
{i}\neq 0]\cdot\Big{(}\frac{1}{\epsilon^{i}}\Big{)}.$$
Since $\Pr[Y_{i}\neq 0]=\epsilon^{i}$, we get
$$\displaystyle\mathbb{E}[\max_{j}\{Y_{j}\}]$$
$$\displaystyle\geq\quad\Pr[Y_{j}=0~{}(\forall j)]\cdot n$$
$$\displaystyle\geq\quad\textstyle\big{(}1-\sum_{j}\Pr[Y_{j}\neq 0]\big{)}\cdot n%
\quad\geq\quad\left(1-n\epsilon\right)\cdot n\quad\geq\quad{n}/{2}$$
for any choice of $\epsilon\leq\frac{1}{2n}$.
This gives an approximation ratio of at least $\frac{n/2}{3}=\frac{n}{6}$ for $\epsilon\leq\frac{1}{2n}$.
∎
D.2 Missing Proofs from Section 5.3
Proof of Claim 5.5.
First, consider supplementing $\textsc{OPT}^{\prime}$ by including tiny elements below $\tau_{c}$ when there is room, i.e. let $Z^{(i)\prime\prime}=Z^{(i)}\mathbf{1}[Z_{i}<\tau_{0}]$ and consider $\textsc{OPT}_{2}^{\prime\prime}=\sum_{j=1}^{r}Z^{(j)\prime\prime}$.
Let $\textsc{OPT}^{\prime\prime}=\textsc{OPT}_{1}^{\prime}+\textsc{OPT}_{2}^{\prime\prime}$.
On a case-by-case basis, $\textsc{OPT}^{\prime\prime}$ differs from $\textsc{OPT}^{\prime}$ by at most $r$ elements, each at most $\tau_{c}\leq\frac{\epsilon}{r}\mathbb{E}[\textsc{OPT}]$.
This proves that $\mathbb{E}[\textsc{OPT}^{\prime}]\geq\mathbb{E}[\textsc{OPT}^{\prime\prime}]-%
\epsilon\mathbb{E}[\textsc{OPT}]$.
We next prove that $\mathbb{E}[\textsc{OPT}^{\prime\prime}]\geq(1-\epsilon)\mathbb{E}[\textsc{OPT}]$, which completes the proof of the claim.
Let $H$ be the event there exists a heavy element, i.e. $\max_{i}Z_{i}\geq\tau_{0}=\mathbb{E}[\textsc{OPT}]/\epsilon$.
Let $p=\Pr[H]$. So,
$$\displaystyle\mathbb{E}[\textsc{OPT}^{\prime\prime}]$$
$$\displaystyle=p\cdot\mathbb{E}[\textsc{OPT}^{\prime\prime}\mid H]+(1-p)\cdot%
\mathbb{E}[\textsc{OPT}^{\prime\prime}\mid\lnot H]$$
$$\displaystyle=p\cdot\mathbb{E}[\textsc{OPT}^{\prime\prime}\mid H]+(1-p)\cdot%
\mathbb{E}[\textsc{OPT}\mid\lnot H]$$
$$\displaystyle\geq p\cdot\mathbb{E}[Z^{(1)}\mid H]+(1-p)\cdot\mathbb{E}[\textsc%
{OPT}\mid\lnot H].$$
(4)
Now, we claim $\mathbb{E}[\textsc{OPT}\mid H]\leq\mathbb{E}[Z^{(1)}\mid H]+\mathbb{E}[\textsc%
{OPT}]$.
Proof: let $M_{i}$ be the event that $i=\arg\max_{i^{\prime}}Z_{i^{\prime}}$ and $Z_{i}\geq\tau_{0}$.
Let $\textsc{OPT}_{-i}$ be the sum of the largest $r-1$ elements excluding $Z_{i}$.
Note that conditioning on all others lying below $Z_{i}$, for any fixed $Z_{i}$, only decreases $\textsc{OPT}_{-i}$, as the variables are independent.
$$\displaystyle\mathbb{E}[\textsc{OPT}\mid H]$$
$$\displaystyle=\mathbb{E}[Z^{(1)}\mid H]+{\textstyle\sum_{i=1}^{n}}\Pr[M_{i}%
\mid H]\cdot\mathbb{E}[\textsc{OPT}_{-i}\mid M_{i}]$$
$$\displaystyle\leq\mathbb{E}[Z^{(1)}\mid H]+{\textstyle\sum_{i=1}^{n}}\Pr[M_{i}%
\mid H]\cdot\mathbb{E}[\textsc{OPT}_{-i}]$$
$$\displaystyle\leq\mathbb{E}[Z^{(1)}\mid H]+{\textstyle\sum_{i=1}^{n}}\Pr[M_{i}%
\mid H]\cdot\mathbb{E}[\textsc{OPT}]\quad=\quad\mathbb{E}[Z^{(1)}\mid H]+%
\mathbb{E}[\textsc{OPT}].$$
Using this,
$$\displaystyle\mathbb{E}[\textsc{OPT}]$$
$$\displaystyle=p\cdot\mathbb{E}[\textsc{OPT}\mid H]+(1-p)\cdot\mathbb{E}[%
\textsc{OPT}\mid\lnot H]$$
$$\displaystyle\leq p\cdot\mathbb{E}[Z^{(1)}\mid H]+p\cdot\mathbb{E}[\textsc{OPT%
}]+(1-p)\cdot\mathbb{E}[\textsc{OPT}\mid\lnot H].$$
This implies
$$\displaystyle(1-p)\mathbb{E}[\textsc{OPT}]~{}~{}\leq~{}~{}p\cdot\mathbb{E}[Z^{%
(1)}\mid H]+(1-p)\cdot\mathbb{E}[\textsc{OPT}\mid\lnot H].$$
Combining with Inequality (4) gives $\mathbb{E}[\textsc{OPT}^{\prime\prime}]\geq(1-p)\mathbb{E}[\textsc{OPT}]$.
We have $p\leq\epsilon$ by Markov’s inequality: $p=\Pr[\max_{i}Z_{i}\geq\tau_{0}]\leq\mathbb{E}[\max_{i}Z_{i}]/\tau_{0}\leq%
\mathbb{E}[\textsc{OPT}]/\tau_{0}=\epsilon$.
This completes the proof.
∎
Proof of Lemma 5.7.
Suppose $r$ is large enough that $\epsilon\leq 0.5$, otherwise the lemma is immediate.
Let $\delta:=\frac{1+c\cdot\beta}{r}$.
Claim D.1.
$\epsilon\geq 2\delta$.
Proof.
Using that $\epsilon\geq r^{-1/4}$, we have $c:=\lceil\frac{1}{\epsilon}\ln\frac{r}{\epsilon^{2}}\rceil\leq\frac{3}{%
\epsilon}\ln(r)$.
Further using that $\epsilon\geq 3r^{-1/4}$, this implies $\ln\frac{c}{\epsilon}\leq\ln\left(r^{1/2}\ln(r)\right)\leq\ln(r)$.
Therefore, $\beta:=3\sqrt{r\ln\frac{c}{\epsilon}}\leq 3\sqrt{r\ln(r)}$.
Then $c\cdot\beta\leq\left(\frac{3}{\epsilon}\ln(r)\right)\left(3\sqrt{r\ln(r)}%
\right)\leq\frac{9\sqrt{r}\left(\ln r\right)^{3/2}}{\epsilon}$.
Using that $\epsilon\geq 9r^{-1/4}\left(\ln r\right)^{3/2}$, this gives $c\cdot\beta\leq r^{3/4}$, so $1+c\cdot\beta\leq 2r^{3/4}$, so $\delta\leq 2r^{-1/4}\leq\epsilon/2$.
∎
Let $K$ be the number of arrivals taken by ALG, a random variable.
Claim D.2.
With at least $1-\epsilon$ probability, $K<r$ (i.e. ALG does not reach its cardinality constraint).
Proof.
The number of arrivals taken by $\text{ALG}^{\prime}$ is at most $\sum_{j=0}^{c}\tilde{r}_{j}=1+c\cdot\beta+\sum_{j=1}^{c}r_{j}$.
Because OPT takes at most $r$ arrivals pointwise, and thus in expectation, we have $\sum_{j=1}^{c}r_{j}\leq r$.
So $\text{ALG}^{\prime}$ takes at most, in the worst case, $K^{\prime}=r+1+c\cdot\beta=r(1+\delta)$ arrivals.
Because ALG keeps each independently with probability $1-\epsilon\leq 1-2\delta<(1-\delta)^{2}$, the chance it reaches $r$ is upper-bounded by the chance that a Binomial($K^{\prime},(1-\delta)^{2}$) variable exceeds $r$.
This is upper-bounded by the chance it exceeds $K^{\prime}(1-\delta)^{2}(1+\delta)=r(1-\delta)^{2}(1+\delta)^{2}=r(1-\delta^{2%
})^{2}<r$.
So by a Chernoff bound,
$$\displaystyle\Pr[K\geq r]\quad\leq\quad\Pr[\text{Binom}(K^{\prime},(1-\delta)^%
{2})\geq K^{\prime}(1-\delta)^{2}(1+\delta)]\quad\leq\quad\exp\left(\frac{-%
\delta^{2}K^{\prime}(1-\delta)^{2}}{3}\right).$$
We have $\delta\leq\epsilon/2\leq 0.25$, and $K^{\prime}\geq r$, so $K^{\prime}(1-\delta)^{2}\geq\frac{r}{2}$.
Also, $\delta\geq\frac{c\cdot\beta}{r}\geq\frac{\beta}{r}$.
$$\displaystyle\Pr[K\geq r]$$
$$\displaystyle~{}\leq~{}\exp\left(\frac{-\delta^{2}r}{6}\right)~{}\leq~{}\exp%
\left(\frac{-\beta^{2}}{6r}\right)~{}\leq~{}\exp\left(-\ln\frac{c}{\epsilon}%
\right)~{}\leq~{}\epsilon.\qed$$
Now, each time $\text{ALG}^{\prime}$ obtains some variable $X_{i}$, ALG also obtains it unless either: it has reached its cardinality constraint; or it independently discards $X_{i}$ (with probability $\epsilon$).
By a union bound over these two events, when $\text{ALG}^{\prime}$ obtains $X_{i}$, ALG also obtains it except with probability $1-2\epsilon$.
So $\mathbb{E}[\text{ALG}]\geq(1-2\epsilon)\mathbb{E}[\text{ALG}^{\prime}]$, which proves Lemma 5.7.
∎
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Band Gap Tuning of DC Reactively Sputtered ZnON Thin Films
††thanks: Citation:
Authors. Title. Pages…. DOI:000000/11111.
Kiran Jose,
National Institute of Technology Calicut
Kozhikode, Kerala, India
[email protected]
\AndJG Anjana
National Institute of Technology Calicut
Kozhikode, Kerala, India
[email protected]
\ANDVenu Anand
National Institute of Technology Calicut
Kozhikode, Kerala, India
[email protected]
\ANDAswathi R Nair
National Institute of Technology Calicut
Kozhikode, Kerala, India
[email protected]
Abstract
Zinc oxynitride (ZnO${}_{x}$N${}_{y}$) has recently emerged as a highly promising band gap-tunable semiconductor material for optoelectronic applications. In this study, a novel DC reactive sputtering protocol was developed to fabricate ZnO${}_{x}$N${}_{y}$ films with varying elemental concentrations, by precisely controlling the working pressure. The band gap was rigorously analyzed using UV-Visible spectroscopy, which was complemented by EDAX spectroscopy to determine the variations in the elemental composition. The correlation between the microstructure and band gap was investigated through the application of AFM, XRD, and Raman spectroscopy, while the Urbach theorem was used to evaluate the defect states. This study revealed the existence of intermediate structures formed during the tuning of the band gap, which can have important implications for future research aimed at developing heterostructures and 2D superlattices for photonics applications.
Keywords Zinc Oxynitride $\cdot$
Thin Film $\cdot$
DC Sputtering $\cdot$
Band Gap Tuning $\cdot$
Urbach’s Tail States $\cdot$
XRD $\cdot$
Raman Spectroscopy $\cdot$
AFM.
1 Introduction
The optoelectronics industry is constantly seeking materials with excellent electrical and optical properties to meet the demands of modern computational algorithms. Amorphous semiconductors are a versatile class of materials that have found wide-spread applications in the field of electronics and flexible electronics, including in the areas of thin-film transistors (TFTs)[1], LCD displays[2], solar cells[3], sensors for gas[4], humidity, temperature, and pressure detection[5], memory devices, neuromorphic devices[6], photodetectors, photovoltaics[7], and spintronics[8]. However, low electron mobility, a wide band gap, and persistent photoconductivity (PPC)[9] of amorphous semiconductors, such as oxides, limit the performance and efficiency of devices such as phototransistors and sensors, making it difficult to meet the demands of modern technology. Researchers have turned to alternative materials like zinc oxynitride (ZnO${}_{x}$N${}_{y}$) which is a band gap tunable semiconductor material with high electron mobility of up to 100V/s[10] for improved performance and efficiency in devices like phototransistors[11]. Additionally, ZnO${}_{x}$N${}_{y}$ has a low persistent photoconductivity (PPC)[12][13], which means it does not store electrical charges for a long time after the light is turned off; this improves the operational stability of the device. These properties make ZnO${}_{x}$N${}_{y}$ a prospective substitute for optoelectronics and photonics applications.
One of the most promising forms of this material is a thin film, which has been the subject of a number of recent research studies[14][15][16]. These studies have investigated the properties and potential applications of ZnO${}_{x}$N${}_{y}$ thin films, including their optical and electronic properties[16-20], as well as their potential use in devices such as light-emitting diodes (LEDs)[17], solar cells, and sensors[18]. Recent research has also focused on developing methods for synthesising high-quality ZnO${}_{x}$N${}_{y}$ thin films, including chemical vapour deposition (CVD)[19] and pulsed laser deposition (PLD) techniques[20].
Sputtering is a well-established technique for depositing thin films of various materials, including ZnO${}_{x}$N${}_{y}$[14]. In reactive sputtering, a target material is bombarded with energetic ions in the presence of various reactive gas ions, which causes the target atoms to be ejected and deposited onto a substrate to form a thin film. One of the key advantages of this technique is the ability to control the stoichiometry and composition of the deposited film by adjusting the sputtering conditions and the composition of the target material[21]. Studies have shown that it is possible to deposit ZnO${}_{x}$N${}_{y}$ thin films with varying nitrogen content, which can affect the film’s electrical and optical properties, making it a powerful technique for depositing ZnO${}_{x}$N${}_{y}$ thin films for various applications[14][17][18].
However, there are challenges associated with reactively sputtering ZnO${}_{x}$N${}_{y}$ thin films. These include a decrease in sputtering yield due to the reacting gas, the formation of a layer of deposited film in the target (target poisoning), and the difference in reactivity of gases when more than one gas is used. Deposition of ZnO${}_{x}$N${}_{y}$ film involves three gases, Argon the sputtering gas, Nitrogen with relatively low reactivity and highly reacting Oxygen, which leads to further complexity. Thus, the presence of residual oxygen can be a major challenge in preparing ZnO${}_{x}$N${}_{y}$ thin films and can even call into question the repetition of the experiment.
The ability to tune the band gap is a significant aspect of ZnO${}_{x}$N${}_{y}$ that draws attention of researchers for its potential use in optoelectronic devices, such as photosensors[22] and high-mobility transistors[23]. The careful selection of the band gap region is of utmost importance, as it has a direct correlation with the morphology, crystal structure, and defect states within the semiconductor, which can greatly impact the device’s performance. Hence, the regulation of the band gap and the comprehension of the related microstructures of ZnO${}_{x}$N${}_{y}$ thin films are imperative for successful utilization in optoelectronic devices.
This work endeavours to address the challenges encountered during the sputtering process of ZnO${}_{x}$N${}_{y}$ thin films, with the objective of finding the optimal conditions for tuning the band gap of the deposited film. The protocol involves adjusting the pressure of the sputtering process by introducing various partial pressures of nitrogen, while keeping the other sputtering parameters constant. Even though a direct current (DC) power source was utilized in this work, the consideration of a radio frequency (RF) power source is also being explored as a potential solution to mitigate target poisoning, which is a prevalent issue in sputtering. This is achieved through the prevention of ion buildup on the target material, facilitated by the alternating polarity of the RF power source.
Once the deposition conditions have been optimized for different band gaps, the study aims to delve deeper into the characterization of the film, examining the evolution of its structural phases, morphology, and defect states as the band gap changes. This comprehensive analysis will be carried out using various state-of-the-art characterization techniques such as Atomic Force Microscopy(AFM), X-ray Diffraction Analysis (XRD), Raman Spectroscopy, and UV-Visible Spectroscopy, to provide a comprehensive understanding of the properties of the ZnO${}_{x}$N${}_{y}$ thin film.
2 Experimental Details
2.1 DC Reactive Sputtering
In this study, ZnO${}_{x}$N${}_{y}$ thin films were prepared using a reactive sputtering technique. A zinc target was sputtered using argon in the presence of both oxygen and nitrogen gases with a DC power source. The major sputtering parameters, including the target material and size, DC power, base pressure, sputtering power, the flow rate of different gases etc., are listed in Table 1. The dc power for all other depositions except for studying the effect of sputtering power was 35 W. The substrates used for characterization were silicon for AFM, Energy Dispersive X-ray Analysis (EDAX), XRD, Raman spectroscopy and Corning Plain Microscope Glass Slide for UV-visible spectroscopy.
The substrates were cleaned by sonicating for 15 minutes in acetone, followed by isopropyl alcohol and doubly deionized water. The target was also cleaned by pre-sputtering using an RF plasma before every sputtering, and the sputtering time was kept to a minimum to reduce the target poisoning.
During the initial phase of the study, the strategy for optimizing the band gap of ZnO${}_{x}$N${}_{y}$ thin film deposition was carefully evaluated. This was achieved by maintaining a constant flow rate of argon at 5 sccm while adjusting the flow rate of nitrogen from 15 to 24 sccm. It is noteworthy that during the deposition process, the chamber was not supplemented with any additional oxygen through the use of mass flow controllers. Instead, the residual oxygen that remained within the chamber after attaining the base pressure was utilized.
2.2 Tauc Plot
The band gap of ZnO${}_{x}$N${}_{y}$ thin films is determined from UV-visible spectroscopy by employing Tauc equation[24],
$$\begin{array}[]{ll}\alpha h\nu=A(h\nu-E_{g})^{n}\end{array}$$
(1)
Where $\alpha$ is the absorption coefficient, $h\nu$ is the photon energy, A is a constant, E${}_{g}$ is the band gap energy, and $n$ is a material constant that is typically taken as 2 for direct band gap semiconductors and 1/2 for indirect band gap semiconductors for allowed transitions. Here, we plot ($\alpha$h$\nu$)${}^{2}$ with h$\nu$[25], and by analyzing the linear portion of the Tauc plot, the direct band gap energy (Eg) is determined by extrapolating to the energy axis.
2.3 Determining Thickness
The deposition rate of the thin film was found by dividing the thickness of the film by the sputtering time. AFM was used to find the thickness of the film. For this, a portion of the silicon wafer was masked, so that no film would be formed in the masked area. By measuring the film height at this step, we estimated the thickness of the film. The roughness parameter of the film was also obtained from the AFM measurement.
2.4 Urbach Energy
The Urbach tail energy can be calculated by analyzing the absorption spectra obtained from UV-visible spectroscopy using Urbach’s rule. The Urbach rule describes the exponential decay of the absorption coefficient as a function of energy and is given by[26]
$$\begin{array}[]{ll}\alpha(E)=\alpha_{0}exp(E/E_{u})\end{array}$$
(2)
where E is the energy and $E_{u}$ is the Urbach tail energy. To calculate the Urbach tail energy, the natural logarithm of the absorption coefficient, ln($\alpha$), is plotted as a function of energy, E. The slope of this plot, just before the band gap energy, is given by -1/$E_{u}$. So the Urbach tail energy, $E_{u}$, can be calculated by taking the reciprocal of the slope[27].
3 Results and Discussion
3.1 Band Gap Engineering
3.1.1 Band Gap Tuning
The way to tune the band gap of ZnO${}_{x}$N${}_{y}$ is by varying the composition of the thin film. A combination of a higher atomic concentration of nitrogen and lower concentration of oxygen will give a low band gap and vice versa[28]. The detailed study of the relation between composition and band gap can be seen in section 3.1.2. Previous research has focused on altering nitrogen-to-oxygen ratios[11] at high sputtering power[29] and working pressures[30] to produce varying compositional stoichiometries. In the present study, we suggest a novel method for accomplishing this with low working pressure, low power, and utilisation of residual oxygen without introducing further oxygen to the process.
In earlier studies, the ZnO${}_{x}$N${}_{y}$ thin films were developed by alternating a 100 to 200 SCCM flow rate of nitrogen with 1 to 10 SCCM of oxygen[13] [30] to balance the disparity in reactivity between the two gases. However, our approach aims to achieve the same result by varying the working pressure inside the chamber. The challenge of incorporating nitrogen in the film is due to the high ionization energy of the nitrogen molecule compared to the oxygen molecule. Since nitrogen atoms are triple-bonded to form a molecule, high energy is required to dissociate this bond[31]. As a result, most of the energy will be used for the ionization of the nitrogen molecule, and the remaining will only be reflected in the kinetic energy.
In order to achieve varying incorporation of nitrogen in the film, we varied the flow rate of nitrogen gas while keeping other parameters constant. The mass flow rate of Argon was kept at 5sccm, and the flow rate of nitrogen was varied from 15 sccm to 24 sccm, with a sputtering power of 35W. The gate valve was kept completely open during all the experiments. The working pressure was monitored and recorded during the experiment; the results are shown in Fig.1. The graph shows an exponential increase in working pressure as the flow rate of nitrogen increases.
It is seen that (Fig.1) the band gap of the film initially decreases and reaches a minimum of 1.6 eV at a flow rate of 20 sccm of nitrogen, after which it drastically increases. This suggests that the incorporation of nitrogen increases as the flow rate of nitrogen increases, reaching a maximum at this point and then falls. The initial trend of increased nitrogen incorporation is a result of the rise in the flow rate of nitrogen, which in turn elevates the partial pressure of nitrogen within the chamber. This finds a optimum point at a certain partial pressure where there is enough amount of nitrogen. However, the fall in the incorporation is due to increased working pressure. After reaching the optimum point, an increase in working pressure leads to a decrease in the number of nitrogen ions that reach the substrate. As discussed earlier, nitrogen ions will have the least kinetic energy among other ions because of the high ionization energy. Despite the high concentration of nitrogen within the chamber, the nitrogen ions may not possess sufficient energy to reach the substrate due to a large number of collisions caused by increased working pressure. This, in turn, leads to a decrease in nitrogen incorporation as the nitrogen flow rate continues to increase.
The trend in the deposition rate of the film as a function of the nitrogen flow rate, as depicted in Fig.1, displays a persistent decline as the concentration of gases within the chamber increases. This initial decline in the deposition rate, which is less pronounced, can be attributed to the conventional phenomenon of reactive sputtering, in which the sputtering rate decreases with increasing amounts of reactive gas. However, after a nitrogen flow rate of 20 sccm, a significant decline in the deposition rate can be observed. This decline is the result of the increased working pressure, as previously discussed, which results in an increase in the number of collisions and a reduction in the probability of the sputtering ions reaching the substrate.
The tuning of the band gap in this study was achieved by controlling the mean free path of ions, specifically nitrogen ions, within the sputtering chamber. The mean free path refers to the average distance travelled by a particle between two consecutive collisions in a gas or solid. In a gaseous medium, it represents the average distance a molecule can travel without encountering another molecule. Before a working pressure of 0.013 mbar (achieved through the application of a nitrogen flow rate of 20 sccm and an argon flow rate of 5 sccm after attaining a base pressure of 4.2 x 10${}^{-6}$ mbar), the ions possessed sufficient path to traverse without collision. At this stage, the mere increase in the amount of reactive gases resulted in a corresponding increase in the incorporation of the element. However, as the nitrogen flow rate was increased to 21 sccm, the working pressure increased to 0.016 mbar, leading to a greater number of collisions and a reduction in the mean free path of the ions. This decrease affected all reactive ions, with a significant drop in deposition rate observed in this region. The impact was particularly pronounced for nitrogen, as a result of its high ionization energy and corresponding lower kinetic energy, leading to a greatly reduced mean free path. As the flow rates were increased further, the working pressure increased nearly exponentially, leading to an exponential decrease in nitrogen incorporation.
The variation of the band gap with respect to the power is presented in Fig.1, reinforcing our proposed theory. As previously noted, the band gap is affected by the mean free path of nitrogen ions in the sputtering chamber. The lower kinetic energy of the nitrogen ions makes it less likely for them to reach the substrate when the working pressure is high, resulting in reduced incorporation of nitrogen. However, by increasing the power, the kinetic energy of the nitrogen ions can also be increased, leading to a higher mean free path and a greater probability of nitrogen ions reaching the substrate.
It was observed that when the flow rate of argon was kept constant at 5 sccm and the nitrogen flow rate was increased to 24 sccm, the band gap was measured to be 3.25 eV at a sputtering power of 35W. This suggests low nitrogen incorporation in the sample. However, increasing the sputtering power to 70W and 105W under the same conditions led to a reduction of the band gap to 2.58 eV and 2.2 eV, respectively (as depicted in Fig.1). This shows that as the power increases, so does the kinetic energy of the nitrogen ions and their mean free path, leading to a higher incorporation of nitrogen into the ZnO${}_{x}$N${}_{y}$ thin films. These results support our theory about controlling the mean free path of nitrogen ions for band gap tuning and suggest that adjusting the power can also be used as an alternative method, in addition to controlling working pressure.
3.1.2 Band Gap and Elemental Composition
The band gap of the ZnO${}_{x}$N${}_{y}$ thin films was characterized using a Tauc plot, derived from the transmittance spectra (as shown in Fig.2). Further insight into the variation in the material’s elemental composition with band gap was obtained through EDAX spectroscopy, as shown in Fig.2. Our observations reveal a clear correlation between the decreasing band gap of the material and the increasing concentration of nitrogen within the film.
ZnO${}_{x}$N${}_{y}$ is an alloy derived form from parental compounds Zinc nitride (Zn${}_{3}$N${}_{2}$) and Zinc oxide (ZnO). The band gap of pure Zn${}_{3}$N${}_{2}$ is reported to be 1.1 eV, while the band gap of pure ZnO is 3.25 eV[12]. One of the main reasons for this difference in band gap is the relative energy levels of the N 2p orbitals and O 2p orbitals[28]. In Zn${}_{3}$N${}_{2}$, the N 2p orbitals, which are observed above the O 2p orbitals, act as the valence band while the conduction band is dominated by the spherical Zn 4s orbital, similar to the case of ZnO[32][30]. This results in a lower band gap of Zn${}_{3}$N${}_{2}$ compared to ZnO. And when it comes to the case of ZnO${}_{x}$N${}_{y}$, as the atomic concentration of nitrogen increases, more electrons occupy the 2P level of nitrogen and start acting as the valence band, which in turn reduces the band gap[33][13].
3.2 Structural Transition
In this section, we aim to gain a comprehensive understanding of the structural evolution in ZnO${}_{x}$N${}_{y}$ thin films as a function of the band gap. Previous studies have established a transition from the cubic structure to a hexagonal wurtzite structure as the band gap increases. However, there remains a significant gap in our understanding of the intermediate phases involved in this structural evolution. To address this, we employ a combined approach of XRD and Raman spectroscopy to provide a thorough characterization of the structural changes. XRD measurements were performed using a Malvern Panalytical system equipped with Cu K$\alpha$ radiation, while Raman spectra were collected after exciting the molecules with a 532 nm wavelength laser on a Labram HR Evolution confocal Raman microscope. The integration of these techniques provides a comprehensive evaluation of the structural properties of the ZnO${}_{x}$N${}_{y}$ thin films and the transitions involved in their formation.
The XRD patterns of ZnO${}_{x}$N${}_{y}$ thin films often showcase distinctive peaks within the 2$\theta$ range of 31${}^{0}$ to 35${}^{0}$, offering a comprehensive understanding of the structural properties of the films. A prominent peak reveals the hexagonal wurtzite structure of ZnO at approximately 34.442${}^{0}$ 2$\theta$ (JCPDS file, card no. 01-074-0534), while a peak indicates the cubic structure of Zn${}_{3}$N${}_{2}$ at approximately 31.3${}^{0}$ 2$\theta$ (JCPDS file, card no. 00-035-0762). As such, the 2$\theta$ range of 31${}^{0}$ to 35${}^{0}$ holds significant importance, as the XRD peaks of ZnO${}_{x}$N${}_{y}$, being an alloy of these two compounds, are expected to be situated between these two peaks.
Additionally, the hexagonal wurtzite structure of ZnO comprises four atoms per unit cell and belongs to the $C^{4}_{6v}(P6_{3}mc)$ space group[34][35]. It exhibits a particular Raman E2 phonon mode at 434 cm${}^{-1}$[36]. On the other hand, Zn${}_{3}$N${}_{2}$ is known to crystallize in the anti-bixbyite structure, which is characterized by a body-centered cubic lattice and belongs to the space group $Ia-3$[37]. A notable feature of pure Zn${}_{3}$N${}_{2}$ cubic lattice is the zero Raman shift observed in its spectrum[38].
Fig.3 and 3 present the structural transition of ZnO${}_{x}$N${}_{y}$ thin films with band gaps through XRD and Raman patterns, respectively. The patterns have been differentiated through the use of various colours, each representing a specific band gap value. The yellow pattern corresponds to a band gap of 1.66 eV, while the purple pattern corresponds to 2.15 eV. The green pattern represents a band gap of 2.7 eV, and the grey pattern corresponds to 3.25 eV.
In the case of ZnO${}_{x}$N${}_{y}$ thin films with the lowest band gap of 1.66 eV, XRD analysis reveals two prominent peaks at 32.89${}^{0}$ and 32.98${}^{0}$, accompanied by Raman peaks at shifts of 239, 352, and 589 cm${}^{-1}$. The XRD peaks are attributed to the formation of an alloy within the film structure, blending the hexagonal wurtzite ZnO and cubic Zn${}_{3}$N${}_{2}$. The Raman peak at 239 cm${}^{-1}$ is related to the Zn-N bonding configurations and is believed to result from local vibrational modes (LVMs) induced by intrinsic lattice defects in the Zn${}_{3}$N${}_{2}$ films[39]. The origin of the Raman band located in the range of 566 to 595 cm${}^{-1}$ has been variously attributed by different researchers to local vibrational modes related to nitrogen, lattice defects within the host, and a fusion of local vibrational modes and disorder-activated scattering [40][38][41]. This peak has been observed in every sample studied. The peak at 353 cm${}^{-1}$, on the other hand, has not been previously identified and is thought to be specific to the alloy structure. This peak decreases as the band gap increases and eventually disappears as the band gap reaches 3.3 eV, corresponding to pure ZnO.
As the band gap increases to 2.23 eV, the XRD patterns become broader, with peaks appearing at 32.77${}^{0}$ and 32.84${}^{0}$. And along with the previous peaks in the Raman spectrum, an additional peak at 445 cm${}^{-1}$ has been detected, which is closely linked to the characteristic wurtzite peak of ZnO. This suggests that as the band gap reaches 2.23 eV, the previously established alloy structure begins to exhibit subtle alterations, leading to the emergence of a distorted ZnO structure.
At 2.78 eV, the XRD and Raman spectra indicate a transformation and progression of the film structure toward a more dominant ZnO character. The XRD peaks that correspond to the alloy phase exhibit a slight shift, whereas a striking characteristic peak of wurtzite ZnO at 34.08${}^{0}$ is detected. Additionally, the Raman peak at 424 cm${}^{-1}$ has shifted to 434 cm${}^{-1}$, providing further validation of the formation of the ZnO phase. Consequently, in this band gap range, a co-existence of both the ZnO${}_{x}$N${}_{y}$ alloy phase and the wurtzite ZnO phase can be observed.
Finally, when the band gap exceeds 3 eV, the alloy phase completely vanishes, yielding a single XRD peak at 33.85${}^{0}$ (JCPDS file, card no. 00-021-1486) of ZnO. In the Raman spectra, the wurtzite peak of ZnO (434 cm${}^{-1}$) becomes the dominant feature, while the peak at 350 cm${}^{-1}$, previously related to the alloy structure, is no longer present. The 239 cm${}^{-1}$ peak is still present with decreased intensity, implying that it is related to some intrinsic defects due to nitrogen. Additionally, the Raman spectra of the 3.3 eV sample reveal several peaks that are closely linked to the wurtzite structure. Of particular significance is the 275 cm${}^{-1}$ peak, which is associated with N-doped ZnO and has been argued to represent a B${}^{low}_{1}$ mode of disorder-activated Raman scattering, a mode that is forbidden in wurtzite ZnO[35],[42],[43]. The 619 and 670 cm${}^{-1}$ peaks, identified by Ribut et al.[44], are related to ZnO deposited on Si substrates. Thus, the results suggest that a wurtzite ZnO structure with a high degree of structural disorder due to nitrogen doping has been formed when the band gap surpasses 3 eV.
3.3 Morphological Evolution
The morphological transition in the thin film was analyzed through the use of XRD and AFM techniques. To determine the average size of crystalline particles, the well-known Scherrer equation was employed in the analysis. The equation calculates crystallite size, represented by D, through a combination of various variables including the Scherrer constant (K), X-ray wavelength ($\lambda$), full width at half maximum (FWHM) of the XRD peak ($\beta$), and the Bragg angle ($\theta$), given by[45],
$$\begin{array}[]{ll}D=K\lambda/\beta cos(\theta)\end{array}$$
(3)
For samples presenting multiple peaks, a weighted average was determined using the intensity of each peak as a weighting factor. AFM allowed for the acquisition of high-resolution images of the thin film surface, enabling a comprehensive evaluation of its morphological features, including grain size, shape, and distribution, as well as roughness.
The effect of the nitrogen flow rate on crystallite size is presented in Fig.4. It was observed that the film with the largest crystallite size was formed at a flow rate of 20 sccm. As the flow rate deviated from this value, either by increasing or decreasing, the crystallite size decreased. As depicted in Figure 3a, the XRD peaks of the alloy phase were found to be very sharp. In the transition states, however, the peaks became broader and the crystallite size started to decrease. When the film reached the wurtzite phase, the crystallite size was significantly reduced due to the high working pressure[46], which resulted in the formation of a disordered phase with numerous nitrogen-induced defects.
The pattern of roughness in the thin film followed a similar trend as the crystallite size (Fig.4). Increased crystallinity of the film resulted in a corresponding increase in roughness, with the maximum roughness observed in the film formed at 20 sccm with the minimum band gap and optimal crystal quality. Films formed at flow rates greater than 20 sccm showed a sharp decrease in roughness, which was attributed to both the high working pressure[47] and the higher degree of amorphous behavior (smaller crystallite size). The variation in film roughness is further visualized through 3D images presented in Fig.5. The AFM images showed that the film became smoother as the band gap increased, although no clear trend was observed with respect to the band gap. The roughness of the film was found to be influenced by both the working pressure and the structural transitions, with both factors contributing equally to the observed variation.
3.4 Urbach Tail States
The Urbach tail is a term used to describe the tail of the density of states (DOS) at the bottom of the conduction band of semiconductors[48][49]. The tail is due to the presence of impurities and defects in the material, which introduces additional energy states near the band gap[50]. In the case of (ZnO${}_{x}$N${}_{y}$) thin films, the Urbach tail can be attributed to the presence of oxygen vacancies and nitrogen interstitials[51][52]. These defects act as shallow acceptors and can introduce energy states within the band gap, which can affect the electronic properties of the material[53].
The Urbach energy is closely related to the band tail width of localized states in semiconductors, and an increase in Urbach energy can be attributed to an increase in defect states[54]. The presence of disorder and defects within thin films can result in the emergence of localized states close to the conduction band level, which causes an expansion of the band tail width ($E_{U}$).
As depicted in Fig. 6, the Urbach energy in ZnO${}_{x}$N${}_{y}$ thin films was found to have a correlation with the band gap. Despite the indirect influence of the nitrogen flow rate on the defect states, which in turn affects the crystallite size, the effect of the structural transition on the Urbach energy was more prominent. The energy initially rises with the increase in the band gap, reaching a peak before decreasing. This upward trend can be attributed to the transition from an alloy state to a mixture of alloy and wurtzite phases, resulting in an increase in structural disorders, dislocations, and vacancies. The peak of Urbach energy corresponds to the coexistence of both phases, where disorder is at its highest. The decline in Urbach energy can be explained by the domination of the wurtzite structure, although the energy remains significantly higher compared to that of ZnO thin films (80-90 meV)[55], indicating the presence of a distorted lattice. It is clear that the evolution of tail states is inextricably linked to structural evolution and follows a similar pattern.
4 Conclusion
In conclusion, a novel strategy for tuning the band gap of ZnO${}_{x}$N${}_{y}$ films has been developed through the manipulation of the working pressure to adjust the mean free path of nitrogen ions during sputtering. The structural evolution of the film with changing band gap was extensively studied, and intermediate structures were identified through XRD and Raman spectroscopy. The initial alloy structure of the film was found to exist from 1.66 eV to 2.15 eV, beyond which a distorted wurtzite structure began to emerge, as indicated by the 451 cm${}^{-1}$ peak in the Raman spectra. At a band gap of 2.74 eV, the peak shifted to 434 cm${}^{-1}$, becoming more prominent and indicating the coexistence of both alloy and wurtzite structures. With an increasing band gap, the wurtzite structure became dominant, completely replacing the alloy structure at 3.25 eV. Additionally, a special Raman peak at 350 cm${}^{-1}$ was identified and thought to be associated with the alloy phase, which disappeared in the fully developed wurtzite phase. The study of the Urbach tails also revealed that the disorder in the film was maximum when two structures coexisted. These findings offer a profound understanding of the structural evolution of ZnO${}_{x}$N${}_{y}$ films and pave the way for further advancements in optoelectronic devices.
5 Acknowledgement
The authors would like to express their gratitude to the various organizations and individuals who supported and contributed to this research. The Department of Science and Technology, SERB provided a Start-Up Grant from 2021 to 2023 and a Faculty Research Grant (FRG) at the National Institute of Technology Calicut (NITC) provided financial support. The Center for Materials Characterization (CMC) at NITC provided access to the Raman and X-ray Diffraction (XRD) facilities and the authors would like to thank Mr. Shintu Varghese for his technical assistance with the Raman facility, Mr. Nithin for his technical support with the XRD analysis, and the staff at CeNS at the Indian Institute of Science (IISc) for their technical support with the AFM facility. The authors also appreciate the assistance of the Department of Chemistry and Mr. Diljith with the UV-Visible Spectrophotometer.
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Monthly Modulation in Dark Matter Direct-Detection Experiments
Vivian Britto
[email protected]
Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada
Joel Meyers
[email protected]
Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada
(January 11, 2021)
Abstract
The count rate in dark matter direct-detection experiments should exhibit modulation signatures due to the Earth’s motion with respect to the Galactic dark matter halo. The annual and daily modulations, due to the Earth’s revolution about the Sun and rotation about its own axis, have been explored previously. Monthly modulation is another such feature present in rate counts, and provides a nearly model-independent method of distinguishing dark matter signal events from background. We study here monthly modulations in detail, examining both the effect of the motion of the Earth about the Earth-Moon barycenter and the gravitational focusing due to the Moon. We show that the former is the dominant source of monthly modulation, and that the amplitude of the monthly modulation varies on an annual cycle. The expected amplitude of monthly modulation is quite small which makes its detection challenging; any such detection however, would provide very strong evidence that candidate events are due to dark matter scattering.
I Introduction
There exists a preponderance of evidence that some form of non-baryonic dark matter makes up a significant fraction of the mass in the universe; its detailed properties however, remain a mystery (for reviews see Jungman et al. (1996); Bertone et al. (2005)). Understanding the nature of dark matter holds considerable importance for particle physics, high energy theory, astrophysics, and cosmology, and there has been a great deal of effort expended toward achieving a better understanding of dark matter through experiment. Several intriguing hints notwithstanding, there so far have been no conclusive results to this end.
Dark matter direct-detection experiments seek to measure the interaction of dark matter particles streaming through the Earth with the ordinary matter that makes up a detector Goodman and Witten (1985); Smith and Lewin (1990); Lewin and Smith (1996): collisions in detector material may result in ionization along with the deposit of heat and/or light, which are measured.111Our discussions will be framed primarily in terms of direct-detection experiments which search for dark matter resembling weakly interacting massive particles. Experiments which search for axion-like particle dark matter Asztalos et al. (2010) also exhibit modulation signals analogous to those discussed throughout this work, though they are manifested in a different way Ling et al. (2004). Many such experiments are currently in operation around the world (see Sec. 24 of Beringer et al. (2012) for a recent survey of the experimental situation).
One of the main challenges these experiments face is the presence of backgrounds: cosmic rays and radioactive decays in the material in and around the experiment, for instance, can create signals which mimic those of dark matter scattering events. Due to such effects, experiments are often carried out underground with large amounts of shielding, and also employ sophisticated methods for distinguishing background events from candidate signal events. Despite these efforts to minimize the number of background events, they can never be completely removed from the data.
One method which has been proposed to distinguish dark matter scattering events from the background is to study the time dependence of the event rate Drukier et al. (1986); Freese et al. (1988, 2013). Specifically, since the motion of a detector relative to the dark matter halo affects the observed event rate by altering the incoming flux of dark matter particles, there should be modulation signatures in the rate count, the details of which are largely independent of dark matter properties and detector physics, though they are sensitive to the local dark matter phase space distribution Drukier et al. (1986); Freese et al. (1988, 2013). Annual modulation is the most prominent example of such an effect, and it arises due to the annual motion of Earthbound detectors around the Sun.
The DAMA experiment has operated for more than a decade and has reported with high significance an annually modulating rate of dark matter candidate events Bernabei et al. (2013). Although the amplitude and phase of the modulation seem to agree well with the modulation expected for dark matter events in the simplest astrophysical scenarios, an interpretation in terms of dark matter seems to be at odds with the null results from several other direct-detection experiments Akerib et al. (2014); Aprile et al. (2012); Angle et al. (2011); Agnese et al. (2013, 2014); Armengaud et al. (2011) (though see Savage et al. (2009)). This disagreement has led some to propose that the annual modulation seen in DAMA data is due to background events which themselves modulate annually (see e.g. Ralston (2010); Blum (2011); Davis (2014)), though some of these proposals have been disputed by the DAMA collaboration Bernabei et al. (2012). Several of these criticisms are grounded on the fact that many physical quantities (temperature, cosmic ray activity, solar neutrino flux) vary on an annual cycle, which could in principle result in an annually modulating event rate. Addressing the veracity of these claims and counter-claims is outside the scope of this paper.
In addition to annual modulation, the daily rotation of the Earth results in a diurnal modulation of the dark matter flux Collar and Avignone (1992); Ling et al. (2004); Lee et al. (2013). While a detection of a diurnal modulation accompanying an annual one would provide great support to the dark matter interpretation of any proposed detection, there are several quantities (similar to those detailed above) which change on a daily cycle, and thus could also conceivably result in a diurnally modulating background event rate. Furthermore, the diurnal modulation of the dark matter event rate results from a combination of the diurnal cycle of the detector velocity, the gravitational focusing of dark matter due to the Earth, and eclipsing of the stream of dark matter particles which pass through the bulk of the Earth, making predictions for the specific form of the diurnal modulation more complicated and model-dependent than the annual modulation. All things considered then, there is a need for a method whereby one can directly confirm or refute the dark matter hypothesis as the source of the annual and/or diurnal modulations. In this paper, we discuss one such method.
The Earth undergoes a monthly motion due to its interaction with the Moon, and this ‘‘wobble’’ of the Earth about the Earth-Moon barycenter results in a monthly modulation of the dark matter event rate. The expected modulation, though unambiguously present and nearly model-independent, is quite small. However, in contrast to the daily and annual modulations, far fewer potential sources of background change over a monthly cycle, making a detection of monthly modulation an extremely convincing confirmation of the detection of dark matter.222It is well known that tides undergo a monthly cycle, and though this seems the most readily identifiable process which could conceivably result in a monthly modulating background, it is not obvious how tides would affect the event rates of dark matter detectors.
In this work, we study in detail the monthly modulation anticipated in the event rate of dark matter direct-detection experiments. Sec. II reviews the relevant background material and defines the notation used throughout the paper. In Sec. III, we examine two sources of monthly modulation: the motion of the Earth about the Earth-Moon barycenter, and gravitational focusing due to the Moon. We demonstrate that the monthly modulation depends on both the sidereal and synodic monthly periods, leading to an annually-varying amplitude for the monthly modulation. Further, we show that the gravitational focusing due to the moon has a negligible impact on the monthly modulation signal. We conclude in Sec. IV. The details of the coordinates and velocities we use are specified in Appendix A.
II Background and Notation
Dark matter direct-detection experiments seek to measure the recoil of nuclei in a detector due to scattering with dark matter particles. The differential scattering rate for such events is given by
$$\frac{\mathop{}\!\mathrm{d}R}{\mathop{}\!\mathrm{d}E_{\mathrm{nr}}}=\frac{\rho%
_{\chi}}{2m_{\chi}\mu^{2}}\sigma(q^{2})\eta(v_{\mathrm{min}},t)\ ,$$
(1)
where $E_{\mathrm{nr}}$ is the nuclear recoil energy, $\rho_{\chi}$ is the local dark matter density, $m_{\chi}$ is the dark matter mass, $\mu$ is the reduced mass of the dark matter-nucleus system, $q$ is the momentum transfer, $\sigma(q^{2})$ is the effective cross section of collision, and
$$\eta(v_{\mathrm{min}},t)=\int_{v_{\mathrm{min}}}^{\infty}\ \!\frac{f(\mathbf{v%
},t)}{v}\,\mathop{}\!\mathrm{d^{3}}v\ ,$$
(2)
where ${f}(\mathbf{v},t)$ is the dark matter velocity distribution in the Lab frame, and $v_{\mathrm{min}}$ is the minimum velocity required of an incoming dark matter particle to produce a recoil energy $E_{\mathrm{nr}}$.
The scattering rate contains several sources of time dependence. The dark matter density and distribution function in the solar neighborhood for instance, are in principle inherently time dependent, though we will assume that any variation in these quantities occurs on time scales much longer than the relevant observations, and can thus be neglected. A more important source of time dependence is the motion of the detector relative to the rest frame of the dark matter halo, which changes the portion of the distribution function which is sampled by the experiment; the differential scattering rate therefore depends upon the velocity of the detector. As stated in Sec. I, the most prominent of this sort of effect is an annual modulation due to the Earth’s motion around the Sun, though there also exists a daily modulation due to the rotation of the Earth and a monthly modulation due to the motion of the Earth about the Earth-Moon barycenter. Also, the gravitational influence of bodies near the detector distorts the local dark matter density and velocity distribution, which gives the rate a dependence upon the position of the detector relative to these bodies.
If gravitational focusing is ignored, the velocity distribution $f(\mathbf{v},t)$ of the dark matter particles in the Lab frame is related to the Dark Matter Halo frame distribution $\tilde{f}(\mathbf{v})$ by the Galilean transformation
$$f(\mathbf{v},t)=\tilde{f}(\mathbf{v}+\mathbf{v}_{\mathrm{obs}}(t))\,,$$
(3)
where $\mathbf{v}_{\mathrm{obs}}(t)$ is the velocity of the detector relative to the Dark Matter Halo frame. If we ignore the effect of the Earth’s rotation, $\mathbf{v}_{\mathrm{obs}}(t)$ is given by
$$\mathbf{v}_{\mathrm{obs}}(t)=\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{es}}(t)\,,$$
(4)
where $\mathbf{v}_{\odot}\approx(11,232,7)$ km/s is the velocity of the Sun in Galactic coordinates Kerr and Lynden-Bell (1986); Schoenrich et al. (2009), and $\mathbf{v}_{\textsc{es}}(t)$ is the velocity of the Earth in the Solar frame.333Throughout this work we use the notation $\mathbf{r}_{\textsc{xy}}$ to denote the position of X as measured from Y, and $\mathbf{v}_{\textsc{xy}}$ to denote the velocity of X relative to Y. The letter S refers to the Sun, E to the Earth, M to the Moon, and B to the Earth-Moon barycenter.
To introduce the effect of gravitational focusing, we use the fact that Liouville’s theorem guarantees the constancy of the phase-space density of dark matter particles along their trajectories Lee et al. (2014): for dark matter particles passing near the Sun and arriving at the Earth, we have
$$\rho_{\chi}f(\mathbf{v},t)=\rho_{\infty}\tilde{f}(\mathbf{v}_{\odot}+\mathbf{v%
}_{\infty,\textsc{s}}[\mathbf{v}_{\textsc{es}}(t)+\mathbf{v}])\ ,$$
(5)
where $\rho_{\infty}$ is the dark matter density asymptotically far away from Sun’s gravitational well. The function $\mathbf{v}_{\infty,\textsc{s}}[\mathbf{v}]$ is derived from the conservation of the Laplace-Runge-Lenz vector Alenazi and Gondolo (2006); Sikivie and Wick (2002) and specifies the velocity $\mathbf{v}_{\infty,\textsc{s}}$ a particle must have far away from the Sun in order to arrive at the Earth with a velocity $\mathbf{v}$ (measured in the Solar frame). As such, it describes the gravitational focusing effect of the Sun and is given by
$$\mathbf{v}_{\infty,\textsc{s}}[\mathbf{v}]=\frac{v_{\infty,\textsc{s}}^{2}\,%
\mathbf{v}+v_{\infty,\textsc{s}}\,u_{\mathrm{esc},\textsc{s}}^{2}\,\mathbf{%
\hat{r}}_{\textsc{es}}/2-v_{\infty,\textsc{s}}\,\mathbf{v}\left(\mathbf{v}%
\cdot\mathbf{\hat{r}}_{\textsc{es}}\right)}{v_{\infty,\textsc{s}}^{2}+u_{%
\mathrm{esc},\textsc{s}}^{2}/2-v_{\infty,\textsc{s}}\left(\mathbf{v}\cdot%
\mathbf{\hat{r}}_{\textsc{es}}\right)}\ ,$$
(6)
where $v_{\infty,\textsc{s}}^{2}=v^{2}-u_{\mathrm{esc},\textsc{s}}^{2}$ from conservation of energy, and $u_{\mathrm{esc},\textsc{s}}=\sqrt{2GM_{\odot}/r_{\textsc{es}}(t)}\approx 40$ km/s is the escape velocity from the Sun near the Earth’s orbit.
Finally, we will assume throughout that the velocity distribution of dark matter in the halo rest frame is given by the Standard Halo Model
$$\tilde{f}(\mathbf{v})=\begin{cases}\frac{1}{N_{\mathrm{esc}}}\left(\frac{1}{%
\pi v_{0}^{2}}\right)^{3/2}\operatorname{e}^{-\mathbf{v}^{2}/v_{0}^{2}}\ ,&|%
\mathbf{v}|<v_{\mathrm{esc}}\,,\\
0\ ,&\mathrm{else}\,,\end{cases}$$
(7)
where
$$N_{\mathrm{esc}}=\operatorname{erf}(z)-\frac{2}{\sqrt{\pi}}z\operatorname{e}^{%
-z^{2}}\,,$$
(8)
and $z\equiv v_{\mathrm{esc}}/v_{0}$. We take $v_{0}=220$ km/s and set the escape velocity from the Galaxy to be $v_{\mathrm{esc}}=550$ km/s Lee et al. (2014); Smith et al. (2007).
III Sources of Monthly Modulation
III.1 Motion of the Earth around the Earth-Moon Barycenter
The barycenter for the Earth-Moon system is located on the line joining their centers at a distance of
$$\frac{r_{\textsc{em}}}{1+\frac{M_{\textsc{e}}}{M_{\textsc{m}}}}\approx 4661\ %
\mathrm{km}\ ,$$
(9)
roughly three-fourths of the Earth’s radius, from the center of the Earth; here $r_{\textsc{em}}$ is the distance between the centers of the Earth and Moon, and $M_{\textsc{e}}$ and $M_{\textsc{m}}$ are their respective masses. The precise description of the orbit of the Earth-Moon system is complicated by the non-negligible effect of the Sun’s gravitational force on the system, but it will be sufficient for our purposes to treat the orbit of the Earth around the barycenter as an ellipse which is slightly inclined (by about $5.2^{\circ}$) relative to the ecliptic, with the orbital period of a sidereal month, $T_{\mathrm{sid}}\approx 27.32$ days. See Appendix A for a more detailed description of the orbits.
It is clear at the outset that including the wobble of the Earth about the Earth-Moon barycenter in the definition of $\mathbf{v}_{\textsc{es}}$ will add some form of a monthly modulation to the differential rate. We can estimate its size to be on the order $v_{\textsc{eb}}/v_{\textsc{bs}}\approx(1.3\times 10^{-2}\,\mathrm{km/s})/(30\,%
\mathrm{km/s})\approx 0.04\%$ of the size of the annual modulation; since the annual motion of the Earth around the Sun itself causes the rate to modulate by about $v_{\textsc{bs}}/(4v_{\odot})\approx 3\%$ for $v_{\textrm{min}}\sim v_{0}$ Lewin and Smith (1996); Lee et al. (2013), the monthly modulation should be approximately $10^{-5}$ times the size of the mean rate for similar detector thresholds. Let us investigate this monthly effect in more detail, beginning by neglecting the effects of gravitational focusing.
In this case, the function $\eta(v_{\mathrm{min}},t)$ defined in Eq. (2), now reads
$$\eta(v_{\mathrm{min}},t)=\int_{v_{\mathrm{min}}}^{\infty}\ \!\frac{\tilde{f}%
\left(\mathbf{v}_{\mathrm{{obs}}}(t)+\mathbf{v}\right)}{v}\,\mathop{}\!\mathrm%
{d^{3}}v\equiv\eta(v_{\mathrm{min}},v_{\mathrm{obs}}(t))\,,$$
(10)
where we have used the fact that for the Standard Halo Model in the absence of gravitational focusing, the time dependence of the mean inverse speed $\eta(v_{\mathrm{min}},t)$ enters only through the speed of the detector relative to the dark matter halo $v_{\mathrm{obs}}(t)$. This integral can be evaluated analytically Lewin and Smith (1996):
$$\eta(v_{\mathrm{min}},v_{\mathrm{obs}}(t))=\begin{cases}\frac{1}{2N_{\mathrm{%
esc}}yv_{0}}\left[\operatorname{erf}(x+y)-\operatorname{erf}(x-y)-\frac{4}{%
\sqrt{\pi}}y\operatorname{e}^{-z^{2}}\right]\ ,&x<z-y\,,\\
\frac{1}{2N_{\mathrm{esc}}yv_{0}}\left[\operatorname{erf}(z)-\operatorname{erf%
}(x-y)-\frac{2(y+z-x)}{\sqrt{\pi}}\operatorname{e}^{-z^{2}}\right]\ ,&z-y<x<z+%
y\,,\\
0\ ,&x>z+y\,,\end{cases}$$
(11)
where we have defined $x\equiv v_{\mathrm{min}}/v_{0}$ and $y\equiv v_{\mathrm{obs}}(t)/v_{0}\,$; recall that $z\equiv v_{\mathrm{esc}}/v_{0}$.
In order to isolate the effects of the Earth’s motion about the Earth-Moon barycenter, we subtract from Eq. 10 the effects of the annual motion of the Earth around the Sun, which can be obtained by computing $\eta$ without the barycentric wobble of the Earth:
$$\Delta\eta(v_{\mathrm{min}},t)=\eta(v_{\mathrm{min}},|\mathbf{v}_{\odot}+%
\mathbf{v}_{\textsc{es}}(t)|)-\eta(v_{\mathrm{min}},|\mathbf{v}_{\odot}+%
\mathbf{v}_{\textsc{bs}}(t)|)\ ,$$
(12)
where $\mathbf{v}_{\textsc{bs}}(t)$ is the velocity of the Earth-Moon barycenter with respect to the Sun (see Appendix A for the specific form of these velocities). Fig. 1 shows a plot of this residual function at $v_{\mathrm{min}}=100\ \mathrm{km/s}$. Note that for the modulation plots throughout this work, we will plot the dimensionless “fractional modulation” on the $y$-axis (defined as $\Delta\eta/\langle\eta\rangle$, where angle brackets refer to the time average) to indicate the size of the modulation as compared to the mean rate, and time measured in days from J2000.0 on the $x$-axis.
First, notice from the figure that the estimate we made above of the relative size of the monthly modulation to the annual is in fact a good one. It is clear however, that the effect of the Earth’s wobble is more complicated than a simple monthly modulation: the amplitude of the monthly modulation itself modulates annually.
In order to explain this behavior, we will treat the velocity of the Earth relative to the Earth-Moon barycenter $\mathbf{v}_{\textsc{eb}}(t)$ as a small perturbation to $\mathbf{v}_{\mathrm{obs}}(t)$ in $\eta$. Using the spherical symmetry of the distribution function, the Taylor expansion of $\eta$ reads
$$\eta\left(v_{\mathrm{min}},\left|\tilde{\mathbf{v}}_{\mathrm{obs}}+\delta%
\mathbf{v}_{\mathrm{obs}}\right|\right)=\eta\left(v_{\mathrm{min}},\left|%
\tilde{\mathbf{v}}_{\mathrm{obs}}\right|\right)+\left.\frac{\partial\eta\left(%
v_{\mathrm{min}},v_{\mathrm{obs}}\right)}{\partial v_{\mathrm{obs}}}\right|_{v%
_{\mathrm{obs}}=\tilde{v}_{\mathrm{obs}}}\left(\hat{\tilde{\mathbf{v}}}_{%
\mathrm{obs}}\cdot\delta\mathbf{v}_{\mathrm{obs}}\right)+\mathcal{O}(\delta v_%
{\mathrm{obs}}^{\,2})\ ;$$
(13)
we can hence approximate the residual function Eq. (12) up to terms of order $v_{\textsc{eb}}^{2}$ as
$$\Delta\eta(v_{\mathrm{min}},t)\simeq\left.\frac{\partial\eta\left(v_{\mathrm{%
min}},v_{\mathrm{obs}}\right)}{\partial v_{\mathrm{obs}}}\right|_{v_{\mathrm{%
obs}}=|\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{bs}}(t)|}\left(\frac{\mathbf{v}_%
{\odot}+\mathbf{v}_{\textsc{bs}}(t)}{|\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{%
bs}}(t)|}\cdot\mathbf{v}_{\textsc{eb}}(t)\right)\ ,$$
(14)
where we have made the identifications $\tilde{\mathbf{v}}_{\mathrm{obs}}(t)=\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{bs%
}}(t)$ and $\delta\mathbf{v}_{\mathrm{obs}}(t)=\mathbf{v}_{\textsc{eb}}(t)$. Fig 1 shows that this approximation provides an excellent fit to $\Delta\eta$, and analytically describes all its discernible features, as we shall now see.
Let us examine the time dependence of the dot product appearing in Eq. (14); see Fig. 2 for a plot. The unit vectors $\hat{\mathbf{v}}_{\textsc{bs}}$ and $\hat{\mathbf{v}}_{\textsc{eb}}$ to zeroth order in eccentricity take the form
$$\displaystyle\hat{\mathbf{v}}_{\textsc{bs}}(t)$$
$$\displaystyle=\bm{\hat{\epsilon}}_{1}\cos{(\omega_{\mathrm{yr}}t-\phi_{1})}+%
\bm{\hat{\epsilon}}_{2}\sin{(\omega_{\mathrm{yr}}t-\phi_{1})}\,,$$
(15)
$$\displaystyle\hat{\mathbf{v}}_{\textsc{eb}}(t)$$
$$\displaystyle=\bm{\hat{\epsilon}}_{1,\textsc{m}}\cos{(\omega_{\mathrm{sid}}t-%
\phi_{2})}+\bm{\hat{\epsilon}}_{2,\textsc{m}}\sin{(\omega_{\mathrm{sid}}t-\phi%
_{2})}\,,$$
(16)
where $\omega_{\mathrm{yr}}=2\pi/T_{\mathrm{yr}}$, $\omega_{\mathrm{sid}}=2\pi/T_{\mathrm{sid}}$; we will leave the phases unspecified in these intermediate steps for clarity, but it is straightforward to retain them throughout the calculation. The dot product of these unit vectors is given by
$$\hat{\mathbf{v}}_{\textsc{bs}}(t)\cdot\hat{\mathbf{v}}_{\textsc{eb}}(t)=\frac{%
1+\cos i_{\textsc{m}}}{2}\cos{((\omega_{\mathrm{sid}}-\omega_{\mathrm{yr}})t-%
\phi_{3})}+\frac{1-\cos i_{\textsc{m}}}{2}\cos{((\omega_{\mathrm{sid}}+\omega_%
{\mathrm{yr}})t-\phi_{4})}\,,$$
(17)
where $i_{\textsc{m}}\approx 5.2^{\circ}$ is the inclination of the orbital plane of the Moon with respect to the ecliptic. We will approximate $\cos i_{\textsc{m}}\approx 1$ for simplicity here. Recognizing the difference between the sidereal and annual frequencies as precisely the synodic frequency, $\omega_{\mathrm{syn}}\equiv\omega_{\mathrm{sid}}-\omega_{\mathrm{yr}}$, we obtain
$$\hat{\mathbf{v}}_{\textsc{bs}}(t)\cdot\hat{\mathbf{v}}_{\textsc{eb}}(t)\simeq%
\cos{(\omega_{\mathrm{syn}}t-\phi_{3})}\,.$$
(18)
We can now use this result to compute the dot product appearing in Eq. (14):
$$\displaystyle\mathcal{P}(t)$$
$$\displaystyle\equiv\left(\frac{\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{bs}}(t)}%
{|\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{bs}}(t)|}\cdot\mathbf{v}_{\textsc{eb}%
}(t)\right)$$
$$\displaystyle\simeq\frac{v_{\textsc{eb}}}{|\mathbf{v}_{\odot}+\mathbf{v}_{%
\textsc{bs}}(t)|}\left[v_{\odot}b_{\textsc{m}}\cos{(\omega_{\textrm{sid}}(t-t_%
{1}))}+v_{\textsc{bs}}\cos{(\omega_{\mathrm{syn}}(t-t_{2}))}\right]$$
$$\displaystyle=\frac{v_{\textsc{eb}}}{|\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{%
bs}}(t)|}\Bigg{[}\left(v_{\odot}b_{\textsc{m}}-v_{\textsc{bs}}\right)\cos{(%
\omega_{\textrm{sid}}(t-t_{1}))}$$
$$\displaystyle\qquad\qquad+2v_{\textsc{bs}}\cos{\left(\frac{\omega_{\mathrm{syn%
}}+\omega_{\mathrm{sid}}}{2}t-\phi_{a}\right)}\cos{\left(\frac{\omega_{\mathrm%
{yr}}}{2}t-\phi_{b}\right)}\Bigg{]}\,,$$
(19)
where $t_{1}$ is the time when $\mathbf{v}_{\textsc{eb}}$ is most nearly parallel to $\mathbf{v}_{\odot}$, $t_{2}$ is the time when $\mathbf{v}_{\textsc{eb}}$ is most nearly parallel to $\mathbf{v}_{\textsc{bs}}$. We have further defined the geometric factor $b_{\textsc{m}}\equiv\sqrt{\left(\hat{\mathbf{v}}_{\odot}\cdot\bm{\hat{\epsilon%
}}_{1,\textsc{m}}\right)^{2}+\left(\hat{\mathbf{v}}_{\odot}\cdot\bm{\hat{%
\epsilon}}_{2,\textsc{m}}\right)^{2}}\simeq 0.45$, which accounts for the alignment of the Moon’s orbital plane with the motion of the Sun. The synodic period $T_{\mathrm{syn}}\approx 29.53\ \mathrm{days}$ is the average period of the Moon’s revolution with respect to the line joining the Sun and Earth, and thus determines the period of the moon’s phases, while $T_{\mathrm{sid}}$ is the period of the Moon as measured against the celestial sphere. From this equation therefore, we see that $\mathcal{P}(t)$ depends both on the synodic and the sidereal period of the Earth’s wobble about the Earth-Moon barycenter, and it is the presence of both these frequencies in the residual function produces that an annual “beat” in the differential scattering rate.
We turn now to the functional dependence of $\Delta\eta$ on $v_{\mathrm{min}}$, which is directly related to detector energy threshold. In Fig. 3 we plot the fractional modulation for four different values of $v_{\textrm{min}}$. Note that both $\Delta\eta$ and $\langle\eta\rangle$ are functions of the threshold speed $v_{\mathrm{min}}$: $\langle\eta\rangle$ is a monotonically decreasing function of $v_{\mathrm{min}}$ (see Fig. 4), while the behavior of $\Delta\eta$ is more complicated and is described by the function
$$\mathcal{A}(v_{\mathrm{min}},t)\equiv\left.\frac{\partial\eta(v_{\mathrm{min}}%
,v_{\mathrm{obs}})}{\partial v_{\mathrm{obs}}}\right|_{v_{\mathrm{obs}}=|%
\mathbf{v}_{\odot}+\mathbf{v}_{\textsc{bs}}(t)|}\,,$$
(20)
which appears in Eq. (14). It can be computed analytically:
$$\mathcal{A}(v_{\mathrm{min}},t)=\begin{cases}\frac{1}{2N_{\mathrm{esc}}y^{2}v_%
{0}^{2}}\left[-\operatorname{erf}(x+y)+\operatorname{erf}(x-y)+\frac{2}{\sqrt{%
\pi}}y\left(\operatorname{e}^{-(x+y)^{2}}+\operatorname{e}^{-(x-y)^{2}}\right)%
\right]\ ,&x<z-y\,,\\
\frac{1}{2N_{\mathrm{esc}}y^{2}v_{0}^{2}}\left[-\operatorname{erf}(z)+%
\operatorname{erf}(x-y)-\frac{2}{\sqrt{\pi}}\left((z-x)\operatorname{e}^{-z^{2%
}}+y\operatorname{e}^{-(x-y)^{2}}\right)\right]\ ,&z-y<x<z+y\,,\\
0\ ,&x>z+y\,;\end{cases}$$
(21)
see Fig. 5 for plots of this function.
First, consider the amplitude of the fractional modulation shown in Fig. 1 and Fig. 3. We see (apart from a decrease in amplitude between $v_{\mathrm{min}}=100$ km/s and $v_{\mathrm{min}}=200$ km/s) that the size of fractional modulation increases with $v_{\mathrm{min}}$, which might seem to imply that modulation is most easily detectable for experiments with high detector threshold. But since the mean rate decreases quite rapidly with $v_{\mathrm{min}}$ (Fig. 4), the small mean rate in this range makes any detection of dark matter more difficult than for experiments with lower thresholds, despite the large fractional modulation at high $v_{\mathrm{min}}$.
Another aspect of $\mathcal{A}$ that is important in describing features of $\Delta\eta$ is that for a fixed $v_{\mathrm{min}}$, $\mathcal{A}$ modulates annually with $t$; see Fig. 4(b). Note that this effect is distinct from, and competes with, the annual beat in $\mathcal{P}$, shown earlier in Fig. 2. For instance, the annual envelope in $\Delta\eta$ for $v_{\mathrm{min}}$ in the range $220-300\ \mathrm{km/s}$ is muted compared to the envelope for $\mathcal{P}$, because the annual feature in $\mathcal{A}$ is out of phase with the annual envelope in $\mathcal{P}$ in this range. Then for $v_{\mathrm{min}}$ in the range $300-330\ \mathrm{km/s}$, the envelope looks much closer to that of $\mathcal{P}$, since the annual feature of $\mathcal{A}$ flattens out for these values. Finally, for $v_{\mathrm{min}}\geq 330\ \mathrm{km/s}$, the characteristics of the envelope in $\Delta\eta$ are pronounced, owing to the fact that the peaks and troughs of $\mathcal{A}$ are now in phase with those of $\mathcal{P}$.
Additionally, because $\mathcal{A}$ is negative for $v_{\mathrm{min}}\mathrel{\mathchoice{\raise 0.0pt\hbox{\scalebox{.8}{\raise 0.%
0pt\hbox{$\displaystyle\lesssim$}}}}{\raise 0.0pt\hbox{\scalebox{.8}{\raise 0.%
0pt\hbox{$\textstyle\lesssim$}}}}{\raise 0.0pt\hbox{\scalebox{.8}{\raise 0.0pt%
\hbox{$\scriptstyle\lesssim$}}}}{\raise 0.0pt\hbox{\scalebox{.8}{\raise 0.0pt%
\hbox{$\scriptscriptstyle\lesssim$}}}}}200\ \mathrm{km/s}$, the fractional modulation has a phase opposite that of $\mathcal{P}$ in this range; compare Figs. 1 and 2. In fact, this flip causes the annual components of $\mathcal{A}$ and $\mathcal{P}$ to line up, accentuating the annual envelope in $\Delta\eta$. The crossing of $\mathcal{A}$ from negative to positive at $v_{\mathrm{min}}\approx 200\ \mathrm{km/s}$ causes a shift in the peak of the annual envelope from early December to early June, which in turn results in the peak of the monthly modulation shifting by about two weeks; this can be used to determine the dark matter mass Lewis and Freese (2004).
III.2 Focusing Effect of the Moon
Just as in the case of the Sun, the gravitational well of the Moon distorts the local distribution of dark matter. As a result, the position of the Moon relative to the Earth affects the scattering rate, and introduces an additional source of monthly modulation. In the absence of other masses, the velocity $\mathbf{v}_{\infty,\textsc{m}}$ that a particle must have infinitely far from the Moon in order to have a velocity $\mathbf{v}$ (in the Lunar frame) at the position of the Earth, is given by straightforward modifications to Eq. (6):
$$\mathbf{v}_{\infty,\textsc{m}}[\mathbf{v}]=\frac{v_{\infty,\textsc{m}}^{2}\,%
\mathbf{v}+v_{\infty,\textsc{m}}\,u_{\mathrm{esc},\textsc{m}}^{2}\,\mathbf{%
\hat{r}}_{\textsc{em}}/2-v_{\infty,\textsc{m}}\,\mathbf{v}\left(\mathbf{v}%
\cdot\mathbf{\hat{r}}_{\textsc{em}}\right)}{v_{\infty,\textsc{m}}^{2}+u_{%
\mathrm{esc},\textsc{m}}^{2}/2-v_{\infty,\textsc{m}}\left(\mathbf{v}\cdot%
\mathbf{\hat{r}}_{\textsc{em}}\right)}\ ,$$
(22)
with $v_{\infty,\textsc{m}}^{2}=v^{2}-u_{\mathrm{esc},\textsc{m}}^{2}\,$, $u_{\mathrm{esc},\textsc{m}}=\sqrt{2GM_{\textsc{m}}/r_{\textsc{em}}(t)}\approx 0%
.01\ \mathrm{km/s}$, and $\mathbf{r}_{\textsc{em}}(t)$ the position of the Earth in the Lunar frame. In reality, however, any particle which passes near the Moon is also necessarily affected by the gravitational pull of the Sun. To account for this, we will make the approximation that the Sun’s gravitational potential does not appreciably change in the vicinity of the Moon, so that we can take the incoming velocity of a particle in the Lunar frame to be given by the result of the Sun’s gravitational deflection at the position of the Moon.444Additionally, one may worry that the Moon is accelerating with respect to the dark matter halo, and thus does not create a steady-state wake of dark matter as does the Sun which moves smoothly with respect the the dark matter halo. Typical dark matter particles cross the gravitational well of the Moon in a matter of a few hours, during which the moon changes its position relative to the Earth by only a small fraction of Earth-Moon distance, and so the effect of the acceleration of the Moon can be safely neglected for our purposes.
Putting all of this together then, the product of the dark matter density and mean inverse speed is given by
$$\rho_{\chi}\int_{v_{\mathrm{min}}}^{\infty}\ \!\frac{f\left(\mathbf{v},t\right%
)}{v}\,\mathop{}\!\mathrm{d^{3}}v=\rho_{\infty}\int_{v_{\mathrm{min}}}^{\infty%
}\ \!\frac{\tilde{f}\left(\mathbf{v}_{\odot}+v_{\infty,\textsc{s}}[\mathbf{v}_%
{\textsc{ms}}+v_{\infty,\textsc{m}}[\mathbf{v}_{\textsc{em}}+\mathbf{v}]]%
\right)}{v}\,\mathop{}\!\mathrm{d^{3}}v\ ,$$
(23)
where $\mathbf{v}_{\textsc{em}}$ is the velocity of the Moon in the Solar frame. Notice that the effects of annual and monthly motion of the Earth, as well as the the gravitational focusing due to the Sun and Moon have been included here.
To estimate the effects of gravitational focusing, we first note that focusing effects peak when the detector is positioned most nearly behind the focusing body with respect to the stream of incoming dark matter particles, and this occurs a quarter period before (or after) the peak of the modulation due to the motion of the detector, which itself occurs when the detector moves most nearly toward (or away from) the stream of incoming particles. Said another way, the two effects peak at different times, so it is their relative size which determines the position of the peak in the actual rate count observed at a detector. In the case of the annual modulation, since the size of the effect of the Sun’s gravitational focusing is similar in magnitude to the effect of the Earth’s motion around the Sun (for low detector thresholds), gravitational focusing results in a phase shift of the annual modulation Lee et al. (2014). Hence, in order to determine whether a similar effect exists for the monthly modulation, we need to estimate the magnitude of the effect of the gravitational focusing effect due to the Moon, and compare it to the size of the effect of the Earth’s barycentric wobble.
The modulation due to the focusing of the Moon scales roughly as $(u_{\mathrm{esc},m}/v)^{2}$ Lee et al. (2014), and so for $v=300\ \mathrm{km/s}$, we expect a monthly modulation on the order of $10^{-7}\%$ of the mean rate due to the gravitational focusing of the Moon alone. We have already estimated that the relative size of the barycentric effect to the annual is $v_{\textsc{eb}}/v_{\textsc{bs}}\approx 0.04\%$, and as stated above, the annual motion of the Earth around the Sun causes the mean rate to modulate by about $v_{\textsc{bs}}/(4v_{\odot})\approx 3\%$. Therefore, we see that relative size of the effect of the Moon’s gravitational focusing to the barycentric motion is approximately
$$\left(\frac{u_{\mathrm{esc},m}}{v}\right)^{2}/\left(\frac{v_{\textsc{eb}}}{v_{%
\textsc{bs}}}\frac{v_{\textsc{bs}}}{4v_{\odot}}\right)=0.008\%\ ,$$
(24)
and so we expect a negligible modification to the monthly modulation as a result of the Moon’s gravitational focusing.
To confirm this estimate, we carried out the full computation as follows. Since the integral appearing in Eq. (23) cannot be evaluated analytically, we computed it numerically, and studied the function
$$\Delta\eta_{\,\textsc{m}}(v_{\mathrm{min}},t)\equiv\int_{v_{\mathrm{min}}}^{%
\infty}\ \!\left[\frac{\tilde{f}\left(\mathbf{v}_{\odot}+v_{\infty,\textsc{s}}%
[\mathbf{v}_{\textsc{ms}}+v_{\infty,\textsc{m}}[\mathbf{v}_{\textsc{em}}+%
\mathbf{v}]]\right)}{v}-\frac{\tilde{f}\left(\mathbf{v}_{\odot}+v_{\infty,%
\textsc{s}}[\mathbf{v}_{\textsc{bs}}+\mathbf{v}]\right)}{v}\right]\,\mathop{}%
\!\mathrm{d^{3}}v\ ,$$
(25)
where the second term includes only the effects of annual modulation compounded by the Sun’s gravitational focusing. The function $\Delta\eta_{\,\textsc{m}}$ then, is a residual of all monthly signatures: it includes the effects of the barycentric motion of the Earth and the gravitational focusing of the Moon, and any compound effects.
We fit $\Delta\eta_{\,\textsc{m}}$ for various $v_{\mathrm{min}}$ and $t$ with functions of the form
$$\Delta\eta_{\,\textsc{m},\textrm{fit}}=C+A\cos{(\omega_{\mathrm{sid}}t-\phi_{%
\textsc{a}})}+B\cos{(\omega_{\mathrm{syn}}t-\phi_{\textsc{b}})}\ \,$$
(26)
and compared the amplitudes, $A$ and $B$, and the phases, $\phi_{\textsc{a}}$ and $\phi_{\textsc{b}}$, with those from analogous fits made to $\Delta\eta$ from Eq. (12). We found that the parameters of the fit were essentially unchanged, confirming that the gravitational focusing of the Moon has a negligible impact on the monthly modulation from the barycentric wobble. One can further see this by computing the function $\Delta\eta_{\,\textsc{m}}-\Delta\eta$, which isolates the effect of the Moon’s gravitational focusing on the rate; see Fig. 6 for a plot. Notice that the graph shares several features with that of $\Delta\eta$, most noticeably the annual envelope, which arises once again from the interaction between $\omega_{\mathrm{sid}}$ and $\omega_{\mathrm{syn}}$. More significant is the fact that the fractional modulation of $\Delta\eta_{\,\textsc{m}}-\Delta\eta$, even though larger for $v_{\mathrm{min}}=100$ km/s than the estimate made above, is significantly smaller than the fractional modulation for $\Delta\eta$ itself. Further, the effect of gravitational focusing decreases with increasing threshold speed, and so it is even less important for higher $v_{\mathrm{min}}$. For all practical purposes then, the gravitational focusing due to the Moon can be ignored when considering monthly modulations in the rate.
In addition to $\Delta\eta_{\,\textsc{m}}$, several other residual functions were computed and examined to check for interplay between the various effects discussed above. For instance, we studied the interaction between the gravitational focusing due to the Sun and the annual envelope of the barycentric wobble signature. The findings from these computations can be summarized simply as follows: (a) the annual modulation compounded by the effects of the Sun’s gravitational focusing is the most dominant feature in the rate, and it is essentially unaffected by the addition of the barycentric wobble to the velocity; (b) the monthly modulation due to the Earth’s motion about the Earth-Moon barycenter, as seen most directly by computing the residual function $\Delta\eta$, is not significantly affected by focusing from either the Sun or the Moon.
IV Discussion and Conclusion
The study of modulation in dark matter direct-detection experiments provides a useful tool for distinguishing signal events from background. Annual modulation is the most prominent and readily detectable type of modulation. However, there are several potential sources of background which also experience annual modulation, which motivates a deeper understanding of the expected time dependence of the dark matter event rate. Let us now briefly examine diurnal modulation, which we have so far ignored.
The rotation of the Earth imparts a velocity to the detector with magnitude $v_{\mathrm{rot}}\approx 0.46\cos{\varphi_{0}}$ km/s relative to the center of the Earth, where $\varphi_{0}$ is the geographical latitude of the detector. This motion results in a daily modulation of the dark matter scattering rate which is approximately $(2.2\cos{\varphi_{0}})\,\%$ of the annual modulation, or about $(0.066\cos{\varphi_{0}})\,\%$ of the mean rate Lee et al. (2013).555Following an analysis similar to that in Sec. III.1 shows that both the sidereal and synodic daily periods of the Earth’s rotation will contribute to the diurnal modulation, leading to an annual cycle in the amplitude of the diurnal modulation. On the other hand, the effect of gravitational focusing due to the Earth is more complicated than for the Sun or Moon: for a detector near the surface of the Earth, many particles will have passed through the bulk of the Earth before arriving at the detector, and a formula like Eq. (6) is no longer sufficient to calculate the effect of the Earth’s gravity on each particle’s velocity; energy conservation still dictates that $v_{\infty,\textsc{e}}^{2}=v^{2}-u_{\mathrm{esc},\textsc{e}}^{2}$, but the angular dependence of the focusing will be complicated. Despite these complexities, we can naively estimate the size of the focusing effect to be $(u_{\mathrm{esc},\textsc{e}}/v)^{2}\approx 0.14\%$ of the mean rate for particles travelling at 300 km/s, which is more than twice as large as the effect of the rotational speed of the Earth. Additionally, the eclipsing of the incoming flux of dark matter particles by the bulk of the Earth (separate from the focusing effect just mentioned) contributes to diurnal modulation, and the size of this effect depends upon the dark matter properties as well as the geographical location of the detector Collar and Avignone (1992); Hasenbalg et al. (1997). Since the diurnal modulation of dark matter scattering rate results from a combination of these three effects, making definite predictions for the expected modulation is challenging. In addition, there are daily cycles which affect potential sources of background. Therefore, despite the larger expected amplitude of daily modulation, a detection of monthly modulation would provide more conclusive evidence in favor of dark matter than would daily modulation.
The DAMA experiment has observed an annual modulation whose magnitude is $(0.0112\pm 0.0012)$ cpd/kg/keV Bernabei et al. (2013). If this modulation is indeed due to dark matter, one should expect a monthly modulation with an amplitude of roughly $4\times 10^{-6}$ cpd/kg/keV, which is unfortunately far below the current sensitivity of the experiment. Thus, it seems as though significant improvements in detector technology and exposure will be required in order to observe monthly modulation.
Nevertheless, given that monthly modulation is observable in principle and can be used to distinguish dark matter events from background, we considered it in some detail in this work. We examined both the motion of the Earth around the Earth-Moon barycenter and the gravitational focusing due to the Moon and found that the former was the dominant contribution to monthly modulation, being almost completely unaffected by the latter. In addition to the expected monthly cycles in the rate count, the annual envelope is a unique marker that would aid in characterizing a detected signal. Though the expected amplitude of monthly modulation is quite small and thus difficult to detect, any detection would provide distinct, model-independent support for an interpretation of a modulating event rate in terms of dark matter.
Acknowledgments
We would like to thank Daniel Green and James Owen for helpful discussions. This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
Appendix A Coordinates
In this appendix, we describe the various position vectors, velocity vectors, and coordinate systems that appear throughout this work. We follow the treatment in Lee et al. (2013), adopting the same conventions and notation. All times are measured in days relative to J2000.0. We describe the orbits in terms of mean orbital elements which we will approximate as constants. This is a reasonable approximation for the description of the orbit of the Earth-Moon barycenter about the Sun, but is a poor approximation for the description of the orbits of the Earth and Moon about the Earth-Moon barycenter. This will not affect the main points presented in the paper, but more accurate results could be obtained by including the evolution of the orbital elements or relying instead on ephemeris data for the positions and velocities of the Earth and Moon.
For a classical Keplerian orbit, given the length of the semi-major axis $a$, and the eccentricity $e$, the central body is located at a distance $f=ae$ from the center of the ellipse. If $t_{\mathrm{p}}$ denotes the time of periapsis (closest approach) of the orbiting body, the mean anomaly $g(t)$ evolves via
$$g(t)=\omega(t-t_{\mathrm{p}})\ ,$$
(27)
where $\omega=2\pi/T$ and $T$ is the orbital period. The true anomaly $\nu$, which is the geometric angle in the plane of the ellipse between the periapsis and the orbital body at time $t$, is given by
$$\nu\simeq g(t)+2e\sin{g(t)}+\frac{5}{4}e^{2}\sin{2g(t)}\ ,$$
(28)
where we have kept terms through order $e^{2}$. The distance between the bodies at a given $\nu$ is specified by the relation
$$r(t)=\frac{a(1-e^{2})}{1+e\cos{\nu}}\ ,$$
(29)
and the longitude $\lambda(t)$ is
$$\lambda(t)=\lambda_{\mathrm{p}}+\nu\ ,$$
(30)
where $\lambda_{\mathrm{p}}$ is the longitude of the periapsis. The position of orbiting body then, is given by
$$\mathbf{r}(t)=r(t)\left(-\sin{\lambda(t)}\,\hat{\textbf{\i}}+\cos{\lambda(t)}%
\,\hat{\textbf{\j}}\right)\ ,$$
(31)
where $\hat{\textbf{\i}}$ and $\hat{\textbf{\j}}$ are orthonormal unit vectors that span the plane of the orbit, and $\hat{\textbf{\j}}$ is taken to be the reference direction.
We now describe the motion of the Earth relative to the Sun in two parts: first we will specify the motion of the Earth-Moon barycenter about the Sun, then we will detail the motion of the Earth (and Moon) about the Earth-Moon barycenter. For the purposes of this calculation, we will assume that the barycenter of the Solar System remains fixed at the center of the Sun.
Using the equations from above, the motion of the Earth-Moon barycenter relative to the Sun is given by
$$\mathbf{r}_{\textsc{bs}}(t)=r_{\textsc{bs}}(t)\left(-\sin{\lambda_{\textsc{bs}%
}(t)}\,\bm{\hat{\epsilon}}_{1}+\cos{\lambda_{\textsc{bs}}(t)}\,\bm{\hat{%
\epsilon}}_{2}\right)\ ,$$
(32)
where $\bm{\hat{\epsilon}}_{1}=(0.9940,0.1085,0.003116)$ and $\bm{\hat{\epsilon}}_{2}=(-0.05173,0.4945,-0.8677)$ are orthonormal unit vectors (given in Galactic coordinates) that span the ecliptic plane, and the relevant orbital elements are $a_{\textsc{bs}}=1.4960\times 10^{8}$ km, $e_{\textsc{bs}}=0.016722$, $T_{\mathrm{yr}}=365.256$ days, $t_{\mathrm{p},\textsc{bs}}=1.70833$ days, and $\lambda_{\mathrm{p},\textsc{bs}}=102.937^{\circ}$ Lee et al. (2013); McCabe (2014).
There are a few additional complications in describing the motion of the Earth about the Earth-Moon barycenter. First, Eq. (29) gives the distance between the orbiting bodies, but we would instead here like to know the position of the Earth relative to the barycenter, not the Moon. This distance is in fact given by
$$r_{\textsc{eb}}(t)=\frac{r_{\textsc{em}}(t)}{1+\frac{M_{\textsc{e}}}{M_{%
\textsc{m}}}}\ ,$$
(33)
and a similar relation describes the distance between the Moon and the barycenter:
$$r_{\textsc{mb}}(t)=\frac{r_{\textsc{em}}(t)}{1+\frac{M_{\textsc{m}}}{M_{%
\textsc{e}}}}\ .$$
(34)
Next, the Earth and Moon orbit about the Earth-Moon barycenter on similar ellipses which lie in a plane which is inclined relative to the ecliptic. To define their orbits then, we will need to define new unit vectors, $\bm{\hat{\epsilon}}_{1,\textsc{m}}$ and $\bm{\hat{\epsilon}}_{2,\textsc{m}}$, which span their barycentric orbital plane. To construct these unit vectors, we begin with $\bm{\hat{\epsilon}}_{1}$ and $\bm{\hat{\epsilon}}_{2}$ and perform two rotations. The first is a clockwise rotation by the longitude of ascending node $\Omega_{\textsc{em}}=125.08^{\circ}$ of the Moon’s orbit with respect to the ecliptic, about the vector $\bm{\hat{\epsilon}}_{3}\equiv\bm{\hat{\epsilon}}_{1}\times\bm{\hat{\epsilon}}_%
{2}$. Then, if we denote as $\bm{\hat{\epsilon}}_{2}^{\prime}$ the vector resulting from the rotation of $\bm{\hat{\epsilon}}_{2}$, the second rotation is anti-clockwise by the angle of inclination $i_{\textsc{em}}=5.16^{\circ}$ about $\bm{\hat{\epsilon}}_{2}^{\prime}$ Standish (2001). In terms of rotation matrices,
$$\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{1,\textsc{m}}\\
\bm{\hat{\epsilon}}_{2,\textsc{m}}\\
\bm{\hat{\epsilon}}_{3,\textsc{m}}\end{array}\right)=\begin{pmatrix}\cos{i_{%
\textsc{em}}}&0&-\sin{i_{\textsc{em}}}\\
0&1&0\\
\sin{i_{\textsc{em}}}&0&\cos{i_{\textsc{em}}}\end{pmatrix}\begin{pmatrix}\cos{%
\Omega_{\textsc{em}}}&\sin{\Omega_{\textsc{em}}}&0\\
-\sin{\Omega_{\textsc{em}}}&\cos{\Omega_{\textsc{em}}}&0\\
0&0&1\end{pmatrix}\left(\begin{array}[]{c}\bm{\hat{\epsilon}}_{1}\\
\bm{\hat{\epsilon}}_{2}\\
\bm{\hat{\epsilon}}_{3}\end{array}\right)\,,$$
(35)
and so we find $\bm{\hat{\epsilon}}_{1,\textsc{m}}=(-0.6025,0.2628,-0.7536)$ and $\bm{\hat{\epsilon}}_{2,\textsc{m}}=(-0.7837,-0.3738,0.4961)$, in Galactic coordinates.
We can now write the position of the Moon relative to the Earth-Moon barycenter in the notation above:
$$\mathbf{r}_{\textsc{mb}}(t)=r_{\textsc{mb}}(t)\left(-\sin{\lambda_{\textsc{em}%
}(t)}\,\bm{\hat{\epsilon}}_{1,\textsc{m}}+\cos{\lambda_{\textsc{em}}(t)}\,\bm{%
\hat{\epsilon}}_{2,\textsc{m}}\right)\ ,$$
(36)
where the orbital period is $T_{\mathrm{sid}}=27.3216$ days, and the orbital elements are $a_{\textsc{em}}=3.8470\times 10^{5}$ km, $e_{\textsc{em}}=0.0554$, $t_{\mathrm{p},\textsc{em}}=18.4493$ days, and $\lambda_{\mathrm{p},\textsc{em}}=318.15^{\circ}$ Standish (2001). The position of the Earth relative to the Earth-Moon barycenter is then constructed from the same orbital elements:
$$\mathbf{r}_{\textsc{eb}}(t)=-r_{\textsc{eb}}(t)\left(-\sin{\lambda_{\textsc{em%
}}(t)}\,\bm{\hat{\epsilon}}_{1,\textsc{m}}+\cos{\lambda_{\textsc{em}}(t)}\,\bm%
{\hat{\epsilon}}_{2,\textsc{m}}\right)\ .$$
(37)
Given Eqs. (32) and (37) then, the description of the Earth’s position in the Solar frame, including its motion around the barycenter, is given by
$$\mathbf{r}_{\textsc{es}}(t)=\mathbf{r}_{\textsc{eb}}(t)+\mathbf{r}_{\textsc{bs%
}}(t)\ ,$$
(38)
and $\mathbf{v}_{\textsc{es}}(t)=\dot{\mathbf{r}}_{\textsc{es}}(t)$. Analogously, the position and velocity of the Moon in the Sun’s frame are given by
$$\mathbf{r}_{\textsc{ms}}(t)=\mathbf{r}_{\textsc{mb}}(t)+\mathbf{r}_{\textsc{bs%
}}(t)\ ;\qquad\mathbf{v}_{\textsc{ms}}(t)=\dot{\mathbf{r}}_{\textsc{ms}}(t)\ .$$
(39)
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IOM 312.F-01-004 (2001). |
Magnetic circular dichroism in EELS: Towards 10 nm resolution
Peter Schattschneider
Cécile Hébert
Stefano Rubino
Michael Stöger-Pollach
Jan Rusz
Pavel Novák
Institute for Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria
University Service Centre for Electron Microscopy, Vienna University of Technology, A-1040 Vienna, Austria
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 16253 Prague 6, Czech Republic
(January 17, 2021)
Abstract
We describe a new experimental setup for the detection of magnetic circular dichroism with fast electrons (emcd). As compared to earlier findings the signal is an order of magnitude higher, while the probed area could be significantly reduced, allowing a spatial resolution of the order of 30 nm.
A simplified analysis of the experimental results is based on the decomposition
of the Mixed Dynamic Form Factor $S(\vec{q},\,\vec{q}^{\prime},E)$ into a real part
related to the scalar product and an imaginary part related to the vector
product of the scattering vectors ${\vec{q}}$ and ${\vec{q}^{\prime}}$. Following the
recent detection of chiral electronic transitions in the electron microscope
the present experiment is a crucial demonstration of the potential of emcd for nanoscale investigations.
1 Introduction
The observation of circular dichroism with electron probes has been considered
impossible except with spin polarized electron probes. In 2003 it was suggested
that this was not the case [1]. The magnetic transitions that give
rise to X-ray magnetic circular dichroism (xmcd) contribute to the imaginary
part of the Mixed Dynamic Form Factor (mdff) [2] for inelastic
electron scattering. Since this quantity can be measured in the transmission
electron microscope (tem) under particular scattering conditions, we predicted
that detection of xmcd should be feasible in the tem. We called the predicted
effect Energy-Loss Magnetic Chiral Dichroism (emcd).
In the first conclusive experimental demonstration of emcd on Fe [3]
it was discovered that the effect is smaller than xmcd. In that experiment the dichroic signal was close to the noise threshold in the then chosen geometry and the area
of investigation was approximately 100 nm in radius.
Here we present a new experimental setup that enhances the count rate by an order of magnitude and
reduces the probed area by another factor of five, thus opening the way to applications of emcd on the nanometric scale.
The mdff $S({\vec{q}},{\vec{q}}^{\prime},E)$ is the essential quantity describing
emcd. It has been used in the description of interference of inelastically
scattered electrons (e. g. [2, 4, 5]).
2 The Mixed Dynamic Form Factor (MDFF)
The semi-relativistic double differential scattering cross section for inelastic
electron scattering (DDSCS) in the plane wave Born approximation
is [6]
$$\frac{\partial^{2}\sigma}{\partial E\partial\Omega}=\frac{4\gamma^{2}}{a_{0}^{%
2}\,q^{4}}\frac{k_{f}}{k_{i}}S({\vec{q}},E)$$
(1)
where $a_{0}$ is the Bohr radius, $k_{i}$ $(k_{f})$ is the wave number of the
incident (outgoing) probe electron, ${\vec{q}}={\vec{k}}_{i}-{\vec{k}}_{f}$ is the
wave vector transfer in the interaction and $E$ the energy loss.
$S({\vec{q}},E)$ is the dynamic form factor (dff).
Interference between inelastically
scattered electrons in the diffraction pattern will occur when the probe
electron consists of two or more mutually coherent plane waves [4, 7]. Experimentally, this can be realized by a
biprism [8] or by any other beam splitter. It was shown
experimentally that the crystal itself can be used as a beam splitter for
inelastic electron scattering [9]. In the crystal the probe
electron is a superposition of Bloch waves which, in turn, are coherent
superpositions of plane waves defined by the allowed Bragg reflections.
For the sake of clarity, we consider the simplest case here, namely the
superposition of two plane waves with complex amplitudes $A_{1,\,2}$, respectively.
Technically, this situation is approximated in electron diffraction by the
two-beam case, the most important plane waves being the incident one and a single
Bragg scattered wave.
The DDSCS is then [2]
$$\frac{\partial^{2}\sigma}{\partial E\partial\Omega}=\frac{4\gamma^{2}}{a_{0}^{%
2}}\frac{k_{f}}{k_{i}}\big{(}|A_{1}|^{2}\frac{S({\vec{q}},E)}{q^{4}}+|A_{2}|^{%
2}\frac{S({\vec{q}}\,^{\prime},E)}{q^{\prime 4}}+2\Re[A_{1}A_{2}^{\ast}\frac{S%
({\vec{q}},{\vec{q}}\,^{\prime},E)}{q^{2}q^{\prime 2}}]\big{)}.$$
(2)
Here,
${\vec{q}}={\vec{k}}_{i}-{\vec{k}}_{f}$, ${\vec{q}}\,^{\prime}={{\vec{k}}^{\prime}}_{i}-{\vec{k}}_{f}$ are
the wave vector transfers from the two incident plane waves ${\vec{k}}_{i},\,{{\vec{k}}^{\prime}}_{i}$ to
${\vec{k}}_{f}$. Since i) the $2p$ $\to$ $3d$ transitions are the dominant ones due to the
shape of the density of states and ii) due to the localized character of $<i|$,
the matrix elements $\langle i|e^{i{\vec{q}}{\vec{R}}}|f\rangle$ are mostly determined by an area within small $R$ values (compared to lattice parameters), we can use the dipole approximation for the mdff [4]
$$S({\vec{q}},{\vec{q}}\,^{\prime},E)_{\text{dip}}=\sum_{if}\langle i|{{\vec{q}}%
{\vec{R}}}|f\rangle\langle f|{{\vec{q}}\,^{\prime}{\vec{R}}}|i\rangle\delta(E+%
E_{i}-E_{f}).$$
(3)
${\vec{R}}$ is the 3-vector operator $(r_{1},r_{2},r_{3})$ of the one-electron
scatterer with initial and final wave functions $|i\rangle$, $|f\rangle$.
Eq. 2 consists of two direct terms, each resembling the angular
scattering distributions centered at the incident and the Bragg scattered plane wave directions, and an interference term. Eq. 2 is formally equivalent to the expression for intensity in the double slit experiment. It should be noted that the diagonal element of the mdff is the dff, $S({\vec{q}},{\vec{q}},E)=S({\vec{q}},E)$.
The mdff describes the mutual coherence of transitions with energy transfer
$E$ and momentum transfer $\hbar{\vec{q}},\hbar{\vec{q}}^{\prime}$ [4]
(Two different momentum transfers can occur in one transition with finite
probability when the incident or the outgoing electron is not a single plane wave. In
the present case $A_{1}|{\vec{k}_{i}}\rangle+A_{2}|{\vec{k}^{\prime}}_{i}\rangle$ is such
a basis function, and the measurement collapses the probe electron into
$|{\vec{k}}_{f}\rangle$ 111It should be noted that
collapsing the probe electron into $|{\vec{k}}_{f}\rangle$ does not
exclude the possibility of interference between outgoing beams with
different wave vectors; any outgoing beam that is Bragg scattered to $|{\vec{k}}_{f}\rangle$ before leaving the crystal can produce interference detectable by the setup described here.).
With the matrix elements
$$r_{jk}=\sum_{if}\langle i|r_{j}|f\rangle\langle f|r_{k}|i\rangle\delta(E+E_{i}%
-E_{f}).$$
(4)
of the transition matrix ${\hat{R}}=\{r_{jk}\}$
the mdff, eq. 3 can be written as
$$S({\vec{q}},{\vec{q}}\,^{\prime},E)_{dip}={\vec{q}}{\hat{R}}{\vec{q}}\,^{%
\prime}\,.$$
(5)
For isotropic systems the transition matrix degenerates to a quantity
proportional to the unity matrix [2]. This case was discussed
in the context of ionisation fine structure and dynamical diffraction [10, 7].
Anisotropy can be induced by a lattice of lower than cubic symmetry, or by
magnetic fields. These can be internal or external, then speaking of natural
or magnetic dichroism [11]. It is well known that with photon
scattering linear as well as circular dichroism can be measured. This
technique is largely applied with external magnetic fields. Linear magnetic
dichroism shows up as an uniaxial anisotropy and can be measured with angle
resolved inelastic electron scattering, tuning $\vec{q}$ parallel or
perpendicular to the anisotropy axis. This is equivalent [12] to the tuning of
linear polarization of the photon in xanes experiments.
It has been thought that circular magnetic dichroism cannot be detected with
electrons except with spin polarized ones. But let us recall that in xanes
the photon does not couple directly to the spin of electrons but to the angular
momentum of the excited atom, and the effect becomes visible by the spin-orbit
coupling [11]. So there is no reason that spin polarized electrons
are needed for detection of circular dichroism in electron energy loss
spectrometry (eels). Rather, in the inelastic electron interaction that is
equivalent to an xmcd experiment, the virtual photon that is exchanged must
be circularly polarized.
The mdff, eq. 5 can be written in a different form when we specify the magnetic field
direction
as the positive $r_{3}$ axis 222In the TEM this is usually also the optical axis. and write
${\vec{q}}=({\vec{q}}_{\perp},q_{3})$. A little algebra shows that
$$S({\vec{q}},{\vec{q}}\,^{\prime},E)_{dip}=\frac{1}{2}({\rm r}_{++}+{\rm r}_{--%
})\,{\vec{q}}_{\perp}\cdot{\vec{q}}_{\perp}\,^{\prime}+{\rm r}_{00}\,q_{3}q^{%
\prime}_{3}+\frac{i}{2}({\rm r}_{++}-{\rm r}_{--})\,({\vec{q}}_{\perp}{\times}%
\,{\vec{q}}_{\perp}\,^{\prime})\cdot{\vec{e}_{3}}$$
(6)
where ${\vec{e}_{3}}$ is the unit vector in direction of the $r_{3}$ axis, and we have used the transition matrix elements in terms of the spherical components $R_{+,\,-,\,0}$ of the 3-space operator
$${\rm r}_{++}=\sum_{if}\langle i|R_{+}|f\rangle\langle f|R_{+}|i\rangle\delta(E%
+E_{i}-E_{f}).$$
(7)
and similar for all other combinations. They relate to the Cartesian components by the transformation rules for vector spherical harmonics [13].
All matrix elements and vector components are real in eq. 6. In this
form we have separated the mdff into a real component proportional to the
scalar product of the wave vector transfers ${\vec{q}}_{\perp},\,{\vec{q}}\,^{\prime}_{\perp}$
and an imaginary part proportional to their vector product. This structure is
equivalent to that of the polarization tensor used in xmcd, which decomposes
into a scalar part (uneffective in dichroic experiments), a second-rank irreducible part detectable by linear dichroism and a pseudovector part
sensitive to magnetic moments [11, 14] and reference
therein.
This form of the MDFF allows a description of the scattering geometry for emcd detection in the tem.
In passing we note that the imaginary part vanishes if the magnetic transitions ($\Delta m=\pm 1$) are degenerate.
Only when the presence of a magnetic field in $r_{3}$ direction lifts the $m$-degeneracy will we see an effect.
For a transition with fixed energy loss $E$ the operators $R_{+}$ and
$R_{-}$ will then contribute with different oscillator strengths, and the mdff eq. 6
will acquire an imaginary part. Its sign depends on which transitions are allowed
by the selection rules.
The imaginary part can be interpreted as the difference in probability to
change the magnetic quantum number by $\pm 1$. It thus describes the
difference in response of the system to left- respectively right-handed
circularly polarized electromagnetic fields 333An imaginary part of the
mdff signifies that time inversion symmetry is broken. In fact this symmetry
breaking relates to the angular momentum operator. Under time inversion its
direction is reversed. In the presence of a magnetic field this is no longer
a symmetry operation..
The scattering vector ${\vec{q}}_{\perp}$ in the diffraction plane is
perpendicular to the magnetic field vector. We assumed already that the
magnetic moments of the scatterer are aligned parallel to the optical
axis $r_{3}$ in the strong magnetic field ($\approx$ 2 T) of the objective lens of the
microscope. We can now evaluate the DDSCS for specimens showing magnetic
circular dichroism.
If in eq. 2 we write the phase shift $\phi$ explicitly as $A_{1}A_{2}^{\ast}=|A_{1}||A_{2}|\cdot e^{-i\phi}$, inspection of eq. 6 then reveals that a
phase shift $\phi\neq n\pi$ between the two incident plane waves is
needed in order to activate the imaginary part of the mdff. A phase shift of $\pm\pi/2$
is recommended since in this case the real part of the mdff disappears
in eq. 2, and only the imaginary part survives.
In a two-beam case with such a phase shift the pseudovector part contributes
with its full magnitude and gives rise to an asymmetry in the scattering cross section of a
magnetic transition such as the L edges of the ferromagnetic d-metals.
3 Experiments and simulations
We performed our experiment on a Co single crystal electropolished sample with a FEI Tecnai F20-FEGTEM S-Twin operating at 200 keV and equipped with a Gatan imaging filter (GIF). The dichroic signal is obtained by first tilting out of the [0001] zone axis to a two-beam case where only the $0000$ and $10\bar{1}0$ reflections are strongly excited. Following the procedure illustrated in [3], a selected area aperture (SAA) is used to delimit a region of about 100 nm radius and 18 $\pm$ 3 nm thickness. The corresponding diffraction pattern is then projected onto the 2 mm spectrometer entrance aperture (SEA).
Drawing a circle with a diameter of G with the 0- and G-reflections on the
left and right side respectively, the strongest dichroic signal can be
expected at the top and bottom points A and B of this Thales circle (fig. 1) where the scalar product ${\vec{q}}\cdot{\vec{q}}\,^{\prime}$ is zero and the pseudovector ${\vec{q}}\times{\vec{q}}\,^{\prime}$ maximises the imaginary part of the mdff.
The camera lenght and projection coils are then adjusted so that the SEA will be centered at these two points (first A, then B) in the Thales circle and determine a collection angle of 2-4 mrad. Two spectra are then acquired sequentially (fig. 2). With an acquisition time of 60 sec per spectrum and an energy dispersion of 0.5 eV/channel the intensity at the $L_{3}$ peak was about 2,500 counts (after background removal).
In order to improve the signal-to-noise ratio a new experimental setup was devised. As detailed above, the sample is tilted out of the [0001] zone axis to a two-beam case where only the $0000$ and $10\bar{1}0$ reflections are observed in the the energy filtered diffraction pattern, which is then projected onto the spectrometer entrance aperture (SEA). Using a rotational sample holder, the reciprocal lattice vector $\vec{G}$ is then aligned parallel to the energy dispersive axis of the CCD camera, so that a q-E diagram can be recorded as depicted in fig. 3. The quadrupoles of the energy filter collapse (integrating the signal in the $q_{x}$ dimension) the circular area to a line in $q_{y}$ when the system is switched to spectroscopy mode. This first modification allows us to record not only both spectra A and B with a single acquisition, but the entire range of spectra with different $q_{y}$ values comprised within the 2 mm SEA. It should be noted however that the integration area in the $q_{x}$ dimension is different for every value of $|q_{y}|$.
A second modification consists in a different method [15] to obtain a spot like inelastic diffraction pattern: the beam (with a convergence semi-angle of $\alpha=2$ mrad) is focussed onto a 18 $\pm$ 3 nm thick area of the Co specimen which is then shifted upwards from the eucentric position by $z=9.25\,\mu$m. The diameter of the illuminated area is
$d=2\alpha z=37$ nm, accurate to 5%, in the present experiment (fig. 4).
Preliminary experiments show that with smaller $z$-shifts the illuminated area can be reduced to less than 10 nm radius, at the moment with untenable distortions of the diffraction pattern. With a C${}_{\text{s}}$ corrector or a monochromator a spatial resolution of 10 nm or less should be attainable.
Similarly to xmcd we define $\Delta\sigma$ as the difference between spectra with opposite helicity and $\bar{\sigma}$ as their average. The dichroic signal is then the relative
difference $\Delta\sigma/\bar{\sigma}$ in the scattering cross section when
the sign of the pseudovector part changes 444Another common definition of the dichroic signal is the ratio between the difference and the sum of spectra with different helicity, which is a factor of 2 smaller than the one used here.. This is obtained by tracing the
spectral intensity at points A and B in fig. 3, and taking their difference, as shown in fig. 5. With an acquisition time of 15 sec and an energy dispersion of 0.3 eV/channel the intensity at the $L_{3}$ peak was about 13,500 counts (after background removal).
The modified scattering geometry provides a count rate per eV
which is an order of magnitude higher than the one achieved in the previous configuration [3], thus improving significantly the signal to
noise ratio. This is essentially caused by the fact that when no SAA is used and the beam is focussed only on the area of interest, all the electrons emitted from the gun contribute to the detected signal. In the formerly used geometry a nearly parallel incident bundle illuminated a large area of the sample of which only a small fraction could be used. This effectively reduced the intensity by which the area of interest is
illuminated, i.e. a large part of the incident electrons did not contribute to the signal.
The increase in the count rate per eV allows us to reduce the acquisition time, thus limiting the effects of beam instability, specimen and energy drift.
The shorter acquisition time, combined with the finer energy dispersion, improves the energy resolution with which the $L_{2,3}$ edges are recorded. In the older setup the $L_{3}$ has a FWHM of 7 eV (fig. 2), compared to the 3.6 eV achieved with the new method (fig. 5).
Ab initio DFT simulations of the dichroic signal were performed using an extension [16] of the WIEN2k package [17] developped for this purpose. In the simulation the effects of thickness, tilt of the incident beam, position of the detector were included as well as the integration over $q_{x}$ in the range dictated by the use of a circular SEA. Up to 8 beams were used for the calculations of the MDFFs. A comparison with the experiment is given in fig. 6 for the $L_{3}$ edge of Cobalt.
The agreement is very good between -0.8 and 0.8 G with some discrepancy appearing at larger scattering angles. This can be due to the faint Bragg spots outside the systematic row (which are neglected in the simulations) and to the fact that the SEA is not exactly in the spectral plane of the energy filter. The error bars correspond to the ($2\sigma$) Poissonian noise calculated for the theoretical signal using the number of electrons contributing to the signal as determined from the experimental data.
4 Conclusions
The strong chiral effect observed in the Co L${}_{2,3}$ edge shows that
emcd can be measured with reasonable collection time in the TEM. The
convergence angle of 2 mrad is obviously not detrimental for the
necessary constant phase shift between the unscattered and the Bragg
scattered electron waves. The $z$-shift of the specimen allows to control
the illuminated area, and with optimized conditions a lower limit of
$\leq$ 10 nm appears realistic. This would define the lateral resolution
in scanning mode. In order to achieve this goal some technical problems
such as the stability of the beam in the magnetic field, constant $z$-shift
or decoupling of the scan coordinate from the positioning of the diffraction
pattern must be solved. The simple concept described above should be an
incentive for novel dichroic experiments in the TEM. It also shows that emcd can be complementary and competitive with traditional or new [18] XMCD techniques.
Acknowledgements:
This work was sponsored by the European Union under contract nr. 508971
(FP6-2003-NEST-A) "Chiraltem". We acknowledge Jo Verbeeck for stimulating discussions.
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Dagomir Kaszlikowski
Nonclassical Phenomena
in Multi-photon Interferometry:
More Stringent Tests Against Local Realism
Praca doktorska napisana
w Instytucie Fizyki Teoretycznej
i Astrofizyki
Uniwersytetu
Gdańskiego
pod kierunkiem
dr hab. Marka Żukowskiego, prof. UG
Gdańsk 2000
So farewell elements of reality!
And farewell in a hurry.
David Mermin
I would like to thank
my thesis supervisor
dr hab. Marek
Żukowski for help, patience
and for
being
my guide
through the maze of quantum world.
I also thank Maja for being with
me
in difficult moments of my life.
This dissertation was supported by The Polish Committee for Scientific Research,
Grant No. 2 P03B 096 15 and by Fundacja Na Rzecz Nauki Polskiej.
Abstract
The present dissertation consists of three parts which are mainly based
on the following papers111The author’s papers will be cited here and
in the main text using the number
from the list in this page in square brackets, whereas other papers will be quoted
by the name of the first author and the year of publication.
:
[1]
M. Żukowski, D. Kaszlikowski and E. Santos,
Phys. Rev. A 60, R2614 (1999)
[2]
M. Żukowski and D. Kaszlikowski,
Acta Phys. Slov. 49, 621 (1999)
[3]
M. Żukowski and D. Kaszlikowski,
Phys. Rev. A 56, R1682 (1997)
[4]
D. Kaszlikowski and M. Żukowski,
Phys. Rev. A 61, 022114 (2000)
[5]
M. Żukowski and D. Kaszlikowski,
Phys. Rev. A 59, 3200 (1999)
[6]
M. Żukowski and D. Kaszlikowski, Vienna Circle Yearbook vol. 7,
editors D. Greenberger, W. L. Reiter,
A. Zeilinger, Kluwer Academic Publishers, Dortrecht (1999)
[7]
M. Żukowski, D. Kaszlikowski, A. Baturo and J. -Å. Larsson,
quant-ph/9910058
[8]
D. Kaszlikowski, P. Gnacinski, M. Żukowski, W. Miklaszewski
and A. Zeilinger, quant-ph/0005028
The first two chapters have an introductory character.
In Chapter 3 it is shown
[1] that the possibility of
distinguishing between
single- and two- photon detection events, usually not met in the actual
experiments, is not a necessary requirement for the proof that the experiments
of Alley and Shih [Shih88] and Ou and Mandel [Ou88] are, modulo
a fair sampling assumption, valid tests of local realism. It is also shown
that
some other interesting phenomena (involving bosonic-type particle
indistinguishability) can be observed during such tests.
Next in Chapter 4 it is shown again
[2] that the possibility of
distinguishing between single and two photon detection events
is not a necessary requirement for the proof that recent operational
realisation of entanglement swapping cannot find a local realistic
description. A simple modification of the experiment is proposed, which
gives a richer set of interesting phenomena.
In Chapter 5 a sequence of Bell inequalities
for $M$ particle systems, which involve three settings of each
of the local measuring apparatuses, is derived [3].
For Greenberger-Horne-Zeilinger
states, quantum mechanics violates these inequalities by factors
exponentially growing with $M$. The threshold visibilities of the
multiparticle sinusoidal interference fringes, for which local realistic
theories are ruled out by these inequalities,
decrease as $(2/3)^{M}$.
In Chapter 6 the Bell
theorem for a pair of two
two-state systems (qubits) in a singlet state for the
entire range of measurement settings is presented [4].
Chapter 7 is devoted to derivation of
a series of Greenberger-Horne-Zeilinger
paradoxes for $M$ qu$N$its (particles described by
an $N$ dimensional Hilbert space) that are fed
into $M$ unbiased $2N$-port spatially separated beam splitters
[5, 6].
In Part III a novel approach to the Bell theorem, via
numerical linear optimisation, is presented [7, 8].
The two-qubit correlation obtained from the quantum state used
in the Bell inequality is sinusoidal, but the standard Bell inequality
only uses two pairs of settings and not the whole
sinusoidal curve. The highest to-date visibility of an
explicit model reproducing
sinusoidal fringes is ${2\over\pi}$. We conjecture from a
numerical approach [7] presented in Chapter 8 that the highest
possible visibility for a local
hidden variable model reproducing
the sinusoidal character of the quantum prediction for
the two-qubit Bell-type interference
phenomena is ${1\over\sqrt{2}}$. In addition,
the approach can be applied directly to experimental data.
In Chapter 9 the approach presented in Chapter 8 is applied
to three qubits in a maximally entangled Greenberger-Horne-Zeilinger
state. For the first time the necessary and sufficient conditions
for violation of local realism for the case in which each
observer can choose from up to 5 settings of the measuring
apparatus are shown.
In Chapter 10 using
the modified approach developed in Chapter 8
it is shown that violations of local realism are stronger
for two maximally entangled qu$N$its, than for two qubits [8].
The magnitude of violation increases with $N$. It is objectively
defined by the required minimal admixture of pure noise
to the maximally entangled state such that a local realistic
description is still possible. Operational realisation of the
two qu$N$it measurement exhibiting strong violations of local
realism involves entangled photons and unbiased multiport
beamsplitters. The approach, extending at present to $N=9$,
neither involves any simplifications, or additional assumptions, nor
does it utilise any symmetries of the problem.
Contents
1 Introduction
2 Some history and basic notions
2.1 Preliminaries
2.2 Entanglement
2.3 Elements of reality
2.4 The Bell theorem
2.5 Experimental tests of the Bell theorem
2.5.1 First experiments
2.5.2 New experiments
2.5.3 Problems encountered in the Bell type experiments
I New theoretical analysis of Alley-Shih, Ou-Mandel
and entanglement swapping experiments
3 Alley-Shih and Ou-Mandel experiments: resolution of the problem
of distinguishability of single and two photon events [1]
3.1 Introduction
3.2 Description of the experiment
3.3 Quantum predictions
3.4 Conditions to violate local realism
3.5 Conclusions
4 Better entanglement swapping [2]
4.1 Introduction
4.2 Description of the experiment
4.3 Quantum predictions
4.4 Conditions to violate local realism
4.5 Proposal of modification of the experiment
4.6 Conclusions
II New generalised Bell inequalities and GHZ paradoxes for qu$N$its
5 Wringing out better Bell inequalities for GHZ experiment [3]
5.1 Introduction
5.2 Geometrical method of finding Bell inequalities
5.3 Quantum and local realistic description of the gedanken experiment
5.4 Derivation of Bell inequalities via the geometrical method
5.4.1 Critical visibility and quantum efficiency of detectors
5.5 Results
5.6 Conclusions
6 Bell inequality for all possible local settings [4]
6.1 Introduction
6.2 Quantum mechanical and local realistic description of the
gedanken experiment
6.3 Derivation of the inequality via the geometrical method
6.4 Results
6.5 Conclusions
7 Greenberger-Horne-Zeilinger paradoxes for qu$N$its [5,6]
7.1 Introduction
7.2 Unbiased multiport beamsplitters
7.3 Quantum mechanical predictions
7.3.1 Bell number assignment
7.3.2 Perfect correlations
7.4 Paradoxes for $N+1$ maximally entangled qu$N$its
7.4.1 Four tritters
7.4.2 General case: $N+1$ maximally entangled qu$N$its
7.5 Paradoxes for $N$ maximally entangled qu$N$its.
7.5.1 Three tritters
7.5.2 General case: N maximally entangled qu$N$its.
7.6 Conclusions
III Extension of Bell Theorem via numerical approach
8 Necessary and sufficient conditions to
violate local realism for two maximally entangled qubits-
extension to more than two local settings [7]
8.1 Introduction
8.2 Quantum mechanical and local realistic
description of the gedanken experiment
8.3 Linear programming and Downhill Simplex Method
8.3.1 Numerical difficulties
8.4 Results
8.4.1 Exemplary numerical model
8.5 Application to experimental data
8.6 Conclusions
9 Necessary and sufficient conditions to
violate local realism for three maximally entangled qubits-
extension to more than two local settings
9.1 Description of the method
9.2 Results
9.3 Conclusions
10 Entangled pairs of qu$N$its: the violation of local
realism increases with $N$ [8]
10.1 Introduction
10.2 Description of the gedanken experiment
10.3 Local realism and joint probability distribution
10.4 Linear programming
10.5 Observables
10.6 Results
10.7 Exemplary analytical model for $N=3$
10.7.1 Explicit model for extremal case
10.8 Conclusions
A Proof of equivalence of the existence of local hidden variables
and a joint probability distribution for incompatible
measurements
Chapter 1 Introduction
Bell’s theorem [Bell64], formulated in 1964, initiated and
revitalised several branches of modern physics. The paper
was the first one to show that quantum entanglement
cannot be in any way simulated by classical correlations.
Within few years, a new branch of experimental physics
emerged: multi-particle quantum interferometry. Since then
it has evolved and extended its field of interest from two-photon
to multi-photon correlations. Recently Bell-EPR
correlations were observed for entangled atoms [Hagley97].
For as much as nearly 20-25 years the paper of Bell was
studied mainly by people interested in the
foundational-interpretational problems of quantum theory.
Suddenly with the discovery of the possibility of employing
Bell-EPR correlations [Eckert91] in ”quantum cryptography”
[Bennett84]
and with the realisation of the importance of entanglement
in the hypothetical quantum computers [Feynman82, Deutsch89],
it turned out that
the paper of Bell can be thought of as the first one in
the field of quantum information.
Studies of quantum information led to a proposal, employing
entanglement, of quantum teleportation [Bennett93]. This phenomenon
was observed in 1997 and seems to be at the moment the crowning
achievement of quantum interferometry [Bouwmeester97]. Interestingly,
the method to obtain quantum teleportation
of photon’s polarisation was developed independently, as a
by product of theoretical and experimental research
towards obtaining Bell-EPR phenomena for particles originating
from independent sources [Yurke92, Żukowski93a, Pan98].
The same method was applied to obtain the first ever observations
of Greenberger-Horne-Zeilinger correlations (GHZ) [Bouwmeester99].
The 1989 theoretical
discovery of GHZ correlations [Greenberger89] and the drastic amplification of
the Bell theorem, which is implied by them, was the event in the research
into foundations of quantum theory which amplified interest in
entanglement (with the interesting sociological consequence:
the earlier terminology- correlated state- was replaced by the
Schrödinger’s term entanglement).
The technological progress of 1980’s and 1990’s has enabled all that
experimental and theoretical activity to flourish. The phenomenon of
parametric down conversion (PDC) turned out to be a versatile source
of entangled photons. The simplicity of the phenomenon of
PDC has lead to an explosion of the number of experiments studying
various aspects of entanglement or the basic phenomena
linked with quantum information and quantum communication (dense coding
[Mattle96],
Bell-state measurement [Michler96], etc.).
Pulsed down conversion enabled to observe
two-photon interferometric effect for independently emitted
photons [Pan98]. The future of experimental quantum information
most probably will be associated with trapped atoms and
microcavity-atom interactions [Hagley97]-
fields in which extensive studies of
entanglement are currently carried out.
Interestingly, all these developments led to studies concerning entanglement
properties of mixed states. In the case of the Bell theorem the pioneering
works were by Werner (1989) [Werner89] and the Horodecki Family (1995)
[Horodecki95]. In recent
years one observes an avalanche of works on the problem of separability
of density matrices initialised by Peres [Peres96] and again
the Horodecki Family (see e.g. [Horodecki96]).
The research into the separability has shown once more (recall
Bell-Kochen-Specker theorem [Kochen67])
the qualitative difference between systems described
by 2-dimensional Hilbert space (qubits) and those described by
Hilbert spaces of higher dimension (qu$N$its).
The present work tries to answer some questions on the relation
of the Bell theorem with various performed or proposed multiparticle
(essentially, multiphoton) quantum interference experiments.
The first part deals with some of performed experiments. Proposals
of improvements and re-interpretations are given [1, 2].
Next, in part two, we study new methods of deriving Bell inequalities both
for the standard two qubit experiments as well as to multi-qubit
GHZ experiments [3, 4]. Derivation of GHZ paradoxes
for gedanken experiments involving $M$ entangled qu$N$its observed beyond
multiport beamsplitters [Żukowski97b] is also shown [5, 6].
Finally the last part is devoted to the Bell theorem without
inequalities via a numerical approach utilising linear optimisation
[7, 8].
Chapter 2 Some history and basic notions
2.1 Preliminaries
According to a prevailing common opinion quantum mechanics is
a fundamental theory which applies to all physical systems. Its
predictive power is astonishing. Up to the present day there has not
been a single experiment which invalidates it.
However, the conclusions that can be drawn from quantum mechanics force us to
entirely abandon the picture of nature implied by classical
physics and the common sense. One of the main sources of difference
between the quantum
world and the classical one, in which we are particularly interested
in this work, is entanglement.
The notion of entanglement was introduced for the first time
by Schrödinger to describe a situation
in which
Maximal knowledge of a total system does not necessarily
include total knowledge of all its parts, not even when these are
fully separated from each other and at the moment are not
influencing each other at all…
The work of Schröedinger [Schröedinger35] was partially
motivated by the seminal paper of Einstein, Podolsky and Rosen (EPR)
[Einstein35] in which the authors used an entangled state
(an EPR state) of
two qubits to show that quantum mechanics could not be considered as a complete
physical theory aiming to describe the phenomena occurring
in micro world. Although they did not state it clearly, EPR effectively
postulated the existence of local hidden variables,
in the form of ”elements of reality”,
which were to play
the same role in quantum mechanics as the positions and velocities of particles
in statistical classical mechanics and that were to solve the
interpretational problems of quantum mechanics. That paper directly influenced the
formulation of the Bell theorem111According to Stapp
[Stapp77], the Bell theorem is one
of the most
important discoveries in modern physics.
[Bell64, Bell66, Bell87].
In his 1964 paper Bell for the very first time showed222A theorem by
von Neumann [vonNeumann32] excluding
the possibility of the existence of hidden variables was formulated
in 1930’s but, as pointed
out by Bell, the assumptions used by von Neumann were
much too restrictive. that the idea of local hidden variables was in
contradiction with quantum mechanics and, what is even more important, that
it could be tested experimentally! That way the subject mainly
discussed by physicists at the parties was brought to the realm of
experimentally verifiable physics.
2.2 Entanglement
To discuss basic features of entanglement let us consider the following
state of two two-state systems (qubits)333If we take as
the qubits two spin ${1\over 2}$ particles
and put $|0\rangle=|-\frac{1}{2}\rangle,|1\rangle=|+\frac{1}{2}\rangle$
the corresponding state
is a rotationally invariant state with the total spin equal zero-
the so called singlet state.
$$\displaystyle|\psi\rangle={1\over\sqrt{2}}(|0\rangle_{1}\otimes|1\rangle_{2}-|%
1\rangle_{1}\otimes|0\rangle_{2}),$$
(2.1)
where $\otimes$ denotes the tensor product444This symbol
will be only used when it makes the notation easier to read. and kets
$|0\rangle_{i},|1\rangle_{i}$ describe two orthogonal states
of the $i$-th qubit.
The above pure state describes a coherent superposition of two
product states that occur with equal probability.
According to quantum mechanics
$|\psi\rangle$ contains all available information about the state of
the qubits. If we write (2.1) in the form of a density matrix
$\rho_{12}$
$$\displaystyle\rho_{12}=|\psi\rangle\langle\psi|={1\over 2}\left[|0\rangle_{1}{%
}_{1}\langle 0|\otimes|1\rangle_{2}{}_{2}\langle 1|-|0\rangle_{1}{}_{1}\langle
1%
|\otimes|1\rangle_{2}{}_{2}\langle 0|\right.$$
$$\displaystyle\left.-|1\rangle_{1}{}_{1}\langle 0|\otimes|0\rangle_{2}{}_{2}%
\langle 1|+|1\rangle_{1}{}_{1}\langle 1|\otimes|0\rangle_{2}{}_{2}\langle 0|%
\right],$$
(2.2)
and trace out one qubit
we obtain a density matrix describing the other qubit, which reads
$$\displaystyle\rho_{k}={1\over 2}\left(|1\rangle_{k}{}_{k}\langle 1|+|0\rangle_%
{k}{}_{k}\langle 0|\right),$$
(2.3)
$k=1,2$.
Such a density matrix describes a situation in which we have a chaotic
mixture of two orthogonal pure states. This is a
situation typical for
the entanglement. We have the full possible information about the
state of two qubits as a
whole but we do not have any information about the state of each
qubit separately, which in fact is not even defined! In addition,
the properties of
the qubits are tightly correlated.
To see this more clearly
let us consider the measurement of two dichotomic observables
with spectrum consisting of $\pm 1$, which spectral decomposition
has the following
form
$$\displaystyle\hat{O}_{k}(\phi_{k})=|+,\phi_{k}\rangle_{k}{}_{k}\langle+,\phi_{%
k}|-|-,\phi_{k}\rangle_{k}{}_{k}\langle-,\phi_{k}|$$
(2.4)
with $k=1,2$ and
$$\displaystyle|+,\phi_{k}\rangle_{k}={1\over\sqrt{2}}\left(|0\rangle_{k}+e^{i%
\phi_{k}}|1\rangle_{k}\right)$$
$$\displaystyle|-,\phi_{k}\rangle_{k}={1\over\sqrt{2}}\left(|0\rangle_{k}-e^{i%
\phi_{k}}|1\rangle_{k}\right),$$
(2.5)
where $\phi_{k}\in[0,2\pi]$. The
mean value $E_{QM}(\phi_{1},\phi_{2})$ of the joint measurement of the
observable
$\hat{O}_{1}(\phi_{1})$ on
the first qubit and the observable $\hat{O}_{2}(\phi_{2})$ on the second one-
the so called correlation function-
reads
$$\displaystyle E_{QM}(\phi_{1},\phi_{2})=\langle\psi|\hat{O}_{1}(\phi_{1})\hat{%
O}_{2}(\phi_{2})|\psi\rangle=-\cos(\phi_{1}+\phi_{2}).$$
(2.6)
It is easy to see that, for example, whenever the sum of the
phases $\phi_{1}$ and
$\phi_{2}$ is $0$ modulo $2\pi$ we observe the so called perfect correlations
between the results of the measurements performed by the
observers; if the first observer obtains $+1$ as the result
of his measurement the second one obtains $-1$ and vice versa.
However, each observer alone measures
$+1$ and $-1$ with equal probability, which can be easily seen
using the density matrix (2.3)
$$\displaystyle\langle O_{k}(\phi_{k})\rangle_{\rho_{k}}=Tr(O_{k}(\phi_{k})\rho_%
{k})=0.$$
(2.7)
The above formulas clearly demonstrate that
all information about the state (2.1) is contained in the joint
properties of the qubits.
2.3 Elements of reality
In their famous paper [Einstein35] EPR used the perfect correlations
observed when measuring local observables on an entangled system of spatially
separated qubits
(2.1) to demonstrate that quantum mechanics is incomplete, i.e., that
one needs some additional parameters to fully describe phenomena occurring
in micro world.
We briefly present their reasoning in the version of Bohm
[Bohm51, Bohm52] to introduce the notion of
local realism, which plays a central part in
the Bell theorem.
EPR reasoning goes as follows. According to quantum mechanics all we know about
the system of two entangled qubits is encoded in the
state (2.1). Quantum mechanics also tells us that we cannot consider
simultaneous measurements
of two non commuting observables and therefore does not even
define predictions for such cases.
Let us imagine that the observers
one and two are spatially separated and
that they simultaneously measure the observables $\hat{O}_{1}(\phi_{1}=0)$ and
$\hat{O}_{2}(\phi_{2}=0)$ (see the picture (2.1)) on
subsystems described by the singlet
state.
For such observables
the
perfect correlations occur, which means that if the first observer has
obtained $+1$ the second one, because of (2.6), must
have obtained
$-1$ and vice versa. Moreover, due to the spatial separation of
the observers and the non superluminal velocity of propagation
of any interaction (information) in nature (locality),
the outcome of the first (second) observer cannot be influenced
by the choice of observable measured
by
the second (first) observer and neither by its outcome. All correlations
between the results of measurements must have been established in
the source.
However, the second observer could have measured
the observable $\hat{O}_{2}(\phi_{2}^{\prime}={\pi\over 2})$ instead of the
previous one characterised by $\phi_{2}=0$. Then, the
result of this would-be measurement, would have enabled him to infer
with probability equal to one
and without disturbing the first qubit
the result of the measurement of
the observable $\hat{O}_{1}(\phi_{1}^{\prime}={\pi\over 2})$ by the first
observer. At this stage of reasoning EPR introduce the notion of
physical reality [Einstein35]:
If, without in any way disturbing a system, we can predict with
certainty (i.e. with probability equal to unity) the value of a
physical quantity, then there exists an element of physical
reality corresponding to this physical quantity.
According to this definition the inference made by the
second observer555The reasoning can be reversed and the
inference can be made by the first observer about the second one.
about the result of the measurement that could have been made by the
first observer has a well defined physical meaning. This would mean
that it is possible to ascribe the definite values to the results of
the measurement
of two non commuting observables $\hat{O}_{i}(\phi_{i}=0)$
and $\hat{O}_{i}(\phi_{i}^{\prime}={\pi\over 2})$ ($i=1,2$), the statement which
makes quantum purist’s hair stand on ends.
Therefore, according to EPR, quantum mechanics is incomplete
and should be completed at least by introducing elements
of reality into the description of quantum state.
2.4 The Bell theorem
The hypothesis of local hidden variables, which effectively
stems from the notion of elements of reality, was proved
to be unacceptable for systems described by quantum
mechanics
by Bell,
[Bell64] who found an inequality which should be
obeyed by any local hidden variable theory, and which
is violated by predictions of quantum mechanics.
Here we present the derivation of the Clauser-Horne-Shimony-Holt (CHSH)
inequality
[Clauser69].
To this end, let us again
consider the experiment in which two spatially separated observers
perform the
measurements
of the observables defined in (2.4) on
the state (2.1). Each of them as a result of their measurement
obtains only one of the two possible values $\pm 1$ (they measure bivalued
observables). Although the different results of the measurement at
each observer’s
side occur with equal probability
the results of joint measurements are correlated, which is
expressed by the formula (2.6).
If deterministic local hidden variables exist they predetermine
the results of each single measurement for each observer at the time
of the emission of each pair of qubits. In other words,
the value of the local hidden variable for a particular emission
of a pair of qubits would allow us to predict the result of the measurement
made by each observer with certainty. However this value is hidden.
To model the probabilistic
nature of quantum experiments
we assume that there exists some probability distribution of local hidden variables
associated with a given quantum mechanical state,
which represents our lack of knowledge about them.
To express the above idea of local hidden variables mathematically we assume that
there is a set of hidden variables $\Lambda$ on which we can define
a probabilistic measure $d\rho(\lambda)$. We also assume the existence
of two bivalued functions $O_{1},O_{2}$ defined on the space
$\Lambda$ which take only the values $\pm 1$. Each of these two functions
must
depend on the local parameters
$\phi_{1}$ and $\phi_{2}$ characterising the experiment performed by the
first and the second observer respectively. Because of the assumption
of locality the function $O_{1}$ can solely depend on the $\phi_{1}$ and the
other one on $\phi_{2}$.
Thus, the correlation function based
on the idea of local hidden variables must have the following form
$$\displaystyle E_{LHV}(\phi_{1},\phi_{2})=\int_{\Lambda}d\rho(\lambda)O_{1}(%
\lambda,\phi_{1})O_{2}(\lambda,\phi_{2}),$$
(2.8)
where $\int_{\Lambda}d\rho(\lambda)=1$.
Now, let us imagine that each observer performs two mutually exclusive
experiments characterised by $\phi_{i},\phi_{i}^{\prime}$ and let us consider
the following expression made out of the four local hidden variables
correlation
functions (2.8)
$$\displaystyle C_{LHV}=E_{LHV}(\phi_{1},\phi_{2})+E_{LHV}(\phi_{1}^{\prime},%
\phi_{2})+E_{LHV}(\phi_{1},\phi_{2}^{\prime})-E_{LHV}(\phi_{1}^{\prime},\phi_{%
2}^{\prime})$$
$$\displaystyle=\int_{\Lambda}d\rho(\lambda)\left[O_{1}\left(\lambda,\phi_{1})(O%
_{2}(\lambda,\phi_{2})+O_{2}(\lambda,\phi_{2}^{\prime})\right)+O_{1}(\lambda,%
\phi_{1}^{\prime})\left(O_{2}(\lambda,\phi_{2})-O_{2}(\lambda,\phi_{2}^{\prime%
})\right)\right].$$
(2.9)
It is easy to see that the modulus of the expression in the square
brackets is either $-2$ or $+2$. Therefore, the hypothesis of local hidden variables implies
that the following inequality (Bell-CHSH inequality) must be valid
$$\displaystyle-2\leq C_{LHV}\leq 2.$$
(2.10)
Is the CHSH inequality always satisfied by quantum predictions for (2.1)?
To answer this question
let us put $\phi_{1}=0,\phi_{1}^{\prime}={\pi\over 2}$ and $\phi_{2}=-\pi/4,\phi_{2}^{\prime}=\pi/4$. For these values of local parameters one has
$$\displaystyle C_{QM}=E_{QM}(\phi_{1},\phi_{2})+E_{QM}(\phi_{1}^{\prime},\phi_{%
2})+E_{QM}(\phi_{1},\phi_{2}^{\prime})-E_{QM}(\phi_{1}^{\prime},\phi_{2}^{%
\prime})$$
$$\displaystyle=-\cos(\phi_{1}+\phi_{2})-\cos(\phi_{1}^{\prime}+\phi_{2})-\cos(%
\phi_{1}+\phi_{2}^{\prime})+\cos(\phi_{1}^{\prime}+\phi_{2}^{\prime})=-2\sqrt{%
2}.$$
(2.11)
Because $-2\sqrt{2}<-2$, we have a contradiction.
The above result, known as the Bell theorem666The inequality
which must be obeyed by any local and realistic theory is usually called the Bell
inequality whereas the violation of such an inequality by quantum mechanics is called the Bell theorem.,
needs some further explanation. In our
reasoning we have made two crucial assumptions without which the theorem would not be valid.
These assumptions are: locality and realism.
The Bell theorem tells us that either notion of locality, or realism, or both are false
in quantum theory [Redhead87].
Another remark is that the Bell-CHSH inequality can be directly applied to any experimental
data. Also, even if quantum mechanics is not valid and we will find another better theory we
can still, using the Bell inequality, verify whether this new theory
fulfils the necessary condition for the local realistic description or not.
The final remark is that one may consider the existence of the so called stochastic
local hidden variables [Clauser74], which do not predict with certainty the results of local measurements but
give merely the probabilities of their occurrence. In such a case instead of functions $O_{n}$
($n=1,2$) appearing in (2.8) we have the probabilities
$P_{1}(m|\lambda,\phi_{1}),P_{2}(m^{\prime}|\lambda,\phi_{2})$
giving the ratio of occurrence of the results $m,m^{\prime}$ when
measuring observables characterised by parameters $\phi_{1},\phi_{2}$ respectively
(obviously they sum up to
one, i.e., $\sum_{m=\pm 1}P_{i}(m|\lambda,\phi_{i})=1$, $i=1,2$). The
relation between $P_{i}(m|\lambda,\phi_{i})$ and $O_{i}(\lambda,\phi_{i})$ ($i=1,2$)
is such that within this description $O_{i}=\pm 1$ have to be replaced by
$$\displaystyle O_{i}(\lambda,\phi_{i})=\sum_{m=\pm 1}mP_{i}(m|\lambda,\phi_{i}).$$
(2.12)
with the values of modulus bounded by 1.
With such
probabilities the idea of stochastic local hidden variables is to
reproduce the quantum probabilities
$P_{QM}(m,m^{\prime}|\phi_{1},\phi_{2})$, i.e., the probabilities of
obtaining the result $m$ and $m^{\prime}$ by the first and the second
observer when measuring
the observable characterised by $\phi_{1}$ and $\phi_{2}$ respectively,
by local hidden variables probabilities of the form
$$\displaystyle P_{HV}(m,m^{\prime}|\phi_{1},\phi_{2})=\int d\rho(\lambda)P_{1}(%
m|\lambda,\phi_{1})P_{2}(m^{\prime}|\lambda,\phi_{2}).$$
(2.13)
It is clear that any deterministic local hidden variables theory
can be always treated as a stochastic one.
Fine [Fine82] proved that a stochastic
local hidden variable theory implies the existence
of an underlying deterministic one777We do not take into account
non Kolmogorovian probability calculus..
In general, all Bell-type inequalities found since the famous Bell
paper [Bell64]
constitute only necessary conditions for the existence of local hidden variables.
The exceptions are
the full set of four Clauser-Horne inequalities (CH) [Clauser74]
and the Bell-CHSH inequality,
which were proved
by Fine [Fine82] to be
also sufficient ones for dichotomic observables888The CHSH inequality
is sufficient if one assumes certain
symmetries of the probabilities. (see also [Peres99]).
In 1989 Greenberger, Horne and Zeilinger (GHZ) [Greenberger89] used
a maximally entangled state of four qubits to show
that the discrepancy between local realism and quantum mechanics
is much stronger than that observed in two qubit
correlations999Later Mermin simplified the proof
and derived GHZ paradox for three qubits [Mermin90a]..
By a clever trick they showed that the idea
of local realism breaks already at the stage of defining elements of
reality.
The Bell theorem
does not have to be restricted to two or three
qubit correlations. The discrepancy
between local realism and quantum mechanics
can be also proved for entangled particles each
described by an $N$ dimensional Hilbert space- so called qu$N$its
(see, for instance,
[Mermin80, Mermin82, Garg82, Gisin92, Peres92, Wódkiewicz94])-
as well as for $M$ entangled qubits (see, for instance,
[Mermin90b]).
2.5 Experimental tests of the Bell theorem
2.5.1 First experiments
The first experimental test of Bell inequality was performed
by Freedman and Clauser [Freedman72] with
photons from atomic cascade decays.
They observed violation of Bell inequality and confirmation
of quantum mechanical predictions. However, in the experiment
the detection efficiency and the angular correlation of the photon pairs
was low, and no care was taken to make
the two polarising settings detection stations to be set
independently [Clauser74, Santos92].
Such care was taken in the most quoted experiment
by Aspect, Grangier and Dalibard [Aspect82].
In this experiment fast switchings of the analyser
position to prevent ”communication” between the source and
the analyser was used. However, as it was pointed out in
[Zeilinger86] the periodic switching was not truly
random and
was predictable after a few periods of the switch101010In 1998
Weihs et. al. [Weihs98] for the first time performed
an experiment in which true locality condition was
enforced. In the experiment two observers were spatially
separated by the distance of 400m, which means that the time
of the measurement had to be shorter than 1.3$\mu$s to prevent
communication with the speed of light between observers.
They succeeded to achieve
the time of measurement within 100 ns and
the violation of the CHSH inequality by 30 standard deviations
was observed.
.
In all experiments
(except one, with systematic errors)
the violation of the Bell inequalities (with certain additional
assumptions)
was
observed. An excellent review concerning these
pioneering experiments can be found
in [Clauser78].
2.5.2 New experiments
Recently the experiments, in which the entangled pairs of
photons are generated in the process of parametric down
conversion (PDC),
dominated the field of laboratory tests of local
realism. In the PDC process the pairs of photons
are spontaneously created. The propagation directions and
the frequencies of created photons (the photons are called
idlers and signals) are highly correlated, which
is used to generate an entangled state. In the type-II
down conversion one has also correlated polarisations.
Among the Bell-type experiments with PDC process
the following ones will be mainly addressed to in this work
•
Alley-Shih and Ou-Mandel experiments [Shih88, Ou88]
•
entanglement swapping [Pan98]
•
GHZ experiment [Bouwmeester99].
First Bell-type experiments with a PDC source of
correlated photons were experiments by
Alley, Shih and Ou, Mandel. In both of them
the violation of the Bell inequality (by 3 standard deviations
in [Shih88] and 6 standard deviations in [Ou88])
and confirmation
of quantum mechanics was reported. However, in both
experiments only coincident counts were measured (half
of the events were discarded). That raised some doubts about
the validity of the experiments as tests of local
realism [Santos92, Garuccio94]. The situation was
clarified in [Popescu97] where it was shown that one
does not need to discard ”wrong” events to test local realism
in the experiments.
In this dissertation we perform
further theoretical analysis of these experiments (see Part I).
In 1993 Żukowski et. al [Żukowski93a] showed
experimental conditions to
entangle particles (photons) originating from
independent sources111111The first proposal was given by Yurke and Stoler
[Yurke92].. Five years
later, in 1998, the first entanglement
swapping experiment was performed [Pan98].
The visibility
of around $65\%$ was observed (the notion of visibility
is explained in the next subsection).
In 1999 Bouwmeester et. al [Bouwmeester99]
reported the first experimental
observation of the GHZ correlations. The experiment was based
on the techniques developed in the teleportation experiment [Bouwmeester97]
and entanglement swapping experiment [Pan98]. In the experiment
pairs of entangled photons (entangled polarisations)
produced in a nonlinear crystal pumped by a short pulse
of ultraviolet light from the laser were used. The applied technique to
obtain GHZ correlations rests upon an observation that when a single
particle from two independent entangled pairs is detected in a manner
such that it is impossible to determine from which pair the single
came, the remaining three particles become entangled.
The high visibility
of around $60\%$ was observed.
The experimental realisation of entanglement does not have to
be restricted to massless particles (photons).
Hagley et. al. reported the experiment in which
entangled atoms were produced [Hagley97]. They demonstrated
the entanglement of pairs of atoms at centimetric distances and measured
their correlations.
The visibility of only around $25\%$ was observed.
2.5.3 Problems encountered in the Bell type experiments
However, in all experiments performed thus far there
has been the problem
with a low quantum efficiency $\eta$ of detectors used to register
incoming particles121212More precisely the quantum efficiency
describes the full detection stations, including all devices that
collect the incoming radiation (lenses, etc.).. The quantum efficiency is
defined as the ratio
of the number of detected particles to the number of the emitted
ones131313In the operational terms it is defined for a detection
station A as the ratio of coincident correlated counts at the pair of
detection stations A and B to the ratio of singles at the station B..
If it lies
below the threshold value $\eta^{tr}_{2}=2\sqrt{2}-2$
there
is no violation of the Bell-CHSH and the CH inequality for maximally
entangled two qubits.
Since the collection efficiency in all experiments done so far was much lower
than $\eta^{tr}_{2}$, all claims about the violations of Bell inequalities in
performed experiments are based on the assumption
that the observed
sub ensemble of particles is a representative one
for the emitted ensemble
(so called
”fair sampling assumption”).
In the real experiment one usually cannot obtain a pure maximally entangled
state of two qubits due to
some
imperfections
in the source producing the state and other difficulties. As a simple
generic model of such experimental imperfections
one can take
$$\displaystyle\hat{\rho}(F_{2})=F_{2}\hat{\rho}_{noise}+(1-F_{2})\hat{\rho}_{%
pure},$$
(2.14)
where for qubits $\hat{\rho}_{noise}={1\over 4}I\otimes I$ ($I$ is a unit $2\times 2$
matrix),
$\hat{\rho}_{pure}=|\psi\rangle\langle\psi|$ (for the definition
of $|\psi\rangle$ see (2.1)) and the real parameter
$F_{2}$ (the index 2 stands for qubit)
lies between zero and one ($0\leq F_{2}\leq 1$).
The $\hat{\rho}_{noise}$ describes a totally chaotic mixture of two qubits.
Results of any measurements carried out on the
$\hat{\rho}_{noise}$ are completely uncorrelated therefore
the parameter $F_{2}$ can be interpreted as the number telling us
how much noise is contained
in the system.
It is easy to
check that for $\hat{\rho}(F_{2})$ the correlation function
defined in (2.6)
reads
$$\displaystyle E_{QM}^{F_{2}}(\phi_{1},\phi_{2})=Tr[\hat{\rho}(F_{2})\hat{O}_{1%
}(\phi_{1})\hat{O}_{2}(\phi_{2})]=-(1-F_{2})\cos(\phi_{1}+\phi_{2}).$$
(2.15)
We see that if $F_{2}>0$ the amplitude of the correlation function is
reduced.
In quantum interferometry the number $1-F_{2}$
is directly linked with
the visibility (contrast)
of interferometric
two-qubit fringes141414The traditional definition of
interference visibility (contrast) is given by
$\displaystyle V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$
(2.16)
where $I$ is the intensity (of light) in an interference
pattern. $I_{max}$ refers in the case
of spatial pattern to the maximum intensity and $I_{min}$
to its neighbouring minimum. The same formula can be used
for an output of a Mach-Zehnder interferometer into one
of its exit arms. In this case $I_{max}$ is the maximum intensity
and $I_{min}$ its neighbouring minimum which occurs after suitable
change of the phase shift. In the case of a quantum process for
a single particle the visibility is defined, in analogy, as
$\displaystyle V=\frac{P_{max}-P_{min}}{P_{max}+P_{min}},$
(2.17)
where $P_{max}$ and $P_{min}$ are the maximal and minimal
probabilities for detection of the particle in a specified
output of an interferometer (also obtainable for certain
different phase settings).
The visibility of two particle interference is again defined
using the same general rule (2.17). However, in this case
$P$’s refer to the probability of coincident counts at a pair
of detectors. In the case when both single particle and two
particle interference occurs in the experiment, the relation
between the two visibilities is quite subtle, and the two-particle
visibility has to be redefined [Jaeger93]. However, throughout the present
work we shall discuss only the cases for which no single particle
interference occurs..
Therefore, sometimes it is convenient
to consider the parameter $V_{2}=1-F_{2}$ instead of $F_{2}$, which is usually
done in the description of real experiments. Throughout the dissertation
both parameters will be used depending on the context.
In the entanglement swapping experiment [Pan98]
and in the
experiments with atoms [Hagley97] the observed visibility
was quite low: around $65\%$ in the entanglement swapping and
$25\%$ in the experiment with entangled atoms.
It is obvious that if there is too much noise in the system, i.e.,
the visibility is low, then one cannot observe violations
of Bell inequalities.
To summarise, if in a Bell-type experiment with
two maximally entangled qubits $V_{2}\leq{1\over\sqrt{2}}$
($F_{2}\geq\frac{2-\sqrt{2}}{2}$)
the Bell-CHSH inequality (and also the CH inequality)
cannot be violated. Furthermore, in the case of
the Bell-CHSH inequality, if $\eta_{2}\leq 2\sqrt{2}-2$, then even if
we have the perfect case, i.e., $V_{2}=1$ ($F_{2}=0$), the violation of
the inequality has only a bona fide status- one has to use
the fair sampling assumption. In all experiments thus far
performed one has $\eta_{2}\ll 2\sqrt{2}-2$. Similar situation (to
be described in more details later) occurs for the GHZ
correlations (with new specific threshold $\eta$’s and $V$’s).
Part I New theoretical analysis of Alley-Shih, Ou-Mandel
and entanglement swapping experiments
*
Some of the performed Bell-type experiments
have a distinguishing trait. Not all event observed follow
the standard pattern assumed in the usual derivations
of Bell inequalities.
To such experiments belong the Alley-Shih-Ou-Mandel
experiment [Shih88, Ou88] and the entanglement swapping
experiment [Pan98].
In all these experiments even in the ideal, gedanken case,
half or more of the emissions do not lead to correlated counts at
spatially separated detector stations. Therefore in order to
prove that such experiments can be indeed considered as tests
of local realism one has to perform an analysis which takes
into account this characteristic trait. Such an analysis
will be presented below for the Alley-Shih, Ou-Mandel
experiments and the entanglement swapping experiment151515The
experiments proposed by Franson (1989) share properties which were
thought to be of a similar nature. By a closer inspection it turns out
that they are different. For explanation see [Aerts99].. Special
care will be taken to discuss the question of whether two versus
single photon counts distinguishability is the necessary requirement
for the studied experiments.
Chapter 3 Alley-Shih and Ou-Mandel experiments: resolution of the problem
of distinguishability of single and two photon events [1]
3.1 Introduction
The first Bell-type experiments which employed parametric
down conversion process as the source of entangled photons
were those reported in refs [Shih88] and [Ou88].
However, the specific traits of those experiments have led to a
protracted dispute
on their validity as tests of local realism.
In this case the issue was not the standard problem of detection efficiency
(which up till now permits a local realistic interpretation of
all performed experiments).
The trait that distinguishes the experiments is that even in the
perfect gedanken situation (which assumes perfect detection)
only in $50\%$ of the detection events
each observer receives a
photon, in the other $50\%$ of events one observer receives both
photons of a pair while the other observer receives none.
The early “pragmatic” approach was to discuss only the events of the
first type (as only such ones lead to spatially
separated coincidences).
Only those were used as the data input to the
Bell inequalities in [Shih88] and [Ou88].
This procedure was soon
challenged (see e.g. [Kwiat94, Kwiat95], and
especially in the theoretical analysis of ref. [Garuccio94]), as it
raises justified doubts whether such
experiments could be ever genuine tests of local realism (as the effective
overall collection efficiency of the photon pairs, $50\%$ in the gedanken
case, is much below what is usually required for tests of local realism).
Ten years after the first experiments of this type were made, finally
the dispute was resolved [Popescu97].
It was proposed, to take into account also those
“unfavourable” cases and to
analyse the entire
pattern of events.
In this way one can indeed show that the experiments
are true test of local realism (namely,
that the CHSH inequalities are violated by quantum predictions
for the idealised case). The idea was based upon
a specific value assignment for the “wrong events”.
However, the scheme presented by
Popescu et al [Popescu97] has one drawback. The authors assumed in
their analysis that the detecting scheme employed in the
experiment should be able to
distinguish between two and one photon detections.
This was not the case in the actual experiments.
The aim of this chapter is to show that even this is unnecessary, all one
needs
is the use of the specific value assignment procedure of [Popescu97].
Finally, we shall also give
prediction of all effects occurring in the experiment.
It is quite often overlooked that a kind of Hong-Ou-Mandel dip phenomenon
[Hong87] can be observed in the experiment.
3.2 Description of the experiment
In the class of experiments we consider
(see (3.1)) [Popescu97]
a type I parametric down-conversion source [Hong85]
is used to generate pairs of
photons which are degenerated in frequency and polarisation
(say $\hat{x}$) but propagate in two different directions.
One of the photons
passes through a wave plate ($WP$) which rotates its polarisation by
$90^{o}$.
The two photons are then directed
onto the two input ports of a (nonpolarising) “$50-50$”
beamsplitter ($BS$). The observation stations
are located in the exit beams of the beamsplitter. Each
local observer
is equipped with a polarising beamsplitter111Following
references [Garuccio94] and [Popescu97]
we assume that both local detection stations
are equipped with polarising beamsplitters, and
each of the output ports is observed by a detector.
In the actual experiments [Shih88] and [Ou88] at each station
only one of the
outputs was monitored., orientated along an
arbitrary axis (which, in principle can be
randomly chosen, in the delayed-choice manner,
just before the photons are supposed to
arrive). Behind each polarising beamsplitter are two detectors, $D_{1}^{+}$,
$D_{1}^{-}$ and $D_{2}^{+}$, $D_{2}^{-}$ respectively, where the lower index indicates
the corresponding observer and the upper index the two exit ports of the
polarised beamsplitter ($``+"$ meaning parallel with the polarisation axis of
the beamsplitter and $``-"$ meaning orthogonal to this axis). All
optical paths are assumed to be equal.
3.3 Quantum predictions
Let us calculate the quantum predictions for the experiment.
We will use the second quantisation
formalism, which is very convenient here, since the whole phenomenon observed
in the experiment rests upon the indistinguishability
of photons.
After the action of the wave-plate one can approximate quantum mechanical
state describing two
photons emerging from a non - linear crystal along the ”signal” and the
”idler”
beam by
$$|\Psi_{0}\rangle=a_{1\vec{x}}^{\dagger}a_{2\vec{y}}^{\dagger}|0\rangle,$$
(3.1)
where $a_{1\vec{x}}^{\dagger}$ and $a_{2\vec{y}}^{\dagger}$ are creation
operators and $|0\rangle$ denotes the vacuum state. Subscripts
$\vec{x},\vec{y}$
decode the polarisation of the photon (either along $\vec{x}$
or $\vec{y}$ axis). The beamsplitter action can be described by
$$\displaystyle a_{1\vec{x}}^{\dagger}={1\over\sqrt{2}}(ic_{\vec{x}}^{\dagger}+d%
_{\vec{x}}^{\dagger})$$
$$\displaystyle a_{2\vec{y}}^{\dagger}={1\over\sqrt{2}}(c_{\vec{y}}^{\dagger}+id%
_{\vec{y}}^{\dagger}),$$
(3.2)
where $c_{\vec{x}}^{\dagger},d_{\vec{x}}^{\dagger},c_{\vec{y}}^{\dagger},d_{\vec{y}}^%
{\dagger}$
are operators describing output modes of the beamsplitter ($c$ stands for
the first observer and $d$ for the second one). Thus
our state $|\Psi_{0}\rangle$ changes to :
$$|\Psi\rangle=\frac{1}{2}(ic_{\vec{x}}^{\dagger}c_{\vec{y}}^{\dagger}-c_{\vec{x%
}}^{\dagger}d_{\vec{y}}^{\dagger}+c_{\vec{y}}^{\dagger}d_{\vec{x}}^{\dagger}+%
id_{\vec{x}}^{\dagger}d_{\vec{y}}^{\dagger})|0\rangle$$
(3.3)
Next comes the action of the polarisers in both beams,
which can be described
as
$$\displaystyle n_{\vec{x}}^{\dagger}=\cos(\theta_{1})n_{\parallel}^{\dagger}+%
\sin(\theta_{1})n_{\perp}^{\dagger}$$
$$\displaystyle n_{\vec{y}}^{\dagger}=\sin(\theta_{1})n_{\parallel}^{\dagger}-%
\cos(\theta_{1})n_{\perp}^{\dagger},$$
where $n^{\dagger}=c^{\dagger}$ or $d^{\dagger}$, and $n_{\parallel}^{\dagger}$ describes
the mode parallel
to
polarizer’s axis and $n_{\perp}^{\dagger}$ describes the mode perpendicular
to polarizer’s axis; $\theta$ is the angle between the $\vec{x}$
axis and polarizer’s axis.
Thus the final state reaching the detector reads
$$\displaystyle|\psi_{final}\rangle=\frac{1}{2}\big{[}\sin(\theta_{1}-\theta_{2}%
)|c_{\parallel},d_{\parallel}\rangle$$
$$\displaystyle+\cos(\theta_{1}-\theta_{2})|c_{\parallel}d_{\perp}\rangle$$
$$\displaystyle-\cos(\theta_{1}-\theta_{2})|c_{\perp},d_{\parallel}\rangle+\sin(%
\theta_{1}-\theta_{2})|c_{\perp},d_{\perp}\rangle$$
$$\displaystyle+i{1\over{\sqrt{2}}}\sin(2\theta_{1})|2c_{\parallel}\rangle+i{1%
\over{\sqrt{2}}}\sin(2\theta_{1})|2c_{\perp}\rangle$$
$$\displaystyle-i\cos(2\theta_{1})|c_{\perp},c_{\parallel}\rangle+i{1\over{\sqrt%
{2}}}\sin(2\theta_{2})|2d_{\parallel}\rangle$$
$$\displaystyle+i{1\over{\sqrt{2}}}\sin(2\theta_{2})|2d_{\perp}\rangle-i\cos(2%
\theta_{2})|d_{\parallel},d_{\perp}\rangle\big{]},$$
where e.g. $|c_{\parallel},d_{\parallel}\rangle$ denotes one photon in the mode $c_{\parallel}$, and
one in
$d_{\parallel}$, whereas $|2c_{\parallel}\rangle={1\over{\sqrt{2}}}{c_{\parallel}^{\dagger}}^{2}|0\rangle$ denotes two photons
in the mode $c_{\parallel}$.
Let us denote by $P(i,\theta_{1};j,\theta_{2})$ the joint
probability for the outcome $i$ to be registered by observer 1 when
her
polariser is oriented along the direction that makes an angle $\theta_{1}$
with the $\vec{x}$ direction and the outcome $j$ to be registered by
observer 2 when her polariser is oriented along the direction that
makes an angle $\theta_{2}$ with the $\vec{x}$ direction. Here
$i,j=1-6$ and have the following meaning [Popescu97]:
1=one photon in $D^{-}$, no photon in $D^{+}$
2=one photon in $D^{+}$, no photon in $D^{-}$
3=no photons
4=one photon in $D^{+}$ and one photon in $D^{-}$
5=two photons in $D^{+}$, no photon in $D^{-}$
6=two photons in $D^{-}$, no photons in $D^{+}$.
The quantum predictions for joint probabilities of those events
are given by:
$$\displaystyle P(1,\theta_{1};1,\theta_{2})=P(2,\theta_{1};2,\theta_{2})={1%
\over 8}[1-\cos 2(\theta_{1}-\theta_{2})],$$
(3.6)
$$\displaystyle P(2,\theta_{1};1,\theta_{2})=P(1,\theta_{1};2,\theta_{2})={1%
\over 8}[1+\cos 2(\theta_{1}-\theta_{2})],$$
(3.7)
$$\displaystyle P(5,\theta_{1};3,\theta_{2})=P(6,\theta_{1};3,\theta_{2})={1%
\over{8}}\sin^{2}(2\theta_{1}),$$
(3.8)
$$\displaystyle P(3,\theta_{1};5,\theta_{2})=P(3,\theta_{1};6,\theta_{2})={1%
\over{8}}\sin^{2}(2\theta_{2}),$$
(3.9)
$$\displaystyle P(4,\theta_{1};3,\theta_{2})={1\over{4}}\cos^{2}(2\theta_{1}),$$
(3.10)
$$\displaystyle P(3,\theta_{1};4,\theta_{2})={1\over{4}}\cos^{2}(2\theta_{2}).$$
(3.11)
Following [Popescu97] we associate with each outcome registered
by the observer
1 and 2
a corresponding value $a_{i}$ and
$b_{j}$ respectively, where $a_{1}=b_{1}=-1$
while all the other values are equal to 1. Let us denote by
$E(\theta_{1},\theta_{2})$ the expectation value of their product
$$E(\theta_{1},\theta_{2})=\sum_{i,j}a_{i}b_{j}P(i,\theta_{1};j,\theta_{2}).$$
(3.12)
After simple calculations one has:
$$\displaystyle E(\psi_{1},\psi_{2})$$
$$\displaystyle=-{1\over{2}}\cos(\psi_{1}+\psi_{2})+\frac{1}{2},$$
(3.13)
where we have put $2\theta_{k}=(-1)^{k-1}\psi_{k}$.
The above formula for the correlation function is valid if one
assumes that it is possible to
distinguish between single and double photon detection. This is
usually not
the case. Thus it is convenient to have a parameter $\alpha$ that
measures the distinguishability of the double and single detection at
one detector ( $0\leq\alpha\leq 1$, and gives
the probability of distinguishing by the employed detecting scheme
of the double counts). The partial distinguishability blurs the
distinction
between events 1 and 6 (2 and 5) and thus part of the events of
the type 6 are interpreted as of type 1 and are ascribed by the local
observer a wrong value, e.g. an event of type 6, if both photons
go to the $``-"$ exit of the polariser, can be interpreted as a firing due
to a single photon and is ascribed a $-1$ value. Please note that
such events like 1 or 2 in station 1 accompanied by 3 (no photon) at station 2
do not contribute to the correlation function because for any
$\alpha$ $P(1,\theta_{1};3,\theta_{2})=P(2,\theta_{1};3,\theta_{2})$.
If the parameter $\alpha$ is taken into account the correlation function
acquires
the following form:
$$\displaystyle E(\psi_{1},\psi_{2};\alpha)$$
$$\displaystyle=-{1\over{2}}\cos(\psi_{1}+\psi_{2})+{1\over{2}}\alpha$$
$$\displaystyle+{1\over{4}}(1-\alpha)(\cos^{2}\psi_{1}+\cos^{2}\psi_{2})$$
(3.14)
3.4 Conditions to violate local realism
After the insertion of the quantum correlation
function (3.14) into the CHSH
inequality,
$$\displaystyle-2\leq E(\psi_{1},\psi_{2};\alpha)+E(\psi_{1}^{\prime},\psi_{2};\alpha)$$
$$\displaystyle+E(\psi_{1},\psi_{2}^{\prime};\alpha)-E(\psi_{1}^{\prime},\psi_{2%
}^{\prime};\alpha))\leq 2,$$
one obtains:
$$\displaystyle-2\leq-{1\over{2}}[\cos(\psi_{1}+\psi_{2})+\cos(\psi_{1}^{\prime}%
+\psi_{2})$$
$$\displaystyle+\cos(\psi_{1}+\psi_{2}^{\prime})-\cos(\psi_{1}^{\prime}+\psi_{2}%
^{\prime})]+\alpha$$
$$\displaystyle+{1\over{2}}(1-\alpha)(\cos^{2}\psi_{1}+\cos^{2}\psi_{2})\leq 2.$$
(3.15)
The interesting feature of this inequality is that it can be violated for all
values of $\alpha$. What is perhaps even more important, it can be
robustly violated even when one is not able to distinguish between
single and double clicks at all ($\alpha=0$). The actual value of
the CHSH expression can reach in this
case $2.33712$ (a numerical result),
which is only slightly less than the maximal value
for $\alpha=1$, which is $\sqrt{2}+1\approx 2.41421$. Therefore we
conclude that in the experiment one can observe violations of
local realism even if one is not able to distinguish between
the double and single counts at one detector. That is, the essential
feature of the method of [Popescu97] to reveal
violations of local realism in the experiment of this type is the
specific value assignment scheme and not the double-single photon
counts distinguishability.
The specific angles at which the maximum violation of
the CHSH inequality is achieved for $\alpha=0$ differ very much from those
for $\alpha=1$ (for which the standard result is reproduced),
and they read (in radians) $\psi_{1}=2.93798$,
$\psi_{1}^{\prime}=4.25513$, $\psi_{2}=-0.20241$ and $\psi_{2}^{\prime}=1.11708$.
Let us notice that with the setup of
(3.1) one is able to observe effects of similar
nature to the famous Hong-Ou-Mandel dip [Hong87]. These are
revealed by the probabilities pertaining to the wrong events (3.8-3.11).
Simply for certain orientations of the polarisers, if the two photons emerge
on
one side of the experiment only, then they must exit the
polarising beamsplitter via a single output port
(this effect is due to the bosonic-type indistinguishability of
photons, see [Hong87]).
Finally let us discuss what is the critical efficiency of the detection of the
experiments of this type.
To this end, in our calculations we will use a very simple model of imperfect
detections:
we insert a beamsplitter with reflectivity $\sqrt{1-\eta}$,
in front of an ideal detector, which observes only the transmitted light. This
results in the
system behaving just like a detector of efficiency $\eta$. If we assume that
the incoming light
is described by a creation operator $a^{\dagger}$ then transmitted
mode is denoted as $t_{a}^{\dagger}$ whereas reflected
mode is denoted as $r_{a}^{\dagger}$ and one has
$$a^{\dagger}=\sqrt{1-\eta}r_{a^{\dagger}}^{\dagger}+\sqrt{\eta}t_{a^{\dagger}}^%
{\dagger}.$$
(3.16)
For instance, if one takes the following part of the state vector
(LABEL:final1):
$$c_{||}^{\dagger}d_{||}^{\dagger}|0\rangle.$$
(3.17)
the beamsplitter model of an imperfect
detector transforms this term into:
$$\displaystyle[(1-\eta)r_{c_{||}}^{\dagger}r_{d_{||}}^{\dagger}+\sqrt{\eta(1-%
\eta)}r_{c_{||}}^{\dagger}t_{d_{||}}^{\dagger}$$
$$\displaystyle+\sqrt{\eta(1-\eta)}t_{c_{||}}^{\dagger}r_{d_{||}}^{\dagger}+\eta
t%
_{c_{||}}^{\dagger}t_{d_{||}}^{\dagger}]|0\rangle.$$
The probabilities now read:
$$\displaystyle P(3,\theta_{1};2,\theta_{2})=P(2,\theta_{1};3,\theta_{2})$$
$$\displaystyle P(1,\theta_{1};3,\theta_{2})=P(3,\theta_{1};1,\theta_{2})=\eta(1%
-\eta)$$
(3.19)
$$\displaystyle P(1,\theta_{1};1,\theta_{2})=P(2,\theta_{1};2,\theta_{2})={1%
\over 4}\eta^{2}[\sin(\theta_{1}-\theta_{2})]^{2}$$
(3.20)
$$\displaystyle P(2,\theta_{1};1,\theta_{2})=P(1,\theta;2,\theta_{2})={1\over 4}%
\eta^{2}[\cos(\theta_{1}-\theta_{2})]^{2}$$
(3.21)
$$\displaystyle P(5,\theta_{1};3,\theta{2})=P(6,\theta_{1};3,\theta_{2})={1\over
8%
}\eta^{2}[\sin(2\theta_{1})]^{2}$$
(3.22)
$$\displaystyle P(3,\theta_{1};5,\theta{2})=P(3,\theta_{1};6,\theta_{2})={1\over
8%
}\eta^{2}[\sin(2\theta_{2})]^{2}$$
(3.23)
$$\displaystyle P(4,\theta_{1};3,\theta_{2})={1\over 4}\eta^{2}[\cos(2\theta_{1}%
)]^{2}$$
(3.24)
$$\displaystyle P(3,\theta_{1};4,\theta_{2})={1\over 4}\eta^{2}[\cos(2\theta_{2}%
)]^{2}$$
(3.25)
The correlation function, which includes the inefficiency of the detection
reads
$$E(\psi_{1},\psi_{2};\eta,\alpha)=\eta^{2}E(\psi_{1},\psi_{2};\alpha)+(1-\eta)^%
{2},$$
(3.26)
where
$E(\psi_{1},\psi_{2};\alpha)$ is given by (3.14).
We have tacitly assumed here that the parameters $\alpha$ and $\eta$
are independent of each other (this assumption may not hold for
specific technical arrangements).
Putting this prediction into CHSH inequality, assuming that $\alpha=1$
(full distinguishability) we obtain a minimum
quantum efficiency needed for violation of local realism equal to $0.91$,
whereas for other values of $\alpha$ we have: for $\alpha=0$ $\eta=0.926$;
for $\alpha=0.5$ $\eta=0.92$;
for $\alpha=0.75$ $\eta=0.92$;
for $\alpha=0.875$ $\eta=0.91$. One should note here that the method of value
assignment of [Popescu97] is in accordance with the method given by Garg and
Mermin [Garg85]
for the optimal estimation of required detector quantum efficiency to violate
local realism
in a Bell-test. Thus the obtained efficiencies are indeed the lowest possible,
and show that
experiments of this type are not good candidates for a ”loophole-free”
Bell-test [Santos92], nevertheless due to the fact that the whole
observable effect is a consequence of quantum principle of
particle indistinguishability
such test are very interesting by themselves - they reveal the
entanglement inherently associated with this principle.
3.5 Conclusions
To conclude, we state that the possibility of distinguishing
between single and two photon detection events, usually not met in
the actual experiments, is not a necessary requirement for the proof
that the experiments of Shih-Alley and Ou-Mandel are, modulo fair
sampling assumption, valid tests of local realism. We also show that
some other interesting phenomena (involving bosonic type particle
indistinguishability) can be observed during such tests.
Chapter 4 Better entanglement swapping [2]
4.1 Introduction
Until recent years it was commonly believed that
particles producing EPR-Bell phenomena have to originate from a single
source, or at least have to interact with each other. However,
under very special conditions, by a suitable monitoring procedure of the
emissions of the independent sources one can pre-select an ensemble of
pairs of particles, which either reveal EPR-Bell correlations, or are in an
entangled state.
The first explicit proposal to use two independent sources of particles in a
Bell test was given by Yurke and Stoler [Yurke92].
However, they did not discuss the importance of very specific
operational requirements necessary to implement such schemes in real
experiments. Such conditions were studied in [Żukowski93a] and
[Żukowski95].
The method of entangling independently radiated photons, which share no
common past, [Żukowski93a] is essentially a pre-selection procedure. The
selected registration acts of the idler photons define the ensemble which
contains entangled signal photons (see next sections). Surprisingly, such a
procedure enables one to realize the Bell’s idea of ”event-ready” detection.
This approach for many years was thought to be completely infeasible and thus
no research was being done in that direction [Clauser78]. This so-called
entanglement swapping technique [Żukowski93a], was
also adopted to observe
experimental quantum states teleportation [Bouwmeester97].
The first entanglement swapping experiment was
performed in 1998 [Pan98]. High visibility (around $65\%$) of two
particle
interference fringes were observed on a pre-selected subset of photons
that never interacted.
This is very close to the usual threshold visibility of
two particle fringes to violate some Bell inequalities, which
is $70.7\%$. Therefore there exists a strong temptation for
breaking this limit, and in this way showing that the two particle fringes
due to entanglement swapping have no local and realistic model.
However, due to the spontaneous nature of the sources involved,
the initial condition for entanglement swapping cannot
be prepared. Simply the probability that the two sources
would produce a pair of entangled states each is of the same order as the
probability that
one of them produces two entangled pairs.
In the latter case no entanglement swapping results. Nevertheless,
such events can excite the trigger detectors (which in the
case of the right initial condition select the antisymmetric
Bell state of the two independent idlers). Therefore they are an unavoidable
feature of the experiment, and have to be taken into account in
any analysis of the possibility of finding a local realistic
description for the experiment.
The aim of this chapter is to perform such an analysis.
We shall show that if all firings of the trigger detectors are
accepted as pre-selecting the events for a Bell-type test111As was the
case in the actual experiment.,
one must necessarily, at least partially, be able to
distinguish between two and single photon events
at the detectors observing the signals to enable demonstrations
of violations of local realism. Whereas, if one accepts additional
selection at the trigger detectors, based on the polarisation
of the idlers, detectors possessing this ability are unnecessary.
We shall present our argumentation assuming that the reader
knows the methods and results of [Żukowski93a, Żukowski95, Pan98].
The analysis will be confined to the gedanken situation
of perfect detection efficiency (the results can be easily generalised
to the non-ideal case).
4.2 Description of the experiment
Consider the set-up of (4.1), which is in
principle the scheme used in the Innsbruck experiment
[Pan98].
Two pulsed type-II
down conversion sources are
emitting their radiation into the
spatial propagation modes $a$ and $b$ (signals), $c$ and $d$ (idlers).
Due to
the statistical properties of the PDC radiation,
the initial state that is fed to the
interferometric set-up has the following form:
$$\displaystyle|\psi\rangle=\sum_{n=0}^{\infty}({\gamma\over\sqrt{2}}(a_{H}^{%
\dagger}d_{V}^{\dagger}+a_{V}^{\dagger}d_{H}^{\dagger}))^{n}$$
$$\displaystyle\times\sum_{m=0}^{\infty}({\gamma\over\sqrt{2}}(e_{H}^{\dagger}b_%
{V}^{\dagger}+e_{V}^{\dagger}b_{H}^{\dagger}))^{m}|0\rangle,$$
(4.1)
where, for instance, $a_{H}^{\dagger}$ denotes the creation operator of
the photon in beam $a$ having “horizontal” polarisation.
As for the entanglement swapping to work one cannot have too excessive
pump powers [Żukowski99],
the $\gamma$ coefficient can be assumed small.
Therefore we select only those terms that are proportional
to $\gamma^{2}$, as these are the lowest order terms
terms that can induce simultaneous firing of both trigger detectors.
They read
$$\displaystyle|\psi^{\prime}\rangle={1\over 2}{\gamma}^{2}((a_{H}^{\dagger}d_{V%
}^{\dagger}+d_{V}^{\dagger}d_{H}^{\dagger})(e_{H}^{\dagger}b_{V}^{\dagger}+e_{%
V}^{\dagger}b_{H}^{\dagger})$$
$$\displaystyle+(a_{H}^{\dagger}d_{V}^{\dagger}+a_{V}^{\dagger}d_{H}^{\dagger})^%
{2}+(e_{H}^{\dagger}b_{V}^{\dagger}+e_{V}^{\dagger}b_{H}^{\dagger})^{2})|0\rangle.$$
(4.2)
The factor
$\frac{1}{2}\gamma^{2}$ simply gives the order of magnitude of
the probability of the two trigger detectors to fire, and therefore we
drop it from further considerations. The action of the
non-polarising beam splitter (BS) is described by
$d_{x}^{\dagger}={1\over\sqrt{2}}({\tilde{d}}_{x}^{\dagger}+i{\tilde{e}}_{x}^{%
\dagger})$ and
$e_{x}^{\dagger}={1\over\sqrt{2}}({\tilde{e}}_{x}^{\dagger}+i{\tilde{d}}_{x}^{%
\dagger})$
where $x=H$ or $x=V$, and $\tilde{e}$ and $\tilde{d}$
represent the modes monitored by the trigger detectors behind
the beam splitter.
Taking into account only the terms in (4.2) that lead to clicks
at two trigger detectors we arrive at
$$\displaystyle|\psi^{\prime\prime}\rangle=(i(a_{H}^{\dagger 2}+b_{H}^{\dagger 2%
}){\tilde{e}_{V}^{\dagger}}{\tilde{d}_{V}^{\dagger}}+i(a_{V}^{\dagger 2}+b_{V}%
^{\dagger 2}){\tilde{e}_{H}^{\dagger}}{\tilde{d}_{H}^{\dagger}}$$
$$\displaystyle+i(a_{H}^{\dagger}a_{V}^{\dagger}+b_{H}^{\dagger}b_{V}^{\dagger})%
({\tilde{e}_{V}}^{\dagger}{\tilde{d}_{H}}^{\dagger}+{\tilde{e}_{H}}^{\dagger}{%
\tilde{d}_{V}}^{\dagger})$$
$$\displaystyle{1\over 2}(a_{H}^{\dagger}b_{V}^{\dagger}-a_{V}^{\dagger}b_{H}^{%
\dagger})({\tilde{e}_{V}}^{\dagger}{\tilde{d}_{H}}^{\dagger}-{\tilde{e}_{H}}^{%
\dagger}{\tilde{d}_{V}}^{\dagger}))|0\rangle.$$
(4.3)
It is convenient to normalise and rewrite the above state into the form:
$$\displaystyle|\psi_{N}\rangle={1\over\sqrt{1}3}\left[i\sqrt{2}({1\over\sqrt{2}%
}a_{H}^{\dagger 2}+{1\over\sqrt{2}}b_{H}^{\dagger 2})|VV\rangle\right.$$
$$\displaystyle\left.+i\sqrt{2}({1\over\sqrt{2}}a_{V}^{\dagger 2}+{1\over\sqrt{2%
}}b_{V}^{\dagger 2})|HH\rangle\right.$$
$$\displaystyle\left.+(i(a_{H}^{\dagger}a_{V}^{\dagger}+b_{H}^{\dagger}b_{V}^{%
\dagger})+{1\over 2}(a_{H}^{\dagger}b_{V}^{\dagger}-a_{V}^{\dagger}b_{H}^{%
\dagger}))|VH\rangle\right.$$
$$\displaystyle\left.+(i(a_{H}^{\dagger}a_{V}^{\dagger}+b_{H}^{\dagger}b_{V}^{%
\dagger})-{1\over 2}(a_{H}^{\dagger}b_{V}^{\dagger}-a_{V}^{\dagger}b_{H}^{%
\dagger})\right]|HV\rangle,$$
(4.4)
where $|VV\rangle={\tilde{e}_{V}^{\dagger}}{\tilde{d}_{V}^{\dagger}}|0\rangle$,
$|HH\rangle={\tilde{e}_{H}^{\dagger}}{\tilde{d}_{H}^{\dagger}}|0\rangle$,
$|VH\rangle={\tilde{e}_{V}}^{\dagger}{\tilde{d}_{H}}^{\dagger}|0\rangle$ and
$|HV\rangle={\tilde{e}_{H}}^{\dagger}{\tilde{d}_{V}}^{\dagger}|0\rangle$.
We see clearly that several processes may lead to
the simultaneous firing of the trigger detectors (which observe
the spatial modes)
$\tilde{e}$ and $\tilde{d}$.
The signal photons enter the polarising
beam splitters. Their action can be
described by the following relations
$$\displaystyle x_{V}^{\dagger}=\cos(\theta_{i})x_{+}^{\dagger}+\sin(\theta_{i})%
x_{-}^{\dagger}$$
$$\displaystyle x_{H}^{\dagger}=-\sin(\theta_{i})x_{+}^{\dagger}+\cos(\theta_{i}%
)x_{-}^{\dagger},$$
(4.5)
with $x=a,b;i=1,2$ respectively and $+$, $-$
denoting the output spatial modes.
4.3 Quantum predictions
The probabilities of various
two-particle processes
that may occur at the spatially separated observation stations, under the
condition of both trigger detectors firing simultaneously, are given by:
$$\displaystyle P(1a_{+},1a_{-};0b_{+},0b_{-})=P(2a_{+},0a_{-};0b_{+},0b_{-})$$
$$\displaystyle=P(0a_{+},2a_{-};0b_{+},0b_{-})=P(0a_{+},0a_{-};1b_{+},1b_{-})$$
$$\displaystyle=P(0a_{+},0a_{-};2b_{+},0b_{-})=P(0a_{+},0a_{-};0b_{+},2b_{-})={2%
\over 13},$$
$$\displaystyle P(1a_{+},0a_{-};1b_{+},0b_{-})=P(0a_{+},1a_{-};0b_{+},1b_{-})={1%
\over 26}\left[\sin(\theta_{1}-\theta_{2})\right]^{2},$$
$$\displaystyle P(1a_{+},0a_{-};0b_{+},1b_{-})=P(0a_{+},1a_{-};1b_{+},0b_{-})={1%
\over 26}\left[\cos(\theta_{1}-\theta_{2})\right]^{2},$$
(4.6)
where, for example, $P(0a_{+},0a_{-};2b_{+},0b_{-})$ denotes
the probability of observing two photons at the output $b_{+}$, and no
photons in the other outputs.
The Bell correlation function
for the product of the measurement results
on the signals at the two sides of the experiment
can be redefined in the way proposed in the previous chapter, i.e.,
all standard Bell-type events are assigned their usual values
whereas all non-standard events are assigned the value of one.
I.e., if no photons are registered at one side, the local value of the
measurement is one, if two photons are registered at one side again
the local measurement value
is one. The latter case includes both the event in which the two photons
end-up at a single detector, as well as those when two detectors
at the local station fire.
Please note, that the experiment considered
is a realization of Bell’s idea of
“event ready detectors” (see e.g. [Clauser78]). Therefore,
non-detection events are operationally well defined (as the simultaneous
firing of the trigger detectors pre-selects the sub-ensemble
of time intervals in which one can expect the signal detectors to fire).
The above value assignment method, as it has been in the previous chapter,
works perfectly if one
assumes that it is possible to
distinguish between single and double photon detection at a single
detector. Therefore, again it is convenient to introduce the parameter
$\alpha$.
The partial distinguishability blurs the
distinction
between events (at one side) in which
there was one photon detected at say the output $\pm$, and
events in which two photons entered a the detector observing output
$\pm$, but the detector failed to distinguish this event
from a single photon count.
In such a case the local event is sometimes ascribed by the local
observer a wrong value
namely $-1$ instead of $1$ (if both photons
go to the $``-"$ exit of the polariser
and the devices fail to inform the experimenter
that it is a two photon event, this is interpreted as a firing due
to a single photon and is ascribed a $-1$ value).
Please note, that if one includes less than perfect detection efficiency
of the detectors this problem is more frequent and more involved
(we shall not study this aspect here).
4.4 Conditions to violate local realism
Under such a value assignment the correlation function reads:
$$\displaystyle E_{\alpha}(\theta_{1},\theta_{2})=-{1\over 13}\cos(2\theta_{1}-2%
\theta_{2})+{4\over 13}(1+2\alpha),$$
(4.7)
where $\alpha$ is the numerical value of the distinguishability.
When we put into the standard CHSH inequality this correlation
function it violates the standard bound of 2, only if the distinguishability
satisfies $\alpha\geq{9-\sqrt{2}\over 8}\approx 0.948$.
Such values are definitely
beyond the current technological limits. As the efficiency
of real detectors makes this problem even more acute, one has to
propose a modification of the experiment that gets rid of this problem.
4.5 Proposal of modification of the experiment
Therefore,
in front of the idler detector $Te$ we propose to put
polarising beam splitter that transmit only vertical polarisation
whereas in front of the idler detector $Td$ one that transmits only
horizontal polarisation. This further reduces the
relevant terms in our state, i.e. those that can
induce firing of the trigger detectors, to the following
ones:
$$\displaystyle|\psi\rangle=\sqrt{2\over 5}\left(i(a_{H}^{\dagger}a_{V}^{\dagger%
}+b_{V}^{\dagger}b_{H}^{\dagger})+{1\over 2}(a_{H}^{\dagger}b_{V}^{\dagger}-a_%
{V}^{\dagger}b_{H}^{\dagger})\right){\tilde{e}}_{V}^{\dagger}{\tilde{d}}_{H}^{%
\dagger}|0\rangle.$$
(4.8)
Again we have normalised the above state.
Using the above formula we can calculate the probabilities of all possible
processes in this interferometric set-up, conditional on
firings of the two trigger detectors:
$$\displaystyle P(1a_{+},1a_{-};0b_{+},0b_{-})={2\over 5}\left[\cos(2\theta_{1})%
\right]^{2},$$
$$\displaystyle P(2a_{+},0a_{-};0b_{+},0b_{-})=P(0a_{+},2a_{-};0b_{+},0b_{-})={1%
\over 5}\left[\sin(2\theta_{1})\right]^{2},$$
$$\displaystyle P(0a_{+},0a_{-};1b_{+},1b_{-})={2\over 5}\cos(2\theta_{2})^{2},$$
$$\displaystyle P(0a_{+},0a_{-};2b_{+},0b_{-})=P(0a_{+},0a_{-};0b_{+},2b_{-})={1%
\over 5}\left[\sin(2\theta_{2})\right]^{2},$$
$$\displaystyle P(1a_{+},0a_{-};1b_{+},0b_{-})=P(0a_{+},1a_{-};0b_{+},1b_{-})={1%
\over 10}\left[\sin(\theta_{1}-\theta_{2})\right]^{2},$$
$$\displaystyle P(1a_{+},0a_{-};0b_{+},1b_{-})=P(0a_{+},1a_{-};1b_{+},0b_{-})={1%
\over 10}\left[\cos(\theta_{1}-\theta_{2})\right]^{2}.$$
(4.9)
Under the earlier defined value assignment the correlation function
for the current version of the experiment reads:
$$\displaystyle E_{\alpha}(\theta_{1},\theta_{2})=-{1\over 5}\cos(2\theta_{1}-2%
\theta_{2})$$
$$\displaystyle+{2\over 5}(1-\alpha)\left[(\cos 2\theta_{1})^{2}+(\cos 2\theta_{%
2})^{2}\right]+{4\over 5}\alpha.$$
(4.10)
When such a correlation functions are inserted into the CHSH inequality
one has:
$$\displaystyle-2\leq-{1\over 5}[\cos 2(\theta_{1}-\theta_{2})+\cos 2(\theta_{1}%
-\theta_{2}^{\prime})$$
$$\displaystyle+\cos 2(\theta_{1}^{\prime}-\theta_{2})-\cos 2(\theta_{1}^{\prime%
}-\theta_{2}^{\prime})]$$
$$\displaystyle+{4\over 5}(1-\alpha)\left[(\cos 2\theta_{1})^{2}+(\cos 2\theta_{%
2})^{2}\right]+{8\over 5}\alpha\leq 2.$$
(4.11)
Please note that some of the terms of the correlation function
which depend only on one local angle cancel upon insertion into CHSH
inequality.
For $\alpha=1$ (perfect distinguishability) the
middle expression in (4.11) reaches
2.16569, i.e. we have a clear violation
of the local realistic bound.
This maximal violation occurs at angles (in radians)
$2\theta_{1}=-1.30278,$ $2\theta_{1}^{\prime}=-2.87435,$ $2\theta_{2}=1.05326,$
$2\theta_{2}^{\prime}=2.62386$.
What is more interesting, for $\alpha=0$, i.e. for a complete lack of
distinguishability
between two and single photon events at one detector, the
expression in (4.11)
reaches a value which is not much lower, namely
2.11453. This can be reached for the orientation angles
$2\theta_{1}=0.0837317,$ $2\theta_{1}^{\prime}=-1.0749,$
$2\theta_{2}=3.05769,$ $2\theta_{2}^{\prime}=4.21568$).
4.6 Conclusions
Therefore we conclude
that the proposed modification of the entanglement swapping experiment,
despite the unwanted additional events
due to the impossibility of controlling the spontaneous emissions
at the two separate sources, makes it possible to consider it
as test of Bell inequalities. The standard configuration can serve
as a test of local realism only under the condition of
extremely high distinguishability between two and single photon counts.
Finally let us mention that the proposed
modification in the configuration
enables one to observe, in the event ready mode,
a bosonic interference effect similar to the
Hong-Ou-Mandel dip [Hong87]. It is described by the first four formulas of
(4.9). E.g. if $\theta_{1}=\pi/4$,
no coincidences between firings of the two detectors
of the station $a$
are allowed. All two photon events at this station
are, under this setting, double counts at a single detector.
Thus, we have two very interesting non-classical
phenomena in one experiment.
Part II New generalised Bell inequalities and GHZ paradoxes for qu$N$its
**
In the first chapter of this part the approach to the Bell theorem
employing Bell inequalities is generalised to GHZ correlations
for which each local observer is allowed to use more then
two settings of his or her measuring apparatus.
In the second chapter the functional Bell inequality is derived.
Although the functional inequalities for two qubits are of less practical
importance they seem to be the first step towards
an analytic search for the critical visibility of two-qubit
sinusoidal interference fringes which violate local realism.
In the third chapter we derive the series of GHZ paradoxes for
$N$ and $N+1$ maximally entangled
qu$N$its observed via unbiased multiport
beamsplitters.
Chapter 5 Wringing out better Bell inequalities for GHZ experiment [3]
5.1 Introduction
Greenberger-Horne-Zeilinger correlations [Greenberger89] lead to a
strikingly more direct refutation of the argument of Einstein Podolsky and
Rosen (EPR), on the possibility of introducing elements of reality to
complete quantum mechanics [Einstein35], than considerations involving only
pairs of qubits. The EPR ideas are based on the observation that for some
systems quantum mechanics predicts perfect correlations of their properties.
However, in the case of three or more qubits, in the entangled GHZ state,
such correlations cannot be consistently used to infer at a distance hidden
properties of the qubits. In contradistinction to the original two
qubit Bell theorem, the idea of EPR, to turn the exact predictions of
quantum mechanics against the claim of its completeness, breaks down already
at the stage of defining the elements of reality.
The reasoning of GHZ involved perfectly correlated qubit systems.
However, the actual data collected in a real laboratory would reveal less
than perfect correlations, and the imperfections of the qubit collection
systems would leave many of the potential events undetected. Therefore the
original GHZ reasoning cannot be ever tested in the laboratory, and one is
forced to make some modifications (already, e.g., in
[Greenberger90]).
To face these difficulties several $M$ qubit Bell inequalities appeared in
the literature [Mermin90b, Roy91, Ardehali92, Belinski93, Żukowski93b].
All these works show that quantum predictions for GHZ states violate these
inequalities by an amount that grows exponentially with $M$. The increasing
number of qubits, in this case, does not bring us closer to the classical
realm, but rather makes the discrepancies between the quantum and the
classical more profound.
The study of three or more qubit
interference effects does not seem to be a good route towards a loophole free
test of the hypothesis of local hidden variables. However,
to interpret the results of such experiments111First observation
of GHZ correlations has been already reported [Bouwmeester99]. one
should know the
borderline between the quantum and the classical (i.e., local realism).
According to current literature (with the exception of [Żukowski93b]) we enter
the non-classical territory when the fringes in a $M$ qubit interference
experiment have visibilities higher than ${2}^{\frac{1}{2}(1-M)}$. The
principal aim of this chapter is to show that, if one allows each of the local
observers to have three measurements to choose from (instead of the
usual two), the actual threshold is lower (for $M>3$).
5.2 Geometrical method of finding Bell inequalities
Let us first explain the method that will be used in the next
two sections
(it is called a geometrical method)
[Żukowski93b] on an example of two maximally
entangled qubits. The method will be presented in the case
of stochastic local hidden variables. Application to deterministic
case is straightforward [Żukowski93b].
We consider the state (2.1) on which
two spatially separated observers $a,b$
measure dichotomic observables $O_{n}(\phi_{n})$ with eigenvalues $\pm 1$
referring to eigenstates
$$\displaystyle|\pm,\phi_{n}\rangle_{n}=\frac{1}{\sqrt{2}}\left(|0\rangle_{n}\pm
e%
^{(i\phi_{n})}|1\rangle_{n}\right)$$
(5.1)
controlled by knobs (local settings) $\phi_{n}$ ($n=a,b$).
Let us further assume that in the experiment observer $a$ chooses between,
say, $N_{a}$ settings of the local apparatus denoted by $\phi^{i}_{a}$,
$i=1,2,\dots,N_{a}$, and the observer $b$ chooses between
$N_{b}$ settings denoted by $\phi_{b}^{j}$ $j=1,2,\dots,N_{b}$. The quantum
prediction for
observer $a$ to obtain the result $m=\pm 1$, and observer $b$ to obtain
$m^{\prime}=\pm 1$ is equal to
$$\displaystyle P_{QM}^{V_{2}}(m,m^{\prime}|\phi_{a}^{i},\phi_{b}^{j})={1\over 4%
}\left(1-V_{2}mm^{\prime}\cos(\phi_{a}^{i}+\phi_{b}^{j})\right),$$
(5.2)
where $V_{2}$ takes into account the possible less than perfect
visibility (see the discussion of the notion of visibility in Introduction
- equation (2.14) and the discussion below).
The question is if this set of probabilities is reproducible
by local hidden variables, i.e., by
$$\displaystyle P_{HV}(m,m^{\prime}|\phi_{a}^{i},\phi_{b}^{j})=\int_{\Lambda}d%
\lambda\rho(\lambda)P_{a}(m|\lambda,\phi_{a}^{i})P_{b}(m^{\prime}|\lambda,\phi%
_{b}^{j})$$
(5.3)
(see also (2.13)).
The set of quantum probabilities (5.2) as well as
the set of local hidden variable probabilities (5.3)
can be treated as the components of $4\times N_{a}\times N_{b}$ dimensional
real vectors $\hat{P}_{QM}^{V_{2}}$ and $\hat{P}_{HV}$ with components
$$\displaystyle P_{QM}^{V_{2}}(mm^{\prime};ij)={1\over 4}\left(1-V_{2}mm^{\prime%
}\cos(\phi_{a}^{i}+\phi_{b}^{j})\right)$$
$$\displaystyle P_{HV}(mm^{\prime};ij)=\int_{\Lambda}d\lambda\rho(\lambda)P_{a}(%
m|\lambda,\phi_{a}^{i})P_{b}(m^{\prime}|\lambda,\phi_{b}^{j})$$
(5.4)
where $i,j,m,m^{\prime}$ enumerate the components of the vectors.
Both $\hat{P}_{QM}^{V_{2}}$ and $\hat{P}_{HV}$ can be considered as
vectors belonging to a real
Hilbert space with
the following scalar product
$$\displaystyle(\hat{F}|\hat{G})=\sum_{m=-1}^{1}\sum_{m^{\prime}=-1}^{1}\sum_{i=%
1}^{N_{a}}\sum_{j=1}^{N_{b}}F(mm^{\prime};ij)G(mm^{\prime};ij),$$
(5.5)
where $\hat{F}$ and $\hat{G}$ are arbitrary vectors from
this space.
In every real Hilbert space any two vectors $\hat{F},\hat{G}$
are equal, i.e. $\hat{F}=\hat{G}$, if and only if
$$\displaystyle(\hat{F}|\hat{G})=||\hat{F}||^{2}=||\hat{G}||^{2}.$$
(5.6)
Thus, if one has two vectors $\hat{F},\hat{G}$ and one knows
the norm of, say, the vector $\hat{F}$,
and $(\hat{F}|\hat{G})<||\hat{F}||^{2}$ then $\hat{F}\neq\hat{G}$.
This simple
observation is especially useful for us because we can always calculate
the norm of the vector $\hat{P}_{QM}^{V_{2}}$, which is not possible for the
vector $\hat{P}_{HV}$. However, as we will see next, one can
estimate the scalar product of $\hat{P}_{QM}^{V_{2}}$ and $\hat{P}_{HV}$.
If $(\hat{P}_{QM}^{V_{2}}|\hat{P}_{HV})<||\hat{P}_{QM}^{V_{2}}||^{2}$ one has the Bell
inequality.
As the simplest example let us show how one can obtain within this approach
the standard threshold value of the visibility (obtained using
the CHSH inequality) for the situation
in which both observers have two local settings to choose from.
To this end let us consider the case in which $N_{a}=N_{b}=2$
and let us choose $\phi_{a}^{1}=0$, $\phi_{a}^{2}={\pi\over 2}$,
and $\phi_{b}^{1}=-{\pi\over 4}$, $\phi_{b}^{2}=+{\pi\over 4}$.
For such local settings one has
$$\displaystyle P^{V_{2}}_{QM}(m,m^{\prime};1,1)={1\over 4}(1-mm^{\prime}{V_{2}%
\over\sqrt{2}})$$
$$\displaystyle P^{V_{2}}_{QM}(m,m^{\prime};1,2)={1\over 4}(1-mm^{\prime}{V_{2}%
\over\sqrt{2}})$$
$$\displaystyle P^{V_{2}}_{QM}(m,m^{\prime};2,1)={1\over 4}(1-mm^{\prime}{V_{2}%
\over\sqrt{2}})$$
$$\displaystyle P^{V_{2}}_{QM}(m,m^{\prime};2,2)={1\over 4}(1+mm^{\prime}{V_{2}%
\over\sqrt{2}})$$
(5.7)
and
the norm of $||\hat{P}_{QM}^{V_{2}}||^{2}$ equals
$1+{1\over 2}V_{2}^{2}$. The next step is to write down
explicitly the scalar product $(\hat{P}_{HV}|\hat{P}_{QM}^{V_{2}})$
$$\displaystyle(\hat{P}_{HV}|\hat{P}_{QM}^{V_{2}})=\int d\lambda\rho(\lambda)[1-%
{1\over 4\sqrt{2}}V_{2}\{I_{a}(\phi_{a}^{1},\lambda)I_{b}(\phi_{b}^{1},\lambda%
)+I_{a}(\phi_{a}^{1},\lambda)I_{b}(\phi_{b}^{2},\lambda)$$
$$\displaystyle+I_{a}(\phi_{a}^{2},\lambda)I_{b}(\phi_{b}^{1},\lambda)-I_{a}(%
\phi_{a}^{2},\lambda)I_{b}(\phi_{b}^{2},\lambda)\}],$$
(5.8)
where, for instance,
$I_{a}(\phi_{a}^{2})=\sum_{m=-1}^{1}mP_{a}(m|\lambda,\phi_{a}^{2})$.
We see that $|I_{a}(\phi_{a}^{2})|\leq 1$ and this immediately
implies that the modulus of expression in the curly brackets
never exceeds $2$.
Thus,
$$\displaystyle(\hat{P}_{HV}|\hat{P}_{QM}^{V_{2}})\leq 1+V_{2}{\sqrt{2}\over 4}.$$
(5.9)
Comparing (5.9) with the norm of $||\hat{P}_{QM}^{V_{2}}||^{2}$ one obtains
that the necessary condition for $\hat{P}_{HV}$ to be equal to
$\hat{P}^{V_{2}}_{QM}$ is $V_{2}\leq{1\over\sqrt{2}}$.
5.3 Quantum and local realistic description of the gedanken experiment
We consider a source emitting $M$ qubits each of
which propagates towards one of $M$ spatially separated measuring devices.
The generic form of a GHZ $M$ qubit state is
$$|\Psi(M)\rangle=\frac{1}{{\sqrt{2}}}(|0\rangle_{1}\dots|0\rangle_{M}+|1\rangle%
_{1}\dots|1\rangle_{M}).$$
(5.10)
Let as assume that the operation of each of the measuring apparatus is
controlled by a knob which sets a parameter $\phi_{l}$, and the $l$-th
apparatus measures a dichotomic observable $O_{l}(\phi_{l})$ with two
eigenvalues $\pm 1$ and the eigenstates defined by
$|\pm,\phi_{l}\rangle_{l}=\frac{1}{\sqrt{2}}\left(|0\rangle_{l}\pm e^{(i\phi_{l%
})}|1\rangle_{l}\right).$ The quantum prediction for obtaining
specific results at the $M$ measurement stations (for the idealised, perfect,
experiment) reads
$$\displaystyle P_{QM}^{(M)}(r_{1},r_{2},\dots,r_{M}|\phi_{1},\dots,\phi_{M})$$
(5.11)
$$\displaystyle={1\over 2^{M}}\left[1+\prod_{l=1}^{M}r_{l}\cos\left(\sum_{k=1}^{%
M}\phi_{k}\right)\right]$$
$$\displaystyle,$$
($r_{l}$ equal to $-1$ or $+1$).
The GHZ correlation function is defined as
$$\displaystyle E^{(M)}(\phi_{1},\dots,\phi_{M})$$
$$\displaystyle=\sum_{r_{1},r_{2},\dots,r_{M}=-1}^{1}\prod_{l=1}^{M}r_{l}P^{(M)}%
(r_{1},\dots,r_{M}|\phi_{1},\dots,\phi_{M}),$$
(5.12)
and in the case of quantum mechanics, i.e. for $P^{(M)}=P_{QM}^{(M)}$, it
reads $E_{QM}^{(M)}=\cos(\sum_{l=1}^{M}\phi_{l}).$
Local realism implies the following structure of probabilities
of specific results (compare with (2.13))
$$\displaystyle P_{HV}^{(M)}(r_{1},\dots,r_{M}|\phi_{1},\dots,\phi_{M})$$
$$\displaystyle=\int_{\Lambda}d\lambda\rho(\lambda)\prod_{l=1}^{M}P_{l}(r_{l}|%
\lambda,\phi_{l})$$
(5.13)
where $P_{l}(r_{l}|\lambda,\phi_{l})$ is the probability to obtain result
$r_{l}$ in the $l$-th apparatus under the condition that the hidden state is
$\lambda$ and the macroscopic variable defining the locally
measured observable is set to the value $\phi_{l}$222
We present our results only for the case of deterministic
local hidden variable theories, i.e., $P_{l}(r_{l}|\lambda,\phi_{l})=1,0$,
but generalisation to
stochastic ones is obvious.. The
locality
of this description is guaranteed by independence of $P_{l}$ on $\phi_{i}$
for all $i\neq l$.
5.4 Derivation of Bell inequalities via the geometrical method
We shall now derive a series of inequalities for the $M$ qubit GHZ
processes based on the already mentioned geometric method
[Żukowski93b].
We assume that each of the $M$ spatially separated observers has three
measurements to choose from. The local phases that they are allowed to set
are $\phi_{1}^{1}=\pi/6,\phi_{2}^{1}=\pi/2,\phi_{3}^{1}=5\pi/6$
(for the first observer) and for all the other $M-1$ observers they are
$\phi_{1}^{i}=0,\phi_{2}^{i}=\pi/3,\phi^{i}_{3}=2\pi/3$,
($i=2,\dots,M$).
Out of the quantum predictions for the $M$ qubit correlation function at
these settings one can construct a matrix endowed with $M$ indices
$$\displaystyle E_{QM}^{(M)}(\phi_{i_{1}}^{1},\dots\phi_{i_{M}}^{M})=\cos\left(%
\sum_{k=1}^{M}\phi_{i_{k}}^{k}\right)=Q_{i_{1},\dots,i_{M}}^{(M)}$$
(5.14)
($i_{k}=1,2,3$).
All that we know about local hidden variable theories is that their
predictions (for the same set of settings as above) must have the following
form:
$$\displaystyle E_{HV}^{(M)}(\phi_{i_{1}}^{1},\dots\phi_{i_{M}}^{M})$$
$$\displaystyle=\int d\lambda\rho(\lambda)\prod_{k=1}^{M}I_{k}(\lambda,\phi_{i_{%
k}}^{k})=H_{i_{1},\dots,i_{M}}^{(M)},$$
(5.15)
where
$$\displaystyle I_{k}(\lambda,\phi_{i_{k}}^{k})=\sum_{r_{k}}r_{k}P_{k}(r_{k}|%
\lambda,\phi_{i_{k}}^{k}).$$
(5.16)
Of course, in the case of a deterministic local hidden variables theory
$I_{k}(\lambda,\phi)=\pm 1$. $H^{(M)}$ is our test matrix.
Please note, that all one knows about $H^{(M)}$ is its structure.
The scalar product of two real matrices is defined by
$$(H^{(M)},Q^{(M)})=\sum_{i_{1},\dots,i_{M}}H_{i_{1},\dots,i_{M}}^{(M)}Q_{i_{1},%
\dots,i_{M}}^{(M)}.$$
(5.17)
Our aim is to show the incompatibility of the local hidden variable
description with the quantum prediction. To this end, we shall show that,
for two or more qubits,
$$\displaystyle({Q}^{(M)},{H}^{(M)})$$
(5.18)
$$\displaystyle\leq 2^{M-1}\sqrt{3}<||{Q}^{(M)}||^{2}=\frac{3^{M}}{2}$$
$$\displaystyle.$$
First, we show that $||{Q}^{(M)}||^{2}=3^{M}/2$. This can be reached in
the following way:
$$\displaystyle||{Q}^{(M)}||^{2}=\sum_{i_{1},\dots,i_{M}}\cos^{2}\left(\sum_{k=1%
}^{M}\phi_{i_{k}}^{k}\right)$$
$$\displaystyle=\frac{1}{2}\sum_{i_{1},\dots,i_{M}}\left[1+\cos\left(2i\sum_{k=1%
}^{M}\phi_{i_{k}}^{k}\right)\right]$$
$$\displaystyle=Re\left\{\sum_{i_{1},\dots,i_{M}}\left[1+\exp\left(2\sum_{k=1}^{%
M}\phi_{i_{k}}^{k}\right)\right]\right\}$$
$$\displaystyle=3^{M}/2+Re\left(\prod^{M}_{k=1}\sum_{i_{k}=1}^{3}\exp(2i\phi_{i_%
{k}}^{k})\right),$$
(5.19)
where $Re$ denotes the real part. Since
$\sum_{l=1}^{3}e^{i(l-1)(2/3)\pi}=0$, the last term vanishes.
The scalar product $(H^{(M)},Q^{(M)})$ is bounded from above by the maximal
possible value of
$$S^{(M)}_{\lambda}=\sum_{i_{1},\dots,i_{M}}\left[\cos\left(\sum^{M}_{k=1}\phi_{%
i_{k}}^{k}\right)\prod_{l=1}^{M}I_{l}(\lambda,\phi_{i_{l}}^{l})\right],$$
(5.20)
and for $M\geq 2$
$$S^{(M)}_{\lambda}\leq{2^{M-1}\sqrt{3}}.$$
(5.21)
To show (5.21), let us first notice that
$$\displaystyle S_{\lambda}^{(M)}=Re\left[\prod_{k=1}^{M}\sum_{i_{k}=1}^{3}I_{k}%
(\lambda|\phi_{i_{k}}^{k})\exp(i\phi_{i_{k}}^{k}))\right].$$
(5.22)
For $k=2,\dots,M$, one has $e^{i\phi_{l}^{k}}=e^{i[(l-1)/3]\pi}$ whereas
for $k=1$, $e^{i\phi_{l}^{1}}=e^{i(\pi/6)}e^{i[(l-1)/3]\pi}$.
Thus, since $I(\lambda|\cdot)=\pm 1$, the possible values for
$$z_{1}^{\lambda}=\sum_{i_{1}=1}^{3}I_{1}(\lambda|\phi_{i_{1}}^{1})\exp(i\phi_{i%
_{1}}^{1})$$
(5.23)
are $0$, $\pm 2e^{i\pi/2}$, $\pm 2e^{-i(\pi/6)}$, or
finally $\pm 2e^{i(\pi/6)}$, whereas for $k=2,\dots,M$ the possible
values of
$$z_{k}^{\lambda}=\sum_{i=1}^{3}I_{k}(\lambda|\phi_{i_{k}}^{k})\exp(i\phi_{i_{k}%
}^{k})$$
(5.24)
have their complex phases shifted by $\pi/6$ with respect to the
previous set; i.e., they are $0$, $\pm 2e^{i(2\pi/3)}$,
$\pm 2$, or finally $\pm 2e^{i(\pi/3)}$. Since
$|z_{1}^{\lambda}\prod_{k=2}^{M}z_{k}^{\lambda}|\leq 2^{M}$ and the minimal
possible overall complex phase (modulo $2\pi$) of
$z_{1}^{\lambda}\prod_{k=2}^{M}z_{k}^{\lambda}$ is
$\pi/6$, one has $Re(z_{1}^{\lambda}\prod_{k=2}^{M}z_{k}^{\lambda})\leq 2^{M}\cos(\pi/6).$
Thus inequalities (5.21) and (5.18) hold.
The left inequality of (5.18) is a Bell inequality for the $M$
qubit
experiment. If one replaces ${H}^{(M)}$ by the quantum prediction
${Q}^{(M)}$
(compare (5.14))
the inequality is violated since
$$({Q^{(M)}},{Q^{(M)}})=\frac{3^{M}}{2}>2^{M-1}\sqrt{3},$$
(5.25)
i.e., (5.18) is violated by the factor
$(3/2)^{M}/\sqrt{3}$ (compare [Mermin90b]).
5.4.1 Critical visibility and quantum efficiency of detectors
The magnitude of violation of a Bell inequality is not a parameter which
can objectively define to what extent local realism is violated.
It is rather the visibility of the
$M$ qubit interference fringes which can be directly observed. Further,
the significance of all Bell-type experiments depends on the efficiency of
the collection of the qubits. Below a certain threshold value for this
parameter experiments cannot be considered as tests of local realism. They
may confirm the quantum predictions but are not falsifications of the
hypothesis of local hidden variables. Therefore we will search for the
critical minimal visibility of $M$ qubit fringes and collection efficiency, which
do not allow anymore a local realistic model.
In a real experiment (under the assumption that quantum mechanics gives
idealised, but correct predictions), the visibility of the $M$ qubit
fringes, $V_{2}(M)$, would certainly be less than 1. Also the probability of
registering all potential events $\eta_{2}(M)$
would be reduced by the overall collection
efficiency. If one assumes that all $M$ local apparata have the same
collection efficiency $\eta$,
and takes into account that these operate
independently of each other333The parameter $\eta$ describes
here the efficiency of a single detector. The assumption of
the independency of detectors gives $\eta_{2}(M)=\eta^{M}$.,
one can model the expected experimental results
by
$$\displaystyle P_{expt}^{(M)}(r_{1},\dots,r_{M}|\phi^{1},\dots,\phi^{M})$$
(5.26)
$$\displaystyle=\eta^{M}{\left(\frac{1}{2}\right)^{M}}\left(1+V_{2}(M)\prod_{l=1%
}^{M}r_{l}\cos\sum_{k=1}^{M}\phi^{k}\right)$$
$$\displaystyle.$$
The full set of events at a given measuring station consists now of the
results $+1$ and $-1$, when we succeed to measure the dichotomic observable,
and a non-detection event (which is, in principle observable, if one uses
event-ready state preparation [Yurke92]) for which one can introduce the
value $0$. The local realistic description requires that the
probabilities of
the possible events should be given by
$$\displaystyle P_{expt}^{HV}(m_{1},\dots,m_{M}|\phi^{1},\dots,\phi^{M})$$
(5.27)
$$\displaystyle=\int d\lambda\rho(\lambda)\prod_{k=1}^{M}P_{k}(m_{k}|\lambda,%
\phi^{k})$$
$$\displaystyle,$$
with $m_{i}=+1,-1$ or $0$. The local hidden
variable correlation function for the experimental results (at the chosen
settings) is now given by
$${E_{expt}^{HV}}_{i_{1},\dots,i_{M}}=\int d\lambda\rho(\lambda)\prod_{k=1}^{M}I%
_{k}^{\prime}(\lambda,\phi_{i_{k}}^{k}),$$
(5.28)
with
$$I_{k}^{\prime}(\lambda,\phi_{i_{k}}^{k})=\sum_{m_{k}=-1,0,+1}m_{k}P_{k}^{HV}(m%
_{k}|\lambda,\phi_{i_{k}}^{k}).$$
(5.29)
For deterministic models one has now $I_{k}^{\prime}(\lambda,\phi_{i_{k}}^{k})=1,0,-1$.
One can impose several symmetries on ${P_{expt}^{HV}}$. These symmetries are
satisfied by the quantum prediction (5.26), and we can expect them to
be satisfied in real experiments, within experimental error. The one that we
impose here is that:
For all sets of results, $\{m_{1},\dots,m_{M}\}$, that have equal number of
zeros (one zero or more) the probability $P_{expt}^{HV}(m_{1},\dots,m_{M})$
has
the same value, and this value is independent of the settings of the local
parameters $\{\phi_{i_{1}}^{1},\dots,\phi_{i_{M}}^{M}\}$.
One can define a function $f_{M}(m)$ which for $m=+1,-1,0$ has the following
values: $f(\pm 1)=\pm 1$, $f(0)=-1$ (compare [Garg85]) and introduce
auxiliary correlation function
$$\displaystyle\tilde{E}_{i_{1},\dots,i_{N}}=\int d\lambda\rho(\lambda)\sum_{m_{%
1},\dots,m_{N}=-1,0,+1}$$
(5.30)
$$\displaystyle\times\prod_{k=1}^{M}[f(m_{k})P_{k}(m_{k}|\lambda,\phi_{i_{k}}^{k%
})]={\tilde{H}}^{(M)}_{i_{1},\dots,i_{M}}$$
$$\displaystyle.$$
Since, due to the symmetry conditions, one has, e.g., $\sum_{m_{2}=1,-1}f(m_{2})P_{expt}^{HV}(0,m_{2},\dots,m_{M})=0$, the
following relation results:
$$\tilde{E}_{i_{1},\dots,i_{M}}={E^{HV}_{expt}}_{i_{1},\dots,i_{M}}+[f(0)]^{M}P(%
0,\dots,0),$$
(5.31)
where
$P(0,\dots,0)$
is the probability that all detectors would fail to register qubits,
and under our assumptions
it is independent of the settings, and equals $(1-\eta)^{M}$.
The auxiliary correlation function must satisfy the original inequality
(5.18); i.e., one has
$$(Q^{(M)},{\tilde{H}}^{(M)})\leq{2^{M-1}}\sqrt{3}.$$
(5.32)
However, this implies that
$$\displaystyle-{2^{M-1}}\sqrt{3}-f(0)^{M}P(0,\dots,0)q_{(M)}$$
$$\displaystyle\leq(Q^{(M)},E_{expt}^{HV})$$
$$\displaystyle\leq{2^{M-1}}\sqrt{3}-f(0)^{M}P(0,\dots,0)q_{(M)},$$
(5.33)
where
$$q_{(M)}=\sum_{i_{1},\dots,i_{M}}Q_{i_{1},\dots,i_{M}}^{(M)}.$$
(5.34)
Therefore, since if $x$ is a possible value for $(Q^{(M)},E_{expt}^{HV})$
then so
is $-x$, one has
$$\displaystyle|({Q^{(M)}},E_{expt}^{HV})|$$
$$\displaystyle\leq{2^{M-1}}\sqrt{3}-P(0,\dots,0)|q_{(M)}|.$$
(5.35)
Thus, we have obtained Bell inequalities of a form which is more suitable
for the analysis of the experimental data.
The prediction (5.26) leads to the following correlation function
$$E_{expt}^{QM}=\eta^{M}V_{2}(M)E^{QM},$$
(5.36)
which, when put into (5.35) in the place of $E_{expt}^{HV}$, gives the
following relation between the threshold visibility, $V^{tr}_{2}(M)$,
and the threshold collection efficiency, $\eta^{tr}$, for the $M$-qubit
experiment:
$${\eta^{tr}}^{M}\frac{3^{M}}{2}V^{tr}_{2}(M)={2^{M-1}}\sqrt{3}-|q_{(M)}|(1-\eta%
^{tr})^{M}.$$
(5.37)
The value of the expression $q_{(M)}$ can be found in the following way:
$$\displaystyle q_{(M)}=\sum_{i_{1},\dots,i_{M}}\cos\left(\sum_{k=1}^{M}\phi_{i_%
{k}}\right)$$
$$\displaystyle=Re\left(\sum_{i_{1},\dots,i_{M}}\prod_{k=1}^{M}\exp(i\phi_{i_{k}%
}^{k})\right)$$
$$\displaystyle=Re\left(\prod_{k=1}^{M}\sum_{i_{k}}\exp(i\phi_{i_{k}}^{k}\right)$$
$$\displaystyle=Re\left[2^{M}i\exp\left(i(M-1)\frac{\pi}{3}\right)\right]=-2^{M}%
\sin\left((M-1)\frac{\pi}{3}\right).$$
(5.38)
5.5 Results
The threshold value of the visibility of the multi-qubit fringes decreases
now faster than in the earlier approaches [Mermin90b]. For perfect
collection efficiency, ($\eta=1$), it has the lowest value, which is
$$V^{tr}_{2}(M)={\sqrt{3}}\left(\frac{2}{3}\right)^{M},$$
(5.39)
and, if $M\geq 4$, it is lower than $(\frac{1}{\sqrt{2}})^{M-1}$. The
specific values for several qubits are $V^{tr}_{2}(2)=76.9\%$,
$V^{tr}_{2}(3)=51.3\%$, $V^{tr}_{2}(4)=34.2\%$, $V^{tr}_{2}(5)=22.8\%$ and
$V^{tr}_{2}(10)=3\%$ (see also figure (5.1)),
whereas the standard methods lead to $V^{old}_{2}(2)=70.7\%$,
$V^{old}_{2}(3)=50.0\%$, $V^{old}_{2}(4)=35.4\%$, $V^{old}_{2}(5)=25.0\%$ and
$V^{old}_{2}(10)=4.4\%$.
The threshold efficiency of the qubit collection also decreases with growing
$M$ (see figure (5.2)), and for perfect visibilities it reads $\eta^{tr}(2)=87.0\%$,
$\eta^{tr}(3)=79.8\%$, $\eta^{tr}(4)=76.5\%$, $\eta^{tr}(5)=74.4\%$ (here the number in the
brackets indicates the number of entangled particles $M$). The gain over
the inequalities [Mermin90b] is in this respect very small, and begins
again at $M=4$. However, for very big $M$ the critical efficiency is close
to $\frac{2}{3}$ (compared with $\frac{1}{\sqrt{2}}$ for [Mermin90b]).
5.6 Conclusions
We conclude that for the original GHZ problem (four
qubits) one should rather aim at making experiments which allow for three
settings at each local observation station. Surprisingly, the measurements
should not be performed for the values for which we have perfect GHZ-EPR
correlations (i.e the values for which the correlation function equals to
$\pm 1$).
Chapter 6 Bell inequality for all possible local settings [4]
6.1 Introduction
The Bell theorem is usually formulated with the help of the Clauser-Horne
[Clauser74] or the CHSH inequality [Clauser69]. These
inequalities are satisfied
by any
local realistic theory and are violated by quantum mechanical predictions.
They involve two apparatus settings at each of the two sides of the experiment.
However, generalisation to more than two settings at each side are possible
[Garuccio80, Braunstein89, Żukowski93b, Gisin99].
There are several motivations for such generalisations. First of all,
new Bell inequalities
may be more appropriate in some experimental situations, e.g., the chained
Bell inequalities
can reveal violation of local realism for the Franson type experiment
[Aerts99].
Also, the academic question, why only two settings at each side, is
that always necessary, is interesting in itself. Further, many of
the currently performed quantum interferometric Bell tests did not
involve stabilisation of the interferometers at specified settings
optimal for the standard Bell inequalities, but rather involved
sample scans of the entire interferometric patterns. Thus it is useful
to have inequalities that are directly applicable to such data.
Here we present a Bell-type inequality that involves all possible settings
of the local measuring apparatus for a pair of two qubits, which is
always
equivalent to two spin ${1\over 2}$ particles.
The method applied is a development of the one given in [Żukowski93b].
However, here we do not restrict ourselves to pairs of coplanar settings
(in the meaning appropriate for two Stern-Gerlach apparatuses).
Our method has two characteristic traits. The first one is that it
indeed involves the entire range of the measurement parameters.
By this, e.g., it distinguishes itself from the limits of infinitely many
settings at each side of the so-called chained inequalities [Braunstein89], in
which not every pair of possible settings is utilised. The second one
is that the method involves the quantum prediction from the very beginning.
As we shall see the quantum prediction determines the structure of our
Bell inequality.
6.2 Quantum mechanical and local realistic description of the
gedanken experiment
As usual one has a source emitting two qubits
each of which propagates towards one of two spatially separated
observers $a$ and $b$. The qubits are described by the maximally entangled
state (2.1).
Let as assume that every observer has a Stern-Gerlach
apparatus, which measures the observable $\vec{n}\cdot\vec{\sigma}$,
where $n=a,b$,
$\vec{n}$ is a unit vector representing direction at which observer
$n$ makes a measurement
and $\vec{\sigma}$ is a vector the components of which are standard Pauli
matrices.
The family of observables $\vec{n}\cdot\vec{\sigma}$
covers all possible dichotomic observables for a two qubit system, endowed
with a spectrum consisting of $\pm 1$.
In each run of the experiment every observer obtains one of the two possible
results of measurement, $\pm 1$. The probability of
obtaining by the observer $a$ the result $m=\pm 1$, when
measuring the observable
$\vec{\sigma}\cdot\vec{a}$, and the result
$m^{\prime}=\pm 1$
by the observer $b$, when measuring the observable
$\vec{\sigma}\cdot\vec{b}$ is equal to
$$\displaystyle P_{QM}^{V_{2}}(m,m^{\prime}|\vec{a},\vec{b})={1\over 4}(1-V_{2}%
mm^{\prime}\vec{a}\cdot\vec{b}).$$
(6.1)
where $0\leq V_{2}\leq 1$ stands for the visibility.
The structure of local hidden variables gives
$$\displaystyle P_{HV}(m,m^{\prime}|\vec{a},\vec{b})=\int_{\Lambda}d\lambda\rho(%
\lambda)P_{a}(m|\lambda,\vec{a})P_{b}(m^{\prime}|\lambda,\vec{b}),$$
(6.2)
with the standard meaning of the used symbols (see (2.13)).
6.3 Derivation of the inequality via the geometrical method
To apply the geometrical method we must define appropriate
Hilbert space. Because we deal with functions
$P_{QM}^{V_{2}}(m,m^{\prime}|\theta_{a},\phi_{a},\theta_{b},\phi_{b})$ and
$P_{HV}(m,m^{\prime}|\theta_{a},\phi_{a},\theta_{b},\phi_{b})$
that
depend on discrete numbers $m,m^{\prime}$
and continuous variables $\theta_{n},\phi_{n}$, where
$\vec{n}=(\sin\theta_{n}\cos\phi_{n},\sin\theta_{n}\sin\phi_{n},\cos\theta_{n})$ it is convenient to define
the scalar product of certain two real functions $f$ and $g$ as
$$\displaystyle\langle f|g\rangle=\sum_{m=-1}^{1}\sum_{m^{\prime}=-1}^{1}$$
$$\displaystyle\int d\Omega_{a}\int d\Omega_{b}f(m,m^{\prime};\theta_{a},\phi_{a%
},\theta_{b},\phi_{b})g(m,m^{\prime};\theta_{a},\phi_{a},\theta_{b},\phi_{b}),$$
(6.3)
where $d\Omega_{n}=\sin\theta_{n}d\theta_{n}d\phi_{n}$
is the rotationally invariant measure
on the sphere of radius one. Our known vector is $P_{QM}^{V_{2}}$, whereas
the test one is $P_{HV}$ (compare with the section describing
geometrical method).
One has
$$\displaystyle||P_{QM}^{V_{2}}||^{2}=\langle P_{QM}^{V_{2}}|P_{QM}^{V_{2}}\rangle$$
$$\displaystyle=(2\pi)^{2}+V_{2}^{2}{{4\pi}^{2}\over 3}.$$
(6.4)
To estimate the scalar product $\langle P_{QM}^{V_{2}}|P_{HV}\rangle$ one has to
use the specific structure of probabilities that are
described by local hidden variables (LHV)
(6.2). Since $P_{HV}$ is a weighted average
over the hidden parameters one can make the following estimate
$$\displaystyle\langle P_{QM}^{V_{2}}|P_{HV}\rangle\leq\max_{\lambda\in\Lambda}%
\left[\sum_{m,m^{\prime}=-1}^{1}\int d\Omega_{a}\int d\Omega_{b}P_{a}(m|%
\lambda,\vec{a})\right.$$
$$\displaystyle\left.\times P_{b}(m^{\prime}|\lambda,\vec{b}){1\over 4}(1-mm^{%
\prime}V_{2}\vec{a}\cdot\vec{b})\right].$$
(6.5)
Since
$$\displaystyle\sum_{m=-1}^{1}P_{a}(m|\lambda,\vec{a})=\sum_{m^{\prime}=-1}^{1}P%
_{b}(m^{\prime}|\lambda,\vec{b})=1,$$
(6.6)
the first term of (6.5) satisfies
$$\displaystyle{1\over 4}\sum_{m,m^{\prime}=-1}\int d\Omega_{a}\int d\Omega_{b}P%
_{a}(m|\lambda,\vec{a})P_{b}(m^{\prime}|\lambda,\vec{b})$$
$$\displaystyle=(2\pi)^{2}.$$
(6.7)
We transform the other term of (6.5) to a more convenient form
$$\displaystyle{1\over 4}\sum_{m,m^{\prime}=-1}^{1}\int d\Omega_{a}\int d\Omega_%
{b}mm^{\prime}P_{a}(m|\lambda,\vec{a})P_{b}(m^{\prime}|\lambda,\vec{b})V\vec{a%
}\cdot\vec{b}$$
$$\displaystyle={1\over 4}\int d\Omega_{a}\int d\Omega_{b}I_{a}(\vec{a},\lambda)%
I_{b}(\vec{b},\lambda)V_{2}\vec{a}\cdot\vec{b},$$
(6.8)
where
$$\displaystyle I_{n}(\vec{n},\lambda)=\sum_{m=-1}^{1}mP_{n}(m|\lambda,\vec{n}),$$
(6.9)
and
one has $|I_{n}(\vec{n},\lambda)|\leq 1$ ($n=a,b$).
The scalar product of two three dimensional vectors $\vec{a}$ and
$\vec{b}$ that appears in (6.8) can be written
as $\vec{a}\cdot\vec{b}=\sum_{k=1}^{3}a_{k}(\theta_{a},\phi_{a})b_{k}(\theta_{b},%
\phi_{b})$,
where
$$\displaystyle\vec{n}=(n_{1},n_{2},n_{3})$$
$$\displaystyle=(\sin\theta_{n}\cos\phi_{n},\sin\theta_{n}\sin\phi_{n},\cos%
\theta_{n}).$$
(6.10)
Therefore
(6.8) reads
$$\displaystyle{V_{2}\over 4}\sum_{k=1}^{3}\int d\Omega_{a}I_{a}(\theta_{a},\phi%
_{a},\lambda)a_{k}(\theta_{a},\phi_{a})$$
$$\displaystyle\times\int d\Omega_{b}I_{b}(\theta_{b},\phi_{b},\lambda)b_{k}(%
\theta_{b},\phi_{b}).$$
(6.11)
We notice here that our expression is a sum of three terms,
each of which is a product of two integrals.
The functions in (6.11) are square integrable, i.e.
integrals
$$\displaystyle\int d\Omega_{n}|I_{n}(\theta_{n},\phi_{n},\lambda)|^{2}$$
(6.12)
and
$$\displaystyle\int d\Omega_{n}|n_{k}(\theta_{n},\phi_{n})|^{2}$$
(6.13)
exist (we remind that
$|I_{n}(\theta_{n},\phi_{n},\lambda)|\leq 1$ which guarantees the
existence of the first integral). This allows us to use formalism of
Hilbert space of square integrable functions on the unit sphere, which
we denote as $L^{2}(S^{3})$.
The functions $n_{k}(\theta_{n},\phi_{n})$ fulfil the orthogonality relation
$\int d\Omega_{n}n_{k}(\theta_{n},\phi_{n})n_{l}(\theta_{n},\phi_{n})={4\pi%
\over 3}\delta_{kl}$.
Thus, if we normalise $n_{k}$ (i.e. we divide them
by their norm, which is $\sqrt{4\pi\over 3}$) we can interpret the integral
$\alpha_{k}^{n}(\lambda)=\sqrt{3\over 4\pi}\int d\Omega_{n}I_{n}(\theta_{n},%
\phi_{n},\lambda)n_{k}(\theta_{n},\phi_{n})$ as
a k-th coefficient of the projection of
$I_{n}(\theta_{n},\phi_{n},\lambda)$ into a three dimensional
subspace of $L^{2}(S^{3})$ spanned by the (normalised) basis functions
$\sqrt{3\over 4\pi}n_{k}(\theta_{n},\phi_{n})$ ($k=1,2,3$).
For later reference we will call this space $\Sigma(3)$.
Therefore (6.8) transforms into
$$\displaystyle V_{2}{\pi\over 3}\sum_{k=1}^{3}\alpha^{a}_{k}(\lambda)\alpha^{b}%
_{k}(\lambda).$$
(6.14)
Denoting the projection of $I_{n}(\theta_{n},\phi_{n},\lambda)$ into
$\Sigma(3)$ by
$I_{n}^{||}(\theta_{n},\phi_{n},\lambda)$ and
using the Schwartz inequality we arrive at
$$\displaystyle{\pi\over 3}\sum_{k=1}^{3}\alpha_{k}^{a}(\lambda)\alpha_{k}^{a}(%
\lambda)\leq{\pi\over 3}||I_{a}^{||}(\cdot,\lambda)||||I_{b}^{||}(\cdot,%
\lambda)||.$$
(6.15)
Therefore, our last step is to calculate the maximal possible value of
the norm $||I_{n}^{||}(\cdot,\lambda)||$. Since the length (norm) of a projection of a
vector into a certain subspace
is equal to the maximal value of its scalar product with any normalised
vector belonging to this subspace,
the norm $||I_{n}^{||}(\cdot,\lambda)||$ is given by
$$\displaystyle||I_{n}^{||}(\cdot,\lambda)||=\max_{|\vec{c}|=1}[\sqrt{3\over 4%
\pi}\int d\Omega_{n}I_{n}(\theta_{n},\phi_{n},\lambda)\sum_{k=1}^{3}c_{k}n_{k}%
(\theta_{n},\phi_{n})],$$
(6.16)
where $\vec{c}=(c_{1},c_{2},c_{3})$ and $|\vec{c}|^{2}=\sum_{k=1}^{3}c_{k}^{2}=1$.
Because
$|I_{n}(\vec{a},\lambda)|\leq 1$ one has
$$\displaystyle||I_{n}^{||}(\cdot,\lambda)||\leq\max_{|\vec{c}|=1}[\sqrt{3\over 4%
\pi}\int d\Omega_{n}|\sum_{k=1}^{3}c_{k}n_{k}(\theta_{n},\phi_{n})|].$$
(6.17)
Every vector
$\vec{c}$ can be obtained by a certain rotation of the versor $\vec{z}$.
Such a rotation is represented by an
orthogonal matrix $\hat{O}$ belonging to the rotation group $SO(3)$.
Therefore, (6.17) can be
rewritten as
$$\displaystyle||I_{n}^{||}(\cdot,\lambda)||\leq\max_{\hat{O}}[\sqrt{3\over 4\pi%
}\int d\Omega_{n}|\hat{O}\vec{z}\cdot\vec{n}(\theta_{n},\phi_{n})|],$$
(6.18)
where the maximum is taken over all possible rotation matrices $\hat{O}$. Since
$|O\hat{z}\cdot\vec{n}(\theta_{n},\phi_{n})|$ is the modulus of the scalar product of
two ordinary three dimensional vectors, it is equal to
$|\vec{z}\cdot\hat{O}^{-1}\vec{n}(\theta_{n},\phi_{n})|$.
An active rotation of the vector $\vec{n}$ is equivalent to a (passive)
change of the spherical coordinates. Utilising the fact that the measure
$d\Omega_{n}$ is rotationally invariant we see that
$$\displaystyle||I_{n}^{||}||\leq\int d\Omega_{n}|\sqrt{3\over 4\pi}\cos\theta_{%
n}|=2\pi\sqrt{3\over 4\pi}.$$
(6.19)
Therefore (6.15) is not greater then ${1\over 4}(2\pi)^{2}$, which
with (6.4) and (6.7) gives us the following
inequalities
$$\displaystyle||P_{QM}^{V_{2}}||^{2}=(2\pi)^{2}+{V_{2}^{2}\over 3}(2\pi)^{2}>(2%
\pi)^{2}+{V_{2}\over 4}(2\pi)^{2}\geq\langle P_{QM}^{V_{2}}|P_{HV}\rangle.$$
(6.20)
6.4 Results
The inequality (6.20) is violated by quantum predictions provided that
the visibility $V_{2}$ is higher then $75\%$. Please notice that
the right hand inequality is a form of a ”functional” Bell inequality.
It simply gives the upper bound for the value of a certain functional
defined on the local realistic probability functions $P_{HV}$. The left
hand inequality shows that the insertion of $P_{QM}^{V_{2}}$ into the
functional Bell inequality leads to its violation provided $V_{2}>0.75$.
The characteristic trait of our functional Bell inequality
is that its form is defined by the quantum prediction $P_{QM}^{V_{2}}$.
6.5 Conclusions
The threshold visibility for two qubit
interference to violate the inequality (6.20) is lower than
in the case of coplanar settings [Żukowski93b], for which the critical
visibility is
${8\over\pi^{2}}$. Also, it is lower than the one given recently
by Gisin [Gisin99]. For
his inequalities involving arbitrary many settings the threshold
visibility equals $V_{2}={\pi\over 4}$. The chained inequalities [Garuccio80],
[Braunstein89],
for evenly spaced settings, with the number of settings going to infinity,
have the property that the critical visibility approaches 1 in the limit
of infinitely many settings.
Taking into account the fact that the necessary condition for the existence
of a local hidden variable model for two local settings at each side of
the experiment implied by the CHSH inequality is $V_{2}\leq{1\over\sqrt{2}}$,
the critical visibility of any hidden variable model
that aims at reproducing quantum correlations for the gedanken experiment
described above for any number of local settings (infinite or not) cannot
be greater than
${1\over\sqrt{2}}$. Therefore, the numerical value of the critical visibility
obtained by the above inequality is overestimated. This is due to the fact
that this inequality is only a necessary condition for the existence
of local hidden variables. However, the inequality (6.20) may be the first step towards finding
the threshold visibility of the local hidden variable model
reproducing quantum mechanical predictions for all positions of
measuring apparatus.
The presented inequality also solves the academic problem of
finding a Bell inequality that involves all possible settings
of the local apparata.
In [Żukowski93b] Żukowski has shown that for GHZ states
involving four or more
particles (and employing all coplanar settings) the functional
Bell inequality approach leads to much more stringent conditions
on the critical visibilities than the approaches of Mermin [Mermin90b] and
Ardehali [Ardehali92]. However, the extension of the presented approach
to GHZ states does not give any improvement with respect to the
standard one. This is due to the fact that original GHZ paradox is
obtainable for co-planar settings only.
Chapter 7 Greenberger-Horne-Zeilinger paradoxes for qu$N$its [5,6]
7.1 Introduction
The Greenberger-Horne-Zeilinger correlations, discovered in 1989,
started a new chapter in the research related with entanglement. To
a great extent this discovery was responsible for the sudden renewal
in the interest in this field, both in theory and experiment.
All these developments finally led to the first actual observation
of three qubit GHZ correlations in 1999 [Bouwmeester99],
and as a by product, since
the experimental techniques involved were of the same kind,
to the famous teleportation experiment [Bouwmeester97].
In this chapter we would like to examine whether GHZ-type paradoxes exist
also in the case of correlations expected in gedanken experiments
involving multiport beam splitters [Klyshko88, Zeilinger93, Zeilinger94],
i.e. for a specific
case of non dichotomic observables (which have properties distinctive
to the dichotomic ones [Gleason57, Bell66, Kochen67]).
To this end, we shall study a
GHZ-Bell type experiment in which one has a source emitting
$M$ qu$N$its in a specific entangled state of the property, that the
qu$N$its propagate towards one of $M$ spatially separated
non conventional measuring devices operated by independent observers.
Each of the devices consists of an unbiased symmetric multiport beam splitter
[Żukowski97b] (with $N$ input and $N$ exit ports), $N$ phase shifters
operated by the observers (one in front of each input), and $N$
detectors (one behind each exit port).
7.2 Unbiased multiport beamsplitters
An unbiased symmetric $2N$-port beam splitter is defined as an $N$-input and
$N$-output interferometric device which has the property that a beam
of light entering via single port is evenly split between all output
ports (see (7.1)). I.e., the unitary matrix defining such a device has the
property that
the modulus of all its elements equals ${1\over\sqrt{N}}$.
An extended
introduction to the physics and theory of such devices is given in
[Żukowski97b], and therefore the reader not acknowledged with those
concepts is kindly asked to consult this reference. Multiport
beam splitters were introduced into the literature on the EPR paradox
in [Klyshko88, Zeilinger93, Zeilinger94] in order to extend two
qubit Bell-phenomena to
observables described as operators in Hilbert spaces of dimension
higher than two. In contradistinction to the higher than 1/2 spin
generalisations of the Bell-phenomena [Mermin80, Garg82, Mermin82, Ardehali91, Agarwal93, Wódkiewicz94, Wódkiewicz95], this type of
experimental devices generalise the ideas of beam-entanglement
[Horne85, Żukowski88, Horne89, Rarity90].
Unbiased symmetric multiport beam splitters are performing unitary
transformations between ”mutually unbiased” bases in the Hilbert space
[Schwinger60, Ivanovic81, Wooters86].
They were tested in several recent experiments
[Mattle95, Reck96], and also various aspects of such devices were
analysed theoretically [Reck94, Jex95].
We shall use here only multiport beam splitters which have the property
that the elements of the unitary transformation which describes their
action are given by
$$U_{m,m^{\prime}}^{N}={1\over\sqrt{N}}\gamma_{N}^{(m-1)(m^{\prime}-1)},$$
(7.1)
where $\gamma_{N}=\exp(i{2\pi\over N})$ and the indices $m$, $m^{\prime}$
denote the input and exit ports. Such devices were called in
[Żukowski97b] the Bell multiports.
7.3 Quantum mechanical predictions
One assumes that the initial $M$ qu$N$it state that feeds $M$ spatially
separated multiports, each of which has $N$ inputs and $N$ outputs,
has the following form:
$$\displaystyle|\psi(M)\rangle={1\over\sqrt{N}}\sum_{m=1}^{N}\prod_{l=1}^{M}|m%
\rangle_{l},$$
(7.2)
where $|m\rangle_{l}$ describes the $l$-th qu$N$it being in the
$m$-th beam, which leads to the $m$-th input of the $l$-th multiport.
Please note, that only one qu$N$it enters each multiport.
However, each of the qu$N$its itself is in a mixed state (with equal
weights), which gives it equal probability to enter the local
multiport via any of the input ports.
The state (7.2) seems to be the most straightforward
generalisation of the GHZ states to the new type of observables. In
the original GHZ states the number of their components (i.e., two) is
equal to the dimension of the Hilbert space describing the relevant
(dichotomic) degrees of freedom of each of the qu$N$its. This
property is shared with the EPR-type states proposed in [Żukowski97b] for
a two-multiport Bell-type experiment - in this case the number of
components equals the number of input ports of each of the multiport
beam splitters. We shall not discuss here the possible methods to
generate such states. However, we briefly mention that the recently
tested entanglement swapping [Żukowski93a, Żukowski95, Pan98]
technique could be used for
this purpose.
As it was mentioned earlier, in front of every input of each multiport
beam splitter one has a tunable phase shifter. The initial state is
transformed by the phase shifters into
$$\displaystyle|\psi(M)^{\prime}\rangle={1\over\sqrt{N}}\sum_{m=1}^{N}\prod_{l=1%
}^{M}\exp(i\phi_{l}^{m})|m\rangle_{l},$$
(7.3)
where $\phi_{l}^{m}$ stands for the setting of the phase shifter in
front of the $m$-th port of the $l$-th multiport.
The quantum prediction for probability to register the first photon
in the output $k_{1}$ of an $2N$ - port device, the second one in the
output $k_{2}$ of the second such device ,…, and the $M$-th one in
the output $k_{M}$ of the $M$-th device is given by:
$$\displaystyle P_{QM}(k_{1},\dots,k_{M}|\vec{\phi_{1}},\dots,\vec{\phi_{M}})=$$
$$\displaystyle({1\over N})^{M+1}|\sum_{m=1}^{N}\exp(i\sum_{l=1}^{M}\phi_{l}^{m}%
)\prod_{n=1}^{M}\gamma_{N}^{(m-1)(k_{n}-1)}|^{2}=$$
$$\displaystyle=({1\over N})^{M+1}\left[N+2\sum_{m>m^{\prime}}^{N}\cos\left(\sum%
_{l=1}^{M}\Delta\Phi_{l,k_{l}}^{m,m^{\prime}}\right)\right],$$
(7.4)
where
$\Delta\Phi_{l,k_{l}}^{m,m^{\prime}}=\phi_{l}^{m}-\phi_{l}^{m^{\prime}}+{2\pi%
\over N}(k_{l}-1)(m-m^{\prime})$. The shorthand symbol
$\vec{\phi}_{k}$ stands for the full set of phase settings in front of
the $k$-th multiport, i.e.
$\phi_{k}^{1},\phi_{k}^{2},\dots,\phi_{k}^{N}$.
7.3.1 Bell number assignment
To efficiently describe the local detection events
let us employ a specific value assignment method (called Bell
number assignment; for a detailed
explanation see again [Żukowski97b]), which ascribes to the detection event
behind the $m$
- th output of a multiport the value $\gamma_{N}^{m-1}$, where
$\gamma_{N}=\exp(i{2\pi\over N})$.
With such a value assignment to the detection events, the Bell-type
correlation function, which is the average of the product of the
expected results, is defined as
$$\displaystyle E(\vec{\phi_{1}},\cdots,\vec{\phi_{M}})=$$
$$\displaystyle=\sum_{k_{1},\cdots,k_{M}=1}^{N}\prod_{l=1}^{M}\gamma_{N}^{k_{l}-%
1}P(k_{1},\cdots,k_{M}|\vec{\phi_{1}},\cdots,\vec{\phi_{M}})$$
(7.5)
and as we shall see for the quantum case it acquires particularly simple
and universal form111This is the main purpose for using this
non-conventional value assignment..
The easiest way to compute the correlation function for the quantum
prediction employs the mid formula of (7.4):
$$\displaystyle E_{QM}(\vec{\phi_{1}},\cdots,\vec{\phi_{M}})=$$
$$\displaystyle=({1\over N})^{M+1}\sum_{k_{1},\cdots,k_{M}=1}^{N}\sum_{m,m^{%
\prime}=1}^{N}\exp\left(i\sum_{n=1}^{M}(\phi_{n}^{m}-\phi_{n}^{m^{\prime}})\right)$$
$$\displaystyle\times\prod_{l=1}^{M}\gamma_{N}{}^{(k_{l}-1)(m-m^{\prime}+1)}=$$
$$\displaystyle=({1\over N})^{M+1}\sum_{m,m^{\prime}=1}^{N}\exp\left(i\sum_{n=1}%
^{N}(\phi_{n}^{m}-\phi_{n}^{m^{\prime}})\right)$$
$$\displaystyle\times\prod_{l=1}^{M}\sum_{k_{l}=1}^{N}\gamma_{N}^{(k_{l}-1)(m-m^%
{\prime}+1)}.$$
(7.6)
Now, one notices that
$\sum_{k_{l}=1}^{N}\gamma_{N}^{(k_{l}-1)(m-m^{\prime}+1)}$
differs from zero
(and equals to N)
only if $m-m^{\prime}+1=0$, modulo N. Therefore we can finally write:
$$\displaystyle E_{QM}(\vec{\phi_{1}},\cdots,\vec{\phi_{M}})$$
$$\displaystyle={1\over N}\sum_{m=1}^{N}\exp(i\sum_{l=1}^{M}\phi^{m,m+1}_{l}),$$
(7.7)
where $\phi^{m,m+1}_{l}=\phi^{m}_{l}-\phi^{m+1}_{l}$ and the above
sum is
understood modulo N, which means that
$\phi^{N+1}_{l}=\phi^{1}_{l}$.
One can notice here a striking simplicity and symmetry of this quantum
correlation function (7.7). It is valid for all possible
values of M (number of qu$N$its) and for all possible values of
$N\geq 2$ (number of ports). For $N=2$, it reduces itself to the usual
two qubit, and for $N=2$, $M\geq 2$ the standard GHZ type
multi-qubit correlation function for beam-entanglement experiments,
namely $\cos(\sum_{l=1}^{M}\phi_{l}^{1,2})$ [Mermin90b]. The Bell -
EPR phenomena discussed in
[Żukowski97b] are described by (7.7) for $M=2,N\geq 3$.
Even for $N=2$, $M=1$ the function (7.7) describes
the following process. Assume that
a traditional four-port 50-50 beam splitter, is fed a single photon
input in a state in which is an equal superposition of being in each
of the two input ports. The value of (7.7) is the average of
expected photo counts behind the exit ports (provided the click at one
of the detectors is described as $+1$ and at the other one as $-1$),
and of course it depends on the relative phase shifts in front of the
beam splitter. In other words, this situation describes a Mach-Zehnder
interferometer with a single photon input at a chosen input port. For
$N=3$, $M=1$ the same interpretation applies to the case of a
generalised three input, three output Mach-Zehnder interferometer
described in [Weihs96], provided one ascribes to firings of
the three detectors respectively
$\gamma_{3}=\alpha\equiv\exp(i{2\pi\over 3})$, $\alpha^{2}$ and
$\alpha^{3}$.
7.3.2 Perfect correlations
The described set of gedanken experiments is rich in EPR-GHZ
correlations (for $M\geq 2$). To reveal the above, let us first analyse
the conditions (i.e. settings) for such correlations. As the
correlation function (7.7) is an average of complex numbers of
unit modulus, one has
$|E_{QM}(\vec{\phi_{1}},\cdots,\vec{\phi_{M})}|\leq 1$. The equality
signals a perfect EPR-GHZ correlation. It is easy to notice that this
may happen only if
$$\exp(i\sum_{l=1}^{M}\phi_{l}^{1,2})=\exp(i\sum_{l=1}^{M}\phi_{l}^{2,3})=\cdots%
=\exp(i\sum_{l=1}^{M}\phi_{l}^{M,1})=\gamma_{N}^{k},$$
where k is an arbitrary
natural number. Under this condition
$E(\vec{\phi_{1}},\cdots,\vec{\phi_{M}})=\gamma_{N}^{k}$. This means that only
those sets of $M$ spatially separated detectors may fire, which are
ascribed such Bell numbers which have the property that their product
is $\gamma_{N}^{k}$. Knowing, which detectors fired in the set of
$M-1$ observation stations, one can predict with certainty which
detector would fire at the sole observation station not in the set.
7.4 Paradoxes for $N+1$ maximally entangled qu$N$its
7.4.1 Four tritters
We shall now present the simplest GHZ-type paradox for such systems.
We take $N=3$ and $M=4$. That is we consider now, the experimental
situation in which one has the source producing the ensemble of four
three-state particles (qutrits) described by the state $|\psi(4)\rangle$
(compare, (7.2)) that feeds four three-port beam splitters (i.e.,
tritters [Żukowski97b]). In this case the quantum correlation function
has the form:
$$\displaystyle E_{QM}(\vec{\phi_{1}},\vec{\phi_{2}},\vec{\phi_{3}},\vec{\phi_{4%
}})$$
$$\displaystyle={1\over 3}\sum_{k=1}^{3}\exp\left(i\sum_{l=1}^{4}(\phi_{l}^{k}-%
\phi_{l}^{k+1})\right).$$
(7.8)
The (deterministic) local hidden variables correlation
function for this type of experiment must have the following structure
[Bell64]:
$$\displaystyle E_{HV}(\vec{\phi_{1}},\vec{\phi_{2}},\vec{\phi_{3}},\vec{\phi_{4%
}})=\int_{\Lambda}\prod_{k=1}^{4}I_{k}(\vec{\phi_{k}},\lambda)\rho(\lambda)d\lambda.$$
(7.9)
The hidden variable function $I_{k}(\vec{\phi_{k}},\lambda)$, which
determines the firing of the detectors behind the k-th multiport,
depends only upon the local set of phases, and takes one of the three
possible values $\alpha$, $\alpha^{2}$, $\alpha^{3}=1$ (these values
indicate which of the detectors is to fire), and $\rho(\lambda)$ is
the distribution of hidden variables.
Consider four gedanken experiments. In the first one our observers,
each of whom operates one of the spatial separated devices, choose
the following phases in front of their three-port beam splitters:
$$\displaystyle\vec{\phi_{1}}\equiv(\phi_{1}^{1},\phi_{1}^{2},\phi_{1}^{3})=(0,{%
2\pi\over 9},{4\pi\over 9})=\vec{\phi_{2}}=\vec{\phi_{3}}=\vec{\phi}$$
$$\displaystyle\vec{\phi_{4}}\equiv(\phi_{4}^{1},\phi_{4}^{2},\phi_{4}^{3})=(0,0%
,0)=\vec{\phi^{\prime}}.$$
(7.10)
In the second experiment, the third observer sets
$\vec{\phi_{3}}=\vec{\phi^{\prime}}$ whereas the other ones set $\vec{\phi}$.
We repeat this swapping of the settings procedure in the next two
experiments until the first observer sets $\vec{\phi^{\prime}}$ and the other
set $\vec{\phi}$. Quantum mechanics predicts that in all four such
experiments
the correlation function
is equal to $\alpha^{2}$ (i.e. we have perfect GHZ correlations).
Namely we have
$$\displaystyle E_{QM}(\vec{\phi},\vec{\phi},\vec{\phi},\vec{\phi}^{\prime})=E_{%
QM}(\vec{\phi},\vec{\phi},\vec{\phi}^{\prime},\vec{\phi})$$
$$\displaystyle=E_{QM}(\vec{\phi},\vec{\phi}^{\prime},\vec{\phi},\vec{\phi})=E_{%
QM}(\vec{\phi}^{\prime},\vec{\phi},\vec{\phi},\vec{\phi})=\alpha^{2}.$$
(7.11)
However, this immediately implies that for any local hidden variables theory that aims
at describing these phenomena one must have for every $\lambda$
$$\displaystyle I_{k}(\vec{\phi^{\prime}},\lambda)\prod_{{l=1},{l\neq k}}^{4}I_{%
l}(\vec{\phi},\lambda)=\alpha^{2},$$
(7.12)
and this must hold for all $k=1,2,3,4$. But, since
$I_{l}(\vec{\phi},\lambda)^{3}=\alpha^{3k}=1$ (where, $k$ represents a
certain integer), then after multiplying these four equations side by
side, one has for every $\lambda$
$$\displaystyle\prod_{l=1}^{4}I_{l}(\vec{\phi^{\prime}},\lambda)=\alpha^{2}.$$
(7.13)
Therefore, if the local hidden variable theory is to agree with the
earlier mentioned quantum predictions (7.11), then one must have
$$\displaystyle E(\vec{\phi^{\prime}},\vec{\phi^{\prime}},\vec{\phi^{\prime}},%
\vec{\phi^{\prime}})=\alpha^{2}=\alpha^{*}.$$
(7.14)
However, the quantum prediction is
$E_{QM}(\vec{\phi^{\prime}},\vec{\phi^{\prime}},\vec{\phi^{\prime}},\vec{\phi^{%
\prime}})=1$. Thus we
have a GHZ-type contradiction that $1=\alpha^{*}$. I.e., hidden
variables predict a different type perfect EPR-GHZ correlation.
I.e. we have a realisation of the GHZ paradox for non-dichotomic
observables.
7.4.2 General case: $N+1$ maximally entangled qu$N$its
We will extend the reasoning to the case when one has $M=N+1$ qu$N$its
(described by the state of the form (7.2)) beamed into multiport
beam splitters with $N$ input and output ports. The quantum prediction for the Bell
correlation function is given by (7.7), with the
appropriate value of $M$.
The local hidden variables correlation function must have
the following structure
$$\displaystyle\int\prod_{k=1}^{M}I_{k}(\vec{\psi_{k}},\lambda)\rho(\lambda)d\lambda,$$
(7.15)
where $k$ now extends from $1$ to $M$ and $\vec{\psi_{k}}$ stands for the
full set of settings in front of the $k$-th multiport, i.e.
$\psi_{k}^{1},\psi_{k}^{2},\cdots,\psi_{k}^{M-1}$, and $I_{k}(\vec{\psi_{k}},\lambda)$
is a hidden variable function depending on the local phase settings,
which has the property that its value, which can be any integer power
of $\gamma_{M-1}$, indicated which local detector is to fire.
Now, as it was in the previous case, we must choose appropriate phases
for each of observers that will be taken in the first experiment. The
appropriate choice is the following one:
$$\displaystyle\vec{\psi}_{1}=\cdots=\vec{\psi}_{M-1}=(0,\delta,2\delta,\cdots,(%
M-2)\delta)=\vec{\psi}$$
$$\displaystyle\vec{\psi}_{M}=(0,\cdots,0)=\vec{\psi^{\prime}},$$
(7.16)
where $\delta={2\pi\over(M-1)^{2}}$. In the next $M-1$ experiments one
applies previously described swapping of the settings procedure until
the first observer sets $\vec{\psi^{\prime}}$ and the other ones set
$\vec{\psi}$. For such choice of phases the quantum correlation
function for every of the $M$ experiment, is equal to
$\gamma_{M-1}^{M-2}=\gamma_{M-1}^{*}$.
(i.e. we have perfect GHZ correlations of the same type for each of
the experiments). But this implies that, for any local hidden variables theory that
aims at describing these phenomena one must have, for every $\lambda$,
$$\displaystyle I_{k}(\vec{\psi^{\prime}},\lambda)\prod_{l\neq k}I_{l}(\vec{\psi%
},\lambda)=\gamma_{M-1}^{*},$$
(7.17)
and that this must hold for all $k=1,\cdots,M$. However, after
multiplying these $M$ equations one has:
$$\displaystyle\prod_{l=1}^{M}I_{l}(\vec{\psi^{\prime}},\lambda)=\gamma_{M-1}^{*},$$
(7.18)
where we have used the property of the Bell numbers generated by
$\gamma_{M-1}$, that each of them to the $M-1$-th power gives 1, and
therefore that $I_{k}(\vec{\psi},\lambda)^{M}=1$. Thus, the local
hidden variable implies that
$$E_{QM}(\vec{\psi^{\prime}},\cdots,\vec{\psi^{\prime}})=\gamma_{M-1}^{*}.$$
However, the quantum prediction is 1. Thus we have the GHZ
contradiction that $1=\gamma_{M-1}^{*}$. I.e. hidden variables predict
a different type perfect EPR - GHZ correlations. In other words the
EPR idea of elements of reality makes no sense for the discussed
experiments, and this hold for an arbitrary number of qu$N$its $M$,
and for suitably related ($M-1$), but in principle arbitrarily high
number of input and exit ports of symmetric multiport beam splitters.
7.5 Paradoxes for $N$ maximally entangled qu$N$its.
In this section we show that the above reasoning can be
as well applied to the case where the number of observers
$M$ equals the number of input ports $N$ of the $2N$ port Bell
multiports.
7.5.1 Three tritters
As the simplest example let us consider the
gedanken experiment with three observers
each of which having tritters (3 input and 3 output ports) as
a measuring device. The correlation function for such
an experiment reads
$$\displaystyle E_{QM}(\vec{\phi}_{1},\vec{\phi}_{2},\vec{\phi}_{3})$$
$$\displaystyle={1\over 3}\sum_{k=1}^{3}\exp\left(i\sum_{l=1}^{3}(\phi_{l}^{k}-%
\phi^{k+1}_{l})\right).$$
(7.19)
In the first run of the experiment we allow the observers
to choose the following settings of the measuring apparatus
$$\displaystyle\vec{\phi}_{1}=(0,{\pi\over 3},{2\pi\over 3})=\vec{\phi}_{2}=\vec%
{\phi}$$
$$\displaystyle\vec{\phi}_{3}=(0,0,0)=\vec{\phi^{\prime}}$$
(7.20)
in the second run they choose
$$\displaystyle\vec{\phi}_{1}=\vec{\phi^{\prime}}$$
$$\displaystyle\vec{\phi}_{2}=\vec{\phi}=\vec{\phi}_{3}$$
whereas in the third run they fix the local settings
of their tritters on
$$\displaystyle\vec{\phi}_{1}=\vec{\phi}=\vec{\phi}_{3}$$
$$\displaystyle\vec{\phi}_{2}=\vec{\phi^{\prime}}.$$
(7.22)
Now, let us calculate the numerical values of
the correlation function for each experimental
situation. We easily find that for all three
experiments this value, due to the special form
of the correlation function, is the same and reads
$$\displaystyle E_{QM}(\vec{\phi},\vec{\phi},\vec{\phi^{\prime}})=E_{QM}(\vec{%
\phi^{\prime}},\vec{\phi},\vec{\phi})=E_{QM}(\vec{\phi},\vec{\phi^{\prime}},%
\vec{\phi})$$
$$\displaystyle=\exp(-i{2\pi\over 3})=\alpha^{2},$$
(7.23)
i.e., we observe perfect correlations.
Proceeding in exactly the same way as in the previous
section we can write the equations that must be fulfilled
by local hidden variables for every $\lambda$, i.e.,
$$\displaystyle I_{1}(\vec{\phi},\lambda)I_{2}(\vec{\phi},\lambda)I_{3}(\vec{%
\phi^{\prime}},\lambda)=\alpha^{2}$$
$$\displaystyle I_{1}(\vec{\phi^{\prime}},\lambda)I_{2}(\vec{\phi},\lambda)I_{3}%
(\vec{\phi},\lambda)=\alpha^{2}$$
$$\displaystyle I_{1}(\vec{\phi},\lambda)I_{2}(\vec{\phi^{\prime}},\lambda)I_{3}%
(\vec{\phi},\lambda)=\alpha^{2}$$
(7.24)
After multiplication of the above equations one arrives at:
$$\displaystyle\prod_{k=1}^{3}I_{k}(\vec{\phi^{\prime}},\lambda)\prod_{k=1}^{3}I%
_{k}(\vec{\phi},\lambda)^{2}=(\alpha^{2})^{3}=1,$$
(7.25)
which can be also written in the following form
$$\displaystyle\prod_{k=1}^{3}I_{k}(\vec{\phi^{\prime}},\lambda)=\prod_{k=1}^{3}%
I_{k}(\vec{\phi},\lambda),$$
(7.26)
where we have used the property of the hidden variable functions,
namely that
$I_{k}(\vec{\phi},\lambda)^{2}=I_{k}(\vec{\phi},\lambda)^{*}$ for
every $\lambda$. However, because $E_{QM}(\vec{\phi^{\prime}},\vec{\phi^{\prime}},\vec{\phi^{\prime}})=1$ we must also have
$$\displaystyle\prod_{k=1}^{3}I_{k}(\vec{\phi^{\prime}},\lambda)=1,$$
(7.27)
which, because of (7.26), gives
$$\displaystyle\prod_{k=1}^{3}I_{k}(\vec{\phi},\lambda)=1$$
(7.28)
for every $\lambda$. This in turn implies that
$$\displaystyle E_{QM}(\vec{\phi},\vec{\phi},\vec{\phi})=1,$$
(7.29)
which means that local hidden variables predict perfect correlations for
the experiment when all observers set their local settings at
$\vec{\phi}$. However, the true quantum prediction is that
$$\displaystyle E_{QM}(\vec{\phi},\vec{\phi},\vec{\phi})=-{1\over 3}.$$
(7.30)
Therefore we have the contradiction: $1=-{1\over 3}$.
This contradiction is of the different
type than the one derived in the previous section although it has been
obtained in the similar way, i.e., the perfect correlations have been
used to derive it- equations (7.24) and (7.25).
Here local hidden variables
imply a certain perfect correlation, which is not predicted by quantum
mechanics.
7.5.2 General case: N maximally entangled qu$N$its.
Now, we employ the above procedure for the case when we have an arbitrary odd
number of multiports and qu$N$its $N=2m+1$. As we have seen in
the previous section the crucial point is to find the
proper phases for the multiports. Let us choose for the
first gedanken experiment the following ones
$$\displaystyle\vec{\phi}_{1}=(0,{\pi\over 2m+1},{2\pi\over 2m+1},...,{2m\pi%
\over 2m+1})=\vec{\phi}$$
$$\displaystyle\vec{\phi}_{2}=\vec{\phi}_{3}=...=\vec{\phi}_{2m}=\vec{\phi}$$
$$\displaystyle\vec{\phi}_{2m+1}=(0,0,...,0)=\vec{\phi^{\prime}}.$$
(7.31)
As before, in the next run of the experiment we choose the same
phases but we change the role of observers such that in the second run
the first
one chooses $\vec{\phi^{\prime}}$ while the rest of them choose $\vec{\phi}$,
in the third run the third one chooses $\vec{\phi^{\prime}}$ while the rest
of them choose
$\vec{\phi}$, etc. Again, the value of the correlation
function for each experiment is the same
$$\displaystyle E_{QM}(\vec{\phi},...,\vec{\phi^{\prime}})=E_{QM}(\vec{\phi^{%
\prime}},\dots,\vec{\phi})=\dots=E_{QM}(\vec{\phi},\dots,\vec{\phi^{\prime}},%
\vec{\phi})$$
$$\displaystyle={1\over 2m+1}\left[\exp\left(-i{2m\pi\over 2m+1}\right)+\exp%
\left(-i{2m\pi\over 2m+1}\right)\right.$$
$$\displaystyle\left.+...+\exp\left(-i{2m\pi\over 2m+1}\right)+\exp\left(i{4m^{2%
}\pi\over 2m+1}\right)\right]$$
$$\displaystyle={1\over 2m+1}\left[2m\exp\left(-i{2m\pi\over 2m+1}\right)+\exp%
\left(i{4m^{2}\pi\over 2m+1}\right)\right]=\gamma_{N}^{-m}$$
(7.32)
Using (7.32), the structure of the hidden
variables correlation function (7.15)
and multiplying the above equations
by each other we arrive at
$$\displaystyle\prod_{k=1}^{2m+1}I_{k}(\vec{\phi^{\prime}},\lambda)=\prod_{k=1}^%
{2m+1}I_{k}(\vec{\phi},\lambda)^{-2m}=\prod_{k=1}^{2m+1}I_{k}(\vec{\phi},\lambda)$$
(7.33)
for every $\lambda$. Because $E_{QM}(\vec{\phi^{\prime}},\dots,\vec{\phi^{\prime}})=1$
one must have
$$\displaystyle\prod_{k=1}^{2m+1}I_{k}(\vec{\phi^{\prime}},\lambda)=1,$$
(7.34)
which gives
$$\displaystyle\prod_{k=1}^{2m+1}I_{k}(\vec{\phi},\lambda)=1$$
(7.35)
for every $\lambda$. Thus, local hidden variables imply
the following perfect correlation
$$\displaystyle E_{QM}(\vec{\phi},...,\vec{\phi})=1,$$
(7.36)
which is untrue because quantum mechanics gives
$$\displaystyle E_{QM}(\vec{\phi},...,\vec{\phi})$$
$$\displaystyle={1\over 2m+1}\left[2m\exp\left(-i\frac{2m+1}{2m+1}\pi\right)\right.$$
$$\displaystyle\left.+\exp\left(i\frac{(2m+1)2m}{2m+1}\pi\right)\right]={-2m+1%
\over 2m+1}.$$
(7.37)
Therefore, for each $m$ one obtains
the untrue identity $1=\frac{-2m+1}{2m+1}$,
which in the limit of $m\longrightarrow\infty$ becomes
$1=-1$.
Now, let us see what happens when one has the even number of qu$N$its
and the multiports $N=2m$ ($m\geq 2$).
As before we must find appropriate phases. Let us make the following
choice:
$$\displaystyle\vec{\phi}_{1}=(0,{\pi\over 2m-1},{2\pi\over 2m-1},...,{(2m-1)\pi%
\over 2m-1})=\vec{\phi}$$
$$\displaystyle\vec{\phi}_{2}=\vec{\phi}_{3}=...=\vec{\phi}_{N-1}=\vec{\phi}$$
$$\displaystyle\vec{\phi}_{N}=(0,0,...,0)=\vec{\phi^{\prime}}.$$
(7.38)
One easily finds that
$$\displaystyle E_{QM}(\vec{\phi},\dots,\vec{\phi^{\prime}})=E_{QM}(\vec{\phi^{%
\prime}},\dots,\vec{\phi})=\dots=E_{QM}(\vec{\phi},\dots,\vec{\phi^{\prime}},%
\vec{\phi})$$
$$\displaystyle={1\over 2m}\left[(2m-1)\exp\left(-i\frac{2m-1}{2m-1}\pi\right)\right.$$
$$\displaystyle\left.+\exp\left(i\frac{(2m-1)^{2}}{2m-1}\pi\right)\right]=-1$$
(7.39)
and
$$\displaystyle E_{QM}(\vec{\phi},..,\vec{\phi})$$
$$\displaystyle={1\over 2m}\left[(2m-1)\exp\left(-i\frac{2m}{2m-1}\pi\right)+%
\exp\left(i\frac{2m(2m+1)}{2m+1}\pi\right)\right]$$
$$\displaystyle={1\over 2m}\left[(2m-1)\exp\left(-i\frac{2m}{2m-1}\pi\right)+1%
\right].$$
(7.40)
Applying the same reasoning as above
one has
$$1={1\over 2m}\left[(2m-1)\exp\left(-i\frac{2m}{2m-1}\pi\right)+1\right].$$
(7.41)
Again, for each $m>2$ one has a contradiction, which
in the limit of $m\longrightarrow\infty$ becomes
$1=-1$.
According to the old quantum rules of thumb one approaches the classical
limit with the growing quantum numbers. The above results (for both the odd
and the even case) once more show
the contrary behaviour.
7.6 Conclusions
The derived paradoxes can be divided into two groups.
To the first group belong the series of paradoxes where one has more
observers than input ports in the unbiased multiport beamsplitters. Within this
group the contradiction manifests itself in different perfect correlations
predicted by quantum mechanics and local realism. As a special case
one obtains the original GHZ paradox.
In the second group one has the series of paradoxes in which the number of
observers and input ports in the unbiased multiport beamsplitters is the same.
Within this group the derivation of the paradoxes relies on the perfect
correlations (as in the first group) but the final result is different.
The resulting contradiction between quantum mechanics and local realism
manifests itself in the fact that local realism predicts a certain perfect
correlation whereas quantum mechanics does not.
An interesting feature of paradoxes within the second group is that they naturally
split into two parts: the even and the odd number of observers (input ports). In each
part the final contradiction has different numerical values.
Furthermore, for the case of two observers one cannot derive the GHZ paradox
with the method presented here.
All paradoxes do not vanish with the growing dimension $N$ of the Hilbert space
describing each qu$N$it.
In conclusion we state that the multiport beam splitters, and the idea
of value assignment based Bell numbers, lead to a strikingly
straightforward generalisation of the GHZ paradox for entangled
qu$N$its.
These properties may possibly find an application in
future quantum information and communication schemes (especially as
GHZ states are now observable in the lab [Bouwmeester99]).
Part III Extension of Bell Theorem via numerical approach
***
In the present part of the dissertation a novel approach
to the Bell theorem, via numerical linear optimisation,
will be presented.
This approach enables one to first of all
solve the old question222See for instance
[Mermin80, Ardehali91, Wódkiewicz91, Gisin92, Wódkiewicz94, Gisin99, Larsson99].
of generalisation of the Bell theorem
to
•
more possible settings at each side of the experiment (for two
and three qubits)
•
entangled pairs of qu$N$its ($N>2$) (in earlier
literature called spins higher than $\frac{1}{2}$)
The early Bell theorem involved only two entangled qubits and two
observers, who could choose among two local mutually incompatible
observables. Generalisations involving more than two observers
were considered by many authors and here in the previous sections.
However, one can ask the following question what the necessary
and sufficient conditions for local realistic description
of situations in which each observer can choose from more than two
observables, and for the case when two observables are not of a
dichotomic nature are.
There were several attempts to unite Bell inequalities
for such problems, however only the stage of necessary
conditions for local realism was reached.
No one has been able to construct the full set of Bell inequalities
for such problems which would provide the sufficient condition
for local realistic description.
This was a serious issue, since necessary conditions are
per se weaker than sufficient ones.
Several papers were written discussing the exploding algebraic
difficulty in finding sufficient sets of Bell inequalities
[Peres99]. For example Peres gives as a number of the so called
Farkas vectors (effectively coefficients in generalised Clauser-Horne
inequalities) for a two qu$N$it experiment with only two observables
(non-degenerate) allowed to be measured at each side [Peres99].
It reads $N_{F}(N)=(2^{N}-2)^{4}$, i.e., one has $N_{F}(2)=16,N_{F}(3)=1296$ etc.
This causes that the computational time grows extremely fast not
allowing
one to calculate anything in reasonable time.
However if one asks directly the question whether the
probabilities given by quantum mechanics are describable via a local
realistic theory, it turns out that this question can be formulated
as a typical linear optimisation problem. Since excellent
numerical methods exist it turns out
that for the number of settings of the order of 10 for two qubits and of
the order of 4 for three qubits, and for
qu$N$its with $N$ up to 10, such problems are solvable
on a standard work-station with the computing time of the
order of one day.
The algorithm to tackle these problems together with
the results will be
presented here.
The threshold noise,
or in other words visibility, below which333In terms of the
visibility above which. there is no
local realistic model for two and three maximally entangled qubits
is of the same value as in the case of standard
Bell inequalities (involving only two settings at each side).
Nevertheless, the presented approach gives a new method of a direct
analysis of experimental data. Such data can be directly tested for
a possibility of a local realistic model of them (without any of the usual
hypothesis about the curves best fitting to the data).
In the case of the second problem, entangled qu$N$its, new surprising
results have been obtained. The computer data reveal growing discrepancy
of the quantum predictions with local realism with the increasing
dimensionality of the Hilbert space of the sub-systems. Earlier
approaches suggested contrary behaviour (see , for instance,
[Mermin80, Mermin82]).
Chapter 8 Necessary and sufficient conditions to
violate local realism for two maximally entangled qubits-
extension to more than two local settings [7]
8.1 Introduction
It is a common wisdom in the quantum optical community that the
threshold visibility of the sinusoidal two-qubit interference
pattern beyond which the Bell inequalities are violated is (for the
case of perfect detectors) $\frac{1}{\sqrt{2}}$ (see, e. g.
[Clauser78]). Most of the experiments exceed that limit (with the usual
“fair sampling assumption”) [Freedman72, Ou88, Rarity90, Kwiat93, Pittman95, Tittel98].
Some difficulties to
reach this threshold were observed in the very early experiments
[Clauser78], as well as in some recent ones involving novel techniques.
Thus far, in atomic interferometry EPR experiments [Hagley97] and
for the phenomenon of entanglement swapping [Żukowski95, Pan98], the
resulting visibility is less than the magic $71\%$.
It is also well known that the Clauser-Horne inequality and the CHSH
inequality are not only necessary conditions for the existence of
local realistic models but also sufficient [Fine82] (in the case of
the CHSH inequality, this requires some simplifying assumptions
[Garg87]). However, the sufficiency proofs used involve only
two pairs of settings of the local macroscopic parameters (e.g.
orientations of the polarisers) that define the measured local
observables. Thus, the constructions are valid for precisely those
settings and nothing more, and there is no guarantee that the models
can be extended to more settings. Consequently one may ask what is
the maximal visibility for a model applicable to all possible
settings of the measuring apparata, that returns sinusoidal
two qubit interference fringes. It is already known that for
perfect detectors, this value cannot be higher than
$\frac{1}{\sqrt{2}}$ or lower than $2/\pi$ (this is the visibility of
the recent ad hoc model by Larsson [Larsson99]; for earlier
models returning visibilities of $50\%$ see e.g. [Wódkiewicz91]).
The knowledge of the maximal visibility of sinusoidal two-qubit
fringes in a Bell-type experiment that still can be fully modelled in a
local realistic way, may help us to distinguish better between ‘local’
and ‘nonlocal’ density matrices. For two two-state systems one can
find precise conditions which have to be satisfied by density matrices
describing the general state, pure or mixed, of the full system, that
enable violation of the CHSH inequalities [Horodecki95]. States
fulfilling such conditions are often called “nonlocal”. However,
since the CHSH inequality is necessary and sufficient only for two
pairs of settings, it is not excluded a priori that some states
that satisfy such inequalities for all possible sets of two pairs of
local dichotomic observables, nevertheless give predictions that in
their entirety cannot be modelled by local hidden variables. Such
models must first of all reproduce the full continuous sinusoidal
variation of two-qubit interference fringes, as well as the other
predictions.
It is clear that the full solution of the question would require a
construction, or a proof of existence, of local hidden variable models
which return sinusoidal fringes of the maximal possible visibility,
that are applicable for all possible settings of the measuring
apparata. Since this seems to be very difficult, we chose a numerical
method of point wise approximation at a finite number of settings at
each side of the experiment. Due to exponential growth of the
computation time when the number of settings increases, we managed to
reach up to 10 settings on each side, i.e. up to 100 measurements points
(which due to a certain symmetry, about which we will say more later,
effectively can be transformed into $20\times 20=400$ points). The
exponential growth hinders any substantial increase in this
number. Such numerical models cannot give a definite
answer concerning the critical visibility of sinusoidal fringes,
however our calculations enable us to put forward a strong conjecture
that this value must be indeed ${1\over\sqrt{2}}$ (see below). The
numerical method presented in this chapter is applied to the correlation
function for two qubits. The alternative approach, much better suited for
general problems is to apply the method to probabilities
of all events that are observed in the experiment. However, for maximally
entangled two qubit state, with the interference visibility reduced by some
noise (the generic problem studied here) application of numerical
approach to correlation function is equivalent to a similar procedure
with probabilities, and what is important due to the fact that
it leads to a less complicated linear optimisation problem
it allows to have numerical results for much more settings on each
side than the method involving probabilities. Nevertheless the algorithm
involving probabilities was also written, and it returns identical
results. It will be presented in the chapter on two entangled qu$N$it
problem.
Experimentally our problem can be formulated in the following way: the
two qubit state produced by the source does not allow for single
qubit interference, and in the experiment less-than-perfect
two-qubit fringes are obtained, due to some fundamental limitations
(like those present in the case of entanglement swapping, e.g.
[Żukowski95, Pan98]) or due to imperfections of the devices. What is the
critical two qubit interference visibility beyond which the
observed process falsifies local realism? We shall ask these questions
assuming, for simplicity, perfect detection efficiencies, which is
possible theoretically, and experimentally thus far amounts to the
usual “fair sampling assumption”.
Furthermore, the problem may be investigated without the use of the
assumption that the observed fringes are of a sinusoidal nature even
though the observed two-qubit fringes in experiments with high
photon counts follow almost exactly the sinusoidal curves
[Freedman72, Ou88, Rarity90, Kwiat93, Pittman95, Tittel98]. In experiments with lower count rates, still with
relatively good level of confidence the recorded data have
approximately the same character, and it is customary to fit them with
sinusoidal curves. It is now a standard procedure to perform the
two-qubit interference experiments by recording many points of the
interference pattern, rather than stabilising the devices at
measurement settings appropriate for the best violation of some Bell
inequality. Further, in some of the experiments, e.g. those involving
optical fibre interferometers it was not possible to
stabilise the phase differences and what is observed is just the
interference pattern changing in time, and the visibility of the
sinusoidal two-qubit fringes is used as the critical parameter
[Tapster94].
Even though the numerical calculations presented here only reach
$20\times 20$ points, this is more than enough in comparison to the
experimental data. The usual experimental scans rarely
involve more than 20 points. Further, our
algorithm can be applied directly to the
measurement data, and in that way one can even avoid the standby
hypothesis that the fringes follow a sinusoidal pattern. The
algorithm can directly answer the question: are the data compatible
with local realism or not? Since physics is an experimental science,
the questions about Nature get their final answers solely in this way.
8.2 Quantum mechanical and local realistic
description of the gedanken experiment
We consider the following gedanken experiment. Two observers
$a$ and $b$ perform measurements of observables
$\vec{a}\cdot\vec{\sigma}$ (observer $a$) and
$\vec{b}\cdot\vec{\sigma}$ (observer $b$) on the state
defined by the equation (2.1). Observer $a$ can choose
between $N_{a}$ settings of the measuring apparatus, which
are defined by vectors $\vec{a}_{i}$ ($i=1,\dots,N_{a}$)
whereas observer $b$ can choose between $N_{b}$ settings
of the measuring apparatus defined by vectors $\vec{b}_{j}$
($j=1,\dots,N_{b}$). Therefore, they perform $N_{a}\times N_{b}$
mutually exclusive experiments. For each experiment
one has the following quantum probabilities (see
the equation (6.1))
$$\displaystyle P_{QM}^{V_{2}}(l,m|\vec{a}_{i},\vec{b}_{j})={1\over 4}(1-lmV_{2}%
\vec{a}_{i}\cdot\vec{b}_{j})$$
(8.1)
and the quantum correlation function
$$\displaystyle E_{QM}^{V_{2}}(\vec{a}_{i},\vec{b}_{j})=-V_{2}\vec{a}_{i}\cdot%
\vec{b}_{j}.$$
(8.2)
There is no one qubit interference, i.e.,
$P_{QM}^{V_{2}}(l|\vec{a}_{i})=P_{QM}^{V_{2}}(m|\vec{b}_{j})={1\over 2}$
for all $i,j$.
If one assumes that the unit vectors which
define the measured observables are always coplanar the correlation
function can be simplified to $E^{V_{2}}_{QM}(\alpha_{i},\beta_{j})=-V_{2}\cos{(\alpha_{i}+\beta_{j})}$
(with the obvious definition of $\alpha_{i}$
and $\beta_{j}$).
Please note that in the considered here
gedanken experiment there is one-to-one equivalence111
This is possible due to ”well” defined values which we ascribe to
the results of measurements (here $\pm 1$) and symmetries
exhibited by the quantum probabilities. In the case of qu$N$its
with $N=3$ we will see that the Bell number assignment (see the chapter
on GHZ paradoxes for qu$N$its) also enables us to use the
quantum correlation
function (7.19) instead of the probabilities.
between the quantum correlation function and quantum
probabilities (see
(8.1) and (8.2)).
Therefore,
we can use either the quantum correlation function
or the quantum probabilities to describe the experiment.
However, the description of the experiment in terms
of the quantum correlation function is more convenient
from the numerical point of view,
which we will be seen
further.
From the numerical values of the quantum correlation
function we form a certain matrix $\hat{Q}_{V_{2}}$
of quantum predictions
with the entries:
$Q_{ij}(V_{2})=E^{V_{2}}_{QM}(\vec{a}_{i},\vec{b}_{j})$.
Within the local hidden variables formalism the correlation
function must have
the following structure
$$E_{LHV}(\vec{a}_{i},\vec{b}_{j})=\int d\lambda\rho(\lambda)A(\vec{a}_{i},%
\lambda)B(\vec{b}_{j},\lambda),$$
(8.3)
where for dichotomic measurements
$$\displaystyle A(\vec{a}_{i},\lambda)=\pm 1$$
(8.4)
and
$$\displaystyle B(\vec{b}_{j},\lambda)=\pm 1,$$
(8.5)
and they represent the values of local
measurements predetermined by the local hidden variables, denoted by $\lambda$, for the
specified local settings. This expression is an average over a certain
local hidden variables distribution $\rho(\lambda)$ of certain factorisable matrices,
namely those with elements given by
$M_{ij}(\lambda)=A(\vec{a}_{i},\lambda)B(\vec{b}_{j},\lambda)$. The
symbol $\lambda$ may hide very many parameters. However, since the
only possible values of $A(\vec{a}_{i},\lambda)$ and
$B(\vec{b}_{j},\lambda)$ are $\pm 1$ there are only $2^{N_{a}}$ different sequences of the values of $(A(\vec{a}_{1},\lambda),A(\vec{a}_{2},\lambda),...,A(\vec{a}_{N_{a}},\lambda))$, and only $2^{N_{b}}$
different sequences of $(B(\vec{b}_{1},\lambda),B(\vec{b}_{2},\lambda),...,B(\vec{b}_{N_{b}},\lambda))$
and consequently they form only $2^{N_{a}+N_{b}}$ matrices $\hat{M}(\lambda)$.
Therefore the structure of local hidden variables models of $E_{LHV}(\vec{a}_{i},\vec{b}_{j})$ reduces to discrete probabilistic models involving the
average of all the $2^{N_{a}+N_{b}}$ matrices $\hat{M}(\lambda)$. In other
words, the local hidden variables can be replaced, without any loss of generality222See, for instance,
[Wigner70, Belinfante73]., by a
certain pair of variables $k$ and $l$ that have integer values
respectively from $1$ to $2^{N_{a}}$ and from $1$ to $2^{N_{b}}$. To each $k$
we ascribe one possible sequence of the possible values of
$A(\vec{a}_{i},\lambda)$, denoted from now on by $A(\vec{a}_{i},k)$,
similarly we replace $B(\vec{b}_{j},\lambda)$ by $B(\vec{b}_{j},l)$.
With this notation the possible local hidden variables models of the correlation function
$E_{LHV}(\vec{a}_{i},\vec{b}_{j})$ acquire the following simple form
$$E_{LHV}(\vec{a}_{i},\vec{b}_{j})=\sum_{k=1}^{2^{N_{a}}}\sum_{l=1}^{2^{N_{b}}}p%
_{kl}A(\vec{a}_{i},k)B(\vec{b}_{j},l),$$
(8.6)
with, of course, the probabilities satisfying $p_{kl}\geq 0$ and
$\sum_{k=1}^{2^{N_{a}}}\sum_{l=1}^{2^{N_{b}}}p_{kl}=1$.
The special case that we study here enables us to simplify the
description further. To satisfy the additional requirement that the
local hidden variables model returns the quantum prediction of equal probability of the
results at the local observation stations, that $P_{QM}^{V_{2}}(l|\vec{a})=P_{QM}^{V_{2}}(m|\vec{b})={1\over 2}$, one can use the following observation.
For each $k$, there must exist a $k^{\prime}\neq k$ with the property that
$A(\vec{a}_{i},k^{\prime})=-A(\vec{a}_{i},k)$, and similarly for each $l$, there
must exist an $l^{\prime}\neq l$ for which
$B(\vec{b}_{j},l^{\prime})=-B(\vec{b}_{j},l)$. Then
$A(\vec{a}_{i},k)B(\vec{b}_{j},l)=A(\vec{a}_{i},k^{\prime})B(\vec{b}_{j},l^{%
\prime})$, and
thus they give exactly the same matrix of local hidden variables predictions. By assuming
$p_{kl}=p_{k^{\prime}l^{\prime}}$ the property of total randomness of local results
will always be reproduced by the local hidden variables models, and the generality will
not be reduced since the contributions of $p_{kl}$ and $p_{k^{\prime}l^{\prime}}$ to
(8.6) cannot be distinguished.
Hence, we will always take only one
representative of the two pairs, reducing in this way the number of
probabilities and matrices of LHV predictions in (8.6) by a
factor of two333The method involving directly probabilities
of all pairs of events avoids this artificiality. However, the
price paid is a much longer computational time. Nevertheless,
we have performed also such a calculation (see the last chapter).
Of course the results of the presented calculation were confirmed..
This is equivalent to taking only half of different
$B(\vec{b}_{j},l)$ (or $A(\vec{a}_{i},k)$ for the situation
is symmetrical), say, first $2^{N_{b}-1}$ ones. Thus, (8.6)
acquires the form
$$\displaystyle E_{LHV}(\vec{a}_{i},\vec{b}_{j})=\sum_{k=1}^{2^{N_{a}}}\sum_{l=1%
}^{2^{N_{b}-1}}\tilde{p}_{kl}A(\vec{a}_{i},k)B(\vec{b}_{j},l),$$
(8.7)
where we still assume that $\sum_{k=1}^{2^{N_{a}}}\sum_{l=1}^{2^{N_{b}-1}}\tilde{p}_{kl}=1$.
Another reduction by a factor of four is given by the fact that in the
coplanar case, the choice of the settings may be limited on each side
to ranges not greater than $\pi$ (i.e. $\phi\leq\alpha_{i}\leq\phi+\pi,\psi\leq\beta_{j}\leq\psi+\pi$). This is due to the simple
observation that a model of the type (8.6) once established
for such settings can be easily extended to settings
$\alpha_{i}^{{}^{\prime}}=\alpha_{j}+\pi,\beta_{j}^{{}^{\prime}}=\beta_{j}+\pi$ by
putting $A(\alpha_{i}^{{}^{\prime}},l)=-A(\alpha_{i},l)$ and
$B(\beta_{j}^{{}^{\prime}},l)=-B(\beta_{j},l)$.
The conditions for local hidden variables to reproduce the quantum prediction with a
final visibility $V_{2}$ can be simplified to the problem of maximising a
parameter $V_{2}$ for which exists a set of $2^{N_{a}+N_{b}-1}$ probabilities
$\tilde{p}_{kl}$, such that
$$\sum_{k=1}^{2^{N_{a}}}\sum_{l=1}^{2^{N_{b}-1}}\tilde{p}_{kl}A(\vec{a}_{i},k)B(%
\vec{b}_{j},l)=Q_{ij}(V_{2}).$$
(8.8)
Because, for the given local settings
(8.8)
imposes linear constraints on the probabilities and the
visibility444The visibility should also
fulfil $V_{2}\leq 1$.,
and we are looking for the maximal $V_{2}$, the problem
can be solved by means of linear programming-
a certain method of optimisation.
8.3 Linear programming and Downhill Simplex Method
Let us briefly describe the idea of linear programming sending more
interested readers to the excellent book by Gass [Gass75].
The set of linear equations (8.8) constitute a certain region
in a $D=2^{N_{a}+N_{b}-1}+1$ dimensional real space- $2^{N_{a}+N_{b}-1}$ probabilities
plus the visibility. The border of the region
consists of hyper planes each defined by one of the equations belonging
to (8.8); thus, if the equations do not contradict each other
the region is a convex set with a certain number of vertices.
On this convex set we define a linear function (cost function)
$f(p_{1},\cdots,p_{2^{N_{a}+N_{b}-1}},V_{2})=V_{2}$, which maximum we seek.
The fundamental
theorem of linear programming states that the cost function
reaches its maximum at one of the vertices. Hence, it suffices to
find numerical values of the cost function calculated at the vertices and
then pick up the largest one. Of course, the algorithmic implementation
of this simple idea is not so easy for we must have a method of finding
the vertices for which the value of the function continually
increases555We
must also know how to find a starting vertex. so
that the program
reaches the optimal solution in the least possible number of steps.
Calculating the value of the cost function
at every vertex would take too much time as there may be too many
of them.
There are lots of excellent algorithms which solve the above optimisation
problem. Here we have used the algorithm invented by Gondzio [Gondzio95]
and implemented in the commercial code HOPDM 2.30 (Higher Order Primal-Dual
Method) written in C programming language.
However, finding the maximal visibility for the given local settings
of the measuring apparata is not enough. We should remember that our main goal
is to find such local setting for which the threshold visibility is the lowest
one. The maximal visibility $V^{max}_{2}$ returned by the HOPDM 2.30 procedure
depends on the local settings entering right hand side of (8.8).
Thus, returned $V^{max}_{2}$
can be treated as the many variable function, which depends on $N_{a}+N_{b}$
angles in the coplanar case and two times more in the non coplanar one, i.e.,
$V_{2}^{max}=V_{2}^{max}(\vec{a}_{1},\dots,\vec{a}_{N_{a}},\vec{b}_{1},\dots,%
\vec{b}_{N_{b}})$.
Hence, we should also have a numerical procedure which finds the minimum
of $V^{max}_{2}$.
Because we do not know much about the structure of $V^{max}_{2}$ as a function
of the local settings
the only reasonable
method of finding the $V^{max}_{2}$ minimum is the Downhill Simplex Method (DSM)
[Nelder65]. The way it looks for the extremum is the following. If the
dimension of the domain of a function is $Dim$ the DSM randomly generates
$Dim+1$ points. This way it creates a starting simplex, which vertices are
these points. Then it calculates the value of a function at the vertices and
starts exploring the space by stretching and contracting the simplex. In every
step when it
finds a vertex where the value of the function is lower then in others it
”goes” in this direction. For more elaborate discussion of this issue see
[Nelder65].
8.3.1 Numerical difficulties
Obviously there are some numerical limitations
of our computer program. First of all it is obvious that the time
needed for computation grows with the number of local settings $N_{a},N_{b}$.
For the given $N_{a},N_{b}$ we have $2^{N_{a}+N_{b}-1}+1$ unknowns and $N_{a}\times N_{b}+2$
linear constraints imposed on them. Furthermore, the dimension
of the domain of $V^{max}_{2}$ is in general $2N_{a}+2N_{b}$, which makes it
harder for the DSM procedure to find a global minimum of $V^{max}_{2}$.
Secondly, the amount of the memory needed to store ”hidden”
matrices grows exponentially with $N_{a}$ and $N_{b}$. All these problems
taken together has limited our research
to the cases where $N_{a}+N_{b}\leq 20$.
It is worth
mentioning at this point about the advantages of using the
quantum correlation function instead of quantum probabilities as the
description of the gedanken experiment. If we are to use
probabilities666For the details concerning this approach see
the chapter about pairs of entangled qu$N$its. we will have
$2^{N_{a}+N_{b}}+1$ unknowns ($2^{N_{a}+N_{b}}$ probabilities plus the
visibility $V_{2}$) and $4\times N_{a}\times N_{b}+1$ constraints.
Because we struggle with
the memory capacity and the computation time every reduction in the number
of unknowns and constraints is a great advantage.
Recently A. Peres [Peres99] has discussed the
algorithms which search for so-called Farkas vectors,
which in turn define coefficients in generalised Bell-inequalities,
the set of which is a sufficient and necessary condition for
classical probabilistic model (here, essentially, local realistic)
to reproduce a certain set of probabilities for pairs of
experiments. However, his method explodes numerically much much
faster then ours. Simply, our method is applied directly to a
certain finite set of specified quantum probabilities777
Instead of the probabilities we use the correlation function but
this does not change anything because
of the mentioned equivalence of these two ways of description of
the considered gedanken experiment.. Whereas
inequalities based on the Farkas lemma apply to all possible sets of probabilities.
8.4 Results
First of all, we have performed calculations in the coplanar case.
We have checked situations
in which the number of settings for observer $a$ and $b$ is the same, i.e.,
$N_{a}=N_{b}=2,3,4,\dots,10$
as well as the cases
in which the number of settings for observer $a$ and $b$ is different, i.e.,
$N_{a}\neq N_{b}$.
Because of mentioned limitations
of the computer program we could
only reach the limit of $N_{a}+N_{b}=20$ in both cases, i.e., for the equal
and unequal number of the settings at each side.
Because of the richness of $N_{a}+N_{b}$ dimensional space (coplanar case)
to
find a global minimum of $V^{max}_{2}$ for the cases where
$N_{a}+N_{b}\leq 12$ we have run the DSM
procedure $30$ times with varied starting points. For the higher
dimensional ones, i.e., where
$13\leq N_{a}+N_{b}\leq 15$ we have run the DSM only a few times whereas
for the extremal cases, i.e., where
$16\leq N_{a}+N_{b}\leq 20$ we have not used the DSM but we simply have
calculated $V^{max}_{2}$ on some
sets of angles containing (or not) the Bell angles (the
definition of the Bell angles is given below).
For $N_{a}+N_{b}\leq 15$ the DSM has always converged to
$V^{max}_{2}={1\over\sqrt{2}}$ within the given numerical precision
of computation.
Furthermore, within the angles
$\alpha_{i},\beta_{j}$ ($i=1\dots N_{a},j=1\dots N_{b}$) for which the optimum has been achieved
there has been always the
subset of four ones (we call the angles belonging to this subset
the Bell angles)
for which the maximal violation of the CHSH inequality
occurs. In other words it means that in the quantum matrix $\hat{Q}$
(matrix of results for the visibility equal one)
with entries calculated for the optimal local settings found by the DSM,
i.e., settings for
which $V_{2}^{max}$ reaches its minimum,
a $2\times 2$ sub matrix $\hat{B}$ appears with moduli of all its
elements equal to ${1\over\sqrt{2}}$, and three of them of the same sign,
e.g.
$$\displaystyle\hat{B}=\left(\begin{array}[]{cc}-{1\over\sqrt{2}}&-{1\over\sqrt{%
2}}\\
-{1\over\sqrt{2}}&+{1\over\sqrt{2}}\end{array}\right).$$
(8.9)
Of course, the elements of $\hat{B}$ can be scattered throughout
the matrix $\hat{Q}$ but by relabelling the indices it is always possible,
for instance,
to have $\hat{B}$ in the left hand corner of $\hat{Q}$.
For the combinations of the number of local settings at each side
of the gedanken experiment such that $16\leq N_{a}+N_{b}\leq 20$
we have calculated $V^{max}_{2}$ on the sets of
local settings including the Bell angles with exactly the same result, i.e.,
$V_{2}^{max}={1\over\sqrt{2}}$ whereas for the sets of the settings not including
the Bell angles $V_{2}^{max}>{1\over\sqrt{2}}$.
We have also tested non coplanar settings with exactly the same result.
The visibility was higher than ${1\over\sqrt{2}}$ if among the
local settings were no the Bell angles (no sub matrix $\hat{B}$)
and it was exactly ${1\over\sqrt{2}}$ otherwise (sub matrix
$\hat{B}$ present).
8.4.1 Exemplary numerical model
Let us show an example of the computer solution for
the $3\times 3$ coplanar case with
the optimisation
over local settings of the measuring apparata (the DSM has been used). The threshold
visibility for the optimal settings found by the program is $V^{max}_{2}=0.707$
whereas the quantum matrix
is given below
$$\displaystyle\hat{Q}_{V^{max}_{2}}=V^{max}_{2}\hat{Q}=V^{max}_{2}\left(\begin{%
array}[]{ccc}0.107&-0.994&-0.399\\
-0.707&-0.707&-0.964\\
-0.707&0.707&-0.266\end{array}\right).$$
(8.10)
The computer hidden variable model reproducing the
above matrix reads
$$\displaystyle\tilde{p}(-1-1-1;+1+1+1)=0.250$$
$$\displaystyle\tilde{p}(+1+1-1;+1-1+1)=0.060$$
$$\displaystyle\tilde{p}(+1+1-1;+1-1-1)=0.135$$
$$\displaystyle\tilde{p}(-1-1+1;+1+1+1)=0.157$$
$$\displaystyle\tilde{p}(+1-1+1;+1+1+1)=0.026$$
$$\displaystyle\tilde{p}(+1-1+1;+1+1-1)=0.067$$
$$\displaystyle\tilde{p}(-1+1-1;+1-1-1)=0.055$$
$$\displaystyle\tilde{p}(+1-1-1;+1-1+1)=0.217$$
$$\displaystyle\tilde{p}(+1-1-1;+1-1-1)=0.033,$$
(8.11)
where, for instance, $\tilde{p}(+1+1-1;+1-1+1)$ denotes
the probability of appearing the factorisable matrix $\hat{M}$
$$\displaystyle\hat{M}=\left(\begin{array}[]{ccc}+1&-1&+1\\
+1&-1&+1\\
-1&+1&-1\end{array}\right)=\left(\begin{array}[]{c}+1\\
+1\\
-1\end{array}\right)\left(\begin{array}[]{c}+1\\
-1\\
+1\end{array}\right)^{T}$$
(8.12)
($T$ denotes ordinary transposition).
This example clearly exhibits the characteristic
trait which has
been mentioned already, namely that in the matrix $\hat{Q}$
there is
a sub matrix $\hat{B}$ corresponding to the matrix obtained for the Bell
angles-
here the elements $Q_{21},Q_{31},Q_{32},Q_{22}$.
The numerical precision of the entries
of the matrix $\hat{Q}$ and the visibility $V^{max}_{2}$ is in this
example $10^{-3}$.
Of course the precision can be increased when necessary but this
makes the time of computation longer. However, in some cases we
have run the program with the numerical precision of order $10^{-6}$
with the same result, i.e., the $V_{2}^{max}=0.707107\pm 10^{-6}$.
8.5 Application to experimental data
To apply our computer program for analysis of experimental data we replace
$Q_{ij}(V_{2})$ in (8.8) by the measured values
$E^{QM}_{ij}(exp)$, and perform the same task. If the critical $V_{2}$
returned by the program888In this case $V_{2}$
does not have the direct interpretation of visibility. Its
value tells us by what factor the observed values of the correlation
function have to be reduced, so that a local hidden variables model
exists. is less than $1$, the data cannot
be reproduced by any local hidden variable model. Note that one even
does not have to know what the settings are.
In a recent Bell-type experiment Weinfurter and Michler999The experimental
results were reported in a different context in [Michler00].
have
obtained the following matrix of results
$$\hat{Q}^{exp}=\left(\begin{array}[]{ccc}-0.894&-0.061&0.761\\
-0.851&0.343&0.765\\
-0.625&0.688&0.516\\
-0.251&0.860&0.103\\
0.226&0.921&-0.389\\
0.530&0.651&-0.648\\
0.855&0.323&-0.832\\
0.852&-0.092&-0.843\\
0.785&-0.539&-0.638\\
0.397&-0.795&-0.253\\
\end{array}\right).$$
(8.13)
The program gives the verdict that the values of all entries to the
matrix of results have to be reduced by the factor of $0.796$ to be
describable by local hidden variables.
Some explanation is needed here. The entries of (8.13)
give the values of the correlation function - there were
three different setting on side A of the experiment and 27 settings
on side B, however only 10 of them are shown here. In the actual
experiment only data from a pair of detectors were collected. To
obtain the matrix we have used the usual assumption that
$E(\alpha,\beta)=4P(+,+;\alpha,\beta)-1.$ We have also renormalised
the numbers of photon pairs counted, so that the average of the
counts over approximately two periods of settings at side B
represents the probability of $\frac{1}{4}$ (in concurrence with the
quantum prediction). To this end we have used data for all 27 settings on
side B.
In the recent long-distance EPR-Bell experiment the following set of
values of the correlation function was obtained101010The numerical
values of the entries
to the matrix were provided by G. Weihs (private communication).:
$$\hat{Q}^{exp}=\left(\begin{array}[]{cc}0.960&-0.102\\
0.903&-0.375\\
0.733&-0.660\\
0.479&-0.809\\
0.191&-0.903\\
-0.120&-0.923\\
-0.429&-0.807\\
-0.666&-0.656\\
-0.842&-0.395\\
-0.951&-0.152\\
-0.953&0.171\\
\end{array}\right).$$
(8.14)
This matrix has to be reduced by the factor of $0.737$ to have a
local realistic description.
8.6 Conclusions
In his paper Peres [Peres99] has conjectured that if the CH
inequalities are satisfied for all possible subsets of two
settings on one side and two settings on the other (out of all
$N_{a}\times N_{b}$ settings) then this is sufficient for a local realistic
model for all $N_{a}\times N_{b}$ settings. Our numerical calculations
strongly support this conjecture.
Furthermore, for the situations characterised by the condition
$N_{a}+N_{b}\leq 12$ we have found the ultimate value of the visibility still
admitting the existence of local hidden variables. Surely, we must be aware of the fact
that the DSM may have not found
a global minimum of $V^{max}_{2}$, in which the case the threshold $V_{2}$ would be
lower than ${1\over\sqrt{2}}$. However, we think that this is highly
improbable, which is strongly supported
by calculations for $N_{a}+N_{b}>12$.
As it has been shown on two examples our computer program
can be also used to analyse directly the raw experimental data.
Here the analysis in terms of the correlation function
has been performed. However, this is not necessary. In the
chapter concerning two maximally entangled qu$N$its (including
also qubits) we
will present a computer program entirely based on the quantum
probabilities rather then on the correlation function. This is
important because the one-to-one equivalence between the correlation
function and probabilities is valid only if one assumes that certain
symmetries concerning probabilities are present
(see (8.1)). In the real experiment one
may expect that these symmetries are not always fulfilled. The program
based on probabilities utterly solves this problem allowing us to
analyse real experiments basing solely on the observed clicks
of detectors.
To conclude, the performed calculations enable us to put forward the
following conjecture:
sinusoidal two-qubit fringes of visibility up to
$\frac{1}{\sqrt{2}}$ are describable by local realistic theories. At
this stage we are not able to give an analytic proof of the above.
However, for finite sets of measurement points the results of data
analysis with the use of our program fully concur with this
hypothesis. One can even say that this is more than enough, since
one cannot experimentally test exact sinusoidal nature of the two
qubit fringes. The real output of an experiment are count rate
sequences at finite number of the measurement points which
follow sinusoidal-like pattern. To such data our program
can be directly applied giving verdict whether the data
admit a local realistic model or not.
Chapter 9 Necessary and sufficient conditions to
violate local realism for three maximally entangled qubits-
extension to more than two local settings
Exactly the same question
on the threshold visibility allowing local and realistic
description for all positions of measuring apparata
can be posed in the case of correlations involving three qubits instead
of two ones. Up to now, the threshold visibility for three qubit
correlations (GHZ correlations) is known to be $V_{2}(3)={1\over 2}$ (in
the brackets we show that we deal with the visibility for three qubits to
distinguish it from the visibility for two qubits $V_{2}$.)
[Mermin90b].
However, this limit has been established for the case when there are two local
settings of the measuring apparatus at each side
of the experiment and it has been obtained in a standard
way, i.e., as
a condition for the violation of an appropriate Bell inequality.
From the previous section we know that there is a more direct method of finding
the numerical value of the threshold visibility, which in addition
gives necessary
and sufficient
conditions for the existence of local hidden variables for the given set of
local settings for each
observer.
In this section we show the application of the presented numerical method to
the GHZ correlations, i.e., to the maximally entangled state of three
qubits. Surprisingly, the obtained results bear great resemblance to
those for two qubits.
9.1 Description of the method
To this end let us consider the following maximally entangled state of
3 qubits
$$\displaystyle|\psi\rangle={1\over\sqrt{2}}(|0\rangle_{1}|0\rangle_{2}|0\rangle%
_{3}+|1\rangle_{1}|1\rangle_{2}|1\rangle_{3})$$
(9.1)
where $|i\rangle_{j}$ is the $i$-th state of the $j$-th qubit.
Each observer measure the observable
$\vec{n}\cdot\vec{\sigma}$, where $n=a,b,c$ ($a$ for the first
observer, $b$ for the second one and $c$ for the third one), $\vec{n}$
is a unit vector
characterising the observable which is measured by observer $n$ and
$\vec{\sigma}$ is a vector the components of which are standard Pauli
matrices. As in the case of two qubits the family of observables
$\vec{n}\cdot\vec{\sigma}$
covers all possible dichotomic observables for a qubit
system, endowed with a spectrum consisting of $\pm 1$.
The probability of obtaining
the result $m=\pm 1$ for the observer $a$, when measuring the
observable characterised by the vector
$\vec{a}$, the result $l=\pm 1$ for the observer $b$, when measuring the
observable characterised by the vector
$\vec{b}$ and the result $k=\pm 1$ for the observer $c$, when measuring the
observable characterised by the vector
$\vec{c}$ is equal to
$$\displaystyle P_{QM}(m,l,k;\vec{a},\vec{b},\vec{c})={1\over 8}(1+mla_{3}b_{3}+%
mka_{3}c_{3}+lkb_{3}c_{3}$$
$$\displaystyle+mlk\sum_{r,p,s=1}^{3}M_{rps}a_{r}b_{p}c_{s}),$$
(9.2)
where $a_{r},b_{p},c_{s}$ are components of vectors $\vec{a},\vec{b},\vec{c}$
and where nonzero elements of the tensor $M_{rps}$ are $M_{111}=1,M_{122}=-1,M_{212}=-1,M_{221}=-1$. In spherical coordinates vectors $\vec{a},\vec{b},\vec{c}$ read
$$\displaystyle\vec{n}=(\cos\phi_{n}\sin\theta_{n},\sin\phi_{n}\sin\theta_{n},%
\cos\theta_{n}),$$
(9.3)
where $\theta_{n}\in[0,\pi],\phi_{n}\in[0,2\pi]$. From now on
we will be
considering only the measurement of the observables characterised
by vectors with the zero third component, which is equivalent to
putting $\theta_{n}=\pi/2$. Thus, the formula (9.2)
acquires simpler form (we have replaced $\phi_{a},\phi_{b},\phi_{c}$
by $\alpha,\beta,\gamma$ respectively)
$$\displaystyle P_{QM}(m,l,k;\alpha,\beta,\gamma)={1\over 8}(1+mlk\sum_{r,p,s=1}%
^{3}M_{rps}a_{r}b_{p}c_{s})$$
(9.4)
in which only a term responsible for three qubit correlations
is present.
The probabilities of obtaining one of the results in the local
stations reveal no dependence on the local parameters,
$P_{QM}(l|\alpha)=P_{QM}(m|\beta)=P_{QM}(n|\gamma)={1\over 2}$. Similarly, the
probabilities describing two qubit correlations do not
reveal dependence on the local parameters either,
$P_{QM}(l,m|\alpha,\beta)=P_{QM}(m,n|\beta,\gamma)=P_{QM}(l,n|\alpha,\gamma)={1%
\over 4}$.
As usual, if $V_{2}(3)<1$ we replace (9.4) by
$$\displaystyle P_{QM}^{V_{2}(3)}(m,l,k|\alpha,\beta,\gamma)={1\over 8}(1+mlkV_{%
2}(3)\sum_{r,p,s=1}^{3}M_{rps}a_{r}b_{p}c_{s}).$$
(9.5)
The quantum prediction
for the correlation function with reduced visibility reads:
$$\displaystyle E_{QM}^{V_{2}(3)}(\alpha,\beta,\gamma)=\sum_{m,l,k=-1}^{1}mlkP(m%
,l,k;\alpha,\beta,\gamma)$$
$$\displaystyle=V_{2}(3)\sum_{r,p,s=1}^{3}M_{rps}a_{r}b_{p}c_{s}=V_{2}(3)\cos(%
\alpha+\beta+\gamma).$$
(9.6)
and there is no single and two qubit interference111We again
use the quantum correlation function instead of quantum probabilities. In
the case considered here these two approaches are equivalent- see the
discussion in the previous chapter..
Now, we proceed analogously to the case of two qubits. Observer
$a$ chooses between $N_{a}$ settings of the measuring
apparata $\alpha_{1},\dots,\alpha_{N_{a}}$, observer
$b$ between $N_{b}$ settings $\beta_{1},\dots,\beta_{N_{b}}$
and ,finally, observer
$c$ between $N_{c}$ settings $\gamma_{1},\dots,\gamma_{N_{c}}$.
For each triple of local
settings we calculate
the quantum correlation function (9.6) $E_{QM}^{V_{2}(3)}(\alpha_{i},\beta_{j},\gamma_{k})$, where $i=1,\dots,N_{a},j=1,\dots,N_{b},k=1,\dots,N_{c}$. Thus we have a tensor222We call it
a tensor for it has three indices. This has nothing in common
with the transformation properties of this object.
$Q_{ijk}(V_{2}(3))=E_{QM}^{V_{2}(3)}(\alpha_{i},\beta_{j},\gamma_{k})$
of quantum predictions.
Within the local hidden variables formalism the correlation
function must have
the following structure
$$E_{LHV}(\alpha_{i},\beta_{j},\gamma_{k})=\int d\rho(\lambda)A(\alpha_{i},%
\lambda)B(\beta_{j},\lambda)C(\gamma_{k},\lambda),$$
(9.7)
where for dichotomic measurements
$$\displaystyle A(\alpha_{i},\lambda)=\pm 1$$
$$\displaystyle B(\beta_{j},\lambda)=\pm 1$$
$$\displaystyle C(\gamma_{k},\lambda)=\pm 1,$$
(9.8)
and they represent the values of local
measurements predetermined by the local hidden variables,
denoted by $\lambda$, for the
specified local settings. This expression is an average over a certain
local hidden variables distribution $\rho(\lambda)$ of certain factorisable tensors,
namely those with elements given by
$T_{ijk}(\lambda)=A(\alpha_{i},\lambda)B(\beta_{j},\lambda)C(\gamma_{k},\lambda)$.
Since the
only possible values of $A(\alpha_{i},\lambda)$,
$B(\beta_{j},\lambda)$ and $C(\gamma_{k},\lambda)$ are $\pm 1$ there are
only $2^{N_{a}}$ different sequences of the values of $(A(\alpha_{1},\lambda),...,A(\alpha_{N_{a}},\lambda))$, $2^{N_{b}}$
different sequences of
$(B(\beta_{1},\lambda),...,B(\beta_{N_{b}},\lambda))$,
$2^{N_{c}}$
different sequences of
$(C(\gamma_{1},\lambda),...,C(\gamma_{N_{c}},\lambda))$
and consequently they form only $2^{N_{a}+N_{b}+N_{c}}$
tensors $T_{ijk}(\lambda)$.
Therefore the structure of local hidden variables models of $E_{LHV}(\alpha_{i},\beta_{j},\gamma_{k})$ reduces to discrete probabilistic models involving the
average of all the $2^{N_{a}+N_{b}+N_{c}}$ tensors $T_{ijk}(\lambda)$.
In other
words, the local hidden variables can be replaced, without any loss of generality, by a
certain triple of variables $l,m,n$ that have integer values
respectively from $1,\dots,2^{N_{a}},1,\dots,2^{N_{b}},1,\dots,2^{N_{c}}$. To each $l$
we ascribe one possible sequence of the possible values of
$A(\alpha_{i},\lambda)$, denoted from now on by $A(\alpha_{i},l)$,
similarly we replace $B(\beta_{j},\lambda)$ by $B(\beta_{j},m)$ and
$C(\gamma_{k},\lambda)$ by $C(\gamma_{k},n)$ .
With this notation the possible local hidden variables models of the correlation function
$E_{LHV}(\alpha_{i},\beta_{j},\gamma_{k})$ acquire the following simple form
$$E_{LHV}(\alpha_{i},\beta_{j},\gamma_{k})=\sum_{l=1}^{2^{N_{a}}}\sum_{m=1}^{2^{%
N_{b}}}\sum_{n=1}^{2^{N_{c}}}p_{lmn}A(\alpha_{i},l)B(\beta_{j},m)C(\gamma_{k},%
n),$$
(9.9)
with, of course, the probabilities satisfying $p_{lmn}\geq 0$ and
$\sum_{l=1}^{2^{N_{a}}}\sum_{m=1}^{2^{N_{b}}}\sum_{n=1}^{2^{N_{c}}}p_{lmn}=1$.
Not all of tensors $T_{ijk}(lmn)$
are different. It is easy to check that
only one fourth of them are. The
situation is similar to that with two
qubits and enables one to simplify the actual computer program.
For the given local settings of the
measuring apparatus at each side of the experiment we want to find
the maximal $V_{2}(3)$ still
admitting the local realistic description in
the form (9.9). Then we want to find such local
settings for which this maximal $V_{2}(3)$ reaches its
minimum. From the previous section we know how to cope with
such a problem. We
use exactly the same numerical approach, i.e., the HOPDM 2.30 and
the DSM procedures.
9.2 Results
We have checked
three experimental situations: $N_{a}=N_{b}=N_{c}=2,3,4,5$ with
the result that the threshold visibility admitting local hidden variables is $V_{2}(3)={1\over 2}$.
This result is in concurrence with
the threshold visibility obtained earlier in [Mermin90b]
with the usage of appropriate Bell inequalities.
Again, because of the complexity of the space being the domain
of the $V^{max}_{2}(3)$ function (defined in analogy to that for
two qubits)- to find a global minimum- for the case $N_{a}=N_{b}=N_{c}=2,3$ we
have run the amoeba procedure
30 times with varied starting points. For $N_{a}=N_{b}=N_{c}=4$
we calculated $V^{max}_{2}(3)$
on $9000$ randomly chosen sets of the local settings whereas in the
case $N_{a}=N_{b}=N_{c}=5$ we have calculated $V^{max}_{2}(3)$ on the following
set of the local settings: $\alpha_{1}=0,\alpha_{2}=\pi/8,\alpha_{3}=\pi/4,\alpha_{4}=3\pi/8,\alpha_{5}=%
\pi/2,\beta_{1}=\gamma_{1}=-\pi/4,\beta_{2}=\gamma_{2}=-\pi/8,\beta_{3}=\gamma%
_{3}=0,\beta_{4}=\gamma_{4}=\pi/8,\beta_{5}=\gamma_{5}=\pi/4$.
An interesting feature of the results is that, as for
two qubits, the
threshold visibility
$V^{max}_{2}(3)={1\over 2}$ is always achieved for such
settings of the measuring
apparata which include as a subset the settings
giving maximal violation of the inequalities presented in
[Mermin90b]. This is in analogy with the two qubit
case in which the threshold visibility of ${1\over\sqrt{2}}$
is always obtained if among the settings one has
a subset which lead to maximal violation of
the CHSH inequality (for the maximally entangled state).
9.3 Conclusions
The presented numerical approach to the three qubit GHZ correlations
gives the sufficient and necessary conditions for the existence
of local hidden variables for the given experimental situation, i.e., for the fixed number
of positions of the measuring apparata at each side of the experiment.
For the cases where $N_{a}=N_{b}=N_{c}=2,3$ we have found such numerical
values of the local settings for which the visibility admitting local hidden variables has the lowest possible value. Up to the possibility that the DSM
procedure has not succeeded in finding the global minimum of
$V^{max}_{2}(3)$ the visibility $V_{2}(3)={1\over 2}$ is the ultimate limit drawing
the borderline between local hidden variables and quantum mechanics for these cases, i.e., for
2 and 3 settings of the measuring apparatus at each side of the experiment.
As
far as I know these are the first results giving the necessary and
sufficient conditions for violation of local realism in the GHZ
case333The inequalities found in [Mermin90b]
give only necessary condition for the existence of local realism..
For $N_{a}=N_{b}=N_{c}=4$ the visibility returned by the program for every
random choice of local settings has been always higher444Sometimes it was
very close to $\frac{1}{2}$. Obviously, it is extremely
difficult to find at random such local settings for which visibility
equals exactly $\frac{1}{2}$.
than $\frac{1}{2}$.
In the extremal case, i.e., for $N_{a}=N_{b}=N_{c}=5$ we have found
the threshold value for local settings including as a subset
settings giving maximal violation of Mermin’s inequality
[Mermin90b] with the result that $V^{max}_{2}(3)={1\over 2}$ (the DSM
has not been used).
Unfortunately, due to the computer time and memory limitations
(of the same origin as in the two qubit case) we
could not check more settings of the measuring apparatus. Nevertheless,
we suppose
that increasing the number of settings will not lead to visibility
lower than $V_{2}(3)={1\over 2}$.
The important aspect of the presented analysis of the GHZ
correlations is that, just like in the case of two qubits, the
numerical approach can be directly applied to measurement data.
We should also mention that the program based on the quantum probabilities
(the idea is presented in the chapter 10)
has also been written for considered here three maximally entangled qubits.
Some calculations
have been performed but due to the time limitations we could not test
so much cases as we did with the program based on the correlation function.
As one expects, the results for the tested cases ($N_{a}=N_{b}=N_{c}=2,3$ with
the DSM
optimisation)
are the same, i.e., the threshold visibility equals $\frac{1}{2}$.
Chapter 10 Entangled pairs of qu$N$its: the violation of local
realism increases with $N$ [8]
10.1 Introduction
John Bell has shown
that no local realistic models
can agree with all quantum mechanical predictions
for the maximally
entangled states of two qubits.
After some years researchers started to ask questions about the
Bell theorem for more complicated systems. The most spectacular
answer came for multiple qubits in the form the
GHZ theorem [Greenberger89]: the conflict
between local realism and quantum mechanics
is much sharper than for two qubits,
and can be shown even at the level
of perfect EPR-type correlations.
The other possible extension are entangled states
of pairs of qu$N$its ($3\leq N$). First results, in 1980-82, suggested that
the conflict between local realism and quantum mechanics
diminishes with growing N [Mermin80, Mermin82, Garg82].
This was felt to be in concurrence with the old
quantum wisdom of higher quantum numbers leading to a
quasi-classical behaviour.
However, the early research was confined to Stern-Gerlach type
measurements performed on pairs of maximally entangled ${N-1\over 2}$
spins [Mermin80, Mermin82, Garg82].
Since operation of a Stern-Gerlach device
depends solely on the orientation of the quantisation axis,
i.e. on only two parameters, devices of this kind
cannot make projections into arbitrary orthogonal
bases of the subsystems. That is, they cannot make full use of
the richness of the $N$-dimensional Hilbert space.
Wódkiewicz [Wódkiewicz94] proposed to employ the measurement of
the observable of a different kind (projection onto a coherent
state) and the Clauser-Horne inequality. Even in this case
in the limit of $N\longrightarrow\infty$ the violation vanishes.
In early 1990’s Peres and Gisin [Peres92, Gisin92]
have shown, that if one
considers certain dichotomic observables applied to maximally
entangled pairs of qu$N$its, the violation of local realism, or
more precisely of the CHSH inequalities, survives
the limit of $N\rightarrow\infty$ and is maximal there.
However, for any dichotomic
quantum observables the CHSH inequalities give violations
bounded by the
Tsirelson limit [Tsirelson80], i.e. limited by the
factor of $\sqrt{2}$.
Therefore, the question
whether the violation of local realism increases with growing $N$
was still left open.
There are some reasons to suspect that violations of local realism
should get stronger with increasing $N$.
For systems described by observables which are at least three valued the
Bell-Kochen-Specker theorem [Gleason57, Kochen67, Bell66]
on non-contextual hidden variable
theories can be applied. This means that any realistic theory of local
observations must be inevitably contextual.
In contradistinction, the original Bell
theorem is formulated for subsystems for which such problems do
not arise.
In this chapter we show that violation of local realism indeed
increases with growing $N$ if one uses non-degenerate observables- already
introduced
unbiased symmetric $2N$ ports.
As a ”measure” of the strength of violation of
local realism in this chapter the threshold noise
admixture $F_{N}$ (see (2.14)) still allowing
a local and realistic description of the gedanken experiment
will be used. Its link with the visibility $V_{N}$ is simple, namely
$F_{N}=1-V_{N}$. However, as the results for objects belonging
to Hilbert spaces of different dimensions will be compared, using
the parameter $F_{N}$ is more objective.
10.2 Description of the gedanken experiment
The general idea is the same as in the previous chapters.
We analyse a Bell-type experiment with two qu$N$its flying towards
two spatially separated observers A and B. We assume that the qu$N$its are prepared in the mixed state $\hat{\rho}(F_{N})$
$$\displaystyle\hat{\rho}(F_{N})=F_{N}\hat{\rho}_{noise}+(1-F_{N})\hat{\rho}_{%
max},$$
(10.1)
where $\hat{\rho}_{noise}={1\over N^{2}}I\otimes I$
($I$ is an $N\times N$ identity matrix)
and
$\hat{\rho}_{max}$ is
the projector $|\psi\rangle\langle\psi|$ on the maximally entangled state
$$\displaystyle|\psi\rangle={1\over\sqrt{N}}\sum_{k=1}^{N}|k\rangle_{A}|k\rangle%
_{B}.$$
(10.2)
Observer A can choose between the non commuting observables
$A_{1},A_{2}$ and
observer B also can choose between the non commuting observables
$B_{1},B_{2}$ (each observable for observer A and B is
characterised by some set of
local parameters (knobs)) . We assume that the spectrum
of each observable consists of
$N$ points, which we enumerate by subsequent natural numbers $k,l=1,2,\dots,N$, where index $k$ refers to observer A and index $l$ to observer B.
Thus, the observers can perform $2\times 2$
mutually exclusive global experiments.
The quantum probability distribution for the specific pairs of results $k$
and $l$,
provided a specific pairs of local observables is
chosen ($A_{i}$ and $B_{j}$), will be denoted by
$P_{QM}^{F_{N}}(k,l|{A}_{i},{B}_{j})$. In the case considered
here the quantum probabilities read
$$\displaystyle P_{QM}^{F_{N}}(k,l|A_{i},B_{j})$$
$$\displaystyle=\frac{1}{N^{2}}F_{N}+(1-F_{N})P_{QM}^{max}(k,l|A_{i},B_{j}),$$
(10.3)
where $P_{QM}^{max}(k,l|A_{i},B_{j})$ is the probability for the given pair
of events for the pure maximally entangled state (10.2). Of course, the
exact form of $P_{QM}^{max}(k,l|A_{i},B_{j})$ depends on the specific choice of
the observables $A_{i},B_{j}$.
According to quantum mechanics the set of $4N^{2}$
such probabilities is the only
information available to the observers.
10.3 Local realism and joint probability distribution
It is well known (see, e. g. [Fine82], [Peres99]) that
the hypothesis of local hidden variables
is equivalent to the existence of a (non-negative) joint
probability distribution involving all
four observables from which it should be possible to
obtain all the quantum predictions as marginals111The simple
proof of this statement is given in the Appendix A..
Let us denote this hypothetical distribution by
$P_{HV}(k,m;l,n|A_{1},A_{2},B_{1},B_{2})$, where $k$ and $m$,
represent the outcome values for observer A observables ($l$ and $n$
for observer B).
In quantum mechanics one cannot even define such objects, since
they involve mutually incompatible measurements.
The local hidden variable probabilities
$P_{HV}(...)$ are defined as
the marginals
$$\displaystyle P_{HV}(k,l|A_{1},B_{1})=\sum_{m}\sum_{n}P_{HV}(k,m;l,n),$$
$$\displaystyle P_{HV}(k,n|A_{1},B_{2})=\sum_{m}\sum_{l}P_{HV}(k,m;l,n),$$
$$\displaystyle P_{HV}(m,l|A_{2},B_{1})=\sum_{k}\sum_{n}P_{HV}(k,m;l,n),$$
$$\displaystyle P_{HV}(m,n|A_{2},B_{2})=\sum_{k}\sum_{l}P_{HV}(k,m;l,n),$$
where $P_{HV}(k,m;l,n)$ is a short hand notation for
$P_{HV}(k,m;l,n|A_{1},A_{2},B_{1},B_{2})$.
The $4\times N^{2}$
equations (LABEL:sumation2)
form the full set of necessary and sufficient conditions
for the existence of
local and realistic
description of the experiment, i.e., for the joint
probability distribution $P_{HV}(k,m;l,n)$.
The Bell theorem says that there are quantum predictions, which for $F_{N}$
below a certain
threshold cannot be
modelled by (LABEL:sumation2), i.e. there exists a critical $F_{N}^{tr}$
below which one cannot have any local realistic model with
$P_{HV}(k,l|A_{i},B_{j})=P_{QM}^{F_{N}}(k,l|A_{i},B_{j})$.
Our goal is to find observables for the two qu$N$its
returning the highest possible critical $F_{N}^{tr}$.
10.4 Linear programming
The set of conditions (LABEL:sumation2) with $P_{QM}^{F_{N}}(k,l|A_{i},B_{j})$
replacing $P_{HV}(k,l|A_{i},B_{j})$ imposes linear constraints on the $N^{4}$
“hidden probabilities” $P_{HV}(k,m;l,n)$
and on the parameter $F_{N}$, which are the nonnegative unknowns. We have more
unknowns ($N^{4}+1$) than equations
($4N^{2}$), and we want to find the
minimal $F_{N}$ for which the set of constraints can still be satisfied. Then
we want to find such local settings characterising the observables that the found
minimal $F_{N}$ reaches its highest possible value222Please, notice
that here we are looking for the threshold value of the noise
admixture $F_{N}$ and not the threshold $V_{N}$ as in the chapters
concerning entangled qubits. Because $V_{N}=1-F_{N}$ to maximal
visibility refers the minimal noise admixture..
From the previous chapters we know how to handle such a problem. We
use our sledge hammer: HOPDM 2.30 and the DSM procedures.
10.5 Observables
In our numerical calculations we have used the observables
defined by unbiased multiport beamsplitters (for the definition
of such a device see chapter devoted
to the GHZ paradoxes for qu$N$its).
The
quantum prediction for the joint probability $P_{QM}^{max}(k,l|A_{i},B_{j})$ to
detect a photon at the $k$-th output of the multiport A characterised
by phase shifters $\vec{\phi}^{i}_{A}=(\phi^{1}_{A}(i),...\phi^{N}_{A}(i))$ ($i=1,2$) and
another one at the $l$-th output of the multiport B characterised
by phase shifters $\vec{\phi}^{j}_{B}=(\phi^{1}_{B}(j),...\phi^{N}_{B}(j))$ ($j=1,2$)
for the maximally entangled state (10.2) can be derived from
(7.4) with $M=2$ and reads:
$$\displaystyle P_{QM}^{max}(k,l|\vec{\phi}^{i}_{A},\vec{\phi}^{j}_{B})$$
$$\displaystyle=(\frac{1}{N^{3}})\left(N+2\sum^{N}_{m>n}\cos{({\bf\Phi}^{m}_{kl}%
(ij)-{\bf\Phi}^{n}_{kl}(ij))}\right),$$
(10.5)
where
${\bf\Phi}^{m}_{kl}(ij)\equiv\phi^{m}_{A}(i)+\phi^{m}_{B}(j)+[m(k+l-2)]\frac{2%
\pi}{N}$.
The counts at a single detector, of course, do not depend upon
the local phase settings:
$P_{QM}(k|A_{i})=P_{QM}(l|B_{j})={1}/{N}$ for all $i,j=1,2$.
10.6 Results
The results are depicted in (10.2).
We see that $F_{N}$ continuously increases with growing $N$ exhibiting
opposite behaviour to the results obtained in earlier works
[Mermin80, Mermin82, Garg82, Peres92, Gisin92, Wódkiewicz94].
For instance, let us compare (10.2) with the results
obtained in [Gisin92], which are depicted in (10.3).
We see that for $N=2$ both results are identical. However, starting from
$N\geq 3$ the results in (10.3) never exceed the
$1-{1\over\sqrt{2}}$. This is due to the fact that in [Gisin92]
dichotomic observables were used and the threshold $F_{N}$ was obtained as
a condition for violation of CHSH inequality. This behaviour
is in agreement with the theorem proved by Tsirelson
[Tsirelson80], which states that the CHSH inequality
for dichotomic observables can be violated only if $F_{N}$ is
lower than $1-{1\over\sqrt{2}}$.
A few words of comment are needed.
One may argue that because of a quite large number
of local macroscopic parameters (the phases)
defining the function to be maximised with the DSM procedure
we
could have missed the global minimum.
While this argument cannot be ruled out in principle, we
stress that in that case the ultimate violation would even be larger.
This would only strengthen our conclusion that two entangled qu$N$it
systems are in stronger conflict with local realism than two entangled
qubits.
An important question is whether unbiased multiports provide us with
a family of observables in maximal conflict with
local realism. For a check of this question
we have also calculated the threshold value
of $F_{3}$ for the case where both observers apply to
the incoming qutrit ($N=3$) the most general
unitary transformation belonging to a full SU(3) group (i.e.
we have any trichotomic observables on each side). Again
we have assumed that each observer chooses between two sets of
local settings. However, in this case each set consists of 8
local settings rather than the three (effectively two) in the tritter case.
The result appears to be the same as for two tritters, which suggests
that tritters (an perhaps generally unbiased
multiports) are optimal devices to test quantum mechanics against
local realism for $N=3$ (for all $N$).
10.7 Exemplary analytical model for $N=3$
The presented above approach based on the idea of marginals
and joint probability distribution compatible with them
is the most general one, i.e., it does not
assume any symmetries within the tested quantum
probabilities (marginals)333Therefore it can be applied for any
marginals not necessarily
quantum ones..
However, this approach has one
drawback. Namely, it involves lots of linear constraints imposed
on the joint distribution probability. In the most general
case, i.e., when observer A chooses $N_{A}$ observables $A_{i}$
and observer B chooses $N_{B}$ observables $B_{j}$ (each observable has
$N$ point spectrum) there are
$N_{A}\times N_{B}\times N^{2}$
linear constraints
that must be fulfilled by the joint probability distribution
(we does not count the constraint that $F_{N}\leq 1$). This
considerably lengthens computation time.
On the
other hand the quantum probabilities (marginals) that
we consider here exhibit some symmetries. Therefore, one
can ask the question if, as it has been with two
and three maximally entangled qubits (see previous chapters),
it is possible to use the properly defined quantum
correlation function
instead of quantum probabilities.
In this section we show that the answer is yes but only
for444We have also performed calculations for $N\geq 4$ with the use of
the quantum correlation function (7.7) with $M=2$. The correlation
function can be reproduced for growing $N$ for lower and lower values
of the noise fraction $F_{N}$. Of course, this
effect has nothing to
do with the possibility of a local realistic description for such
cases. Simply the correlation functions defined with Bell numbers
for $N\geq 4$ start to wash out the details of the full set
of probabilities describing the experiment.
$N=3$.
Considering the quantum
correlation function will benefit in calculating the threshold $F_{3}$
for more local settings of the measuring apparata than with
the program presented in this chapter.
To this end let us consider two maximally entangled qutrits
(qu$N$its with $N=3$) and the quantum correlation function
defined in
(7.7) with $M=2$ and $N=3$. We remember that
it has been obtained by ascribing to the local results
of measurements subsequent powers of
$\alpha=\exp(\frac{2\pi i}{3})$. We can prove that
there is one-to-one equivalence between quantum probabilities
(7.4) and the quantum correlation function (7.7).
According to (7.4) for $N=3$ and for
any local settings $\vec{\phi}_{A},\vec{\phi}_{B}$
quantum probabilities $P^{QM}(kl|\vec{\phi}_{A},\vec{\phi}_{B})$
can be divided into three groups
$$\displaystyle P_{QM}^{1}=P_{QM}(12)=P_{QM}(21)=P_{QM}(33)$$
$$\displaystyle P_{QM}^{2}=P_{QM}(11)=P_{QM}(23)=P_{QM}(32)$$
$$\displaystyle P_{QM}^{3}=P_{QM}(13)=P_{QM}(31)=P_{QM}(22),$$
(10.6)
where we do not show the dependence on local settings to shorten
the notation.
This allows us to write the correlation function
$E_{QM}(\vec{\phi}^{i}_{A},\vec{\phi}^{j}_{B})$ in the form (for
convenience we put $F_{3}=0$)
$$\displaystyle E_{QM}=3(P_{QM}^{1}+\alpha^{2}P_{QM}^{2}+\alpha P_{QM}^{3}),$$
(10.7)
where again we do not show the dependence of the correlation
function on local settings.
Using the identity ${1\over 3}=P_{QM}^{1}+P_{QM}^{2}+P_{QM}^{3}$ and
$\alpha+\alpha^{2}+\alpha^{3}=0$ we arrive at
$$\displaystyle{1\over 3}(E_{QM}-1)=(\alpha-1)P_{QM}^{3}-(2+\alpha)P_{QM}^{2},$$
(10.8)
which allows us to express all quantum probabilities through
the correlation function (now $F_{3}$ is arbitrary)
$$\displaystyle P_{QM}^{1}={1\over 9}+{2\over 9}ReE_{QM}^{F_{3}}$$
$$\displaystyle P_{QM}^{2}={1\over 9}-{1\over 9}(\sqrt{3}ImE_{QM}^{F_{3}}+ReE_{%
QM}^{F_{3}})$$
$$\displaystyle P_{QM}^{3}={1\over 9}+{1\over 9}(\sqrt{3}ImE_{QM}^{F_{3}}-ReE_{%
QM}^{F_{3}}).$$
(10.9)
Therefore, we can apply the approach based on the quantum
correlation function presented in the chapter concerning
two and three qubits.
For every measurement of the pair of observables $A_{i}$ and $B_{j}$
($i=1,\dots,N_{a};j=1,\dots,N_{b}$)
defined
by the tritter and the phase shifters we calculate the quantum
correlation function obtaining this way the
quantum matrix of predictions
$\hat{Q}$ with entries
$Q_{ij}(F_{3})=E_{QM}^{F_{3}}(\vec{\phi}^{i}_{A},\vec{\phi}^{j}_{B})$.
Note that now the observer A has $N_{A}$ observables to choose
from (B has $N_{B}$).
The hidden variable model
in this case reads
$$\displaystyle E_{LHV}(\vec{\phi}^{i}_{A},\vec{\phi}^{j}_{B})=\sum_{k=1}^{N^{N_%
{a}}}\sum_{l=1}^{N^{N_{b}-1}}p_{kl}A(\vec{\phi}^{i}_{A},k)B(\vec{\phi}^{j}_{B}%
,l),$$
(10.10)
where the functions $A(\vec{\phi}^{i}_{A},k)=\alpha^{m}$ and
$B(\vec{\phi}^{j}_{B},l)=\alpha^{n}$ ($m,n=1,2,3$). If it is to
reproduce the quantum correlation function it must fulfil
the following set of $N_{a}\times N_{b}$ linear constraints555The number
of relevant $p_{kl}$ can be reduced by a factor of $3$ using a development
of the trick that has led to the representation of the correlation function
for two qubits in the form of (8.7).
$$\displaystyle Q_{ij}(F_{3})=\sum_{k=1}^{N^{N_{a}}}\sum_{l=1}^{N^{N_{b}-1}}p_{%
kl}A(\vec{\phi}^{i}_{A},k)B(\vec{\phi}^{j}_{B},l).$$
(10.11)
Using HOPDM 2.30 and the DSM we can find the threshold $F_{3}$ still
admitting a local and realistic description of the gedanken experiment.
The results are identical with those obtained by the previous
method. However, since the algorithm is now much faster
we were able to perform calculations involving up to 5
different phase settings on each side (i.e. allowing
each observer to have a choice of up to 5
different observables to measure). Just as in the
two qubit case the optimal $F_{3}$ stayed unchanged at the value
obtained for the problem with $N_{a}=N_{b}=2$.
10.7.1 Explicit model for extremal case
Exemplary optimal settings for violation of local realism
in the experiment with two qutrits and $N_{a}=N_{b}=2$ are
$$\displaystyle\vec{\phi_{A}}^{1}=(0,\frac{\pi}{3},-\frac{\pi}{3})$$
$$\displaystyle\vec{\phi_{A}}^{2}=(0,0,0)$$
$$\displaystyle\vec{\phi_{B}}^{1}=(0,\frac{\pi}{6},-\frac{\pi}{6})$$
$$\displaystyle\vec{\phi_{B}}^{2}=(0,-\frac{\pi}{6},\frac{\pi}{6}).$$
(10.12)
For such settings the quantum matrix
$\hat{Q}$ (with $F_{3}=0$) reads666We present here result of an algebraic
re-calculation of the computer output. Perhaps, the most exciting aspect of
such exercise is the exact value of $F_{3}=\frac{11-6\sqrt{3}}{2}$.
$$\displaystyle\hat{Q}=\left(\begin{array}[]{cc}Q_{1}&Q_{1}^{*}\\
Q_{2}&Q_{1}\\
\end{array}\right),$$
(10.13)
where
$Q_{1}=\frac{2\sqrt{3}+1}{6}-i\frac{2-\sqrt{3}}{6}$ and
$Q_{2}=-{1\over 3}(1+2i)$. Please, note that the all entries
of (10.13) have the same modulus equal $\frac{\sqrt{5}}{3}$.
The hidden variables can only reproduce the matrix of
correlation function given by $(1-F_{3})\hat{Q}$ with
$F_{3}\leq\frac{11-6\sqrt{3}}{2}$.
For the threshold maximal $F_{3}$ the explicit model is given by
$$\displaystyle\hat{Q}^{F_{3}}=p\left(\begin{array}[]{cc}\alpha^{3}&\alpha^{3}\\
\alpha^{3}&\alpha^{3}\\
\end{array}\right)+p\left(\begin{array}[]{cc}\alpha^{2}&\alpha^{3}\\
\alpha^{2}&\alpha^{3}\\
\end{array}\right)+$$
$$\displaystyle p\left(\begin{array}[]{cc}\alpha^{3}&\alpha^{3}\\
\alpha^{2}&\alpha^{2}\\
\end{array}\right)+p\left(\begin{array}[]{cc}\alpha^{3}&\alpha\\
\alpha^{2}&\alpha^{3}\\
\end{array}\right)+$$
$$\displaystyle q\left(\begin{array}[]{cc}\alpha^{3}&\alpha^{2}\\
\alpha&\alpha^{3}\\
\end{array}\right)+q\left(\begin{array}[]{cc}\alpha&\alpha^{3}\\
\alpha^{2}&\alpha\\
\end{array}\right)=(1-F_{3})\hat{Q},$$
(10.14)
where $p=\frac{4-2\sqrt{3}}{3}$,
$q=\frac{8\sqrt{3}-13}{6}$.
This model has to be understood in the following
way. Consider the first term. The rank-one matrix
$$\displaystyle\left(\begin{array}[]{cc}\alpha^{3}&\alpha^{3}\\
\alpha^{3}&\alpha^{3}\\
\end{array}\right)$$
(10.15)
can be factorised into column and row matrices built out of
powers of $\alpha$, in the following three ways:
$$\displaystyle\left(\begin{array}[]{cc}\alpha^{3}&\alpha^{3}\\
\alpha^{3}&\alpha^{3}\\
\end{array}\right)=\left(\begin{array}[]{c}\alpha^{3}\\
\alpha^{3}\end{array}\right)\left(\begin{array}[]{c}1\\
1\end{array}\right)^{T}=\left(\begin{array}[]{c}\alpha^{2}\\
\alpha^{2}\end{array}\right)\left(\begin{array}[]{c}\alpha\\
\alpha\end{array}\right)^{T}=\left(\begin{array}[]{c}\alpha\\
\alpha\end{array}\right)\left(\begin{array}[]{c}\alpha^{2}\\
\alpha^{2}\end{array}\right)^{T}.$$
(10.16)
Therefore with probability $p^{\prime}={1\over 3}p$ each of this
factorisations is present in the model of $\hat{Q}^{F_{3}}$
(compare (10.10) and (10.11)).
10.8 Conclusions
It is evident, that indeed two entangled
qu$N$its violate local realism stronger than two entangled qubits,
and that the violation increases with $N$. It is important to stress that
the values were obtained using four independently written codes,
one of them employing a different linear optimisation procedure
(from the NAG Library).
It is interesting to compare our results
with the limit [Horodecki99]
for the non separability of the density matrices [Werner89]
of the two entangled systems.
The fact that this limit, $\frac{N}{N+1}$, is always higher than
ours indicates that that requirement of having local
quantum description of the two subsystems is a much more stringent
condition than our requirement of admitting any possible
local realistic model.
It will be interesting to consider within our approach
different families of states, generalisations to more than two qu$N$its,
extensions of the families of observables,
to see if a wider choice of experiments than can be performed on one
side (i.e., more than two) can lead to even stronger violations
of local realism,
and finally to see experimental realizations of such schemes.
Finally we should mention that this approach can be extended
to the research into the threshold efficiency of detectors
for such experiments. The computer program for imperfect
detectors has been already written.
Appendix A Proof of equivalence of the existence of local hidden variables
and a joint probability distribution for incompatible
measurements
This appendix is a re-derivation of
known results (see e.g. [Fine82]).
We consider the experiment with two observers
A and B. Observer A can choose to measure $N_{A}$ observables $A_{i}$
($i=1,2,\dots,N_{A}$) whereas observer can choose to measure
$N_{B}$ observables $B_{j}$ ($j=1,2,\dots,N_{B}$).
In the measurement of the observable $A_{i}$ observer A
can obtain $N$ results, which we
denote by $k_{i}$ ($i=1,\dots,N$).
Similarly, in the measurement of the observable $B_{j}$ observer B
can obtain $N$ results,
which we denote by $l_{j}$ ($j=1,\dots,N$). In the experiment
the observers can only measure joint probabilities
$P(k_{i},l_{j}|A_{i},B_{j})$, i.e., probabilities of obtaining the
result $k_{i}$ when measuring the observable $A_{i}$ and the
result $l_{j}$ when measuring the observable $B_{j}$.
First, let us show that the existence of stochastic local hidden
variables recovering the marginals $P(k_{i},l_{j}|A_{i},B_{j})$
implies the existence of a joint probability
distribution
$$\displaystyle P_{HV}(k_{1},\dots,k_{N_{A}},l_{1},\dots,l_{N_{B}}|A_{1},\dots,A%
_{N_{A}},B_{1},\dots,B_{N_{B}})$$
(A.1)
compatible with the given marginals.
The existence of local hidden variable space $\Lambda$
and the ”hidden” probabilities $P_{i}(k_{i}|\lambda,A_{i})$
and $P_{j}(l_{j};\lambda,B_{j})$ (see (2.13)) allows one
to define the joint probability distribution
(A.1) in the following
way
$$\displaystyle P_{HV}(k_{1},\dots,k_{N_{A}},l_{1},\dots,l_{N_{B}}|A_{1},\dots,A%
_{N_{A}},B_{1},\dots,B_{N_{B}})$$
$$\displaystyle=\int_{\Lambda}d\rho(\lambda)\prod_{i=1}^{N_{A}}P_{i}(k_{i}|%
\lambda,A_{i})\prod_{j=1}^{N_{B}}P_{j}(l_{j}|\lambda,B_{j}).$$
(A.2)
It is evident that (A.2) returns as marginals $P(k_{i},l_{j}|A_{i},B_{j})$.
Now let us assume that (A.1) exists and is
compatible with the marginals
$P(k_{i},l_{j}|A_{i},B_{j})$
We can define the local hidden variables as follows.
To each sequence of possible results
$(\tilde{k}_{1},\tilde{k}_{2},\dots,\tilde{k}_{N_{A}},\tilde{l}_{1},\tilde{l}_{%
2},\dots,\tilde{l}_{N_{B}})$
we ascribe a hidden variable, which we denote as
$\lambda(\tilde{k}_{1},\tilde{k}_{2},\dots,\linebreak\tilde{k}_{N_{A}},\tilde{l%
}_{1},\tilde{l}_{2},\dots,\tilde{l}_{N_{B}})$. Therefore, the local hidden variable space $\Lambda$ consists
of $N^{N_{A}+N_{B}}$ different hidden variables. On this space we define
the discrete probability
distribution of local hidden variables $\rho(\lambda)$
in the following way
$$\displaystyle\rho(\lambda(\tilde{k}_{1},\tilde{k}_{2},\dots,\tilde{k}_{N_{A}},%
\tilde{l}_{1},\tilde{l}_{2},\dots,\tilde{l}_{N_{B}}))$$
$$\displaystyle=P_{HV}(\tilde{k}_{1},\dots,\tilde{k}_{N_{A}},\tilde{l}_{1},\dots%
,\tilde{l}_{N_{B}}|A_{1},\dots,A_{N_{A}},B_{1},\dots,B_{N_{B}}).$$
(A.3)
The hidden probabilities
$P_{i}(k_{i}|\lambda,A_{i})$
and
$P_{j}(l_{j}|\lambda,B_{j})$
we define as
$$\displaystyle P_{i}(k_{i}|\lambda(\tilde{k}_{1},\tilde{k}_{2},\dots,\tilde{k}_%
{N_{A}},\tilde{l}_{1},\tilde{l}_{2},\dots,\tilde{l}_{N_{B}}),A_{i})=\delta_{%
\tilde{k}_{i},k_{i}}$$
$$\displaystyle P_{j}(l_{j}|\lambda(\tilde{k}_{1},\tilde{k}_{2},\dots,\tilde{k}_%
{N_{A}},\tilde{l}_{1},\tilde{l}_{2},\dots,\tilde{l}_{N_{B}}),B_{j})=\delta_{%
\tilde{l}_{j},l_{j}},$$
(A.4)
where $\delta$ denotes the Kronecker delta function. Then the marginals
are recovered in the following way
$$\displaystyle P(k_{i},l_{j}|A_{i},B_{j})$$
$$\displaystyle=\sum_{\lambda}\rho(\lambda(\tilde{k}_{1},\tilde{k}_{2},\dots,%
\tilde{k}_{N_{A}},\tilde{l}_{1},\tilde{l}_{2},\dots,\tilde{l}_{N_{B}}))\delta_%
{\tilde{k}_{i},k_{i}}\delta_{\tilde{l}_{j},l_{j}},$$
(A.5)
where we sum over all $N^{N_{A}+N_{B}}$ hidden variables $\lambda$.
Using the above definitions (A.5) can be rewritten as
$$\displaystyle P(k_{i},l_{j}|A_{i},B_{j})=\sum_{\lambda}\rho(\lambda)P_{i}(k_{i%
}|\lambda,A_{i})P_{j}(l_{j}|\lambda,B_{j}),$$
(A.6)
i.e. we have a typical structure of a local hidden variable
model (for hidden variables forming a discrete set).
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LongitudinalspinSeebeckeffectcontributionintransversespinSeebeckeffectexperimentsinPt/YIGandPt/NFO
D. Meier
[email protected]
www.spinelectronics.de
D. Reinhardt
M. vanStraaten
C. Klewe
DepartmentofPhysics,CenterforSpinelectronicMaterialsandDevices,BielefeldUniversity,33501Bielefeld,Germany
M. Althammer
Walther-Meissner-Institut,BayerischeAkademiederWissenschaften,Walther-Meissner-Strasse8,85748Garching,Germany
M. Schreier
Walther-Meissner-Institut,BayerischeAkademiederWissenschaften,Walther-Meissner-Strasse8,85748Garching,Germany
Physik-Department,TechnischeUniversitätMünchen,85748Garching,Germany
S.T.B. Goennenwein
Walther-Meissner-Institut,BayerischeAkademiederWissenschaften,Walther-Meissner-Strasse8,85748Garching,Germany
NanosystemsInitiativeMunich(NIM),Schellingstraße4,80799München,Germany
A. Gupta
CenterforMaterialsforInformationTechnology,UniversityofAlabama,TuscaloosaAlabama35487,USA
M. Schmid
C.H. Back
InstituteofExperimentalandAppliedPhysics,UniversityofRegensburg,93040,Germany
J.-M. Schmalhorst
T. Kuschel
G. Reiss
DepartmentofPhysics,CenterforSpinelectronicMaterialsandDevices,BielefeldUniversity,33501Bielefeld,Germany
(January 12, 2021)
Abstract
WeinvestigatetheinversespinHallvoltageofa$10\text{\,}\mathrm{nm}$thinPtstripdepositedonthemagneticinsulators\ceY3Fe5O12(YIG)and\ceNiFe2O4(NFO)withatemperaturegradientinthefilmplane.WeobservecharacteristicstypicalofthespinSeebeckeffect,althoughwedonotobserveachangeofsignofthevoltageatthePtstripwhenitismovedfromhottocoldside,whichisbelievedtobethemoststrikingfeatureofthetransversespinSeebeckeffect.Therefore,werelatetheobservedvoltagestothelongitudinalspinSeebeckeffectgeneratedbyaparasiticout-of-planetemperaturegradient,whichcanbesimulatedbycontacttipsofdifferentmaterialandheatconductivitiesandbytipheating.ThisworkgivesnewinsightsintotheinterpretationoftransversespinSeebeckeffectexperiments,whicharestillunderdiscussion.
SpincaloritronicsisanactivebranchinspintronicsBauer et al. (2012); Wolf et al. (2001).Theinterplaybetweenheat,chargeandspintransportopensanewareaoffascinatingissuesinvolvingtheuseofwasteheatinelectronicdevices.OnepotentiallyusefuleffectforheatharvestingKirihara et al. (2012)isthespinSeebeckeffect(SSE)Uchida et al. (2008)whichwasobservedin2008.Itwasreportedthataspincurrentperpendiculartoanappliedtemperaturegradientcanbegeneratedinaferromagneticmetal(FMM)bythetransversespinSeebeckeffect(TSSE)Uchida et al. (2008).Anadjacentnormalmetal(NM)convertsthespincurrentviatheinversespinHalleffect(ISHE)Saitoh et al. (2006)intoatransversevoltage,whichisantisymmetricwithrespecttotheexternalmagneticfield$H$($V(H)\!=\!-V(-H)$,cf.Fig. 1 (a)).Inthisgeometry,thetemperaturegradientistypicallyalignedin-plane($\nabla T_\text{x}$)andcanalsoinduceaplanarNernsteffect(PNE)inFMMwithmagneticanisotropyAvery et al. (2012)whichisduetotheanisotropicmagneticthermopowerandsymmetricwithrespectto$H$($V(H)\!=\!V(-H)$).Forpure$\nabla T_\text{x}$inaferro(i)magneticinsulator(FMI)thereisnoPNE,sincetherearenofreechargecarriersavailable.However,iftheNMmaterialisclosetotheStonercriterion,astaticmagneticproximityeffectcouldinduceasocalledproximityPNE,whichingeneralispresentinspinpolarizedNMadjacenttoaFMMandcouldalsooccurinaNM-FMIcontact(cf.Fig. 1 (a)).
ForthelongitudinalSSE(LSSE)Uchida et al. (2010a)thespincurrentflowsdirectlyfromtheFMintoanadjacentNMparalleltothetemperaturegradient(cf.Fig. 1 (b)),whichistypicallyalignedout-of-plane($\nabla T_\text{z}$).InNM/FMMbilayerstheanomalousNernsteffect(ANE)canoccur,butisabsentintheFMI.InsemiconductingmaterialstheANEcontributestotheLSSEasalreadyshownforPt/NFOatroomtemperatureMeier et al. (2013a).Additionally,iftheNMwouldbespinpolarizedbytheproximitytotheFM,anadditionalproximityANEcouldoccurGuo et al. (2014)(cf.Fig. 1 (b)).WewouldliketopointoutthatFigs. 1 (a)and1 (b)includeresultsofpreviousexperimentswithpure$\nabla T_\text{x}$onNM/FMMUchida et al. (2008); Jaworski et al. (2010); Avery et al. (2012); Meier et al. (2013b)andonNM/FMIUchida et al. (2010a)aswellaspure$\nabla T_\text{z}$onNM/FMMMeier et al. (2013a)andonNM/FMIUchida et al. (2010b, c); Huang et al. (2012); Weiler et al. (2012); Qu et al. (2013); Kehlberger et al. (2014); Siegel et al. (2014); Agrawal et al. (2014).AssummarizedinFig. 1 (c)anunintended$\nabla T_\text{z}$canhampertheevaluationofTSSEexperimentswithapplied$\nabla T_\text{x}$.HeatflowintothesurroundingareaorthroughtheelectricalcontactscaninduceanadditionalANEinNM/FMMbilayersandNM/magneticsemiconductorsasdiscussedinliteratureBosu et al. (2011); Huang et al. (2011); Schmid et al. (2013); Meier et al. (2013b); Bui and Rivadulla (2014); Soldatov et al. (2014).Butsince,inprinciple,alltheeffectsofanLSSEexperimentcanbepresentintheTSSEexperimentwithunintended$\nabla T_\text{z}$,proximityNernsteffectsandespeciallyparasiticLSSEcanalsobepresentinNM/FMIbilayersasalreadymentionedrecentlySchreier et al. (2013).Thisleadstofourpossibleeffectswhichareantisymmetricwithrespecttotheexternalmagneticfield,whenthetemperaturegradientisnotcontrolledverycarefully(cf.Fig. 1 (c)).Thesephenomenaanddiscussionsofside-effectsinTSSEexperimentshavenotbeentreatedsystematicallyintheliteratureuptonowforNM/FMIbilayersandwillbeinvestigatedinthisstudy.TheYIGfilmsinourexperimentshadathicknessof$t_\text{YIG}=$180\text{\,}\mathrm{nm}$$andweredepositedongadoliniumgalliumgarnet(\ceGd3Ga5O12,GGG)(111)-orientedsinglecrystalsubstrateswithwidthandlength$w=l=$5\text{\,}\mathrm{mm}$$bypulsedlaserdeposition(PLD)fromastoichiometricpolycrystallinetarget.Thefilmsshowacoercivefieldofabout$100\text{\,}\mathrm{O}\mathrm{e}$andasaturationmagnetizationof$M_\text{S}=$120\text{\,}\mathrm{kA}\text{/}\mathrm{m}$$.The\ceNFOfilmswithathicknessofabout$t_\text{NFO}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$weredepositedon$10\text{\times}5\text{\,}\mathrm{mm}$\ceMgAl2O4(\ceMAO)(100)-orientedsubstratesbydirectliquidinjectionchemicalvapordeposition(DLI-CVD)Li et al. (2011); Meier et al. (2013a).Afteravacuumbreakandcleaningwithethanolinanultrasonicbatha$t_\text{Pt}=$10\text{\,}\mathrm{nm}$$thinPtstripwasdepositedbydcmagnetronsputteringinanAratmosphereof$1.5\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$througha$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$widesplit-maskononesamplesideofthe\ceYIGand\ceNFOfilmswithalengthof$l_\text{Pt}=$5\text{\,}\mathrm{mm}$$.FortheexperimentsweusedthesamesetupandtechniquedescribedinRef. Meier et al. (2013b).TheexperimentsareconductedunderambientconditionsandtheendsofthePtstripwerecontactedwithamicroprobesystemwithAuandWtipsofdifferentdiameters.Furthermore,onesetofAutipswasequippedwitha$1.5\text{\,}\mathrm{k\SIUnitSymbolOhm}$resistorforheatingthetiptointentionallyinducea$\nabla T_\text{z}$(cf.Fig. 2 (a))Meier et al. (2013b).Fig. 2 (b)showsthenearlylinearrelationbetweenthepower$P_{\text{needle}}$dissipatedintheresistorandthetiptemperature$T_{\text{needle}}$asdeterminedwithatype-Kthermocouplegluedtothetipinacalibrationmeasurement.Thevoltage$V$atthePtstripwasmeasuredwithaKeithley2182Ananovoltmeter.Anexternalmagneticfield$H$wasappliedalongxinarangeof$\pm 600\text{\,}\mathrm{O}\mathrm{e}$for\ceYIGandof$\pm 1000\text{\,}\mathrm{O}\mathrm{e}$for\ceNFOfilms.
First,theISHEvoltagefromthePt/YIGsamplewasmeasuredforvarious$\Delta T_\text{x}$.ThePtstripwasonthehotsideandcontactedwithWtips.Thevoltageshowsnosignificantvariationwithinthesensitivitylimitwhen$H$isvaried(Fig. 3 (a)).NoNernsteffectsareobservedduetotheinsulatingmagneticlayerandnoevidenceforanadditional$\nabla T_\text{z}$canbedetected.Therefore,theclampingandheatingofthesampleinoursetupandthecontactingwiththinWtipsresultsinapure$\nabla T_\text{x}$asalreadyshowninRef. Meier et al. (2013b).TSSEisnotobservablealthoughthePtstripislocatednearthehotsideoftheYIGfilmandfarawayfromthecenterwheretheTSSEshouldvanish.
InthenextstepwecreatedasecondheatsinkbyusingthickerAutips(Fig. 3 (b)).Thediameterofthecontactareaisofabout$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$incontrasttothecontactareawithWtipsofabout$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$.Itcanbeseenthatthemeasuredvoltagenowshowsanantisymmetricbehaviorwithrespecttomagneticfieldinversion(Fig. 3 (b)-(e)).Next,weappliedavoltagetotheAutipresistortoheattheneedleandchangetheout-of-planeheatflow.TheheatedAuneedleislabeled$T_{\text{needle}}$ = RT + xwithroomtemperatureRT = $296\text{\,}\mathrm{K}$.Themagnitude$V_{\text{sat}}$(thevoltageinsaturation)iscalculatedwith$(V_+-V_-)/2$fortheaveragevoltage$V_\pm$intheregionof$$500\text{\,}\mathrm{O}\mathrm{e}$<H<$600\text{\,}\mathrm{O}\mathrm{e}$$and$$-600\text{\,}\mathrm{O}\mathrm{e}$<H<$-500\text{\,}\mathrm{O}\mathrm{e}$$forPt/YIGaswellas$$900\text{\,}\mathrm{O}\mathrm{e}$<H<$1000\text{\,}\mathrm{O}\mathrm{e}$$and$$-1000\text{\,}\mathrm{O}\mathrm{e}$<H<$-900\text{\,}\mathrm{O}\mathrm{e}$$forPt/NFO,respectively.InFig. 3 (b)for$T_{\text{needle}}=\text{RT}$asmallantisymmetriceffectofabout$V_{\text{sat}}=$-50\text{\,}\mathrm{nV}$$isobtainedwhenthePtstripisatthecoldsideoftheYIGfilm.Whentheneedleisheatedto$T_{\text{needle}}=\text{RT}+$12\text{\,}\mathrm{K}$$theISHEvoltagechangesitssigntoavalueof$V_\text{sat}=$+95\text{\,}\mathrm{nV}$$andchangesfurtherto$V_\text{sat}=$+590\text{\,}\mathrm{nV}$$for$T_{\text{needle}}=\text{RT}+$24\text{\,}\mathrm{K}$$.ByusingtheseAuneedleswithlargecontactareasanadditionalsmallout-of-planeheatflowisgeneratedevenatthecoldsideofthesample.Thisheatflowchangesitssignwithincreasing$T_{\text{needle}}$whichcanbedetectedbythesignreversalofthemeasuredvoltage.When$\nabla T_\text{x}$isincreased($\Delta T_\text{x}=$30\text{\,}\mathrm{K}$$inFig. 3 (c))theISHEvoltageatthePtis$V_\text{sat}=$-170\text{\,}\mathrm{nV}$$for$T_{\text{needle}}=\text{RT}$andthereforethreetimeslargerthanfor$\Delta T_\text{x}=$10\text{\,}\mathrm{K}$$.TheISHEvoltageagainincreaseswithincreasing$T_{\text{needle}}$andchangessign.
ForaPtstripatthehotside$V_{\text{sat}}$withouttipheatingislargerthanatthecoldside.For$\Delta T_\text{x}=$5\text{\,}\mathrm{K}$$themagnitudeisabout$V_{\text{sat}}=$-130\text{\,}\mathrm{nV}$$(Fig. 3 (d))whichcanbedecreasedto$V_{\text{sat}}=$-300\text{\,}\mathrm{nV}$$for$\Delta T_\text{x}=$10\text{\,}\mathrm{K}$$(Fig. 3 (e)).Thesignandthemagnitudeof$V_{\text{sat}}$canalsobecontrolledby$T_\text{needle}$and,therefore,by$\nabla T_\text{z}$.When$T_\text{needle}$isfixedat$\text{RT}+$31\text{\,}\mathrm{K}$$,$V_{\text{sat}}$isabout$+180\text{\,}\mathrm{nV}$for$\Delta T_\text{x}=$5\text{\,}\mathrm{K}$$and$+90\text{\,}\mathrm{nV}$for$\Delta T_\text{x}=$10\text{\,}\mathrm{K}$$.Foraverificationofthisbehaviorinanothermaterialsystem,\ceNFOfilmswithaPtstripwereusedandcontactedwithAuneedles.When$\Delta T_\text{x}=$0\text{\,}\mathrm{K}$$andnotipheatingisappliednosignificantchangein$V$asafunctionof$H$isobserved(Fig. 4 (a)).ThisbehavioristhesameforPtstripsonbothsidesofthe\ceNFOfilm(Fig. 4 (c)).Whena$\nabla T_\text{x}$isapplied,thesamesignof$V_{\text{sat}}$onbothsidesisachieved(Fig. 4 (b),(d)).Theeffectcanagainbemanipulatedbyapplyinganeedletemperatureof$T_{\text{needle}}=\text{RT}+$31\text{\,}\mathrm{K}$$.ThediscrepancyforPt/NFOmeasurementsonhotside(smaller$|V_{\text{sat}}|$,Fig. 4 (b))andcoldside(larger$|V_{\text{sat}}|$,Fig. 4 (d))comparedtoPt/YIGonhotside(larger$|V_{\text{sat}}|$,Fig. 3 (e))andcoldside(smaller$|V_{\text{sat}}|$,Fig. 3 (b))canbeexplainedbycontactingtheneedlesagainafterreversingthesampleto”move”thePtstripfromthehottothecoldside.TherealcontactareabetweenthetipsandthePtcanbedifferentwhenthesampleisremounted.Furthermore,$T_\text{needle}$wasvariedfor$\Delta T_\text{x}=$12\text{\,}\mathrm{K}$$(Fig. 4 (e)).$V_\text{sat}$at$T_\text{needle}=\text{RT}$isabout$+690\text{\,}\mathrm{nV}$.Theabsolutevalueismorethantwotimessmallerthanfor$\Delta T_\text{x}=$30\text{\,}\mathrm{K}$$($V_{\text{sat}}=$-1550\text{\,}\mathrm{nV}$$)withthechangeof$\Delta T_\text{x}$.$V_\text{sat}$alsovanishesfor$T_\text{needle}=\text{RT}+$12\text{\,}\mathrm{K}$$andchangessignforincreasing$T_\text{needle}$.
$V_{\text{sat}}$measuredforPt/YIGatthehotandcoldsideisplottedasafunctionof$T_\text{needle}$fordifferent$\Delta T_\text{x}$inFig. 5.Anon-heatedAuneedleresultsinthesamesignof$V_{\text{sat}}$forall$\Delta T_\text{x}$,while$|V_{\text{sat}}|$issmalleronthecoldsidecomparedtothehotside.Weexplainthisbehaviourof$V_\text{sat}$byanunintendedheatfluxthroughtheAuneedlesleadingtoaverticaltemperaturegradient$\nabla T_\text{z}$and,therefore,toaLSSEinducedspincurrentintothePt.Wepointout,thatthesignoftheresultingvoltageatthePtstripisquantitativelyconsistentwiththemagnitudeoftherecentLSSEreportedbySchreieretal.Schreier et al. (2014)fromcomparativeexperimentsperformedindifferentgroupsifweconsideranunintendedout-of-plane$\nabla T_\text{z}$pointinginto-zdirection.ForaheatedAuneedle$V_{\text{sat}}$increasesandcrosseszero(cf.Fig. 5).Here,thetipheatingcompensatestheout-of-planeheatfluxinducedby$\Delta T_\text{x}$($\nabla T_\text{z}=0$).Afterthesignchangeof$V_{\text{sat}}$(andtherefore,$\nabla T_\text{z}$),thevaluesincreaseswithalarger(smaller)slopeforthecold(hot)side.Xiaoetal.Xiao et al. (2010)discussedthetemperaturedifference$\Delta T_\text{me}$betweenthemagnontemperatureintheFMandtheelectrontemperatureintheNMastheoriginofthethermallyinducedspincurrent.$\Delta T_\text{me}$canbeinferredfromtherecordedvoltageasXiao et al. (2010); Schreier et al. (2013)
$$\Delta T_\text{me}=\frac{V_\text{sat}\,\pi\,M_\text{S}\,V_\text{a}\,t_\text{Pt%
}}{g_\text{r}\,\gamma\,k_\text{B}\,e\,\Theta_\text{SH}\,\rho_\text{Pt}\,l_%
\text{Pt}\,\lambda\,\tanh(t/2\lambda)}.$$
(1)
Here,$V_\text{a}$isthemagneticcoherencevolume,$g_\text{r}$istherealpartofthespinmixingconductance,$\gamma$isthegyromagneticratio,$k_\text{B}$istheBoltzmannconstant,$e$istheelementarycharge,$\Theta_\text{SH}$isthespinHallangle,and$\lambda$isthespindiffusionlengthoftheNMmaterial.Thetwotemperaturemodelhasprovensuccessfulinrelating$\Delta T_\text{me}$tothephonontemperature,accessibleinexperimentsSchreier et al. (2013); Flipse et al. (2012).Wesimulatethephononandmagnontemperaturesassuming1Dtransportinourfilmsanddisregardtheinfluenceofthermalcontactresistanceotherthanthecouplingbetweenmagnonsandelectrons.Thisyieldsavalue$\Delta T_\text{z}$,the(phonon)temperaturedropacrosstheYIGfilm,forwhichtheexperimentallymeasured$V_\text{sat}$isobtained.Weassume$\rho_\text{Pt}=$40\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,%
}\mathrm{cm}$$.AllmaterialdependentYIGparametersweretakenfromRef. Schreier et al. (2013).Wecalculated$\Delta T_\text{me}$forthelargestandsmallest$V_\text{sat}$takenfromFig. 5atRT(redcurvesinFig. 3 (b)and(e))andthecorresponding$\Delta T_\text{z}$.For$|V_\text{sat}|=$50\text{\,}\mathrm{nV}$$(cf.Fig. 3 (b))weobtain$\Delta T_\text{me}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro K}$$andacorresponding$\Delta T_\text{z}=$2\text{\,}\mathrm{mK}$$.For$|V_\text{sat}|=$300\text{\,}\mathrm{nV}$$(cf.Fig. 3 (e))thecorrespondingvaluesare$\Delta T_\text{me}=$3.2\text{\,}\mathrm{\SIUnitSymbolMicro K}$$and$\Delta T_\text{z}=$12\text{\,}\mathrm{mK}$$.Theobtained$\Delta T_\text{z}$areintheorderofafewmillikelvins.Itisreasonabletoassumethatsuchvaluescanbeinducedbye.g.thickcontacttips,especiallyconsideringthatourinitialsimplificationsshouldleadtoanoverestimationof$\Delta T_\text{z}$Schreier et al. (2013).ThetransversespinSeebeckconfigurationwasinvestigatedindetailinRef. Schreier et al. (2013).Itwasfoundthatfor$\Delta T_\text{x}=$20\text{\,}\mathrm{K}$$theobtained$\Delta T_\text{me}$iswellbelow$1\text{\,}\mathrm{\SIUnitSymbolMicro K}$,evenattheveryedgeofthesamplewhere$\Delta T_\text{me}$ismaximized.Thisfurthersupportsthenotionthatspuriousout-of-planegradientsareresponsibleforthevoltagesobservedinoursamples.ThedifferentmagnitudesinvoltageforPt/NFOcomparedtoPt/YIGcanbeexplainedbydifferentcontactareasofthetips,differentthicknessesandthermalconductivitiesbetweenNFOandYIGaswellasdifferentspinmixingconductancesWeiler et al. (2013); Qiu et al. (2013).However,forbothsamplesystemsthespinmixingconductanceshouldbelargeenoughtoobserveathermallydrivenspincurrentacrosstheNM/FMIinterface,sinceweclearlyobserveanLSSEduetotheAutipheatinginduced$\Delta T_\text{z}$.InadditiontotheLSSE,wenowdiscussotherparasiticeffectsliketheANEandproximityANE,whichcanbeproducedbyanunintended$\Delta T_\text{z}$(seeFig. 1).WecanexcludeanANEforYIGduetothelackofchargecarriers.ForNFOweobservedanANEwhichisoneorderofmagnitudesmalleratRTthantheLSSEMeier et al. (2013a).ThisANEcanbeexplainedbytheweakconductanceofNFOatRTduetothermalactivationenergiesofafewhundredmeVdependingonthepreparationtechniqueMeier et al. (2013a); Klewe et al. (2014).TheproximityANEinPt/NFOcanalsobeexcluded,sincenospinpolarizationwasfoundusingx-rayresonantmagneticreflectivity(XRMR)measurements,whichareverysensitivetotheinterfacespinpolarizationKuschel et al. (2014).IncaseofPt/YIGGeprägsetal.presentedx-raymagneticcirculardichroismmeasurements(XMCD)withnoevidenceforanyspinpolarizationinPtGeprägs et al. (2012),whileLuetal.couldshowXMCDmeasurementsindicatingmagneticmomentsinPtontheirYIGsamplesLu et al. (2013).FutureinvestigationswithXRMRcangivemoreinsighttothisdiscrepancy.However,Kikkawaetal.couldshowthatapotentialcontributionofaproximityANEadditionaltotheLSSEisnegligiblysmallKikkawa et al. (2013).Thissupportsourconclusion,thatthemainantisymmetriccontributioninourmeasurementsonbothPt/YIGandPt/NFOistheLSSE,whichisdrivenbyanout-of-planetemperaturegradient.Wedonotobserveanysymmetriccontributionfor$\nabla T_\text{x}$withouttipheating.Therefore,PNEandproximityPNEcontributionscanalsobeexcluded.Nevertheless,wefindasmallsymmetriccontributionforstrongtipheatingasdemonstratedinFig. 3 (c)for$\Delta T_{\text{needle}}=\text{RT}+$31\text{\,}\mathrm{K}$$.Intheregionof$H_C$smallpeaksarevisibleundersymmetrizationofthevoltage.Thishintsattheexistenceofanadditionalmagnetothermopowereffectpotentiallyinducedbyatemperaturegradient$\nabla T_\text{y}$andwillbepartoffutureinvestigations.Recently,Wegroweetal.Wegrowe et al. (2014)usedanisotropicheat-transportasaninterpretationforthemeasuredvoltagesusingin-planetemperaturegradients.Intheirwork,theyderivedtheanisotropicfield-dependenttemperaturegradientinFMMandFMIfromtheOnsagerreciprocityrelations.Therefore,thethermocoupleeffectbetweentheFM,theNMandthecontactingtipscangeneratefield-dependentvoltagesifthereisadifferenceintheSeebeckcoefficients.Inourinvestigatedsystems,theSeebeckcoefficientsareindeeddifferentforFM,Ptandthecontacttips.However,sincewedonotobserveafield-dependentvariationoftheISHEvoltagebyusingWtips,anyanisotropicfield-dependentheat-transportcanbeexcludedasthereasonfortheobservedvoltages.Insummary,weinvestigatedtherelevanceofTSSEinPt/YIGandPt/NFOsystems.WefoundnosignificantISHEvoltagesuponapplyinganin-planetemperaturegradientandusingsharpWtips($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$contactdiameter)fortheelectricalcontacting.However,uponusingtipswithmorethan$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$diametercontactarea,whichinduceanadditionalout-of-planetemperaturegradient,anantisymmetriccontributiontotheISHEvoltageofthePtstripcouldbeintroducedatwill.ThisantisymmetriceffectcanbeidentifiedasLSSEwhichwasverifiedbycontrollingtheneedletemperatureandvaryingtheout-of-planetemperaturegradient.Takentogether,inallourexperiments,wethusonlyobserveLSSE-typesignatures.TheseLSSEvoltagescanbereminiscentofaTSSE-typeresponseifanunintentional(orintentional)$\nabla T_\text{z}$ispresent.ThisshowsthatutmostcareisrequiredifoneistointerpretmagnetothermopowereffectsintermsoftheTSSE.WethanktheDFG(SPP1538)andtheEMRPJRPEXL04SpinCalforfinancialsupport.TheEMRPisjointlyfundedbytheEMRPparticipatingcountrieswithinEURAMETandtheEU.
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gr-qc/0309126
SU–GP–03/6–3
Indecomposable Ideals in Incidence Algebras$\,$††${}^{\star}$ To appear in a special issue of Modern Physics Letters A
devoted to the proceedings of “Balfest”,
held May, 2003, in Vietri sul Mare, Italy
Rafael D. Sorkin
Department of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A.
internet email address: [email protected]
Abstract
The elements of a finite partial order $P$ can be identified with the
maximal indecomposable two-sided ideals of its incidence algebra A,
and then for two such ideals, $I\prec J\iff IJ\not=0$. This offers
one way to recover a poset from its incidence algebra.
In the course of proving the above, we classify all of the two-sided
ideals of A.
In contemporary physical theory, the concept of a “space” or, more
formally, of a set with structure, plays a central role. Most notably,
spacetime itself is conceived of in this manner – as a differentiable
manifold. However, one can observe a certain tension between two ways
of conceptualizing such structures and working with them. Tangent
vectors, for example, can be
thought of either as infinitesimal
displacements or as sets of numbers obeying a linear
transformation law. In present day language, the two opposed
tendencies
of thought
can
to some extent be characterized by the words “geometrical” and
“algebraic”, although neither term is really suitable. Perhaps,
“intrinsic” vs “coordinate based” comes closer;
and sometimes the words “synthetic” and “analytic” have been used to
convey the same opposition (as in synthetic versus analytic
geometry).$\,$††${}^{\dagger}$ In the philosophical world, these two attitudes manifest themselves to
some degree as “materialism” and “instrumentalism” although the
correspondence is obviously very imperfect (cf. the oft stated idea that
an instrument reading is always a number.)
Consider, for example, Minkowski spacetime ${\hbox{\openface M}}^{4}$. From the
“intrinsic” side it can be understood, on one hand, as a topological
space of dimension four supporting such concepts as straight line
(inertial motion), light cone, and parallelogram. Or by focusing on its
causal relationships rather than its metric and topological ones, one
can understand ${\hbox{\openface M}}^{4}$ as a certain partially ordered set
(poset), a second “intrinsic” characterization that is nevertheless
very different from the first. In contrast to both these
characterizations, ${\hbox{\openface M}}^{4}$ would be described from the
“coordinate based” side by four real variables $t,x,y,z$ which
geometrically have the meaning of numerical functions on spacetime. Here,
the algebraic relationships among the four variables take center stage,
while the actual elements of the space (the points) withdraw into the
background.
Of course the “intrinsic” and “coordinate based” descriptions of
${\hbox{\openface M}}^{4}$ are mathematically equivalent. The most highly developed
and general instance of this sort of equivalence is the
Gel’fand isomorphism, which implies in particular
that any manifold can be recovered,
as a topological space,
from the $C^{*}$-algebra of scalar functions that
it supports (in effect its coordinate functions). However, a manifold
per se is not yet a spacetime because it lacks metrical information. To
recover that as well, one can proceed as in [1] or following
the more detailed scheme of [2]. Neither approach captures in
any essential manner the Lorentzian character of the metric,
however. (Indeed, the latter scheme is actually incompatible with
Lorentzian signature.) The question thus arises whether there exists a
similar “algebraization” of spacetime based not on its topological and
metrical attributes but on its causal order.
The finding of such a correspondence could be expected to hold interest
for more than one reason. On one hand, some workers, going back to
[3], have viewed algebraization as potentially a means by which
to introduce a fundamental spacetime discreteness, a view which appears
to account for much of the current interest in “non-commutative
geometry”. On the other hand, algebraization has from the outset been
one of the royal roads to the “quantization” of a theory, so that one
might hope that any new equivalence between intrinsic and algebraic
descriptions of spacetime – or of whatever hypothetical substratum one
takes to replace spacetime – would open up new avenues for building a
theory of quantum gravity. It is this second prospect that primarily
animates the considerations of the present paper, which are inspired by
the hypothesis that the deep structure of spacetime is that of a causal set [4] [5]. Since this structure is already inherently
discrete, there is no need to introduce discreteness and therefore
no reason to appeal to algebraization on that score. Nonetheless, one
may still feel it useful to attempt various algebraic reformulations of
causal set kinematics in the hope that one of them might help lead us to
the correct quantum theory of causal set dynamics.
I will have a bit more to say about this in the conclusion, but for now
let us turn to the mathematical question that will primarily concern us
herein: that of finding a suitable “algebraization” of the poset
concept.
As I remarked earlier, a relativistic spacetime is inherently a
partial order$\,$††${}^{\flat}$ Order,
partial order,
partially ordered set,
poset,
and
ordered set
are all synonyms.
(at least to the extent that one can count on the impossibility of
“time travel”).
A causal set is also a partial order, but with the crucial difference
that it is locally finite.$\,$††${}^{\star}$ A poset is called locally finite if all its intervals are
finite. If this is strengthened to the requirement that its “upward”
and “downward” sets be separately finite, the poset becomes suitable
to represent a “finitary topological space”. Posets of this type
have been the subject of much work by the dedicatee and his
co-workers, but in a spatial context rather than a spacetime one
[6]. (See also [7] [8]).
The results to be proven below will of course apply
equally well in this context.
Rather than
seeking an algebra to capture the structure of an arbitrary poset, then,
let us confine ourselves to the simpler case of
locally finite orders; in fact let us simplify still further
to the case of orders whose cardinality is strictly finite.
So let $C$ be a poset of finite cardinality.
Does there exist an algebra A naturally associated to $C$ and from
which, conversely, $C$ can be recovered? One algebra that people have
studied in this connection is the so called incidence algebra of
$C$, which one might view as an algebra of retarded (or, dually,
advanced) functions on $C{\times}C$, the product being given by
convolution. In the finite case, this is just a matrix algebra, or more
accurately, it is the algebra of all matrices with zeros in certain
specified locations which reflect directly the defining order relation
$\prec$ of $C$.
It is known that the incidence algebra does indeed capture the structure of
$C$, at least if one interprets $\prec$ reflexively in the
sense that the diagonals of the matrices representing the elements of
A are left free. Remarkably, the nature of the
space $\leftrightarrow$ algebra correspondence
in this case
can be arranged to be
closely analogous to that of the Gel’fand isomorphism,
even though the algebra which figures
in the latter
is commutative and semisimple,
whereas an
incidence algebra is not only non-commutative,
but almost nilpotent
(which is about as far from semi-simplicity as one can get).
Despite these differences, one can in both cases
choose to
identify the
elements of the underlying space with the maximal two-sided ideals of
A, as explained in [9] and [10].
As we will see in a moment, however, there is quite a different way to
set up a correspondence between $C$ and A in the poset case, and this
is perhaps fortunate, in that one might feel uncomfortable with certain
features of the Gel’fand-like scheme as adapted to partial orders. A
first concern arises from the circumstance that it is not the
fundamental order
relation $\prec$ as such that one directly recovers between elements
regarded as maximal ideals, but rather the “nearest neighbor” or link relation$\,$††${}^{\dagger}$ Also called “covering relation” in the mathematical literature.
[9] [11].
The full precedence relation must then be re-generated via transitive
closure.
In the purely discrete case, this is always possible, so viewing one
relation as more or less “fundamental” than the other seems largely a
matter of taste.
However, the failure to recover $\prec$ directly could bode ill in a
continuum context where the simple precedence relation continues to make
sense but links no longer exist (since between any two causally related
points, one can always interpolate a third).
Moreover, even for finite posets $C$, the correspondence between
elements and maximal ideals falls apart if one adopts the irreflexive
convention$\,$††${}^{\flat}$ defined below for $C$, and this is
disturbing because one would hope that a choice of convention would not
influence the underlying relationships so strongly. Again, one might
hope that some other scheme would be more robust in this regard.
In view of such doubts, it seems worth exploring other ways to recover a
poset from its incidence algebra, a task we begin here by showing that
it is equally possible to identify the poset elements, not with the maximal ideals, but with certain indecomposable ideals; and
by doing so
one obtains the precedence relation directly as a relation
between the corresponding ideals. Of course, it would also make sense
to explore alternatives to the incidence algebra itself, but I will not
attempt that in this paper.
Some Definitions
A finite order is a set $C$ comprising a finite number of
elements and carrying a “precedence relation” $\prec$ which is
transitive and asymmetric. That is,
for arbitrary elements $x,y,z$ of $C$
we always have $x\prec y\prec z\Rightarrow x\prec z$,
and we never have $x\prec y\prec x$ when $x$ and $y$ are distinct.
In addition we will
always assume, unless stated otherwise, that $\prec$ is reflexive,
i.e. that every element $x$ precedes itself: $x\prec x$. Although, this
is in some sense merely a convention that one can adopt or reject at
will, it turns out to influence profoundly the structure of the
incidence algebra.$\,$††${}^{\star}$ For example, with the reflexive convention, A has in general more
idempotents than $C$ has elements, whereas with the opposite,
irreflexive convention, A has no idempotents at all.
We can now define the incidence algebra of $C$ by introducing for
each related pair $x{\prec}y$ a generator $[xy]$ and taking A to be
the set of formal linear combinations of these generators. The order
structure of $C$ is then further encoded in the rule for
multiplication of algebra elements as given by the relation
$[xy]\cdot[yz]=[xz]$, for all triples $x\prec y\prec z$.
For definiteness, we will take the field of scalars to be the complexes
ℂ, although nothing will depend on this.
Two other representations of the incidence algebra are useful
and (for finite $C$) strictly equivalent to the definition just given.
First one may think of A as an algebra of $n\times n$ matrices,
where $n=|C|$ is the cardinality of $C$ and the generator $[xy]$ corresponds
to the matrix with a single $1$ in row $x$, column $y$ and
zeros everywhere else.$\,$††${}^{\dagger}$ In the Dirac notation, this correspondence reads
$[xy]=|{x}\!><\!{y}|$,
a notation that was used in [9].
The asymmetry of the relation $\prec$
then translates into the fact that
if one chooses a suitable labeling for the elements of $C$
(a so called natural labeling)
then
all the matrices representing members of A are upper triangular
(but not strictly so, inasmuch as diagonal generators like $[xx]$ are
also part of A thanks to our standing assumption that $\prec$ is
reflexive).
A slightly different
representation of A comes from
thinking of the entries of the matrix as the values of a “two-point
function” $f:C\times C\mathop{\rightarrow}{\hbox{\openface C}}$. With this representation a member
of the incidence algebra is an arbitrary advanced function and the
algebra product is convolution: $f=g\cdot h\iff f(x,z)=\sum_{y}g(x,y)h(y,z)$.
Of course this is just the formula for matrix multiplication in a slightly
different notation.
By an ideal $I$ of A, I will always mean, in this paper, a
two-sided ideal, in other words a nonempty
subset of A closed under addition,
scalar multiplication,
and left or right multiplication by an arbitrary element of A.
Note in this connection that, by virtue of our choice of reflexive
convention for $C$, A is automatically “unital”: it has an identity
element given by $1=[xx]+[yy]+[zz]+\cdots$, where the sum extends over
all elements of $C$. Consequently there is no need to distinguish, for
example, “regular ideals” from irregular ones [12].
The sum $I_{1}+I_{2}$ of two ideals $I_{1}$ and $I_{2}$ is the
collection of all sums $a_{1}+a_{2}$ where $a_{1}\in I_{1}$ and $a_{2}\in I_{2}$.
Equivalently $I_{1}+I_{2}$ is the least ideal containing both $I_{1}$ and
$I_{2}$: it is their “join” in the lattice of ideals of A.
An indecomposable ideal
in A is then a
non-zero
ideal that cannot be
expressed as the sum of two ideals distinct from itself.
(In the language of lattice theory, such an ideal is said to be “join
irreducible”. A closely related notion was employed in the definitions
of $TIP$ and $TIF$ in [13].)
By a maximal indecomposable ideal
I will mean an indecomposable ideal
that is contained properly in no other indecomposable ideal.
One also defines simply a maximal ideal $I\not={{\hbox{\german A}}}$ as one which
cannot be enlarged without coinciding with A.
(Thus, a maximal ideal is
a maximal element in the family of all ideals not equal to A,
while a maximal indecomposable ideal is
a maximal element in the family of all indecomposable ideals.)
Similarly, we define the product of two ideals $I$ and $J$
as the collection of products of members of $I$ with members of $J$:
$IJ=\left\{xy\,|\,x,y\in{{\hbox{\german A}}}\right\}$. From the definitions it is immediate that
$IJ$ is also an ideal and is contained in both $I$ and $J$. (It is,
however, not in general equal to their intersection $I\cap J$, which
would be their “meet” in the lattice of ideals.)
Finally, we define within an arbitrary order $P$ a downward set as a
subset $D\subseteq P$ that is closed under “taking of pasts”:
$x\prec y\in D\Rightarrow x\in D$.
An upward set $U$ is defined dually.
(In spacetime language, these could be called, respectively, “past
sets” and “future sets”.)
Recovery of the poset from its incidence algebra
Our main result can be stated as a theorem:
THEOREM Let $C$ be a finite order (with reflexive convention) and let
A be its incidence algebra. Then the elements $x$ of $C$ correspond
bijectively with the maximal indecomposable ideals $I$ of A, and
under this correspondence the relation $x_{1}\prec{x_{2}}$ goes over to
$I_{1}I_{2}\not=0$.
Before proving the theorem, let us notice a sense in which this (or any
other) equivalence between a poset and its incidence algebra is somewhat
less satisfactory than the Gel’fand isomorphism mentioned earlier, the
flaw being that A has in general more automorphisms than $C$
has.$\,$††${}^{\flat}$ Because the algebra A is not commutative, it will in general possess
continuous families of inner automorphisms, whereas $C$, being a finite
set, can have at most a finite number of automorphisms. (On the other
hand, it looks as if the superfluity due to the inner automorphisms
might be the only one. That is, it looks as if we might have
$\mathop{\rm Aut}\nolimits(C)\simeq\mathop{{\rm Outer}}\nolimits({{\hbox{%
\german A}}}):=(\mathop{\rm Aut}\nolimits{{\hbox{\german A}}})/(\mathop{{\rm
Inner%
}}\nolimits{{\hbox{\german A}}})$, where $(\mathop{{\rm Inner}}\nolimits{{\hbox{\german A}}})$ is
the group of inner automorphisms of A.)
Although the isomorphism equivalence class of A can be deduced from
that of $C$ and conversely, there is thus some “looseness” in the
correspondence that is not present in the Gel’fand case.
Possibly related is the failure, pointed out in [14], of
the correspondence between a poset and its incidence algebra to be
functorial between the category of finite orders with isotone mappings
and the category of algebras with algebra homomorphisms.$\,$††${}^{\star}$ This is not necessarily a “failing” in itself. For example, the
association to a manifold $M$ of its tensor algebra is not
functorial, nor (as remarked to me by Chris Isham) is the association
to $M$ of its diffeomorphism group. Nevertheless, for certain purposes
the lack of functoriality can be a problem, e.g. if one wished to
reproduce the limiting process of [7] in terms of incidence
algebras.
The ideals of A
We will prove the above theorem by classifying the ideals of A. To
this end, let us notice that every member of A has a unique
expression as a sum of multiples of the generators $[xy]$
that we defined
earlier. To convey the fact that one particular such generator $[xy]$
occurs with a nonzero coefficient in some member of the ideal $I$,
let me say, for lack of a better word, that $[xy]$ “figures in $I$”.
In
contrast, the statement that $[xy]$ “is an element of $I$”
(in symbols, $[xy]\in I$)
means that
some $A\in{I}$ literally coincides with $[xy]$. In the matrix
representation of A, a generator $[xy]$ corresponds to a particular
location in the matrix. It then “figures in” $I$ if some matrix of
$I$ has a nonzero coefficient in that location, whereas it is an element
of $I$ if some matrix of $I$ has a $1$ in that
location and zeros everywhere else. In these terms we can now state a
key lemma.
LEMMA If some $[xy]$ “figures in” the ideal $I$ then it is actually
an element of $I$
PROOF By assumption there is $A\in I$ such that $A=\alpha[xy]+B$ where
$\alpha$ is a nonzero scalar and
$B$ is a sum of multiples of pairs $[uv]$ such that either $u$ differs from
$x$ or $v$ from $y$. But this means that $[xx]B[yy]=0$ because the same
holds for every one of its constituent
pairs
$[uv]$.
Hence
$[xx]A[yy]=\alpha[xx][xy][yy]=\alpha[xy]$
is a multiple of $[xy]$, and this implies
immediately that $[xy]$ itself belongs to $I$ by the definition of an
ideal.
(Namely $[xy]=(1/\alpha)[xx]A[yy]$ must be an element of $I$ if
$A$ is one.)
COROLLARY Every ideal $I$ of A is the set of all linear combinations of
some unique set
${\cal U}(I)$
of generators $[xy]$.
That is, we have for every ideal of the incidence algebra,
$I=\mathop{\rm span}\nolimits{\cal U}(I)$,
where
${\cal U}(I)=\{[x_{1}y_{1}],[x_{2}y_{2}],[x_{3}y_{3}],\cdots[x_{n}y_{n}]\}$,
with the
$[x_{j}y_{j}]$
being uniquely determined by $I$ and conversely.
It is moreover, easy to figure out which
sets
of pairs can belong to an
ideal in this way.
Let $S={\cal U}(I)$ be such a set. If
$[xy]\in S$ and $u\prec x\prec y\prec v$ then $[uv]=[ux][xy][yv]\in{I}$
and therefore $[uv]\in{S}$. Thus $S$ is necessarily closed under the
process of “passing to nested pairs”. To express this succinctly let
us formally introduce this “nesting” as an order relation among pairs.
DEFINITION $[xy]\ll[uv]\iff u\prec x$ and $y\prec v$
This definition makes the set of all pairs
$[xy]$ into a poset $\Gamma$,
and with reference to this auxiliary poset, we see that the sets
${\cal U}(I)$
are precisely the upward sets
of $\Gamma$. Thus we have proved:
THEOREM The ideals of A correspond bijectively with the upward sets of
the poset $\Gamma$ of pairs $[x,y]$
Moreover,
the relation of inclusion between ideals obviously mirrors exactly
the relation of inclusion between the corresponding upward sets.
In particular, we have
LEMMA ${\cal U}(I+J)={\cal U}(I)\cup{\cal U}(J)$
PROOF Recall that $I+J$ is precisely the smallest ideal including both
$I$ and $J$; and notice that the union of two upward sets is also an upward
set.
It follows immediately that
an ideal $I$ is indecomposable iff its corresponding upward set, ${\cal U}(I)$,
is not the union of two upward sets distinct from itself.
But in any (finite) poset
an upward set has this property iff it is
(empty or)
what might be called a
“principal upward set”, that is iff it has the form
$\left\{\eta\,|\,\xi\ll\eta\right\}$ for some $\xi$. We have thus shown:
THEOREM The indecomposable ideals of A correspond bijectively
with the pairs $[xy]$, $x,y\in C$, $x\prec{y}$
From this we can easily conclude that the
maximal indecomposable ideals correspond precisely with the
principal upward sets of minimal elements of $\Gamma$,
which in turn are clearly
the diagonal pairs $[xx]$, for $x{\in}C$.
In other words:
LEMMA An ideal $I\subseteq{{\hbox{\german A}}}$ is maximal indecomposable
$\iff I={{\hbox{\german A}}}[xx]{{\hbox{\german A}}}$ for some $x\in C$.
This substantiates the first assertion of our main theorem.$\,$††${}^{\dagger}$ Very similar reasoning yields the “Gel’fand” correspondence between
maximal ideals and elements of $C$, since a maximal upward set in $\Gamma$ is
precisely the complement of a single minimal element of $\Gamma$, i.e. (as
we have just observed) of a single pair $[xx]$. However, if our only
purpose were to classify the maximal ideals of A, it would be
simpler just to adopt the representation of an element of A as a
matrix and pay attention to the set of zeros on its diagonal.
In order to complete the proof, we have only to verify
that $I_{x}I_{y}\not=0$ iff $x{\prec}y$,
where I’ve written $I_{x}$ for ${{\hbox{\german A}}}[xx]{{\hbox{\german A}}}$.
But this is completely
straightforward. In fact, we have:
LEMMA Let ${{\hbox{\german I}}}(x,y)={{\hbox{\german A}}}[xy]{{\hbox{\german A}}}$ denote the principal ideal generated by $[xy]$.
Then the product ${{\hbox{\german I}}}(x,y){{\hbox{\german I}}}(u,v)$ equals ${{\hbox{\german I}}}(x,v)$ when $y\prec u$, and
zero otherwise.
PROOF Write B for the product in question. Because A is unital,
${{\hbox{\german A}}}{{\hbox{\german A}}}={{\hbox{\german A}}}$, whence ${{\hbox{\german B}}}={{\hbox{\german A}}}[xy]{{\hbox{\german A}}}{{\hbox{%
\german A}}}[uv]{{\hbox{\german A}}}={{\hbox{\german A}}}[xy]{{\hbox{\german A%
}}}[uv]{{\hbox{\german A}}}$. Now,
$[xy]{{\hbox{\german A}}}[uv]\not=0\iff[yu]\in{{\hbox{\german A}}}\iff y\prec u$, and in
that
case,
clearly, ${{\hbox{\german B}}}={{\hbox{\german A}}}[xy][yu][uv]{{\hbox{\german A}}}={{%
\hbox{\german A}}}[xv]{{\hbox{\german A}}}$.
COROLLARY ${{\hbox{\german I}}}(x,x){{\hbox{\german I}}}(y,y)$ equals ${{\hbox{\german I}}}(x,y)$ when $x{\prec}y$, and zero otherwise.
Our main theorem is thus demonstrated. Along the way, we have seen that
every indecomposable ideal is a principal ideal (of some pair $[xy]$).
We have also found (in view of the most recent lemma) an equivalent way to
characterize the maximal indecomposable ideals: An indecomposable ideal
$I$ is maximal iff it is “idempotent” in the sense that $II=I$
(which happens iff $II\not=0$).
Finally, it bears remarking that the last lemma actually furnishes an
order on the full set of indecomposable ideals. Since every such ideal
has the form ${{\hbox{\german A}}}[xy]{{\hbox{\german A}}}$
for some $[xy]$,
the substrate of this order can be taken to be the set of pairs$\,$††${}^{\flat}$ Thus, the order in question shares its substrate with the order
$\Gamma$. The two precedence relations are obviously very different,
however.
$x\prec y$ of elements of $C$ and its precedence relation is
then
$[xy]\prec[uv]\iff y\prec u$.
(Such an order is often called an “interval order”.)
It is actually this
order that we recover most directly from the algebra A.
Interestingly enough, it is neither reflexive nor
irreflexive.$\,$††${}^{\star}$ providing a counter-argument to the opinion that the choice between the
two possibilities is always purely conventional!
Rather, it is precisely its reflexive elements that correspond to
elements of the underlying poset $C$.
Remarks
With the proof of our main theorem, we are in the curious position of
being able to discern two very different algebraic images of our
original poset $C$ within its incidence algebra A. On one hand, we
can identify the element $x\in C$ with the ideal generated by
$[xx]$, on the other hand with the ideal of all algebra elements omitting $[xx]$. Depending on which possibility one selects, one
recovers from the ideals either the relation $\prec$ or its associated
link relation, respectively. Either way, one can conclude that the full
structure of $C$ is captured by A.
However, with the second scheme, this conclusion rests heavily on our
assumption that $C$ is finite, which (trivially) guarantees enough
discreteness so that any two related elements of $C$ can be joined by a
chain of links. In a continuum such as a Lorentzian manifold (or for a
countable dense set thereof), this is assuredly not the case because no
links are present at all. Hence the construction which recovers $\prec$
directly (that based on maximal indecomposable ideals) seems more robust
from this point of view. It might be interesting to test whether this
is indeed true by generalizing A to, say, a globally hyperbolic
spacetime $M$ and asking whether some suitable analog of the
construction of this paper would give back the metric of $M$.
(This would require one to recover the volume density $\sqrt{-g}$
in addition to the causal order of $M$, but that is not obviously
impossible, since $\sqrt{-g}$ does enter into A, via the
definition of the convolution product.)
Of course, it would also be interesting to investigate possible
inter-relationships between the two schemes for recovering the elements
of $C$. Perhaps they could already be seen to be equivalent at the
algebraic level, or perhaps they complement each other in some way.
Concerning physical applications of the duality between a poset and its
incidence algebra, there seems not much to say at present. If one is
thinking in terms of quantum gravity and causal sets, then trading a
causal set $C$ for its incidence algebra A is not necessarily a step
in the right direction, because a quantal “sum over causal sets” seems
easier to imagine (as in [5]) than a corresponding “sum over
algebras A”. On the other hand, the physical causal set $C$ is not
the only poset to figure in the theory. The “poset of stems” played
an important role in the considerations of [5], where it served as
a fixed arena allowing one to define conveniently the Markov process of
“classical sequential growth”. (This poset is illustrated in Figure 1
of [5]. Its elements are the finite orders, and its precedence
relation is “inclusion as a downward set”.) Perhaps the incidence
algebra of this poset could play a role in the search for a
physically appropriate dynamics of “quantal sequential growth”. Such
a dynamics would provide the quantal analog for causal sets of the
classical Einstein equations for continuum Lorentzian manifolds. That
is, it would provide a theory of quantum gravity.
Returning to the realm of pure mathematics for a moment, one can ask
whether algebras other than the incidence algebra might have a role to
play in the “algebraization” of the poset concept. If so, the new
algebra might be constructed either from the incidence algebra or
directly from the elements of the poset. Here, just to illustrate the
type of thing one could consider, is an algebra of the second sort. It
might even be trivial as far as I know, but at least it is not as
obviously trivial as some similar possibilities I played with first! One
takes for generators the elements of $C$ itself, and one imposes three
sets of relations:
(i) $a\prec b\prec c\Rightarrow abc=ac$;
(ii) $a\not=b\Rightarrow aba=0$;
(iii) $a\not=b,\,a\prec b\Rightarrow ba=0$.
If this construction is worthy of further consideration, one might begin
by asking whether it is functorial and how the resulting algebra’s
automorphism group relates to that of $C$.
I would like to thank Prakash Panangaden, Ioannis Raptis and Kamesh Wali
for inspiration and encouragement.
This research was partly supported by NSF grant PHY-0098488 and by a
grant from the Office of Research and Computing of Syracuse University.
References
[1]
R. Geroch,
“Einstein Algebras”,
Comm. Math. Phys. 26: 271-275 (1972)
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L. Bombelli, J. Lee, D. Meyer and R.D. Sorkin,
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[5]
David P. Rideout and Rafael D. Sorkin,
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Phys. Rev. D 61: 024002 (2000)
$\langle$gr-qc/9904062$\rangle$
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G. Sparano, and P. Teotonio-Sobrinho,
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$\langle$hep-th/9510217$\rangle$
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$\langle$gr-qc/9904079$\rangle$ |
Anisotropic Weyl symmetry and cosmology
Taeyoon
[email protected] ,
Phillial [email protected], and
Jongsu [email protected]
Department of Physics and
Institute of Basic Science,
Sungkyunkwan University, Suwon 440-746 Korea
Abstract
We construct an anisotropic Weyl invariant theory in the ADM
formalism and discuss its cosmological consequences. It extends the
original anisotropic Weyl invariance of Hořava-Lifshitz gravity
using an extra scalar field. The action is invariant under the
anisotropic transformations of the space and time metric components
with an arbitrary value of the critical exponent $z$. One of the
interesting features is that the cosmological constant term
maintains the anisotropic symmetry for $z=-3$. We also include the
cosmological fluid and show that it can preserve the anisotropic
Weyl invariance if the equation of state satisfies $P=z\rho/3$.
Then, we study cosmology of the Einstein-Hilbert-anisotropic Weyl
(EHaW) action including the cosmological fluid, both with or without
anisotropic Weyl invariance. The correlation of the critical
exponent $z$ and the equation of state parameter $\bar{\omega}$
provides a new perspective of the cosmology. It is also shown that
the EHaW action admits a late time accelerating universe for an
arbitrary value of $z$ when the anisotropic conformal invariance is
broken, and the anisotropic conformal scalar field is interpreted as
a possible source of dark energy.
1 Introduction
Gravity theory with a local Weyl invariance was proposed as an
alternative theory of gravity [1] and various aspects have
been investigated for a long time. Among them, there are two main
avenues. The first one is conformal gravity where the theory is
built on the local conformal invariance and the general covariance.
In this theory, the conformal invariant action can be realized by
introducing the quadratic Weyl curvature tensor. This theory has the
dimensionless gravitational coupling constant and the property that
it is renormalizable [2], asymptotically free
[3] and could be potentially unitary [4]. The
other way of achieving conformal invariance is to introduce an extra
gauge scalar field to compensate the non-invariance of the
Einstein-Hilbert action [5]. Ever since, many attempts of
incorporating conformal invariance in the theory of general
relativity have been carried out in the diverse areas of theoretical
physics [6, 7, 8, 9, 10, 11, 12, 13].
One application of conformal symmetry in the latter case is to consider the conformal
scalar field which is non-minimally coupled to the curvature scalar with a
coupling constant [14]. In such a theory, the gravity
sector is described by the Einstein-Hilbert action, and the
conformal scalar field is treated as conformal matter. The exact
conformal symmetry is imposed with a specific choice of the
non-minimal coupling constant. In particular, this approach found
wide applications in cosmological models
[14, 15, 16] in which it was first proposed to
describe decaying cosmological constant [17].
The original conformal invariance is isotropic in the sense that the
time and space components of the metric transform in the same
manner. On the other hand, the canonical ADM formalism
[18] decomposes space-time into space and time. In this
background, one could envisage an anisotropic Weyl transformation
in which the space and time components of the metric transform
differently, but still leaves the “action” invariant. Especially,
such an attempt is well motivated by the recent upsurge [19] of
interest in the Hořava-Lifshitz gravity [20] which has
the feature of anisotropy between the space and time. In fact, in
[20] it is shown that classical action can have an
anisotropic (local) Weyl invariance for specific values of the
critical exponent $z=3$ and the free parameter of the metric on the
space of metrics. In this case, each component of the metric
transforms as $g_{00}\rightarrow e^{6\omega(t,x)}g_{00}~{},g_{0i}\rightarrow e^{2\omega(t,x)}%
g_{0i}~{},g_{ij}\rightarrow e^{2\omega(t,x)}g_{ij}.$
As was mentioned before, however, one can also construct a
conformally invariant gravity with curvature scalar by introducing
an extra scalar field. Therefore, it seems natural to attempt to
extend the anisotropic Weyl invariance of Hořava-Lifshitz
gravity in this way. The purpose of this work is to show that this
type of extension is indeed possible. We first review the
anisotropic Weyl invariance of Hořava-Lifshitz gravity in the
ADM formalism. Then, we extend the analysis to anisotropic Weyl
invariant theory by introducing a scalar field with a suitable
conformal weight which is given by the critical exponent.
It turns out that the conformal invariance can be preserved
for any value of the critical exponent, especially under the
transformation $g_{00}\rightarrow e^{2z\omega(t,x)}g_{00}$.
We will treat the resulting anisotropic conformal scalar field as
describing the conformal matter and couple to the Einstein-Hilbert
action444 In this paper, we only pay attention to the
anisotropic Weyl invariance at the lowest curvature level and do not
include the higher derivative terms such as the Cotton tensor, $R^{2}$
term, etc. in the action.. For $z=1$, the action reduces to the
conformal matter theory of Ref. [14, 15, 16].
Otherwise, the local Lorentz invariance is explicitly broken.
One of the motivations for considering an arbitrary value of $z$ is
that we look for the role of the critical exponent in the cosmology
and interpret the anisotropic conformal scalar field as a possible
source of dark energy [21]. In order to describe the
cosmology in this context, we first break the conformal invariance
[22, 23, 24] explicitly by considering an arbitrary
potential term for the scalar field. Then, we search for
cosmological solutions paying attention to the possible role of $z$
in the evolution of the universe. For example, we compare with the
cosmological model with $z=1$, and show that accelerating phase can
exist in vacuum with an arbitrary value of $z$. We also introduce a
cosmological fluid with equation of state $\bar{\omega}$, and study
the cosmology. If the conformal invariance is imposed on the fluid,
we find that the critical exponent and the equation of state must be
correlated. When the conformal invariance is broken, there is no
correlation and the critical exponent $z$ remains as a free
parameter. We will present
the conformal preserving case also, because it seems that the
cosmology in this case also shows some interesting feature. For
example, the cosmological constant term enjoys anisotropic Weyl
invariance with $z=-3$, in which case a cosmological solution which
extrapolates between the matter dominated epoch and the accelerating
phase exists.
This paper is organized as follows. In Sec.2, we briefly review the
anisotropic Weyl invariance in the Hořava-Lifshitz gravity. In
Sec.3, we consider the Einstein-Hilbert action with an (isotropic)
conformal symmetry in the ADM formalism and extend to the action
with an anisotropic Weyl invariance by introducing a scalar field.
In particular, we show that cosmological fluid can preserve the
anisotropic Weyl invariance if the equation of state satisfies $P=z\rho/3$. In Sec.4, we study cosmology with the FRW metric and apply
it to the EHaW action with an arbitrary potential including also the
cosmological fluid with or without anisotropic Weyl invariance. We
briefly summarize the results and discuss them in Sec.5.
2 Anisotropic Weyl invariance in the Hořava-Lifshitz gravity
Let us consider an action of $z=3$ gravity theory in $1+3$
dimensions [20]:
$$\displaystyle S_{aH}$$
$$\displaystyle=$$
$$\displaystyle\int dt\,d^{3}x\,\sqrt{g}\,N\left\{\frac{2}{\kappa^{2}}\left(K_{%
ij}K^{ij}-\lambda K^{2}\right)-\frac{\kappa^{2}}{2w^{4}}C_{ij}C^{ij}\right\},$$
(2.1)
where $\kappa,~{}w$ are dimensionless constant
parameters, $K_{ij}$ is the extrinsic curvature which is defined by
$$\displaystyle K_{ij}=-\frac{1}{2N}(\partial_{t}g_{ij}-\nabla_{i}N_{j}-\nabla_{%
j}N_{i})$$
(2.2)
and
$$\displaystyle C^{ij}=\epsilon^{ik\ell}\nabla_{k}\left(R^{j}{}_{\ell}-\frac{1}{%
4}R\delta_{\ell}^{j}\right)$$
(2.3)
is the Cotton tensor. Here, $R$ is the curvature scalar in 3 space
dimensions. Under anisotropic Weyl transformation,
$$\displaystyle N\rightarrow e^{3\omega(t,x)}N,~{}~{}N_{i}\rightarrow e^{2\omega%
(t,x)}N_{i},~{}~{}g_{ij}\rightarrow e^{2\omega(t,x)}g_{ij},$$
(2.4)
the action (2.1) transforms to
$$\displaystyle S_{aH}$$
$$\displaystyle\rightarrow$$
$$\displaystyle S_{aH}=\int dt\,d^{3}x\,\sqrt{g}\,N\left\{\frac{2}{\kappa^{2}}%
\left[K_{ij}K^{ij}-\lambda K^{2}-2(1-3\lambda)K\left(\frac{\dot{\omega}-\nabla%
_{i}\omega N^{i}}{N}\right)\right.\right.$$
(2.5)
$$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+%
\left.\left.3(1-3\lambda)\left(\frac{\dot{\omega}-\nabla_{i}\omega N^{i}}{N}%
\right)^{2}\right]-\frac{\kappa^{2}}{2w^{4}}C_{ij}C^{ij}\right\}.$$
In the above equation (2.5), the third and fourth terms vanish
with $\lambda=1/3$, and the action (2.1) has an anisotropic
Weyl invariance. As was shown in [20], this symmetry is an
additional gauge symmetry supplementing the foliated
diffeomorphisms.
It is interesting to note that under general
anisotropic Weyl transformation,
$$\displaystyle N\rightarrow e^{z\omega(t,x)}N,~{}~{}N_{i}\rightarrow e^{2\omega%
(t,x)}N_{i},~{}~{}g_{ij}\rightarrow e^{2\omega(t,x)}g_{ij},$$
(2.6)
the extrinsic curvature terms become
$$\displaystyle K_{ij}K^{ij}-\frac{1}{3}K^{2}\rightarrow e^{-2z\omega}(K_{ij}K^{%
ij}-\frac{1}{3}K^{2}).$$
(2.7)
Since the volume element transforms as $\sqrt{g}N\rightarrow e^{(3+z)\omega}\sqrt{g}N$ under the transformation $(\ref{KK1})$, it
has no anisotropic Weyl invariance unless $z=3$. One may, however,
write an anisotropic Weyl invariant action by introducing some
scalar field which can compensate the conformal weight $3-z$. With a
proper power of this scalar field, the conformal weight $z-3$ coming
from the volume element and the Cotton tensor term which transforms
as $C_{ij}C^{ij}\rightarrow e^{-6\omega}C_{ij}C^{ij}$, can also be
compensated. In the next section, we will explicitly construct
anisotropic Weyl action which is invariant for an arbitrary $z$
including curvature scalar. As was mentioned before, we will drop
the Cotton tensor term and include only terms up to second
derivatives.
3 Anisotropic Weyl invariant gravity for general $z$
Let us first consider a conformally invariant action in four
dimensions. This is given by
$$\displaystyle S_{C}=\int\,d^{4}x\sqrt{-g}\phi^{2}\left(R_{(4)}-6\frac{\nabla_{%
\gamma}\nabla^{\gamma}\phi}{\phi}\right).$$
(3.1)
The above action is invariant under [5, 10, 25]
$$\displaystyle g_{\mu\nu}\rightarrow e^{2\omega}g_{\mu\nu},~{}~{}\phi%
\rightarrow e^{-\omega}\phi,$$
(3.2)
where $\omega=\omega(t,x)$.
Considering the ADM decomposition and rearranging terms, it can be
rewritten as [10]
$$\displaystyle S_{C}=\int\,dtd^{3}xN\sqrt{g}~{}\varphi^{4}\left(R-8\frac{\nabla%
_{i}\nabla^{i}\varphi}{\varphi}+B_{ij}B^{ij}-B^{2}\right),$$
(3.3)
where $\varphi^{2}=\phi$ and
$$\displaystyle B_{ij}$$
$$\displaystyle=$$
$$\displaystyle K_{ij}-\frac{2}{N\varphi}g_{ij}(\dot{\varphi}-\nabla_{i}\varphi N%
^{i})$$
(3.4)
$$\displaystyle\equiv$$
$$\displaystyle K_{ij}+\frac{\theta}{2N}g_{ij},$$
(3.5)
with
$\theta=-4(\dot{\varphi}-\nabla_{i}\varphi N^{i})/\varphi$. The
above action (3.3) is invariant under
$$\displaystyle N$$
$$\displaystyle\rightarrow$$
$$\displaystyle e^{\omega}N,~{}~{}~{}N_{i}\rightarrow e^{2\omega}N_{i},$$
$$\displaystyle g_{ij}$$
$$\displaystyle\rightarrow$$
$$\displaystyle e^{2\omega}g_{ij},~{}~{}\varphi\rightarrow e^{-\frac{\omega}{2}}\varphi.$$
(3.6)
We can extend the above procedure to an anisotropic Weyl
invariance. One can show that the following action
$$\displaystyle S_{\varphi}=\int\,dtd^{3}xN\sqrt{g}~{}\left[~{}\varphi^{2z+2}%
\left(R-8\frac{\nabla_{i}\nabla^{i}\varphi}{\varphi}\right)+\varphi^{-2z+6}B_{%
ij}B^{ij}-\varphi^{-2z+6}B^{2}~{}\right],$$
(3.7)
is invariant with respect to
$$\displaystyle N$$
$$\displaystyle\rightarrow$$
$$\displaystyle e^{z\omega}N,~{}~{}~{}N_{i}\rightarrow e^{2\omega}N_{i},$$
$$\displaystyle g_{ij}$$
$$\displaystyle\rightarrow$$
$$\displaystyle e^{2\omega}g_{ij},~{}~{}\varphi\rightarrow e^{-\frac{\omega}{2}}\varphi.$$
(3.8)
Note that we
have a factor $z$ in the above equations which turns out to be
coincident with the critical exponent. That is, beside the local
transformations (3.8), the above action
(3.7) is also invariant with respect to a global
transformation
$$\displaystyle t\rightarrow b^{z}t,~{}~{}x\rightarrow bx,~{}~{}N_{i}\rightarrow
b%
^{1-z}N_{i},~{}~{}\varphi\rightarrow b^{-\frac{1}{2}}\varphi,$$
(3.9)
with $N$ and $g_{ij}$ being unchanged.
When $z=1$, the above action (3.7) reduces to the ADM
decomposition of the conformally invariant action (3.1). On the
other hand, in the same manner, considering the transformation law
(2.7) in the previous section, the extrinsic curvature terms
with an anisotropic Weyl invariance can be written as
$$\displaystyle S_{aK}$$
$$\displaystyle=$$
$$\displaystyle\int dt\,d^{3}x\,\sqrt{g}\,N\left\{\varphi^{-2z+6}\left(K_{ij}K^{%
ij}-\frac{1}{3}K^{2}\right)\right\}.$$
(3.10)
From the action (3.7), (3.10), one can construct the general
action with anisotropic Weyl invariance as follows
$$\displaystyle S_{aW}$$
$$\displaystyle=$$
$$\displaystyle\int\,dtd^{3}xN\sqrt{g}~{}\left[~{}\varphi^{2z+2}\eta\left(R-8%
\frac{\nabla_{i}\nabla^{i}\varphi}{\varphi}\right)+\right.$$
(3.11)
$$\displaystyle\left.\varphi^{-2z+6}(\eta+\xi)B_{ij}B^{ij}-\varphi^{-2z+6}(\eta+%
\frac{\xi}{3})B^{2}-V(\varphi)\right],$$
where $\eta,~{}\xi$ are some constants and for the anisotropic Weyl
invariance, $V(\varphi)=\alpha\varphi^{2(z+3)}$ with some constant
$\alpha$. Note that in the case of $z=1$ the transformation law of
$N$ becomes isotropic Weyl transformation, i.e., ordinary conformal
transformation. As expected, for $\eta=1$, $\xi=\alpha=0$ and
$\eta=\alpha=0$, $\xi=1$ the action (3.11) becomes
(3.7) and (3.10) respectively. We also note that in
the case of $z=-3$, the potential term becomes a cosmological
constant.
The most general action which includes the cosmological fluid will
be given by
$$\displaystyle S_{aWf}=S_{aW}+S_{f},$$
(3.12)
where $S_{f}$ is the action which has anisotropic Weyl invariance
without $\varphi$ term. We show in the following lines that the
equation of the state for this cosmological fluid satisfies
$$\displaystyle P=\frac{z}{3}\rho.$$
(3.13)
In order to see this, we first vary for $N$, $N^{i}$, $g^{ij}$,
$\varphi$, and obtain the equations of motion:
$$\displaystyle\delta_{N}S_{aWf};$$
$$\displaystyle\varphi^{2z+2}\{\eta\left(R-8\frac{\nabla_{i}\nabla^{i}\varphi}{%
\varphi}\right)-\varphi^{-4z+4}(\eta+\xi)B_{ij}B^{ij}+\varphi^{-4z+4}(\eta+%
\frac{\xi}{3})B^{2}\}$$
(3.14)
$$\displaystyle-\alpha\varphi^{2(z+3)}-\rho=0,$$
$$\displaystyle\delta_{N^{i}}S_{aWf};$$
$$\displaystyle 2\varphi^{-2z+6}\{-(\eta+\xi)\nabla_{j}B^{j}_{i}+2(z-3)(\eta+\xi%
)\frac{\nabla_{j}\varphi}{\varphi}B^{j}_{i}\}$$
(3.15)
$$\displaystyle+2\varphi^{-2z+6}\{(2(1-z)\eta+\frac{2}{3}(3-z)\xi)\frac{\nabla_{%
i}\varphi}{\varphi}B+(\eta+\frac{\xi}{3})\nabla_{i}B\}=0,$$
$$\displaystyle\delta_{g^{ij}}S_{aWf};$$
$$\displaystyle N\varphi^{2z+2}\{\eta A_{ij}^{(1)}+\varphi^{-4z+4}(\eta+\xi)A_{%
ij}^{(2)}-\varphi^{-4z+4}(\eta+\frac{\xi}{3})A_{ij}^{(3)}\}$$
(3.16)
$$\displaystyle+\frac{N}{2}\alpha\varphi^{2(z+3)}g_{ij}-\frac{N}{2}g_{ij}P=0,$$
where $\rho=-\left.\frac{1}{\sqrt{g}}\frac{\delta S_{f}}{\delta N}\right.,~{}Pg_{ij}=%
-\left.\frac{2}{N\sqrt{g}}\frac{\delta S_{f}}{\delta g^{ij}}\right.$,
$$\displaystyle A_{ij}^{(1)}$$
$$\displaystyle=$$
$$\displaystyle R_{ij}-\frac{1}{2}g_{ij}R-\frac{\nabla_{i}\nabla_{j}N}{N}-4(z-1)%
\frac{\nabla_{i}N}{N}\frac{\nabla_{j}\varphi}{\varphi}+4z\frac{\nabla_{j}N}{N}%
\frac{\nabla^{j}\varphi}{\varphi}g_{ij}$$
$$\displaystyle-2(2z+1)(z-3)\frac{\nabla_{i}\varphi\nabla_{j}\varphi}{\varphi^{2%
}}+\frac{\nabla_{k}\nabla^{k}N}{N}g_{ij}+2(z-1)(2z+1)\frac{\nabla_{k}\varphi%
\nabla^{k}\varphi}{\varphi^{2}}g_{ij}-$$
$$\displaystyle 2(z+1)\frac{\nabla_{i}\nabla_{j}\varphi}{\varphi}+2(z+1)\frac{%
\nabla_{k}\nabla^{k}\varphi}{\varphi}g_{ij},$$
$$\displaystyle A_{ij}^{(2)}$$
$$\displaystyle=$$
$$\displaystyle\frac{-N_{i}\nabla_{k}B_{j}^{k}}{N}-\frac{N_{j}\nabla_{k}B_{i}^{k%
}}{N}+\frac{\nabla_{i}N^{k}B_{jk}}{N}+\frac{\nabla_{j}N^{k}B_{ik}}{N}+\frac{N_%
{k}\nabla^{k}B_{ij}}{N}-2B_{i}^{k}B_{jk}-\frac{1}{2}B_{kl}B^{kl}g_{ij}+$$
$$\displaystyle BB_{ij}-\frac{\dot{B_{ij}}}{N}+(1-\frac{z}{2})\frac{\theta B_{ij%
}}{N}+4(z-3)\frac{\nabla_{k}\varphi}{\varphi}\frac{N_{i}B^{k}_{j}}{N}+\frac{4%
\nabla_{i}\varphi}{\varphi}\frac{N_{j}B}{N},$$
$$\displaystyle A_{ij}^{(3)}$$
$$\displaystyle=$$
$$\displaystyle\frac{B^{2}}{2}g_{ij}-\frac{\nabla_{j}BN_{i}}{N}-\frac{\nabla_{i}%
BN_{j}}{N}+2z\frac{\nabla_{j}\varphi}{\varphi}\frac{N_{i}B}{N}+2z\frac{\nabla_%
{i}\varphi}{\varphi}\frac{N_{j}B}{N}-\frac{z\theta}{2N}Bg_{ij}+\frac{\nabla_{k%
}BN^{k}g_{ij}}{N}-\frac{\dot{B}g_{ij}}{N},$$
and
$$\displaystyle\delta_{\varphi}S_{aWf};$$
$$\displaystyle N\varphi^{2z+1}[(2z+2)\eta R-16(2z+1)\eta\frac{\nabla_{i}\nabla^%
{i}\varphi}{\varphi}-16z(2z+1)\eta\frac{\nabla_{i}\varphi\nabla^{i}\varphi}{%
\varphi^{2}}+$$
(3.17)
$$\displaystyle\{2(\eta-\xi)+2z(\eta+\frac{\xi}{3})\}B^{2}\varphi^{-4z+4}-4z\eta
NB%
\theta\varphi^{-4z+4}+8\eta\nabla_{i}BN^{i}\varphi^{-4z+4}+$$
$$\displaystyle(-2z+6)(\eta+\xi)B_{ij}B^{ij}\varphi^{-4z+4}-8\eta\nabla_{i}%
\nabla^{i}N-16(2z+1)\eta\nabla_{i}N\frac{\nabla^{i}\varphi}{\varphi}-8\eta\dot%
{B}\varphi^{-4z+4}]$$
$$\displaystyle-2(z+3)N\alpha\varphi^{2z+5}=0.$$
In the Appendix, we show that Eq. (3.13) must be satisfied in
order to be consistent with Eqs. (3.14)$\sim$(3.17).
Note that this condition is only for cosmological fluid with
anisotropic Weyl invariance555When $z=1$, this condition
corresponds to $T^{\mu}_{\mu}=0$ which represents the condition of
isotropic conformally invariant fluid. That is,
the above condition (3.13) is
equivalent to $P=\frac{1}{3}\rho$.
It is pointed out that the
above condition (3.13) also can be obtained by dimensional
analysis as shown in [26].. It can be violated, if we do not insist on
the symmetry. In
the next section, we will consider cosmological consequences of both
unbroken and broken cases. In the broken case, we consider an
arbitrary potential $V(\varphi)$ breaking anisotropic Weyl
invariance in Eq. (3.11).
4 Cosmological solutions
In order to investigate cosmological consequences, we first recall
that the Einstein-Hilbert action in the ADM formalism is given by
$$\displaystyle S_{EH}=\int\,dtd^{3}xN\sqrt{g}~{}\frac{1}{2\kappa^{2}}(R+K_{ij}K%
^{ij}-K^{2}),$$
(4.1)
and consider
$$\displaystyle S_{EHaW}=S_{EH}+S_{aWf},$$
(4.2)
where $S_{aWf}$ is the anisotropic Weyl action given in Eq.
(3.12) with an arbitrary potential $V(\varphi)$. Let us
introduce the Friedmann-Robertson-Walker metric via
$$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-\bar{\kappa}r^{2}}+%
r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\right],$$
(4.3)
where $\bar{\kappa}=-1,0,1$ corresponding to an open, flat, closed
universe respectively. Note that for $z=1$, $\eta=1$, and $\xi=0$,
the action (4.2) is reduced to the conformal
quintessence [14, 15, 16]. In this background, one
finds the following results,
$$\displaystyle K_{ij}$$
$$\displaystyle=$$
$$\displaystyle-Hg_{ij}~{},~{}~{}K=-3H,$$
(4.4)
$$\displaystyle R_{ij}$$
$$\displaystyle=$$
$$\displaystyle\frac{2\bar{\kappa}}{a^{2}}g_{ij}~{},~{}~{}R=\frac{6\bar{\kappa}}%
{a^{2}}.$$
(4.5)
For the above action (4.2), the equations of motion
become
$$\displaystyle\delta_{N}S_{EHaW};$$
$$\displaystyle\frac{1}{2\kappa^{2}}(R-K_{ij}K^{ij}+K^{2})+\delta_{N}S_{aWf},$$
(4.6)
$$\displaystyle\delta_{N^{i}}S_{EHaW};$$
$$\displaystyle\frac{1}{2\kappa^{2}}(-2\nabla_{j}K^{j}_{i}+2\nabla_{i}K)+\delta_%
{N^{i}}S_{aWf},$$
(4.7)
$$\displaystyle\delta_{g^{ij}}S_{EHaW};$$
$$\displaystyle\frac{N}{2\kappa^{2}}A_{ij}^{(0)}+\delta_{g^{ij}}S_{aWf},$$
(4.8)
where
$$\displaystyle A_{ij}^{(0)}$$
$$\displaystyle=$$
$$\displaystyle R_{ij}-\frac{1}{2}g_{ij}R-\frac{\nabla_{i}\nabla_{j}N}{N}+\frac{%
\nabla_{k}\nabla^{k}N}{N}g_{ij}-\frac{N_{i}\nabla_{k}K_{j}^{k}}{N}-\frac{N_{j}%
\nabla_{k}K_{i}^{k}}{N}+\frac{\nabla_{i}N^{k}K_{jk}}{N}+$$
(4.9)
$$\displaystyle\frac{\nabla_{j}N^{k}K_{ik}}{N}+\frac{N_{k}\nabla^{k}K_{ij}}{N}-2%
K_{i}^{k}K_{jk}-\frac{1}{2}K_{kl}K^{kl}g_{ij}+KK_{ij}-\frac{\dot{K_{ij}}}{N}$$
$$\displaystyle-\frac{K^{2}}{2}g_{ij}+\frac{\nabla_{j}KN_{i}}{N}+\frac{\nabla_{i%
}KN_{j}}{N}-\frac{\nabla_{k}KN^{k}g_{ij}}{N}+\frac{\dot{K}g_{ij}}{N}.$$
Assuming a homogeneous scalar field $\varphi(x)=\varphi(t)$, one
obtains the following equations from Eqs. (4.6) $\sim$
(4.8) and (3.17):
$$\displaystyle(\frac{1}{2\kappa^{2}}\varphi^{2z-6}+\eta)H^{2}=\eta H\theta-\eta%
\frac{\theta^{2}}{4}+\frac{1}{6}{\varphi}^{2z-6}V+\frac{\rho}{6}\varphi^{2z-6}%
-\frac{\bar{\kappa}}{a^{2}}\left(\frac{\varphi^{2z-6}}{2\kappa^{2}}+\eta%
\varphi^{4z-4}\right),$$
(4.10)
$$\displaystyle(\frac{1}{2\kappa^{2}}\varphi^{2z-6}+\eta)\dot{H}=-\frac{\varphi^%
{2z-6}}{4}(\rho+P)+\eta\frac{\dot{\theta}}{2}+\frac{\eta z}{2}(-H\theta+\frac{%
\theta^{2}}{2})+\frac{\bar{\kappa}}{a^{2}}\left(\frac{\varphi^{2z-6}}{2\kappa^%
{2}}+\eta\varphi^{4z-4}\right),$$
(4.11)
and
$$\displaystyle\eta(z+1)\frac{\bar{\kappa}}{a^{2}}\varphi^{4z-4}+\eta(z+3)H^{2}-%
3\eta H\theta-\frac{\eta(z-3)}{4}\theta^{2}+2\eta\dot{H}-\eta\dot{\theta}-%
\frac{1}{12}\varphi^{2z-5}V^{\prime}=0,$$
(4.12)
where ${}^{\prime}$ denotes differentiation with respect to $\varphi.$
Note that Eq. (4.7) is satisfied trivially. When $\eta=V=0$,
one can check that Eqs. (4.10), (4.11) are equivalent to
the ordinary Friedmann equations. We assume flat universe with
$\bar{\kappa}=0$ from here on. Then, Eqs.
(4.10)$\sim$(4.12) can combine into the following
equation,
$$\displaystyle(z+3)H^{2}+2\dot{H}$$
$$\displaystyle=$$
$$\displaystyle\kappa^{2}\left(\frac{z}{3}\rho-P\right)-\frac{\kappa^{2}}{3}%
\left(\frac{\varphi}{2}V^{\prime}(\varphi)-(z+3)V(\varphi)\right).$$
(4.13)
In the next subsections, we discuss possible solutions of these
equations for both with or without anisotropic Weyl invariance.
4.1 Case with anisotropic
invariance
In the case of cosmological fluid with anistotropic
invariance, introducing $P=\bar{\omega}\rho$ then $z=3\bar{\omega}$.
Here $\bar{\omega}$ is the equation of state parameter. Since the
right hand side of Eq. (4.13) vanishes in this case, we have
$$\displaystyle(z+3)H^{2}+2\dot{H}=0.$$
(4.14)
In the above equation, one can check easily that in the case of $z\neq-3$, $H$
behaves as $1/t$ which correspond the power law solutions
and for $z=-3$, $H$ is a constant value which represents the exponentially
accelerating solution.
To see these behaviors in more detail, we find explicit solutions of Eqs.
(4.10)$\sim$(4.12). For power law solutions, we have
$$\displaystyle\rho=\rho_{0}a^{-3(1+\bar{\omega})},~{}~{}a=a_{0}t^{\frac{2}{3(1+%
\bar{\omega})}},~{}~{}\varphi=\varphi_{0}t^{-\frac{1}{3(1+\bar{\omega})}},~{}~%
{}\alpha=0,$$
(4.15)
where $\rho_{0},a_{0},\varphi_{0}$ are positive constants which satisfy
the relations
$\rho_{0}a_{0}^{-3(1+\bar{\omega})}=\frac{4}{3\kappa^{2}(1+\bar{\omega})^{2}}$.
This solution preserves the standard cosmology. Most of all, in this
case, it should be remarked that $z=1,~{}0$ correspond to
$\bar{\omega}=1/3$ (radiation dominance),$~{}0$ (matter dominance)
respectively. In the region $-3<z<-1$, it has an accelerating phase,
whereas decelerating phase in $z>-1$. In particular, for $z=-3$
the solution is
$$\displaystyle\rho=P=0,~{}~{}a=a_{0}e^{Ht},~{}~{}\varphi=\varphi_{0}e^{-\frac{H%
}{2}t},$$
(4.16)
where $H^{2}=\kappa^{2}\alpha/3$.
Note that with the anisotropic Weyl invariant fluid with
$P=z\rho/3$, the universe is described by different values of the
critical exponent in radiation-dominated, matter-dominated, and
vacuum-dominated epochs, but still can maintain the anisotropic Weyl
invariance666Note that the critical exponent decreases as
the universe evolves. One possible interpretation of the result
might be that this running behavior of the parameter $z$ in the
Lagrangian is due to the renormalization properties of the scalar
field theory and could be viewed as being reasonable in the sense of
the renormalization group. However, a demonstration of this
running behavior is beyond the scope of the present paper, and since
the macroscopic equations of state in each epoch is involved with
different cosmological fluids, it may be hard that it could actually
be realized at the microscopic level. Another interpretation is
that there are several anisotropic conformal invariant sectors
described by $z$ and each epoch corresponds to one of these sectors
via cosmological fluid contents at that epoch..
However, if we break the anisotropic Weyl invariance of the
cosmological fluid, while preserving it for the potential part, we
can describe the extrapolation from matter dominance to vacuum
dominance of the universe with a single value of the critical
exponent.
In this case, with $P=\bar{\omega}\rho$, Eq.(4.13) becomes
$$\displaystyle(z+3)H^{2}+2\dot{H}=\frac{\kappa^{2}}{3}(z-3\bar{\omega})\rho.$$
(4.17)
In particular for $z=-3$,
$$\displaystyle\dot{H}=-\frac{\kappa^{2}}{2}(1+\bar{\omega})\rho.$$
(4.18)
Then, from Eqs. (4.10)$\sim$(4.12) we find the following
solution
$$\displaystyle H=\sqrt{\frac{\kappa^{2}\alpha}{3}}\coth[At],~{}~{}\varphi=B%
\left(\sinh[At]\right)^{-\frac{1}{3(1+\bar{\omega})}},~{}~{}\rho=\alpha(\sinh[%
At])^{-2},$$
(4.19)
where $A=(1+\bar{\omega})\sqrt{3\kappa^{2}\alpha}/2$, $B$ is a constant
and $a(t)=a_{0}(\sinh[At])^{2/3(1+\bar{\omega})}$.
For early universe with small t, $a(t)\sim t^{2/3(1+\bar{\omega})}$, and it describes the standard
power law expansion with equation of state parameter with $\bar{\omega}$.
For late time with large $t$, $a(t)\sim e^{\sqrt{\frac{\Lambda}{3}}t}$ with $\Lambda=\kappa^{2}\alpha$. This
solution describes an extrapolation from matter-dominance into
vacuum dominance.
4.2 Case without anisotropic invariance
In this case, $P=\bar{\omega}\rho$ and $V(\varphi)$ is different from the anisotropic invariant
potential $\alpha\varphi^{2(z+3)}$. Eq.(4.13) can be solved in three cases.
In the case $\rho=0$, there are two potential forms that are
interesting from the cosmological point of view.
i) $V(\varphi)=A_{1}\varphi^{-2z+6}+A_{2}\varphi^{-2z+10}$, where
$A_{1},~{}A_{2}$ are some constants. Then
Eqs.(4.10)$\sim$(4.12) and Eq.(4.13) yields the
following solution:
$$\displaystyle H=B_{1}\tanh[B_{2}t],~{}~{}\varphi=(-2\eta\kappa^{2})^{\frac{1}{%
2z-6}},$$
(4.20)
where $B_{1}=\sqrt{A_{2}/(3\eta(z+3))}(-2\eta\kappa^{2})^{1/(z-3)},~{}B_{2}=\sqrt{A_{%
2}(z+3)/(12\eta)}(-2\eta\kappa^{2})^{1/(z-3)}$ and
$A_{1}=-(-2\eta\kappa^{2})^{2/(z-3)}A_{2}$. Here $\eta<0$, $z~{}(\neq 3)>-3$ and $A_{2}<0$.
It is interesting to note that in this case, the effective equation
of state is given by
$$\displaystyle\omega(t)=-1-\frac{z+3}{3}{\rm csch}^{2}[B_{2}t]~{}<-1$$
(4.21)
corresponding to the phantom model [21].
ii) $V(\varphi)=\left(\frac{\varphi^{2z-6}}{1+\frac{1}{2\eta\kappa^{2}}\varphi^{2z-%
6}}\right)^{\frac{z+3}{z-3}}.$
With $\varphi=\varphi_{0}=const$, we find the de Sitter solution, $a\sim e^{Ht}$, where
$$\displaystyle H=\sqrt{\frac{1}{6\eta}}\left(\frac{\varphi_{0}^{2z-6}}{1+\frac{%
1}{2\eta\kappa^{2}}\varphi_{0}^{2z-6}}\right)^{\frac{z}{z-3}}.$$
(4.22)
In the case of $z=3$ the above solution diverges and is replaced
with
$$\displaystyle V(\varphi)=\varphi^{\frac{12}{1/2\eta\kappa^{2}+1}},~{}~{}H=%
\frac{1}{6\eta(1+1/2\eta\kappa^{2})}\varphi_{0}^{\frac{12}{1/2\eta\kappa^{2}+1%
}}.$$
When $\rho\neq 0$, the available solution is given when
$V=V_{0}=const$ and $z\neq-3$:
$$\displaystyle H=\sqrt{\frac{\kappa^{2}V_{0}}{3}}\coth[Ct],~{}~{}\varphi=D\left%
(\sinh[Ct]\right)^{-\frac{1}{3(1+\bar{\omega})}},~{}~{}\rho=V_{0}(\sinh[Ct])^{%
-2},$$
(4.23)
where $C=(1+\bar{\omega})\sqrt{3\kappa^{2}V_{0}}/2$, $D$ is a constant
and $a(t)=a_{0}(\sinh[Ct])^{2/3(1+\bar{\omega})}$. This is also an
extrapolating solution from matter dominance to vacuum dominance.
5 Conclusion and discussion
In this paper, we were able to construct an anisotropic Weyl
invariant action in the ADM formalism with the help of extra scalar
field, generalizing the original work of Hořava. Among possible
applications of the result, we treated this action as an anisotropic
Weyl matter field and considered the EHaW action adding the
cosmological fluid sector, and studied cosmological consequences. It
is found that if the anisotropic Weyl invariance is imposed, there
must be a correlation of the critical exponent $z$ and the equation
of state parameter $\bar{\omega}$ satisfying $\bar{\omega}=z/3$.
According to this condition, it is possible to reinterpret
cosmological evolution not by $\bar{\omega}$ but by $z$. In the
early universe, radiation and matter dominance correspond to $z=1$
and $z=0$ anisotropic Weyl invariance respectively. At late times,
it has $z=-3$ which is de-Sitter phase, i.e., $\bar{\omega}=-1$. It
is also shown that for particular value of $z=-3$, the potential
term for the scalar field becomes cosmological constant and there
exists an extrapolating solution from matter dominated epoch into a
late time accelerating universe in the broken cosmological fluid
case. We also found de Sitter solution in the case where the
anisotropic conformal invariance is broken by the potential term,
and especially in the polynomial potential (case i) of Sec. 4.2),
the effective equation of state parameter is less than -1. The
compatibility of the cosmology considered in this work with the
observed CMB anisotropies and structure formation remains to be
seen.
We recall that in the standard cosmology, the cosmological fluid
sector breaks conformal invariance, unless $\bar{\omega}=1/3$, i.e.,
radiation dominated. In the present anisotropic Weyl invariance,
cosmological fluid sector can maintain the invariance for an
arbitrary value of $\bar{\omega}$ due to the constraint
$\bar{\omega}=z/3$. However, for realistic cosmology, the
anisotropic conformal invariance has to be broken, and the parameter
$z$ is free. The physical significance of this parameter, in general
including the cosmological case is yet to be explored. One example
of anisotropic local conformal invariance appears in the condensed
matter system[27], and considering the AdS/CMT
correspondence might shed some light on this.
We conclude with a couple of remarks on the issues related to
Hořava-Lifshitz gravity in the case of the anisotropic Weyl
invariant action. The first one is the question on the possible
existence of the scalar graviton which shows pathological behavior.
In the original Hořava-Lifshitz gravity, this was pointed out to
cause serious problems, but, subsequently it was shown that this
could be cured via a natural extension of the Hořava-Lifshitz
gravity by abandoning the projectability and adding suitable space
dependent lapse functions [28]. Since local Lorentz
invariance is broken when $z\neq 1$, the action (3.11)
confronts the same problem and the scalar graviton persists in the
anisotropic Weyl action (3.11). To see this more closely,
we first fix the gauge by choosing a constant value for the field
$\varphi$. Then, one can show that the $\eta$ term in the action
(3.11) does not produce any scalar graviton mode since
the ratio of $B_{ij}B^{ij}$ to $B^{2}$ is equal to $1$. Also, for the
$\xi$ term with Weyl invariance of the original Hořava-Lifshitz
gravity, it has been shown explicitly that this $\xi$ term also does
not produce graviton mode [29]. Even though these terms do
not propagate the scalar graviton separately, their sum do propagate
the extra mode and the conformal action (3.11) turns out
to coincide with the low-energy limit of the non-projectable
Hořava-Lifshitz gravity which was shown to propagate extra
graviton mode [30]. Unfortunately, this scalar
graviton problem is not cured for the full Einstein-Hilbert
anisotropic Weyl action (4.2). In this case, conformal
symmetry is not present and we cannot gauge fix $\varphi$ away.
Assuming that the action admits a flat background with a constant
$\varphi=\varphi_{0}$ which would require the potential to have a
minimum with $V(\varphi_{0})=0$, we can consider a perturbation around
$\varphi_{0}$, $\varphi=\varphi_{0}+\tilde{\varphi}$. It turns out that
the Einstein-Hilbert action has the effect of changing the
coefficient of the scalar graviton mode, but the perturbed action
reduces to the theory where the $\tilde{\varphi}$ is coupled with the
low-energy limit of the non-projectable Hořava-Lifshitz gravity
previously mentioned. To investigate further the nature of this
scalar-tensor-type theory and especially to check whether the scalar
graviton problem can be cured along the line of Ref. [28] are left
as open problems. The other is to check whether the anisotropic
Weyl action (3.11) could be derived using the detailed
balance condition777In Ref.[29] it was argued that the
anisotropic Weyl invariant action of Hořava-Lifshitz gravity
might be derived from the detailed balance condition. of Ref.
[20]. This condition may also restrict the diverse terms
which will appear when the anisotropic Weyl invariance is exploited
to construct the higher curvature terms not considered in this work.
This remains as a future study.
Appendix
In the action (3.12), $S_{aWf}=S_{aW}+S_{f}$ is invariant
with respect to
$$\displaystyle N$$
$$\displaystyle\rightarrow$$
$$\displaystyle e^{z\omega}N,~{}~{}~{}N_{i}~{}~{}\rightarrow~{}~{}e^{2\omega}N_{%
i},$$
$$\displaystyle g_{ij}$$
$$\displaystyle\rightarrow$$
$$\displaystyle e^{2\omega}g_{ij},~{}~{}\varphi~{}~{}~{}\rightarrow~{}e^{-\frac{%
\omega}{2}}\varphi.$$
For the infinitesimal
$\omega$, one can find the followings,
$$\displaystyle\delta N$$
$$\displaystyle=$$
$$\displaystyle z\omega N,~{}~{}~{}~{}~{}\delta N^{i}~{}~{}=~{}~{}0,$$
$$\displaystyle\delta g^{ij}$$
$$\displaystyle=$$
$$\displaystyle-2\omega g^{ij},~{}~{}\delta\varphi~{}~{}~{}=~{}-\frac{\omega}{2}\varphi.$$
And for
the above transformation, $\delta S_{aWf}$ is
$$\displaystyle\delta S_{aWf}(N,g^{ij},N^{i},\varphi)~{}~{}=~{}~{}0$$
$$\displaystyle=$$
$$\displaystyle\frac{\delta S_{aWf}}{\delta N}\delta N+\frac{\delta S_{aWf}}{%
\delta g^{ij}}\delta g^{ij}+\frac{\delta S_{aWf}}{\delta N^{i}}\delta N^{i}+%
\frac{\delta S_{aWf}}{\delta\varphi}\delta\varphi$$
(5.1)
$$\displaystyle=$$
$$\displaystyle\omega\left(zN\frac{\delta S_{aWf}}{\delta N}-2g^{ij}\frac{\delta
S%
_{aWf}}{\delta g^{ij}}-\frac{\varphi\delta S_{aWf}}{2\delta\varphi}\right).$$
Substituting (3.14) $\sim$ (3.17) into (5.1) and
after some tedious calculations one can find the following
condition,
$$\displaystyle 2zN(-\rho)+6NP=0$$
$$\displaystyle\rightarrow$$
$$\displaystyle P=\frac{z}{3}\rho.$$
Acknowledgments
We thank the anonymous referee for valuable suggestions. We also
like to thank Seyen Kouwn, Joohan Lee, Tae Hoon Lee, and Won-Il
Myeong for useful comments. This work was supported by the National
Research Foundation of Korea(NRF) grant funded by the Korea
government(MEST) through the Center for Quantum Spacetime(CQUeST) of
Sogang University with grant number 2005-0049409.
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Measurements of $C\!P$-Violating Asymmetries in
the Decay $B^{0}\rightarrow K^{+}K^{-}K^{0}$
B. Aubert
M. Bona
D. Boutigny
Y. Karyotakis
J. P. Lees
V. Poireau
X. Prudent
V. Tisserand
A. Zghiche
Laboratoire de Physique des Particules, IN2P3/CNRS et Université de Savoie, F-74941 Annecy-Le-Vieux, France
J. Garra Tico
E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
L. Lopez
A. Palano
Università di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy
G. Eigen
B. Stugu
L. Sun
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
G. S. Abrams
M. Battaglia
D. N. Brown
J. Button-Shafer
R. N. Cahn
Y. Groysman
R. G. Jacobsen
J. A. Kadyk
L. T. Kerth
Yu. G. Kolomensky
G. Kukartsev
D. Lopes Pegna
G. Lynch
L. M. Mir
T. J. Orimoto
M. T. Ronan
K. Tackmann
W. A. Wenzel
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
P. del Amo Sanchez
C. M. Hawkes
A. T. Watson
University of Birmingham, Birmingham, B15 2TT, United Kingdom
T. Held
H. Koch
B. Lewandowski
M. Pelizaeus
T. Schroeder
M. Steinke
Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany
D. Walker
University of Bristol, Bristol BS8 1TL, United Kingdom
D. J. Asgeirsson
T. Cuhadar-Donszelmann
B. G. Fulsom
C. Hearty
T. S. Mattison
J. A. McKenna
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A. Khan
M. Saleem
L. Teodorescu
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
V. E. Blinov
A. D. Bukin
V. P. Druzhinin
V. B. Golubev
A. P. Onuchin
S. I. Serednyakov
Yu. I. Skovpen
E. P. Solodov
K. Yu. Todyshev
Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia
M. Bondioli
S. Curry
I. Eschrich
D. Kirkby
A. J. Lankford
P. Lund
M. Mandelkern
E. C. Martin
D. P. Stoker
University of California at Irvine, Irvine, California 92697, USA
S. Abachi
C. Buchanan
University of California at Los Angeles, Los Angeles, California 90024, USA
S. D. Foulkes
J. W. Gary
F. Liu
O. Long
B. C. Shen
L. Zhang
University of California at Riverside, Riverside, California 92521, USA
H. P. Paar
S. Rahatlou
V. Sharma
University of California at San Diego, La Jolla, California 92093, USA
J. W. Berryhill
C. Campagnari
A. Cunha
B. Dahmes
T. M. Hong
D. Kovalskyi
J. D. Richman
University of California at Santa Barbara, Santa Barbara, California 93106, USA
T. W. Beck
A. M. Eisner
C. J. Flacco
C. A. Heusch
J. Kroseberg
W. S. Lockman
T. Schalk
B. A. Schumm
A. Seiden
D. C. Williams
M. G. Wilson
L. O. Winstrom
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
E. Chen
C. H. Cheng
F. Fang
D. G. Hitlin
I. Narsky
T. Piatenko
F. C. Porter
California Institute of Technology, Pasadena, California 91125, USA
R. Andreassen
G. Mancinelli
B. T. Meadows
K. Mishra
M. D. Sokoloff
University of Cincinnati, Cincinnati, Ohio 45221, USA
F. Blanc
P. C. Bloom
S. Chen
W. T. Ford
J. F. Hirschauer
A. Kreisel
M. Nagel
U. Nauenberg
A. Olivas
J. G. Smith
K. A. Ulmer
S. R. Wagner
J. Zhang
University of Colorado, Boulder, Colorado 80309, USA
A. M. Gabareen
A. Soffer
W. H. Toki
R. J. Wilson
F. Winklmeier
Q. Zeng
Colorado State University, Fort Collins, Colorado 80523, USA
D. D. Altenburg
E. Feltresi
A. Hauke
H. Jasper
J. Merkel
A. Petzold
B. Spaan
K. Wacker
Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany
T. Brandt
V. Klose
M. J. Kobel
H. M. Lacker
W. F. Mader
R. Nogowski
J. Schubert
K. R. Schubert
R. Schwierz
J. E. Sundermann
A. Volk
Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany
D. Bernard
G. R. Bonneaud
E. Latour
V. Lombardo
Ch. Thiebaux
M. Verderi
Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France
P. J. Clark
W. Gradl
F. Muheim
S. Playfer
A. I. Robertson
Y. Xie
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
M. Andreotti
D. Bettoni
C. Bozzi
R. Calabrese
A. Cecchi
G. Cibinetto
P. Franchini
E. Luppi
M. Negrini
A. Petrella
L. Piemontese
E. Prencipe
V. Santoro
Università di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy
F. Anulli
R. Baldini-Ferroli
A. Calcaterra
R. de Sangro
G. Finocchiaro
S. Pacetti
P. Patteri
I. M. Peruzzi
Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy
M. Piccolo
M. Rama
A. Zallo
Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy
A. Buzzo
R. Contri
M. Lo Vetere
M. M. Macri
M. R. Monge
S. Passaggio
C. Patrignani
E. Robutti
A. Santroni
S. Tosi
Università di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy
K. S. Chaisanguanthum
M. Morii
J. Wu
Harvard University, Cambridge, Massachusetts 02138, USA
R. S. Dubitzky
J. Marks
S. Schenk
U. Uwer
Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany
D. J. Bard
P. D. Dauncey
R. L. Flack
J. A. Nash
M. B. Nikolich
W. Panduro Vazquez
M. Tibbetts
Imperial College London, London, SW7 2AZ, United Kingdom
P. K. Behera
X. Chai
M. J. Charles
U. Mallik
N. T. Meyer
V. Ziegler
University of Iowa, Iowa City, Iowa 52242, USA
J. Cochran
H. B. Crawley
L. Dong
V. Eyges
W. T. Meyer
S. Prell
E. I. Rosenberg
A. E. Rubin
Iowa State University, Ames, Iowa 50011-3160, USA
A. V. Gritsan
Z. J. Guo
C. K. Lae
Johns Hopkins University, Baltimore, Maryland 21218, USA
A. G. Denig
M. Fritsch
G. Schott
Universität Karlsruhe, Institut für Experimentelle Kernphysik, D-76021 Karlsruhe, Germany
N. Arnaud
J. Béquilleux
M. Davier
G. Grosdidier
A. Höcker
V. Lepeltier
F. Le Diberder
A. M. Lutz
S. Pruvot
S. Rodier
P. Roudeau
M. H. Schune
J. Serrano
V. Sordini
A. Stocchi
W. F. Wang
G. Wormser
Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France
D. J. Lange
D. M. Wright
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
I. Bingham
C. A. Chavez
I. J. Forster
J. R. Fry
E. Gabathuler
R. Gamet
D. E. Hutchcroft
D. J. Payne
K. C. Schofield
C. Touramanis
University of Liverpool, Liverpool L69 7ZE, United Kingdom
A. J. Bevan
K. A. George
F. Di Lodovico
W. Menges
R. Sacco
Queen Mary, University of London, E1 4NS, United Kingdom
G. Cowan
H. U. Flaecher
D. A. Hopkins
S. Paramesvaran
F. Salvatore
A. C. Wren
University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom
D. N. Brown
C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
J. Allison
N. R. Barlow
R. J. Barlow
Y. M. Chia
C. L. Edgar
G. D. Lafferty
T. J. West
J. I. Yi
University of Manchester, Manchester M13 9PL, United Kingdom
J. Anderson
C. Chen
A. Jawahery
D. A. Roberts
G. Simi
J. M. Tuggle
University of Maryland, College Park, Maryland 20742, USA
G. Blaylock
C. Dallapiccola
S. S. Hertzbach
X. Li
T. B. Moore
E. Salvati
S. Saremi
University of Massachusetts, Amherst, Massachusetts 01003, USA
R. Cowan
D. Dujmic
P. H. Fisher
K. Koeneke
G. Sciolla
S. J. Sekula
M. Spitznagel
F. Taylor
R. K. Yamamoto
M. Zhao
Y. Zheng
Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA
S. E. Mclachlin
P. M. Patel
S. H. Robertson
McGill University, Montréal, Québec, Canada H3A 2T8
A. Lazzaro
F. Palombo
Università di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy
J. M. Bauer
L. Cremaldi
V. Eschenburg
R. Godang
R. Kroeger
D. A. Sanders
D. J. Summers
H. W. Zhao
University of Mississippi, University, Mississippi 38677, USA
S. Brunet
D. Côté
M. Simard
P. Taras
F. B. Viaud
Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7
H. Nicholson
Mount Holyoke College, South Hadley, Massachusetts 01075, USA
G. De Nardo
F. Fabozzi
Also with Università della Basilicata, Potenza, Italy
L. Lista
D. Monorchio
C. Sciacca
Università di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy
M. A. Baak
G. Raven
H. L. Snoek
NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop
J. M. LoSecco
University of Notre Dame, Notre Dame, Indiana 46556, USA
G. Benelli
L. A. Corwin
K. Honscheid
H. Kagan
R. Kass
J. P. Morris
A. M. Rahimi
J. J. Regensburger
Q. K. Wong
Ohio State University, Columbus, Ohio 43210, USA
N. L. Blount
J. Brau
R. Frey
O. Igonkina
J. A. Kolb
M. Lu
R. Rahmat
N. B. Sinev
D. Strom
J. Strube
E. Torrence
University of Oregon, Eugene, Oregon 97403, USA
N. Gagliardi
A. Gaz
M. Margoni
M. Morandin
A. Pompili
M. Posocco
M. Rotondo
F. Simonetto
R. Stroili
C. Voci
Università di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy
E. Ben-Haim
H. Briand
G. Calderini
J. Chauveau
P. David
L. Del Buono
Ch. de la Vaissière
O. Hamon
Ph. Leruste
J. Malclès
J. Ocariz
A. Perez
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France
L. Gladney
University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
M. Biasini
R. Covarelli
E. Manoni
Università di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy
C. Angelini
G. Batignani
S. Bettarini
M. Carpinelli
R. Cenci
A. Cervelli
F. Forti
M. A. Giorgi
A. Lusiani
G. Marchiori
M. A. Mazur
M. Morganti
N. Neri
E. Paoloni
G. Rizzo
J. J. Walsh
Università di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy
M. Haire
Prairie View A&M University, Prairie View, Texas 77446, USA
J. Biesiada
P. Elmer
Y. P. Lau
C. Lu
J. Olsen
A. J. S. Smith
A. V. Telnov
Princeton University, Princeton, New Jersey 08544, USA
E. Baracchini
F. Bellini
G. Cavoto
A. D’Orazio
D. del Re
E. Di Marco
R. Faccini
F. Ferrarotto
F. Ferroni
M. Gaspero
P. D. Jackson
L. Li Gioi
M. A. Mazzoni
S. Morganti
G. Piredda
F. Polci
F. Renga
C. Voena
Università di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy
M. Ebert
T. Hartmann
H. Schröder
R. Waldi
Universität Rostock, D-18051 Rostock, Germany
T. Adye
G. Castelli
B. Franek
E. O. Olaiya
S. Ricciardi
W. Roethel
F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
R. Aleksan
S. Emery
M. Escalier
A. Gaidot
S. F. Ganzhur
G. Hamel de Monchenault
W. Kozanecki
G. Vasseur
Ch. Yèche
M. Zito
DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France
X. R. Chen
H. Liu
W. Park
M. V. Purohit
J. R. Wilson
University of South Carolina, Columbia, South Carolina 29208, USA
M. T. Allen
D. Aston
R. Bartoldus
P. Bechtle
N. Berger
R. Claus
J. P. Coleman
M. R. Convery
J. C. Dingfelder
J. Dorfan
G. P. Dubois-Felsmann
W. Dunwoodie
R. C. Field
T. Glanzman
S. J. Gowdy
M. T. Graham
P. Grenier
C. Hast
T. Hryn’ova
W. R. Innes
J. Kaminski
M. H. Kelsey
H. Kim
P. Kim
M. L. Kocian
D. W. G. S. Leith
S. Li
S. Luitz
V. Luth
H. L. Lynch
D. B. MacFarlane
H. Marsiske
R. Messner
D. R. Muller
C. P. O’Grady
I. Ofte
A. Perazzo
M. Perl
T. Pulliam
B. N. Ratcliff
A. Roodman
A. A. Salnikov
R. H. Schindler
J. Schwiening
A. Snyder
J. Stelzer
D. Su
M. K. Sullivan
K. Suzuki
S. K. Swain
J. M. Thompson
J. Va’vra
N. van Bakel
A. P. Wagner
M. Weaver
W. J. Wisniewski
M. Wittgen
D. H. Wright
A. K. Yarritu
K. Yi
C. C. Young
Stanford Linear Accelerator Center, Stanford, California 94309, USA
P. R. Burchat
A. J. Edwards
S. A. Majewski
B. A. Petersen
L. Wilden
Stanford University, Stanford, California 94305-4060, USA
S. Ahmed
M. S. Alam
R. Bula
J. A. Ernst
V. Jain
B. Pan
M. A. Saeed
F. R. Wappler
S. B. Zain
State University of New York, Albany, New York 12222, USA
W. Bugg
M. Krishnamurthy
S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
R. Eckmann
J. L. Ritchie
A. M. Ruland
C. J. Schilling
R. F. Schwitters
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen
X. C. Lou
S. Ye
University of Texas at Dallas, Richardson, Texas 75083, USA
F. Bianchi
F. Gallo
D. Gamba
M. Pelliccioni
Università di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy
M. Bomben
L. Bosisio
C. Cartaro
F. Cossutti
G. Della Ricca
L. Lanceri
L. Vitale
Università di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy
V. Azzolini
N. Lopez-March
F. Martinez-Vidal
Also with Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
D. A. Milanes
A. Oyanguren
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
J. Albert
Sw. Banerjee
B. Bhuyan
K. Hamano
R. Kowalewski
I. M. Nugent
J. M. Roney
R. J. Sobie
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
J. J. Back
P. F. Harrison
J. Ilic
T. E. Latham
G. B. Mohanty
M. Pappagallo
Also with IPPP, Physics Department, Durham University, Durham DH1 3LE, United Kingdom
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
H. R. Band
X. Chen
S. Dasu
K. T. Flood
J. J. Hollar
P. E. Kutter
Y. Pan
M. Pierini
R. Prepost
S. L. Wu
University of Wisconsin, Madison, Wisconsin 53706, USA
H. Neal
Yale University, New Haven, Connecticut 06511, USA
(November 20, 2020)
Abstract
We analyze the decay $B^{0}\rightarrow K^{+}K^{-}K^{0}$ using 383 million $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ events collected by
the BABAR detector at SLAC to extract $C\!P$ violation parameter values
over the Dalitz plot. Combining all $K^{+}K^{-}K^{0}$ events,
we find ${A}_{C\!P}=-0.015\pm 0.077\pm 0.053$ and $\beta_{\mathit{eff}}=0.352\pm 0.076\pm 0.026\,\rm\,rad$,
corresponding to a $C\!P$ violation significance of $4.8\sigma$.
A second solution near
$\pi/2-\beta_{\mathit{eff}}$ is disfavored with a significance of
$4.5\sigma$. We also report ${A}_{C\!P}$ and $\beta_{\mathit{eff}}$ separately for decays to
$\phi(1020)K^{0}$, $f_{0}(980)K^{0}$, and $K^{+}K^{-}K^{0}$ with $m_{K^{+}K^{-}}>1.1{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$.
pacs: 13.25.Hw, 12.15.Hh, 11.30.Er
††preprint: BABAR-PUB-07/029††preprint: SLAC-PUB-12625
BABAR-PUB-07/029
SLAC-PUB-12625
arXiv:0706.3885 [hep-ex]
[10mm]
††thanks: Deceased
The BABAR Collaboration
In the Standard Model (SM),
the phase in the
Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix CKM is the
sole source of $C\!P$ violation in the quark sector.
Due to interference between decays with and without mixing,
this phase yields observable
time-dependent $C\!P$ asymmetries in $B^{0}$ meson decays.
In particular, significant $C\!P$ asymmetries in $b\rightarrow s\overline{s}s$ decays, such as $B^{0}\rightarrow K^{+}K^{-}K^{0}$ conjugate , are expected Chen:2006nk ; Aubert:2004zt .
Deviations from the predicted $C\!P$ asymmetry behavior for $B^{0}\rightarrow K^{+}K^{-}K^{0}$ are expected to depend weakly on Dalitz plot (DP)
position Beneke:2005pu ; Buchalla:2005us .
Since the $b\rightarrow s\overline{s}s$
amplitude is dominated by loop contributions,
heavy virtual particles beyond the SM
might contribute significantly Buchalla:2005us ; newphysics .
This sensitivity motivates
measurements of $C\!P$ asymmetries in multiple $b\rightarrow s\overline{s}s$
decays Chen:2006nk ; Aubert:2005ja ; Aubert:2006wv ; Chen:2005dr .
Previous measurements of $C\!P$ asymmetries in $B^{0}\rightarrow K^{+}K^{-}K^{0}$
have been performed separately
for events with $K^{+}K^{-}$ invariant mass ($m_{K^{+}K^{-}}$) in the
$\phi$ mass assume1020 region,
and for events excluding the $\phi$ region,
neglecting interference effects among intermediate
states Chen:2006nk ; Chen:2005dr ; Aubert:2005ja .
In this Letter we describe a
time-dependent DP analysis of $B^{0}\rightarrow K^{+}K^{-}K^{0}$
decay from which we extract the values of the
$C\!P$ violation parameters ${A}_{C\!P}$ and $\beta_{\mathit{eff}}$ by
taking into account the complex amplitudes describing the entire $B^{0}$ and
$\kern 1.8pt\overline{\kern-1.8ptB}{}^{0}$ Dalitz plots.
We first extract the values of the parameters of the amplitude model,
and measure the
average $C\!P$ asymmetry in $B^{0}\rightarrow K^{+}K^{-}K^{0}$ decay over the entire DP.
Using this model, we then measure the $C\!P$ asymmetries for
the $\phi K^{0}$ and $f_{0}K^{0}$ decay channels, from a
“low-mass” analysis of events with $m_{K^{+}K^{-}}<1.1{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$. Finally, we perform a “high-mass” analysis to determine the
average $C\!P$ asymmetry for events with $m_{K^{+}K^{-}}>1.1{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$.
The data sample for this analysis was collected with the BABAR detector ref:babar at the PEP-II asymmetric-energy $e^{+}e^{-}$ collider at SLAC.
Approximately $383\times 10^{6}$ $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ pairs recorded at the $\mathchar 28935\relax{(4S)}$ resonance were used.
We reconstruct $B^{0}\rightarrow K^{+}K^{-}K^{0}$ decays by combining two oppositely-charged kaon
candidates with a $K^{0}$ reconstructed as
$K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{+}\pi^{-}$ ($B^{0}_{\scriptscriptstyle(+-)}$) Aubert:2004ta , $K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{0}\pi^{0}$ ($B^{0}_{\scriptscriptstyle(00)}$), or $K^{0}_{\scriptscriptstyle L}$ ($B^{0}_{\scriptscriptstyle(L)}$).
Each $K^{0}_{\scriptscriptstyle S}\rightarrow\pi^{0}\pi^{0}$ candidate is formed from two
$\pi^{0}\rightarrow\gamma\gamma$ candidates.
Each photon has
$E_{\gamma}>50\mathrm{\,Me\kern-1.0ptV}$ and transverse shower shape
consistent with an electromagnetic shower.
Both $\pi^{0}$ candidates satisfy
$100<m_{\gamma\gamma}<155{\mathrm{\,Me\kern-1.0ptV\!/}c^{2}}$ and yield
an invariant mass $m_{\pi^{0}\pi^{0}}$ in the range $-20<m_{\pi^{0}\pi^{0}}-m_{K^{0}_{\scriptscriptstyle S}}<30{\mathrm{\,Me\kern-1%
.0ptV\!/}c^{2}}$.
A $K^{0}_{\scriptscriptstyle L}$ candidate is defined by an unassociated energy deposit in the electromagnetic
calorimeter or an isolated signal in the Instrumented Flux Return Aubert:2005ja .
For each fully reconstructed $B^{0}$ meson ($B_{C\!P}$), we use the remaining
tracks in the event to reconstruct the decay vertex of the other $B$ meson ($B_{\mathrm{tag}}$), and to identify its flavor $q_{\mathrm{tag}}$ Aubert:2004zt .
For each event we calculate
the difference ${\rm\Delta}t\equiv t_{C\!P}-t_{\mathrm{tag}}$ between the proper
decay times of the $B_{C\!P}$ and $B_{\mathrm{tag}}$ mesons, and its uncertainty $\sigma_{{\rm\Delta}t}$.
We characterize $B^{0}_{\scriptscriptstyle(+-)}$ and $B^{0}_{\scriptscriptstyle(00)}$ candidates using two kinematic
variables: the beam-energy-substituted mass $m_{\rm ES}$ and the energy
difference $\Delta E$ Aubert:2005ja .
The signal region (SR) is defined as $\mbox{$m_{\rm ES}$}>5.26$ ${\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$,
and $|\mbox{$\Delta E$}|<0.06$ $\mathrm{\,Ge\kern-1.0ptV}$ for $B^{0}_{\scriptscriptstyle(+-)}$, or $-0.120<\mbox{$\Delta E$}<0.06$
$\mathrm{\,Ge\kern-1.0ptV}$ for $B^{0}_{\scriptscriptstyle(00)}$.
For $B^{0}_{\scriptscriptstyle(L)}$ the SR is defined by $-0.01<\mbox{$\Delta E$}<0.03\mathrm{\,Ge\kern-1.0ptV}$ Aubert:2005ja ,
and the missing
momentum for the entire event is required to be consistent with the calculated
$K^{0}_{\scriptscriptstyle L}$ laboratory momentum.
The main source of background is
continuum $e^{+}e^{-}\rightarrow q\overline{q}~{}(q=u,d,s,c)$ events.
We use event-shape variables to exploit the jet-like structure of these
events in order to remove much of this background Aubert:2005ja .
We perform an unbinned maximum likelihood fit to the selected $K^{+}K^{-}K^{0}$ events using the likelihood
function defined in Ref. Aubert:2005ja .
The probability density function (PDF), ${\cal P}_{i}$,
is given by
$$\displaystyle{\cal P}_{i}\equiv{\mathcal{P}}(\mbox{$m_{\rm ES}$})\cdot{%
\mathcal{P}}(\mbox{$\Delta E$})\cdot{\mathcal{P}}_{\mathrm{Low}}$$
$$\displaystyle\cdot~{}{\cal P}_{DP}(m_{K^{+}K^{-}},\cos\theta_{H},{\rm\Delta}t,%
q_{\mathrm{tag}})\otimes{\cal R}({\rm\Delta}t,\sigma_{{\rm\Delta}t}),$$
where $i$ = (signal, continuum, $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ background), and
${\mathcal{R}}$ is the ${\rm\Delta}t$ resolution function Aubert:2004zt .
For $B^{0}_{\scriptscriptstyle(L)}$, ${\mathcal{P}}(\mbox{$m_{\rm ES}$})$ is not used.
${\mathcal{P}}_{\mathrm{Low}}$ is a PDF used only in the
low-mass fit, which depends on the event-shape
variables and, for $B^{0}_{\scriptscriptstyle(L)}$ only, the missing momentum in the event Aubert:2005ja .
We characterize $B^{0}$ ($\kern 1.8pt\overline{\kern-1.8ptB}{}^{0}$) events on the DP in terms of
$m_{K^{+}K^{-}}$ and $\cos\theta_{H}$, the cosine of the helicity angle between the $K^{+}$ ($K^{-}$) and the
$K^{0}$ ($\kern 2.0pt\overline{\kern-2.0ptK}{}^{0}$) in the rest frame of the $K^{+}K^{-}$ system.
The DP PDF for signal events is
$${\mathcal{P}}_{DP}=d\Gamma\cdot\varepsilon(m_{K^{+}K^{-}},\cos\theta_{H})\cdot%
|J|,$$
(2)
where $d\Gamma$ is the time- and flavor-dependent decay rate over the
DP, $\varepsilon$ is the efficiency, and
$J$ is the Jacobian of the transformation
to our choice of DP coordinates.
The time- and flavor-dependent decay rate is
$$\displaystyle\frac{d\Gamma}{d{\rm\Delta}t}\propto\frac{e^{-|{\rm\Delta}t|/\tau%
}}{2\tau}$$
$$\displaystyle\times$$
$$\displaystyle\Big{[}~{}\left|{\cal A}\right|^{2}+\left|\bar{{\cal A}}\right|^{2}$$
$$\displaystyle+$$
$$\displaystyle~{}q_{\mathrm{tag}}~{}2\mathrm{Im}\left(\xi\bar{\cal A}{\cal A}^{%
*}\right)\sin{\rm\Delta}m_{d}{\rm\Delta}t$$
$$\displaystyle-$$
$$\displaystyle~{}q_{\mathrm{tag}}\left(\left|{\cal A}\right|^{2}-\left|\bar{{%
\cal A}}\right|^{2}\right)\cos{\rm\Delta}m_{d}{\rm\Delta}t~{}\Big{]},$$
where $\tau$ and ${\rm\Delta}m_{d}$ are the lifetime and mixing frequency of the $B^{0}$ meson, respectively PDG .
The parameter $\xi=\eta_{C\!P}e^{-2i\beta}$, where $\beta=\arg(-V_{cd}V_{cb}^{*}/V_{td}V_{tb}^{*})$ and $V_{qq^{\prime}}$ are CKM
matrix elements CKM .
The $C\!P$ eigenvalue $\eta_{C\!P}=1~{}(-1)$ for the $K^{0}_{\scriptscriptstyle S}$ ($K^{0}_{\scriptscriptstyle L}$) mode.
We define the amplitude
( )
[-.7ex]$\kern-1.8pt{\cal A}$
for
( )
[-.7ex]$\kern-1.8ptB^{0}$
decay as a sum of isobar amplitudes PDG ,
$$\displaystyle\kern 1.8pt\shortstack{{\tiny(\rule[1.72pt]{10.0pt}{0.284528pt})}%
\\
[-.7ex]$\kern-1.8pt{\cal A}$}{}(m_{K^{+}K^{-}},\cos\theta_{H})=\sum\limits_{r}%
{\kern 1.8pt\shortstack{{\tiny(\rule[1.72pt]{10.0pt}{0.284528pt})}\\
[-.7ex]$\kern-1.8pt{\cal A}$}{}}_{r}$$
$$\displaystyle=\sum\limits_{r}c_{r}(1\mp b_{r})e^{i(\varphi_{r}\mp\delta_{r})}%
\cdot f_{r}(m_{K^{+}K^{-}},\cos\theta_{H}),$$
where
the minus signs are associated with the $\overline{\cal A}$,
the parameters $c_{r}$ and $\varphi_{r}$ are the magnitude and phase of the
amplitude of component $r$, and we allow for different isobar
coefficients for $B^{0}$ and $\kern 1.8pt\overline{\kern-1.8ptB}{}^{0}$ decays through the asymmetry
parameters $b_{r}$ and $\delta_{r}$.
Our isobar model includes resonant amplitudes $\phi$,
$f_{0}$, $\chi_{c0}(1P)$, and
$X_{0}(1550)$ Aubert:2006nu ; Garmash:2004wa ;
non-resonant terms; and
incoherent terms for $B^{0}$ decay to $D^{-}K^{+}$
and $D^{-}_{s}K^{+}$.
For each resonant term,
the function $f_{r}=F_{r}\times T_{r}\times Z_{r}$ describes the dynamical
properties, where $F_{r}$ is the Blatt-Weisskopf centrifugal barrier factor for the
resonance decay vertex blatt , $T_{r}$ is the resonant mass-lineshape, and
$Z_{r}$ describes the angular distribution in the decay Zemach:1963bc .
The barrier factor
$F_{r}=1/\sqrt{1+(Rq)^{2}}$ blatt for the $\phi$, where $\vec{q}$ is the
$K^{+}$ momentum in the $\phi$ rest frame and $R=1.5~{}{\mathrm{\,Ge\kern-1.0ptV}}^{-1}$;
$F_{r}=1$ for the scalar resonances.
For $\phi$ decay $Z_{r}\sim\vec{q}\cdot\vec{p}$, where
$\vec{p}$ is the momentum of the $K^{0}$ in the $\phi$
rest frame, while $Z_{r}=1$ for the scalar decays.
We describe the $\phi$, $X_{0}(1550)$, and
$\chi_{c0}(1P)$ with relativistic Breit-Wigner lineshapes PDG .
For the $\phi$ and $\chi_{c0}(1P)$ parameters we use average
measurements PDG . For the $X_{0}(1550)$ resonance, we use
parameters from our analysis of the $B^{+}\rightarrow K^{+}K^{-}K^{+}$ decay Aubert:2006nu .
The $f_{0}$ resonance is described by a coupled-channel amplitude Flatte:1976xu ,
with the parameter values of Ref. Ablikim:2004wn .
We include three non-resonant (NR) amplitudes
parameterized as $f_{\mathit{NR},k}=\exp(-\alpha m^{2}_{k})$,
where the parameter $\alpha=0.14\pm 0.01~{}c^{4}/\mathrm{\,Ge\kern-1.0ptV}^{2}$ is taken from
measurements of $B^{+}\rightarrow K^{+}K^{-}K^{+}$ decays with larger signal samples Garmash:2004wa ; Aubert:2006nu .
We include a complex isobar coefficient for each component $k=(K^{+}K^{-},K^{+}K^{0},K^{-}K^{0})$.
PDFs for $q\overline{q}$ background in $B^{0}\rightarrow K^{+}K^{-}K^{0}_{\scriptscriptstyle S}$
are modeled using
events in the region $5.2<\mbox{$m_{\rm ES}$}<5.26{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$. The region
$0.02<\mbox{$\Delta E$}<0.04\mathrm{\,Ge\kern-1.0ptV}$ is used for $B^{0}_{\scriptscriptstyle(L)}$.
Simulated $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ events are used to define $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ background PDFs.
We use two-dimensional histogram PDFs to model the DP distributions for $q\overline{q}$ and $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ backgrounds.
We compute the $C\!P$ asymmetry parameters for component $r$ from the asymmetries in amplitude ($b_{r}$)
and phase ($\delta_{r}$) given in Eq. (Measurements of $C\!P$-Violating Asymmetries in
the Decay $B^{0}\rightarrow K^{+}K^{-}K^{0}$).
The rate asymmetry is
$${A}_{C\!P,r}=\frac{|\bar{\cal A}_{r}|^{2}-|{\cal A}_{r}|^{2}}{|\bar{\cal A}_{r%
}|^{2}+|{\cal A}_{r}|^{2}}=\frac{-2b_{r}}{1+b_{r}^{2}},$$
(5)
and $\beta_{\mathit{eff},r}=\beta+\delta_{r}$ is the phase asymmetry.
The selection criteria yield 3266 $B^{0}_{\scriptscriptstyle(+-)}$, 1611 $B^{0}_{\scriptscriptstyle(00)}$, and 27513 $B^{0}_{\scriptscriptstyle(L)}$ candidates
which we fit to obtain
the event yields, the isobar coefficients of the DP model,
and the $C\!P$ asymmetry parameters averaged over the DP. The
parameters $b_{r}$ and $\delta_{r}$ are constrained to be the same for
all model components, so in this case ${A}_{C\!P,r}={A}_{C\!P}$ and $\beta_{\mathit{eff},r}=\beta_{\mathit{eff}}$.
We find $947\pm 37$ $B^{0}_{\scriptscriptstyle(+-)}$, $144\pm 17$ $B^{0}_{\scriptscriptstyle(00)}$, and $770\pm 71$ $B^{0}_{\scriptscriptstyle(L)}$ signal
events.
Isobar coefficients and fractions are reported in
Table 1, and $C\!P$ asymmetry results are
summarized in Table 2.
The fraction ${\cal F}_{r}$ for resonance $r$ is computed as in Ref. Aubert:2006nu .
Note that there is a $\pm\pi\rm\,rad$ ambiguity in the $\chi_{c0}(1P)K^{0}$ phase.
In Fig. 1, we plot twice the change in the negative
logarithm of the likelihood as a function of $\beta_{\mathit{eff}}$.
We find that
the $C\!P$-conserving case of $\beta_{\mathit{eff}}=0$ is excluded at
$4.8\sigma$ ($5.1\sigma$), including statistical and systematic errors (statistical errors only).
Also, the interference between $C\!P$-even and $C\!P$-odd
amplitudes leads to the exclusion of the $\beta_{\mathit{eff}}$ solution near $\pi/2-\beta$
at $4.5\sigma$ ($4.6\sigma$).
We also measure $C\!P$ asymmetry parameters for events with $m_{K^{+}K^{-}}<1.1$ ${\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$.
In this region, we find 1359 $B^{0}_{\scriptscriptstyle(+-)}$, 348 $B^{0}_{\scriptscriptstyle(00)}$,
and 7481 $B^{0}_{\scriptscriptstyle(L)}$ candidates.
The fit yields $282\pm 20$, $37\pm 9$ and $266\pm 36$ signal events,
respectively.
The most significant contributions in this region are from
$\phi K^{0}$ and $f_{0}K^{0}$ decays, with a smaller contribution from
the low-mass tail from non-resonant decays.
In this fit we vary the amplitude asymmetries $b_{r}$ and
$\delta_{r}$ for
the $\phi$ and $f_{0}$, while the other components are fixed to the SM
expectations of
$\beta_{\mathit{eff}}=0.370~{}\rm\,rad$ and ${A}_{C\!P}=0$ hfag . We also vary the isobar coefficient for the $\phi$,
while fixing the others to the results from the whole DP fit.
There are two solutions with likelihood difference of only $\Delta\log L=0.1$.
Solution (1) is consistent with the SM, while in
Solution (2) $\beta_{\mathit{eff}}$ for the $f_{0}$ differs significantly from the SM
value (Table 2). The solutions also differ significantly
in the values of the $\phi$ isobar coefficient.
There is also a mathematical ambiguity of $\pm\pi\rm\,rad$ on
$\beta_{\mathit{eff}}$ for the $\phi$, with a corresponding change of $\pm\pi\rm\,rad$ in the
solution for $\varphi_{\phi}$. This ambiguity is present for both solutions.
The fit correlation between the $\phi$ and $f_{0}$ in $\delta_{r}$ is $0.71$ epaps .
Finally, we perform a fit to extract the average $C\!P$ asymmetry parameters
in the high-mass region. In the 2384 $B^{0}_{\scriptscriptstyle(+-)}$, 1406 $B^{0}_{\scriptscriptstyle(00)}$, and 20032
$B^{0}_{\scriptscriptstyle(L)}$ selected events with $m_{K^{+}K^{-}}>1.1~{}{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$, we find signal yields of
$673\pm 31$, $87\pm 14$ and $462\pm 56$ events, respectively; the
$C\!P$ asymmetry results are shown in Table 2.
We find that
for this fit the $C\!P$-conserving case of $\beta_{\mathit{eff}}=0$ is excluded at
$5.1\sigma$, including statistical and systematic errors.
Figure 2 shows distributions of the DP variables $m_{K^{+}K^{-}}$ and $\cos\theta_{H}$ obtained using the method described in Pivk:2004ty . Figure 3 shows
the ${\rm\Delta}t$-dependent asymmetry between $B^{0}$- and $\kern 1.8pt\overline{\kern-1.8ptB}{}^{0}$-tagged events.
Systematic errors on the $C\!P$-asymmetry parameters are listed in Table 3.
The fit bias uncertainty includes effects of detector resolution and possible
correlations among the fit variables determined from full-detector
simulations.
We also account for uncertainties due to the isobar model:
experimental precision of resonance parameter values; alternate $X_{0}(1550)$
parameter values Garmash:2004wa ; and, in the low- and high-mass fits,
the statistical uncertainties on the isobar coefficients determined in
the fit to the whole DP.
Other uncertainties common to many BABAR time-dependent analyses, including
those due to fixed PDF parameters, and possible $C\!P$ asymmetries in the
$B\kern 1.8pt\overline{\kern-1.8ptB}{}$ background are also taken into
account Aubert:2005ja ; ref:tagint . Uncertainties due to fixed
PDF parameters are evaluated by shifting the fixed parameters and
refitting the data.
As a cross-check, we perform the analysis using $B^{0}_{\scriptscriptstyle(+-)}$ alone and find
results consistent with those in Table 2.
In summary, in a sample of $383\times 10^{6}$ $B\kern 1.8pt\overline{\kern-1.8ptB}{}$ meson pairs we
simultaneously analyze the
DP distribution and measure the
time-dependent $C\!P$ asymmetries for $B^{0}\rightarrow K^{+}K^{-}K^{0}$ decays.
The values of $\beta_{\mathit{eff}}$ and ${A}_{C\!P}$ are consistent with the SM expectations of
$\beta\simeq 0.370~{}\rm\,rad,~{}{A}_{C\!P}\simeq 0$ hfag . The
signficance of $C\!P$ violation is $4.8\sigma$, and we reject the solution near
$\pi/2-\beta$ at $4.5\sigma$.
We also
measure $C\!P$ asymmetries for the decays $B^{0}\rightarrow\phi K^{0}$ and $B^{0}\rightarrow f_{0}K^{0}$, where we find $\beta_{\mathit{eff}}$ lower than the SM expectation by about $2\sigma$.
The $C\!P$ parameters in the high-mass region are compatible with SM
expectations, and we observe $C\!P$ violation at the
level of $5.1\sigma$.
We are grateful for the excellent luminosity and machine conditions
provided by our PEP-II colleagues,
and for the substantial dedicated effort from
the computing organizations that support BABAR.
The collaborating institutions wish to thank
SLAC for its support and kind hospitality.
This work is supported by
DOE
and NSF (USA),
NSERC (Canada),
CEA and
CNRS-IN2P3
(France),
BMBF and DFG
(Germany),
INFN (Italy),
FOM (The Netherlands),
NFR (Norway),
MIST (Russia),
MEC (Spain), and
STFC (United Kingdom).
Individuals have received support from the
Marie Curie EIF (European Union) and
the A. P. Sloan Foundation.
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Prime rings with PI rings of constants
V. K. Kharchenko
J.Keller
S. Rodríguez-Romo
Centre of Theoretical Research
UNAM, Campus Cuautitlán
Apdo. Postal 95, Cuautitlán Izcalli
Estado de México, 54768, México
The first author wishes to thank
CONACYT–México for its support, Catedra Patrimonial number 940411-R96 and also
Russian Found of Fundamental Research, grant 95-01-01356. The third author wishes to thank
CONACYT–México for its support under grant 4336E9406.
()
Abstract
It is shown that if the ring of constants of a restricted
differential Lie algebra with a quasi-Frobenius inner part satisfies a
polynomial identity (PI) then the original prime ring has a generalized
polynomial identity (GPI). If additionally the ring of constants is
semiprime then the original ring is PI. The case of a
non-quasi-Frobenius inner part is also considered.
in memory of S.A.Amitsur
1 Introduction
Rings of constants of restricted differential Lie algebras with an outer
action on prime and semiprime rings were investigated in detail in
papers [Kh82], [Po83], [Pi86] (see also [Kh91,Ch.4, Ch6(6.4)]). In the
present paper we are going to consider actions with a nontrivial inner
part. In the papers [Ko91] and [Kh81] it is shown that the minimal restriction
required is that the inner part should be quasi-Frobenius (selfinjective).
We are interested in the structure of a prime ring $R$ provided it is known
that its ring of constants satisfies a polynomial identity. I.V.L’vov’s
example [Lv93] shows that in this case the ring $R$ does not need to
be a PI-ring. We will show that in this case $R$ satisfies a generalized
polynomial identity.
The notion of a generalized polynomial identity was introduced by
S.A.Amitsur in [Am65]. In his paper S.A.Amitsur proved a structure
theorem for primitive rings with generalized polynomial identities.
Later W.S.Martindale [Ma69] generalized this result to arbitrary
prime rings. Using this theorem
we will prove that if the ring of constants is a semiprime PI-ring
and the inner part is quasi-Frobenius, then the ring $R$ is a PI-ring.
2 Preliminaries
Recall that a derivation of a ring $R$ is an additive mapping
$d:R\rightarrow R$ satisfying the condition
$(xy)^{d}=x^{d}y+xy^{d}.$
If $d_{1},d_{2}$ are derivations then it is easy to see that the
commutator $[d_{1},d_{2}]=d_{1}d_{2}-d_{2}d_{1}$ is also a derivation. Therefore
the set $DerR$ of all derivations of $R$ is a Lie subring in the ring
of endomorphisms of the abelian group $(R,+)$. Moreover, if $z$ is a central
element, then the composition of $d$ with the multiplication by $z$
is a derivation
$$(xy)^{dz}=z(xy)^{d}=(zx^{d})y+x(zy^{d})$$
In this case the operators of multiplication may not commute with derivations:
$x^{zd}\stackrel{{\scriptstyle\rm def}}{{=}}(zx)^{d}=z^{d}x+zx^{d}$ or
$$zd=dz+z^{d}.$$
(1)1( 1 )
Thus the set $DerR$ is a right module over the center $Z.$ The
module structure of $DerR$
is connected with the commutator operation by the formula
$$[dz,d_{1}]=[d,d_{1}]z+dz^{d_{1}}.$$
(2)2( 2 )
Note that $z^{d_{1}}$ is again a central element:
$[z^{d_{1}},x]=[z^{d_{1}},x]+[z,x^{d_{1}}]=[z,x]^{d_{1}}$=0.
Finally, if the characteristic $p$ of the ring $R$ is nonzero, $pR=0,$
then the $p$th power of any derivation will be a derivation by the Leibniz
formula
$$(xy)^{d^{p}}=\sum_{k=0}^{k=p}C_{p}^{k}x^{d^{k}}y^{d^{p-k}}=x^{d^{p}}y+xy^{d^{p%
}}.$$
Now it is natural to formulate the following definition.
2.1. Definition. A set of derivations is called a differential
restricted Lie $Z$-algebra, or shortly a Lie $\partial$-algebra, if
it is a right $Z$-submodule of $DerR$ closed with respect to the operations
$[d_{1},d_{2}]=d_{1}d_{2}-d_{2}d_{1}$ and $d^{[p]}=d^{p}.$
Note that the notion of a Lie $\partial$-algebra can be formalized
abstractly as a restricted Lie ring with a structure of right
$Z$-module connected with the main operations by formula (2) and the
following formula
$$(dz)^{[p]}=d^{[p]}z^{p}+d\cdot(\ldots((\overbrace{z^{d}z)^{d}z)^{d}\ldots)^{d}%
}^{p-1}z$$
(3)3( 3 )
which follows from (1) (see details in [Kh91, pp. 6-11]; for a slightly more
general approach see in [Pa87]).
Now let $R$ be a prime ring. Denote by $R_{\cal F}$ its left Martindale ring
of quotients (see, for example, [Kh91 pp.19-24]), by $Q$ the symmetric Martindale
ring of quotients. Recall that the center $C$ of $R_{\cal F}$ is called the
extended (or generalized) centroid of $R$ and it is a field (see [Ma69]).
All derivations of $R$ can be uniquely extended to derivations of $Q$ and
of $R_{\cal F}.$ The extended derivations are characterized in $DerQ$ by the
property $R^{d}\subseteq R$ but the linear combinations over $C$ of extended
derivations do not satisfy this property. Therefore we have to consider more
general objects.
2.2. Definition. A derivation $d$ of $Q$ is called $R$-continuous
if there
exists a nonzero two-sided ideal $I$ of $R$ such that $I^{d}\subseteq R.$
It is easy to see that the set ${\cal D}(R)$ of all $R$-continuous derivations
is a differential restricted Lie $C$-subalgebra of $DerQ.$
In the present paper we consider Lie
$\partial$-algebras of $R$-continuous derivations which are finite
dimensional over $C.$
Let us fix the notations $R,C,Q,R_{\cal F},{\cal D}(R)$
for a prime ring, its extended centroid, the symmetric Martindale ring of
quotients, the left Martindale ring of quotients
and the Lie $\partial$-algebra of $R$-continuous derivations,
respectively. Throughout the paper $L$ denotes a
restricted differential Lie $C$-algebra of $R$-continuous derivations,
$L\subseteq{\cal D}(R),$ finite dimensional over $C,$
and $R^{L}=\{r\in R:\forall\mu\in L\ \ \ r^{\mu}=0\}$ is its ring of constants.
3 The inner part of a Lie $\partial$-algebra
If $a$ is an element of $Q$ then the map $a^{-}:x\rightarrow xa-ax$
is an $R$-continuous derivation, i.e. $Q^{-}\subseteq{\cal D}(R).$
3.1. Definition. The space $K(L)$ generated over $C$ by all
$q\in Q$ such that $q^{-}\in L$ is called the inner linear part of $L.$
It is clear that $C^{-}=0,$ therefore $K(L)$ contains $C$ and in particular
it contains the unit of $Q.$
3.2. Lemma. The space $K(L)$ is a restricted Lie subalgebra of the
adjoint restricted Lie algebra $Q^{(-)}.$
Recall that $Q^{(-)}$ is a restricted Lie algebra defined on the $C$-space
$Q$ with the operations $[q_{1},q_{2}]=q_{1}q_{2}-q_{2}q_{1},\ \ q^{[p]}=q^{p}.$
For the proof of the lemma it is enough to show that $K(L)$ is closed
with respect to these operations. This fact immediately follows from the
formulae
$$[a,b]^{-}=[a^{-},b^{-}]$$
(4)4( 4 )
$$(a^{p})^{-}=(a^{-})^{[p]}.$$
(5)5( 5 )
3.3. Lemma. $K(L)^{-}$ is equal to the subalgebra $L_{int}$
of all inner derivations of $L.$
The proof is evident.
3.4. Definition. The associative subalgebra ${\cal B}(L)$
generated in $Q$ by $K(L)$ is called the inner associative part of $L.$
3.5. Lemma. The algebra ${\cal B}(L)$ is of finite dimension
over $C.$
Proof. By the definition of operations in $K(L),$ the identity map
$id$ is a homomorphism of restricted Lie algebras
$id:K(L)\rightarrow{\cal B}(L)^{(-)}.$ Therefore ${\cal B}(L)$ as an associative
envelope of ${\cal B}(L)^{(-)}$ is a homomorphic image of the universal
restricted associative envelope $U_{p}(K(L)).$ The latter has dimension
$(\dim K(L))^{p}.$ The lemma is proved.
3.6. Lemma. The algebra ${\cal B}(L)$ is stable under the action of
$L,$ i.e. ${\cal B}(L)^{\mu}\subseteq{\cal B}(L)$ for all $\mu\in L.$
The proof follows from the formula
$$(q^{\mu})^{-}=[q^{-},\mu].$$
(6)6( 6 )
4 Differential operators
Denote by $\Phi(L)$ the associative
subring generated in the endomorphism ring
of the abelian group $(Q,+)$ by $L$ and by the operators of left and
right multiplications by elements from ${\cal B}(L).$ By formula
(1) the ring $\Phi(L)$ may not be an algebra over $C.$ Of course
$\Phi(L)$ is an algebra over the subfield of central constants
$$F=C^{L}\stackrel{{\scriptstyle\rm def}}{{=}}\{c\in C:\forall l\in L\ \ c^{l}=0\}.$$
Nevertheless $\Phi(L)$ is a left and a right space over $C$
while the subring of left multiplications, ${\cal B}(L)^{l},$ and that of
right multiplications, ${\cal B}(L)^{r},$ are algebras over $C.$
4.1. Let us fix derivations $\mu_{1},\ldots,\mu_{m}\in L$ such
that $\mu_{1}+K(L)^{-},\ldots,\mu_{m}+K(L)^{-}$
form a basis for the right $C$-space $L/K(L)^{-}.$ An operator $\Delta$
is called correct if it is of the form:
$$\Delta=\mu^{s_{1}}_{1}\mu^{s_{2}}_{2}\ldots\mu^{s_{m}}_{m},$$
where $0\leq s_{i}<p$ and we suppose that $\mu^{0}=1$ is the identity
operator.
Let $U$ be a right linear space generated by all correct operators. By
formula (1) this set will be a left space over $C,$ also.
4.2. Proposition. The ring $\Phi(L)$ of differential operators
is isomorphic as a left and a right space over $C$ to a
tensor product over $C:$
$$\Phi(L)\simeq{\cal B}(L)^{r}\otimes{\cal B}(L)^{l}\otimes U\simeq U\otimes{%
\cal B}(L)^{l}\otimes{\cal B}(L)^{r},$$
(7)7( 7 )
where $U$ is the linear space generated by correct operators over $C$.
Proof. It is enough to show that each differential operator
$d\in\Phi(L)$ has a unique representation in the form
$$d=\sum_{i,j,k}\alpha^{(k)}_{ij}a^{r}_{ik}a^{l}_{jk}\Delta_{k}$$
(8)8( 8 )
and a unique representation in the form
$$d=\sum_{i,j,k}\Delta_{k}a^{l}_{ik}a^{r}_{jk}\alpha^{(k)}_{ij},$$
(9)9( 9 )
where $a_{ik},a_{jk}\in A$ and $A$ is some fixed basis
of ${\cal B}(L)$ over $C$
(recall that by associativity, $a^{r}_{ik}a^{l}_{jk}=a^{l}_{jk}a^{r}_{ik}$)
and the $\Delta_{k}$’s
are correct words in $\{\mu_{1},\ldots,\mu_{m}\}.$
The existence of this presentation follows from the relations
$$\mu a^{r}=a^{r}\mu-(a^{\mu})^{r}$$
(10)10( 10 )
$$\mu a^{l}=a^{l}\mu-(a^{\mu})^{l}$$
(11)11( 11 )
$$\mu^{p}=\mu_{1}c_{1}+\ldots+\mu_{m}c_{m}+b^{r}-b^{l}$$
(12)12( 12 )
$$\mu_{i}\mu_{j}=\mu_{j}\mu_{i}+\mu_{1}c_{1}+\ldots+\mu_{m}c_{m}+b^{r}-b^{l},$$
(13)13( 13 )
where in formula (12) $\mu_{1}c_{1}+\ldots+\mu_{m}c_{m}+b^{-}$
is a representation
of $\mu^{p}\in L$ as a linear combination of $\mu_{i}$’s modulo $K(L)^{-}$
and in (13) $\mu_{1}c_{1}+\ldots+\mu_{m}c_{m}+b^{-}$ is the
corresponding representation
of $[\mu_{i},\mu_{j}]\in L.$
The transformations of the left hand sides to the right hand sides
(in the last formula
only if $i>j$) allow us to reduce the operator to the form (8).
If we write formulae (10), (11) in the form
$$a^{r}\mu=\mu a^{r}+(a^{\mu})^{r}$$
(14)14( 14 )
$$a^{l}\mu=\mu a^{l}+(a^{\mu})^{l}$$
(15)15( 15 )
then in the same way the operator is reduced to the form (9).
For the proof of the uniqueness it is possible to use
the following results on differential
identities (see [Kh91, theorem 2.2.2, corollary 2.5.8] or [Kh78]).
4.3. Proposition. If the derivations $\mu_{1},\ldots,\mu_{m}\in{\cal D}(R)$ are linearly independent modulo $Q^{-},$ and if
the ring $R$ satisfies an identity of the type
$$\sum^{p^{n}}_{k=1}\sum_{i}a_{ki}x^{\Delta_{k}}b_{ki}=0,$$
where $\Delta_{1},\ldots,\Delta_{p^{n}}$ — are all correct operators and the
coefficients $a_{ki},b_{ki}$ belong to $R_{\cal F},$ then
$\sum_{i}a_{ki}\otimes b_{ki}=0$ in $R_{\cal F}\otimes_{C}R_{\cal F}$
for all $k,1\leq k\leq p^{n}.$ In the same way if the identity
$$\sum^{p^{n}}_{k=1}(\sum_{i}a_{ki}xb_{ki})^{\Delta_{k}}=0$$
is valid then $\sum_{i}a_{ki}\otimes b_{ki}=0,1\leq k\leq p^{n}.$
Since ${\cal D}(I)={\cal D}(R)$ and $Q(I)=Q(R)$ for each nonzero ideal $I$
of $R$ (see [Kh91, Lemma 1.8.4]), then proposition 4.3 shows that the restriction of a nonzero
differential operator $d\in\Phi(L)$ to $I$ is nonzero. This note is
important due to the following lemma:
4.4. Lemma. For each differential operator $d\in\Phi(L)$
there exists a nonzero two sided ideal $I$ of $R$ such that
$I^{d}\subseteq R.$
The proof is easily obtained by induction from the formula
$(I^{2})^{\mu}=I^{\mu}I+II^{\mu}\subseteq I$
which is valid for the ideal $I$ such that $I^{\mu}\subseteq R.$
5 Quasi-Frobenius algebras
Recall that a finite dimensional algebra $B$
over a field $C$ is called quasi-Frobenius if
one of the following equivalent conditions is valid (see [CR62]).
(Q1) For each left ideal $\lambda$ and right ideal $\rho$ of $B$
the following equalities hold:
$$l(r(\lambda))=\lambda,\ \ \ r(l(\rho))=\rho,$$
where $l(A)=\{b\in B:bA=0\}$ is the left annihilator,
$r(A)=\{b\in B:Ab=0\}$ is the right annihilator.
(Q2)The left regular module ${}_{B}B$ is injective.
(Q3) Modules ${}_{B}B$ and $(B_{B})^{*}=Hom(B,C)$ have the same indecomposable
components.
Recall that for any left (right) module $M$ the set of all linear
functionals $M^{*}$ has a structure of right (left) module defined by the
formula $(m^{*}b)(m)=m^{*}(bm)$ (respectively $m(bm^{*})=(mb)m^{*}$).
The modules $M$ and $N$ for $N\simeq M^{*}$ are called
conjugated modules. If the
module $M$ is of finite dimension then $(M^{*})^{*}\simeq M$ and the
conjugacy of modules (left and right), $M$ and $N,$ can be
characterized by the existence of a nondegenerate associative bilinear form
$(\ ,\ ):N\times M\rightarrow C.$ In this case for every basis
$a_{1},\ldots,a_{n}$ of $M$ there exists a dual basis
$a_{1}^{*},\ldots,a_{n}^{*}$ of $N$ which is characterized by the
following properties
$(a_{i}^{*},a_{i})=1,(a_{i}^{*},a_{j})=0,i\neq j.$
Condition (Q3) implies the following condition which is important for us:
(Q4) The sum of all right ideals $\rho$ of $B$
conjugated to left ideals of $B$ is equal to
$B.$
It can be proved that this condition is also equivalent to $B$ being
quasi-Frobenius. Moreover, as (Q1) is left-right symmetric then the left
analog of (Q4) is also valid.
(Q5) The sum of all left ideals ${\lambda}$ of $B$ conjugated to right
ideals of $B$ is equal to $B.$
The most important subclass of the class of quasi-Frobenius algebras is
the class of Frobenius algebras. These algebras are defined by one of the
following equivalent conditions ([CR62]).
(F1) For each left ideal $\lambda$ and right ideal $\rho$ of $B$
the following equalities hold:
$$l(r(\lambda))=\lambda,\ \ \dim r(\lambda)+\dim\lambda=\dim B$$
$$r(l(\rho))=\rho,\ \ \dim l(\rho)+\dim\rho=\dim B.$$
(F2) There exists an element
$\varepsilon\in B^{*}$ whose kernel contains
no nonzero onesided ideals of $B.$
(F3) There exists a nondegenerate associative bilinear form
$B\times B\rightarrow C.$
(F4) The modules ${}_{B}B$ and $(B_{B})^{*}$ are isomorphic.
Classical examples of Frobenius algebras are: group algebras of finite
groups over a field of arbitrary characteristic, universal restricted
enveloping algebras of finite dimensional Lie $p$-algebras, finite dimensional
Hopf algebras, Clifford algebras. Finite dimensional semisimple algebras
evidently satisfy (F1) therefore they are Frobenius.
6 Universal constants
Let $\lambda$ and $\rho$ be left and right conjugated ideals of
${\cal B}(L).$ Let us choose a basis $a_{1},\ldots,a_{n}$ of $\lambda$
and let $a_{1}^{*},\ldots,a_{n}^{*}$ be the dual basis of $\rho.$
It is well-known that the element $c=\sum a_{i}\otimes a_{i}^{*}$ of the
tensor product ${\cal B}\otimes_{C}{\cal B}$ commutes with the elements of
${\cal B},\ \ bc=cb$ for all $b\in{\cal B}.$ This implies that the
set of values of the operator $c_{\lambda,\rho}=\sum a_{i}^{l}(a_{i}^{*})^{r}$
is contained in the centralizer of ${\cal B}.$ In particular for any
$\mu\in K(L)^{-}$ we have
$$c_{\lambda,\rho}(x)^{\mu}=0.$$
(16)16( 16 )
Let $U(L)$ be the associative subring of $\Phi(L)$ generated by $L$
and by the operators of multiplication by central elements. It is clear that
$U(L)$ is both a left and a right space over $C$ and an algebra over the field
of central constants $F=C^{L}.$
Consider the right ideal $I=K(L)^{-}\cdot U(L)$ of $U(L).$
First of all the formula
$\mu a^{-}=a^{-}\mu-(a^{\mu})^{-}$ shows that $I$ is a two sided ideal of $U(L).$
The same formula and formulae (12), (13)
show that the identity operator and operators of the
form $a_{1}^{-}a_{2}^{-}\ldots a_{s}^{-}\Delta,$
where $\Delta$ is a correct operator, $a_{i}\in K(L),\ s\geq 0,$
generate $U(L)$ as a left space over $C.$
6.1. Proposition. The factor-algebra $U(L)/I=\overline{U}$
is Frobenius as
an algebra over $F=C^{L}.$
Proof. By the well-known R.Baer theorem [Ba27] the dimension of $C$
over $F$ is finite and therefore $\overline{U}=U(L)/I$ has a finite dimension
over $F.$ Since $K(L)^{-}\subseteq I,$ the elements $\bar{\mu}_{1}=\mu_{1}+I,\ldots,\bar{\mu}_{m}=\mu_{m}+I$ generate $\overline{U}$ as a ring over $C.$
Moreover the relations $\bar{\mu}_{i}\bar{\mu}_{j}=\overline{[\mu_{i},\mu_{j}]}+\bar{\mu}_{j}\bar{\mu}%
_{i}$ show that the
images of
correct words
$\bar{\Delta}_{k}$ generate $\overline{U}$ as a left vector space over $C.$
The main note is that the elements $\bar{\Delta}_{k}$
are linearly independent over $C.$ If
$$\sum_{k}c_{k}\Delta_{k}=\sum_{k}d_{k}\Delta_{k}\in I,$$
where $d_{k}$ are linear combinations of products of the type
$a_{1}^{-}\cdots a_{s}^{-},$ then taking into account that $a^{-}=a^{r}-a^{l}$
and using Proposition 4.3, we have $c^{r}_{k}=d_{k}$ for all $k,$
which is impossible since $c_{k}^{r}(1)=c_{k},\ d_{k}(1)=0.$
Thus $\bar{\Delta}_{k}$ are linearly independent.
Now let us define Berkson’s linear map
(see [Be64]) $\varphi:\overline{U}\rightarrow C$ which
corresponds to
the element $\sum c_{k}\bar{\Delta}_{k}$
the coefficient of $\bar{\Delta}_{p^{m}}=\bar{\mu}^{p-1}_{1}\ldots\bar{\mu}^{p-1}_{m}.$ The kernel of this linear map contains neither left nor right
nonzero ideals, since the product
$$(\bar{\mu}_{1}^{s_{1}}\ldots\bar{\mu}_{m}^{s_{m}})(\bar{\mu}^{p-s_{1}-1}_{1}%
\ldots\bar{\mu}^{p-s_{m}-1}_{m})$$
written as a linear combination of correct words
contains a unique member $\bar{\Delta}_{p^{m}}$ with a coefficient
equal to 1.
If $\psi:C\rightarrow F$ is any projection, then the linear functional
$\varepsilon:d\mapsto\psi(\varphi(d))$ satisfies (F2) and therefore
$\overline{U}$ is a Frobenius algebra. The proposition is proved.
Let us consider the right subspace $\hat{U}$ of $\overline{U}$ over $C$
generated by all nonempty words $\bar{\Delta}_{k}.$ This space does not
contain the unit (the identity operator) and it is a right
(but, possibly, not
a left) ideal because by formula (14) one has
$$\bar{\Delta}c\bar{\mu}=\bar{\Delta}\bar{\mu}c+\bar{\Delta}c^{\mu}.$$
By formula (13), the product $\bar{\Delta}\bar{\mu}$ can be
written as a linear combination $\sum\bar{\Delta}_{k}c_{k},$ where
$\bar{\Delta}_{k}$ are nonidentity correct operators.
Thus, the left annihilator $A=l(\hat{U})$ in the algebra $\overline{U}$ is not
equal to zero. Moreover, by (F1) its dimension over $F$ is connected with
the dimension of $\hat{U}$ by the formula
$\dim_{F}\overline{U}=\dim_{F}\hat{U}+\dim_{F}A.$ On the other hand
$\dim_{F}\overline{U}=\dim_{F}\hat{U}+\dim_{F}C$ i.e. the dimensions of $A$
and $C$ over $F$ coincide. It means that $A$ is one dimensional over $C$
i.e. $A=C\bar{f}$ (but possibly $A\neq\bar{f}C$ as $A$ may not be a
right $C$-space), where $\bar{f}=\sum\bar{\Delta}_{k}c_{k}=\sum c_{k}^{\prime}\bar{\Delta}_{k}$ is a nonzero element of $\overline{U}.$
Thus, we have obtained that $\bar{f}\bar{\mu}_{i}=\bar{0}$ in $\overline{U}.$
In the ring of differential operators this means that
$f\mu_{i}\in K(L)^{-}\cdot U(L).$ We have also that
$fK(L)^{-}\subseteq K(L)^{-}\cdot U(L)$ as $I=K(L)^{-}\cdot U(L)$
is a two sided ideal. Thus
$$fL\subseteq f(\sum(\mu_{i}C+K(L)^{-}))\subseteq K(L)^{-}\cdot U(L)$$
which, using formula (16), implies
$$((c_{\lambda,\rho}(x))^{f})^{\mu}=0$$
(17)17( 17 )
for all ${\mu\in L}.$ Let us formulate the obtained result as a lemma
(see also Lemma 4.6, [Kh95]).
6.2. Lemma. There exists a differential operator $f$ of the type
$\sum\Delta_{k}c_{k}=\sum c^{\prime}_{k}\Delta_{k},$ such that for each
conjugated left ideal $\lambda$ and right ideal $\rho$ of ${\cal B}$
with dual bases $a_{1},\ldots,a_{n}$ and $a_{1}^{*},\ldots,a_{n}^{*},$ the
operator
$$u_{\lambda,\rho}=\sum_{i}a_{i}^{l}(a_{i}^{*})^{r}f$$
(18)18( 18 )
has values only in the ring of constants $Q^{L}.$ There exists a nonzero
ideal $I$ of $R$ such that
$$0\neq u_{\lambda,\rho}(I)\subseteq R^{L}$$
(19)19( 19 )
Proof. The representation of
$f$ in the form $\sum c^{\prime}_{k}\Delta_{k}$
follows from (10).
Formula (19) follows from formula (17), proposition 4.3 and lemma 4.4.
7 PI rings of constants
In this secton we will prove the theorem about a generalized polynomial
identity and discuss
its generalization to the case when the inner part is not quasi-Frobenius.
7.1. Theorem. Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ Suppose that the inner associative
part ${\cal B}(L)$ of $L$ is quasi-Frobenius. If the ring of constants $R^{L}$
is PI then $R$ is GPI.
Proof. Let $f(x_{1},\ldots,x_{n})=0$ be a multilinear identity
of $R^{L}.$ Let us choose arbitrary left ideals $\lambda_{1},\ldots,\lambda_{n}$
of ${\cal B}(L)$ having conjugated right ones $\rho_{1},\ldots,\rho_{n}.$
By Lemma 6.2 for every $j,1\leq j\leq n$ there exists an operator
$$u_{j}=u_{\lambda_{j},\rho_{j}}=\sum_{i}a^{l}_{ij}(a_{ij}^{*})^{r}f_{j}=\sum_{i%
,k}a_{ij}^{l}(a_{ij}^{*})^{r}c_{k}^{\prime}\Delta_{k}$$
and a nonzero ideal $I_{j}$ of $R,$ such that $0\neq u_{j}(I_{j})\subseteq R^{L}.$
If $I=\cap I_{j}$ then $u_{j}(x)\in R^{L}$ for all $x\in I$
and therefore the following differential identity holds in $I$
$$f(u_{1}(x_{1}),\ u_{2}(x_{2}),\ldots,\ u_{n}(x_{n}))=0.$$
Let us fix some values of $x_{2}=b_{2},\ldots,x_{n}=b_{n}$ in $I.$ We have
$$f(\sum_{i,k}(c_{k}^{\prime}a_{i1}x_{1}a_{i1}^{*})^{\Delta_{k}},\ u_{2}(b_{2}),%
\ldots,\ u_{n}(b_{n}))=0.$$
(20)20( 20 )
By Leibnitz formula any expression of the type $(axb)^{\Delta}$
can be written in the form
$$(axb)^{\Delta}=ax^{\Delta}b+\sum_{s}a_{s}x^{\Delta_{s}}b_{s},$$
where $\Delta_{s}$ are subwords of $\Delta.$ In particular
$$(c_{k}^{\prime}a_{i1}x_{1}a_{i1}^{*})^{\Delta_{k}}=c_{k}^{\prime}a_{i1}x_{1}^{%
\Delta_{k}}a_{i1}^{*}+\sum_{s}a_{s}x_{1}^{\Delta_{s}}b_{s}.$$
(21)21( 21 )
If $\Delta_{k_{0}}$ is the greatest operator such that $c^{\prime}_{k_{0}}$
is not zero, then this formula allows us to represent (20) in the form
$$\sum^{k_{0}}_{k=1}\sum_{i}v_{ki}x_{1}^{\Delta_{k}}w_{ki}=0,$$
here we suppose that $\Delta_{1}<\Delta_{2}<\ldots<\Delta_{p^{m}}$ is the
lexicographic ordering of all correct operators.
By Proposition 4.3 applied to the prime ring $I$ we have
$$\sum_{i}v_{k_{0}i}\otimes w_{k_{0}i}=0$$
in the tensor product $I_{\cal F}\otimes_{C(I)}I_{\cal F},$ where
$C(I)$ is the generalized centroid of $I$ and $I_{\cal F}$ is the left
Martindale ring of quotients of $I.$ It is well-known and it is easy to
see that $I_{\cal F}=R_{\cal F}$ and $C(I)=C(R).$ Therefore for any
$x_{1}\in R_{\cal F}$ we have the identity
$$\sum_{i}v_{k_{0}i}x_{1}w_{k_{0}i}=0.$$
This identity with (21) and (20) implies that the identity
$$c_{k_{0}}^{\prime}f(\sum_{i}a_{i1}x_{1}a_{i1}^{*},u_{2}(b_{2}),\ldots,u_{n}(b_%
{n}))=0$$
(22)22( 22 )
is valid for each $x_{1}\in R_{\cal F}.$
Since the values $b_{2},\ldots,b_{n}$ are arbitrary from $I$, we have an identity
of the form
$$f(\sum_{i}a_{i1}x_{1}a_{i1}^{*},u_{2}(x_{2}),\ldots,u_{n}(x_{n}))=0,$$
(23)23( 23 )
where $x_{1}\in R_{\cal F},x_{2}\in I,\ldots,x_{n}\in I.$
Now let us fix values
$x_{1}\in R_{\cal F},x_{3}=b_{3}\in I,\ldots,x_{n}=b_{n}\in I.$
Then in the same way we obtain
$$f(\sum_{i}a_{i1}x_{1}a_{i1}^{*},\sum_{i}a_{i2}x_{2}a_{i2}^{*},\ldots,u_{n}(x_{%
n}))=0,$$
where $x_{1},x_{2}\in R_{\cal F},x_{3},\ldots x_{n}\in I.$
Continuing this process we will obtain the following identity on $R_{\cal F}:$
$$f(\sum_{i}a_{i1}x_{1}a_{i1}^{*},\sum_{i}a_{i2}x_{2}a_{i2}^{*},\ldots,\sum_{i}a%
_{in}x_{n}a_{in}^{*})=0,$$
(24)24( 24 )
This is a generalized identity valid in $R_{\cal F}\supseteq R.$ All we need
is to prove that for some $\lambda_{1},\ldots,\lambda_{n};\rho_{1},\ldots,\rho_{n}$ this is not a trivial identity. It means that the left hand side
of (24) is not zero in the free product $R_{\cal F}*_{C}C\langle x_{1},\ldots,x_{n}\rangle$
or, in other words, this identity does not follow from the trivial
generalized identities
$xc=cx,$ where $c\in C.$
Otherwise assume all these identities are trivial.
Any application
of a trivial
identity does not change the order of the indeterminates, therefore
all the generalized monomials (i.e. sums of all monomials with fixed order
of sequence of the indeterminates) in the identities (24) should be (trivial)
identities. These generalized monomials have the form
$$\alpha_{\pi}(\sum_{i}a_{i\pi(1)}x_{\pi(1)}a^{*}_{i\pi(1)})(\sum_{i}a_{i\pi(2)}%
x_{\pi(2)}a^{*}_{i\pi(2)})\cdots(\sum_{i}a_{i\pi(n)}x_{\pi(n)}a^{*}_{i\pi(n)}),$$
where $\pi$ is a permutation and
$$f(x_{1},\ldots x_{n})=\sum_{\pi}\alpha_{\pi}x_{\pi(1)}\cdots x_{\pi(n)}.$$
Since one of the coefficients $\alpha_{\pi}$ is equal to one
(let $\alpha_{1}=1$),
$$(\sum_{i}a_{i1}x_{1}a^{*}_{i1})(\sum_{i}a_{i2}x_{2}a^{*}_{i2})\cdots(\sum_{i}a%
_{in}x_{n}a^{*}_{in})=0$$
(25)25( 25 )
Let us fix some values of $x_{2},\ldots,x_{n}$ in $R$ and apply Proposition 4.3
to (25), where we suppose $x=x_{1},$ and all coefficients $a_{ki},\ \ k=2,3,\ldots p^{m}$ are zero. We have
$$(\sum_{i}a_{i1}\otimes a_{i1}^{*})(\sum_{i}a_{i2}x_{2}a^{*}_{i2})\cdots(\sum_{%
i}a_{in}x_{n}a^{*}_{in})=0.$$
The set $\{a_{i1}\}$ is a basis of the ideal $\lambda_{1},$
i.e. this is a linearly independent set, therefore
$$a_{i1}^{*}(\sum_{i}a_{i2}x_{2}a^{*}_{i2})\cdots(\sum_{i}a_{in}x_{n}a^{*}_{in})=0$$
for all $a_{i1}^{*}$ from the dual basis $\{a_{i1}^{*}\}$ of the conjugated
ideal $\rho_{1}.$ This implies that
$$\rho_{1}(\sum_{i}a_{i2}x_{2}a^{*}_{i2})\cdots(\sum_{i}a_{in}x_{n}a^{*}_{in})=0.$$
Since the pair $(\lambda_{1},\rho_{1})$ was chosen in an arbitrary way,
$$(\sum_{\rho^{*}\simeq\ a\ left\ ideal\ of\ {\cal B}}\rho)(\sum_{i}a_{i2}x_{2}a%
^{*}_{i2})\cdots(\sum_{i}a_{in}x_{n}a^{*}_{in})=0.$$
(26)26( 26 )
By Property (Q5) of quasi-Frobenius algebras
$$1\in{\cal B}=(\sum_{\rho^{*}\simeq\ a\ left\ ideal\ of\ {\cal B}}\rho)$$
and
therefore
$$(\sum_{i}a_{i2}x_{2}a^{*}_{i2})\cdots(\sum_{i}a_{in}x_{n}a^{*}_{in})=0.$$
Now the evident induction works. The theorem is proved.
The same proof can be applied also for some cases when the inner
part is not quasi-Frobenius but has enough pairs of conjugated
one-sided ideals. Indeed, let us denote by ${\cal B}_{r}$ the sum
of all right ideals of a finite dimensional algebra ${\cal B}$
conjugated to left ones.
7.2. Lemma. ${\cal B}_{r}$ is a two-sided ideal of ${\cal B}.$
Proof. Let $\rho$ be a right ideal such that the dual left
module $\rho^{*}={\rm Hom}(\rho,C)$ is isomorphic to a left ideal
$\lambda.$ If $b\in{\cal B}$ then we have an exact sequence of homomorphisms
of right ideals $\rho\rightarrow b\rho\rightarrow 0.$ The conjugated
sequence has the form $\rho^{*}\leftarrow(b\rho)^{*}\leftarrow 0,$
therefore the right ideal $b\rho$ has a conjugated module $(b\rho)^{*}$
which is isomorphic to a left subideal of $\lambda\simeq\rho^{*}.$
Thus $b\rho\subseteq{\cal B}_{r}$ and ${\cal B}_{r}$ is a two-sided ideal.
The lemma is proved.
In the same way one can define an ideal ${\cal B}_{l}$ — the sum of all
left ideals conjugated to right ones.
7.3. Theorem. Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ If the algebra of
constants $R^{L}$ satisfies a multilinear polynomial identity of degree
$n$ and ${\cal B}(L)^{n}_{r}\neq 0,$ then $R$ is a GPI-ring.
Proof. In the same way as in the proof of Theorem 7.1 we have
identities (24). If all of these identities are trivial then we also have
the identities (26)
which can be written in the form
$${\cal B}(L)_{r}(\sum a_{i2}x_{2}a_{i2}^{*})\cdots(\sum a_{in}x_{n}a_{in}^{*})=0.$$
(27)27( 27 )
If $b$ is an arbitrary element of ${\cal B}(L),$ then
$b(\sum_{i}a_{ik}x_{k}a_{ik}^{*})=(\sum_{i}a_{ik}x_{k}a_{ik}^{*})b.$
Therefore for $b\in{\cal B}(L)_{r},$ identity (27) implies
$$(\sum a_{i2}x_{2}a_{i2}^{*})\cdots(\sum a_{in}x_{n}a_{in}^{*})b=0.$$
(28)28( 28 )
By Proposition 4.3 we have
$$(\sum a_{i2}\otimes a_{i2}^{*})\cdots(\sum a_{in}x_{n}a_{in}^{*})b=0,$$
as in the proof of Theorem 7.1 we have
$${\cal B}(L)_{r}(\sum a_{i3}x_{3}a_{i3}^{*})\cdots(\sum a_{in}x_{n}a_{in}^{*})b%
=0,$$
thus
$$(\sum a_{i3}x_{3}a_{i3}^{*})\cdots(\sum a_{in}x_{n}a_{in}^{*}){\cal B}(L)_{r}^%
{2}=0.$$
Now the evident induction implies ${\cal B}(L)_{r}^{n}=0.$ Hence
one of the GPI’s (24) is not trivial. The theorem is proved.
In a symmetrical way one can prove that the condition ${\cal B}(L)^{n}_{l}\neq 0$
also implies that one of the identities (24) is not trivial. It can be proved
that ${\cal B}^{n}_{r}=0$ iff ${\cal B}^{n}_{l}=0:$
7.4. Proposition. Let ${\cal B}$ be a finite dimensional algebra.
Then all $n+1$ conditions ${\cal B}_{r}^{k}{\cal B}_{l}^{n-k}=0,\ \ k=0,\ldots,n$
are equivalent to each other.
Proof. It is enough to show that the conditions for $k$
and $k+1$ are equivalent. The condition ${\cal B}_{r}^{k}{\cal B}_{l}^{n-k}=0$
is equivalent to
${\cal B}_{r}^{k}{\cal B}_{l}^{n-k-1}\lambda=0$ for
all pairs of conjugated ideals $\rho,\lambda.$ Since the form
$(\ ,\ ):\rho\times\lambda\rightarrow C$ is
nondegenerate the last condition
for given $\lambda,\rho$ is equivalent to
$(\rho,{\cal B}_{r}^{k}{\cal B}_{l}^{n-k-1}\lambda)=0.$ By
associativity of the form this is equivalent to
$(\rho{\cal B}_{r}^{k}{\cal B}_{l}^{n-k-1},\lambda)=0$ and since
the form is nondegenerate
this is equivalent to $\rho{\cal B}_{r}^{k}{\cal B}_{l}^{n-k-1}=0.$
The last conditions for all pairs of conjugated ideals $\lambda,\rho$
are equivalent to ${\cal B}_{r}^{k+1}{\cal B}_{l}^{n-k-1}=0.$ The proposition
is proved.
Now it is a question of interest whether the condition ${\cal B}(L)^{n}_{r}=0$
implies that all identities (24) are trivial generalized polynomial
identities. The answer is yes:
7.5. Proposition. If under the conditions of theorem 7.3
${\cal B}(L)_{r}^{n}=0,$ then all identities (24) are trivial.
Proof. It is enough to show that all the generalized monomials (25)
are trivial identities. We will prove by inverse induction on $k$
that for arbitrary
$b_{1},\ldots,b_{k}\in{\cal B}(L)_{r}$ the generalized polynomial
$$(\sum_{i}a_{i\ k+1}x_{k+1}a_{i\ k+1}^{*})\cdots(\sum_{i}a_{in}x_{n}a_{in}^{*})%
b_{k}b_{k-1}\cdots b_{1}=0$$
(29)29( 29 )
is a trivial generalized identity.
If $k=n$ then (29) has the form $b_{n}b_{n-1}\cdots b_{1}=0$ that is a trivial
identity as ${\cal B}(L)^{n}_{r}=0.$
Assume that (29) is a trivial identity. The identities
$$b(\sum_{i}a_{is}xa_{is}^{*})=(\sum_{i}a_{is}xa_{is}^{*})b,\ \ b\in{\cal B}(L)$$
(30)30( 30 )
are trivial generalized polynomial identities (as well as any
linear generalized identity). Let $b_{k}=a_{ik}^{*},$ then from (29) and (30)
we have the following trivial identity
$$a_{ik}^{*}(\sum_{i}a_{i\ k+1}x_{k+1}a_{i\ k+1}^{*})\cdots(\sum_{i}a_{in}x_{n}a%
_{in}^{*})b_{k-1}\cdots b_{1}=0.$$
Multiplication of this equality on the left by $a_{ik}x_{k}$ and summation
over $i$ gives the equality (29) with a smaller $k.$ The proposition is
proved.
8 Semiprime PI-rings of constants
In this secton we will prove under the conditions of Theorem 7.1,
that if
the ring of constants $R^{L}$ is a semiprime PI-ring, then $R$ is also PI.
8.1. Theorem. Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ Suppose that the inner associative
part ${\cal B}(L)$ of $L$ is quasi-Frobenius. If the ring of constants $R^{L}$
is a semiprime PI-ring, then $R$ is PI.
Proof. By Theorem 7.1 the ring $R$ satisfies a generalized polynomial
identity. Moreover all generalized polynomial identities (24) hold
in its left Martindale ring of quotients $R_{\cal F}.$ In particular they hold
in the central closure $RC\subseteq R_{\cal F}$ of the ring $R.$
By the Martindale structure theorem
[Ma69] this central closure has an idempotent
$e,$ such that $D=eRCe$ is a skew field of finite dimension
over $C.$ (Note that formally Martindale
theorem can be applied only if the
coefficients of the identity belong to $R.$ In our case they belong to
$R_{\cal F}$ but may not belong to $R.$ Nevetheless Martindale’s original
proof is correct for our case too; see, for instance, [Kh91, Theorem 1.13.4]
or the special investigation in [La86].)
Thus, by the Martindale theorem, $RC$ is a primitive ring with a nonzero socle.
The N. Jacobson structure theorem [Ja64] shows that $RC$ is a dense subring in
the finite topology in the complete ring ${\cal E}$ of linear transformations
of the left space $V=eRCe$ over the skew field $D.$
Moreover, the left Martindale quotient ring $(RC)_{\cal F}$ is equal to
${\cal E}$ (see, [Ha82, Lemma 1.1] and [Ha87, Remark 4.9] or
[Kh91, Theorem 1.15.1]). It is easy to see
that $R_{\cal F}\subseteq(RC)_{\cal F}={\cal E}.$ (Indeed, if $q\in R_{\cal F}$ and $Iq\subseteq R$ for a nonzero ideal $I$ of $R,$ then we can extend
$q$ to the ideal $IC$ of $RC$ by the obvious formula
$(\sum i_{\alpha}c_{\alpha})q=\sum(i_{\alpha}q)c_{\alpha}.$
This is well-defined. Indeed, if $\sum i_{\alpha}c_{\alpha}=0$
and $J$ is a nonzero
ideal of $R$
such that $Jc_{\alpha}\subseteq R$ then $\sum(jc_{\alpha})i_{\alpha}=0$
for all $j\in J.$ Therefore $\sum(jc_{\alpha})(i_{\alpha}q)=0;$ i.e.
$J(\sum c_{\alpha}(i_{\alpha}q))=0$ and $\sum(i_{\alpha}q)c_{\alpha}=0.$)
Now all the coefficients of (24) belong to
${\cal E}$ and since addition and multiplication are continuous in the finite
topology, the identities (24) hold in ${\cal E}.$ (Here one can use
also Corollary 2.3.2 from [Kh91] which
allows us to extend identities from $RC$ to
$(RC)_{\cal F}.$)
Now we are going to prove that the space $V$ is finite dimensional over $D.$
In that case the dimension of ${\cal E}$ over $C$ will also be finite:
$d=\dim_{C}{\cal E}=(\dim_{D}V)^{2}\cdot\dim_{C}D$ and ${\cal E}$ (and
therefore $R$),
like any $d$-dimensional algebra, will
satisfy the standard polynomial identity:
$$S_{d}(x_{1},\ldots,x_{d+1})\equiv\sum(-1)^{\pi}x_{\pi(1)}\cdots x_{\pi(d+1)}=0.$$
On the contrary, suppose that $V$ has infinite dimension $\dim V=\beta.$
Let $M$ be the set of all linear transformations whose rank is less then $\beta.$
(Recall that the rank of a transformation $l$ is the
dimension over $D$ of
its image.) It is well-known that $M$ is a maximal ideal of ${\cal E}.$
So the factor ring $\bar{\cal E}={\cal E}/M$ is a simple ring with a unit.
8.2. Lemma. The ring $\bar{\cal E}$ is not Artinian.
Proof. Let $\{e_{i},i\in I\}$ be a basis of $V$ over $D.$ and
$$I_{1}\supset I_{2}\supset\ldots\supset I_{n}\supset\ldots$$
be a chain of
subsets such that $|I_{k}\setminus I_{k+1}|=\beta,$ and let
$$A_{n}=\{l\in{\cal E}:e_{i}l=0\ \ \forall i\in I\setminus I_{n}\}.$$
Then
$$(A_{1}+M)/M\supset A_{2}+M/M\supset\ldots\supset A_{n}+M/M\supset\ldots$$
is an infinite descending chain of right ideals of $\bar{\cal E}.$
Indeed, if $A_{n}+M=A_{n+1}+M,$ then for the transformation $w,$ defined by
$$e_{i}w=\left\{\begin{array}[]{ll}e_{i}&\mbox{if $i\in I_{n}\setminus I_{n+1}$}%
\\
0&\mbox{otherwise}\end{array}\right.,$$
we should get a presentation $w=a+m,$ where $a\in A_{n+1},\ m\in M.$
Let $V_{1}$ be a subspace generated by $\{e_{i}:i\in I_{n}\setminus I_{n+1}\}.$
Then $V_{1}=V_{1}w\subseteq V_{1}a+V_{1}m=V_{1}m.$ However, $\dim_{D}V_{1}=\beta,$
while $\dim_{D}V_{1}m\leq\dim Vm<\beta,$ which is a contradiction. The lemma is
proved.
8.3. Lemma. The ring $\bar{\cal E}$ does not satisfy a non trivial
generalized polynomial identity.
Proof. Like any simple ring with a unit, the ring $\bar{\cal E}$ is
primitive. If it satisfies a GPI, then by the S.A. Amitsur structure theorem
[Am65] it has a nonzero socle $S,$ which is a two-sided ideal and
therefore $S=\bar{\cal E}.$ In N. Jacobson presentation of $\bar{\cal E}$
as a dense ring of linear transformations, the socle consists of all
transformations of finite rank. This means that the unit has finite
rank and therefore the space has finite dimension. Thus $\bar{\cal E}$
is the ring of all linear transformations of a finite dimensional space
over a skew field. In particular $\bar{\cal E}$ is Artinian; this is a
contradiction
to Lemma 8.2. The lemma is proved.
Let us consider now identities (24). We have seen that all these identities
hold in ${\cal E}.$ If we apply the natural homomorphism
$\varphi:{\cal E}\rightarrow\bar{\cal E}={\cal E}/M$ we obtain
the following identities of the ring $\bar{\cal E}$
$$f(\sum_{i}\bar{a}_{i1}x_{1}\bar{a}^{*}_{i1},\ldots,\sum_{i}\bar{a}_{in}x_{n}%
\bar{a}^{*}_{in})=0,$$
(31)31( 31 )
where $\bar{a}=\varphi(a)=a+M.$
By Lemma 8.3 all we need is to prove that
one of the identities (31) is a nontrivial
GPI of $\bar{\cal E}.$
First of all we have to calculate the generalized centroid of $\bar{\cal E}.$
As $\bar{\cal E}$ is a simple ring with a unit, it equals its left Martindale
quotient ring and therefore the generalized centroid is equal to the center.
8.4. Lemma. The center of
$\bar{\cal E}$ is canonically isomorphic
to $C,\ \ C(\bar{\cal E})=\varphi(C).$
Proof. See [Ro58, Corollary 3.3].
We will need the following result which gives a criterium
for determining when the
ring of constants is semiprime (see Theorem 5.1 [Kh95]).
8.5. Theorem. Under the conditions of theorem 8.1, the ring of
constants is semiprime if and only if ${\cal B}(L)$ is differentially
semisimple, i.e. it has no nonzero differential (with respect to
action of $L$) ideals with zero
multiplication or, equivalently, it is a sum of a finite number of
differentially simple algebras.
By this theorem we have that in our situation the algebra ${\cal B}(L)$
is differentially semisimple.
8.6. Lemma. The ideal $M$ is a differential ideal with
respect to $L,$ i.e. $M^{\mu}\subseteq M$ for each $\mu\in L.$
Proof. Note that $M$ is a differential ideal with respect to
each derivarion of ${\cal E}.$ Indeed, if $l\in M$ than $l$ is a
transformation of rank less then $\beta$ and the projection
$e:V\rightarrow{\rm im}l$ also has rank less than $\beta,$
in which case $l=le.$ We have $l^{\mu}=l^{\mu}e+le^{\mu}\in M$
for each derivation $\mu\in{\rm Der}({\cal E}).$
By proposition 1.8.1 [Kh91] any $R$-continuous derivation has a unique
extension to $R_{\cal F}.$ In particular each derivation from $L$
is defined on $RC.$ Again by the same proposition we have that
the elements of $L$ have a unique extensions
to $(RC)_{\cal F}={\cal E}.$ Thus we have
obtained that the ideal $M$ is differential with respect to $L.$
The lemma is proved.
As a consequence we have that the intersection $M_{0}=M\cap{\cal B}(L)$ is
a differential
ideal of ${\cal B}(L),$ which is not equal to ${\cal B}(L)$
(it does not contain 1). The left
annihilator $l(M_{0})$ of $M_{0}$ in ${\cal B}(L)$ is also a differential
ideal, therefore $l(M_{0})\cap M_{0}$ is a differential ideal with zero
mulitiplication. By theorem 8.5, $l(M_{0})\cap M_{0}=0.$ In the same way
the left annihilator of the sum $l(M_{0})+M_{0}$ is zero (it is contained in
$l(M_{0})$ and, therefore, has a zero multiplication). Now property (Q1)
of quasi-Frobenius algebras implies that
$l(M_{0})+M_{0}=r(l(l(M_{0})+M_{0}))=r(0)={\cal B}(L)$ and, finally
$${\cal B}(L)=l(M_{0})\oplus M_{0}=e{\cal B}(L)\oplus(1-e){\cal B}(L),$$
(37)37( 37 )
where $e$ is a central idempotent defined by the corresponding decomposition
of the unit $1=e\oplus(1-e).$
Let us return to identities (31). Suppose that in these identities
$\{a_{ij}\}$ and $\{a_{ij}^{*}\}$ are bases of conjugated ideals
$\lambda_{j},\rho_{j}$ contained in $l(M_{0}).$ In that case the sets
$A_{j}=\{\bar{a}_{ij},i=1,\ldots m\}$ are linearly independent over
the center of $\bar{\cal E}$ (see lemma 8.4). Moreover, the $C$-space
generated by all possible $a_{ij}^{*}$’s contains the
unit $e$ of $l(M_{0})$ because for each
conjugated pair of ideals $\lambda,\rho$
the one-sided ideals $e\lambda,e\rho$ are also conjugated with
respect to the same
form (note that $e$ is a central idempotent of ${\cal B}(L)$).
This implies that the linear space over the center of $\bar{\cal E}$
generated by all $\bar{a}_{ij}^{*}$’s contains the unit $\bar{e}$ of
$\bar{\cal E}.$ This fact allows us to prove that one of the identities
(31) is nontrivial in the same manner as it was done in the end of
the proof of Theorem 7.1. By Lemma 8.3, Theorem 8.1 is proved.
In this proof we used the fact that the
inner part ${\cal B}(L)$ is differentially
semisimple and that it has enough pairs of conjugated ideals.
Therefore in the way analogous to Theorem 7.3 we can formulate a
slightly more general result.
8.7. Theorem. Let $L$ be a finite dimensional restricted
differential Lie $C$-algebra of $R$-continuous derivations of a prime
ring $R$ of positive characteristic $p>0.$ Suppose that the inner
part ${\cal B}(L)$ is a direct sum of differentially simple ideals
$${\cal B}(L)=B_{1}\oplus B_{2}\oplus\ldots\oplus B_{m}.$$
If the algebra of constants $R^{L}$ satisfies a multilinear polynomial
identity of degree $n$ and $(B_{i})^{n}_{r}\neq 0,i=1,\ldots,m,$ then
$R$ is a PI-ring.
The only place where we have used that ${\cal B}(L)$ is quasi-Frobenius
is decomposition (37). Therefore it is enough to show that
each differential ideal of
the direct sum of differentially simple algebras with units
is a direct summand.
If $B=B_{1}\oplus B_{2}\oplus\ldots\oplus B_{m}$ is a direct sum of
differentially simple algebras then for any differential ideal $A$
we have that $AB_{i}$ is a differential ideal of $B_{i}.$ This implies
that either $B_{i}\subseteq A$ or $AB_{i}=0.$ In the same way
either $B_{i}\subseteq A$ or $B_{i}A=0.$
Let $l(A)$ be the left annihilator of $A,$ then $l(A)\cap A$ is
a differential ideal with zero multiplication, so its product
with each $B_{i}$ is zero. This is possible only if the intersection
is zero. In the same way the left annihilator of the sum $l(A)+A$
has a zero multiplication and therefore it is equal to zero. It means that
$l(A)+A$ contains all the components $B_{i}$ and $l(A)\oplus A=B.$
ACKNOWLEDGMENT
The authors are grateful to Professor Dalit Baum for her help.
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Reduction of surface spin-induced electron spin relaxations in nanodiamonds
Zaili Peng
Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
Jax Dallas
Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
Susumu Takahashi
[email protected]
Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA
Department of Physics & Astronomy, University of Southern California, Los Angeles, California 90089, USA
(November 18, 2020)
Abstract
Nanodiamonds (NDs) hosting nitrogen-vacancy (NV) centers are promising for applications of quantum sensing.
Long spin relaxation times ($T_{1}$ and $T_{2}$) are critical for high sensitivity in quantum applications.
It has been shown that fluctuations of magnetic fields due to surface spins strongly influences $T_{1}$ and $T_{2}$ in NDs. However, their relaxation mechanisms have yet to be fully understood.
In this paper, we investigate the relation between surface spins and $T_{1}$ and $T_{2}$ of single-substitutional nitrogen impurity (P1) centers in NDs.
The P1 centers located typically in the vicinity of NV centers are a great model system to study the spin relaxation processes of the NV centers.
By employing high-frequency electron paramagnetic resonance (EPR) spectroscopy, we verify that air annealing removes surface spins efficiently and significantly reduces their contribution to $T_{1}$.
I Introduction
Diamond is a fascinating material in physics, chemistry and biology.
For example, a negatively charged nitrogen-vacancy (NV) center in diamond is a promising platform for fundamental sciences and applications of quantum sensing because of its unique magnetic and optical properties as well as a long coherence time at room temperature. Gruber et al. (1997); Wrachtrup and Jelezko (2006); Epstein et al. (2005); Gaebel et al. (2006); Childress et al. (2006); Takahashi et al. (2008); Degen (2008); Maze et al. (2008); Balasubramanian et al. (2008); Hall et al. (2009); Maletinsky et al. (2012)
Magnetic sensing using a single NV center has been utilized to improve the sensitivity of electron paramagnetic resonance (EPR) spectroscopy to the level of a single spin. de Lange et al. (2012); Grinolds et al. (2013); Tetienne et al. (2013); Mamin et al. (2013); Grinolds et al. (2014); Kaufmann et al. (2013); Sushkov et al. (2014); Shi et al. (2015); Abeywardana et al. (2016)
NV-detected EPR allows the detection of external spins existing around the NV center within several nanometers.
NV-based sensing is also useful to detect electric field, temperature, strain and pH value in a nanoscale volume. Dolde et al. (2011); Jarmola et al. (2012); Cai, Jelezko, and Plenio (2014); Fujisaku et al. (2019)
In NV-detected magnetic sensing, a magnetic field is detected through the measurement of the spin relaxation times of NVs such as $T_{2}$ and $T_{1}$.
For example, in NV-based AC magnetic sensing measurement using a spin echo sequence, the detectable magnetic field is proportional to $1/\sqrt{T_{2}}$. Taylor et al. (2008)
A small number of Gd${}^{3+}$ spins has been detected through sensing of fluctuating magnetic fields from Gd${}^{3+}$ spins. Tetienne et al. (2013)
In this case, the detection is achieved by measuring changes of $T_{1}$ relaxation time and the detectable magnetic field is proportional to $1/T_{1}$.
Thus, long $T_{1}$ and $T_{2}$ times are desired for high detection sensitivity.
In NV-based magnetic sensing applications, it is also critical to position the NV center near a target of the magnetic field sensing.
NVs located near the diamond surface and NVs in nanodiamonds (NDs) will therefore be an ideal platform for the applications.
However, $T_{1}$ and $T_{2}$ relaxation times of those NVs are often significantly reduced by surface defects and impurities including dangling bonds, graphite layers and transition metals. De Sousa (2007); Tisler et al. (2009); Kaufmann et al. (2013); Song et al. (2014); Myers et al. (2014a); Rosskopf et al. (2014); Ofori-Okai et al. (2012); Tetienne et al. (2013); Iakoubovskii et al. (2000); Shames et al. (2002); Soltamova et al. (2009); Dubois et al. (2009); Peng et al. (2019)
For instance, it has been reported that shorter $T_{1}$ and $T_{2}$ were observed from shallow NVs. Ofori-Okai et al. (2012); Myers et al. (2014b)
It has also been reported that $T_{1}$ of NVs in NDs is shorter than $T_{1}$ in bulk diamond.
The recent study showed that $T_{1}$ of NV centers is shorter in a smaller size of NDs and the result implies a decoherence process due to surface impurities although the surface impurities were not measured in the study. Tetienne et al. (2013)
Moreover, control of the diamond surface enables the determination of spin relaxation mechanisms, subsequently improving the sensitivity of the NV-based magnetic sensing techniques. The recent experiment by Tsukahara et al. showed that air annealing efficiently removes graphite layers compared with tri-acid cleaning and increases the $T_{2}$ time 1.4 times longer. Tsukahara et al. (2019)
In this paper, we investigate the relation between surface spins and $T_{1}$ and $T_{2}$ of single substitutional nitrogen impurity (P1) centers in NDs using high-frequency (HF) EPR spectroscopy.
Our previous study on NDs suggested that the surface spins are dangling bonds located in the surface shell with a thickness of $\sim 9$ nm. Peng et al. (2019)
Therefore, the present study aims to remove the surface spins by etching of NDs more than 9 nm and improve the spin relaxation times.
Although $T_{1}$ and $T_{2}$ of NV centers are the primary interest for the quantum sensing applications, there are a few advantages to study the spin relaxation on P1 centers over NV centers.
First, NV centers are located near P1 centers, shown by the detection of their magnetic dipole coupling via double electron-electron resonance spectroscopy. de Lange et al. (2012); Abeywardana et al. (2016); Stepanov and Takahashi (2016)
Therefore, their $T_{1}$ and $T_{2}$ times are similar and the relaxation mechanisms are often common. Takahashi et al. (2008)
Second, as shown in the previous study, Peng et al. (2019)
EPR signals of both P1 and surface spins are observable in the same measurement.
This allows us to determine the amount of surface spins and to study the spin relaxations using the same ND samples.
In the experiment, we employ air annealing to etch the diamond surface efficiently.
The performance of the air annealing is confirmed by dynamic light scattering (DLS) and 230 GHz EPR experiments.
The result of the DLS characterization shows a uniform etching and a linear etching rate of ND samples. We also confirm the reduction of the surface spins after the annealing process with high resolution 230 GHz EPR spectral analysis.
Then, we investigate $T_{1}$ of P1 centers after the annealing using 115 GHz pulsed EPR spectroscopy.
The 115 GHz EPR configuration is advantageous for pulsed EPR experiment because of its higher output power.
The temperature and size dependence study elucidates surface spin-induced $T_{1}$ process.
From the result, we find that air annealing significantly reduces the presence of surface spins, but a small fraction remains,
even after the thickness of NDs is reduced more than 9 nm.
We also find that the surface spin contribution on $T_{1}$ is suppressed by a factor of $7.5\pm 5.4$ after annealing at 550 ${}^{\circ}$C for 7 hours.
With the same annealing condition, $T_{2}$ is improved by a factor of $1.2\pm 0.2$.
EPR signals of both P1 and surface spins are observable in the same measurement.
This allows us to determine the amount of surface spins and to study the spin relaxations using sample samples.
In the experiment, we employ air annealing to etch the diamond surface efficiently.
The performance of the air annealing is confirmed by dynamic light scattering (DLS) and 230 GHz EPR experiments.
The result of the DLS characterization shows a uniform etching and a linear etching rate of ND samples.
We also confirm the reduction of the surface spins after the annealing process with high resolution 230 GHz EPR spectral analysis.
Then, we investigate $T_{1}$ of P1 centers after the annealing using 115 GHz pulsed EPR spectroscopy.
The 115 GHz EPR configuration is advantageous for pulsed EPR experiment because of its higher output power.
The temperature and size dependence study elucidates surface spin-induced $T_{1}$ process.
From the result, we find that air annealing significantly reduces the presence of surface spins, but a small fraction remains, even after the thickness of NDs is reduced more than $9$ nm.
We also find that the surface spin contribution on $T_{1}$ is suppressed by a factor of $7.5\pm 5.4$ after annealing at 550 ${}^{\circ}$C for 7 hours.
With the same annealing condition, $T_{2}$ is improved by a factor of $1.2\pm 0.2$.
II Materials and Methods
II.1 Nanodiamond
Five different sizes of diamond powders were investigated in the present study.
The samples include micron-sized diamond powders ($10\pm 1$ $\mu$m) (Engis Corporation), and four different sizes of NDs (Engis Corporation and L.M. Van Moppes and Sons SA).
The mean diameters of the ND samples specified by the manufacturers are $550\pm 100$ nm, $250\pm 80$ nm, $100\pm 30$ nm, and $50\pm 20$ nm.
All diamond powders were manufactured by mechanical milling or grinding of type-Ib diamond crystals.
The concentration of nitrogen related impurities in the ND powders is in the order of 10 to 100 parts per million (ppm) carbon atoms.
II.2 Air annealing
The air annealing process was performed using a tube furnace (MTI Corporation) where a ND sample was positioned in a quartz tube located in the cylindrical access of the furnace.
For the preparation of the annealing process, the ND sample was placed in a 5 ml of acetone.
The ND sample in acetone was then mixed by utilizing ultrasound sonication for 10 min at room temperature in order to achieve uniform dispersion.
After the ultrasound sonication, the sample solution was placed in a crucible and kept in a fume hood overnight (without application of heating) in order to evaporate acetone from the crucible.
In the air annealing process, the temperature of the furnace was first stabilized at the annealing temperature (550 ${}^{\circ}$C in the present case), and then the ND sample in the crucible was inserted at the center of the quartz tube.
In order to improve homogeneity of the application of the air annealing over the ND powders, the NDs were mixed by a lab spatula periodically during the annealing (typically mixed for 30 seconds every 10 minutes).
We also limited the initial amount of ND samples to be approximately 30 mg for the homogeneous application of the air annealing.
II.3 Dynamic light scattering
The size of a diamond powder sample was characterized by dynamic light scattering (DLS) (Wyatt Technology).
A diamond powder sample of $\sim 1$ mg was suspended in methanol and sonicated for two hours before the measurement of DLS.
The DLS measurement was performed with a 632 nm incident laser and $163.5^{\circ}$ of detection angle.
The second correlation data was analyzed using the constrained regularization method to obtain particle sizes.
II.4 HF EPR spectroscopy
HF (230 GHz and 115 GHz) EPR experiments were performed using a home-built system at University of Southern California.
The HF EPR spectrometer consists of a high-frequency high-power solid-state source, quasioptics, a corrugated waveguide, a 12.1 Tesla superconducting magnet, and a superheterodyne detection system.
The output power of the source system is 100 mW at 230 GHz and 480 mW at 115 GHz, respectively.
A sample on a metallic end-plate at the end of the corrugated waveguide is placed at the center of the 12.1 Tesla EPR superconducting magnet.
Details of the system have been described elsewhere. Cho, Stepanov, and Takahashi (2014)
In the present study, the diamond powder sample was placed in a Teflon sample holder (5 mm diameter), typically containing the diamond powders of 5 mg. Cho et al. (2015)
For cw EPR experiments, the microwave power and magnetic field modulation strength were adjusted to maximize the intensity of EPR signals without distortion of the signals. Peng et al. (2019)
A typical modulation amplitude was 0.02 mT at a modulation frequency of 20 kHz.
III Results and Discussion
We employed air annealing for the removal of the surface spins in the present study.
In the air annealing the surface removal is caused by etching by oxygen where oxygen molecules oxidize carbon and form gaseous CO and CO${}_{2}$.
We first compared the weight of the ND sample before and after the annealing process.
Figure 1(a) shows the ND normalized weight as a function of the annealing time.
In the experiment, the annealing was done at an annealing temperature of 550 ${}^{\circ}$C.
The result shows linear reduction in ND weight with increased annealing time.
The size of the ND samples was then characterized using DLS.
As shown in Fig. 1(b), the ND size decreased from $d_{peak}=53.4$ nm to 22.4 nm after the annealing for 9 hours.
The observed reduction and narrow distribution of the size indicates a successful and uniform application of the annealing to NDs.
Figure 1(c) shows the ND size as a function of the annealing duration.
We observed a linear relationship between the size reduction and the annealing duration.
A reduction rate of $3.5$ nm/hour was obtained from the linear fit.
Next, we characterized paramagnetic spins existing in NDs using 230 GHz EPR spectroscopy.
Figure 2(a) shows 230 GHz continuous-wave EPR spectra on 50-nm ND samples before and after the air annealing.
The measurements were performed at room temperature.
As shown in Fig. 2(a), all spectra contain a pronounced and broad EPR signal at 8.206 Tesla and a narrow EPR signal at $\sim$8.207 Tesla. From the EPR spectral analysis shown in the inset of Fig. 2(a), we identified that the EPR signal at 8.207 Tesla is from P1 centers while the signal at 8.206 Tesla is from the surface spins (dangling bonds).
The result is consistent with the previous HF EPR study. Peng et al. (2019)
As shown in Fig. 2(a), the intensity of the EPR signals from the surface spins decreases significantly after the annealing.
In general, the EPR intensity is related to the spin population, we therefore analyzed the EPR intensity ratio between the surface spins ($I_{S}$, where S represents surface spins) and P1 ($I_{P1}$) to determine their spin population ratio.
For example, we obtained $I_{S}/I_{P1}$ to be 61 and 5 with no annealing and 9 hour annealing, respectively.
The result from the EPR intensity and DLS analyses was summarized in Fig. 2(b).
Since our previous HF EPR study of the non-annealed NDs showed the core-shell structure with the shell thickness ($t$) of 9 nm, Peng et al. (2019) we first consider the core-shell model to understand the size dependence of the EPR intensity.
In the core-shell model, the EPR intensity ratio,
$$\left(\frac{I_{X}}{I_{P1}}\right)_{coreshell}=\frac{\rho_{X}V_{X}}{\rho_{P1}V_%
{P1}}=\frac{\rho_{X}[4\pi/3\{(d/2)^{3}-(d/2-t)^{3}\}]}{\rho_{P1}[4\pi/3(d/2-t)%
^{3}]},$$
where $\rho_{X}$ ($\rho_{P1}$) is the density of the surface spins (P1 spins) and $V_{X}$ ($V_{P1}$) is the volume of the surface spin (P1 spin) locations.
The calculated $(I_{S}/I_{P1})_{coreshell}$ is shown in Fig. 2(b).
However, we observed a poor agreement with the experimental data in the range of $d<35$.
There may be two possible reasons to explain the result.
First, as reported previously Gaebel et al. (2012); Wolfer et al. (2009); De Theije et al. (2000); Dallek, Kabacoff, and Norr (1991); Xie et al. (2018), the etching rate of the air annealing depends on a crystallographic axis.
It has been shown that the etching rate of the (111) plane is a couple of times faster than the (100) plane. Sun and Alam (1992)
However, this can explain only the dependence of EPR, but not the dependence of DLS.
Another possible reason is the creation of a small amount of surface spins during air annealing.
For instance, it has been reported that dangling bonds were created by air annealing, especially when the surface termination was dominated by C-H bonds. Wenjun et al. (1992)
In the latter scenario, the surface spins in the non-annealed NDs (dangling bonds) are located in the shell, and then air annealing removes the dangling bonds in the shell as well as creates a small amount of dangling bonds on the surface (see Fig. 2(a)).
To take into account the surface spins created by air annealing, we added a contribution from the surface spin model with which,
$$\left(\frac{I_{X}}{I_{P1}}\right)_{surface}=\frac{\rho_{s}[4\pi(d/2)^{2}]}{%
\rho_{P1}[4\pi/3(d/2)^{3})]}\propto\frac{\rho_{s}}{d}.$$
$\rho_{s}$ is the surface spin density, treating as a constant here.
As shown in Fig. 2(b), the sum of the core-shell and surface models agrees with the experimental result, supporting the latter case.
Next, we measured the spin relaxation times ($T_{1}$ and $T_{2}$) of the ND samples.
The experimental results of the 50-nm ND sample is shown in Fig. 3(a).
The measurements of the $T_{1}$ and $T_{2}$ relaxation times of P1 centers were carried out using the inversion recovery and the spin echo sequences, respectively.
The $T_{1}$ and $T_{2}$ measurements of P1 centers were performed at a microwave frequency of 115 GHz and 4.103 Tesla, corresponding to the center peak of the P1 EPR spectrum.
By fitting the change of the spin echo intensity with a single exponential function, we obtained $T_{1}$ to be $0.382\pm 0.080$ ms, and $T_{2}$ is $0.413\pm 0.007$ $\mu$s as shown in the inset of Fig. 3(a) (see Supplementary Material for the description of the $T_{1}$ and $T_{2}$ determination).
Moreover, we measured temperature dependence of $T_{1}$ and $T_{2}$.
Figure 3 (b) and Table 1 summarize the result of the $T_{1}$ measurements as a function of temperature.
We observed that $T_{1}$ times increase drastically by decreasing temperature.
In addition, the temperature dependence is strongly correlated with the size of the diamond powder.
To understand the temperature dependence, we first considered a contribution of the spin-lattice relaxation observed in bulk diamond.
According to the previous studies of $T_{1}$ of bulk diamond, the temperature dependence of $T_{1}$ is well explained by a spin-orbit induced tunneling model, Reynhardt, High, and Van Wyk (1998); Takahashi et al. (2008) in which a spin flip event occurs due to the tunneling between P1’s molecular axis orientations.
Using the spin-orbit induced tunneling model, we write that
$1/T_{1}$ is proportional to $T^{5}$, namely, $1/T_{1}=CT^{5}$, where the $T$-linear term in the spin-orbit induced tunneling model Reynhardt, High, and Van Wyk (1998) is omitted because of its negligible contribution in the present temperature range.
By fitting the experimental data of the 10-$\mu$m diamond to the $T^{5}$ model, we obtained $C=(2.96\pm 0.52)\times 10^{-10}$ $s^{-1}K^{-5}$, which is in a good agreement with previous finding. Takahashi et al. (2008); Reynhardt, High, and Van Wyk (1998)
The result was obtained from a weighted fit analysis in order to take into account the uncertainty in $T_{1}$ values (See Supplementary Material for the details).
Furthermore, in cases of smaller diamond samples (from 550-nm to 50-nm NDs in Fig. 3(b)), we observed a strong deviation from the $T^{5}$ model and found that $1/T_{1}$ at low temperatures highly correlates with the size of NDs.
Recent investigation of shallow NV centers as well as NV centers in a single ND showed that $T_{1}$ in NDs is attributed to surface spins. Tetienne et al. (2013); Ofori-Okai et al. (2012); Kaufmann et al. (2013); Rosskopf et al. (2014)
Since the surface spins were also detected from the same ND samples in our experiment, it is likely that the surface spins also influence $T_{1}$ of P1 centers in NDs.
In order to take into account relaxation processes from both the surface spins and the spin-orbit induced tunneling, we consider the following for $1/T_{1}$,
$$\frac{1}{T_{1}}=CT^{5}+\Gamma_{s},$$
(1)
where $\Gamma_{s}$ is the $1/T_{1}$ contribution from surface spins, originated by fluctuations of the magnetic dipole fields from the surface spins.
$\Gamma_{s}$ is assumed to be independent of temperature in a temperature range of the present experiment.
In this $1/T_{1}$ model, when temperature increases, the first term (the spin-orbit induced tunneling contribution) increases.
Therefore, when a sample has a significant contribution from the surface spin relaxation, $1/T_{1}$ has less pronounced temperature dependence.
We performed a weighted fit analysis on 50-nm, 100-nm, 250-nm and 550-nm ND samples to determine their $\Gamma_{s}$ (See Supplementary Material for the details).
As shown in Fig 3 (b), we found a good agreement between the temperature dependence of $1/T_{1}$ and the model.
For example, we obtained that $\Gamma_{s}$ of the 50-nm ND sample was 2430 $\pm$ 650 $s^{-1}$.
Furthermore, we investigated the temperature- and size-dependence of $T_{2}$ relaxation time of P1 centers.
In contrast to the result of $T_{1}$, $T_{2}$ of P1 centers in the studied NDs does not show noticeable temperature dependence (see Fig. 3(c)).
Table 2 shows the summary of the temperature- and size-dependence of $T_{2}$ as well as the mean $T_{2}$ ($\overline{T_{2}}$) which was obtained from a weighted fit analysis in order to take into account the errors in the $T_{2}$ values (See Supplementary Material for the details of the $\overline{T_{2}}$ analysis).
$\overline{T_{2}}$ for 50-nm ND and 550-nm samples were $0.474\pm 0.060$ $\mu$s and $2.03\pm 0.10$ $\mu$s, respectively.
Therefore, $T_{2}$ of the 50-nm NDs is approximately 4.3 times shorter than that of 550-nm NDs.
The result indicates the effect of the surface spins on $T_{2}$.
On the other hand, $T_{2}$ of 550-nm and 10-$\mu$m are similar ($\sim$2 $\mu$s).
This is probably because couplings to neighboring P1 centers dominates their $T_{2}$ processes.
Finally, we study the spin relaxation times ($T_{1}$ and $T_{2}$) of the annealed NDs.
As shown in Fig. 4(a), $T_{1}$ times in the annealed diamond became longer after the annealing in the measured temperature range.
In addition, as shown in Fig. 4(a), the $T_{1}$ times of the annealed NDs are still shorter than that of bulk diamond, implying the existence of remaining surface spins.
To extract the contribution of the surface spins, we employed Eq. (1) to determine $\Gamma_{s}$.
From the analysis, we indeed found that $\Gamma_{s}$ in the annealed NDs are smaller than that of the non-annealed samples.
The obtained $\Gamma_{s}$ are 531 $\pm$ 217 $s^{-1}$ and 325 $\pm$ 217 $s^{-1}$ for the NDs annealed at $550^{\circ}$C for 5 hours and 7 hours, respectively, which are 4.6 $\pm$ 2.2 and 7.5 $\pm$ 5.4 times smaller than that of the non-annealed 50-nm NDs as shown in Fig. 4(b) (see Supplementary Material for the calculation of the $\Gamma_{s}$ improvement factor).
We next discuss a model of the surface spin-induced $T_{1}$ ($\Gamma_{s}$).
As reported previously, Steinert et al. (2013); Tetienne et al. (2013); Rosskopf et al. (2014) by considering fluctuating magnetic fields ($B_{dip}$) from surface spins, $\Gamma_{s}$ is proportional to the variance ($\langle B_{dip}^{2}\rangle$) and the spin density ($\rho_{s}$).
By assuming that surface spins cover the whole surface uniformly, $B_{dip}^{2}(\overrightarrow{r}_{P1})\propto\int_{S}\rho_{s}b_{dip}^{2}(%
\overrightarrow{r}-\overrightarrow{r}_{P1})dS$, where the radius vectors ($\overrightarrow{r}$ and $\overrightarrow{r}_{P1}$) define the locations of the surface and P1 spins relative to the center of the ND, respectively.
$b_{dip}(\overrightarrow{r}$) is the magnetic dipole field from the surface spins.
By taking into account the quantization axis of P1 and the surface spins along the external magnetic field and considering a spherical shape of NDs and a spatially uniform $\rho_{s}$, $b_{dip}^{2}(\overrightarrow{r})$ is proportional to $1/d^{6}$ and the surface integral is proportional to $d^{2}$, where $d$ is a diameter of a ND, the magnetic field fluctuations ($B^{2}_{dip}(\overrightarrow{r})$) is therefore proportional to $\rho_{s}/d^{4}$ and $\Gamma_{s}$ is also proportional to $\rho_{s}/d^{4}$.
The $\Gamma_{s}$ values obtained from the temperature dependence $T_{1}$ in Fig. 3(b) were plotted as a function of the ND size in Fig. 4(b).
We found a good agreement between the obtained $\Gamma_{s}$ value and the $1/d^{4}$ size dependence.
Thus, the result supports the $T_{1}$ relaxation mechanism in NDs due to the surface spins.
Furthermore, as shown in Fig. 4(b), $\Gamma_{s}$ of the annealed NDs are very different from the $1/d^{4}$ line of the non-annealed NDs, indicating significant reduction of the surface spin density.
Using the same model ($\Gamma_{s}\propto\rho_{s}/d^{4}$), we estimated that $\rho_{s}$ for the annealed NDs is $\sim$100 times smaller than that of the non-annealed NDs (Fig. 4(b)).
In addition the $T_{2}$ of the annealed diamond was studied.
As shown in Fig. 4(c), similarly to the non-annealed NDs, $T_{2}$ of the annealed diamond showed no temperature dependence.
The mean $T_{2}$ times were $0.675\pm 0.274$ $\mu$s and $0.589\pm 0.048$ $\mu$s after the 5 and 7 hour annealing, respectively, showing that the extension of $T_{2}$ by a factor of $1.4\pm 0.6$ and $1.2\pm 0.2$, respectively (see Supplementary Material for the calculation of the $T_{2}$ improvement factor).
This improvement is due to the reduction of the surface spins.
The observed $T_{2}$ improvement is comparable with the previously reported result. Tsukahara et al. (2019)
By considering the $T_{2}$ results of the non-annealed NDs, we speculate that the $T_{2}$ relaxation in the annealed NDs is caused by couplings to residual surface spins and P1 centers.
IV Summary
In summary, we investigated the relationship between the surface spins and the spin relaxation times ($T_{1}$ and $T_{2}$) of P1 centers in NDs.
We reduced the amount of the surface spins using air annealing.
The amount of the surface spins was characterized by HF EPR analysis.
The pulsed HF EPR experiment extracted the contribution of the surface spins on the $T_{1}$ relaxation successfully.
We found clear correlation between the amount of the surface spins and $T_{1}$.
In addition, the present study showed the improvement of $T_{1}$ and $T_{2}$ by removing the surface spins.
The finding of the present investigation sets the basis to suppress the spin relaxation process due to the surface spins in NDs which is critical for NV-based sensing applications.
The present method is also potentially applicable to improve spin and optical properties of other nanomaterials.
V Supplementary Material
See the supplementary material for the $T_{1}$ and $T_{2}$ determination method, the EPR spectral analysis and the analyses of the temperature- and size-dependent $T_{1}$ and $T_{2}$.
Acknowledgements.
We thank Benjamin Fortman for useful discussion of the EPR data analyses.
This work was supported by the National Science Foundation (DMR-1508661 and CHE-1611134), the USC Anton B. Burg Foundation and the Searle scholars program. This material is also based upon work supported by the Chemical Measurement and Imaging program in the National Science Foundation Division of Chemistry under Grant No. CHE-2004252 (with partial co-funding from the Quantum Information Science program in the Division of Physics) (ST).
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author
upon reasonable request.
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Yang- Mills on Quantum Heisenberg Manifolds
Partha Sarathi Chakraborty
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai
600113
[email protected]
and
Satyajit Guin
The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai
600113
[email protected]
(Date:: January 13, 2021)
Abstract.
In the noncommutative geometry program of Connes there are two variations of the concept of Yang-Mills action functional. We show that for the quantum Heisenberg manifolds they agree.
Key words and phrases:Yang-Mills, Quantum Heisenberg Manifolds, Connection, Curvature
2000 Mathematics Subject Classification: Primary 46L87, 58B34
1. Introduction
Quantum Heisenberg manifolds (QHM) were introduced by Rieffel in [6] as strict deformation quantization of Heisenberg manifolds. He introduced a parametric family of deformations and for generic parameter values these are simple $C^{*}$-algebras with an ergodic action of the Heisenberg group of $3\times 3$ upper triangular matrices with ones on the diagonal. They admit a unique invariant trace. Connes has showed in [2] that whenever one has a $C^{*}$-dynamical system with dynamics governed by a Lie group and an invariant trace one can extend the basic notions of geometry. Leter Connes and Rieffel ([3]) has introduced the concepts of Yang-Mills action functional and quantum Heisenberg manifold presents an ideal case for such considerations. Recently Kang [5] has studied Yang-Mills for the QHM. However the popular formulation of noncommutative geometry today is through spectral triples. In this approach as well Connes ([4]) defined the concept of the Yang-Mills action functional. Now starting with a $C^{*}$-dynamical system with an invariant trace there is a general prescription that produces a candidate for a spectral triple, but there is no general theorem and in each case one has to verify the self-adjointness and the compact resolvent of the Dirac operator. It was shown in [1] that in the case of QHM the general principle gives rise to an honest spectral triple. A natural question in this context is whether even in this case these two notions of YM coincide and that is the content of this paper. We show that the notions agree in the context of QHM. This parallels proposition 13 in the last chapter of [4] where similar results were obtained for noncommutative two torus.
2. The Quantum Heisenberg Algebra
Notation: for $x\in\mathbb{R}$, $e(x)$ stands for $e^{2\pi ix}$ where $i=\sqrt{-1}.$
Definition 2.1.
For any positive integer $c$, let $S^{c}$ denote the space of smooth functions
$\Phi:\mathbb{R}\times\mathbb{T}\times\mathbb{Z}\rightarrow C$ such that
•
$\Phi(x+k,y,p)=e(ckpy)\Phi(x,y,p)$ for all $k\in\mathbb{Z}$,
•
for every polynomial $P$ on $\mathbb{Z}$ and every partial differential operator $\widetilde{X}=\frac{\partial^{m+n}}{\partial x^{m}\partial y^{n}}$ on $\mathbb{R}\times\mathbb{T}$ the function $P(p)(\widetilde{X}\Phi)(x,y,p)$ is bounded on $K\times\mathbb{Z}$ for any compact subset $K$ of $\mathbb{R}\times\mathbb{T}$.
For each $\hbar,\mu,\nu\in\mathbb{R},\mu^{2}+\nu^{2}\neq 0$, let ${\mathcal{A}}^{\infty}_{\hbar}$ denote $S^{c}$ with product and involution defined by
(2.1)
$$\displaystyle(\Phi\star\Psi)(x,y,p)=\sum_{q}\Phi(x-\hbar(q-p)\mu,y-\hbar(q-p)%
\nu,q)\Psi(x-\hbar q\mu,y-\hbar q\nu,p-q)$$
(2.2)
$$\displaystyle\Phi^{*}(x,y,p)=\bar{\Phi}(x,y,-p).$$
Then, $\pi:{\mathcal{A}}^{\infty}_{\hbar}\rightarrow\mathcal{B}(L^{2}(\mathbb{R}%
\times\mathbb{T}\times\mathbb{Z}))$ given by
(2.3)
$$\displaystyle(\pi(\Phi)\xi)(x,y,p)=\sum_{q}\Phi(x-\hbar(q-2p)\mu,y-\hbar(q-2p)%
\nu,q)\xi(x,y,p-q)$$
gives a faithful representation of the involutive algebra ${\mathcal{A}}_{\hbar{}}^{\infty}$.
${\mathcal{A}}^{c,\hbar}_{\mu,\nu}=$ norm closure of $\pi({\mathcal{A}}_{\hbar{}}^{\infty})$ is called the Quantum Heisenberg Manifold.
We will identify ${\mathcal{A}}_{\hbar{}}^{\infty}$ with $\pi({\mathcal{A}}_{\hbar{}}^{\infty})$ without any mention.
Since we are going to work with fixed parameters $c,\mu,\nu,\hbar$ we will drop them altogether and denote ${\mathcal{A}}^{c,\hbar}_{\mu,\nu}$ simply by $\mathcal{A}_{\hbar}$ here the subscript remains merely as a reminiscent of Heisenberg only to distinguish it from a general algebra.
Action of the heisenberg group: Let $c$ be a positive integer. Let us consider the group structure on $G=\mathbb{R}^{3}=\{(r,s,t):r,s,t\in\mathbb{R}\}$ given by the multiplication
(2.4)
$$\displaystyle(r,s,t)(r^{\prime},s^{\prime},t^{\prime})=(r+r^{\prime},s+s^{%
\prime},t+t^{\prime}+csr^{\prime}).$$
Later we will give an explicit isomorphism between $G$ and $H_{3}$, the Heisenberg group of $3\times 3$ upper triangular matrices with real entries and ones on the diagonal. Through this identification we can identify $G$ with the Heisenberg group.
For $\Phi\in S^{c},(r,s,t)\in\mathbb{R}^{3}\equiv G$,
(2.5)
$$\displaystyle(L_{(r,s,t)}\phi)(x,y,p)=e(p(t+cs(x-r)))\phi(x-r,y-s,p)$$
extends to an ergodic action of the Heisenberg group on ${\mathcal{A}}^{c,\hbar}_{\mu,\nu}$.
The Trace:
The linear functional $\tau:{\mathcal{A}}_{\hbar{}}^{\infty}\rightarrow\mathbb{C}$, given by $\tau(\phi)=\int^{1}_{0}\int_{\mathbb{T}}\phi(x,y,0)dxdy$
is invariant under the Heisenberg group action. So, the group action can be lifted to $L^{2}({\mathcal{A}}_{\hbar{}}^{\infty})$.
We will denote the action at the Hilbert space level by the same symbol.
3. Yang-Mills in the dynamical system approach
In ([3]) Connes and Rieffel introduced Yang-Mills functional in the setting of $C^{*}$-dynamical systems. We will recall their definition in the context of QHM. Here the dynamics is governed by the Lie group $G$. We can identify $G$ with $H_{3}$ through the isomorphism that identifies $(r,s,t)\in G$ with the matrix $\begin{pmatrix}1&cs&t\cr 0&1&r\cr 0&0&1\end{pmatrix}$. Let $\mathfrak{g}$ be the Lie-algebra of $G$. We can identify $\mathfrak{g}$ with the Lie-algebra of $H_{3}$, which is given by $3\times 3$ upper triangular matrices with real entries with zeros on the diagonal. Fix a real number $\alpha$ greater than one. This number will remain fixed throughout and we will comment about it later. In this approach one has to fix an inner product structure on the Lie algebra of the underlying Lie group and in our case we do so by declaring the following basis,
(3.1)
$$\displaystyle X_{1}=\left(\begin{matrix}0&0&0\cr 0&0&1\cr 0&0&0\cr\end{matrix}%
\right),X_{2}=\left(\begin{matrix}0&c&0\cr 0&0&0\cr 0&0&0\cr\end{matrix}\right%
),X_{3}=\left(\begin{matrix}0&0&c\alpha\cr 0&0&0\cr 0&0&0\cr\end{matrix}\right)$$
as orthonormal. Their Lie bracket is given by,
(3.2)
$$\displaystyle[X_{1},X_{3}]=[X_{2},X_{3}]=0,[X_{1},X_{2}]=-\frac{1}{\alpha}X_{3}.$$
The exponential map from $\mathfrak{g}$ to $G$ acts on these elements as follows
$$\displaystyle exp(rX_{1})=(r,0,0),exp(sX_{2})=(0,s,0),\mbox{ and }exp(tX_{3})=%
(0,0,c\alpha t).$$
For $X\in\mathfrak{g}$, let $d_{X}$ be the
derivation of ${\mathcal{A}}_{\hbar{}}^{\infty}$ given by $d_{X}(a)=\frac{d}{dt}\mid_{t=0}L_{exp(tX)}(a)$. Let us denote the $d_{X_{j}}$’s, for $j=1,2,3$ by $d_{j}$. Then they are given by
(3.3)
$$\displaystyle d_{1}(f)$$
$$\displaystyle=$$
$$\displaystyle-\frac{\partial f}{\partial x},$$
(3.4)
$$\displaystyle d_{2}(f)$$
$$\displaystyle=$$
$$\displaystyle 2\pi icpxf(x,y,p)-\frac{\partial f}{\partial y},$$
(3.5)
$$\displaystyle d_{3}(f)$$
$$\displaystyle=$$
$$\displaystyle 2\pi ipc\alpha f(x,y,p).$$
We now recall the Hermitian structure on finitely generated projective modules. This is needed to define the Yang-Mills action functional. Let $\mathcal{H}$ be a Hilbert space and $\mathcal{A}$ be a unital involutive subalgebra of $\mathcal{B}(\mathcal{H})$, the algebra of bounded operators on $\mathcal{H}$, closed under holomorphic function calculus.
Let $\mathcal{E}$ be a finitely generated projective $\mathcal{A}$ module.
Define $\mathcal{E}^{*}$ as the space of $\mathcal{A}$ linear mappings from $\mathcal{E}$ to $\mathcal{A}$. Clearly $\mathcal{E}^{*}$ is a right $\mathcal{A}$ module.
Definition 3.1.
A Hermitian structure on $\mathcal{E}$ is an $\mathcal{A}$-valued positive-definite inner product
$\langle\,\,,\,\rangle_{\mathcal{A}}$ such that,
(a)
$\langle\xi,\xi^{\prime}\rangle_{\mathcal{A}}^{*}=\langle\xi^{\prime},\xi%
\rangle_{\mathcal{A}}\,,\,\,\,\forall\,\xi,\xi^{\prime}\in\mathcal{E}$.
(b)
$\langle\xi,\xi^{\prime}.a\rangle_{\mathcal{A}}=(\langle\xi,\xi^{\prime}\rangle%
_{\mathcal{A}}).a\,,\,\,\,\forall\,\xi,\xi^{\prime}\in\mathcal{E},\,\,\forall%
\,a\in\mathcal{A}$.
(c)
The map $\xi\longmapsto\Phi_{\xi}$ from $\mathcal{E}$ to $\mathcal{E}^{*}\,$, given by $\Phi_{\xi}(\eta)=\langle\xi,\eta\rangle_{\mathcal{A}}\,,\,\forall\eta\in%
\mathcal{E}\,$, gives an $\mathcal{A}$-module isomorphism between $\mathcal{E}$ and $\mathcal{E}^{*}$. This property will be referred as the self-duality of $\mathcal{E}$.
Let $\mathcal{E}$ be a finitely generated projective ${\mathcal{A}}_{\hbar{}}^{\infty}$ module with a hermitian structure. A connection is a map
(3.6)
$$\displaystyle\nabla$$
$$\displaystyle:$$
$$\displaystyle\mathcal{E}\rightarrow\mathcal{E}\otimes{\mathfrak{g}}^{*},\mbox{%
such that }$$
(3.7)
$$\displaystyle\nabla_{X}(\xi.a)$$
$$\displaystyle=$$
$$\displaystyle\nabla_{X}(\xi).a+\xi.d_{X}(a)\forall\xi\in\mathcal{E},\,\forall a%
\in{{\mathcal{A}}_{\hbar{}}^{\infty}}.$$
We shall say that $\nabla$ is compatible with respect to the Hermitian structure on $\mathcal{E}$
iff :
(3.8)
$$\displaystyle\langle\nabla_{X}\,\xi\,,\xi^{\prime}\rangle_{\mathcal{A}}\,+\,%
\langle\xi\,,\nabla_{X}\,\xi^{\prime}\rangle_{\mathcal{A}}=d_{X}(\langle\,\xi,%
\xi^{\prime}\,\rangle_{\mathcal{A}})\,,\quad\forall\,\xi\,,\xi^{\prime}\in%
\mathcal{E},\,\,\forall\,X\in\mathfrak{g}\,.$$
We will denote the set of compatible connections by $C(\mathcal{E})$.
The curvature $\Theta_{\nabla}$ of a connection $\nabla$ is the alternating bilinear $End(\mathcal{E})$-valued form on $\mathfrak{g}$ defined by,
$$\displaystyle\Theta_{\nabla}(X\wedge Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}%
,\forall X,Y\in\mathfrak{g}.$$
Proposition 3.2.
Let $\mathcal{E}$ be a finitely generated projective ${\mathcal{A}}_{\hbar{}}^{\infty}$ module. Then the space $C(\mathcal{E})$
of compatible connections is given by triples of linear maps $\nabla_{j}:\mathcal{E}\rightarrow\mathcal{E},j=1,2,3$ such that
(3.9)
$$\displaystyle\nabla_{j}(\xi.a)$$
$$\displaystyle=\nabla_{j}(\xi).a+\xi.d_{j}(a),$$
$$\displaystyle j=1,2,3$$
(3.10)
$$\displaystyle d_{j}(\langle\,\xi,\xi^{\prime}\,\rangle_{\mathcal{A}})$$
$$\displaystyle=\langle\nabla_{j}\,\xi\,,\xi^{\prime}\rangle_{\mathcal{A}}\,+\,%
\langle\xi\,,\nabla_{j}\,\xi^{\prime}\rangle_{\mathcal{A}},$$
$$\displaystyle\forall\,\xi\,,\xi^{\prime}\in\mathcal{E},$$
$$\displaystyle j=1,2,3\,.$$
Proof.
Given a compatible connection $\nabla$ let $\nabla_{j}=\nabla_{X_{j}}$ for $j=1,2,3.$. The condition (3.9,
3.10) holds because $\nabla$ satisfies conditions (3.6,3.8). Conversely if
(3.9, 3.10) holds and we define $\nabla$ by specifying its components on the basis 3.1 such that $\nabla_{X_{j}}=\nabla_{j}$, then clearly the conditions of a compatible connection (3.6,3.8) are satisfied.
$\Box$
For QHM the curvature is given by
$$\displaystyle\Theta_{\nabla}(X_{1}\wedge X_{3})$$
$$\displaystyle=$$
$$\displaystyle[\nabla_{X_{1}},\nabla_{X_{3}}],$$
$$\displaystyle\Theta_{\nabla}(X_{2}\wedge X_{3})$$
$$\displaystyle=$$
$$\displaystyle[\nabla_{X_{2}},\nabla_{X_{3}}],$$
$$\displaystyle\Theta_{\nabla}(X_{1}\wedge X_{2})$$
$$\displaystyle=$$
$$\displaystyle[\nabla_{X_{1}},\nabla_{X_{2}}]+\frac{1}{\alpha}\nabla_{X_{3}}.$$
Here the third equality uses the relation $[X_{1},X_{2}]=-\frac{1}{\alpha}X_{3}$ from (3.2).
Definition 3.3.
Let $\mathcal{E}$ be a finitely generated projective ${\mathcal{A}}_{\hbar{}}^{\infty}$ module with a hermitian structure. Then the Yang-Mills action functional for a compatible connection $\nabla\in C(\mathcal{E})$ is given by
(3.11)
$$\displaystyle YM(\nabla)=-\widetilde{\tau}({([\nabla_{X_{1}},\nabla_{X_{3}}])}%
^{2}+{([\nabla_{X_{2}},\nabla_{X_{3}}])}^{2}+([\nabla_{X_{1}},\nabla_{X_{2}}]+%
\frac{1}{\alpha}\nabla_{X_{3}})^{2})).$$
4. Yang-Mills for spectral triples
In ([4]), Connes gave a second approach to Yang-Mills for spectral triples. In this approach one begins with a spectral triple. Recall that a spectral triple is given by a triple $(\mathcal{A},\mathcal{H},D)$ where
(i)
$\mathcal{H}$ is a separable Hilbert space,
(ii)
$\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ is a unital involutive sub-algebra closed under holomorphic function calculus,
(iii)
$D$ is a self-adjoint operator with compact resolvent such that $[D,\mathcal{A}]\subseteq\mathcal{B}(\mathcal{H}).$
A spectral triple is $d+$-summable if the Dixmier trace of ${|D|}^{-d}$ is finite. Starting with a $d+$-summable spectral triple Connes defines a complex as follows.
Definition 4.1.
Let $(\mathcal{A},\mathcal{H},D)$ be a $d+$-summable spectral triple. Then the space of universal $k$-forms is given by
$\Omega^{k}(\mathcal{A})=\{\sum_{i=1}^{N}a_{0}^{i}\delta a_{1}^{i}\ldots\delta a%
_{k}^{i}|n\in\mathbb{N},a_{j}^{i}\in\mathcal{A}\}$. The direct sum of all these spaces $\Omega^{\bullet}(\mathcal{A})=\oplus_{0}^{\infty}\Omega^{k}(\mathcal{A})$
is the unital graded algebra of universal forms. Here $\delta$ is an abstract linear operator with $\delta^{2}=0,\delta(ab)=\delta(a)b+a\delta(b)$. $\Omega^{\bullet}(\mathcal{A})$ becomes a *algebra under the involution ${(\delta a)}^{*}=-\delta(a^{*})\forall a\in\mathcal{A}$. Let $\pi:\Omega^{\bullet}(\mathcal{A})\rightarrow\mathcal{B}(\mathcal{H})$ be the $\star$-representation given by $\pi(a)=a,\pi(\delta a)=[D,a].$
Let $J_{k}=ker\pi|_{\Omega^{k}(\mathcal{A})}$ The unital graded differential $\star$-algebra of differential forms
$\Omega^{\bullet}_{D}(\mathcal{A})$ is defined by
$$\Omega_{D}^{\bullet}(\mathcal{A})=\oplus_{0}^{\infty}\Omega_{D}^{k}(\mathcal{A%
}),\Omega_{D}^{k}(\mathcal{A})=\Omega^{k}(\mathcal{A})/(J_{k}+\delta J_{k-1})%
\cong\pi(\Omega^{k}(\mathcal{A})/\pi(\delta J_{k-1}).$$
The abstract differential $\delta$ induces a differential $\tilde{d}$ on the complex $\Omega^{\bullet}_{D}(\mathcal{A})$ so that we get a chain complex $(\Omega^{\bullet}_{D}(\mathcal{A}),\tilde{d})$ and a chain map $\pi_{D}:\Omega^{\bullet}(\mathcal{A})\rightarrow\Omega^{\bullet}_{D}(\mathcal{%
A})$ such that the following diagram
$\pi_{D}$$\Omega^{\bullet}(\mathcal{A})$$\Omega^{\bullet}_{D}(\mathcal{A})$$\pi_{D}$$\Omega^{\bullet+1}(\mathcal{A})$$\Omega^{\bullet+1}_{D}(\mathcal{A})$$\tilde{d}$$\delta$
commutes.
Definition 4.2.
Let $\mathcal{E}$ be a Hermitian, finitely generated projective module over $\mathcal{A}$. A compatible connection on $\,\mathcal{E}\,$ is a linear mapping
$\,\widetilde{\nabla}:\mathcal{E}\longrightarrow\mathcal{E}\,\otimes_{\mathcal{%
A}}\Omega_{D}^{1}\,$ such that,
(a)
$\widetilde{\nabla}(\xi a)=(\widetilde{\nabla}\xi)a+\xi\otimes\tilde{d}a,\,\,\,%
\,\,\forall\xi\in\mathcal{E},a\in\mathcal{A}$;
(b)
$(\,\xi,\widetilde{\nabla}\eta\,)-(\,\widetilde{\nabla}\xi,\eta\,)=\tilde{d}%
\langle\,\xi,\eta\,\rangle_{\mathcal{A}}\,\,\,\,\,\,\,\forall\xi,\eta\in%
\mathcal{E}$.
The meaning of the last equality in $\Omega_{D}^{1}$ is,
if $\widetilde{\nabla}(\xi)=\sum\xi_{j}\otimes\omega_{j}$, with $\xi_{j}\in\mathcal{E}\,,\,\omega_{j}\in\Omega_{D}^{1}(\mathcal{A})$,
then $(\widetilde{\nabla}\xi,\eta)=\sum\omega_{j}^{*}\langle\xi_{j},\eta\rangle_{%
\mathcal{A}}$.
Also, any two compatible connections can only differ by an element of
Hom${}_{\mathcal{A}}(\mathcal{E}\,,\,\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{1%
}(\mathcal{A}))$. That is, the space of all
compatible connections on $\,\mathcal{E}$, which we denote by $\widetilde{C}(\mathcal{E})$, is an affine space with associated vector
space $Hom_{\mathcal{A}}(\mathcal{E}\,,\,\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{%
1}(\mathcal{A}))$.
To define the curvature $\varTheta$ of a connection $\widetilde{\nabla}$, one first extends $\widetilde{\nabla}$ to a unique linear mapping $\widetilde{\nabla}$ from
$\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{1}$ to $\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{2}$ such that,
(4.1)
$$\displaystyle\widetilde{\nabla}(\xi\otimes\omega)=(\widetilde{\nabla}\xi)%
\omega+\xi\otimes\tilde{d}\omega,\,\,\,\forall\,\,\xi\in\mathcal{E},\,\,\omega%
\in\Omega_{D}^{1}.$$
It can be easily checked that $\widetilde{\nabla}$, defined above, satisfies the Leibniz rule.
It follows that $\varTheta=\widetilde{\nabla}\circ\widetilde{\nabla}$ is an element of $Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{2})$. Recall that $\Omega_{D}^{2}\cong\pi(\Omega^{2})/\pi(dJ_{1})$. Let $\mathcal{H}_{2}$ be the Hilbert space completion
of $\pi(\Omega^{2})$ with respect to the inner-product
(4.2)
$$\displaystyle\langle T_{1},T_{2}\rangle=Tr_{\omega}(T_{1}^{*}T_{2}|D|^{-d}),\,%
\forall\,T_{1},T_{2}\in\pi(\Omega^{2}).$$
Let $\widetilde{\mathcal{H}}_{2}$ be the Hilbert space completion of $\pi(dJ_{1})$ with the above inner-product. Clearly $\widetilde{\mathcal{H}}_{2}\subseteq\mathcal{H}_{2}$. Let $P$ be the orthogonal projection of $\mathcal{H}_{2}$ onto the orthogonal complement of $\widetilde{\mathcal{H}}_{2}$. Now define $\langle\,[T_{1}],[T_{2}]\,\rangle_{\Omega_{D}^{2}}=\langle PT_{1},PT_{2}%
\rangle,\,$ for all $\,[T_{i}]\in\Omega_{D}^{2}$. This gives a well-defined inner-product on $\Omega_{D}^{2}$.
Now the inner-product on $Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{2})$ is described as follows. Suppose $\mathcal{E}=p{\mathcal{A}}^{q}$, where $p\in M_{q}(\mathcal{A})$ is a projection.
Then we have the embedding
$$Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{2})=%
Hom_{\mathcal{A}}(pA^{q},pA^{q}\otimes_{\mathcal{A}}\Omega_{D}^{2})\cong Hom_{%
\mathcal{A}}(pA^{q},p(\Omega_{D}^{2})^{q})\subseteq Hom_{\mathcal{A}}(\mathcal%
{A}^{q},(\Omega_{D}^{2})^{q}).$$
The inner product between $\phi,\psi\in Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E}\otimes_{\mathcal{A}}%
\Omega_{D}^{2})$ is given by
$$\langle\langle\phi,\psi\rangle\rangle=\displaystyle\sum_{j,k}\langle{(\phi(e_{%
j}))}_{k},{(\psi(e_{j}))}_{k}\rangle_{\Omega_{D}^{2}}$$
where $\{e_{1},\ldots,e_{q}\}$ is the standard basis of $\mathcal{A}^{q}$ and ${(\phi(e_{j}))}_{k}$, (respectively ${(\psi(e_{j}))}_{k}$ denote the $k$-th component of $\phi(e_{j})$ $(\psi(e_{j}))$.
Remark 4.3.
Let us assume that $\Omega_{D}^{2}$ is free of rank $n$ and the inner product described above between two $n$-tuples $\underline{a},\underline{b}\in\Omega_{D}^{2}={\mathcal{A}}^{n}$ is given by $<\underline{a},\underline{b}>=\sum_{j}{\rm Tr}_{\omega}(a_{j}^{*}b_{j}){|D|}^{%
-d}$. If we use the embedding
$$Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E}\otimes_{\mathcal{A}}\Omega_{D}^{2})%
\cong\oplus_{k=1}^{n}Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E})\subseteq\oplus%
_{k=1}^{n}M_{q}(\mathcal{A})\cong\oplus_{k=1}^{n}\mathcal{A}\otimes M_{q}(%
\mathbb{C}).$$
Then $\phi,\psi\in Hom_{\mathcal{A}}(\mathcal{E},\mathcal{E}\otimes_{\mathcal{A}}%
\Omega_{D}^{2})$ can be identified with two $n$-tuples $\phi=(\phi_{1},\cdots,\phi_{n}),$ and $\psi=(\psi_{1},\cdots,\psi_{n})$, with each $\phi_{j},\psi_{j}\in\mathcal{A}\otimes M_{q}(\mathbb{C})$. Let $\tau^{\prime}$ be the trace on $\mathcal{A}$ given by $a\mapsto{\rm Tr}_{\omega}a{|D|}^{-d}$ then
$$\langle\langle\phi,\psi\rangle\rangle=\sum_{j=1}^{n}\tau^{\prime}\otimes{\rm{%
Trace}}\phi_{j}^{*}\psi_{j}.$$
Definition 4.4.
The functional on $\widetilde{C}(\mathcal{E})$ given by $\widetilde{\textit{YM}}\,(\widetilde{\nabla})=\langle\langle\,\varTheta,%
\varTheta\,\rangle\rangle$ is called the Yang-Mills functional.
5. Equivalence of the two approaches
In this section we will show that for the quantum Heisenberg manifolds there is a correspondence between the set of compatible connections so that the corresponding Yang-Mills functionals agree. To that end one must construct a spectral triple on this algebra. There is a general recipe that begins with $(A,G,\alpha,\tau)$ a $C^{*}$-dynamical system with an invariant trace. Of course one also requires that the dynamics is governed by a Lie group. Let us assume that the Lie group has dimension $n$. Then by fixing a basis $X_{1},\cdots,X_{n}$ of the Lie-algebra of the Lie group one produces a densely defined operator $D$ on the Hilbert space $\mathcal{H}=L^{2}(A,\tau)\otimes{\mathbb{C}}^{N},N=2^{\lfloor n/2\rfloor}$. There is a natural representation of the algebra on $\mathcal{H}$ and $D$ produces bounded commutators with the image of $\mathcal{A}$, the smooth algebra of the system. However, in general one does not know whether $D$ admits a self-adjoint extension with compact resolvent. It was shown in ([1]) that for QHM indeed $D$ admits a self-adjoint extension with compact resolvent provided one chooses the Lie algebra basis considered in (3.1). For our present purpose it is enough to recall the operators $[D,\phi]$ for $\phi\in S^{c}$. Note that here the dimension of the associated Lie group is three.
Let $\sigma_{1},\sigma_{2},\sigma_{3}$ be $2\times 2$ self-adjoint trace-less matrices given by
$$\displaystyle\sigma_{1}=\left(\begin{matrix}1&0\cr 0&-1\cr\end{matrix}\right),%
\,\,\sigma_{2}=\left(\begin{matrix}0&-1\cr-1&0\cr\end{matrix}\right),\,\,%
\sigma_{3}=\left(\begin{matrix}0&i\cr-i&0\cr\end{matrix}\right).$$
Then,
$$\displaystyle\sigma_{1}\sigma_{2}=i\sigma_{3},\,\,\sigma_{2}\sigma_{3}=i\sigma%
_{1},\,\,\sigma_{3}\sigma_{1}=i\sigma_{2}.$$
Let $\phi\in S^{c}$, then
(5.1)
$$\displaystyle[D,\phi]=\sum\delta_{j}(\phi)\otimes\sigma_{j}\mbox{ where }%
\delta_{j}(\phi)=id_{j}(\phi)$$
and the derivations $d_{j}$ are given by (3.3,3.4,3.5). The $\delta_{j}$’s satisfy the following commutation relations
(5.2)
$$\displaystyle[\delta_{1},\delta_{3}]=[\delta_{2},\delta_{3}]=0,[\delta_{1},%
\delta_{2}]=-\frac{i}{\alpha}\delta_{3}.$$
These forms were computed in [1]. In the following proposition we recall the description of the space of forms as ${\mathcal{A}}_{\hbar{}}^{\infty}-{\mathcal{A}}_{\hbar{}}^{\infty}$-bimodules.
Proposition 5.1.
(i)
The space of one forms as an ${\mathcal{A}}_{\hbar{}}^{\infty}-{\mathcal{A}}_{\hbar{}}^{\infty}$-bimodule is given by
$$\displaystyle\Omega^{1}_{D}({\mathcal{A}}_{\hbar{}}^{\infty})$$
$$\displaystyle=$$
$$\displaystyle\{\sum a_{j}\otimes\sigma_{j}|a_{j}\in{\mathcal{A}}_{\hbar{}}^{%
\infty},\sigma_{j}^{\prime}s\mbox{ as above }\}\subseteq{\mathcal{A}}_{\hbar{}%
}^{\infty}\otimes M_{2}(\mathbb{C})\subseteq\mathcal{B}(\mathcal{H})$$
$$\displaystyle\cong$$
$$\displaystyle{\mathcal{A}}_{\hbar{}}^{\infty}\oplus{\mathcal{A}}_{\hbar{}}^{%
\infty}\oplus{\mathcal{A}}_{\hbar{}}^{\infty}.$$
(ii)
$\pi(\Omega^{k}({\mathcal{A}}_{\hbar{}}^{\infty}))={\mathcal{A}}_{\hbar{}}^{%
\infty}\otimes M_{2}(\mathbb{C})={\mathcal{A}}_{\hbar{}}^{\infty}\oplus{%
\mathcal{A}}_{\hbar{}}^{\infty}\oplus{\mathcal{A}}_{\hbar{}}^{\infty}\oplus{%
\mathcal{A}}_{\hbar{}}^{\infty}.$
(iii)
$\pi(\delta J_{1})={\mathcal{A}}_{\hbar{}}^{\infty}\otimes I_{2}\subseteq{%
\mathcal{A}}_{\hbar{}}^{\infty}\otimes M_{2}(\mathbb{C})\subseteq\mathcal{B}(%
\mathcal{H})$.
(iv)
The space of two forms as an ${\mathcal{A}}_{\hbar{}}^{\infty}-{\mathcal{A}}_{\hbar{}}^{\infty}$-bimodule is given by
$$\displaystyle\Omega^{2}_{D}({\mathcal{A}}_{\hbar{}}^{\infty})$$
$$\displaystyle=$$
$$\displaystyle\{\sum a_{j}\otimes\sigma_{j}|a_{j}\in{\mathcal{A}}_{\hbar{}}^{%
\infty},\sigma_{j}^{\prime}s\mbox{ as above }\}\subseteq{\mathcal{A}}_{\hbar{}%
}^{\infty}\otimes M_{2}(\mathbb{C})\subseteq\mathcal{B}(\mathcal{H})$$
$$\displaystyle\cong$$
$$\displaystyle{\mathcal{A}}_{\hbar{}}^{\infty}\oplus{\mathcal{A}}_{\hbar{}}^{%
\infty}\oplus{\mathcal{A}}_{\hbar{}}^{\infty}.$$
(v)
The product map from $\Omega^{1}_{D}({\mathcal{A}}_{\hbar{}}^{\infty})\times\Omega^{1}_{D}({\mathcal%
{A}}_{\hbar{}}^{\infty})$ to $\Omega^{2}_{D}({\mathcal{A}}_{\hbar{}}^{\infty})$ is given by
$$(a\otimes\sigma_{j})\cdot(b\otimes\sigma_{k})=(1-\delta_{jk})ab\otimes\sigma_{%
j}\sigma_{k},\forall j,k=1,2,3.$$
Here $\delta_{jk}$ is the Kronecker delta.
Proof.
Only (v) was not mentioned in ([1]). This follows because the space of forms $\Omega^{1}_{D}({\mathcal{A}}_{\hbar{}}^{\infty}),\Omega^{2}_{D}({\mathcal{A}}_%
{\hbar{}}^{\infty})$ are identified with subspaces of ${\mathcal{A}}_{\hbar{}}^{\infty}\otimes M_{2}(\mathbb{C})$ and the multiplication is induced from the multiplication on ${\mathcal{A}}_{\hbar{}}^{\infty}\otimes M_{2}(\mathbb{C})$.
$\Box$
We also recall proposition 14 from [1].
Proposition 5.2.
If $1,\hbar\mu,\hbar\nu$ are independent over $\mathbb{Q}$ then the positive linear functional on ${\mathcal{A}}_{\hbar{}}^{\infty}\otimes M_{2}({\mathbb{C}})$ given by
$\tau^{\prime}:a\mapsto tr_{\omega}a{|D|}^{-3}$ coincides with $\frac{1}{2}(tr_{\omega}{|D|}^{-3})\tau\otimes tr$ where $tr_{\omega}$ is a Dixmier trace. Thus $\tau^{\prime}=\frac{1}{2}(tr_{\omega}{|D|}^{-3})\widetilde{\tau}$, where $\widetilde{\tau}$, is the trace on $End({\mathcal{E}})$ used in definition (3.3).
Proposition 5.3.
(i)
The differential $\tilde{d}:{\mathcal{A}}_{\hbar{}}^{\infty}\rightarrow\Omega_{D}^{1}({\mathcal{%
A}}_{\hbar{}}^{\infty})$ satisfies $\tilde{d}(a)=\sum_{j=1}^{3}\delta_{j}(a)\otimes\sigma_{j}.$
(ii)
The differential $\tilde{d}:\Omega_{D}^{1}({\mathcal{A}}_{\hbar{}}^{\infty})\rightarrow\Omega_{D%
}^{2}({\mathcal{A}}_{\hbar{}}^{\infty})$ satisfies
(5.3)
$$\displaystyle\tilde{d}(a\otimes\sigma_{1})$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=2,3}\delta_{j}(a)\otimes\sigma_{j}\sigma_{1},$$
(5.4)
$$\displaystyle\tilde{d}(a\otimes\sigma_{2})$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1,3}\delta_{j}(a)\otimes\sigma_{j}\sigma_{2},$$
(5.5)
$$\displaystyle\tilde{d}(a\otimes\sigma_{3})$$
$$\displaystyle=$$
$$\displaystyle\delta_{1}(a)\otimes\sigma_{1}\sigma_{3}+\delta_{2}(a)\otimes%
\sigma_{2}\sigma_{3}-\frac{i}{\alpha}a\otimes\sigma_{1}\sigma_{2}.$$
Proof.
(i) This follows from $\tilde{d}(a)=[D,a]=\sum_{j}\delta_{j}(a)\otimes\sigma_{j}$.
(ii) The differential $\tilde{d}:\Omega_{D}^{1}({\mathcal{A}}_{\hbar{}}^{\infty})\rightarrow\Omega_{D%
}^{2}({\mathcal{A}}_{\hbar{}}^{\infty})$ is defined in such a way that the following diagram commutes.
$\pi_{D}$$\Omega^{1}(\mathcal{{\mathcal{A}}_{\hbar{}}^{\infty}})$$\Omega^{1}_{D}(\mathcal{{\mathcal{A}}_{\hbar{}}^{\infty}})$$\pi_{D}$$\Omega^{2}(\mathcal{{\mathcal{A}}_{\hbar{}}^{\infty}})$$\Omega^{2}_{D}(\mathcal{{\mathcal{A}}_{\hbar{}}^{\infty}})$$\tilde{d}$$\delta$
Therefore to see how it acts on an element of $\Omega^{1}_{D}(\mathcal{{\mathcal{A}}_{\hbar{}}^{\infty}})$ we pick an element and lift that to $\Omega^{1}(\mathcal{{\mathcal{A}}_{\hbar{}}^{\infty}})$ and then follow the diagram. Let $\phi_{mn}\in S^{c}$ be the function $\phi_{m,n}(x,y,p)=e(mx+ny)\delta_{p0}$. These functions are eigenfunctions for $\delta_{j}$’s and satisfy
$$\displaystyle\delta_{1}(\phi_{10})$$
$$\displaystyle=2\pi\phi_{10},$$
$$\displaystyle\delta_{2}(\phi_{10})$$
$$\displaystyle=0,$$
$$\displaystyle\delta_{3}(\phi_{10})$$
$$\displaystyle=0,$$
$$\displaystyle\delta_{1}(\phi_{01})$$
$$\displaystyle=0,$$
$$\displaystyle\delta_{2}(\phi_{10})$$
$$\displaystyle=2\pi\phi_{01},$$
$$\displaystyle\delta_{3}(\phi_{01})$$
$$\displaystyle=0.$$
Let $\tilde{a}=\frac{1}{2\pi}a\phi_{10}^{*}\delta(\phi_{10})\in\Omega^{1}$, then,
$$\displaystyle\pi_{D}(\tilde{a})$$
$$\displaystyle=\frac{1}{2\pi}(a\phi_{10}^{*}\otimes I_{2})(\sum_{j=1}^{3}\delta%
_{j}(\phi_{10})\otimes\sigma_{j}))$$
$$\displaystyle=\frac{1}{2\pi}(a\phi_{10}^{*}2\pi\phi_{10})\otimes\sigma_{1}$$
$$\displaystyle=a\otimes\sigma_{1}.$$
Therefore,
$$\displaystyle\tilde{d}(a\otimes\sigma_{1})$$
$$\displaystyle=\pi_{D}(\delta(\tilde{a}))$$
$$\displaystyle=\frac{1}{2\pi}(\sum_{j=1}^{3}\delta_{j}(a\phi_{10}^{*})\otimes%
\sigma_{j})(2\pi\phi_{10}\otimes\sigma_{1})$$
$$\displaystyle=\sum_{j\neq 1}\delta_{j}(a)\otimes\sigma_{j}\sigma_{1}.$$
Similarly
$$\displaystyle\tilde{d}(a\otimes\sigma_{2})$$
$$\displaystyle=\sum_{j\neq 2}\delta_{j}(a)\otimes\sigma_{j}\sigma_{2}.$$
To see (5.5) observe that,
$$\displaystyle\tilde{d}(a\delta_{3}(b)\otimes\sigma_{3})$$
$$\displaystyle=$$
$$\displaystyle\tilde{d}(a\sum_{j}\delta_{j}(b)\otimes\sigma_{j})-\tilde{d}(a%
\delta_{1}(b)\otimes\sigma_{1})-\tilde{d}(a\delta_{2}(b)\otimes\sigma_{2})$$
$$\displaystyle=$$
$$\displaystyle\tilde{d}(\pi_{D}(a\delta(b)))-\tilde{d}(a\delta_{1}(b)\otimes%
\sigma_{1})-\tilde{d}(a\delta_{2}(b)\otimes\sigma_{2})$$
$$\displaystyle=$$
$$\displaystyle\pi_{D}(\delta(a)\delta(b))-\tilde{d}(a\delta_{1}(b)\otimes\sigma%
_{1})-\tilde{d}(a\delta_{2}(b)\otimes\sigma_{2})$$
$$\displaystyle=$$
$$\displaystyle\sum(\delta_{j}(a)\otimes\sigma_{j})(\delta_{k}(a)\otimes\sigma_{%
k})-\tilde{d}(a\delta_{1}(b)\otimes\sigma_{1})-\tilde{d}(a\delta_{2}(b)\otimes%
\sigma_{2})\mbox{ mod }\pi(\delta J_{1})$$
$$\displaystyle=$$
$$\displaystyle\delta_{1}(a)\delta_{3}(b)\otimes\sigma_{1}\sigma_{3}+\delta_{2}(%
a)\delta_{3}(b)\otimes\sigma_{2}\sigma_{3}+a[\delta_{2},\delta_{1}](b)\otimes%
\sigma_{1}\sigma_{2}$$
$$\displaystyle-a\delta_{3}(\delta_{1}(b))\otimes\sigma_{3}\sigma_{1}-a\delta_{3%
}(\delta_{2}(b))\otimes\sigma_{3}\sigma_{2}$$
$$\displaystyle=$$
$$\displaystyle\delta_{1}(a\delta_{3}(b))\otimes\sigma_{1}\sigma_{3}+\delta_{2}(%
a\delta_{3}(b))\otimes\sigma_{2}\sigma_{3}+a[\delta_{2},\delta_{1}](b)\otimes%
\sigma_{1}\sigma_{2}$$
$$\displaystyle=$$
$$\displaystyle\delta_{1}(a\delta_{3}(b))\otimes\sigma_{1}\sigma_{3}+\delta_{2}(%
a\delta_{3}(b))\otimes\sigma_{2}\sigma_{3}+\frac{i}{\alpha}a\delta_{3}(b)%
\otimes\sigma_{1}\sigma_{2}.$$
The last equality uses $[\delta_{2},\delta_{1}]=\frac{i}{\alpha}\delta_{3}.$
Since span of elements of the form $a\delta_{3}(b)$ forms an ideal in ${\mathcal{A}}_{\hbar{}}^{\infty}$ and ${\mathcal{A}}_{\hbar{}}^{\infty}$ is simple, (5.5) follows.
$\Box$
Corollary 5.4.
$\tilde{d}(1\otimes\sigma_{j})=\begin{cases}0&\mbox{ if }j=1,2;\\
\frac{-1}{\alpha}\otimes\sigma_{3}&\mbox{ if }j=3.\end{cases}$
Now we have all the ingredients to describe the space $\widetilde{C}({\mathcal{E}})$ of compatible connections on a finitely
generated projective ${\mathcal{A}}_{\hbar{}}^{\infty}$-module ${\mathcal{E}}$ with a Hermitian structure.
Proposition 5.5.
Let $\mathcal{E}$ be a finitely generated projective ${\mathcal{A}}_{\hbar{}}^{\infty}$ module. Then the space
$\widetilde{C}(\mathcal{E})$
of compatible connections is given by triples of linear maps $\widetilde{\nabla}_{j}:\mathcal{E}\rightarrow\mathcal{E},j=1,2,3$ such that
(5.6)
$$\displaystyle\widetilde{\nabla}_{j}(\xi.a)$$
$$\displaystyle=\nabla_{j}(\xi).a+\xi.\delta_{j}(a),$$
$$\displaystyle j=1,2,3$$
(5.7)
$$\displaystyle\delta_{j}(\langle\,\xi,\xi^{\prime}\,\rangle_{\mathcal{A}})$$
$$\displaystyle=\langle\xi\,,\widetilde{\nabla}_{j}\,\xi^{\prime}\rangle-\langle%
\widetilde{\nabla}_{j}\,\xi\,,\xi^{\prime}\rangle\,\,,$$
$$\displaystyle\forall\,\xi\,,\xi^{\prime}\in\mathcal{E},$$
$$\displaystyle j=1,2,3\,.$$
Proof.
By proposition (5.1) we can identify ${\mathcal{E}}\otimes_{{\mathcal{A}}_{\hbar{}}^{\infty}}\Omega_{D}^{1}({%
\mathcal{A}}_{\hbar{}}^{\infty})$ with the
subspace $\sum_{j}{\mathcal{E}}\otimes\sigma_{j}\subseteq{\mathcal{E}}\otimes M_{2}({%
\mathbb{C}})$. Thus any compatible connection
$\widetilde{\nabla}$ is prescribed by
three maps $\widetilde{\nabla}_{j}:{\mathcal{E}}\rightarrow{\mathcal{E}}$ such that
$$\widetilde{\nabla}(\xi)=\sum_{j=1}^{3}\widetilde{\nabla}_{j}(\xi)\otimes\sigma%
_{j}.$$
Then
$$\displaystyle\widetilde{\nabla}(\xi.a)$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{3}\widetilde{\nabla}_{j}(\xi.a)\otimes\sigma_{j}$$
$$\displaystyle=$$
$$\displaystyle\widetilde{\nabla}(\xi).a+\xi\otimes\widetilde{d}(a)$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{3}\widetilde{\nabla}_{j}(\xi).a\otimes\sigma_{j}+\sum%
_{j=1}^{3}\xi.\delta_{j}(a)\otimes\sigma_{j}.$$
Thus comparing coefficients of $\sigma_{j}$ we get,
$$\widetilde{\nabla}_{j}(\xi.a)=\nabla_{j}(\xi).a+\xi.\delta_{j}(a),\,j=1,2,3.$$
For (5.7) note that,
$$\displaystyle\sum_{j=1}^{3}\delta_{j}(\langle\xi,\xi^{\prime}\rangle)\otimes%
\sigma_{j}$$
$$\displaystyle=$$
$$\displaystyle\widetilde{d}\langle\xi,\xi^{\prime}\rangle)$$
$$\displaystyle=$$
$$\displaystyle(\xi,\widetilde{\nabla}\xi^{\prime})-(\widetilde{\nabla}\xi,\xi^{%
\prime})$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{3}(\langle\xi,\widetilde{\nabla}_{j}\xi^{\prime}%
\rangle-\langle\widetilde{\nabla}_{j}\,\xi\,,\xi^{\prime}\rangle)\otimes\sigma%
_{j}.$$
$\Box$
Theorem 5.6.
Let $\mathcal{E}$ be a finitely generated projective ${\mathcal{A}}_{\hbar{}}^{\infty}$ module with a Hermitian structure. Then
$\Phi:C(\mathcal{E})\rightarrow\widetilde{C}({\mathcal{E}})$ given by $\Phi(\nabla)=\widetilde{\nabla}$, where
$\widetilde{\nabla}(\xi)=i\nabla(\xi)$ is well defined and
$$\frac{1}{2}(tr_{\omega}{|D|}^{-3})\rm{YM}(\nabla)=\widetilde{\rm{YM}}(\Phi(%
\nabla)).$$
Proof.
Let $\nabla\in C(\mathcal{E})$ be a compatible connection and $\nabla_{j},j=1,2,3$ be its components as defined in the proof of proposition (3.2). If we define ${\widetilde{\nabla}}_{j}=i\nabla_{j},j=1,2,3$, then ${\widetilde{\nabla}}_{j}$’s satisfy (5.6) because $\delta_{j}=i.d_{j},j=12,3$ and (3.9) holds. Similarly (5.7) follows from (3.10). Thus the triple ${\widetilde{\nabla}}_{j},j=1,2,3$ defines a compatible connection $\widetilde{\nabla}\in\widetilde{C}({\mathcal{E}})$. This proves the map $\Phi$ is well defined with the given domain and range. In fact it is an isomorphism.
Let $\widetilde{\nabla}$ denote the extended connection as defined on (4.1) then using propositions, (5.1,5.3), we get,
$$\displaystyle\widetilde{\nabla}(\xi\otimes\sigma_{1})$$
$$\displaystyle=$$
$$\displaystyle\sum_{j\neq 1}{\widetilde{\nabla}}_{j}(\xi)\otimes\sigma_{j}%
\sigma_{1}$$
$$\displaystyle\widetilde{\nabla}(\xi\otimes\sigma_{2})$$
$$\displaystyle=$$
$$\displaystyle\sum_{j\neq 2}{\widetilde{\nabla}}_{j}(\xi)\otimes\sigma_{j}%
\sigma_{2}$$
$$\displaystyle\widetilde{\nabla}(\xi\otimes\sigma_{3})$$
$$\displaystyle=$$
$$\displaystyle\sum_{j\neq 3}{\widetilde{\nabla}}_{j}(\xi)\otimes\sigma_{j}%
\sigma_{3}-\frac{1}{\alpha}\xi\otimes\sigma_{3}.$$
The curvature $\varTheta$ of the connection $\widetilde{\nabla}$ is given by $\varTheta=\widetilde{\nabla}\circ\widetilde{\nabla}$. Which turns out to be,
$$\varTheta(\xi)=i[{\widetilde{\nabla}}_{2},{\widetilde{\nabla}}_{3}](\xi)%
\otimes\sigma_{1}+i[{\widetilde{\nabla}}_{3},{\widetilde{\nabla}}_{1}](\xi)%
\otimes\sigma_{2}+(i[{\widetilde{\nabla}}_{1},{\widetilde{\nabla}}_{2}]-\frac{%
1}{\alpha}{\widetilde{\nabla}}_{3})(\xi)\otimes\sigma_{3}.$$
Repeated application of 5.7 gives,
(5.8)
$$\displaystyle\delta_{k}(\langle\xi,{\widetilde{\nabla}}_{j}(\eta)\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle\xi,{\widetilde{\nabla}}_{k}({\widetilde{\nabla}}_{j}(\eta%
))\rangle-\langle{\widetilde{\nabla}}_{k}(\xi),{\widetilde{\nabla}}_{j}(\eta)\rangle,$$
(5.9)
$$\displaystyle\delta_{j}(\langle\xi,{\widetilde{\nabla}}_{k}(\eta)\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle\xi,{\widetilde{\nabla}}_{j}({\widetilde{\nabla}}_{k}(\eta%
))\rangle-\langle{\widetilde{\nabla}}_{j}(\xi),{\widetilde{\nabla}}_{k}(\eta)\rangle,$$
(5.10)
$$\displaystyle\delta_{j}(\langle{\widetilde{\nabla}}_{k}(\xi),\eta\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle{\widetilde{\nabla}}_{k}(\xi),{\widetilde{\nabla}}_{j}(%
\eta)\rangle-\langle{\widetilde{\nabla}}_{j}({\widetilde{\nabla}}_{k}(\xi)),%
\eta\rangle,$$
(5.11)
$$\displaystyle\delta_{k}(\langle{\widetilde{\nabla}}_{j}(\xi),\eta\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle{\widetilde{\nabla}}_{j}(\xi),{\widetilde{\nabla}}_{k}(%
\eta)\rangle-\langle{\widetilde{\nabla}}_{k}({\widetilde{\nabla}}_{j}(\xi)),%
\eta\rangle.$$
Now, (5.8)-(5.9)+(5.10)-(5.11) gives,
(5.12)
$$\displaystyle\langle\xi,[{\widetilde{\nabla}}_{k},{\widetilde{\nabla}}_{j}](%
\eta)\rangle-\langle[{\widetilde{\nabla}}_{j},{\widetilde{\nabla}}_{k}](\xi),\eta\rangle$$
$$\displaystyle=$$
$$\displaystyle[\delta_{k},\delta_{j}]\langle\xi,\eta\rangle).$$
Combining (5.2) and (5.12) we get,
$$\displaystyle\langle\xi,[{\widetilde{\nabla}}_{1},{\widetilde{\nabla}}_{3}](%
\eta)\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle[{\widetilde{\nabla}}_{3},{\widetilde{\nabla}}_{1}](\xi),\eta\rangle$$
$$\displaystyle\langle\xi,[{\widetilde{\nabla}}_{2},{\widetilde{\nabla}}_{3}](%
\eta)\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle[{\widetilde{\nabla}}_{3},{\widetilde{\nabla}}_{2}](\xi),\eta\rangle$$
$$\displaystyle\langle\xi,(i[{\widetilde{\nabla}}_{1},{\widetilde{\nabla}}_{2}]-%
\frac{1}{\alpha}{\widetilde{\nabla}}_{3})(\eta)\rangle$$
$$\displaystyle=$$
$$\displaystyle\langle(i[{\widetilde{\nabla}}_{1},{\widetilde{\nabla}}_{2}]-%
\frac{1}{\alpha}{\widetilde{\nabla}}_{3})(\xi),\eta\rangle$$
These relations give,
$$\displaystyle\widetilde{\textit{YM}}\,(\widetilde{\nabla})$$
$$\displaystyle=$$
$$\displaystyle\langle\langle\,\varTheta,\varTheta\,\rangle\rangle$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}(tr_{\omega}{|D|}^{-3})\widetilde{\tau}(-{([{%
\widetilde{\nabla}}_{1},{\widetilde{\nabla}}_{3}])}^{2}-{([{\widetilde{\nabla}%
}_{2},{\widetilde{\nabla}}_{3}])}^{2}+{(i[{\widetilde{\nabla}}_{1},{\widetilde%
{\nabla}}_{2}]-\frac{1}{\alpha}{\widetilde{\nabla}}_{3})}^{2})$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{2}(tr_{\omega}{|D|}^{-3})\widetilde{\tau}({([\nabla_{1}%
,\nabla_{3}])}^{2}+{([\nabla_{2},\nabla_{3}])}^{2}+([\nabla_{1},\nabla_{2}]+%
\frac{1}{\alpha}\nabla_{3})^{2}))$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}(tr_{\omega}{|D|}^{-3})\textit{YM}(\nabla).$$
$\Box$
References
[1]
Chakraborty, Partha Sarathi; Sinha, Kalyan B. : Geometry on the quantum Heisenberg manifold. J. Funct. Anal., 203, (2003), no. 2, 425-452.
[2]
A. Connes, $C^{*}$-algèbres et géométrie differentielle, C.R. Acad. Sc. Paris Ser. A-B 290 (1980), no. 13, A599$-$A604.
[3]
A. Connes, M.A. Rieffel, Yang-Mills for non-commutative two-tori, Contemp. Math. 62 (1987) 237-266.
[4]
Connes, A. : Noncommutative Geometry, Academic
Press (1994).
[5]
Kang, Sooran : The Yang-Mills functional and Laplace’s equation on quantum Heisenberg manifolds. J. Funct. Anal., 258, (2010), no. 1, 307-327.
[6]
Rieffel, M. Deformation quantization of Heisenberg manifolds,Communications in Math. Phys.,122, 531-562,(1989) |
On Rate-Splitting by a Secondary Link in Multiple Access Primary Network
John Tadrous and Mohammed Nafie
Authors are with the Wireless Intelligent Networks Center (WINC), Nile University, Cairo, Egypt.
E-mail: [email protected], [email protected]
Abstract
An achievable rate region is obtained for a primary multiple access network coexisting with a secondary link of one transmitter and a corresponding receiver. The rate region depicts the sum primary rate versus the secondary rate and is established assuming that the secondary link performs rate-splitting. The achievable rate region is the union of two types of achievable rate regions. The first type is a rate region established assuming that the secondary receiver cannot decode any primary signal, whereas the second is established assuming that the secondary receiver can decode the signal of one primary receiver. The achievable rate region is determined first assuming discrete memoryless channel (DMC) then the results are applied to a Gaussian channel. In the Gaussian channel, the performance of rate-splitting is characterized for the two types of rate regions. Moreover, a necessary and sufficient condition to determine which primary signal that the secondary receiver can decode without degrading the range of primary achievable sum rates is provided. When this condition is satisfied by a certain primary user, the secondary receiver can decode its signal and achieve larger rates without reducing the primary achievable sum rates from the case in which it does not decode any primary signal. It is also shown that, the probability of having at least one primary user satisfying this condition grows with the primary signal to noise ratio.
Rate-splitting, Cognitive radios, Discrete memoryless channels.
I Introduction
Apotential benefit of allowing secondary users to share primary bands is the enhancement of the spectrum utilization. As introduced in [1] and [2], cognitive radios, or secondary users, are frequency agile devices that can utilize unused spectrum bands through dynamic spectrum access. In dynamic spectrum access secondary users should sense the spectrum and identify unused bands, or spectrum holes. If a band is sensed and found to be in low use by primary users, i.e., underutilized, a secondary user may opportunistically access this band by adjusting its transmit parameters to fully utilize this band without causing excessive interference on the primary users. However, a secondary user has to leave this band and switch to another if the demand by primary users increases.
The notion of dynamic spectrum access has opened research in different problems regarding the new functionalities that a secondary user should perform, e.g., spectrum sensing, spectrum sharing, spectrum mobility and spectrum management [2] and [3]. Moreover, information theoretic bounds on potential achievable rates by cognitive radio networks are being investigated. In most of those works cooperation between primary and secondary transmitters is considered. In [4] an achievable rate region of primary versus secondary users’ rates is introduced when a cognitive transmitter has full knowledge of the primary message in a two-transmitter two-receiver interference channel and the primary user cooperates with the secondary link through rate-splitting introduced in [7]. In [5] and [6] the notion of conferencing is introduced for the interference channel where the cognitive link is assumed to know part or all of the message of the primary transmitter.
In this paper we consider a multiple access channel (MAC) of two transmitters and a common receiver shared by a secondary link of single transmitter and a corresponding receiver. The secondary transmitter is assumed to employ rate-splitting by dividing its signal into two parts: one part is decodable by the secondary receiver and treated as noise by the primary receiver, whereas the other part is decodable at both receivers. Based on this scheme we:
•
Establish an achievable rate region, $\mathcal{R}^{o}$, for the primary sum rate versus the secondary rate in a discrete memoryless channel (DMC) setup assuming that all of the primary signals are treated as noise at the secondary receiver.
•
Establish another achievable rate region, $\mathcal{R}^{r}_{i}$, for which the signal of primary transmitter $i$ is to be fully decodable at the secondary receiver besides being decodable at the primary receiver. For this scheme we show that there exists a case for which $\mathcal{R}^{r}_{i}$ includes $\mathcal{R}^{o}$.
•
Provide an overall achievable rate region
$$\mathcal{R}=\mathcal{R}^{o}\bigcup\left(\cup_{i\in\{1,2\}}\mathcal{R}_{i}^{r}%
\right).$$
•
Apply the results obtained in DMC case in a Gaussian setup where the effect of rate-splitting on the achievable rate region is analyzed. A necessary and sufficient condition is established for obtaining the overall rate region without rate-splitting.
•
Derive a necessary and sufficient condition so that the secondary receiver can decode the signal of one primary user without affecting the range of achievable primary sum rates, but only enhances the range of achievable secondary rates. We call this condition primary decodability condition for Gaussian (PDCG) channel.
•
Show, numerically, that the probability of having at least one primary user satisfying PDCG monotonically increases with the signal-noise-ratio of the primary users.
We have provided some of the results in this paper in a conference paper version [9].
The introduced network model of MAC primary network shared by secondary operations has been addressed in some resource allocation frameworks without rate-splitting by secondary users [10]-[14]. Rate-splitting by a secondary link, however, has been introduced in [8] where the secondary user is assumed to know the codebook of a primary transmitter and opportunistically splits its rate into two parts and decodes it in the following way. It decodes the first part treating both the primary signal and the second part as noise, decodes and cancels the primary signal and then decodes the second part. This scheme is generalized in this paper as we consider the cases when the signal of one primary transmitter is decodable at the secondary receiver and when all the primary signals are treated as noise.
The rest of this paper is organized as follows. In Section II the discrete memoryless channel (DMC) models are defined. In Section III the achievable rate regions are established for the defined DMC models. Then, obtained results are applied in a Gaussian channel setup in Section IV and the paper is conncluded in Section V.
II Channel Model
In our formulation we denote random variables by $X$, $Y$, $\cdots$ with realizations $x$, $y$, $\cdots$ from sets $\mathcal{X}$, $\mathcal{Y}$, $\cdots$ respectively. The communication channel is considered to be discrete and memoryless.
II-A Basic Channel Model
We consider a basic channel $C_{B}$ defined by a tuple $(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{s},\omega,\mathcal{Y}_{p},%
\mathcal{Y}_{s})$, where $\mathcal{X}_{1}$, $\mathcal{X}_{2}$ are two finite input alphabet sets of the primary transmitters and $\mathcal{X}_{s}$ is a finite input alphabet set of the secondary transmitter. Sets $\mathcal{Y}_{p}$ and $\mathcal{Y}_{s}$ are two finite output alphabet sets at the primary and secondary receivers respectively, and $\omega$ is a collection of conditional channel probabilities $\omega(y_{p}y_{s}|x_{1}x_{2}x_{s})$ of $(y_{p},y_{s})\in{\mathcal{Y}_{p}\times{\mathcal{Y}_{s}}}$ given $(x_{1},x_{2},x_{s})\in{\mathcal{X}_{1}\times\mathcal{X}_{2}\times\mathcal{X}_{%
s}}$, with marginal conditional distributions:
$$\displaystyle\omega_{p}(y_{p}|x_{1}x_{2}x_{s})=\sum_{y_{s}\in\mathcal{Y}_{s}}{%
\omega(y_{p}y_{s}|x_{1}x_{2}x_{s})},$$
$$\displaystyle\omega_{s}(y_{s}|x_{1}x_{2}x_{s})=\sum_{y_{p}\in\mathcal{Y}_{p}}{%
\omega(y_{p}y_{s}|x_{1}x_{2}x_{s})}.$$
Since the channel is memoryless, the conditional probability $\omega^{n}(\textbf{y}_{p}\textbf{y}_{s}|\textbf{x}_{1}\textbf{x}_{2}\textbf{x}%
_{s})$ is given by
$$\omega^{n}(\textbf{y}_{p}\textbf{y}_{s}|\textbf{x}_{1}\textbf{x}_{2}\textbf{x}%
_{s})=\prod_{t=1}^{n}{\omega(y_{p}^{(t)}y_{s}^{(t)}|x_{1}^{(t)}x_{2}^{(t)}x_{s%
}^{(t)})},$$
where
$$\displaystyle\textbf{x}_{a}=$$
$$\displaystyle(x_{a}^{(1)},\cdots,x_{a}^{(n)})\in{\mathcal{X}_{a}^{n}},$$
$$\displaystyle a=1,2,s,$$
$$\displaystyle\textbf{y}_{a}=$$
$$\displaystyle(y_{a}^{(1)},\cdots,y_{a}^{(n)})\in{\mathcal{Y}_{a}^{n}},$$
$$\displaystyle a=p,s.$$
The same also holds for the marginal conditional distributions $\omega^{n}_{p}(\textbf{y}_{p}|\textbf{x}_{1}\textbf{x}_{2}\textbf{x}_{s})$ and $\omega^{n}_{s}(\textbf{y}_{s}|\textbf{x}_{1}\textbf{x}_{2}\textbf{x}_{s})$.
Let $\mathcal{M}_{1}=\{1,\cdots,M_{1}\}$, $\mathcal{M}_{2}=\{1,\cdots,M_{2}\}$ be message sets for primary transmitters 1 and 2 respectively, and $\mathcal{M}_{s}=\{1,\cdots,M_{s}\}$ be a message set for the secondary transmitter. A code $(n,M_{1},M_{2},M_{s},\epsilon)$ is a collection of $M_{1}$, $M_{2}$ and $M_{s}$ codewords such that:
1.
Sender $a$, $a=1,2,s$, has an encoding function $\phi_{a}:i\rightarrow\textbf{x}_{ai}$, $i\in\mathcal{M}_{a}$ and $\textbf{x}_{ai}\in\mathcal{X}^{n}$.
2.
The primary receiver has $M_{1}M_{2}$ disjoint decoding sets $\mathcal{D}_{pij}\subseteq{\mathcal{Y}_{p}^{n}}$, $ij\in\mathcal{M}_{1}\times\mathcal{M}_{2}$, and a decoding function $\psi_{p}:\textbf{y}_{p}\rightarrow ij$ if $\textbf{y}_{p}\in\mathcal{D}_{pij}$, where $ij\in\mathcal{M}_{1}\times\mathcal{M}_{2}$.
3.
The secondary receiver has $M_{s}$ disjoint decoding sets $\mathcal{D}_{sk}\subseteq{\mathcal{Y}_{s}^{n}}$, $k\in\mathcal{M}_{s}$, and a decoding function $\psi_{s}:\textbf{y}_{s}\rightarrow{k}$ if $\textbf{y}_{s}\in\mathcal{D}_{sk}$, where $k\in\mathcal{M}_{s}$ (see Fig.1).
4.
Probability of error for the primary network and the secondary link are less than $\epsilon$, that is, $Pe_{p}\leq{\epsilon}$ and $Pe_{s}\leq{\epsilon}$ respectively, where
$$\displaystyle Pe_{p}=\frac{1}{M_{1}M_{2}M_{s}}\sum_{i,j,k}\omega_{p}^{n}(%
\textbf{y}_{p}\notin\mathcal{D}_{pij}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x}%
_{sk}),$$
(1)
$$\displaystyle Pe_{s}=\frac{1}{M_{1}M_{2}M_{s}}\sum_{i,j,k}\omega_{s}^{n}(%
\textbf{y}_{s}\notin\mathcal{D}_{sk}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x}_%
{sk}).$$
(2)
A rate tuple $(R_{1},R_{2},R_{s})$ of nonnegative real values is achievable if for any $\eta>0$, $0<\epsilon<1$ there exists a code such that
$$\frac{1}{n}\log{M_{a}}\geq{R_{a}-\eta},\ \ a=1,2,s,$$
(3)
with sufficiently large $n$.
II-B Rate-Splitting Channel
Rate-splitting channel, $C_{RS}$, is a modified version of the basic channel $C_{B}$, where $C_{RS}$ is defined by a tuple $(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{s},\omega,\mathcal{Y}_{p},%
\mathcal{Y}_{s})$ with its elements are as defined in $C_{B}$. Moreover, the input message sets for the primary transmitters are also $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ exactly as in $C_{B}$. However, the secondary user is assumed to have two finite message sets $\mathcal{L}_{s}=\{1,\cdots,L_{s}\}$, $\mathcal{N}_{s}=\{1,\cdots,N_{s}\}$. Hence, a code $(n,M_{1},M_{2},L_{s},N_{s},\epsilon)$ over the channel $C_{RS}$ is a collection of $M_{1}$, $M_{2}$, $L_{s}N_{s}$ codewords such that:
1.
Primary transmitter $a$, $a=1,2$, has an encoding function $\phi_{a}:i\rightarrow\textbf{x}_{ai}$, $i\in\mathcal{M}_{a}$, $\textbf{x}_{ai}\in\mathcal{X}_{a}^{n}$.
2.
The secondary transmitter has an encoding function $\phi_{s}:kl\rightarrow\textbf{x}_{skl}$, $kl\in\mathcal{L}_{s}\times\mathcal{N}_{s}$, $\textbf{x}_{skl}\in\mathcal{X}_{s}^{n}$.
3.
The primary receiver has $M_{1}M_{2}N_{s}$ disjoint decoding sets $\mathcal{D}_{pijl}\subseteq{\mathcal{Y}_{p}^{n}}$, $ijl\in\mathcal{M}_{1}\times\mathcal{M}_{2}\times\mathcal{N}_{s}$ and a decoding function $\psi_{p}:\textbf{y}_{p}\rightarrow ijl$ if $\textbf{y}_{p}\in{\mathcal{D}_{pijl}}$, where $ijl\in\mathcal{M}_{1}\times\mathcal{M}_{2}\times\mathcal{N}_{s}$.
4.
The secondary receiver has $L_{s}N_{s}$ disjoint decoding sets $\mathcal{D}_{skl}\subseteq{\mathcal{Y}_{s}^{n}}$, $kl\in{\mathcal{L}_{s}\times\mathcal{N}_{s}}$, and a decoding function $\psi_{s}:\textbf{y}_{p}\rightarrow kl$ if $\textbf{y}_{p}\in\mathcal{D}_{skl}$, where $kl\in\mathcal{L}_{s}\times\mathcal{N}_{s}$ (see Fig.2).
5.
Probability of error for primary network and secondary link are less than $\epsilon$, that is $Pe_{p}^{o}\leq{\epsilon}$ and $Pe_{s}^{o}\leq{\epsilon}$ respectively, where
$$\displaystyle Pe_{p}^{o}=\\
\displaystyle\frac{1}{M_{1}M_{2}L_{s}N_{s}}\sum_{i,j,k,l}\omega_{p}^{n}(%
\textbf{y}_{p}\notin\mathcal{D}_{pijl}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x%
}_{skl}),$$
(4)
$$\displaystyle Pe_{s}^{o}=\\
\displaystyle\frac{1}{M_{1}M_{2}L_{s}N_{s}}\sum_{i,j,k,l}\omega_{s}^{n}(\text{%
y}_{s}\notin\mathcal{D}_{skl}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x}_{skl}).$$
(5)
A rate tuple $(R_{1},R_{2},S,T)$ of non-negative real values is achievable over the channel $C_{RS}$ if there exists a code $(n,M_{1},M_{2},L_{s},N_{s},\epsilon)$ such that for any arbitrary $0<\epsilon<1$ and $\eta>0$
$$\displaystyle\frac{1}{n}\log{M_{1}}\geq{R_{1}-\eta},$$
(6)
$$\displaystyle\frac{1}{n}\log{M_{2}}\geq{R_{2}-\eta},$$
(7)
$$\displaystyle\frac{1}{n}\log{L_{s}}\geq{S-\eta},$$
(8)
$$\displaystyle\frac{1}{n}\log{N_{s}}\geq{T-\eta},$$
(9)
with sufficiently large $n$.
Lemma 1
If a rate tuple $(R_{1},R_{2},S,T)$ is achievable for $C_{RS}$, then a rate tuple $(R_{1},R_{2},R_{s})$ where $R_{s}=S+T$ is achievable for $C_{B}$.
Proof:
It is sufficient to show that, if $(n,M_{1},M_{2},L_{s},N_{s},\epsilon)$ is a code for $C_{RS}$ then $(n,M_{1},M_{2},L_{s}N_{s},\epsilon)$ is a code for $C_{B}$. To do so, let $\mathcal{D}_{pij}=\cup_{l=1}^{N_{s}}\mathcal{D}_{pijl}$. Then
$$\omega_{p}^{n}(\textbf{y}_{p}\notin\mathcal{D}_{pij}|\textbf{x}_{1i}\textbf{x}%
_{2j}\textbf{x}_{skl})\leq\omega_{p}^{n}(\textbf{y}_{p}\notin\mathcal{D}_{pijl%
}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x}_{skl}).$$
(10)
So, if $(n,M_{1},M_{2},L_{s},N_{s},\epsilon)$ is a code for $C_{RS}$ then $Pe_{p}^{o}\leq{\epsilon}$ and $Pe_{s}^{o}\leq\epsilon$, hence, from (10) $Pe_{p}\leq{\epsilon}$ and $Pe_{s}\leq\epsilon$ when $k$ and $M_{s}$ of (1) and (2) are replaced with $kl$ and $L_{s}N_{s}$ respectively, meaning that $(n,M_{1},M_{2},L_{s}N_{s},\epsilon)$ is a code for $C_{B}$.
∎
II-C Rate-Splitting Channel with Decodable Primary Signal at the Secondary Receiver
We introduce another channel, $C_{RS}^{p}$, in which the secondary user splits its set of messages into two sets, exactly as the case of $C_{RS}$. However, we assume that the signal of one primary transmitter is decodable at the secondary receiver. Without loss of generality, assume this this is the first primary transmitter. Thus, $C_{RS}^{p}$ is defined by a tuple $(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{s},\omega,\mathcal{Y}_{p},%
\mathcal{Y}_{s})$ with its elements defined as in $C_{B}$ and $C_{RS}$. A code $(n,M_{1},M_{2},L_{s},N_{s})$ over the channel $C_{RS}^{p}$ is a collection of $M_{1}$, $M_{2}$, $L_{s}N_{s}$ codewords such that conditions 1), 2) and 3) of the same code but in $C_{RS}$ are satisfied besides the following two conditions:
1.
Secondary receiver has $M_{1}L_{s}N_{s}$ disjoint decoding sets $\mathcal{D}_{sikl}\subseteq\mathcal{Y}_{s}^{n}$, and a decoding function $\psi_{s}:\textbf{y}_{s}\rightarrow ikl$ if $\textbf{y}_{s}\in\mathcal{D}_{sikl}$, where $ikl\in\mathcal{M}_{1}\times\mathcal{L}_{s}\times\mathcal{N}_{s}$.
2.
Probability of error for the primary network and the secondary link are less than $\epsilon$, that is, $Pe_{p}^{r}\leq{\epsilon}$ and $Pe_{s}^{r}\leq{\epsilon}$ respectively, where
$$\displaystyle Pe_{p}^{r}=\\
\displaystyle\frac{1}{M_{1}M_{2}L_{s}N_{s}}\sum_{i,j,k,l}\omega_{p}^{n}(%
\textbf{y}_{p}\notin\mathcal{D}_{pijl}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x%
}_{skl}),$$
(11)
$$\displaystyle Pe_{s}^{r}=\\
\displaystyle\frac{1}{M_{1}M_{2}L_{s}N_{s}}\sum_{i,j,k,l}\omega_{s}^{n}(%
\textbf{y}_{s}\notin\mathcal{D}_{sikl}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x%
}_{skl}).$$
(12)
A rate tuple $(R_{1},R_{2},S,T)$ of non-negative real values is achievable over the channel $C_{RS}^{p}$ if for any arbitrary $\eta>0$ and $0<\epsilon<1$ the inequalities (6)-(9) are satisfied for sufficiently large $n$.
Lemma 2
If a rate tuple $(R_{1},R_{2},S,T)$ is achievable for $C_{RS}^{p}$, then a rate tuple $(R_{1},R_{2},R_{s})$ where $R_{s}=S+T$ is achievable for $C_{B}$.
Proof:
The proof follows exactly as the proof of Lemma 1 noting that, if $\mathcal{D}_{skl}=\cup_{i=1}^{M_{1}}\mathcal{D}_{sikl}$, then
$$\omega_{s}^{n}(\textbf{y}_{s}\notin\mathcal{D}_{skl}|\textbf{x}_{1i}\textbf{x}%
_{2j}\textbf{x}_{skl})\leq\omega_{s}^{n}(\textbf{y}_{s}\notin\mathcal{D}_{sikl%
}|\textbf{x}_{1i}\textbf{x}_{2j}\textbf{x}_{skl}).$$
(13)
∎
III Achievable Rate Region
In this section we consider the characterization of the achievable rate region for $C_{B}$. In order to do so, we first establish two achievable rate regions, one for $C_{RS}$ and another for $C_{RS}^{p}$. Then, we define the achievable rate region for $C_{B}$.
We consider the random variables $U$, $W$ and $Q$ defined over the finite sets $\mathcal{U}$, $\mathcal{W}$ and $\mathcal{Q}$ respectively, where $Q$ is a time sharing parameter. Let the set $\mathcal{P}^{*}$ contains all $Z=QUWX_{1}X_{2}X_{s}Y_{p}Y_{s}$ such that:
•
$X_{1}$, $X_{2}$, $U$ and $W$ are conditionally independent given $Q$,
•
$X_{s}=f(UW|Q)$,
Since $X_{s}=f(UW|Q)$, then $\mathcal{U}$ and $\mathcal{W}$ can be considered as input sets to the channels $C_{RS}$ and $C_{RS}^{p}$. We establish achievable rate regions for $C_{RS}$ and $C_{RS}^{p}$ as follows.
III-A Achievable Rate Region for $C_{RS}$
Theorem 1
For any $Z\in\mathcal{P}^{*}$, $\delta^{o}(Z)$ is the set of achievable rate tuples $(R_{1},R_{2},S,T)$ for $C_{RS}$ if the following inequalities are satisfied:
$$\displaystyle R_{1}\leq{I(Y_{p};X_{1}|WX_{2}Q)},$$
(14)
$$\displaystyle R_{2}\leq{I(Y_{p};X_{2}|WX_{1}Q)},$$
(15)
$$\displaystyle T\leq{I(Y_{p};W|X_{1}X_{2}Q)},$$
(16)
$$\displaystyle R_{1}+R_{2}\leq{I(Y_{p};X_{1}X_{2}|WQ)},$$
(17)
$$\displaystyle T+R_{1}\leq{I(Y_{p};WX_{1}|X_{2}Q)},$$
(18)
$$\displaystyle T+R_{2}\leq{I(Y_{p};WX_{2}|X_{1}Q)},$$
(19)
$$\displaystyle T+R_{1}+R_{2}\leq{I(Y_{p};WX_{1}X_{2}|Q)};$$
(20)
$$\displaystyle S\leq{I(Y_{s};U|WQ)},$$
(21)
$$\displaystyle T\leq{I(Y_{s};W|UQ)},$$
(22)
$$\displaystyle S+T\leq{I(Y_{s};UW|Q)}.$$
(23)
Proof:
Please refer to Appendix A
∎
Corollary 1
For $\delta^{o}=\cup_{Z\in\mathcal{P}^{*}}\delta^{o}(Z)$, any rate tuple of $\delta^{o}$ is achievable.
In the defined network we focus on the achievable rates by the primary network $R_{p}=R_{1}+R_{2}$ and the secondary link $R_{s}=S+T$. Let $\mathcal{R}^{o}(Z)$ be the set of all rate tuples $(R_{s},R_{p})$ having $(R_{1},R_{2},S,T)$ satisfy (14)-(23) for all $Z\in\mathcal{P}^{*}$, then we determine $\mathcal{R}^{o}(Z)$ in the following theorem.
Theorem 2
For any $Z\in{\mathcal{P}^{*}}$ the achievable rate region $\mathcal{R}^{o}(Z)$ of the defined channel $C_{RS}$ consists of all rate pairs $(R_{s},R_{p})$ that satisfy
$$R_{p}\leq{\rho_{p}^{o}},\quad R_{s}\leq{\rho_{s}^{o}},\quad R_{s}+R_{p}\leq{%
\rho_{sp}^{o}}$$
(24)
where
$$\displaystyle\rho_{p}^{o}=I(Y_{p};X_{1}X_{2}|WQ),$$
(25)
$$\displaystyle\rho_{s}^{o}=I(Y_{s};U|WQ)+\sigma^{*},$$
(26)
$$\displaystyle\begin{split}\displaystyle\rho_{sp}^{o}=&\displaystyle\rho_{p}^{o%
}+I(Y_{s};U|WQ)\\
&\displaystyle+\min\{I(Y_{s};W|Q),I(Y_{p},W|Q)\}\end{split}$$
(27)
and
$$\sigma^{*}=\min\{I(Y_{p};W|X_{1}X_{2}Q),I(Y_{s};W|Q)\}.$$
(28)
Proof:
To proof the theorem it is sufficient to determine the rate tuples $(R_{s},R_{p})$ of the corner points of $\mathcal{R}^{o}(Z)$. To do so, we refer to Fig. 3.
•
Point A:
$R_{s}^{A}=0$, i.e., $S^{A}=T^{A}=0$. Thus the maximum rate at which the primary network can operate is determined from (17) as:
$$R_{p}^{A}={I(Y_{p};X_{1}X_{2}|WQ)}=\rho_{p}^{o}$$
(29)
•
Point B:
At this point we find the maximum possible rate at which the secondary user can transmit when the primary rate is $R_{p}^{B}=\rho_{p}^{o}$. In this case the relations of (14)-(23) are reduced to
$$\displaystyle T\leq{I(Y_{p};W|Q)},$$
(30)
$$\displaystyle\rho_{p}^{o}+T\leq{I(Y_{p};WX_{1}X_{2}|Q)};$$
(31)
$$\displaystyle T\leq{I(Y_{s};W|UQ)},$$
(32)
$$\displaystyle S\leq{I(Y_{s};U|WQ)},$$
(33)
$$\displaystyle S+T\leq{I(Y_{s};UW|Q)}.$$
(34)
Since $T$ is irrelevant in (33), then $S$ can be set to
$$S^{B}=I(Y_{s};U|WQ).$$
(35)
Hence, using chain rule in (31) and (34), the maximum value for $T$ would be
$$T^{B}=\min\{I(Y_{p};W|Q),I(Y_{s};W|Q)\}$$
(36)
and $R_{s}^{B}=S^{B}+T^{B}$.
•
Point D:
$R_{1}^{D}=R_{2}^{D}=R_{p}^{D}=0$, then (14)-(23) are reduced to
$$\displaystyle T\leq{I(Y_{p};W|X_{1}X_{2}Q)};$$
(37)
$$\displaystyle S\leq{I(Y_{s};U|WQ)},$$
(38)
$$\displaystyle T\leq{I(Y_{s};W|UQ)},$$
(39)
$$\displaystyle S+T\leq{I(Y_{s};UW|Q)}.$$
(40)
Since $T$ is irrelevant in (38), $S$ can be set to
$$S^{D}=I(Y_{s};U|WQ).$$
(41)
Then,
$$T^{D}=\sigma^{*}=\min\{I(Y_{s};W|Q),I(Y_{p};W|X_{1}X_{2}Q)\}$$
(42)
and $R_{s}^{D}=S^{D}+T^{D}=\rho_{s}^{o}$.
•
Point C:
At $R_{s}^{C}=\rho_{s}^{o}$, the maximum possible primary rate $R_{p}=R_{1}+R_{2}$ has to satisfy
$$\displaystyle R_{p}\leq{I(Y_{p};X_{1}X_{2}|WQ)},$$
(43)
$$\displaystyle R_{p}\leq{I(Y_{p};WX_{1}X_{2}|Q)-\sigma^{*}}.$$
(44)
Using chain rule, (44) can be rewritten as
$$R_{p}\leq{I(Y_{p};X_{1}X_{2}|WQ)+I(Y_{p};W|Q)-\sigma^{*}}.$$
(45)
Thus, if $I(Y_{p};W|Q)-\sigma^{*}>0$ then (45) will be dominated by (43). Otherwise, (45) dominates (43). So, $R_{p}^{C}$ will be given by,
$$R_{p}^{C}=I(Y_{p};X_{1}X_{2}|WQ)-\left[\sigma^{*}-I(Y_{p};W|Q)\right]^{+}$$
(46)
where $[x]^{+}=\max\{0,x\}$.
The following is to show that both points $(R_{s}^{B},R_{p}^{B})$ and $(R_{s}^{C},R_{p}^{C})$ lie on the line $R_{s}+R_{p}={\rho_{sp}^{o}}$:
For Point B, using direct substitution with
$$R_{s}^{B}=I(Y_{s};U|WQ)+\min\{I(Y_{p};W|Q),I(Y_{s};W|Q)\}$$
and
$$R_{p}^{B}=\rho_{p}^{o}$$
it is clear that $R_{s}^{B}+R_{p}^{B}=\rho_{sp}^{o}$.
For Point C, we consider the following two possibilities:
•
$\sigma^{*}\geq{I(Y_{p};W|Q)}$:
Here $\min\{I(Y_{s};W|Q),I(Y_{p},W|Q)\}=I(Y_{p};W|Q)$. Consequently,
$$\rho_{sp}^{o}=I(Y_{s};U|WQ)+I(Y_{p};WX_{1}X_{2}|Q)$$
and
$$R_{s}^{C}+R_{p}^{C}=I(Y_{s};U|WQ)+I(Y_{p};WX_{1}X_{2}|Q).$$
•
$\sigma^{*}<I(Y_{p};W|Q)$:
Since $I(Y_{p};W|X_{1}X_{2}Q)\geq{I(Y_{p};W|Q)}$, therefore
$$I(Y_{s};W|Q)<I(Y_{p};W|Q).$$
Consequently,
$$\rho_{sp}^{o}=I(Y_{s};UW|Q)+I(Y_{p};X_{1}X_{2}|WQ)$$
and
$$R_{s}^{C}+R_{p}^{C}=I(Y_{s};UW|Q)+I(Y_{p};X_{1}X_{2}|WQ).$$
Therefore, both rate tuples $(R_{s}^{B},R_{p}^{B})$ and $(R_{s}^{C},R_{p}^{C})$ lie on the line $R_{s}+R_{p}=\rho_{sp}^{o}$.
∎
Note that, in the Appendix of [7] Han and Kobayashi argued that part of the achievable rate region by their introduced scheme was bounded by lines of slopes $-0.5$ and $-2$. Although from (14)-(23) reducing $T$ by a value of $r$ may result in increase of $R_{p}$ by $2r$, the proof that point $(R_{s}^{C},R_{p}^{C})$ lie on the line $R_{s}+R_{p}=\rho_{sp}^{o}$ means that a bound of slope $-2$ does not exist for $\mathcal{R}^{o}(Z)$.
Corollary 2
Any rate tuple $(R_{s},R_{p})$ of the region
$$\mathcal{R}^{o}=\mbox{closure of}\bigcup_{Z\in\mathcal{P}^{*}}\mathcal{R}^{o}(Z)$$
(47)
is achievable.
III-B Achievable Rate Region for $C_{RS}^{p}$
Since in $C_{RS}^{p}$ the signal of one primary user has to be decodable at the secondary receiver, the model of $C_{RS}^{p}$ can be considered as the modified interference channel model, $C_{m}$, introduced in [7]. The signals of the two primary users can be treated as if they are produced from single source splitting its signal into two parts and encoding each part separately such that, one part is decodable at both receivers while the other is decodable only at the primary receiver. For this channel, we define the set $\delta_{i}^{r}(Z)$ as the set of all achievable rate tuples $(R_{1},R_{2},S,T)$ when the signal of primary transmitter $i$, $i\in\{1,2\}$, is decodable by the secondary receiver. Without loss of generality, we assume that $i=1$. Then, we define an achievable rate region for $C_{RS}^{p}$ in the following theorem.
Theorem 3
For any $Z\in\mathcal{P}^{*}$, $\delta_{1}^{r}(Z)$ is the set of achievable rate tuples $(R_{1},R_{2},S,T)$ over the channel $C_{RS}^{p}$ if the following inequalities are satisfied:
$$\displaystyle R_{1}\leq I(Y_{p};X_{1}|WX_{2}Q),$$
(48)
$$\displaystyle R_{2}\leq I(Y_{p};X_{2}|WX_{1}Q),$$
(49)
$$\displaystyle T\leq I(Y_{p};W|X_{1}X_{2}Q),$$
(50)
$$\displaystyle R_{1}+R_{2}\leq I(Y_{p};X_{1}X_{2}|WQ),$$
(51)
$$\displaystyle R_{1}+T\leq I(Y_{p};WX_{1}|X_{2}Q),$$
(52)
$$\displaystyle R_{2}+T\leq I(Y_{p};WX_{2}|X_{1}Q),$$
(53)
$$\displaystyle R_{1}+R_{2}+T\leq I(Y_{p};WX_{1}X_{2}Q);$$
(54)
$$\displaystyle S\leq I(Y_{s};U|WX_{1}Q),$$
(55)
$$\displaystyle T\leq I(Y_{s};W|UX_{1}Q),$$
(56)
$$\displaystyle R_{1}\leq I(Y_{s};X_{1}|UWQ),$$
(57)
$$\displaystyle S+T\leq I(Y_{s};UW|X_{1}Q),$$
(58)
$$\displaystyle R_{1}+S\leq I(Y_{s};UX_{1}|WQ),$$
(59)
$$\displaystyle R_{1}+T\leq I(Y_{s};WX_{1}|UQ),$$
(60)
$$\displaystyle R_{1}+S+T\leq I(Y_{s};UWX_{1}|Q).$$
(61)
Proof:
The proof follows exactly as the proof of Theorem 3.1 in [7].
∎
Corollary 3
For $\delta_{1}^{r}=\cup_{Z\in\mathcal{P}^{*}}\delta_{1}^{r}(Z)$, any rate tuple of $\delta_{1}^{r}$ is achievable.
For $C_{RS}^{p}$ we define the region $\mathcal{R}_{i}^{r}(Z)$ as the set of rate tuples $(R_{s},R_{p})$ where $R_{s}=S+T$, $R_{p}=R_{1}+R_{2}$ and $(R_{1},R_{2},S,T)$ is an element of $\delta_{i}^{r}(Z)$ for any $Z\in\mathcal{P}^{*}$, $i\in\{1,2\}$.
Theorem 4
For any $Z\in\mathcal{P}^{*}$ the achievable rate region $\mathcal{R}_{1}^{r}(Z)$ for the channel $C_{RS}^{p}$ consists of all rate pairs $(R_{s},R_{p})$ that satisfy
$$\displaystyle R_{s}\leq\rho_{s}^{r},$$
$$\displaystyle R_{p}\leq\rho_{p}^{r},$$
$$\displaystyle R_{s}+R_{p}\leq\rho_{sp}^{r},$$
(62)
$$\displaystyle 2R_{s}+R_{p}\leq\rho_{2p}^{r},$$
$$\displaystyle R_{s}+2R_{p}\leq\rho_{s2}^{r}$$
where
$$\displaystyle\rho_{s}^{r}=I(Y_{s};U|WX_{1}Q)+\sigma_{s}^{*},$$
(63)
$$\displaystyle\rho_{p}^{r}=I(Y_{p};X_{2}|WX_{1}Q)+\sigma_{p}^{*},$$
(64)
$$\begin{split}\displaystyle\rho_{sp}^{r}=&\displaystyle I(Y_{s};U|WX_{1}Q)+I(Y_%
{p};X_{2}|WX_{1}Q)+\\
&\displaystyle+\min\{I(Y_{p};WX_{1}|Q),I(Y_{s};WX_{1}|Q),\\
&\displaystyle I(Y_{p};W|X_{1}Q)+I(Y_{s};X_{1}|WQ),\\
&\displaystyle I(Y_{p};X_{1}|WQ)+I(Y_{s};W|X_{1}Q)\},\end{split}$$
(65)
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2I(Y_{s};U|WX_{1}Q)+2%
\sigma_{s}^{*}+I(Y_{p};X_{2}|WX_{1}Q)\\
&\displaystyle-\left[\sigma_{s}^{*}-I(Y_{p};W|X_{1}Q)\right]^{+}+\min\{I(Y_{s}%
;X_{1}|WQ),\\
&\displaystyle I(Y_{s};WX_{1}|Q)-\sigma_{s}^{*},I(Y_{p};X_{1}|Q)\\
&\displaystyle+\left[I(Y_{p};W|X_{1}Q)-\sigma_{s}^{*}\right]^{+},I(Y_{p};X_{1}%
|WQ)\},\end{split}$$
(66)
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2I(Y_{p};X_{2}|WX_{1}Q)%
+2\sigma_{p}^{*}+I(Y_{s};U|WX_{1}Q)\\
&\displaystyle-\left[\sigma_{p}^{*}-I(Y_{s};X_{1}|WQ)\right]^{+}+\min\{I(Y_{p}%
;W|X_{1}Q),\\
&\displaystyle I(Y_{p};WX_{1}|Q)-\sigma_{p}^{*},I(Y_{s};W|Q)\\
&\displaystyle+\left[I(Y_{s};X_{1}|WQ)-\sigma_{p}^{*}\right]^{+},I(Y_{s};W|X_{%
1}Q)\},\end{split}$$
(67)
and
$$\displaystyle\sigma_{s}^{*}=\min\{I(Y_{s};W|X_{1}Q),I(Y_{p};W|X_{1}X_{2}Q)\},$$
(68)
$$\displaystyle\sigma_{p}^{*}=\min\{I(Y_{p};X_{1}|WQ),I(Y_{s};X_{1}|UWQ)\}$$
(69)
as shown in Fig. 4
Proof:
From the similarity between $C_{RS}^{p}$ and the modified interference channel of Han and Kobayashi [7], the derivation of the achievable rate region can be found in the Appendix of [7]. The analysis basically goes as that done for $\mathcal{R}^{o}(Z)$ in $C_{RS}$. In this proof we directly mention the corner points of the $\mathcal{R}_{1}^{r}(Z)$ shown in Fig. 4 as follows.
•
Point A:
$$R_{s}^{A}=0,$$
(70)
$$R_{p}^{A}=\rho_{p}^{r}=I(Y_{p};X_{2}|X_{1}WQ)+\sigma_{p}^{*}.$$
(71)
•
Point B:
$$\begin{split}\displaystyle R_{s}^{B}=&\displaystyle I(Y_{s};U|WX_{1}Q)-[\sigma%
_{p}^{*}-I(Y_{s};X_{1}|WQ)]^{+}\\
&\displaystyle+\min\{I(Y_{p};W|X_{1}Q),I(Y_{p};WX_{1}|Q)-\sigma_{p}^{*},\\
&\displaystyle I(Y_{s};W|Q)+\left[I(Y_{s};X_{1}|WQ)-\sigma_{p}^{*}\right]^{+},%
\\
&\displaystyle I(Y_{s};W|X_{1}Q)\},\end{split}$$
(72)
$$R_{p}^{B}=\rho_{p}^{r}=I(Y_{p};X_{2}|X_{1}WQ)+\sigma_{p}^{*}.$$
(73)
•
Point C:
$$R_{s}^{C}=2\rho_{sp}^{r}-\rho_{s2}^{r},\\
$$
(74)
$$R_{p}^{C}=\rho_{s2}^{r}-\rho_{sp}^{r}.$$
(75)
•
Point D:
$$R_{s}^{D}=\rho_{2p}^{r}-\rho_{sp}^{r},\\
$$
(76)
$$R_{p}^{D}=2\rho_{sp}^{r}-\rho_{sp}^{r}.$$
(77)
•
Point E:
$$R_{s}^{E}=I(Y_{s};U|WX_{1}Q)+\sigma_{s}^{*},$$
(78)
$$\begin{split}\displaystyle R_{p}^{E}=&\displaystyle I(Y_{p};X_{2}|WX_{1}Q)-[%
\sigma_{s}^{*}-I(Y_{p};W|X_{1}Q)]^{+}\\
&\displaystyle+\min\{I(Y_{s};X_{1}|WQ),I(Y_{s};WX_{1}|Q)-\sigma_{s}^{*},\\
&\displaystyle I(Y_{p};X_{1}|Q)+\left[I(Y_{p};W|X_{1}Q)-\sigma_{s}^{*}\right]^%
{+},\\
&\displaystyle I(Y_{p};X_{1}|WQ)\}.\end{split}$$
(79)
•
Point F:
$$R_{s}^{rF}=\rho_{s}^{r}=I(Y_{s};U|WX_{1}Q)+\sigma_{s}^{*},$$
(80)
$$R_{p}^{F}=0.$$
(81)
∎
Corollary 4
Any rate tuple $(R_{s},R_{p})$ of the region
$$\mathcal{R}_{1}^{r}=\mbox{closure}\bigcup_{Z\in\mathcal{P}^{*}}\mathcal{R}_{1}%
^{r}(Z)$$
(82)
is achievable.
Constraining the signal of one primary user to be decodable at the secondary receiver might result in a degradation in the achievable primary rate especially when the secondary rate is very small. In general $\mathcal{R}^{o}$ and $\mathcal{R}_{i}^{r}$ do not necessarily include one another. However, there exists a case for which $\mathcal{R}^{o}\subseteq\mathcal{R}_{i}^{r}$. To characterize this case we introduce the following theorem.
Theorem 5
For a given $Z\in\mathcal{P}^{*}$, $\mathcal{R}^{o}(Z)\subseteq\mathcal{R}_{i}^{r}(Z)$ if and only if
$$I(Y_{p};X_{i}|WQ)\leq I(Y_{s};X_{i}|UWQ).$$
(83)
Proof:
Please refer to Appendix B.
∎
Corollary 5
If for all $Z\in\mathcal{P}^{*}$ condition (83) is satisfied, then $\mathcal{R}^{o}\subseteq\mathcal{R}_{i}^{r}$, where $\mathcal{R}_{i}^{r}=\cup_{Z\in\mathcal{P}^{*}}\mathcal{R}_{i}^{r}(Z)$.
Theorem 5 shows that when a primary user encodes its messages at a rate decodable at both receivers, the primary network may achieve the same rate range when none of the signal of its users is decodable at the secondary receiver. Moreover, at every primary rate the secondary rate is enhanced (see Fig.10). Hence, we conclude the following Proposition.
Proposition 1
If for any $Z\in\mathcal{P}^{*}$ condition (83) is satisfied, then allowing the secondary receiver to decode the signal of primary user $i$ at this $Z$ enhances the range of the secondary achievable rates without affecting the range of the achievable primary sum rates.
We call Corollary 5 Primary Decodability Condition (PDC).
III-C Achievable Rate Region for the Channel $C_{B}$
From $C_{RS}$ and $C_{RS}^{p}$ we define
$$\mathcal{R}_{i}(Z)=\mathcal{R}^{o}(Z)\cup\mathcal{R}_{i}^{r}(Z),\quad Z\in%
\mathcal{P}^{*},i\in\{1,2\},$$
(84)
and
$$\mathcal{R}_{i}=\text{closure}\bigcup_{Z\in\mathcal{P}^{*}}\mathcal{R}_{i}(Z),%
\quad i\in\{1,2\}.$$
(85)
Hence, an achievable rate region for the channel $C_{B}$
$$\mathcal{R}=\mathcal{R}_{1}\cup\mathcal{R}_{2},$$
(86)
or equivalently,
$$\mathcal{R}=\mathcal{R}^{o}\cup\mathcal{R}_{1}^{r}\cup\mathcal{R}_{2}^{r}.$$
(87)
Note that, inequalities (16) and (50) used in $\delta^{o}(Z)$ and $\delta_{1}^{r}(Z)$, assuming $i=1$, respectively, to limit the error in decoding the public part of the secondary signal at the primary receiver while the primary signals are decoded successfully. In fact, the primary receiver may not be interested in limiting the probability of such error event. Similarly, inequality (57) in $\delta_{1}^{r}(Z)$ may not be relevant as the secondary receiver is not interested in limiting the probability of error in decoding the primary signal when the two parts of its signal are decoded successfully. However, removing (16) from the definition of $\delta^{o}(Z)$ and (50) and (57) from the definition of $\delta_{1}^{r}(Z)$ does not enhance the achievable rate region $\mathcal{R}$.
To demonstrate this fact, we define $\delta^{\prime}(Z)$ exactly as $\delta(Z)$ but without the constraint of (16), and $\delta_{1}^{\prime r}(Z)$ exactly as $\delta_{1}^{r}(Z)$ but without the constraints (50) and (57). Let $\mathcal{R}^{\prime o}(Z)$ and $\mathcal{R}_{1}^{\prime r}(Z)$ be two sets of rate tuples $(R_{s},R_{p})$ such that $R_{s}=S+T$ and $R_{p}=R_{1}+R_{2}$ and the rate tuple $(R_{1},R_{2},S,T)$ is an element of $\delta^{\prime o}(Z)$ and $\delta_{1}^{\prime r}(Z)$, respectively. Also we define
$$\mathcal{R}_{1}^{\prime}(Z)=\mathcal{R}^{\prime o}(Z)\cup\mathcal{R}_{1}^{%
\prime r}(Z).$$
Theorem 6
If $\mathcal{R}_{1}^{\prime}=\bigcup_{Z\in\mathcal{P}^{*}}\mathcal{R}_{1}^{\prime}%
(Z)$, then $\mathcal{R}_{1}^{\prime}=\mathcal{R}_{1}$.
Proof:
Please refer to Appendix E.
∎
Corollary 6
For
$$\mathcal{R}^{\prime}=\text{closure of }\mathcal{R}^{\prime}_{1}\cup\mathcal{R}%
^{\prime}_{2},$$
then
$$\mathcal{R}^{\prime}=\mathcal{R}.$$
IV Gaussian Channel
In this section we quantify the obtained achievable rate regions in a Gaussian channel model. A memoryless Gaussian channel of the introduced system is defined by a tuple $(\mathcal{X}_{1},\mathcal{X}_{2},\mathcal{X}_{s},\omega,\mathcal{Y}_{p},%
\mathcal{Y}_{s})$ with $\mathcal{X}_{1}=\mathcal{X}_{2}=\mathcal{X}_{s}=\mathcal{Y}_{p}=\mathcal{Y}_{s%
}=\Re$ (the field of real numbers), and a channel probability $\omega$ specified by,
$$\displaystyle y_{p}=\sqrt{g_{1}^{p}}x_{1}+\sqrt{g_{2}^{p}}x_{2}+\sqrt{g_{s}^{p%
}}x_{s}+n_{p},$$
(88)
$$\displaystyle y_{s}=\sqrt{g_{1}^{s}}x_{1}+\sqrt{g_{2}^{s}}x_{2}+\sqrt{g_{s}^{s%
}}x_{s}+n_{s}$$
(89)
for $x_{1}\in\mathcal{X}_{1}$, $x_{2}\in\mathcal{X}_{2}$, $x_{s}\in\mathcal{X}_{s}$, $y_{p}\in{\mathcal{Y}_{p}}$ and $y_{s}\in\mathcal{Y}_{s}$, where $n_{p}$ and $n_{s}$ are independent Gaussian additive noise samples with zero mean and variance $N_{0}$, and $g_{1}^{p}$, $g_{2}^{p}$, $g_{s}^{p}$, $g_{1}^{s}$, $g_{2}^{s}$ and $g_{s}^{s}$ are the channel power gains. Power constraints are imposed on codewords $\textbf{x}_{1}(i)$, $\textbf{x}_{2}(j)$, $\textbf{x}_{s}(k)$ ($i\in{\mathcal{M}_{1}}$, $j\in{\mathcal{M}_{2}}$, $k\in\mathcal{M}_{s}$):
$$\displaystyle\frac{1}{n}\sum_{t=1}^{n}{(x_{1}(i)^{(t)})^{2}}={P_{1}},$$
(90)
$$\displaystyle\frac{1}{n}\sum_{t=1}^{n}{(x_{2}(j)^{(t)})^{2}}={P_{2}},$$
(91)
$$\displaystyle\frac{1}{n}\sum_{t=1}^{n}{(x_{s}(k)^{(t)})^{2}}={P_{s}}.$$
(92)
For computation, we define a subclass $\mathcal{G}(P_{1},P_{2},P_{s})$ of $\mathcal{P}^{*}$ as follows: $Z=\phi UWX_{1}X_{2}X_{s}Y_{p}Y_{s}\in{\mathcal{G}(P_{1},P_{2},P_{s})}$ if and only if $Z\in{\mathcal{P}^{*}}$, $\sigma^{2}(X_{1})={P_{1}}$, $\sigma^{2}(X_{2})={P_{2}}$ and $\sigma^{2}(X_{s})={P_{s}}$ with $X_{1}$, $X_{2}$, $U$ and $W$ are zero mean Gaussian and $X_{s}=U+W$.
Hence, we have the following rate regions achievable:
$$\displaystyle\mathcal{R}^{o}_{g}$$
$$\displaystyle=$$
$$\displaystyle\text{closure of }\bigcup_{Z\in\mathcal{G}(P_{1},P_{2},P_{s})}%
\mathcal{R}^{o}(Z),$$
(93)
$$\displaystyle\mathcal{R}^{r}_{ig}$$
$$\displaystyle=$$
$$\displaystyle\text{closure of }\bigcup_{Z\in\mathcal{G}(P_{1},P_{2},P_{s})}%
\mathcal{R}^{r}_{i}(Z),i\in\{1,2\},$$
(94)
$$\displaystyle\mathcal{R}_{ig}$$
$$\displaystyle=$$
$$\displaystyle\text{closure of }\bigcup_{Z\in\mathcal{G}(P_{1},P_{2},P_{s})}%
\mathcal{R}_{i}(Z),i\in\{1,2\},$$
(95)
$$\displaystyle\mathcal{R}_{g}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{R}^{o}_{g}\bigcup\left(\cup_{i\in\{1,2\}}\mathcal{R}^{r}%
_{ig}\right)=\mathcal{R}_{1g}\bigcup\mathcal{R}_{2g}.$$
(96)
Assume the secondary user splits its power into $\lambda P_{s}$ and $\bar{\lambda}P_{s}$ such that $0\leq\lambda\leq 1$ and $\lambda+\bar{\lambda}=1$. The part of secondary signal decodable at the primary and secondary receivers is encoded with power $\bar{\lambda}P_{s}$ where the other part is encoded with power $\lambda P_{s}$. Let $\tau(x)=0.5\log_{2}(1+x)$, the relevant quantities in Theorems 2 and 4 will be given by:
$$\displaystyle I(Y_{p};X_{1}X_{2}|W)=\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_%
{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};X_{1}X_{2})=\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2%
}}{g_{s}^{p}P_{s}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};X_{2}|WX_{1})=\tau\left(\frac{g_{2}^{p}P_{2}}{g_{s}^{p}%
\lambda P_{s}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};X_{1}|W)=\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}%
\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};W|X_{1}X_{2})=\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s%
}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};W|X_{1})=\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_%
{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};WX_{1})=\tau\left(\frac{g_{1}^{p}P_{1}+g_{s}^{p}\bar{%
\lambda}P_{s}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{p};W)=\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p%
}\lambda P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}+N_{0}}\right)$$
$$\displaystyle I(Y_{p};X_{1})=\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}P_{s}+g_%
{2}^{p}P_{2}+N_{0}}\right);$$
$$\displaystyle I(Y_{s};U|WX_{1})=\tau\left(\frac{g_{s}^{s}\lambda P_{s}}{g_{2}^%
{s}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{s};U|W)=\tau\left(\frac{g_{s}^{s}\lambda P_{s}}{g_{1}^{s}P_%
{1}+g_{2}^{s}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{s};W|X_{1})=\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_%
{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{s};WX_{1})=\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}+g_{1%
}^{s}P_{1}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{s};W)=\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s%
}\lambda P_{s}+g_{1}^{s}P_{1}+g_{2}^{s}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{s};X_{1}|W)=\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}%
\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right),$$
$$\displaystyle I(Y_{s};X_{1}|UW)=\tau\left(\frac{g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}%
+N_{0}}\right).$$
IV-A Performance of Rate-Splitting
In this subsection we study the effect of rate-splitting by the secondary link on the achievable rate regions $\mathcal{R}^{o}_{g}$ and $\mathcal{R}^{r}_{ig}$, $i\in\{1,2\}$ and hence $\mathcal{R}_{ig}$. For each region there exists a case for which no rate-splitting determines the overall region, i.e., each achievable rate region is obtained at $\lambda=0$ or $\lambda=1$. We say that rate-splitting does not affect an achievable rate region $\mathcal{A}$ if $\mathcal{A}(Z)$ coincides on $\mathcal{A}$ at $\lambda=0$ or $\lambda=1$, $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$, where $\mathcal{A}=\bigcup_{Z\in\mathcal{G}(P_{1},P_{2},P_{s})}\mathcal{A}(Z)$, meaning that either decoding the whole secondary signal at the primary receiver or not decoding it at all determines $\mathcal{A}$.
IV-A1 For $\mathcal{R}_{g}^{o}$
The region $\mathcal{R}^{o}_{g}$ is obtained when the secondary receiver is assumed to treat the primary interference as noise. The following theorem determines the effect of rate-splitting on $\mathcal{R}^{o}_{g}$.
Theorem 7
For $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$, an achievable rate region $\mathcal{R}^{o}(Z)$ coincides on $\mathcal{R}^{o}_{g}$ if and only if $\lambda=0$ and
$$I(Y_{s};W)\leq{I(Y_{p};W|X_{1}X_{2})}$$
(97)
or equivalently,
$$g_{s}^{s}N_{0}\leq g_{s}^{p}(g_{1}^{s}P_{1}+g_{2}^{s}P_{2}+N_{0}).$$
(98)
Proof:
Please refer to Appendix C.
∎
Theorem 7 shows that rate-splitting does not affect the achievable rate region $\mathcal{R}^{o}_{g}$ when inequality (98) is satisfied. Hence, a primary receiver decoding all the secondary signal is preferable at this case. Fig. 5a depicts this case for different values of $\lambda$. It is clear that $\mathcal{R}^{o}(Z)$ at smaller $\lambda$ contains $\mathcal{R}^{o}(Z)$ at larger $\lambda$. This figure was obtained at $g_{1}^{p}=2.5664$, $g_{2}^{p}=3.7653$, $g_{1}^{s}=0.1812$, $g_{2}^{s}=0.1784$, $g_{s}^{p}=2.3620$ and $g_{s}^{s}=8.6065$, and at the following power setup. The noise variance $N_{0}=1$ unit power and $\frac{P_{1}}{N_{0}}=\frac{P_{2}}{N_{0}}=\operatorname{SNR}_{p}=10$ dB and $\frac{P_{s}}{N_{0}}=\operatorname{SNR}_{s}=10$ dB. Note that, in this case the maximum secondary throughput does not depend on $\lambda$, so the best performance from the primary rate point of view is to decode all the secondary signal by setting $\lambda=0$.
Moreover, when inequality (98) is not satisfied, rate-splitting affects $\mathcal{R}^{o}_{g}$ as for any two different values of $\lambda$ the corresponding $\mathcal{R}^{o}(Z)$s do not contain one another. Hence, $\mathcal{R}^{o}_{g}$ is obtained by varying $\lambda$ from $0$ to $1$. Fig. 5b represents the case when (98) is not satisfied for the following parameters. $g_{1}^{p}=1.5066$, $g_{2}^{p}=0.8290$, $g_{1}^{s}=0.1902$, $g_{2}^{s}=0.0122$, $g_{s}^{p}=1.1953$ and $g_{s}^{s}=10.3229$ with the same power setup of Fig. 5a.
Also, it is shown in [9] that when (98) is not satisfied, then the sum throughput of the whole network, i.e., $R_{s}+R_{p}$ increases with $\lambda$. That is, as $\lambda$ increases the primary sum rate decreases but the secondary rate gains an increase larger than the decrease in rate encountered by the primary network. Fig. 6 depicts $R_{s}+R_{p}$ for the same simulation parameters of Fig. 5b. It is clear that the increase in the total sum rate, $R_{s}+R_{p}$, is accompanied by a decrease in the sum primary rate $R_{p}$. Hence, the sum primary rate has to be protected above a minimum limit.
IV-A2 For $\mathcal{R}^{r}_{ig}$, $i\in\{1,2\}$
The region $\mathcal{R}^{r}_{ig}$ is obtained when the secondary receiver can decode the signal of primary user $i$. Rate-splitting effect on this region is determined in the following theorem.
Theorem 8
For $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$ and $i\in\{1,2\}$, an achievable rate region $\mathcal{R}^{r}_{i}(Z)$ coincides on $\mathcal{R}^{r}_{ig}$ if and only if $\lambda=0$ and
$$I(Y_{s};W|X_{i})\leq{I(Y_{p};W|X_{1}X_{2})}$$
(99)
or equivalently,
$$g_{s}^{s}N_{0}\leq g_{s}^{p}(g_{j}^{s}P_{j}+N_{0}),\quad j\in\{1,2\},j\neq i.$$
(100)
Proof:
Please refer to Appendix D
∎
Hence, if inequality (100) is satisfied, $\mathcal{R}^{r}_{ig}$ is obtained without rate-splitting, specifically, when $\lambda=0$.
Fig. 7 shows the performance of rate-splitting under same power setup used with Fig. 5, where it is assumed that the secondary receiver can decode the signal of primary user $1$. In Fig. 7a the achievable rate region $\mathcal{R}^{r}_{1g}$ coincides on $\mathcal{R}^{r}_{1}(Z)$ when inequality (100) is satisfied. The parameters for this scenario are $g_{1}^{p}=5.5303$, $g_{2}^{p}=4.2865$, $g_{1}^{s}=0.6542$, $g_{2}^{s}=0.8121$, $g_{s}^{p}=3.9334$ and $g_{s}^{s}=8.1575$.
In Fig. 7b the opposite scenario is considered where inequality (100) is not satisfied. It is obvious that the overall rate region $\mathcal{R}^{r}_{1g}$ is obtained by varying $\lambda$ from $0$ to $1$ as a consequence of the fact that rate regions corresponding to different values of $\lambda$ do not include one another if inequality (100) is not satisfied. The channel gains for Fig. 7b are $g_{1}^{p}=9.566$, $g_{2}^{p}=14.5045$, $g_{1}^{s}=0.0808$, $g_{2}^{s}=0.2894$, $g_{s}^{p}=0.7032$ and $g_{s}^{s}=16.6226$.
Consequently, the achievable rate region $\mathcal{R}_{ig}$ coincides on $\mathcal{R}_{ig}(Z)$ at $\lambda=0$ if and only if (100) is satisfied.
IV-B On Decoding One Primary Signal
In Subsection III-B we introduce an achievable rate-region for the DMC case assuming that the signal of one primary transmitters has to be reliably decoded by the secondary receiver. Although this may impose a constraint on the range of achievable sum rates by the primary network, we showed in Theorem 5 and Corollary 5 that there exists a condition for which this constraint only enhances the achievable rates for the secondary link without degrading the range of achievable rates by the primary network. This condition is called PDC. When applying this condition to the given Gaussian channel the PDC would be: If for all $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$ $I(Y_{p};X_{i}|W)\leq I(Y_{s};X_{i}|UW)$ then $\mathcal{R}^{o}_{g}\subseteq\mathcal{R}^{r}_{ig}$. Equivalently, the following inequality must hold,
$$\displaystyle\tau\left(\frac{g_{i}^{p}P_{i}}{g_{s}^{p}\lambda P_{s}+g_{j}^{p}P%
_{j}+N_{0}}\right)\leq\tau\left(\frac{g_{i}^{s}P_{i}}{g_{j}^{s}P_{j}+N_{0}}%
\right),\\
\displaystyle\forall\lambda:0\leq\lambda\leq 1,\quad j\neq i,\quad i,j\in\{1,2\}.$$
(101)
But since $I(Ys;X_{i}|UW)$ does not depend on $\lambda$, then a necessary and sufficient condition to have (101) satisfied is
$$\frac{g_{i}^{p}}{g_{j}^{p}P_{j}+N_{0}}\leq\frac{g_{i}^{s}}{g_{j}^{s}P_{j}+N_{0%
}},\quad j\neq i,\quad i,j\in\{1,2\}.$$
(102)
We call inequality (102) primary decodability condition for Gaussian channel (PDCG).
Fig. 8 shows a scenario for which three rate regions are obtained: $\mathcal{R}^{o}_{g}$, $\mathcal{R}^{r}_{1g}$ and $\mathcal{R}^{r}_{2g}$. It is clear that $\mathcal{R}^{o}_{g}\subseteq\mathcal{R}^{r}_{1g}$ meaning that primary user $1$ satisfies the PDCG described in (102), whereas primary user $2$ does not. By decoding the signal of primary user $1$ at the secondary receiver, the range of achievable primary rates in $\mathcal{R}^{o}_{g}$ remains the same for $\mathcal{R}^{r}_{1g}$ while the secondary link can achieve higher rate at a given primary rate in $\mathcal{R}^{r}_{1g}$ than in $\mathcal{R}^{o}_{g}$. The power setup used to produce this figure is the same as that of Fig. 5 and the channel gains are $g_{1}^{p}=0.3413$, $g_{2}^{p}=10.2047$, $g_{1}^{s}=0.2821$, $g_{2}^{s}=0.3782$, $g_{s}^{p}=0.2495$ and $g_{s}^{s}=6.3337$.
Note that, a primary user that satisfies PDCG does not always exist, so we evaluate the probability of PDCG as the probability of finding at least one primary user satisfying (102). We assume $N_{0}=1$ unit power and $g_{1}^{s}$ and $g_{2}^{s}$ are i.i.d. exponentially distributed with mean $\mu_{s}$, whereas $g_{1}^{p}$ and $g_{2}^{p}$ are i.i.d. exponentially distributed with mean $\mu_{p}$, where $g_{1}^{s}$, $g_{2}^{s}$, $g_{1}^{p}$ and $g_{2}^{p}$ are mutually independent. A closed form formula for the probability of PDCG is difficult to obtain, so we evaluate it numerically by generating $10^{7}$ different values for each channel gain element and calculating the average number of times at which neither primary user satisfies (102) at a given $P_{1}$ and $P_{2}$, then by subtracting it from $1$ we get a numerical estimate for the probability of PDCG. A simulation has been done in which we assume that $\frac{P_{1}}{N_{0}}=\frac{P_{2}}{N_{0}}=\operatorname{SNR}_{p}$. We vary $\operatorname{SNR}_{p}$ and evaluate the corresponding probability of PDCG. This simulation is done for the following pairs of $(\mu_{p},\mu_{s})$: $(1,1)$, $(1,5)$, $(5,1)$ and $(5,5)$. The result is shown in Fig. 9, where it is obvious that the probability of PDCG increases with $\operatorname{SNR}_{p}$, and that the increase in $\mu_{s}$ yields more increase in probability of PDCG.
V Conclusion
In this work we established an achievable rate region for a primary multiple access network coexisting with a secondary link that comprises one transmitter and a corresponding receiver. The achievable rate regions are obtained for the sum primary rate versus the secondary rate. We first considered DMC where the secondary link employs rate-splitting, and established two types of achievable rate regions: one type is when the secondary receiver cannot decode any of the primary signals, whereas the second is when the secondary is able to decode the signal of only one primary transmitter. The overall achievable rate region is the union of those two types of regions. Moreover, we showed that there exists a case for which allowing the secondary receiver to decode a primary signal results in an achievable rate region that includes the achievable rate region obtained when the secondary receiver does not decode the primary signal. Then, we investigated the performance of rate-splitting in the Gaussian channel where it was found that rate-splitting by the secondary user is useless when the channel between the secondary transmitter and the primary receiver supports larger rate than the channel between the two secondary nodes. Furthermore, on decoding the signal of a primary transmitter at the secondary receiver, a necessary and sufficient condition has been provided to allow the secondary user decode the primary signal without reducing the range of achievable primary sum rates but only increases the range of achievable secondary rates. Finally, we showed numerically that the probability of finding at least one primary user that satisfies this condition increases with the signal to noise ratio of the primary users.
Appendix A Proof of Theorem 1
It is sufficient to show that there exists at least one code for which if the rate tuple $(R_{1},R_{2},S,T)$ satisfies (14)-(23) then the rate tuple is achievable. We use the following random code.
A-A Random Code Generation
A random code $\mathcal{C}$ is generated as follows. Let $\textbf{q}=(q^{(1)},\cdots,q^{(n)})$ be a random i.i.d sequence of $\mathcal{Q}^{n}$, $\textbf{u}_{k}=(u_{k}^{(1)},\cdots,u_{k}^{(n)})$, $k\in\mathcal{L}_{s}$ a sequence of random variables of $\mathcal{U}^{n}$ that are i.i.d given q. Moreover, $\textbf{u}_{k}$ and $\textbf{u}_{k^{\prime}}$ are independent $\forall k\neq k^{\prime}$, $k,k^{\prime}\in\mathcal{L}_{s}$. Similarly, generate $\textbf{w}_{l}$, $l\in\mathcal{N}_{s}$, $\textbf{x}_{1i}$, $i\in\mathcal{M}_{1}$ and $\textbf{x}_{2j}$, $j\in\mathcal{M}_{2}$.
A-B Encoding
For primary user $1$ to send a message $i\in\mathcal{M}_{1}$, it sends $\textbf{x}_{1i}$. Similarly, for primary user $2$ to send a message $j\in\mathcal{M}_{2}$, it sends $\textbf{x}_{2j}$. For the secondary user to send a message $kl\in\mathcal{L}_{s}\times\mathcal{N}_{s}$, it sends $f^{n}(\textbf{u}_{k}\textbf{w}_{l}|\textbf{q})=\left(f^{(1)}(u^{(1)}_{k}w^{(1)%
}_{l}|q^{(1)}),\cdots,f^{(n)}(u^{(n)}_{k}w^{(n)}_{l}|q^{(n)})\right)$, where q is known at the transmitters.
A-C Decoding: Jointly-Typical Decoding
We use the concept of jointly typical sequences and the properties of typical sets introduced in Chapter 15 of [15] to implement the decoding functions. Let $A_{\epsilon}^{(n)}$ denote the set of typical $(\textbf{q},\textbf{x}_{1},\textbf{x}_{2},\textbf{w}_{l},\textbf{y}_{p})$ sequences, then the primary receiver decides $ijl$ if $(\textbf{q},\textbf{x}_{1i},\textbf{x}_{2j},\textbf{w}_{l},\textbf{y}_{p})\in A%
_{\epsilon}^{(n)}$. Also, for $B_{\epsilon}^{(n)}$ is the set of typical $(\textbf{q},\textbf{u},\textbf{w},\textbf{y}_{s})$ sequences, the secondary receiver decides $kl$ if $(\textbf{q},\textbf{u}_{k},\textbf{w}_{l},\textbf{y}_{s})\in B_{\epsilon}^{(n)}$.
A-D Probability of Error Analysis
By the symmetry of the random code generation, the conditional probability of error does not depend on the transmitted messages. Hence, the conditional probability of error is the same as the average probability of error. So, let $ijkl=1111$ are sent. An error occurs if the transmitted codewords are not typical with the received sequences.
A-D1 For the Primary Receiver
Let the event
$$E_{p}(ijl)=\left\{(\textbf{q},\textbf{x}_{1i},\textbf{x}_{2j},\textbf{w}_{l},%
\textbf{y}_{p})\in A_{\epsilon}^{(n)}\right\},$$
hence the probability of error averaged over the random code $\mathcal{C}$ is
$$\bar{P}e_{p}^{o}=P\left(E_{p}^{c}(111)\bigcup\cup_{ijl\neq 111}E_{p}(ijl)%
\right),$$
where $E_{p}^{c}(111)$ denotes the complement of $E_{p}(111)$. Using union bound we have
$$\begin{split}\displaystyle\bar{P}e_{p}^{o}&\displaystyle\leq P\left(E_{p}^{c}(%
111)\right)+P\left(\cup_{ijl\neq 111}E_{p}(ijl)\right)\\
&\displaystyle\leq P\left(E_{p}^{c}(111)\right)+(M_{1}-1)P(E_{p}(211))\\
&\displaystyle\quad+(M_{2}-1)P(E_{p}(121))+(N_{s}-1)P(E_{p}(112))\\
&\displaystyle\quad+(M_{1}-1)(M_{2}-1)P(E_{p}(221))\\
&\displaystyle\quad+(M_{1}-1)(N_{s}-1)P(E_{p}(212))\\
&\displaystyle\quad+(M_{2}-1)(N_{s}-1)P(E_{p}(122))\\
&\displaystyle\quad+(M_{1}-1)(M_{2}-1)(N_{s}-1)P(E_{p}(222)).\end{split}$$
From the properties of jointly typical sequences [15], $P(E_{p}^{c}(111))\rightarrow\epsilon$ as $n\rightarrow\infty$, and
$$\begin{split}\displaystyle P(E_{p}(211))&\displaystyle=2^{-n(H(X_{1}|Q)-H(X_{1%
}|X_{2}WY_{p}Q))+6\epsilon}\\
&\displaystyle=2^{-n(I(X_{1};X_{2}WY_{p}|Q))+6\epsilon}\\
&\displaystyle=2^{-n(I(Y_{p};X_{1}|WX_{2}Q))+6\epsilon},\end{split}$$
where the last equality holds from the assumption that $X_{1}$, $X_{2}$, $U$ and $W$ are independent and conditionally independent given $Q$. Similarly for other $E_{p}(ijl\neq{111})$ and applying Equations (6)-(9) we get
$$\begin{split}\displaystyle\bar{P}e_{p}^{o}&\displaystyle\leq 2^{-n(I(Y_{p};X_{%
1}|WX_{2}Q)-R_{1}+\eta-6\epsilon)}\\
&\displaystyle\quad+2^{-n(I(Y_{p};X_{2}|WX_{1}Q)-R_{2}+\eta-6\epsilon)}\\
&\displaystyle\quad+2^{-n(I(Y_{p};W|X_{1}X_{2}Q)-T+\eta-6\epsilon)}\\
&\displaystyle\quad+2^{-n(I(Y_{p};X_{1}X_{2}|WQ)-(R_{1}+R_{2})+\eta-6\epsilon)%
}\\
&\displaystyle\quad+2^{-n(I(Y_{p};WX_{1}|X_{2}Q)-(T+R_{1})+\eta-6\epsilon)}\\
&\displaystyle\quad+2^{-n(I(Y_{p};WX_{2}|X_{1}Q)-(T+R_{2})+\eta-6\epsilon)}\\
&\displaystyle\quad+2^{-n(I(Y_{p};X_{1}X_{2}W|Q)-(T+R_{1}+R_{2})+\eta-6%
\epsilon)}.\end{split}$$
Thus if (14)-(20) are satisfied, $\bar{P}e_{p}^{o}\rightarrow\epsilon$ as $n\rightarrow\infty$.
A-D2 For the Secondary Receiver
Let the event
$$E_{s}(kl)=\left\{(\textbf{q},\textbf{u}_{k},\textbf{w}_{l},\textbf{y}_{s})\in B%
_{\epsilon}^{(n)}\right\}$$
hence the probability of decoding error averaged over the random code $\mathcal{C}$ is
$$\bar{P}e_{s}^{o}=P\left(E_{s}^{c}(11)\bigcup\cup_{kl\neq 11}E_{p}(kl)\right),$$
where $E_{s}^{c}(11)$ denotes the complement of $E_{s}(11)$. Using union bound we have
$$\begin{split}\displaystyle\bar{P}e_{s}^{o}\leq&\displaystyle P(E_{s}^{c}(11))+%
(L_{s}-1)P(E_{s}(21))\\
&\displaystyle+(N_{s}-1)P(E_{s}(12))\\
&\displaystyle+(L_{s}-1)(N_{s}-1)P(E_{s}(22)).\end{split}$$
Since $P(E_{s}^{c}(11))\rightarrow\epsilon$ as $n\rightarrow\infty$, then
$$\begin{split}\displaystyle\bar{P}e_{s}^{o}\leq&\displaystyle 2^{-n(I(Y_{s};U|%
WQ)-S+\eta-6\epsilon)}\\
&\displaystyle+2^{-n(I(Y_{s};W|UQ)-T+\eta-6\epsilon)}\\
&\displaystyle+2^{-n(I(Y_{s};UW|Q)-(S+T)+\eta-6\epsilon)}\end{split}$$
So, if (21)-(23) are satisfied, $\bar{P}e_{s}^{o}\rightarrow\epsilon$ as $n\rightarrow\infty$.
This concludes the proof.
Appendix B Proof of Theorem 5
B-A Sufficiency Part
Suppose (83) is satisfied, we use Fig. 10 to prove that $\mathcal{R}^{o}(Z)\subseteq\mathcal{R}_{i}^{r}(Z)$. It is sufficient to show that $R_{p}^{A^{o}}=R_{p}^{A^{r}}$, $R_{s}^{B^{o}}\leq R_{s}^{B^{r}}$, $R_{s}^{D^{o}}\leq R_{s}^{F^{r}}$ and that lines $2R_{s}+R_{p}=\rho_{2p}^{r}$ and $R_{s}+R_{p}=\rho_{sp}^{o}$ intersect at a point $(R_{s}^{*},R_{p}^{*})$ for which $R_{s}^{*}\geq R_{s}^{D^{o}}$, i.e., the intersection between the two lines is outside $\mathcal{R}^{o}(Z)$. Consider the primary user whose signal is not decodable at the secondary receiver is indexed by $j$, $j\in\{1,2\}$ and $i\neq j$.
B-A1 Proof of $R_{p}^{A^{o}}=R_{p}^{A^{r}}$
From the analysis of the channels $C_{RS}$ and $C_{RS}^{p}$ in Section III we have
$$\displaystyle R_{p}^{A^{o}}=$$
$$\displaystyle I(Y_{p};X_{1}X_{2}|WQ),$$
$$\displaystyle R_{p}^{A^{r}}=$$
$$\displaystyle I(Y_{p};X_{j}|WX_{i}Q)+\sigma_{p}^{*}.$$
From (83), $\sigma_{p}^{*}=I(Y_{p},X_{i}|WQ)$. Therefore,
$$R_{p}^{A^{r}}=I(Y_{p};X_{1}X_{2}|WQ)=R_{p}^{A^{o}}.$$
B-A2 Proof of $R_{s}^{B^{o}}\leq R_{s}^{B^{r}}$
From the proof of Theorem 2
$$R_{s}^{B^{o}}=I(Y_{s};U|WQ)+\min\{\overbrace{I(Y_{p};W|Q)}^{o_{1}},\overbrace{%
I(Y_{s};W|Q)}^{o_{2}}\},$$
(103)
and from the proof of Theorem 4
$$\begin{split}\displaystyle R_{s}^{B^{r}}=&\displaystyle I(Y_{s};U|WX_{i}Q)-[I(%
Y_{p};X_{i}|WQ)\\
&\displaystyle-I(Y_{s};X_{i}|WQ)]^{+}+\min\{I(Y_{p};W|Q),\\
&\displaystyle I(Y_{s};W|Q)+[I(Y_{s};X_{i}|WQ)-I(Y_{p};X_{i}|WQ)]^{+},\\
&\displaystyle I(Y_{s};W|X_{i}Q)\}.\end{split}$$
If $I(Y_{p};X_{i}|WQ)\leq I(Y_{s};X_{i}|WQ)$
$$\begin{split}\displaystyle R_{s}^{B^{r}}=&\displaystyle I(Y_{s};U|WX_{i}Q)+%
\min\{\overbrace{I(Y_{p};W|Q)}^{\nu_{1}},\\
&\displaystyle\overbrace{I(Y_{s};W|Q)+I(Y_{s};X_{i}|WQ)-I(Y_{p};X_{i}|WQ)}^{%
\nu_{2}},\\
&\displaystyle\overbrace{I(Y_{s};W|X_{i}Q)}^{\nu_{3}}\}.\end{split}$$
(104)
Note that, $\nu_{1}=o_{1}$.
•
If $o_{1}\leq o_{2}$ in (103)
$$R_{s}^{B^{o}}=I(Y_{s};U|WQ)+I(Y_{p};W|Q),$$
$$\begin{split}\displaystyle R_{s}^{B^{r}}&\displaystyle=I(Y_{s};U|WX_{i}Q)+I(Yp%
;W|Q)\\
&\displaystyle\geq R_{s}^{B^{o}}.\end{split}$$
•
If $o_{2}\leq o_{1}$ in (103)
$$\begin{split}\displaystyle R_{s}^{B^{o}}=&\displaystyle I(Y_{s};U|WQ)+I(Y_{s};%
W|Q)\\
\displaystyle=&\displaystyle I(Y_{s};UW|Q).\end{split}$$
When $\nu_{1}=\min\{\nu_{1},\nu_{2},\nu_{3}\}$ in (104), then
$$\begin{split}\displaystyle R_{s}^{B^{r}}&\displaystyle=I(Y_{s};U|WX_{i}Q)+%
\overbrace{I(Y_{p};W|Q)}^{\geq o_{2}}\\
&\displaystyle\geq R_{s}^{B^{o}}.\end{split}$$
When $\nu_{2}=\min\{\nu_{1},\nu_{2},\nu_{3}\}$ in (104), then
$$\begin{split}\displaystyle R_{s}^{B^{r}}&\displaystyle=I(Y_{s};U|WX_{i}Q)+I(Y_%
{s};W|Q)+I(Y_{s};X_{i}|WQ)\\
&\displaystyle\quad-I(Y_{p};X_{i}|WQ)\\
&\displaystyle\geq I(Y_{s};U|WX_{i}Q)+I(Y_{s};W|Q)\\
&\displaystyle\geq R_{s}^{B^{o}}.\end{split}$$
When $\nu_{3}=\min\{\nu_{1},\nu_{2},\nu_{3}\}$ in (104), then
$$\begin{split}\displaystyle R_{s}^{B^{r}}&\displaystyle=I(Y_{s};U|WX_{i}Q)+I(Y_%
{s};W|X_{i}Q)\\
&\displaystyle=I(Y_{s};UW|X_{i}Q)\\
&\displaystyle\geq R_{s}^{B^{o}}.\end{split}$$
If $I(Y_{s};X_{i}|WQ)\leq I(Y_{p};X_{i}|WQ)$
$$\begin{split}\displaystyle R_{s}^{B^{r}}=&\displaystyle I(Y_{s};U|WX_{i}Q)+I(Y%
_{s};X_{i}|WQ)-I(Y_{p};X_{i}|WQ)\\
&\displaystyle+\min\{\overbrace{I(Y_{p};W|Q)}^{\nu_{4}},\overbrace{I(Y_{s};W|Q%
)}^{\nu_{5}}\}.\end{split}$$
(105)
Note that, $o_{1}=\nu_{4}$ and $o_{2}=\nu_{5}$.
•
If $o_{1}\leq o_{2}$ in (103)
$$\begin{split}\displaystyle R_{s}^{B^{o}}=I(Y_{s};U|WQ)+I(Y_{p};W|Q),\end{split}$$
$$\begin{split}\displaystyle R_{s}^{B^{r}}&\displaystyle=I(Y_{s};UX_{i}|WQ)-I(Y_%
{p};X_{i}|WQ)+I(Y_{p};W|Q)\\
&\displaystyle=I(Y_{s};U|WQ)+I(Y_{p};W|Q)\\
&\displaystyle\quad+\overbrace{I(Y_{s};X_{i}|UWQ)-I(Y_{p};X_{i}|WQ)}^{\geq 0%
\mbox{ from \eqref{eq:cond}}}\\
&\displaystyle\geq R_{s}^{B^{o}}.\end{split}$$
•
If $o_{2}\leq o_{1}$ in (103)
Proof follows exactly as the case of $o_{1}\leq o_{2}$.
B-A3 Proof of $R_{s}^{F^{r}}\geq R_{s}^{D^{o}}$
$$\begin{split}\displaystyle R_{s}^{F^{r}}=&\displaystyle I(Y_{s};U|WX_{i}Q)+%
\min\{I(Y_{s};W|X_{i}Q),\\
&\displaystyle I(Y_{p};W|X_{1}X_{2}Q)\}.\end{split}$$
$$\begin{split}\displaystyle R_{s}^{D^{o}}=&\displaystyle I(Y_{s};U|WQ)+\min\{I(%
Y_{s};W|Q),\\
&\displaystyle I(Y_{p};W|X_{1}X_{2}Q)\}.\end{split}$$
It is obvious that each term in $R_{s}^{F^{r}}$ is greater than or equal to its corresponding term in $R_{s}^{D^{o}}$. Hence, $R_{s}^{F^{r}}\geq R_{s}^{D^{o}}$.
B-A4 Proof of the intersection point between the two lines $2R_{s}+R_{p}=\rho_{2p}^{r}$ and $R_{s}+R_{p}=\rho_{sp}^{o}$ occurs at a point $(R_{s}^{*},R_{p}^{*})$ where $R_{s}^{*}\geq R_{s}^{D^{o}}$
The secondary rate of the intersection point is $R_{s}^{*}=\rho_{2p}^{r}-\rho_{sp}^{o}$. From Theorems 2 and 4
$$R_{s}^{D^{o}}=I(Y_{s};U|WQ)+\sigma^{*},$$
(106)
$$\begin{split}\displaystyle R_{s}^{*}=&\displaystyle 2I(Y_{s};U|WX_{i}Q)+2%
\sigma_{s}^{*}+I(Y_{p};X_{j}|WX_{i}Q)\\
&\displaystyle-[\sigma_{s}^{*}-I(Y_{p};W|X_{i}Q)]^{+}+\min\{I(Y_{s};X_{i}|WQ),%
\\
&\displaystyle I(Y_{s};WX_{i}|Q)-\sigma_{s}^{*},I(Y_{p};X_{i}|Q)+[I(Y_{p};W|X_%
{i}Q)\\
&\displaystyle-\sigma_{s}^{*}]^{+},I(Y_{p};X_{i}|WQ)\}-I(Y_{p};X_{1}X_{2}|WQ)%
\\
&\displaystyle-I(Y_{s};U|WQ)-\min\{I(Y_{s};W|Q),I(Y_{p};W|Q)\}.\end{split}$$
(107)
Hence, it is required to show that $R_{s}^{*}\geq R_{s}^{D^{o}}$
If $\sigma_{s}^{*}=I(Y_{s};W|X_{i}Q)\leq I(Y_{p};W|X_{1}X_{2}Q)$
•
If $I(Y_{s};W|X_{i}Q)\leq I(Y_{p};W|X_{i}Q)$,
from (106) and (107) we have
$$\begin{split}\displaystyle R_{s}^{D^{o}}=I(Y_{s};U|WQ)+I(Y_{s};W|Q)=I(Y_{s};UW%
|Q),\end{split}$$
(108)
$$\begin{split}\displaystyle R_{s}^{*}=&\displaystyle 2I(Y_{s};U|WX_{i}Q)+2I(Y_{%
s};W|X_{i}Q)-I(Y_{s};U|WQ)\\
&\displaystyle+I(Y_{p};X_{j}|WX_{i}Q)+\min\{\overbrace{I(Y_{s};X_{i}|Q)}^{\nu_%
{6}},\\
&\displaystyle\overbrace{I(Y_{p};X_{i}|Q)+I(Y_{p};W|X_{i}Q)-I(Y_{s};W|X_{i}Q)}%
^{\nu_{7}},\\
&\displaystyle\overbrace{I(Y_{p};X_{i}|WQ)}^{\nu_{8}}\}-I(Y_{p};X_{1}X_{2}|WQ)%
-\min\{\nu_{4},\nu_{5}\}.\end{split}$$
(109)
When $\nu_{6}=\min\{\nu_{6},\nu_{7},\nu_{8}\}$ in (109), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+\overbrace{I(Y_{s};UW|Q)}^{=R_{s}^{D^{o}}}\\
&\displaystyle\quad+I(Y_{s};W|X_{i}Q)-\min\{\nu_{4},\nu_{5}\}+I(Y_{p};X_{j}|WQ%
)\\
&\displaystyle\quad+\overbrace{I(Y_{s};X_{i}|UWQ)}^{\geq I(Y_{p};X_{i}|WQ)%
\mbox{ from \eqref{eq:cond}}}-I(Y_{p};X_{1}X_{2}|WQ)\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
When $\nu_{7}=\min\{\nu_{6},\nu_{7},\nu_{8}\}$ in (109), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+\overbrace{I(Y_{s};UW|X_{i}Q)}^{\geq R_{s}^{D^{o}}}\\
&\displaystyle\quad+\nu_{4}-\min\{\nu_{4},\nu_{5}\}\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
When $\nu_{8}=\min\{\nu_{6},\nu_{7},\nu_{8}\}$ in (109), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+\overbrace{I(Y_{s};UW|X_{i}Q)}^{\geq R_{s}^{D^{o}}}\\
&\displaystyle\quad+I(Y_{s};W|X_{i}Q)-\min\{\nu_{4},\nu_{5}\}\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
•
If $I(Y_{s};W|X_{i}Q)\geq I(Y_{p};W|X_{i}Q)$,
$R_{s}^{D^{o}}$ will remain the same as (108) and $R_{s}^{*}$ will be given by
$$\begin{split}\displaystyle R_{s}^{*}=&\displaystyle 2I(Y_{s};U|WX_{i}Q)+2I(Y_{%
s};W|X_{i}Q)-I(Y_{s};U|WQ)\\
&\displaystyle+I(Y_{p};X_{j}|WX_{i}Q)+I(Y_{p};W|X_{i}Q)\\
&\displaystyle-I(Y_{s};W|X_{i}Q)+\min\{\overbrace{I(Y_{s};X_{i}|Q)}^{\nu_{9}},%
\overbrace{I(Y_{p};X_{i}|Q)}^{\nu_{10}}\}\\
&\displaystyle-I(Y_{p};X_{1}X_{2}|WQ)-\min\{\nu_{4},\nu_{5}\}.\end{split}$$
(110)
When $\nu_{9}=\min\{\nu_{9},\nu_{10}\}$ in (110), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+\overbrace{I(Y_{s};UW|Q)}^{=R_{s}^{D^{o}}}\\
&\displaystyle\quad+I(Y_{p};W|X_{i}Q)-\min\{\nu_{4},\nu_{5}\}+I(Y_{s};X_{i}|%
UWQ)\\
&\displaystyle\quad+I(Y_{p};X_{j}|WX_{i}Q)-I(Y_{p};X_{1}X_{2}|WQ)\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
When $\nu_{10}=\min\{\nu_{9},\nu_{10}\}$ in (110), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+\overbrace{I(Y_{s};UW|X_{i}Q)}^{\geq R_{s}^{D^{o}}}\\
&\displaystyle\quad+I(Y_{p};W|X_{i}Q)-\min\{\nu_{4},\nu_{5}\}\\
&\displaystyle\geq R_{s}^{D^{o}}\end{split}$$
If $\sigma_{s}^{*}=I(Y_{p};W|X_{1}X_{2}Q)\leq I(Y_{s};W|X_{i}Q)$
from (106) and (107) we have
$$R_{s}^{D^{o}}=I(Y_{s};U|WQ)+\min\{\overbrace{I(Y_{s};W|Q)}^{o_{2}},\overbrace{%
I(Y_{p};W|X_{1}X_{2}Q)}^{o_{3}}\},$$
(111)
$$\begin{split}\displaystyle R_{s}^{*}=&\displaystyle 2I(Y_{s};U|WX_{i}Q)-I(Y_{s%
};U|WQ)+I(Y_{p};W|X_{i}Q)\\
&\displaystyle+I(Y_{p};X_{j}|WX_{i}Q)+I(Y_{p};W|X_{1}X_{2}Q)\\
&\displaystyle+\min\{\overbrace{I(Y_{p};X_{i}|Q)}^{\nu_{10}},\overbrace{I(Y_{s%
};X_{i}|WQ)}^{\nu_{11}},\\
&\displaystyle\overbrace{I(Y_{s};WX_{i}|Q)-I(Y_{p};W|X_{1}X_{2}Q)}^{\nu_{12}}%
\}-\min\{\nu_{4},\nu_{5}\}\\
&\displaystyle-I(Y_{p};X_{1}X_{2}|WQ)\end{split}$$
(112)
•
If $o_{2}\leq o_{3}$ in (111),
$$R_{s}^{D^{o}}=I(Y_{s};U|WQ)+I(Y_{s};W|Q)=I(Y_{s};UW|Q).$$
When $\nu_{10}=\min\{\nu_{10},\nu_{11},\nu_{12}\}$ in (112), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+\nu_{4}\\
&\displaystyle\quad-\min\{\nu_{4},\nu_{5}\}+\overbrace{I(Y_{s};U|WX_{i}Q)+o_{3%
}}^{\geq R_{s}^{D^{o}}}.\end{split}$$
Since $o_{2}\leq o_{3}$, then $\nu_{11}$ cannot be smaller than $\nu_{12}$.
When $\nu_{12}=\min\{\nu_{10},\nu_{11},\nu_{12}\}$, then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+I(Y_{p};W|X_{i}Q)\\
&\displaystyle\quad-\min\{\nu_{4},\nu_{5}\}+\overbrace{I(Y_{s};UW|Q)}^{=R_{s}^%
{D^{o}}}+I(Y_{s};X_{i}|UWQ)\\
&\displaystyle\quad+I(Y_{p};X_{j}|WX_{i}Q)-I(Y_{p};X_{1}X_{2}|WQ)\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
•
If $o_{2}\geq o_{3}$
$$R_{s}^{D^{o}}=I(Y_{s};U|WQ)+I(Y_{p};W|X_{1}X_{2}Q).$$
When $\nu_{10}=\min\{\nu_{10},\nu_{11},\nu_{12}\}$ in (112), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)\\
&\displaystyle\quad+\overbrace{I(Y_{s};U|WX_{i}Q)+I(Y_{p};W|X_{1}X_{2}Q)}^{%
\geq R_{s}^{D^{o}}}\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
When $\nu_{11}=\min\{\nu_{10},\nu_{11},\nu_{12}\}$ in (112), then
$$\begin{split}\displaystyle R_{s}^{*}&\displaystyle=I(Y_{s};U|WX_{i}Q)-I(Y_{s};%
U|WQ)+I(Y_{s};X_{i}|UWQ)\\
&\displaystyle\quad+I(Y_{p};X_{j}|WX_{i}Q)-I(Y_{p};X_{1}X_{2}|WQ)-\nu_{4}\\
&\displaystyle\quad+I(Y_{p};W|X_{i}Q)+\overbrace{I(Y_{s};U|WQ)+I(Y_{p};W|X_{1}%
X_{2}Q)}^{=R_{s}^{D^{o}}}\\
&\displaystyle\geq R_{s}^{D^{o}}.\end{split}$$
Since $o_{2}\geq o_{3}$ then $\nu_{12}$ cannot be smaller than $\nu_{11}$.
B-B Necessity Part
Suppose $\mathcal{R}^{o}(Z)\subseteq\mathcal{R}_{i}^{r}(Z)$ then $R_{s}^{A^{o}}$ must be not larger than $R_{s}^{A^{r}}$ which necessitates the satisfaction of (83).
This concludes the proof.
Appendix C Proof of Theorem 7
C-A Sufficiency Part
We refer to Fig. 3 to determine the effect of varying $\lambda$ on $\mathcal{R}^{o}(Z)$ where $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$.
•
Point A:
$$R_{p}^{A}=\rho_{p}^{o}=\tau{\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{%
p}\lambda P_{s}+N_{0}}\right)}$$
•
Point D:
$$R_{s}^{D}=\rho_{s}^{o}=\tau{\left(\frac{P_{s}g_{s}^{s}}{g_{1}^{s}P_{1}+g_{2}^{%
s}P_{2}+N_{0}}\right)}$$
•
$R_{s}+R_{p}$:
$$\rho_{sp}^{o}=\tau{\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda
P%
_{s}+N_{0}}\right)}+\tau\left(\frac{g_{s}^{s}P_{s}}{P_{1}g_{1}^{s}+P_{2}g_{2}^%
{s}+N_{0}}\right).$$
It is obvious that if (98) is satisfied, then $\rho_{p}^{o}$, $\rho_{s}^{o}$ and $\rho_{sp}^{o}$ increase as $\lambda$ decreases. Consequently, $\mathcal{R}^{o}(Z)$ at $\lambda=0$ includes all other $\mathcal{R}^{o}(Z)$ obtained at $0<\lambda\leq 1$. Hence, $\mathcal{R}^{o}(Z)$ coincides on $\mathcal{R}^{o}_{g}$ at $\lambda=0$.
C-B Necessity Part
Here we prove that the condition in (98) is necessary for $\mathcal{R}^{o}(Z)$ to coincide on $\mathcal{R}^{o}_{g}$ at $\lambda=0$ and $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$. We do so by showing that, if (98) is not satisfied, then for any two different values of $\lambda$ the corresponding rate regions do not contain one another. Assume that (98) is not satisfied, then by referring to Fig. 3 we have:
•
Point A:
$$R_{p}^{A}=\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s%
}+N_{0}}\right)$$
i.e., the $R_{p}^{A}$ decreases as $\lambda$ increases.
•
Point D:
$$R_{s}^{D}=\tau\left({\frac{g_{s}^{s}\lambda P_{s}}{g_{1}^{s}P_{1}+g_{2}^{s}P_{%
2}+N_{0}}}\right)+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p}%
\lambda P_{s}+N_{0}}\right)$$
then by substituting with $\bar{\lambda}=1-\lambda$ and differentiating $R_{s}^{D}$ with respect to $\lambda$ we get,
$$\frac{\partial{R_{s}^{D}}}{\partial{\lambda}}=\frac{\frac{1}{2\ln(2)}P_{s}(g_{%
s}^{s}N_{0}-g_{s}^{p}(P_{1}g_{1}^{s}+P_{2}g_{2}^{s}+N_{0}))}{(\lambda P_{s}g_{%
s}^{p}+N_{0})(P_{1}g_{1}^{s}+P_{2}g_{2}^{s}+\lambda P_{s}g_{s}^{s}+N_{0})}$$
(113)
and since the condition (98) is not satisfied, the numerator of (113) is always positive, therefore, $R_{s}^{D}$ increases as $\lambda$ increases.
Since $R_{p}^{A}$ decreases and $R_{s}^{D}$ increases as $\lambda$ increases, then for any two different values of $\lambda$ the corresponding rate regions will never contain one another. Hence the overall rate region $\mathcal{R}_{g}^{o}$ does not coincide on a certain $\mathcal{R}^{o}(Z)$ at a certain $\lambda$.
This concludes the proof.
Appendix D Proof of Theorem 8
For the proof, we consider $i=1$, i.e., the secondary user is assumed to be able to decode the signal of primary user $1$.
D-A Sufficiency part
In this part we show that, if inequality (100) is satisfied then $\mathcal{R}_{1g}^{r}$ coincides on $\mathcal{R}^{r}_{1}(Z)$ at $\lambda=0$. We refer to Fig. 4 and determine the effect of varying $\lambda$ on $\mathcal{R}^{r}_{1}(Z)$, $Z\in\mathcal{G}(P_{1},P_{2},P_{s})$ as follows.
D-A1 At Point A
$$\begin{split}\displaystyle R_{p}^{rA}=&\displaystyle\tau\left(\frac{g_{2}^{p}P%
_{2}}{g_{s}^{p}\lambda{P_{s}}+N_{0}}\right)+\min\left\{\tau\left(\frac{g_{1}^{%
s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)\right.,\\
&\displaystyle\left.\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{%
2}^{p}P_{2}+N_{0}}\right)\right\}.\end{split}$$
Therefore, $R_{s}^{rA}$ increases as $\lambda$ decreases.
D-A2 At Point F
$$\begin{split}\displaystyle R_{s}^{rF}=\tau\left(\frac{g_{s}^{s}P_{s}}{g_{2}^{s%
}P_{2}+N_{0}}\right).\end{split}$$
Hence, $R_{s}^{rF}$ does not depend on $\lambda$.
D-A3 $R_{s}^{r}+R_{p}^{r}=\rho_{sp}^{r}$
$$\begin{split}\displaystyle\rho_{sp}^{r}=&\displaystyle\tau\left(\frac{g_{s}^{s%
}\lambda{P_{s}}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{2}^{p}P_{2}}{%
g_{s}^{p}\lambda{P_{s}}+N_{0}}\right)+\min\biggl{\{}\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g_{1}^{p}%
P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)}^{\mu_{1}},%
\overbrace{\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}+g_{1}^{s}P_{1}}{g_{s}^{%
s}\lambda{P_{s}}+g_{2}^{s}P_{2}+N_{0}}\right)}^{\mu_{2}}\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p%
}\lambda{P_{s}}+g_{2}^{p}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{1}^{s}P_{1}}{g%
_{s}^{s}\lambda{P_{s}}g_{2}^{s}+P_{2}+N_{0}}\right)}^{\mu_{3}},\\
&\displaystyle\overbrace{\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda{P_{s%
}}+g_{2}^{p}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g%
_{s}^{s}\lambda{P_{s}}+g_{s}^{s}P_{2}+N_{0}}\right)}^{\mu_{4}}\biggr{\}}\end{split}$$
(114)
When $\mu_{1}=\min\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\}$ in (114)
$$\rho_{sp}^{r}=\tau\left(\frac{g_{s}^{s}\lambda{P_{s}}}{g_{2}^{s}P_{2}+N_{0}}%
\right)+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_%
{2}}{g_{s}^{p}\lambda{P_{s}}+N_{0}}\right).$$
$$\begin{split}\displaystyle\frac{\partial\rho_{sp}^{r}}{\partial\lambda}&%
\displaystyle=-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}N_{0}-g_{s}^{s}%
N_{0})}{\ln 2(g_{s}^{p}\lambda{P_{s}}+N_{0})(g_{s}^{s}\lambda P_{s}+g_{2}^{s}P%
_{2}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Hence, $\rho_{sp}^{r}$ decreases with $\lambda$. Note that, $\bar{\lambda}=1-\lambda$.
When $\mu_{2}=\min\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\}$ in (114)
$$\tau\left(\frac{g_{s}^{s}P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)+%
\tau\left(\frac{g_{2}^{p}P_{2}}{g_{s}^{p}\lambda{P_{s}}+N_{0}}\right),$$
i.e., $\rho_{sp}^{r}$ decreases with $\lambda$.
When $\mu_{3}=\min\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\}$ in (114)
$$\rho_{sp}^{r}=\tau\left(\frac{g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}}{g_{s}^{s}%
P_{2}+N_{0}}\right)+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g_{2}^{p}P_{2}%
}{g_{s}^{p}\lambda P_{s}+N_{0}}\right).$$
$$\begin{split}\displaystyle\frac{\partial\rho_{sp}^{r}}{\partial\lambda}&%
\displaystyle=-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}g_{1}^{s}P_{1}+%
g_{s}^{p}N_{0}-g_{s}^{s}N_{0})}{\ln 2(g_{s}^{p}\lambda{P_{s}}+N_{0})(g_{s}^{s}%
\lambda P_{s}+g_{2}^{s}P_{2}+g_{1}^{s}P_{1}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Thus, $\rho_{sp}$ decreases with $\lambda$.
When $\mu_{4}=\min\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\}$ in (114)
$$\rho_{sp}^{r}=\tau\left(\frac{g_{s}^{s}P_{s}}{g_{s}^{s}P_{2}+N_{0}}\right)+%
\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}%
\right).$$
Therefore, $\rho_{sp}^{r}$ decreases with $\lambda$.
D-A4 $R_{s}^{r}+2R_{p}^{r}=\rho_{s2}^{r}$
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{2}^%
{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+2\sigma_{p}^{*}+\tau\left(\frac%
{g_{s}^{s}\lambda P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle-\left[\sigma_{p}^{*}-\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}%
\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)\right]^{+}+\min\biggl{\{}\\
&\displaystyle\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p}\lambda P_%
{s}+g_{2}^{p}P_{2}+N_{0}}\right),\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g%
_{1}^{p}P_{1}}{g_{s}^{p}\lambda{P_{s}}+g_{2}^{p}P_{2}+N_{0}}\right)-\sigma_{p}%
^{*},\\
&\displaystyle\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P_%
{s}+g_{2}^{s}P_{2}+N_{0}}\right),\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{%
g_{s}^{s}\lambda P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\left[\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}\lambda P_{s}+g_%
{2}^{s}P_{2}+N_{0}}\right)-\sigma_{p}^{*}\right]^{+}\biggr{\}},\\
\displaystyle\sigma_{p}^{*}=&\displaystyle\min\left\{\tau\left(\frac{g_{1}^{p}%
P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right),\tau\left(\frac{g_{%
1}^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)\right\}.\end{split}$$
At $\sigma_{p}^{*}=\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p%
}P_{2}+N_{0}}\right)\leq\tau\left(\frac{g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)$
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{1}^%
{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\tau\left(\frac{%
g_{s}^{s}\lambda P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle-\left[\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_%
{2}^{p}P_{2}+N_{0}}\right)-\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}\lambda P_%
{s}+g_{2}^{s}P_{2}+N_{0}}\right)\right]^{+}\\
&\displaystyle+\min\biggl{\{}\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s%
}^{p}\lambda P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}+N_{0}}\right),\\
&\displaystyle\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P_%
{s}+g_{1}^{s}P_{1}+g_{2}^{s}P_{2}+N_{0}}\right)+\biggl{[}\tau\left(\frac{g_{1}%
^{s}P_{1}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle-\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p%
}P_{2}+N_{0}}\right)\biggr{]}^{+},\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}%
{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)\biggr{\}}.\end{split}$$
•
If $\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}%
\right)\leq\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{%
2}+N_{0}}\right)$
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{1}^%
{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\tau\left(\frac{%
g_{s}^{s}\lambda P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\min\biggl{\{}\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p%
}\lambda P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}+N_{0}}\right)}^{\mu_{5}},%
\overbrace{\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P_{s}%
+g_{2}^{s}P_{2}+N_{0}}\right)}^{\mu_{6}},\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}+g_{1}^{s}%
P_{1}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)-\tau\left(\frac{g_{%
1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)}^{\mu_{7}}%
\biggr{\}}.\end{split}$$
(115)
When $\mu_{5}=\min\{\mu_{5},\mu_{6},\mu_{7}\}$ in (115) we have
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle\tau\left(\frac{g_{s}^{p%
}\bar{\lambda}P_{s}+g_{1}^{p}P_{1}+g_{2}pP_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}%
\right)+\tau\left(\frac{g_{s}^{s}\lambda P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda
P%
_{s}+N_{0}}\right).\end{split}$$
(116)
Note that, the third term in (116) is decreasing with $\lambda$, and the first derivative of the first two terms with respect to $\lambda$ is given by,
$$\begin{split}&\displaystyle-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}N_%
{0}-g_{s}^{s}N_{0})}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0})(g_{s}^{s}\lambda P_{s%
}+g_{2}^{s}P_{2}+N_{0})}\\
&\displaystyle-\frac{0.5g_{s}^{p}P_{s}(g_{2}^{p}P_{2}+g_{1}^{p}P_{1})}{\ln 2(g%
_{s}^{p}\lambda P_{s}+N_{0})(g_{s}^{p}\lambda P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_%
{2}+N_{0})}.\end{split}$$
Since inequality (100) is satisfied for user $1$, then the derivative is negative and consequently $\rho_{s2}^{r}$ is decreasing with $\lambda$.
When $\mu_{6}=\min\{\mu_{5},\mu_{6},\mu_{7}\}$ in (115), we have
$$\rho_{s2}^{r}=2\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda
P%
_{s}+N_{0}}\right)+\tau\left(\frac{g_{s}^{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right),$$
i.e., $\rho_{s2}^{r}$ is decreasing with $\lambda$.
When $\mu_{7}=\min\{\mu_{5},\mu_{6},\mu_{7}\}$ in (115), we have
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{1}^%
{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)-\tau\left(\frac{%
g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{s}P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N%
_{0}}\right).\end{split}$$
Hence, $\rho_{s2}^{r}$ is decreasing with $\lambda$.
•
If $\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}%
\right)\leq\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{%
2}+N_{0}}\right)$
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{1}^%
{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)-\tau\left(\frac{%
g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\min\biggl{\{}\overbrace{\tau\left(\frac{g_{s}^{p}\bar{\lambda}%
P_{s}}{g_{s}^{p}\lambda P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}+N_{0}}\right)}^{%
\mu_{5}},\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s%
}\lambda P_{s}+g_{1}^{s}P_{1}+g_{2}^{s}P_{2}+N_{0}}\right)}^{\mu_{8}}\biggr{\}%
}+\tau\left(\frac{g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}%
\right)\end{split}$$
(117)
When $\mu_{5}=\min\{\mu_{5},\mu_{8}\}$ in (117), then
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle\tau\left(\frac{g_{1}^{p%
}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)-\tau\left(\frac{g_%
{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda
P%
_{s}+N_{0}}\right)+\tau\left(\frac{g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}}{g_{2%
}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p}\lambda P%
_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}+N_{0}}\right).\end{split}$$
(118)
For all values of $0\leq\lambda\leq 1$, the difference between the first two terms in (118) is always positive and decreasing as $\lambda$ increases. The first derivative of the last three terms in (118) with respect to $\lambda$ is given by,
$$\begin{split}&\displaystyle-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}g_%
{1}^{s}P_{1}+g_{s}^{p}N_{0}-g_{s}^{s}N_{0})}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0%
})(g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+g_{1}^{s}P_{1}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Therefore, $\rho_{s2}^{r}$ is decreasing with $\lambda$.
When $\mu_{8}=\min\{\mu_{5},\mu_{8}\}$ in (117), then
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{1}^%
{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)-\tau\left(\frac{%
g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{s}P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N%
_{0}}\right).\end{split}$$
In the above formula, the difference between the first two terms is always positive and decreasing as $\lambda$ increases. The third term does not depend on $\lambda$. Hence, $\rho_{s2}^{r}$ is decreasing with $\lambda$.
At $\sigma_{p}^{*}=\tau\left(\frac{g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)%
\leq\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0%
}}\right)$
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{2}^%
{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\tau\left(\frac{g_{1}^{s}P_{1}}%
{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{s}^{s}\lambda P_{s}+g_{1}^{s}%
P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle+\min\biggl{\{}\overbrace{\tau\left(\frac{g_{s}^{s}\bar{\lambda}%
P_{s}}{g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}+g_{2}^{s}P_{2}+N_{0}}\right)}^{%
\mu_{8}},\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p%
}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)}^{\mu_{9}},\\
&\displaystyle\overbrace{\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g_{1}^{p}%
P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)-\tau\left(\frac{g_{%
1}^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right)}^{\mu_{10}}\biggr{\}}.\end{split}$$
(119)
When $\mu_{8}=\min\{\mu_{8},\mu_{9},\mu_{10}\}$ in (119), we have
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle 2\tau\left(\frac{g_{2}^%
{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\tau\left(\frac{g_{1}^{s}P_{1}}%
{g_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{s}P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s}P_{2}+N%
_{0}}\right).\end{split}$$
That is, $\rho_{s2}^{r}$ is decreasing with $\lambda$.
When $\mu_{9}=\min\{\mu_{8},\mu_{9},\mu_{10}\}$ in (119), we have
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle\tau\left(\frac{g_{2}^{p%
}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\tau\left(\frac{g_{1}^{s}P_{1}}{g%
_{2}^{s}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s%
}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g_{2}^{p}P_{2%
}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right).\end{split}$$
(120)
The first term in (120) is decreasing with $\lambda$ for all values of $\lambda$. The first derivative of the other terms with respect to $\lambda$ is given by
$$\begin{split}&\displaystyle-\frac{0.5P_{s}(g_{s}^{p}g_{s}^{s}P_{2}+g_{s}^{p}g_%
{1}^{s}P_{1}+g_{s}^{p}N_{0}-g_{s}^{s}N_{0})}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0%
})(g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+g_{1}^{s}P_{1}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Hence, $\rho_{s2}^{r}$ is decreasing with $\lambda$.
When $\mu_{10}=\min\{\mu_{8},\mu_{9},\mu_{10}\}$ in (119), we have
$$\begin{split}\displaystyle\rho_{s2}^{r}=&\displaystyle\tau\left(\frac{g_{2}^{p%
}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\tau\left(\frac{g_{s}^{p}\bar{%
\lambda}P_{s}+g_{1}^{p}P_{1}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}%
\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}}{g_{2}^{s%
}P_{2}+N_{0}}\right).\end{split}$$
(121)
The first term in (121) is decreasing with $\lambda$, and the first derivative of the other three terms with respect to $\lambda$ is given by,
$$\begin{split}&\displaystyle-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}g_%
{1}^{s}P_{1}+g_{s}^{p}N_{0}-g_{s}^{s}N_{0})}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0%
})(g_{s}^{s}\lambda P_{s}+g_{1}^{s}P_{1}+g_{2}^{s}P_{2}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Thus, $\rho_{s2}^{r}$ is decreasing with $\lambda$.
D-A5 $2R_{s}^{r}+R_{p}^{r}=\rho_{2p}^{r}$
From (100),
$$\sigma_{s}^{*}=\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P%
_{s}+g_{2}^{s}P_{2}+N_{0}}\right).$$
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2\tau\left(\frac{g_{s}^%
{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{2}^{p}P_{2}}{g_{s}^{%
p}\lambda P_{s}+N_{0}}\right)\\
&\displaystyle-\left[\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}%
\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)-\tau\left(\frac{g_{s}^{p}\bar{%
\lambda}P_{s}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)\right]^{+}%
\\
&\displaystyle+\min\biggl{\{}\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}P_{s}+g_%
{2}^{s}P_{2}+N_{0}}\right),\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}P_{s}+g_{2%
}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\left[\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p}%
\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)-\tau\left(\frac{g_{s}^{s}\bar{%
\lambda}P_{s}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)\right]^{+},%
\\
&\displaystyle\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}%
P_{2}+N_{0}}\right)\biggr{\}}.\end{split}$$
If $\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P%
_{2}+N_{0}}\right)\leq\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p}%
\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)$
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2\tau\left(\frac{g_{s}^%
{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{2}^{p}P_{2}}{g_{s}^{%
p}\lambda P_{s}+N_{0}}\right)\\
&\displaystyle+\min\biggl{\{}\overbrace{\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^%
{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)}^{\mu_{11}},\overbrace{\tau\left%
(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)}^{%
\mu_{12}},\\
&\displaystyle\overbrace{\tau\left(\frac{g_{1}^{p}P_{1}+g_{s}^{p}\bar{\lambda}%
P_{s}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)-\tau\left(\frac{g_{%
s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)}%
^{\mu_{13}}\biggr{\}}.\end{split}$$
(122)
•
When $\mu_{11}=\min\{\mu_{11},\mu_{12},\mu_{13}\}$ in (122), then
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2\tau\left(\frac{g_{s}^%
{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{2}^{p}P_{2}}{g_{s}^{%
p}\lambda P_{s}+N_{0}}\right)\\
&\displaystyle\tau\left(\frac{g_{1}^{s}P_{1}}{g_{s}^{s}\lambda P_{s}+g_{2}^{s}%
P_{2}+N_{0}}\right).\end{split}$$
It is clear that, $\rho_{2p}^{r}$ is decreasing with $\lambda$.
•
When $\mu_{12}=\min\{\mu_{11},\mu_{12},\mu_{13}\}$ in (122), then
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2\tau\left(\frac{g_{s}^%
{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{2}^{p}P_{2}}{g_{s}^{%
p}\lambda P_{s}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p%
}P_{2}+N_{0}}\right).\end{split}$$
It is also clear that $\rho_{2p}^{r}$ is decreasing with $\lambda$.
•
When $\mu_{13}=\min\{\mu_{11},\mu_{12},\mu_{13}\}$ in (122), then
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2\tau\left(\frac{g_{s}^%
{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{%
p}P_{s}+g_{2}^{p}P_{2}+N_{0}}\right)\\
&\displaystyle+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}+g_{2}^{p}P_{2}}{g_{%
s}^{p}\lambda P_{s}+N_{0}}\right)-\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}%
{g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right).\end{split}$$
$$\begin{split}\displaystyle\frac{\partial\rho_{2p}^{r}}{\partial\lambda}=&%
\displaystyle-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}N_{0}-g_{s}^{s}N%
_{0})}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0})(g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{%
2}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Thus, $\rho_{2p}^{r}$ is decreasing with $\lambda$.
If $\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}P%
_{2}+N_{0}}\right)\leq\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}%
\lambda P_{s}+g_{2}^{s}P_{2}+N_{0}}\right)$
$$\begin{split}\displaystyle\rho_{2p}^{r}=&\displaystyle 2\tau\left(\frac{g_{s}^%
{s}P_{s}}{g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P%
_{s}+g_{2}^{p}P_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)\\
&\displaystyle-\tau\left(\frac{g_{s}^{s}\bar{\lambda}P_{s}}{g_{s}^{s}\lambda P%
_{s}+g_{2}^{s}P_{2}+N_{0}}\right)+\min\biggl{\{}\tau\left(\frac{g_{1}^{s}P_{1}%
}{g_{s}^{s}P_{s}+g_{2}^{s}P_{2}+N_{0}}\right),\\
&\displaystyle\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}P_{s}+g_{2}^{p}P_{2}+N_%
{0}}\right)\biggr{\}}.\end{split}$$
$$\begin{split}\displaystyle\frac{\partial\rho_{2p}^{r}}{\partial\lambda}=&%
\displaystyle-\frac{0.5P_{s}(g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}N_{0}-g_{s}^{s}N%
_{0})}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0})(g_{s}^{s}\lambda P_{s}+g_{2}^{s}P_{%
2}+N_{0})}\\
&\displaystyle\leq 0\quad\text{from \eqref{eq:cond3}}.\end{split}$$
Therefore, $\rho_{2p}^{r}$ is decreasing with $\lambda$.
Thus, since we showed that if (100) is satisfied, assuming that the secondary receiver can decode the signal of primary user $1$, then $\rho_{p}^{r}$, $\rho_{sp}^{r}$, $\rho_{s2}^{r}$ and $\rho_{2p}^{r}$ decrease with $\lambda$, whereas $\rho_{s}^{r}$ does not depend on $\lambda$, hence, $\mathcal{R}^{r}_{1}(Z)$ at $\lambda=0$ coincides on $\mathcal{R}^{r}_{1g}$. And for any $\lambda_{1}$ and $\lambda_{2}$ such that $\lambda_{1}>\lambda_{2}$, $\mathcal{R}^{r}_{1}(Z)$ at $\lambda_{1}$ is a subset of $\mathcal{R}^{r}_{1}(Z)$ at $\lambda_{2}$.
D-B Necessity Part
In this part of the proof we show that, if condition (100) is not satisfied then $\mathcal{R}^{r}_{1g}$ does not coincide on any $\mathcal{R}^{r}_{1}(Z)$ for all values of $\lambda$. So, assume that (100) is not satisfied, i.e.,
$$N_{0}g_{s}^{s}>g_{s}^{p}g_{2}^{s}P_{2}+g_{s}^{p}N_{0}.$$
(123)
By referring to Fig. 4, the effect of $\lambda$ on $\mathcal{R}^{r}_{1}(Z)$ at points $A$ and $F$ is determined as follows.
D-B1 At Point A
$$\begin{split}\displaystyle R_{p}^{rA}=&\displaystyle\tau\left(\frac{g_{2}^{p}P%
_{2}}{g_{s}^{p}\lambda P_{s}+N_{0}}\right)+\min\biggl{\{}\tau\left(\frac{g_{1}%
^{s}P_{1}}{g_{2}^{s}P_{2}+N_{0}}\right),\\
&\displaystyle\tau\left(\frac{g_{1}^{p}P_{1}}{g_{s}^{p}\lambda P_{s}+g_{2}^{p}%
P_{2}+N_{0}}\right)\biggr{\}}.\end{split}$$
It is clear that $R_{p}^{rA}$ is decreasing with $\lambda$.
D-B2 At Point F
$$\begin{split}\displaystyle R_{s}^{rF}=\tau\left(\frac{g_{s}^{s}\lambda P_{s}}{%
g_{2}^{s}P_{2}+N_{0}}\right)+\tau\left(\frac{g_{s}^{p}\bar{\lambda}P_{s}}{g_{s%
}^{p}\lambda P_{s}+N_{0}}\right).\end{split}$$
$$\begin{split}\displaystyle\frac{\partial R_{s}^{rF}}{\partial\lambda}&%
\displaystyle=\frac{0.5P_{s}(g_{s}^{s}N_{0}-(g_{s}^{p}g_{s}^{s}P_{2}+g_{s}^{p}%
N_{0}))}{\ln 2(g_{s}^{p}\lambda P_{s}+N_{0})(g_{s}^{s}\lambda P_{s}+g_{2}^{s}P%
_{2}+N_{0})}\\
&\displaystyle>0\quad\text{from \eqref{eq:cond3_n}}.\end{split}$$
Consequently, $R_{s}^{rF}$ is increasing with $\lambda$.
So, for any two different values of $\lambda$, the corresponding rate regions $\mathcal{R}^{r}_{1}(Z)$ do not include one another, thus $\mathcal{R}^{r}_{1g}$ does not coincide on $\mathcal{R}^{r}_{1}(Z)$ at any value of $\lambda$.
Appendix E Proof of Theorem 6
From the definition of $\delta^{\prime o}(Z)$ and $\delta_{1}^{\prime r}(Z)$ it is clear that $\delta^{o}(Z)\subseteq\delta^{\prime o}(Z)$ and $\delta_{1}^{r}(Z)\subseteq\delta_{1}^{\prime r}(Z)$. Consequently, $\mathcal{R}^{o}(Z)\subseteq\mathcal{R}^{\prime o}(Z)$, $\mathcal{R}_{1}^{r}\subseteq\mathcal{R}_{1}^{\prime r}(Z)$ and $\mathcal{R}_{1}(Z)\subseteq\mathcal{R}_{1}^{\prime}(Z)$. However, we show that if there exists $Z\in\mathcal{P}^{*}$ such that a rate tuple $(R_{s},R_{p})$ belongs to $\mathcal{R}_{1}^{\prime}(Z)$ but does not belong to $\mathcal{R}_{1}(Z)$, then there exists another $Z^{\prime}\in\mathcal{P}^{*}$ for which $(R_{s},R_{p})$ belongs to $\mathcal{R}_{1}(Z^{\prime})$.
Following a similar procedure to that used in the proof of Theorem 2, the region $\mathcal{R}^{\prime o}(Z)$ is defined by:
$$R_{p}\leq I(Y_{p};X_{1}X_{2}|WQ),$$
(124)
$$\begin{split}\displaystyle R_{s}\leq&\displaystyle I(Y_{s};U|WQ)+\min\{I(Y_{s}%
;W|Q),\\
&\displaystyle I(Y_{p};WX_{1}|X_{2}Q),I(Y_{p};WX_{2}|X_{1}Q)\},\end{split}$$
(125)
$$\begin{split}\displaystyle R_{s}+R_{p}\leq&\displaystyle I(Y_{s};U|WQ)+I(Y_{p}%
;X_{1}X_{2}|WQ)+\\
&\displaystyle\min\{I(Y_{s};W|Q),I(Y_{p};W|Q)\}.\end{split}$$
(126)
E-A For $\mathcal{R}^{o}(Z)$
Suppose that at a certain $Z\in\mathcal{P}^{*}$, $R_{s}^{\prime}>I(Y_{s};U|WQ)+I(Y_{p};W|X_{1}X_{2}Q)$, hence, the rate tuple $(R_{s}^{\prime},R_{p}^{\prime})\in\mathcal{R}^{\prime o}(Z)$ but $(R_{s}^{\prime},R_{p}^{\prime})\notin\mathcal{R}^{o}(Z)$. From (124)-(126), $(R_{s}^{\prime},R_{p}^{\prime})$ has to satisfy
$$\displaystyle R_{s}\leq I(Y_{s};UW|Q)=I(Y_{s};X_{s}|Q),$$
(127)
$$\displaystyle R_{p}<I(Y_{p};X_{1}X_{2}|Q).$$
(128)
Now, assume another $Z^{\prime}\in\mathcal{P}^{*}$ such that $W=\phi$, i.e., no rate-splitting. At this $Z^{\prime}$, $\mathcal{R}^{o}(Z^{\prime})$ is given by
$$\displaystyle R_{s}\leq I(Y_{s};X_{s}|Q),$$
(129)
$$\displaystyle R_{p}\leq I(Y_{p};X_{1}X_{2}|Q).$$
(130)
Then it is clear that $(R_{s}^{\prime},R_{p}^{\prime})\in\mathcal{R}^{o}(Z^{\prime})$. Thus,
$$\mathcal{R}^{\prime o}(Z)\subseteq\mathcal{R}^{o}(Z)\cup\mathcal{R}^{o}(Z^{%
\prime}).$$
E-B For $\mathcal{R}_{1}^{\prime r}(Z)$
First, for a point $(R_{s}^{\prime\prime},R_{p}^{\prime\prime})$ such that $R_{s}^{\prime\prime}>I(Y_{s};U|WQ)+I(Y_{p};W|X_{1}X_{2}Q)$ at a specific $Z\in\mathcal{P}^{*}$, a similar argument as in the above subsection (Subsection E-A), or in Lemma 2 of [16], can show that there exists $Z^{\prime\prime}\in\mathcal{P}^{*}$ such that $(R_{s}^{\prime\prime},R_{p}^{\prime\prime})\in\mathcal{R}_{1}^{r}(Z^{\prime%
\prime})$.
Second, for another point $(R_{s}^{**},R_{p}^{**})$ such that $R_{p}^{**}>I(Y_{p};X_{2}|WX_{1}Q)+I(Y_{s};X_{1}|UWQ)$, or in other words $R_{1}^{**}>I(Y_{s};X_{1}|UWQ)$, in this case, $\delta_{1}^{\prime r}(Z)\subset\delta^{\prime o}(Z)$. And since $\mathcal{R}^{\prime o}(Z)$ is the set of $(R_{s},R_{p})$ corresponding to $\delta^{\prime o}(Z)$ for which $R_{s}=S+T$ and $R_{p}=R_{1}+R_{2}$, then $\mathcal{R}_{1}^{\prime r}(Z)\subset\mathcal{R}^{\prime o}(Z)$. Moreover, it has been shown in the above subsection (Subsection E-A) that $\mathcal{R}^{\prime o}(Z)\subseteq\mathcal{R}^{o}(Z)\cup\mathcal{R}^{o}(Z^{%
\prime})$. Therefore,
$$\mathcal{R}_{1}^{\prime r}(Z)\subseteq\mathcal{R}_{1}^{r}(Z)\cup\mathcal{R}_{1%
}^{r}(Z^{\prime\prime})\cup\mathcal{R}^{o}(Z)\cup\mathcal{R}^{o}(Z^{\prime}).$$
Consequently,
$$\mathcal{R}_{1}^{\prime}=\mathcal{R}_{1}.$$
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Exploiting weak ties in trust-based recommender systems using regular equivalence
Tomislav Duricic
Graz University of TechnologyGraz, Austria
[email protected]
,
Emanuel Lacic
Know-Center GmbHGraz, Austria
[email protected]
,
Dominik Kowald
Know-Center GmbHGraz, Austria
[email protected]
and
Elisabeth Lex
Graz University of TechnologyGraz, Austria
[email protected]
Abstract.
User-based Collaborative Filtering (CF) is one of the most popular approaches to create recommender systems. CF, however, suffers from data sparsity and the cold-start problem since users often rate only a small fraction of available items. One solution is to incorporate additional information into the recommendation process such as explicit trust scores that are assigned by users to others or implicit trust relationships that result from social connections between users. Such relationships typically form a very sparse trust network, which can be utilized to generate recommendations for users based on people they trust.
In our work, we explore the use of regular equivalence applied to a trust network to generate a similarity matrix that is used for selecting $k$-nearest neighbors used for item recommendation. Two vertices in a network are regularly equivalent if their neighbors are themselves equivalent and by using the iterative approach of calculating regular equivalence, we can study the impact of strong and weak ties on item recommendation. We evaluate our approach on cold-start users on a dataset crawled from Epinions and find that by using weak ties in addition to strong ties, we can improve the performance of a trust-based recommender in terms of recommendation accuracy.
Problem & objective. Ever since their introduction, user-based Collaborative Filtering (CF) approaches have been one of the most widely adopted and studied algorithms in the recommender systems literature (Schafer et al., 2007). CF is based on the intuition that those users, who have shown similar item rating behavior in the past, will likely give similar ratings to items in the future.
The basis of CF is to retrieve the $k$-nearest neighbors of a target user for whom the recommendations are generated and to recommend items from these $k$ neighbors, which were rated highly by them but have not yet been rated by the target user. An issue of CF is the cold-start user problem, i.e., novel users, who have rated zero or only a small number of items (Schein
et al., 2002) and whose ratings cannot thus be exploited to find similar users.
As a remedy, trust-based CF methods exploit trust statements expressed by users on platforms such as Epinions (Massa and Avesani, 2007). Such trust statements can be explicit, i.e., users assign trust scores to others or implicit, i.e., users engage in social connections with others they trust. Based on explicit and implicit trust statements, we can generate trust networks and recommend items for users based on people they trust (Lathia
et al., 2008). Since trust networks are often sparse, a particular property of trust, namely transitivity, can be exploited to propagate trust in the network by forming weak ties between users. In this way, new connections are established between users, who do not share a direct link, but are weakly connected via intermediary users (Golbeck, 2005; Massa and Avesani, 2007).
In our work, we focus on the first step of CF, i.e., finding the $k$-nearest neighbors. We explore the power of weak ties to find similar neighbors by utilizing a similarity measure from network science referred to as ”Katz similarity” (KS) (Newman, 2010). Although Katz himself never discussed it, KS captures regular equivalence of nodes in a network and can be applied in many different settings (Hasani-Mavriqi et al., 2018; Helic, 2014).
Approach & method. Firstly, we utilize the trust connections to create an adjacency matrix where each entry represents a directed trust link between two users.
Secondly, we apply the KS measure on the created trust adjacency matrix. More specifically, we calculate the pairwise similarities between users by using the iterative approach for calculating KS:
(1)
$$\displaystyle\bm{\sigma}^{(l_{max}+1)}=\sum\limits_{l=0}^{l_{max}}(\alpha\bm{A%
})^{l}$$
The iterative approach provides the possibility to set the maximum used path length ($l_{max}$). This approach effectively gives us the ability to define the maximum path length used for forming weak ties between users who are not directly connected. We use the resulting similarity matrix and apply various row normalization ($L_{1}$, $L_{2}$, $max$) and degree normalization techniques (in-degree, out-degree, and combined degree normalization) to get a better distribution of similarity values and better evaluation results concerning recommendation accuracy in return. Lastly, we apply an additional method to increase the similarity values derived from weak ties (Duricic
et al., 2018):
(2)
$$\displaystyle\bm{\sigma}_{boost}=\bm{A}+\hat{\bm{\sigma}}_{norm}$$
where $\hat{\bm{\sigma}}_{norm}$ is calculated by setting the values of strong tie similarities in $\bm{\sigma}$ to $0$, normalizing the resulting matrix and then setting them to $1$. With this approach, we achieve that each entry in $\bm{\sigma}_{boost}$ has a similarity value of $1$ between pairs of nodes for which there exists an explicit trust connection in $\bm{A}$ while also increasing the importance of similarity values derived from weak ties. We evaluate these approaches on the Epinions dataset presented in (Massa and Avesani, 2007) and compare results for $l_{max}=1$ (using only strong ties) and $l_{max}=2$ (using strong ties in combination with weak ties derived from paths of length $2$).
Results & discussion. In our study, we evaluate $33$ approaches for various combinations of $l_{max}$ values and normalization techniques. We compare these approaches with three different baselines: $MP$ (recommending most popular items), $Trust_{exp}$ (CF using trust connections for finding top $k$ similar neighbors) and $Trust_{jac}$ (CF using Jaccard coefficient on explicit trust values for finding top $k$ similar neighbors). However, in Table 1, we only report the results for a subset of these approaches that provide the most insightful findings. All of the evaluation results are reported for $n=10$, i.e., for $10$ recommended items.
The $Trust_{exp}$ baseline uses only strong ties (explicit trust connections) for making recommendations and one of our main finding from the results of the conducted experiments is that by incorporating weak ties using paths of maximum length $2$ from the target node (i.e. similar to adding friends of friends into the neighborhood), we can improve the quality of the recommendations in terms of recommendation accuracy with the best approach being the $KS_{PCMB}$. In the best performing approach, we set $l_{max}$ to $2$, apply combined degree (sum of in and out degrees) normalization, remove the strong ties, then perform $max$ row normalization and then add the strong ties with the similarity value of $1$. We also find that if we don’t employ weak ties, i.e., $l_{max}$ is set to $1$, we achieve better results when we do not apply degree and row normalization (i.e., basically the $Trust_{exp}$ baseline). However, if $l_{max}$ is set to $2$, we can observe improvements in almost all of the cases except when no row normalization is applied, e.g., in the case of $KS_{PNNN}$.
Finally, in Figure 1, we show the performance of all approaches listed in Table 1 in form of Recall-Precision plots for different number of recommended items (i.e., $n=1-10$). The results clearly show that the best performing algorithm (i.e., $KS_{PCMB}$) again outperforms all of the other approaches also for a smaller number of recommended items (i.e., for $n<10$).
Conclusion & future work.
In this paper, we explored the use of Katz similarity (KS), a similarity measure of regular equivalence in networks, for selecting $k$-nearest neighbors in a Collaborative Filtering (CF) algorithm for cold-start users. We used an iterative approach to compute KS since it provides the ability to restrict the length of paths in the network used for similarity calculation. Consequently, we can investigate weak ties of arbitrary length. We found that KS can be a useful measure for neighbor selection if used with degree normalization and row normalization. In summary, with our work, we aimed to shed light on how to exploit weak ties in social networks to increase the performance of trust-based recommender systems. For future work, we plan to run additional experiments using different values for $l_{max}$ and to explore the use of recently popularized node embeddings (e.g., Node2Vec (Grover and
Leskovec, 2016)) to identify weak ties between users.
Keywords. Weak ties; Katz similarity; Trust-based recommenders
††journalyear: 2019††copyright: none††conference: ESCSS 2019; September 02-04, 2019; Zurich, Switzerland††price: 15.00††doi: xx.xxx/xxxxxx.xxxxxx
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Computational characteristics of feedforward neural networks for solving a stiff differential equation
Toni Schneidereit and Michael BreuÃ
Applied Mathematics Group
Brandenburg University of Technology Cottbus-Senftenberg
Platz der Deutschen Einheit 1, 03046 Cottbus, Germany
{Toni.Schneidereit,breuss}@b-tu.de
(January 10, 2021)
Abstract
Feedforward neural networks offer a promising approach for solving differential equations.
However, the reliability and accuracy of the approximation still represent delicate
issues that are not fully resolved in the current literature.
Computational approaches are in general highly dependent on a variety
of computational parameters as well as on the choice of optimisation methods,
a point that has to be seen together with the structure of the cost function.
The intention of this paper is to make a step towards resolving these open issues.
To this end we study here the solution of a simple but fundamental stiff ordinary differential equation
modelling a damped system. We consider two computational approaches for solving differential equations
by neural forms. These are the classic but still actual method of trial solutions defining the cost function,
and a recent direct construction of the cost function related to the trial solution method.
Let us note that the settings we study can easily be applied more generally, including solution of partial
differential equations.
By a very detailed computational study we show that it is possible to identify preferable choices
to be made for parameters and methods. We also illuminate some interesting effects that are observable in
the neural network simulations.
Overall we extend the current literature in the field by showing what can be done
in order to obtain reliable and accurate results by the neural network approach.
By doing this we illustrate the importance of a careful choice of the computational setup.
\SelectInputMappings
germandbls=Ã
Keywords: feedforward neural networks, ordinary differential equations, trial solution, ADAM, backpropagation
1 Introduction
Differential equations occur in many fields of science and engineering and represent a useful description of many physical phenomena.
They are usually formulated as initial or boundary value problems, where conditions at the beginning of a process
or at boundary points are given to obtain one specific solution. It is useful to approximate differential equations
by numerical methods [1, 2] like finite difference methods, and also neural networks have been applied to this end, see e.g. [3, 4, 5].
The variety of possible neural network architectures is immense [6]. Already in classic works
in the field, feedforward neural networks have proven to be useful for solving differential equations
[7, 8].
Within this framework of feedforward neural networks (from now on denoted here simply as neural networks),
two particular approaches have been investigated in the literature within the last decades that appear to be
very promising.
The trial solution (TS) method has been proposed for the approximation of a
given differential equation, which we abbreviate here as TSM [9]. The TS, also called neural form
in [9, 10], has to contain the neural network output and has to satisfy given initial or boundary conditions by construction.
Under the latter conditions there are multiple different possible forms of the TS for the approximation of a differential equation.
Recently a systematic construction approach for the TS has been proposed [10]. However, as indicated in the latter work, the TS construction
may become difficult to realise for complex problems. We will follow here the original approach proposed in [9].
Let us note that the same TS structure is used for example in the recent Legendre neural network [11].
It appears evident that our investigation may also be useful in the context of such extensions.
Recently published in 2019, an approach has been proposed to avoid finding a TS [12] as this may be an intricate
ingredient of the TSM method. Because the corresponding
method is motivated and technically related to the TSM method, we call it here modified trial solution method (mTSM).
Instead of building the cost function by use of the TS that meets conditions imposed on a differential equation,
the approximating solution function is set in [12] to be the neural network output directly.
The latter does not satisfy given initial or boundary conditions by construction as in TSM, but these are added
as additional terms in the cost function.
In the mentioned works, both TSM and mTSM have proven to be capable of solving ordinary (ODEs) and partial differential equations (PDEs)
as well as systems of ODEs and PDEs. This has been demonstrated for several examples and even complex simulations,
showing the potential of the methods to obtain high-quality results. This has motivated us to consider in higher detail some of the computational
issues that arise in the application of these methods in a first study [13].
Let us mention here also a recent complementary work where the activation functions are subject of a
computational study [14].
Despite these promising developments, there are still many open questions related to both TSM and mTSM. First of all, in the
original work [12] an emphasis was layed on the new construction principle and the application of the proposed method
in a cosmological context. However, one may wonder about the direct comparison of the two methods TSM and mTSM in terms of quality
of results as well as in the related computational aspects. Let us stress in this context, that the original works
[9, 10, 12] mainly describe the network architecture and elaborate on the TS respectively
mTSM construction, but they do not contain more details of the computational characteristics of the methods. Yet it turns out that it
is not trivial to define a computational framework that gives competitive results.
Our contribution.
In this work we build upon our first parameter study in [13] and extend the investigations in several directions.
Following the basic line of the first work, we study here the variance between the exact solution of an ordinary differential equation and the
approximations provided by TSM and mTSM. As one apparent difference to the proceeding in our previous conference paper, we extend here
the investigation w.r.t. the number of training data, and we give many more details in the evaluation. We also give here additional and
as it turns out meaningful experiments concerned with the roles of neural network weight initialisation, number of hidden layers and number of hidden layer neurons.
We perform several experiments on the variety of parameters related to the differential equation, neural network and optimisation methods.
Let us stress that the amount of parameters for the differential equation, neural network and optimisation is numerous. Our contribution in
the main part of this paper is a study of the variation on (i) Weight initialisation methods, (ii) Number of hidden layer neurons,
(iii) Number of hidden layers, (iv) Number of training epochs, (v) Stiffness parameter and domain size,
(vi) Optimisation methods, and especially their mutual dependence. Let us note that it has turned out to be a nontrivial task to set up a meaningful proceeding that gives an account of the latter aspect. We consider the evaluation presented here as a number of carefully chosen experiments that are in many respects related to each other.
For investigating the computational characteristics, we consider a simple while important stiff ODE model equation [15]
with a damping behaviour for studying the stability and reliability of both methods. We also present here as another contribution
a detailed study of the influence of the stiffness parameter contained in the ODE. Let us note that a similar solution behaviour
is to be expected when resolving for instance parabolic PDEs.
2 Neural network architecture and optimisation
Neural networks are usually pictured as neurons (circles) and connecting weights (lines). Fig. 1 shows the standard neural network architecture for our experiments. It consists of three layers and features one input layer neuron for $x\in D\subset\mathbb{R}$ (where $D$ denotes the domain) with one bias neuron which can be considered as an offset, five hidden layer neurons and one linear output layer neuron. In experiments on the number of hidden layers and the number of hidden layer neurons, the architecture is extended. Each
neuron is connected with every single neuron in the next layer by the weights $w_{j}$, $u_{j}$ and $v_{j}$, $j$=1,$\ldots$,5, which are stored in the weight vector $\vec{p}$. The input layer passes the domain data $x$, weighted by $w_{j}$ and $u_{j}$ to the hidden layer for processing. The processed data is then, now weighted by $v_{j}$, sent to the output layer in order to generate the neural network output $N(x,\vec{p})$. That means in detail, the hidden layer receives the weighted sum $z_{j}=w_{j}x+u_{j}$ as input and processes this data by the sigmoid activation function $\sigma_{j}=\sigma(z_{j})$=$1/(1+e^{-z_{j}})$. Since the output layer consists of a linear neuron, the neural network output is generated by the linear combination
$$N(x,\vec{p})=\sum_{j=1}^{5}v_{j}\sigma(z_{j}).$$
The sigmoid activation function is a continuous and arbitrarily often differentiable function with values between 0 and 1.
Let us note, that in order to solve differential equations of order $n$ with neural networks, it is important to choose an activation function, which is at least $(n+1)$ times continuously differentiable, since the later shown solution approaches require the $n$-th activation function derivative and the optimisation methods require another differentation.
The universal approximation theorem [16] states, that one hidden layer with a finite number of sigmoidal activation functions is able to approximate every continuous function on a subset of $\mathbb{R}$.
In general, it is common to initialise the weights with small random values [17], therefore the first computation of $N(x,\vec{p})$ returns a random value. This value is used to compute the cost or loss function $E[\vec{p}]$ which is then subject to optimisation.
With the first random output of $N(x,\vec{p})$, the optimisation will return different weight updates
when starting several computations with exactly the same computational parameters (but random weight initialisation).
Another option is to choose the initialisation to be constant when starting several computations.
That is, $N(x,\vec{p})$ first returns always the same value, and therefore with
the weight updates to be constant as well, computations with same parameter setting return equal results.
For supervised learning, where both input data $x_{i}$ (representing the discrete domain or grid) and correct output data $d_{i},~{}i$=$1,\ldots,n$, are known, the cost function may be chosen as the squared $l_{2}$-norm
$$E[\vec{p}]=\frac{1}{2}\big{\lVert}N(x_{i},\vec{p})-d_{i}\big{\rVert}_{2}^{2},$$
while in case of unsupervised learning, where no correct output data is known, the cost function is part of the modelling process. Our approach follows the latter track.
3 Solution approaches
In this section, we will describe the trial solution construction for the TSM and mTSM methods more in detail, as well as the approaches on how to make use of neural networks in order to solve ordinary differential equations (ODEs) in form of
$$G\left(x,u(x),\frac{d}{dx}u(x)\right)=0,~{}~{}x\in D\subset\mathbb{R},$$
(1)
with given initial or boundary conditions. In Eq. (1), $u(x)$ denotes the exact solution function with $x$ as independent variable. Although $G$ denotes a first order ODE, let us note again that it is also possible to solve higher order ordinary or partial differential equations (PDEs), as well as
systems of ODEs or PDEs, cf. [9, 12].
3.1 Trial solution method (TSM)
Let us now recall the approach from [9]. In order to satisfy initial or boundary conditions, the TS is constructed to satisfy these conditions and is therefore written as a sum of two terms
$$u_{t}(x,\vec{p})=A(x)+B(x)N(x,\vec{p}).$$
(2)
In Eq. (2), $A(x)$ is supposed to satisfy the initial or boundary conditions at the initial or boundary points, while $B(x)$ is constructed to become zero at these points to eliminate the impact of $N(x,\vec{p})$ there. That is, the TS may be defined in many possible forms for one differential equation, satisfying the mentioned conditions. Especially the choice of $B(x)$ determines the impact of $N(x,\vec{p})$ over the domain. Now, the TS transforms Eq. (1) into
$$G\left(x,u_{t}(x,\vec{p}),\frac{\partial}{\partial x}u_{t}(x,\vec{p})\right)=0,$$
(3)
so that the partial derivative of the trial solution with respect to input $x$ which we have to consider is
$$\frac{\partial}{\partial x}u_{t}(x,\vec{p})=A^{\prime}(x)+B^{\prime}(x)N(x,%
\vec{p})+B(x)\frac{\partial}{\partial x}N(x,\vec{p})$$
with
$$\frac{\partial}{\partial x}N(x,\vec{p})=\sum_{j=1}^{5}v_{j}w_{j}\sigma^{\prime%
}(z_{j})$$
In order to generate training data for the neural network, we discretise the domain $D$ by a uniform grid
with $n$ gridpoints $x_{i}$. Over this discrete domain, Eq. (3) is now solved by an unconstrained
optimisation problem using the cost function
$$E[\vec{p}]=\frac{1}{2}\left\lVert G\left(x_{i},u_{t}(x_{i},\vec{p}),\frac{%
\partial}{\partial x}u_{t}(x_{i},\vec{p})\right)\right\rVert_{2}^{2}.$$
3.2 Modified trial solution method (mTSM)
This method, proposed in [12], introduces
$$u_{t}(x,\vec{p})=N(x,\vec{p}),$$
as a TS directly for all differential equations. Therefore $u_{t}$ does not satisfy initial or boundary conditions by construction
as in (2), they rather appear in the cost function as additional terms
$$\displaystyle E[\vec{p}]$$
$$\displaystyle=\frac{1}{2}\left\lVert G\left(x_{i},u_{t}(x_{i},\vec{p}),\frac{%
\partial}{\partial x}u_{t}(x_{i},\vec{p})\right)\right\rVert_{2}^{2}$$
$$\displaystyle+\frac{1}{2}\big{\lVert}u_{t}(x_{m},\vec{p})-K(x_{m})\big{\rVert}%
_{2}^{2},$$
where $K(x_{m}),~{}m$=$1,\ldots,l$, denote the initial or boundary conditions.
The modelled cost function is now subject to optimisation with respect to the adjustible neural network weights $\vec{p}$.
4 Optimisation
For cost function minimisation we use first order methods, based on gradient descent. A commonly employed, simple optimisation technique is
backpropagation, which uses the cost function gradient with respect to the neural network weights to determine their influence on $N(x,\vec{p})$
and to update them. It is well-known that backpropagation enables to find a local minimum in the weight space.
The training is usually done several times with
all training data. After one complete iteration through all input data, one epoch of training is done and for efficient training
(finding a minimum in the weight space), several epochs of training are performed.
For the k-th epoch, backpropagation with momentum update rule [18] reads as
$$\vec{p}(k+1)=\vec{p}(k)\underbrace{-\alpha\frac{\partial E[\vec{p}(k)]}{%
\partial\vec{p}(k)}+\beta\Delta\vec{p}(k-1)}_{\Delta\vec{p}(k)}.$$
(4)
Since only the neural network output $N(x,\vec{p})$ and the derivative w.r.t. $x$ depend on $\vec{p}$ and as both expressions are given,
the corresponding derivatives used in the gradient of the cost function are computed as
$$\displaystyle\frac{\partial}{\partial w_{j}}N(x,\vec{p})=v_{j}x\sigma^{\prime}%
(z_{j})$$
$$\displaystyle\frac{\partial}{\partial u_{j}}N(x,\vec{p})=v_{j}\sigma^{\prime}(%
z_{j})$$
$$\displaystyle\frac{\partial}{\partial v_{j}}N(x,\vec{p})=\sigma(z_{j})$$
and
$$\displaystyle\frac{\partial}{\partial w_{j}}\left(\frac{\partial}{\partial x}N%
(x,\vec{p})\right)=v_{j}\sigma^{\prime}(z_{j})+v_{j}w_{j}x\sigma^{\prime\prime%
}(z_{j})$$
$$\displaystyle\frac{\partial}{\partial u_{j}}\left(\frac{\partial}{\partial x}N%
(x,\vec{p})\right)=v_{j}w_{j}\sigma^{\prime\prime}(z_{j})$$
$$\displaystyle\frac{\partial}{\partial v_{j}}\left(\frac{\partial}{\partial x}N%
(x,\vec{p})\right)=w_{j}\sigma^{\prime}(z_{j})$$
The momentum term in Eq. (4), with momentum parameter $\beta$, uses impact from last epoch to reduce the chance of getting
stuck too early during training in a local minimum or at a saddle point.
The learning rate $\alpha$ in general, is a scaling factor for the
gradient and has major influence on the update. A very basic approach is to choose $\alpha$ as a constant learning rate (cBP).
In order to prevent the optimiser from oscillating around a minimum one may employ a variable learning rate (vBP) as an alternative.
Different approaches for learning rate control exist [19], we opt to employ the linear decreasing model
$$\alpha(k)=\left\{\begin{array}[]{ll}\alpha_{0}-\frac{\displaystyle{\alpha_{0}-%
\alpha_{e}}}{\displaystyle{k_{c}}}k,&~{}k\leq k_{c}\\
\alpha_{e},&~{}k>k_{c}\end{array}\right.$$
with an initial learning rate $\alpha_{0}$, a final learning rate $\alpha_{e}$ and an epoch cap $k_{c}$.
In our experiments we will also consider ADAM (adaptive moment estimation) which is an adaptive optimisation method. It uses
estimations of first (mean) and second (uncentered variance) moments of the gradient, see [20] for details.
An advantage of ADAM is the potential for achieving rapid training speed. While backpropagation scales the gradient uniformly
in every direction in weight space (by $\alpha$), ADAM computes an individual learning rate for every weight.
5 Experiments and results
For experiments on both solution approaches with different parameter variations, as well as optimisation with ADAM and backpropagation,
we make use of the model problem
$$\frac{d}{dx}u(x)=\lambda u(x),~{}~{}u(0)=1,$$
(5)
a homogenous first order ordinary differential equation with $\lambda$$\in$$\mathbb{R}$, $\lambda<0$. The ODE (5) has the
exact solution $u(x)=e^{\lambda x}$ and respresents a simple model for stiff phenomena involving a damping mechanism.
The numeric error $\Delta u$ shown in subsequent diagrams is defined as the $l_{1}$-norm of the difference between the exact solution and the
corresponding trial solution
$$\Delta u=\big{\lVert}u(x_{i})-u_{t}(x_{i},\vec{p})\big{\rVert}_{1}~{}.$$
With $u_{t}(x,\vec{p})=1+xN(x,\vec{p})$ we take the form of the trial solution for TSM
proposed in [9]
to construct the cost function
$$\displaystyle E[\vec{p}]=\frac{1}{2}\bigg{\lVert}N(x$$
$${}_{i},\vec{p})+x_{i}\frac{\partial}{\partial x}N(x_{i},\vec{p})$$
$$\displaystyle-\lambda\left(1+x_{i}N(x_{i},\vec{p})\right)\bigg{\rVert}_{2}^{2}%
~{}.$$
For mTSM the trial solution $u_{t}(x,\vec{p})=N(x,\vec{p})$ results in the cost function
$$\displaystyle E[\vec{p}]=\frac{1}{2}$$
$$\displaystyle\left\lVert\frac{\partial}{\partial x}N(x_{i},\vec{p})-\lambda N(%
x_{i},\vec{p})\right\rVert_{2}^{2}$$
$$\displaystyle+\frac{1}{2}$$
$$\displaystyle\big{\lVert}N(x_{1},\vec{p})\big{|}_{x_{1}=0}-1\big{\rVert}_{2}^{%
2}~{}.$$
In subsequent experiments we study $\Delta u$ with respect to several, meaningful variations of computational parameters.
The main parameters respectively values of the computational settings are defined as in Table 1. We use our own Fortran implementation for the neural network, the solution approaches and the optimiser, by following the proposed methods in the corresponding papers, without the use of deep learning libraries. Therefore we have total control over the computations and are able to perform investigations related to every aspect of the methods and the code.
In most subsequent experiments we used cBP instead of vBP, to reduce the amount of parameters. The learning rate for cBP is $\alpha$=1e-3 with $\beta$=9e-1.
Only in optimisation comparison, vBP appears with $\alpha_{0}$=1e-2, $\alpha_{e}$=1e-3, $k_{c}$=1e4 and $\beta$=9e-1 as well. ADAM parameters are, as employed in [20], $\alpha$=1e-3, $\beta_{1}$=9e-1, $\beta_{2}$=9.99e-1 and $\epsilon$=1e-8. In addition, some experiments show averaged graphs to see the general trend with a reduced influence of fluctuations. If we do not say otherwise in the subsequent experiments, the computational parameters are fixed with one hidden layer, five hidden layer neurons, number of maximal epochs $k_{max}$=1e5, domain data $x$$\in$[0,2] and stiffness parameter $\lambda$=-5.
Concerning the following experiments, let us stress again that these are not considered to be separate or independent of each other.
We will consequently follow a line of argumentation that enables us (i) to reduce step by step the degrees of freedom in the choice of
computational settings, and (ii) to clarify the influence of individual computational parameters. In doing this we also demonstrate how
to achieve tractable results. We consider this as an important part of our work since this makes the whole approach more meaningful.
5.1 Experiment 1: Weight initialisation
This experiment illustrates differences between the two weight initialisation methods, employing either $\vec{p}^{~{}init}_{const}$ or $\vec{p}^{~{}init}_{rnd}$.
We averaged 1e2 iterations for displaying each point in the graph depicting $\overline{\Delta u}_{rnd}$, implying that for the values given at the lower axis
we perform computations with 1e2 overlayed random perturbations, with random numbers in range of 1e-2, as initialisation around each point.
The averaging is important to mention, because every iteration with $\vec{p}^{~{}init}_{rnd}$ and exactly the same parameter setup, is expected to return different results.
Let us first comment on our choice of $\vec{p}^{~{}init}_{const}$. Evidently, one has to choose here some fixed value, and by
further experiments not documented here in detail, the value zero appears to be a suitable generic choice for mTSM.
Let us now consider the experiments documented in Figure 2.
In general, TSM with both cBP and ADAM (see illustrations (a)-(f)) does not return reliable results for $\vec{p}^{~{}init}_{const}$ with the current parameter setup.
All experiments for TSM with $\vec{p}^{~{}init}_{const}$ give here uniformly a very high error (depicted by orange/solid lines), even when increasing the
number of training data.
Turning to mTSM, the overall clearly best results for $\vec{p}^{~{}init}_{const}$ are provided by using ADAM and ntD=40 in terms of the largest stable region.
The results demonstrated in all of the experiments for mTSM with $\vec{p}^{~{}init}_{const}$ show that the ADAM solver provides a reliable proceeding, virtually
independently of the number of training data.
When considering $\vec{p}^{~{}init}_{rnd}$, the ADAM solver gives also for TSM reasonable results in terms of
the numerical error with a large stable region. A similar but less clear error behaviour can be observed for using ADAM with mTSM.
As a general trend in all experiments with $\vec{p}^{~{}init}_{rnd}$, we observe that weight initialisation in a small range around zero seems to work best.
Let us also comment on illustrations (j)-(l), that we observe here the behaviour that both $\vec{p}^{~{}init}_{const}$ and $\vec{p}^{~{}init}_{rnd}$ around zero seem to work reasonably
with cBP. One may conjecture for other example ODEs, that there could be some constant initialisation and a range of random fluctuations around it
that may work well.
Since a suitable choice of $\vec{p}^{~{}init}_{const}$ and $\vec{p}^{~{}init}_{rnd}$ is important in all subsequent experiments,
we decided as a consequence of the experiments discussed here to initialise $\vec{p}^{~{}init}_{const}$
with zeros and $\vec{p}^{~{}init}_{rnd}$ with random values in range of 0 to 1e-2 from now on.
5.2 Experiment 2: Number of hidden layer neurons
The behaviour of $\Delta$u${}_{const}$ and $\overline{\Delta u}_{rnd}$ when increasing the number of hidden layer neurons is subject to this experiment, where $\overline{\Delta u}_{rnd}$ is averaged over 1e2 computations for every tested number of hidden layer neurons.
There is almost no difference between the experiments for TSM, they all show a similar saturating behaviour.
As discussed in the previous experiment, it is clear that we have to focus here on the random initialisation, and
for this setup we observe here reliable results for about five or more neurons.
Turning to mTSM, a higher number of hidden layer neurons leads to an increase in accuracy for ADAM for larger numbers of training data explored here (ntD=40).
For smaller numbers of training data (ntD=10,20) we observe here that the number of hidden layer neurons and thus the degrees of freedom introduced
by the neural network should be in a relatively small range, e.g. about half the amount of training data.
Also for cBP the saturation value of the error is affected by increasing the amount of hidden layer neurons.
The general trend for $\overline{\Delta u}_{rnd}$ is that slightly higher accuracy is provided in this way, and that the saturation
level is visible already when using a small number of neurons.
Generally in all cases, one can clearly observe the benefit of introducing two to three or more neurons, as this leads to a significant drop
in all computed numerical errors.
As a consequence of these investigations, we employ five hidden layer neurons in the other experiments (note that this setting has also been used in
the previous experiment) as this appears to be justified by the stable solutions and the amount of computational time.
5.3 Experiment 3: Number of hidden layers
In order to focus on the impact of the number of hidden layers, we decided here to keep the number of neurons in the
hidden layers constant, employing five neurons plus an additional bias neuron in each layer.
As in previous experiments, $\overline{\Delta u}_{rnd}$ is averaged by 1e2 iterations.
Results show that one hidden layer is not always enough to provide reliable results, especially for $\Delta$u${}_{const}$ and TSM.
Increasing the number of training data (ntD) changes the number of hidden layers that give
the best approximation in some cases, but it does not seem to have in general a highly beneficial
influence.
Turning to the most important aspect of our investigation in this experiment, one has to distinguish the effect
of increasing the number of hidden layers with respect to the individual methods mTSM and TSM. For the mTSM
method we find that one or two layers are sufficient to obtain – together with ADAM optimisation –
accurate and reliable results. Considering TSM our study shows a very different result, namely that each increase
in the number of hidden layers up to about three respectively four makes up one order of accuracy gain,
for $\vec{p}^{~{}init}_{const}$ respectively $\vec{p}^{~{}init}_{rnd}$.
The latter result appears to be to some degree surprising, as the universal approximation theorem
should imply that one hidden layer could be enough to give here experimentally an accurate approximation of our solution function.
Let us recall in this context Experiment 5.2, where we have seen that an increase of the number of neurons
in one hidden layer leads to a saturation in the accuracy for $\vec{p}^{~{}init}_{rnd}$, while we observe here a clear improvement.
Increasing the number of neurons and using $\vec{p}^{~{}init}_{const}$ did not lead to reasonable results there, while $\vec{p}^{~{}init}_{const}$ here in combination with more hidden layers gives good results plus a significant improvement in the current study.
As a consequence of this investigation, we decided to use one hidden layer for all computations in the other
experiments, having in mind that TSM may allow an accuracy gain for more hidden layers.
5.4 Experiment 4: Number of epochs
In this experiment we aim to investigate if it is possible to fix the maximal number of training epochs to a convenient
value. This relates to the question if one could bound the computational load by employing in general a small number of
training cycles. To this end, we consider the convergence of the training as a function of an increasing maximal number
of epochs $k_{max}$. In addition we illuminate the influence of the number of training data.
More precisely, we increased $k_{max}$ from 1 to 1e5 and averaged 1e2 iterations for one and
the same $k_{max}$. Put in other words, and to make clear the meaning of the lower axis in Figure 4, one
entry of the number $k_{max}$ relates to 1e2 corresponding complete optimisations of the neural network.
Let us note again, that in the case of $\vec{p}^{~{}init}_{rnd}$, the convergence behaviour can only be evaluated
by average values, and that each computation was done with a new $\vec{p}^{~{}init}_{rnd}$.
As can be seen by Figure 4, best results are returned by mTSM with ADAM for both $\vec{p}^{~{}init}_{rnd}$
(especially ntD=20) and $\vec{p}^{~{}init}_{const}$ (especially ntD=40). Except for TSM and ntD=10, the ADAM optimiser clearly
reaches a saturation regime showing convergence for TSM and mTSM with $\vec{p}^{~{}init}_{rnd}$.
For cBP, $\Delta$u${}_{const}$ and $\overline{\Delta u}_{rnd}$, still may decrease for even higher $k_{max}$ as evaluated here.
However, let us note here that we employed in cBP a constant learning rate, for decreasing learning rates as often used
for training we may expect that a saturation regime may be observed.
However, with ADAM, $\overline{\Delta u}_{rnd}$ shows a small fluctuating behaviour in the convergence regime, so that results
for non-averaged computations with $\vec{p}^{~{}init}_{rnd}$ may be not satisfying. The cBP optimiser together with both TSM and
mTSM shows very minor fluctuations, but also provides less good approximations. However, these tend to get better with higher
ntD.
In the context of our results, let us note that in [12] the authors employed 5e4 epochs. Our investigation
shows that the corresponding results are supposed to be in the convergence regime.
In conclusion, we find that $k_{max}$=1e5 as used for all other experiments is suitable to obtain reliable approximations.
5.5 Experiment 5: Stiffness parameter $\lambda$ and domain size of $D$
Let us now investigate the solution behaviour with respect to interesting choices of the stiffness parameter $\lambda$, and
it turns out that it makes sense to do this together with an investigation of the domain size of $D$.
Let us note that informally speaking, these parameters also impact the general trend of the exact solution in a similar way
so that it appears also from this point of view natural to evaluate them together in one experiment.
As shown in Figure 6, the influence of different domains with increasing ntD is the objective of this experiment.
Intervals used for computations are given in terms of $x$$\in$$[0,x_{end}]$, with the smallest interval being $x$$\in$[0,5e-2] and then increasing
in steps of 5e-2. As also in the first experimental part here, $\overline{\Delta u}_{rnd}$ is averaged by 1e2 iterations for each domain.
Turning to the results, first we want to point out that for TSM, cBP and ntD=20 there are values displayed as $\overline{\Delta u}_{rnd}$=9e0,
to visualise them. In reality, these values were Not a Number (NaN), which means, that at this point at least one of the 1e2 averaged
iterations diverged for small values of $\lambda$, respectively large domains.
Furthermore, the solution accuracy for TSM and cBP is strictly decreasing for smaller $\lambda$ and larger domains until it saturates in
unstable regions. While increasing the number of training data from ntD=10 to ntD=20 some iterations diverged, another increase to ntD=40 enlarges
the unstable region with a stabilisation inbetween.
In the total, we observe that there seems to be a relation between the experiments that one may roughly formulate as a relation between
$\lambda$ and domain size given by $x_{end}$ as a factor of $-2$. We also conjecture, that the higher the values of $-\lambda$ and $x_{end}$,
the more neurons or layers are required for a convenient solution.
As a consequence of these experiments, we decided to fix $\lambda$=-5 and $x$$\in$[0,2] for all computations in the other experiments.
5.6 Experiment 6: Optimisation methods
The final experiment in this paper compares ADAM, cBP and vBP optimisation for TSM and mTSM, depending on ntD=10,20,40 with the other computational parameters fixed to one hidden layer, five hidden layer neurons, $k_{max}$=1e5, $\lambda$=-5 and $x$$\in$[0,2].
Figure 7 shows 1e5 (non-averaged) computed results for each parameter setup and weight initialisation.
Previous experiments led to the conclusion, that TSM in combination with $\vec{p}^{~{}init}_{const}$ only provides unstable solutions for the
chosen parameter setup. Therefore, when evaluating TSM, we will only refer to the non-averaged numeric error $\Delta$u${}_{rnd}$ for $\vec{p}^{~{}init}_{init}$.
To start the evaluation with TSM and ADAM, there are almost no visible differences between ntD=10 and ntD=40, with a large difference between
the best and the least good approximation, see first row in Figure 7. Only for ntD=20 the solutions tend to be more similar.
In contrast, the difference between the best and the least good approximation for TSM and cBP grows by one order of magnitude with a higher number of training data
while simultaneously the accuracy for the best approximations increases, cf. second row in the figure.
The reason we show results on vBP only in this final experiment (see third row in the figure) is, that the efficiency of an adaptive stepsize method may be in general highly
dependent on the used stepsize model and parameters. However, the results turn out to be interesting. In combination with ntD=10, vBP and TSM reveal several minima far away
from the best approximation. Even more minima appear for a training data increase to ntD=20. However, another increase to ntD=40 stabilises the solutions.
In addition, ntD=40 provides the best approximations for TSM and vBP. One may conjecture here, that either one has here to reach a critical number of training data, or
that the weight initialisation here is not adequate together with lower ntD.
Now we turn to mTSM and ADAM, see first row in Figure 8. We find $\vec{p}^{~{}init}_{const}$ to show reliable results ($\Delta$u${}_{const}$)
and a small gain in accuracy for higher ntD. For $\vec{p}^{~{}init}_{rnd}$, we find the best approximations throughout the whole experiment to be provided by ntD=10.
However, most of the 1e5 computed results appear around a less good (but still reasonable) accuracy with only a few results peaking further in accuracy.
Increasing the number of training data to ntD=20 and ntD=40 results in a drop of accuracy from the former best solutions, while overall the results become
more similar.
For mTSM and cBP, see second row in the figure, we find a similar behaviour of $\Delta$u${}_{const}$, similarly to the case mTSM and ADAM. The solutions become
slightly more accurate and similar with higher ntD. However both weight initialisation methods can not compete with the combination mTSM and ADAM.
Now for mTSM and vBP as displayed by the last row in the figure, we find stable results for all ntD, which is in sharp contrast to TSM and vBP.
Again, $\Delta$u${}_{const}$ behaves like the other computations for mTSM, and we find similarities in the overall behaviour of $\Delta$u${}_{rnd}$ compared to TSM and cBP.
Increasing ntD leads to slightly better approximations, while the difference between the best and the least good approximation grows.
Concluding, the overall best performance related to the numeric error shows mTSM and ADAM for both $\vec{p}^{~{}init}_{const}$ and $\vec{p}^{~{}init}_{rnd}$.
Although TSM and vBP appear to have some stability flaws for lower ntD, it stabilises for ntD=40. Overall, both vBP and cBP can not compete with ADAM and mTSM.
6 Conclusion and future work
When solving the stiff model ODE with feedforward neural networks, the solution reliability depends on a variety of parameters. We find the weight
initialisation to have a major influence. While the initialisation with zeros does not provide reasonable appromixations for TSM with one hidden layer, it is capable
to work reasonably well for mTSM. First setting the weights to small random values shows the best results with ADAM and mTSM, although the use of more training data
may yield less reliable results. This may indicate an overfitting and could be resolved by employing more neurons or other adjustments.
This may be a subject for a future study.
However, our work also indicates that all the investigated issues may have to be considered together as a complete package i.e. the investigated aspects may
not be evaluated completely independent of each other. Even after a detailed investigation as provided here it seems not to be possible to single out an individual aspect
that dominates the overall accuracy and reliability.
We tend to favour the combination of ADAM and mTSM in further computationally oriented research, since it provides the best approximations for both weight initialisation
methods. Future research may also include theoretical work e.g. on sensitivity and different trial solution forms for TSM. One main goal in this context is to decrease the
variation of possible solutions together with an increase of the solution accuracy.
Moreover, our third experiment has shown that it may make sense to investigate deep networks,
since these could result in a significant accuracy gain, reminding of higher order effects in
classic numerical analysis.
Acknowledgement
This publication was funded by the Graduate Research School (GRS) of the Brandenburg University of Technology Cottbus-Senftenberg. This work is part of the
Research Cluster Cognitive Dependable Cyber Physical Systems.
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Asymptotic dimension of coarse spaces via maps to simplicial complexes
M. Cencelj
IMFM, Pedagoška fakulteta, Jadranska ulica 19,
SI-1111 Ljubljana,
Slovenija
[email protected]
,
J. Dydak
University of Tennessee, Knoxville, TN 37996, USA
[email protected]
and
A. Vavpetič
Fakulteta za Matematiko in Fiziko,
Univerza v Ljubljani,
Jadranska ulica 19,
SI-1111 Ljubljana,
Slovenija
[email protected]
(Date:: November 26, 2020)
Abstract.
It is well-known that a paracompact space $X$ is of covering dimension at most $n$
if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton
$K^{(n)}$. We use the same idea to characterize asymptotic dimension in the coarse category
of arbitrary coarse spaces. Continuity of the map $f$ is replaced by variation of $f$ on elements
of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.
Key words and phrases:asymptotic dimension, coarse geometry, Lipschitz maps, Property A
2000 Mathematics Subject Classification: Primary 54F45; Secondary 55M10
This research was supported by the Slovenian Research
Agency grants P1-0292-0101, J1-6721-0101, J1-5435-0101
Contents
1 Introduction
1. Introduction
It is well-known (see [Dyd]) that the covering dimension $\mathrm{dim}(X)$
of a paracompact space can be defined as the smallest integer $n$
with the property that any commutative diagram |
On nonexistence and existence of positive global solutions to heat equation with a potential term on Riemannian manifolds
Qingsong Gu
Department of Mathematics and Statistics, Memorial University of Newfoundland, A1C 5S7, NL, Canada.
[email protected]
,
Yuhua Sun
School of Mathematical Sciences and LPMC, Nankai University, 300071
Tianjin, P. R. China
[email protected]
and
Fanheng Xu
School of Mathematical Sciences and LPMC, Nankai University, 300071
Tianjin, P. R. China
[email protected]
(Date:: January 05, 2019)
Abstract.
We reinvestigate nonexistence and existence of global positive solutions to heat equation with
a potential term
on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.
Key words and phrases:heat equation with potential term;
Riemannian manifolds; sharp volume growth
1991 Mathematics Subject Classification: Primary: 35J61, Secondary: 58J05
Sun was supported by the National Natural Science Foundation of China (No.11501303, No.11871296,
No.11761131002),
and
also by the Fundamental Research Funds for the Central Universities.
1. Introduction
In this paper we investigate nonexistence and existence of global positive solutions to the
following problem
$$\left\{\begin{array}[]{ll}{{\partial_{t}u}=\Delta u-V(x)u+{u^{p}}}\quad\mbox{%
in $M\times(0,\infty)$},\\
{u(x,0)={u_{0}(x)}}\quad\mbox{in $M$},\end{array}\right.$$
(1.1)
where $p>1$, and $M$ is
a connected non-compact
geodesically complete Riemannian manifold with $dimM\geq 3$,
$\Delta$ is the Laplace-Beltrami operator on $M$, $V(x)$ is a smooth function and can be allowed to
be negative,
and ${u_{0}}$ is a nonnegative function which is not identically zero.
The main objective of this paper is to illustrate the following questions:
1.
What are the influences of potential $V$ and $p$ on the
nonexistence and existence of global positive solutions to problem (1.1)?
2.
Are these influences of $p$ sharp in some kind of sense
for different potential $V$?
Before answering these questions, let us firstly recall some history in this area.
When $M=\mathbb{R}^{N}$, problem (1.1) and its variations have been investigated widely in different respects, see [2, 3, 10, 31, 32], and also a very good survey paper by Levine [21].
Among these literatures,
the first celebrated result on problem (1.1) is due to Fujita’s famous paper [9] dealing
with the case when $M=\mathbb{R}^{N}$ and $V(x)\equiv 0$.
He proved that
(1)
If $1<p<1+\frac{2}{N}$, and $u_{0}>0$, then (1.1) possesses no global positive solution.
(2)
If $p>1+\frac{2}{N}$, and $u_{0}$
is smaller than a small Gaussian, then (1.1) has global solutions.
Here the number $1+\frac{2}{N}$ is called the Fujita exponent, and usually denoted
by $p^{*}$. The question of whether $p^{*}=1+\frac{2}{N}$
belongs to the blow-up case is much more difficult.
The case $p=1+\frac{2}{N}$ was decided by Hayakawa [16] for $N=1,2$ and by Kobayashi, Sirao and Tanaka [18] for general $N$. One can also see the papers
[1],[32] for different methods and further developments.
Zhang investigated problem (1.1) when
$V(x)$ has the asymptotic behavior like
$\frac{\omega}{1+|x|^{b}}$
for some $\omega\neq 0$ and $b>0$. He showed that
Theorem 1.1.
[35, Zhang]
Let $M=\mathbb{R}^{N}$ with $N\geq 3$.
(1)
If, for some $b>2$ and $\omega>0$, $0\leq V(x)\leq\frac{\omega}{1+|x|^{b}}$ holds, then
$p^{*}=1+\frac{2}{N}$;
(2)
If, for some $b\in(0,2)$ and $\omega>0$, $V(x)\geq\frac{\omega}{1+|x|^{b}}$ holds, then $p^{*}=1$ and there exists global solutions for all $p>1$;
(3)
If, for some $b>2$ and $\omega<0$ with
$|\omega|$ small enough, $\frac{\omega}{1+|x|^{b}}\leq V(x)\leq 0$ holds, then $p^{*}=1+\frac{2}{N}$;
(4)
If, for some $b\in(0,2)$ and $\omega<0$, $V(x)\leq\frac{\omega}{1+|x|^{b}}$ holds,
then $p^{*}=\infty$, which means there exist no
global solutions to (1.1) for any $p>1$.
When $V(x)$ behaves like $\frac{\omega}{|x|^{2}}$ for some $\omega>0$ and large $|x|$, Ishige proved that
Theorem 1.2.
[20, Ishige]
Let $M=\mathbb{R}^{N}$ with $N\geq 3$. Assume that $V(x)\geq 0$. Let $\omega>0$.
(1)
If $V(x)\geq\frac{\omega}{|x|^{2}}$ for large $x$, then for $p>p^{*}(\omega)$, there exists global positive
solution to (1.1);
(2)
If $V(x)\leq\frac{\omega}{|x|^{2}}$ for large $x$, then for $1<p\leq p^{*}(\omega)$, there exists no global positive solution to (1.1);
where
$$p^{*}(\omega)=1+\frac{2}{N+\alpha(\omega)},$$
(1.2)
and
$$\alpha(\omega)=\frac{-(N-2)+\sqrt{(N-2)^{2}+4\omega}}{2}$$
(1.3)
is the larger root of the equation $\alpha(\alpha+N-2)=\omega$.
When $V(x)$ behaves like $\frac{\omega}{|x|^{2}}$, and $\omega$ can be allowed to be negative satisfying $-\frac{(N-2)^{2}}{4}\leq\omega<0$, Pinsky obtained that
Theorem 1.3.
[27, Pinsky]
Let $M=\mathbb{R}^{N}$ with $N\geq 3$.
(1)
If $V(x)\geq\frac{\omega}{|x|^{2}}$, then there exists global solution to (1.1)
when $p>p^{*}(\omega)$;
(2)
If $V(x)\leq\frac{\omega}{|x|^{2}}$, for large $|x|$, then there are no global solutions
to (1.1) when $1<p\leq p^{*}(\omega)$.
Now let us transfer our attentions from Euclidean space to
manifold. We make a rough assumption on manifold: assume that $M$ is a connected non-compact
geodesically complete Riemannian manifold, $d$ is the geodesic distance on $M$, and $\mu_{0}$ is the Riemannian
measure of $M$. Fix a reference point $x_{0}\in M$, let $B(x_{0},r)$ denote the geodesic ball
on $M$ centered at $x_{0}$ with radii $r>0$.
The study of nonlinear parabolic equations on manifolds become more and more
intriguing, not only because that it has so many applications in geometry and many other areas,
but also because usually the approach which is applied for the manifold case is quite different from the Euclidean ones.
In [34], Zhang provided a unified approach to obtain
blow-up results for several variations of problem (1.1) when $V(x)=0$.
To cite his result more precisely, let us introduce his assumptions on the manifold
(i).
$\mu_{0}(B(x,r))\leq Cr^{\alpha}$, when $r$ is large and for all $x\in M$.
(ii).
$\frac{\partial\log g^{\frac{1}{2}}}{\partial r}\leq\frac{C}{r}$, where $r=d(x_{0},x)$ is smooth. Here $x_{0}$ is a fixed reference point, and $g^{\frac{1}{2}}$ is the volume density of the manifold.
Zhang obtained Fujita exponent of problem (1.1) when $V(x)=0$.
Theorem 1.4.
[34, Zhang]
Assume conditions (i) and (ii) on manifold are satisfied, and $\alpha\geq 1$. If $1<p\leq 1+\frac{2}{\alpha}$, then problem (1.1)
possesses no global positive solution to (1.1).
The approach applied by Zhang in [34] is quite powerful, and even very effective to nonlinear homogeneous and
inhomogeneous equations, semilinear parabolic equations and porous medium equations with nonlinear source, even to the blow-up problems in exterior domains [36].
Zhang’s approach is by first constructing a suitable integral functional to show that the integral functional in selected
fixed domain will blow-up or will be identically equal to zero, then one can derive the blow-up results of nonlinear parabolic equations on manifolds.
However, after a very careful examination of Zhang’s paper [34], one can find that the assumptions (i) and (ii) on manifold are essential in his approach, either can not
be relaxed or can not be dropped,
and also, the paper [34]
needs to deal with the critical case in a separate way to obtain the blow-up results.
In [23], Mastrolia, Monticelli and Punzo investigated the problem (1.1)
with $V(x)\equiv 0$
$$\left\{\begin{array}[]{ll}\partial_{t}u=\Delta u+u^{p}\quad\mbox{in $M\times(0%
,\infty)$},\\
{u(x,0)={u_{0}(x)}}\quad\mbox{in $M$},\end{array}\right.$$
(1.4)
They showed that Zhang’s result can be improved: assumption (ii) can be dropped and assumption (i) can be relaxed to
a milder version
$$\mu_{0}(B(x_{0},r))\leq Cr^{\alpha}\ln^{\frac{\alpha}{2}}r,\quad\mbox{for %
large enough $r$},$$
(1.5)
for some reference point $x_{0}$, the same result still holds.
Their technique is to multiply the equation (1.1) by $u^{a}\varphi^{b}$, and to obtain an
integral estimate involving $u$ to show the nonexistence results.
This technique is called the nonlinear capacity method,
which is systematically studied by Mitidieri and Pohozaev to deal with the elliptic inequality and parabolic differential inequalities.
Let us refer to [4, 5, 24, 25] for more details.
Here we point out that their proof relies on a very delicate choice of test function $\varphi$. Moreover, the sharpness of $\frac{\alpha}{2}$ is not shown in their paper [23].
In this paper, the purpose of the paper is threefold:
the first one is to provide a sufficient condition for
the nonexistence of global solution to problem (1.1) with general $V$; the second one is to
attempt to
show a unified approach to deal with the parabolic equation with the potential term, moreover,
we present a totally different test function $\varphi$ from the one used in [23]; the third one is to
show the sharpness of the (general) volume assumption of $\frac{\alpha}{2}$, which has not been shown before.
The idea of using the upper bound of volume of geodesic ball to derive Liouville’s
uniqueness type result has already been widely used
in literature. It originated from the celebrated work of Cheng and Yau [7]. They proved that if
on a geodesically complete
Riemannian manifold $M$, for some reference point $x_{0}\in M$, the following
$$\mu_{0}(B(x_{0},r))\leq Cr^{2},$$
holds for all large enough $r$, then any non-negative superharmonic function on $M$ is identically constant.
For other
related studies in this area we refer the readers to [11, 13, 23, 30].
Our paper is inspired by the elliptic results in [14], [15] and [29], and
parabolic results in [23].
In the paper [14], Grigor’yan and the second author investigated the following differential inequality on $M$
$$\Delta u+u^{\sigma}\leq 0,$$
(1.6)
and proved that if, for some reference point $x_{0}\in M$ and $\alpha>2$, the following
$$\mu_{0}(B(x_{0},r))\leq Cr^{\alpha}\ln^{\frac{\alpha-2}{2}}r,$$
(1.7)
holds for all large enough $r$, then, for any $\sigma\leq\frac{\alpha}{\alpha-2}$,
the only nonnegative solution to (1.6) is identically equal to zero.
They also showed the exponents $\alpha$ and $\frac{\alpha-2}{2}$
in (1.7) are sharp, and can not be relaxed.
Otherwise, there exists some model manifold which satisfies (1.7) and admits positive solution
to (1.6).
The main technique applied in [14] relies on a very delicate choice of test function on manifolds.
Recently in [15], Grigor’yan, the second author and Verbitsky generalized the above results to the integrated
form, they obtained the necessary and sufficient condition for the existence of positive solutions
in terms of Green function of $\Delta$. Especially, when $M$ has nonnegative Ricci curvature, they showed that problem (1.7) admits a positive $C^{2}\text{-}$solution
if and only if
$$\int_{r_{0}}^{\infty}\frac{r^{\sigma-1}}{[\mu_{0}(B(x_{0},r))]^{\sigma-1}}dr<\infty,$$
(1.8)
for some reference point $x_{0}$ and $r_{0}>0$.
Further in [29], the second author used two different test functions to show
that if the volume of geodesic ball satisfies some suitable growth, then the uniqueness result
of nonnegative solutions for semi-linear elliptic differential inequalities holds.
Throughout the paper, we require that $V$ admits a smooth positive solution to
$$\Delta h=Vh,$$
(1.9)
on $M$.
Actually, such a solution $h$ exists widely, for example,
Lemma 1.5.
[12, Lemmas 10.1 and 10.3] For any smooth
non-negative function $\Psi$ on $M$, there exists a smooth positive
function $h$ such that
$$\Delta h=\Psi h\quad\text{on }M.$$
(1.10)
If in addition $\Psi$ is Green bounded, namely,
$$\sup_{x\in M}\int_{M}G(x,y)\Psi(y)d\mu_{0}(y)<\infty,$$
(1.11)
then the equation (1.10) has a solution $h\asymp 1$ on $M$.
Here, $G(x,y)$ is a finite positive Green function with respect to $\Delta$ on $M$, and
the sign $\asymp$ means the ratio of the left-hand and right-hand is bounded from above and below
by two positive constants.
We then apply the technique of Doob’s $h\text{-}$transform. Consider the weighted manifold $(M,\mu)$,
where $\mu$ is a measure on $M$ defined by
$$d\mu:=h^{2}d\mu_{0}.$$
(1.12)
The weighted Laplacian $\tilde{\Delta}$ of $(M,\mu)$ is defined by
$$\tilde{\Delta}:=\frac{1}{h^{2}}\mathop{\mathrm{div}}\nolimits(h^{2}\nabla).$$
In particular, if $h\equiv 1$ then $\tilde{\Delta}$ is the Laplace-Beltrami
operator $\Delta$ on $M$.
By using $\Delta h=Vh$, for any smooth function $v(x)$, we know
$$\tilde{\Delta}v+Vv=(\Delta v+2\frac{\nabla h\cdot\nabla v}{h})+\frac{\Delta h}%
{h}v=\frac{1}{h}(h\Delta v+2\nabla h\cdot\nabla v+\Delta hv)=\frac{\Delta(hv)}%
{h},$$
Whence
$$\tilde{\Delta}v=\frac{1}{h}(\Delta(hv)-Vhv),$$
and
$$\tilde{\Delta}=\frac{1}{h}\circ(\Delta-V)\circ h.$$
Let $u$ be a smooth positive solution to (1.1) and let $u=hv$, we know from the above $v$
is a smooth positive global solution to the following Cauchy problem
$$\left\{\begin{array}[]{ll}{{\partial_{t}v}=\tilde{\Delta}v+{h^{p-1}v^{p}}}%
\quad\mbox{in $M\times(0,\infty)$},\\
{v(x,0)={v_{0}(x)}}\quad\mbox{in $M$},\end{array}\right.$$
(1.13)
where $v_{0}(x)=\frac{u_{0}}{h}(x)$.
Conversely, if $v$ is a smooth positive solution to problem (1.13), then
$u=hv$ is a solution to (1.1) with $u_{0}=hv_{0}$. Hence, the two problems
(1.1) and (1.13) are equivalent in the classical sense so that we only need to deal with
(1.13) in the following. Actually, problems (1.1)
and (1.13) can also be seen equivalent from the weak sense in the below.
Denote by $W_{loc}^{1,2}\left(M,d\mu\right)$ the space of functions $f\in L_{loc}^{2}\left(M,d\mu\right)$ whose weak gradient $\nabla f$ is also in $L_{loc}^{2}\left(M,d\mu\right)$. Denote by $W_{c}^{1,2}\left(M,d\mu\right)$ the subspace of $W_{loc}^{1,2}\left(M,d\mu\right)$ of functions with compact support. Spaces
$W_{loc}^{1,2}(M\times[0,\infty),d\mu dt),W_{c}^{1,2}(M\times[0,\infty),d\mu dt)$
are defined similarly.
Definition 1.6.
$v$ is called a global weak solution to (1.13)
if $v$ is a nonnegative $W_{loc}^{1,2}(M\times[0,\infty),d\mu dt)$ function,
and for any nonnegative function $\psi\in W_{c}^{1,2}(M\times[0,\infty),d\mu dt)$, the following holds
$$\displaystyle\int_{M}\psi(x,0)v_{0}d\mu+\int_{0}^{\infty}\int_{M}[v\partial_{t%
}\psi-(\nabla v,\nabla\psi)+h^{p-1}v^{p}\psi]d\mu dt=0.$$
(1.14)
Remark 1.7.
From Definition 1.6, we know if $v$ is a weak solution to (1.1), and $v_{0}$ is nonnegative, we obtain, for any
nonnegative function $\psi\in W_{c}^{1,2}(M\times[0,\infty),d\mu dt)$
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\psi d\mu dt\leq\int_{0}^{%
\infty}\int_{M}(\nabla v,\nabla\psi)d\mu dt-\int_{0}^{\infty}\int_{M}v\partial%
_{t}\psi d\mu dt.$$
(1.15)
Before presenting the main results, we introduce some notations.
Let us define
$$\displaystyle P:=\frac{2}{{p-1}},\quad Q:=\frac{1}{{p-1}},$$
(1.16)
and a new measure $\nu$ on $M$ by
$$d\nu=h^{-1}d\mu=hd\mu_{0}.$$
(1.17)
We say that condition $(H)$ holds: if $\Delta h=Vh$ admits a smooth positive solution
$h$ and
there exist two nonnegative constants $\delta_{1},\delta_{2}$, and some reference point $x_{0}$ such that
$$cr^{-\delta_{1}}\leq h(x)\leq Cr^{\delta_{2}},\quad\mbox{for large enough $r=d%
(x,x_{0})$}.$$
($$H$$)
Our main result is the following.
Theorem 1.8.
Assume that condition $(H)$ is satisfied on $M$. If the following
$$\displaystyle\nu(B(x_{0},r))\leq C{r^{P}}{\ln^{Q}}r,$$
(1.18)
holds for all large enough $r$,
then problem (1.1) admits no global positive solution.
Here $P$ and $Q$ are defined as in (1.16).
In particular, when $V\equiv 0$, we choose $h\equiv 1$, and hence condition $(H)$ is satisfied. By Theorem
1.8, we have
Corollary 1.9.
For $V\equiv 0$, if, for some reference point ${x_{0}}\in M$, the following
$$\displaystyle\mu_{0}(B(x_{0},r))\leq C{r^{P}}{\ln^{Q}}r,$$
(1.19)
holds for all large enough $r$,
then problem (1.1)
admits no global positive solution either.
Remark 1.10.
Theorem 1.8 and Corollary 1.9 provide us an affirmative answer to the following question: how much could we relax the assumption on the volume growth of geodesic balls to ensure that problem
(1.1)
admits no global positive solution
when the nonlinear term $u^{p}$ is fixed? In Section 4, we show the sharpness of (1.19), which means that if we relax $P,Q$ a little,
there exists a global positive solution to (1.1) on $M$ for small $u_{0}$.
Our method is to multiply the equation (1.1) by $v^{a}\varphi^{b}$
( here $a,b$ are variable parameters).
By building suitable integral estimates of $v$
and choosing suitable test function $\varphi$, we can obtain the blow-up results.
Actually, the test function $\varphi$ we use here can be considered as a parabolic version
used in [29].
Corollary 1.9 can be presented in another equivalent form
Corollary 1.11.
For $V\equiv 0$, if, for some reference point ${x_{0}}\in M$ and $\alpha>0$, the following
$$\displaystyle\mu_{0}(B(x_{0},r))\leq Cr^{\alpha}\ln^{\frac{\alpha}{2}}r,$$
(1.20)
holds for all large enough $r$. If $1<p\leq 1+\frac{2}{\alpha}$, then problem (1.1) admits no global positive solution.
Remark 1.12.
Corollary 1.11 tells us if we know the upper bound of the volume of geodesic ball, then we can determine the range of $p$ to suffice that problem
(1.1) admits no global positive solution. Here the volume upper bound
condition (1.20)
is also sharp, and can not be relaxed either, please see Theorems 1.14 and 1.15.
Corollary 1.11 is a generalization of Zhang’s result, please see Theorem 1.4.
Corollary 1.11 was first obtained by Mastrolia, Monticelli, and Punzo in [23].
We then turn to study the existence of global solutions to problem (1.1).
For that, we need slightly
strengthen our assumptions on $M$. Let $\tilde{P}_{t}(x,y)$ be the smallest fundamental solution of the heat equation
$$\partial_{t}v=\tilde{\Delta}v\quad\mbox{on $M$}.$$
We know $\tilde{P}_{t}(x,y)$ is called the heat kernel of $\tilde{\Delta}$, and has the following properties
•
Symmetry: $\tilde{P}_{t}(x,y)=\tilde{P}_{t}(y,x)$, for all $x,y\in M,t>0$.
•
Markovian property:
$\tilde{P}_{t}(x,y)\geq 0$, for all $x$, $y\in M$ and $t>0$, and
$$\int_{M}\tilde{P}_{t}(x,y)d\mu(y)\leq 1,\quad\mbox{for all $x\in M$ and $t>0$}.$$
(1.21)
•
The semigroup identity: for all $x$, $y\in M$ and $t$, $s>0$,
$$\tilde{P}_{t+s}(x,y)=\int_{M}\tilde{P}_{t}(x,z)\tilde{P}_{s}(z,y)d\mu(z).$$
(1.22)
•
Approximation of identity: for any $f\in L^{2}(M,d\mu)$,
$$\left\|\int_{M}\tilde{P_{t}}(x,y)f(y)d\mu(y)-f\right\|_{L^{2}(M,d\mu)}\to 0,%
\quad\mbox{as $t\to 0_{+}$}.$$
(1.23)
Let $P_{t}^{V}(x,y)$ denote the heat kernel of $-\Delta+V$ on $(M,\mu_{0})$.
When $V=0$, we denote by $P_{t}(x,y):=P_{t}^{0}(x,y)$ the heat kernel of $\Delta$.
When $M$ has nonnegative Ricci curvature, by famous Li-Yau estimate in [22], we have
$$P_{t}(x,y)\asymp\frac{C}{\mu_{0}(x,\sqrt{t})}\exp{\left(-\frac{d^{2}(x,y)}{ct}%
\right).}$$
(1.24)
Especially, when
$M=\mathbb{R}^{N}$
$$P_{t}(x,y)=\frac{1}{(4\pi t)^{\frac{N}{2}}}\exp\left(-\frac{|x-y|^{2}}{4t}%
\right).$$
The questions to obtain the lower bound and upper bound of heat kernels $\tilde{P}_{t}(x,y)$ and $P_{t}^{V}(x,y)$ under different geometric
conditions on the underlying manifold have been extensively studied in the past few decades, let us
refer to the papers [6, 8, 12, 13, 28].
We say $P_{t}^{V}$ satisfies the condition $(DUE)$, if $P_{t}^{V}$ has the following upper estimate
$$P_{t}^{V}(x,y)\leq\frac{C_{1}}{\mu_{0}(x,\sqrt{t})},$$
($$DUE$$)
for some constant $C_{1}$.
The heat kernels $P_{t}^{V}$ and $\tilde{P}_{t}$ are bridged by the following lemma.
Lemma 1.13.
[12, Lemma 4.7]
The heat kernels $P_{t}^{V}$ and $\tilde{P}_{t}$ have the following relation:
$$P_{t}^{V}(x,y)=\tilde{P}_{t}(x,y)h(x)h(y).$$
(1.25)
If condition $(DUE)$ is satisfied on $M$, and $V$ is Green bounded and nonnegative, by Lemma 1.13,
we have
$$\tilde{P}_{t}(x,y)\leq\frac{C}{\mu_{0}(x,\sqrt{t})}.$$
(1.26)
for some constant $C$.
Our existence result is stated as follows.
Theorem 1.14.
Assume that $V\geq 0$ is Green bounded and $P_{t}^{V}$ satisfies condition $(DUE)$.
If, for some $\varepsilon>0$, the following inequality
$$\mu_{0}(B(x_{0},r))\geq cr^{P}\ln^{Q+\varepsilon}r,$$
(1.27)
holds for all large enough $r$, then there exists a global positive solution to
(1.1) for some small $u_{0}$. Here $P,Q$ are defined as in (1.16).
Theorem 1.14 also has an equivalent form.
Theorem 1.15.
Assume that $V\geq 0$ is Green bounded and $P_{t}^{V}$ satisfies condition $(DUE)$.
Assume also, for some $\varepsilon>0$, the following inequality
$$\mu_{0}(B(x_{0},r))\geq cr^{\alpha}\ln^{\frac{\alpha}{2}+\varepsilon}r,$$
(1.28)
holds for all large enough $r$. If $p>1+\frac{2}{\alpha}$, then
there exists a global positive solution to
(1.1) for some small $u_{0}$.
The paper is organized as follows: In Section 2, we present some examples to see the applications of our main
result.
In Section 3, we give the proof of Theorem
1.8. In Section 4, we present the proof of
Theorem 1.14.
Notation.
The letters $C,C^{\prime},C_{0},C_{1},c_{0},c_{1}...$ denote positive
constants whose values are unimportant and may vary at different occurrences.
2. Some examples
In this section we present several examples to show the applications of Theorem 1.8 and
Corollary 1.11.
First, let us make some preliminary works.
Define the Riesz potential on $\mathbb{R}^{N}$ for $0<\alpha<N$ by
$$I_{\alpha}f(x)=c(N,\alpha)\int_{\mathbb{R}^{N}}\frac{f(y)}{|x-y|^{N-\alpha}}dy,$$
(2.1)
where $f\in L_{loc}^{1}(\mathbb{R}^{N})$, and $\int_{|x|\geq 1}|x|^{N-\alpha}|f(x)|dx<\infty$, and
$$c(N,\alpha)=\frac{\Gamma(\frac{N-\alpha}{2})}{\pi^{\frac{N}{2}}2^{\alpha}%
\Gamma(\frac{\alpha}{2})}.$$
Here $\Gamma(\cdot)$ is the Gamma function.
Lemma 2.1.
[17, Corollary 2.9]
If $V\leq 0$, and there exists some constant $C_{2}(N)$ such that
$$I_{1}[(I_{1}V)^{2}](x)\leq-C_{2}(N)I_{1}V(x),$$
(2.2)
then there exists a positive solution $h$ to
$$\Delta h=Vh\quad\mbox{in $\mathbb{R}^{N}$}.$$
Moreover, if
$I_{2}(-V)<\infty$, then the solution $h$ satisfies
$$\exp(-I_{2}V)\leq h\leq\exp(-C_{3}I_{2}V),$$
(2.3)
for some constant $C_{3}=C_{3}(N)>0$.
Proposition 2.2.
[33, Proposition 2.1]
Let $V(x)=\frac{1}{1+|x|^{b}}$ for some $b>2$. Then
$$\sup_{x\in\mathbb{R}^{N}}I_{2}V(x)<\infty.$$
(2.4)
Proposition 2.3.
Let $V(x)=\frac{1}{1+|x|^{b}}$ for some $b>2$. Then there exists a constant $C(N,b)>0$ such that for all $x\in\mathbb{R}^{N}$,
$$I_{1}[(I_{1}V)^{2}](x)\leq C(N,b)I_{1}V(x).$$
(2.5)
Proof.
We divide the proof into two steps.
Step 1.
We show the following estimate
$$I_{1}V(x)\asymp\left\{\begin{array}[]{ll}1,&\hbox{$|x|\leq 1$;}\\
|x|^{1-b}+|x|^{1-N}\left(1+\int_{1}^{|x|}r^{N-b-1}dr\right),&\hbox{$|x|>1$.}%
\end{array}\right.$$
(2.6)
By definition of $I_{1}V$, we have
$$I_{1}V(x)=C(N)\int_{\mathbb{R}^{N}}\frac{dy}{(1+|x-y|^{b})|y|^{N-1}}.$$
(2.7)
Firstly, we deal with the case that $|x|\leq 1$. The integral of the right hand side of (2.7) can be written as
$$\displaystyle\int_{\mathbb{R}^{N}}\frac{dy}{(1+|x-y|^{b})|y|^{N-1}}$$
$$\displaystyle=$$
$$\displaystyle\int_{|y|\leq 2|x|}\frac{dy}{(1+|x-y|^{b})|y|^{N-1}}+\int_{|y|>2|%
x|}\frac{dy}{(1+|x-y|^{b})|y|^{N-1}}$$
(2.8)
$$\displaystyle=:$$
$$\displaystyle J_{1}+J_{2}.$$
Then for $|y|\leq 2|x|$, we have $|y-x|\leq 3|x|\leq 3$, and $1+|x-y|^{b}\asymp 1$. Using polar coordinates,
we obtain
$$J_{1}\asymp\int_{|y|\leq 2|x|}\frac{dy}{|y|^{N-1}}\asymp\int_{0}^{2|x|}\frac{r%
^{N-1}}{r^{N-1}}dr\asymp|x|.$$
For $|y|>2|x|$, we have $|y|/2\leq|y-x|\leq 3|y|/2$, and $1+|x-y|^{b}\asymp 1+|y|^{b}$, then by the fact $b>2$,
we obtain
$$J_{2}\asymp\int_{|y|\geq 2|x|}\frac{dy}{(1+|y|^{b})|y|^{N-1}}=C(N)\int_{2|x|}^%
{\infty}\frac{dr}{1+r^{b}}\asymp 1.$$
By substituting the two estimates to (2.8), we obtain
$$I_{1}V(x)\asymp 1+|x|\asymp 1,$$
which is the first estimate in (2.6).
Secondly, when $|x|>1$, let us write the integral in (2.7) as
$$\int_{\mathbb{R}^{N}}\frac{dy}{(1+|x-y|^{b})|y|^{N-1}}=\int_{|y|\leq|x|/2}+%
\int_{|x|/2<|y|\leq 2|x|}+\int_{|y|>2|x|}=:K_{1}+K_{2}+K_{3}.$$
(2.9)
Then we estimate $K_{1},K_{2},K_{3}$ respectively.
For $|y|\leq|x|/2$, we have $|y-x|\asymp|x|$. Thus
$$K_{1}\asymp\frac{1}{|x|^{b}}\int_{|y|\leq|x|/2}\frac{dy}{|y|^{N-1}}\asymp|x|^{%
1-b}.$$
For $|x|/2<|y|\leq 2|x|$, we have $|y|\asymp|x|$. Thus
$$K_{2}\asymp\frac{1}{|x|^{N-1}}\int_{|x|/2<|y|\leq 2|x|}\frac{dy}{1+|x-y|^{b}}.$$
Noting that $\{y:\ |y-x|\leq|x|/2\}\subseteq\{y:\ |x|/2<|y|\leq 2|x|\}\subseteq\{y:\ |y-x|%
\leq 3|x|\}$, we have
$$\displaystyle\int_{|x|/2<|y|\leq 2|x|}\frac{dy}{1+|x-y|^{b}}\leq\int_{|z|\leq 3%
|x|}\frac{dz}{1+|z|^{b}}\asymp\int_{0}^{3}r^{N-1}dr+\int_{3}^{3|x|}r^{N-b-1}dr$$
$$\displaystyle\asymp 1+\int_{1}^{|x|}r^{N-b-1}dr,$$
and similarly,
$$\int_{|x|/2<|y|\leq 2|x|}\frac{dy}{1+|x-y|^{b}}\geq\int_{|z|\leq|x|/2}\frac{dz%
}{1+|z|^{b}}\asymp 1+\int_{1}^{|x|}r^{N-b-1}dr.$$
Combining the above estimates, we obtain
$$K_{2}\asymp|x|^{1-N}\left(1+\int_{1}^{|x|}r^{N+b-1}dr\right).$$
For $|y|>2|x|$, we have $|y-x|\asymp|y|$. Thus
$$K_{3}\asymp\int_{|y|>2|x|}\frac{dy}{(1+|y|^{b})|y|^{N-1}}\asymp\int_{|y|>2|x|}%
\frac{dy}{|y|^{b+N-1}}\asymp|x|^{1-b}.$$
By substituting the estimates of $K_{1},K_{2},K_{3}$ into (2.9), we obtain the second estimate in (2.6).
Step 2.
Now we apply (2.6) to show (2.5) with $V=\frac{1}{1+|x|^{b}}$. We separate the proof into two cases.
Case of $N\geq b$. We show that there is a constant $C=C(N,b)>0$ such that for all $x\in\mathbb{R}^{N}$,
$$(I_{1}V(x))^{2}\leq CV(x),$$
(2.10)
and (2.5) follows immediately by taking $I_{1}$ on both sides of (2.10).
When $|x|\leq 1$, (2.10) is true, since we have
$$(I_{1}V)(x)\asymp V(x)\asymp 1.$$
When $|x|>1$, by (2.6), we have
$$(I_{1}V)(x)\asymp\left\{\begin{array}[]{ll}(1+\log|x|)|x|^{1-b},&\hbox{$N=b$,}%
\\
|x|^{1-b},&\hbox{$N>b$.}\end{array}\right.$$
Noting $b>2$, we have
$$(I_{1}V(x))^{2}\leq C(1+\log|x|)^{2}|x|^{2-2b}\leq C|x|^{-b}\leq CV(x),$$
which proves (2.10).
Case of $N<b$. By (2.6), we have
$$I_{1}V(x)\asymp\left\{\begin{array}[]{ll}1,&\hbox{$|x|\leq 1$,}\\
|x|^{1-N},&\hbox{$|x|>1$.}\end{array}\right.$$
(2.11)
By definition
$$I_{1}[(I_{1}V)^{2}](x)=C(N)\int_{\mathbb{R}^{N}}\frac{(I_{1}V)^{2}(y)dy}{|x-y|%
^{N-1}}.$$
(2.12)
Let us first consider $|x|\leq 1$. Applying (2.11), we obtain
$$\displaystyle\int_{\mathbb{R}^{N}}\frac{(I_{1}V)^{2}(y)dy}{|x-y|^{N-1}}$$
$$\displaystyle=$$
$$\displaystyle\int_{|y|\leq 2}\frac{(I_{1}V)^{2}(y)dy}{|x-y|^{N-1}}+\int_{|y|>2%
}\frac{(I_{1}V)^{2}(y)dy}{|x-y|^{N-1}}$$
$$\displaystyle\asymp$$
$$\displaystyle\int_{|y|\leq 2}\frac{dy}{|x-y|^{N-1}}+\int_{|y|>2}|y|^{2-2N}%
\frac{dy}{|x-y|^{N-1}}$$
$$\displaystyle\leq$$
$$\displaystyle\int_{|z|\leq 3}\frac{dz}{|z|^{N-1}}+\int_{|y|>2}\frac{dy}{|y|^{3%
N-3}}$$
$$\displaystyle\asymp$$
$$\displaystyle 1,$$
which together with $I_{1}V(x)\asymp 1$, implies for $|x|\leq 1$,
$$I_{1}[(I_{1}V)^{2}](x)\leq CI_{1}V(x).$$
Then we consider $|x|>1$. Rewrite the integral in (2.12) as
$$\displaystyle\int_{\mathbb{R}^{N}}\frac{(I_{1}V)^{2}(y)dy}{|x-y|^{N-1}}$$
$$\displaystyle=$$
$$\displaystyle\left(\int_{|y|\leq 1/2}+\int_{1/2<|y|\leq|x|/2}+\int_{|x|/2<|y|%
\leq 2|x|}+\int_{|y|>2|x|}\right)\frac{(I_{1}V)^{2}(y)dy}{|x-y|^{N-1}}$$
$$\displaystyle=:$$
$$\displaystyle L_{1}+L_{2}+L_{3}+L_{4}.$$
We estimate $L_{i}(i=1,2,3,4)$ as follows.
For $|y|\leq 1/2$, we have by (2.11) that $I_{1}V(y)\asymp 1$, thus
$$L_{1}\asymp\int_{|y|\leq 1/2}\frac{dy}{|x-y|^{N-1}}\asymp\int_{|y|\leq 1/2}%
\frac{dy}{|x|^{N-1}}\asymp|x|^{1-N}.$$
For $1/2<|y|\leq|x|/2$, we have by (2.11) that $(I_{1}V(y))^{2}\asymp|y|^{2-2N}$, and $|x-y|\asymp|x|$, thus
$$L_{2}\asymp\frac{1}{|x|^{N-1}}\int_{1/2<|y|\leq|x|/2}{|y|^{2-2N}dy}\asymp\frac%
{1-|x|^{2-N}}{|x|^{N-1}}.$$
For $|x|/2<|y|\leq 2|x|$, we have $(I_{1}V(y))^{2}\asymp|y|^{2-2N}\asymp|x|^{2-2N}$, thus
$$L_{3}\asymp{|x|^{2-2N}}\int_{|x|/2<|y|\leq 2|x|}\frac{dy}{|x-y|^{N-1}}\asymp|x%
|^{3-2N}.$$
For $|y|>2|x|$, we have $(I_{1}V(y))^{2}\asymp|y|^{2-2N}$, and $|x-y|\asymp|y|$, thus
$$L_{4}\asymp\int_{|y|>2|x|}\frac{|y|^{2-2N}dy}{|y|^{N-1}}\asymp{|x|^{3-2N}}.$$
Combing the above estimates, we obtain
$$\displaystyle I_{1}[(I_{1}V)^{2}](x)$$
$$\displaystyle\asymp L_{1}+L_{2}+L_{3}+L_{4}$$
$$\displaystyle\asymp|x|^{1-N}+\frac{1-|x|^{2-N}}{|x|^{N-1}}+|x|^{3-2N}+{|x|^{3-%
2N}}$$
$$\displaystyle\asymp|x|^{1-N}.$$
Thus applying (2.11), we obtain for $|x|>1$,
$$I_{1}[(I_{1}V)^{2}](x)\leq CI_{1}V(x).$$
Hence, (2.5) also holds for the case of $N<b$. The proof is complete.
$\square$
Lemma 2.4.
If $V(x)=\frac{\omega}{1+|x|^{b}}$ for some $\omega<0$ and $b>2$, then there exists a
positive solution $h$ to
$$\Delta h=V(x)h.$$
Moreover, $h\asymp 1$.
Proof.
Combining Lemma 2.1 and Proposition 2.3, we obtain there exists a positive solution
$h$ to
$$\Delta h=Vh,$$
and by Proposition 2.2, we have
$$\sup_{x\in\mathbb{R}^{N}}I_{2}(-V)<\infty.$$
(2.13)
Hence, from (2.14), we obtain
$$h\asymp 1.$$
$\square$
Lemma 2.5.
[19, Lemma 2.2]
Assume that $V$ satisfies the following conditions for some $\omega>0$ and $\theta>0$
(1)
$V=V(|x|)\in C^{1}(\mathbb{R}^{N})$, and $V(r)\geq 0$ on $[0,\infty)$,
(2)
$\sup\limits_{r\geq 1}r^{2+\theta}|V(r)-\frac{\omega}{r^{2}}|<\infty$,
(3)
$\sup\limits_{r\geq 1}|r^{3}V^{\prime}(r)|<\infty$.
Then there exists a unique $C^{2}$ solution $h(r)>0$ to
$$\Delta h=hV,\quad\mbox{in $\mathbb{R}^{N}$},$$
such that
$$h(r)\asymp r^{\alpha(\omega)},\quad\mbox{for large enough $r$}.$$
(2.14)
where
$\alpha(\omega)$ is defined as in (1.3).
Example 2.6.
Let $V(x)=0$, and $M=\mathbb{R}_{g}^{k}\times\mathbb{S}^{l}$ be endowed with product metric.
Here $\mathbb{R}_{g}^{k}=(\mathbb{R}^{k},g)$ is a model manifold with induced metric
$g=dr^{2}+\psi(r)^{2}d\theta^{2}$, where $(r,\theta)$ is the polar coordinates in $\mathbb{R}^{k}$, and
$\psi(r)$ is a smooth, positive function on $(0,\infty)$ such that
$$\psi(r)=\left\{\begin{array}[]{ll}r,\quad\mbox{for small $r$},\\
\left(r^{\alpha-1}\ln^{\frac{\alpha}{2}}r\right)^{\frac{1}{k-1}},\quad\mbox{%
for large $r$}.\end{array}\right.$$
If $V(x)=0$, we could choose $h=1$, and hence, in (1.17) $d\nu=d\mu=d\mu_{0}$.
Then the volume of the ball $B_{r}:=B_{r}(0)$ in $\mathbb{R}_{g}^{k}$ can be determined by
$$\displaystyle\mu_{0}(B_{r})=\int_{0}^{r}S(\tau)d\tau,$$
where
$S$ is the surface area defined by
$$S(r)=\left\{\begin{array}[]{ll}r^{k-1},\quad\mbox{for small $r$},\\
r^{\alpha-1}\ln^{\frac{\alpha}{2}}r,\quad\mbox{for large $r$}.\end{array}\right.$$
Hence, we obtain
$$\displaystyle\mu_{0}(B_{r})\leq Cr^{\alpha}\ln^{\frac{\alpha}{2}}r,\quad\mbox{%
for large enough $r$}.$$
If follows that the geodesic ball $B(0,r)$ in M satisfies
$$\displaystyle\mu_{0}(B(0,r))\leq Cr^{\alpha}\ln^{\frac{\alpha}{2}}r,\quad\mbox%
{for large enough $r$}.$$
Applying Corollary 1.11, we derive that when $p\leq 1+\frac{2}{\alpha}$, then
(1.1) on $\mathbb{R}_{g}^{k}\times\mathbb{S}^{l}$ admits no global positive solution. Especially, when $\mathbb{R}_{g}^{k}=\mathbb{R}^{k}$, we know that the critical exponent
for $\mathbb{R}^{k}\times\mathbb{S}^{l}$ is
$1+\frac{2}{k}$.
Example 2.7.
When $M=\mathbb{R}^{N}$, we consider the following classes of $V(x)$.
(1)
If $0\leq V(x)\leq\frac{\omega}{1+|x|^{b}}$ for some $b>2$ and $\omega>0$, we know
by Proposition 2.2
$$\sup_{x\in\mathbb{R}^{N}}I_{2}V<\infty,$$
which means that $V(x)$ is Green bounded. By Lemma 1.5, we know that
$$\Delta h=Vh,$$
admits a solution
$h\asymp 1$.
Noting that
$$\nu(B(0,r))=\int_{B(0,r)}hdx\asymp r^{N}.$$
(2.15)
By Theorem 1.8, we know if
$$\nu(B(0,r))\leq Cr^{P}\ln^{Q}r,$$
or more precisely, when
$$p\leq 1+\frac{2}{N}.$$
there exists no global solution to (1.1).
This result also covers the result (1) of Theorem 1.1.
(2)
When $0\leq V(x)\leq\frac{\omega}{|x|^{2}}$, for large $|x|$, by employing Comparison principle, we can replace $V(x)$ by $\frac{\omega}{|x|^{2}}(1+|x|^{-\theta})$ for large $|x|$ still denoted by $V(x)$.
If we can show that (1.1) admits no global positive solution with
$V(x)=\frac{\omega}{|x|^{2}}(1+|x|^{-\theta})$ for large $|x|$, then the original problems admits
no global positive solution by Comparison principle.
Applying Lemma 2.5, we know the following problem with $V(x)=\frac{\omega}{|x|^{2}}(1+|x|^{-\theta})$ for large $|x|$
$$\Delta h=Vh,\quad\mbox{in $\mathbb{R}^{N}$},$$
(2.16)
admits a unique solution $h>0$ such that
$$h(x)\asymp|x|^{\alpha(\omega)},\mbox{for large $|x|$}.$$
(2.17)
where $\alpha(\omega)$ is defined as in (1.3).
For large $r$, we obtain that
$$\nu(B(0,r))=\int_{B(0,r)}hdx\asymp r^{N+\alpha(\omega)},\quad\mbox{for large $%
r$}.$$
(2.18)
Applying Theorem 1.8, we obtain that
when $p\leq 1+\frac{2}{N+\alpha(\omega)}$, there exists no global positive solution to (1.1).
In this case, the result is also in accordance with the (2) in Theorem 1.2.
(3)
When $\frac{\omega}{1+|x|^{b}}\leq V(x)\leq 0$ for some $\omega<0$, and $b>2$.
By Lemma 2.4, we obtain that $h\asymp 1$. Applying Theorem 1.8, we obtain that
when $p\leq 1+\frac{2}{N}$, there exists no global positive solution to (1.1).
In this case, the result is also in accordance with the (3) in Theorem 1.1. Actually, we remove the restriction that $\omega$ is small enough, and we improve the result obtained in Theorem 1.1.
(4)
When $V=\frac{\alpha(\omega)N+\omega|x|^{2}}{(1+|x|^{2})^{2}}$ for $\omega\in[-\frac{(N-2)^{2}}{4},0)$, we know
$$V(x)\leq\frac{\omega}{1+|x|^{2}},$$
and $\Delta h=Vh$ admits a solution
$h(x)=(1+|x|^{2})^{\frac{\alpha(\omega)}{2}}$.
By Theorem 1.8, we know if $p\leq 1+\frac{2}{N+\alpha(\omega)}$, there exists no global positive solution to (1.1).
Remark 2.8.
Here we can not cover the case of $V(x)\leq\frac{\omega}{1+|x|^{2}}\leq 0$, the difficulty is that we
do not know the
asymptotic behavior of $h$ when $|x|\to\infty$. However, we conjecture that when $V(x)$ behaves like $\frac{\omega}{1+|x|^{2}}$, $\Delta h=Vh$ admits a solution $h\asymp|x|^{\alpha(\omega)}$
for large $|x|$.
Example 2.9.
Assume that $M$ satisfies
$$\mu_{0}(B(x_{0},r))\leq Cr^{\alpha},\quad\mbox{for large enough $r$},$$
(2.19)
and
$$G(x,y)\asymp d(x,y)^{2-\alpha},\quad\mbox{for large enough $d(x,y)$}.$$
Let $V(x)\asymp\frac{\omega}{1+|x|^{b}}$ for $b>2$, and $\omega>0$, we know
$$\sup_{x}\int_{M}G(x,y)V(y)dy<\infty,$$
(2.20)
hence $V(x)$ is Green bounded. Hence by Lemma 1.5, we know there exists a function
$h(x)\asymp 1$ satisfying $\Delta h=Vh$. Applying Theorem 1.8, we know if
$$p\leq 1+\frac{2}{\alpha}$$
(2.21)
then there is no positive global solution to (1.1).
3. Nonexistence of global positive solution
Proof of Theorem 1.8.
Let $\varphi\in W_{c}^{1,2}(M\times[0,+\infty),d\mu dt)$ be a function satisfying $0\leq\varphi\leq 1$, $\varphi\equiv 1$ in a neighborhood of $D_{R}:=\overline{B(x_{0},R)}\times[0,{R^{2}}]$. Define
$$\displaystyle\psi(x,t)=v(x,t)^{-a}\varphi(x,t){{}^{b}},$$
(3.1)
where $a$ will take arbitrarily small positive value near zero, and $b$ will be chosen to be a large enough fixed constant.
Without loss of generality, let us assume that $1/v$ is locally bounded, otherwise we can replace $v$ by $v+\varepsilon$ for $\varepsilon>0$, at last we can let $\varepsilon\to 0_{+}$. From (3.1), we know
that $\psi$ has compact support and is bounded.
Note that
$$\displaystyle\nabla\psi=bv^{-a}\varphi^{b-1}\nabla\varphi-av^{-a-1}\varphi^{b}%
\nabla v,$$
(3.2)
and
$$\displaystyle{\partial_{t}}\psi=bv^{-a}\varphi^{b-1}\partial_{t}\varphi-av^{-a%
-1}\varphi^{b}\partial_{t}v.$$
(3.3)
Thus
$$\psi\in W_{c}^{1,2}(M\times[0,+\infty),d\mu dt).$$
Substituting (3.2) into (1.15), we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\psi d\mu dt$$
$$\displaystyle+$$
$$\displaystyle a\int_{0}^{\infty}\int_{M}v^{-a-1}\left|{\nabla v}\right|^{2}%
\varphi^{b}d\mu dt$$
(3.4)
$$\displaystyle\leq$$
$$\displaystyle\int_{0}^{\infty}\int_{M}(\nabla v,bv^{-a}\varphi^{b-1}\nabla%
\varphi)d\mu dt-\int_{0}^{\infty}\int_{M}v\partial_{t}\psi d\mu dt.$$
Applying the Young’s inequality to the first term in the right-hand side of (3.4), we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}(\nabla v,bv^{-a}\varphi^{b-1}\nabla%
\varphi)d\mu dt$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}\int_{M}(a^{\frac{1}{2}}v^{\frac{-a-1}{2}}%
\varphi^{\frac{b}{2}}\nabla v,ba^{-\frac{1}{2}}v^{\frac{1-a}{2}}\varphi^{\frac%
{b}{2}-1}\nabla\varphi)d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{a}{2}\int_{0}^{\infty}\int_{M}v^{-a-1}\left|\nabla v\right|%
^{2}\varphi^{b}d\mu dt+\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{1-a}\varphi%
^{b-2}\left|\nabla\varphi\right|^{2}d\mu dt.$$
Substituting the above into (3.4), we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\psi d\mu dt+\frac{a}{2}\int%
_{0}^{\infty}\int_{M}v^{-a-1}\left|\nabla v\right|^{2}\varphi^{b}d\mu dt$$
(3.5)
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{1-a}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt-\int_{0}^{\infty}\int_{M}v\partial_{t}%
\psi d\mu dt.$$
Combining (3.5) with (3.3), we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}v^{-a-1}\left|\nabla v\right|^{2}\varphi%
^{b}d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{a^{2}}\int_{0}^{\infty}\int_{M}v^{1-a}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt$$
$$\displaystyle-\frac{2}{a}\int_{0}^{\infty}\int_{M}v[bv^{-a}\varphi^{b-1}%
\partial_{t}\varphi-av^{-a-1}\varphi^{b}\partial_{t}v]d\mu dt,$$
which is
$$\displaystyle\int_{0}^{\infty}\int_{M}v^{-a-1}\left|\nabla v\right|^{2}\varphi%
^{b}d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{a^{2}}\int_{0}^{\infty}\int_{M}v^{1-a}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt$$
(3.6)
$$\displaystyle-\frac{2}{a}\int_{0}^{\infty}\int_{M}(bv^{1-a}\varphi^{b-1}%
\partial_{t}\varphi-av^{-a}\varphi^{b}\partial_{t}v)d\mu dt.$$
Let us use another feasible test function $\psi(x,t)=\varphi(x,t)^{b}$.
Substituting $\psi=\varphi^{b}$ into (1.15), we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\varphi^{b}d\mu dt\leq\int_{%
0}^{\infty}\int_{M}(\nabla v,b\varphi^{b-1}\nabla\varphi)d\mu dt-b\int_{0}^{%
\infty}\int_{M}v\varphi^{b-1}\partial_{t}\varphi d\mu dt.$$
(3.7)
Let us estimate the first term in the right-hand side of (3.7) via the Young’s inequality
$$\displaystyle\int_{0}^{\infty}\int_{M}(\nabla v,b\varphi^{b-1}\nabla\varphi)d%
\mu dt$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}\int_{M}(a^{\frac{1}{2}}v^{\frac{-a-1}{2}}%
\varphi^{\frac{b}{2}}\nabla v,ba^{-\frac{1}{2}}v^{\frac{a+1}{2}}\varphi^{\frac%
{b}{2}-1}\nabla\varphi)d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{a}{2}\int_{0}^{\infty}\int_{M}v^{-a-1}\left|\nabla v\right|%
^{2}\varphi^{b}d\mu dt+\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{a+1}\varphi%
^{b-2}\left|\nabla\varphi\right|^{2}d\mu dt.$$
Combining the above with (3.7), we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\varphi^{b}d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{a}{2}\int_{0}^{\infty}\int_{M}v^{-a-1}\left|\nabla v\right|%
^{2}\varphi^{b}d\mu dt$$
$$\displaystyle+\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{a+1}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt-b\int_{0}^{\infty}\int_{M}v\varphi^{b-1}%
\partial_{t}\varphi d\mu dt.$$
Substituting (3.6) into the above, we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\varphi^{b}d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{1-a}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt$$
(3.8)
$$\displaystyle-\int_{0}^{\infty}\int_{M}(bv^{1-a}\varphi^{b-1}\partial_{t}%
\varphi-av^{-a}\varphi^{b}\partial_{t}v)d\mu dt$$
$$\displaystyle+\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{a+1}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt$$
$$\displaystyle-b\int_{0}^{\infty}\int_{M}v\varphi^{b-1}\partial_{t}\varphi d\mu
dt.$$
For convenience, let us denote
$$\displaystyle I$$
$$\displaystyle:=$$
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}\varphi^{b}d\mu dt,$$
$$\displaystyle K_{1}$$
$$\displaystyle:=$$
$$\displaystyle\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{1-a}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt,$$
$$\displaystyle K_{2}$$
$$\displaystyle:=$$
$$\displaystyle-\int_{0}^{\infty}\int_{M}(bv^{1-a}\varphi^{b-1}\partial_{t}%
\varphi-av^{-a}\varphi^{b}\partial_{t}v)d\mu dt,$$
$$\displaystyle K_{3}$$
$$\displaystyle:=$$
$$\displaystyle\frac{b^{2}}{2a}\int_{0}^{\infty}\int_{M}v^{a+1}\varphi^{b-2}%
\left|\nabla\varphi\right|^{2}d\mu dt,$$
$$\displaystyle K_{4}$$
$$\displaystyle:=$$
$$\displaystyle-b\int_{0}^{\infty}\int_{M}v\varphi^{b-1}\partial_{t}\varphi d\mu
dt.$$
Then (3.8) can be written as follows
$$\displaystyle I\leq K_{1}+K_{2}+K_{3}+K_{4}.$$
(3.9)
Before estimating (3.9), let us introduce some notations
$$\displaystyle J(\theta_{1},\theta_{2}):=\int_{0}^{\infty}\int_{M}h^{\theta_{1}%
}\left|\nabla\varphi\right|^{\theta_{2}}d\nu dt,\quad L(\theta_{1},\theta_{2})%
:=\int_{0}^{\infty}\int_{M}h^{\theta_{1}}\left|\partial_{t}\varphi\right|^{%
\theta_{2}}d\nu dt.$$
(3.10)
Noting $\varphi\equiv 1$ in a neighborhood of $D_{R}$, and applying the Hölder’s inequality,
we obtain
$$\displaystyle K_{1}$$
$$\displaystyle=$$
$$\displaystyle\frac{b^{2}}{2a}\iint_{D_{R}^{c}}v^{1-a}\varphi^{b-2}\left|\nabla%
\varphi\right|^{2}d\mu dt$$
$$\displaystyle=$$
$$\displaystyle\frac{b^{2}}{2a}\iint_{D_{R}^{c}}(h^{\frac{(p-1)(1-a)}{p}}v^{1-a}%
\varphi^{\frac{b(1-a)}{p}})(h^{-\frac{(p-1)(1-a)}{p}}\varphi^{-\frac{b(1-a)}{p%
}+b-2}\left|\nabla\varphi\right|^{2})d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{2a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d%
\mu dt\right)^{\frac{1-a}{p}}$$
$$\displaystyle\times\left(\int_{0}^{\infty}\int_{M}h^{-\frac{(p-1)(1-a)}{p+a-1}%
}\varphi^{[-\frac{b(1-a)}{p}+b-2]\frac{p}{p+a-1}}\left|\nabla\varphi\right|^{%
\frac{2p}{p+a-1}}d\mu dt\right)^{\frac{p+a-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{2a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d%
\mu dt\right)^{\frac{1-a}{p}}$$
$$\displaystyle\times\left(\int_{0}^{\infty}\int_{M}h^{\frac{ap}{p+a-1}}\varphi^%
{[-\frac{b(1-a)}{p}+b-2]\frac{p}{p+a-1}}\left|\nabla\varphi\right|^{\frac{2p}{%
p+a-1}}d\nu dt\right)^{\frac{p+a-1}{p}}.$$
Here we have used that $D_{R}^{c}=M\times[0,\infty)\setminus D_{R}$, and
$d\nu=h^{-1}d\mu$.
Noting that $0\leq\varphi\leq 1$, and by choosing sufficiently large $b$, we obtain
$$\displaystyle{K_{1}}\leq\frac{C}{a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^%
{b}d\mu dt\right)^{\frac{1-a}{p}}J\left(\frac{ap}{p+a-1},\frac{2p}{p+a-1}%
\right)^{\frac{p+a-1}{p}}.$$
(3.11)
Applying integration by parts to $K_{2}$, we obtain
$$\displaystyle K_{2}$$
$$\displaystyle=$$
$$\displaystyle-\int_{0}^{\infty}\int_{M}(bv^{1-a}\varphi^{b-1}\partial_{t}%
\varphi-av^{-a}\varphi^{b}\partial_{t}v)d\mu dt$$
$$\displaystyle=$$
$$\displaystyle-\int_{0}^{\infty}\int_{M}v^{1-a}\partial_{t}(\varphi^{b})-\frac{%
a}{1-a}\partial_{t}(v^{1-a})\varphi^{b}d\mu dt$$
$$\displaystyle=$$
$$\displaystyle-\int_{0}^{\infty}\int_{M}v^{1-a}\partial_{t}(\varphi^{b})d\mu dt%
-\frac{a}{1-a}\int_{M}v_{0}^{1-a}\varphi(x,0)^{b}d\mu$$
$$\displaystyle-\frac{a}{1-a}\int_{0}^{\infty}\int_{M}v^{1-a}\partial_{t}(%
\varphi^{b})d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle-\frac{1}{1-a}\int_{0}^{\infty}\int_{M}v^{1-a}\partial_{t}(%
\varphi^{b})d\mu dt$$
$$\displaystyle=$$
$$\displaystyle-\frac{b}{1-a}\iint_{D_{R}^{c}}v^{1-a}\varphi^{b-1}\partial_{t}%
\varphi d\mu dt.$$
Using Hölder’s inequality again and by similar arguments as in $K_{1}$, we obtain
$$\displaystyle K_{2}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b}{1-a}\iint_{D_{R}^{c}}(h^{\frac{(p-1)(1-a)}{p}}v^{1-a}%
\varphi^{\frac{1-a}{p}b})(h^{-\frac{(p-1)(1-a)}{p}}\varphi^{b-1-\frac{1-a}{p}b%
}\left|\partial_{t}\varphi\right|)d\mu dt$$
(3.12)
$$\displaystyle\leq$$
$$\displaystyle\frac{b}{1-a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt%
\right)^{\frac{1-a}{p}}$$
$$\displaystyle\times\left(\int_{0}^{\infty}\int_{M}h^{-\frac{(p-1)(1-a)}{p+a-1}%
}\varphi^{[b-1-\frac{1-a}{p}b]\frac{p}{p+a-1}}\left|\partial_{t}\varphi\right|%
^{\frac{p}{p+a-1}}d\mu dt\right)^{\frac{p+a-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{C}{1-a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt%
\right)^{\frac{1-a}{p}}L\left(\frac{ap}{p+a-1},\frac{p}{p+a-1}\right)^{\frac{p%
+a-1}{p}}.$$
Similarly, we obtain
$$\displaystyle K_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{b^{2}}{2a}\iint_{D_{R}^{c}}v^{a+1}\varphi^{b-2}\left|\nabla%
\varphi\right|^{2}d\mu dt$$
(3.13)
$$\displaystyle=$$
$$\displaystyle\frac{b^{2}}{2a}\iint_{D_{R}^{c}}(h^{\frac{(p-1)(a+1)}{p}}v^{a+1}%
\varphi^{\frac{a+1}{p}b})(h^{-\frac{(p-1)(a+1)}{p}}\varphi^{b-2-\frac{a+1}{p}b%
}\left|\nabla\varphi\right|^{2})d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle\frac{b^{2}}{2a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d%
\mu dt\right)^{\frac{a+1}{p}}$$
$$\displaystyle\times\left(\int_{0}^{\infty}\int_{M}h^{-\frac{(p-1)(a+1)}{p-a-1}%
}\varphi^{[b-2-\frac{a+1}{p}b]\frac{p}{p-a-1}}\left|\nabla\varphi\right|^{%
\frac{2p}{p-a-1}}d\mu dt\right)^{\frac{p-a-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{C}{a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt%
\right)^{\frac{a+1}{p}}J\left(-\frac{ap}{p-a-1},\frac{2p}{p-a-1}\right)^{\frac%
{p-a-1}{p}},$$
and
$$\displaystyle K_{4}$$
$$\displaystyle=$$
$$\displaystyle-b\iint_{D_{R}^{c}}v\varphi^{b-1}\partial_{t}\varphi d\mu dt$$
(3.14)
$$\displaystyle\leq$$
$$\displaystyle b\iint_{D_{R}^{c}}(h^{\frac{p-1}{p}}v\varphi^{\frac{b}{p}})(h^{-%
\frac{p-1}{p}}\varphi^{b-1-\frac{b}{p}}\left|\partial_{t}\varphi\right|)d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle b\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt\right)^{%
\frac{1}{p}}\left(\int_{0}^{\infty}\int_{M}h^{-1}\varphi^{[b-1-\frac{b}{p}]%
\frac{p}{{p-1}}}\left|\partial_{t}\varphi\right|^{\frac{p}{p-1}}d\mu dt\right)%
^{\frac{p-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt\right)^{%
\frac{1}{p}}L\left(0,\frac{p}{p-1}\right)^{\frac{p-1}{p}}.$$
Substituting (3.11), (3.12), (3.13) and (3.14) into (3.8), we obtain
$$\displaystyle I$$
$$\displaystyle\leq$$
$$\displaystyle\frac{C}{a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt%
\right)^{\frac{1-a}{p}}J\left(\frac{ap}{p+a-1},\frac{2p}{p+a-1}\right)^{\frac{%
p+a-1}{p}}$$
(3.15)
$$\displaystyle+\frac{C}{1-a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu
dt%
\right)^{\frac{1-a}{p}}L\left(\frac{ap}{p+a-1},\frac{p}{p+a-1}\right)^{\frac{p%
+a-1}{p}}$$
$$\displaystyle+\frac{C}{a}\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt%
\right)^{\frac{a+1}{p}}J\left(-\frac{ap}{p-a-1},\frac{2p}{p-a-1}\right)^{\frac%
{p-a-1}{p}}$$
$$\displaystyle+C\left(\iint_{D_{R}^{c}}h^{p-1}v^{p}\varphi^{b}d\mu dt\right)^{%
\frac{1}{p}}L\left(0,\frac{p}{p-1}\right)^{\frac{p-1}{p}},$$
which is
$$\displaystyle I$$
$$\displaystyle\leq$$
$$\displaystyle\frac{C}{a}I^{\frac{1-a}{p}}J\left(\frac{ap}{p+a-1},\frac{2p}{p-1%
+a}\right)^{\frac{p-1+a}{p}}+\frac{C}{1-a}I^{\frac{1-a}{p}}L\left(\frac{ap}{p+%
a-1},\frac{p}{p-1+a}\right)^{\frac{p-1+a}{p}}$$
(3.16)
$$\displaystyle+\frac{C}{a}I^{\frac{1+a}{p}}J\left(-\frac{ap}{p-a-1},\frac{2p}{p%
-1-a}\right)^{\frac{p-1-a}{p}}+CI^{\frac{1}{p}}L\left(0,\frac{p}{p-1}\right)^{%
\frac{p-1}{p}}.$$
We claim that there exists a constant $C_{0}>0$ such that
$$\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}d\mu dt\leq C_{0}<\infty.$$
(3.17)
We divide the proof into two cases:
Case 1: if
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}d\mu dt\leq 1,$$
then we let $C_{0}=1$, and it follow that (3.17) is true.
Case 2: If Case 1 is not satisfied, then we obtain
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}d\mu dt>1,$$
Hence, we can find a large enough $R$ such that
$$\displaystyle\iint_{D_{R}}h^{p-1}v^{p}d\mu dt>1.$$
(3.18)
Recall $p>1$, and choose a positive constant $\beta$ satisfying
$$\frac{{1+\beta}}{p}<1.$$
Let $a$ satisfy $0<a\ll\min\{1,\beta\}$. Combining (3.15) and (3.18), we obtain
$$\displaystyle I$$
$$\displaystyle\leq$$
$$\displaystyle CI^{\frac{1+\beta}{p}}\left[\frac{1}{a}J\left(\frac{ap}{p+a-1},%
\frac{2p}{p+a-1}\right)^{\frac{p+a-1}{p}}+L\left(\frac{ap}{p+a-1},\frac{p}{p+a%
-1}\right)^{\frac{p+a-1}{p}}\right.$$
$$\displaystyle\left.+\frac{1}{a}J\left(-\frac{ap}{p-a-1}\frac{2p}{p-a-1}\right)%
^{\frac{p-a-1}{p}}+L\left(0,\frac{p}{p-1}\right)^{\frac{p-1}{p}}\right].$$
It follows that
$$\displaystyle I^{1-\frac{1+\beta}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C\left[\frac{1}{a}J\left(\frac{ap}{p+a-1},\frac{2p}{p+a-1}\right%
)^{\frac{p+a-1}{p}}+L\left(\frac{ap}{p+a-1},\frac{p}{p+a-1}\right)^{\frac{p+a-%
1}{p}}\right.$$
(3.19)
$$\displaystyle+\left.\frac{1}{a}J\left(-\frac{ap}{p-a-1},\frac{2p}{p-a-1}\right%
)^{\frac{p-a-1}{p}}+L\left(0,\frac{p}{p-1}\right)^{\frac{p-1}{p}}\right].$$
Let $g\in{{C}^{\infty}}[0,\infty)$ be a nonnegative function satisfying
$g(t)=1$ on $\left[0,1\right]$; $g(t)=0$ on $\left[2,\infty\right)$; $\left|g^{\prime}\right|\leq C_{1}<\infty$.
Let $\{\eta_{k}\}_{k\in\mathbb{N}},\{\gamma_{k}\}_{k\in\mathbb{N}}\in C^{\infty}[0,\infty)$ be two sequences of functions defined respectively by
$$\displaystyle\eta_{k}(t)=g\left(\frac{t}{2^{2k}}\right),$$
(3.20)
and
$$\displaystyle\gamma_{k}(x)=g\left(\frac{r(x)}{2^{k}}\right),$$
(3.21)
where $r(x)=d({x_{0}},x)$.
From (3.20) and (3.21), we have
$$\displaystyle\left|{\partial_{t}{{\eta}_{k}}}\right|\left\{\begin{array}[]{ll}%
\leq\frac{C}{{{2}^{2k}}},\quad t\in[{{2}^{2k}},{{2}^{2k+1}}],\\
=0,\quad\text{otherwise},\end{array}\right.$$
(3.22)
and
$$\displaystyle\left|\nabla{{\gamma}_{k}}\right|\left\{\begin{array}[]{ll}\leq%
\frac{C}{2^{k}},\ x\in B(x_{0},2^{k+1})\setminus B(x_{0},2^{k}),\\
=0,\quad\text{otherwise}.\end{array}\right.$$
(3.23)
Let us define a sequence of functions $\{\varphi_{i}(x,t)\}_{i\in\mathbb{N}}$ by
$$\displaystyle\varphi_{i}(x,t)=\frac{1}{i}\sum\limits_{k=i+1}^{2i}{{{\eta}_{k}}%
(t){{\gamma}_{k}}(x)},$$
(3.24)
It follows that $\varphi_{i}(x,t)=1$ when $(x,t)\in\overline{B(x_{0},2^{i})}\times[0,2^{2i}]$.
Moreover, for distinct $k$, noting that $\text{supp}({\partial_{t}{{\eta}_{k}}})$ and $\text{supp}(\nabla{{\gamma}_{k}})$ are disjoint respectively,
we obtain for any $\theta>0$
$$\displaystyle\left|{\partial_{t}{{\varphi}_{i}}}\right|^{\theta}=i^{-\theta}%
\sum\limits_{k=i+1}^{2i}\left|\gamma_{k}\partial_{t}(\eta_{k})\right|^{\theta}.$$
(3.25)
and
$$\displaystyle\left|\nabla\varphi_{i}\right|^{\theta}=i^{-\theta}\sum\limits_{k%
=i+1}^{2i}\left|\eta_{k}\nabla\gamma_{k}\right|^{\theta}.$$
(3.26)
Hence
$$\displaystyle{\varphi_{i}}\in W_{c}^{1,2}(M\times[0,+\infty)).$$
Let
$$\displaystyle a=\frac{1}{i}.$$
(3.27)
Substituting the above with $\varphi=\varphi_{i}$ into (3.19), we obtain
$$\displaystyle I^{1-\frac{1+\beta}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C\left[iJ\left(\frac{p/i}{p+1/i-1},\frac{2p}{p+1/i-1}\right)^{%
\frac{p+1/i-1}{p}}+L\left(\frac{p/i}{p+1/i-1},\frac{p}{p+1/i-1}\right)^{\frac{%
p+1/i-1}{p}}\right.$$
(3.28)
$$\displaystyle\left.+iJ\left(-\frac{p/i}{p-1/i-1}\frac{2p}{p-1/i-1}\right)^{%
\frac{p-1/i-1}{p}}+L\left(0,\frac{p}{p-1}\right)^{\frac{p-1}{p}}\right].$$
Substituting (3.24) into (3.10), and combining (3.23) and (3.26), noting $\eta_{k}\leq 1$, we obtain
$$\displaystyle iJ\left(\frac{p/i}{p+1/i-1},\frac{2p}{p+1/i-1}\right)^{\frac{p+1%
/i-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle i\left(\int_{0}^{\infty}\int_{M}h^{\frac{p/i}{p+1/i-1}}i^{-\frac%
{2p}{p+1/i-1}}\sum\limits_{k=i+1}^{2i}\left|\eta_{k}\nabla\gamma_{k}\right|^{%
\frac{2p}{p+1/i-1}}d\nu dt\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle i^{-1}\left(\sum\limits_{k=i+1}^{2i}\int_{0}^{2^{2k+1}}\eta_{k}^%
{\frac{2p}{p+1/i-1}}\int_{B(x_{0},2^{k+1})\backslash B(x_{0},2^{k})}h^{\frac{p%
/i}{p+1/i-1}}\left|\nabla\gamma_{k}\right|^{\frac{2p}{p+1/i-1}}d\nu dt\right)^%
{\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle Ci^{-1}\left(\sum\limits_{k=i+1}^{2i}2^{2k+1}(2^{k+1})^{\frac{%
\delta_{2}p/i}{p+1/i-1}}2^{-\frac{2pk}{p+1/i-1}}\nu(B(x_{0},2^{k+1}))\right)^{%
\frac{p+1/i-1}{p}},$$
where we have used the condition $(H)$.
Applying volume condition $\nu(B(x_{0},r))\leq Cr^{P}\ln^{Q}r$ with $P=\frac{2}{p-1}$ and
$Q=\frac{1}{p-1}$, we obtain
$$\displaystyle iJ\left(\frac{p/i}{p+1/i-1},\frac{2p}{p+1/i-1}\right)^{\frac{p+1%
/i-1}{p}}$$
(3.29)
$$\displaystyle\leq$$
$$\displaystyle Ci^{-1}\left(\sum\limits_{k=i+1}^{2i}2^{2k-\frac{2pk}{p+1/i-1}}2%
^{\frac{k\delta_{2}p/i}{p+1/i-1}}2^{kP}k^{Q}\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle Ci^{-1+Q\frac{p+1/i-1}{p}}\left(\sum\limits_{k=i+1}^{2i}2^{k[2-%
\frac{2p}{p+1/i-1}+\frac{\delta_{2}p/i}{p+1/i-1}+P]}\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle Ci^{-1+Q\frac{p+1/i-1}{p}+\frac{p+1/i-1}{p}}.$$
Here we have used that
$$\displaystyle\left(\sum\limits_{k=i+1}^{2i}2^{k[2-\frac{2p}{p+1/i-1}+\frac{%
\delta_{2}p/i}{p+1/i-1}+P]}\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle 2^{2\delta_{2}p}\left(\sum\limits_{k=i+1}^{2i}2^{\frac{2kp}{i(p-%
1)(p+1/i-1)}}\right)^{\frac{p+1/i-1}{p}}$$
(3.30)
$$\displaystyle\leq$$
$$\displaystyle Ci^{\frac{p+1/i-1}{p}}.$$
Noting that
$$\limsup\limits_{i\to\infty}i^{-1+Q\frac{p+1/i-1}{p}+\frac{p+1/i-1}{p}}=\limsup%
\limits_{i\to\infty}i^{\frac{1}{i(p-1)}}=1,$$
we obtain
$$\displaystyle iJ\left(\frac{p/i}{p+1/i-1},\frac{2p}{p+1/i-1}\right)^{\frac{p+1%
/i-1}{p}}\leq C.$$
(3.31)
Substituting (3.24) into (3.10), applying (3.22) and (3.25), noting $\gamma_{k}\leq 1$, we obtain
$$\displaystyle L\left(\frac{p/i}{p+1/i-1},\frac{p}{p+1/i-1}\right)^{\frac{p+1/i%
-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle\left(i^{-\frac{p}{p+1/i-1}}\sum\limits_{k=i+1}^{2i}\int_{2^{2k}}%
^{2^{2k+1}}\left|\partial_{t}\eta_{k}\right|^{\frac{p}{p+1/i-1}}dt\int_{B(x_{0%
},2^{k+1})}h^{\frac{p/i}{p+1/i-1}}\gamma_{k}^{\frac{p}{p+1/i-1}}d\nu\right)^{%
\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C\left(i^{-\frac{p}{p+1/i-1}}\sum\limits_{k=i+1}^{2i}2^{-\frac{2%
kp}{p+1/i-1}}2^{2k}(2^{k+1})^{\frac{\delta_{2}p/i}{p+1/i-1}}\nu(B(x_{0},2^{k+1%
}))\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C\left(i^{-\frac{p}{p+1/i-1}}\sum\limits_{k=i+1}^{2i}2^{-\frac{2%
kp}{p+1/i-1}}2^{2k}2^{\frac{k\delta_{2}p/i}{p+1/i-1}}2^{kP}k^{Q}\right)^{\frac%
{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C^{\prime}\left(i^{-\frac{p}{p+1/i-1}+Q}\sum\limits_{k=i+1}^{2i}%
2^{k[2-\frac{2p}{p+1/i-1}+P]}\right)^{\frac{p+1/i-1}{p}},$$
where the term $2^{\frac{k\delta_{2}p/i}{p+1/i-1}}$ has been absorbed into constant $C^{\prime}$.
Using (3.30) again, we have
$$\left(L\left(\frac{p/i}{p+1/i-1},\frac{p}{p+1/i-1}\right)\right)^{\frac{p+1/i-%
1}{p}}\leq C\left(i^{-\frac{p}{p+1/i-1}+Q+1}\right)^{\frac{p+1/i-1}{p}}.$$
Since
$$\mathop{\limsup}\limits_{i\to\infty}\left(i^{-\frac{p}{p+1/i-1}+Q+1}\right)^{%
\frac{p+1/i-1}{p}}=1,$$
we obtain
$$\displaystyle\left(L\left(\frac{p/i}{p+1/i-1},\frac{p}{p+1/i-1}\right)\right)^%
{\frac{p+1/i-1}{p}}\leq C.$$
(3.32)
Similarly,
$$\displaystyle iJ\left(-\frac{p/i}{p-1/i-1},\frac{2p}{p-1/i-1}\right)^{\frac{p-%
1/i-1}{p}}$$
$$\displaystyle=i^{-1}\left(\sum\limits_{k=i+1}^{2i}\int_{0}^{2^{2k+1}}\eta_{k}^%
{\frac{2p}{p-1/i-1}}dt\int_{B(x_{0},2^{k+1})\backslash B(x_{0},2^{k})}h^{-%
\frac{p/i}{p-1/i-1}}\left|\nabla\gamma_{k}\right|^{\frac{2p}{p-1/i-1}}d\nu%
\right)^{\frac{p-1/i-1}{p}}$$
$$\displaystyle\leq Ci^{-1}\left(\sum\limits_{k=i+1}^{2i}2^{2k+1}2^{-k\frac{2p}{%
p-1/i-1}}(2^{k})^{\frac{\delta_{1}p/i}{p-1/i-1}}\nu(B(x_{0},2^{k+1}))\right)^{%
\frac{p-1/i-1}{p}}$$
$$\displaystyle\leq C^{\prime}i^{-1+Q\frac{-1/i-1+p}{p}}\left(\sum\limits_{k=i+1%
}^{2i}2^{k(2-\frac{2p}{p-1/i-1}+P)}\right)^{\frac{p-1/i-1}{p}}$$
$$\displaystyle\leq Ci^{-1+Q\frac{p-1/i-1}{p}+\frac{p-1/i-1}{p}}$$
$$\displaystyle\leq Ci^{-\frac{1}{i(p-1)}}<\infty.$$
(3.33)
and
$$\displaystyle\left(L\left(0,\frac{p}{p-1}\right)\right)^{\frac{p-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle\left(i^{-\frac{p}{p-1}}\sum\limits_{k=i+1}^{2i}\int_{2^{2k}}^{2^%
{2k+1}}\left|\partial_{t}\eta_{k}\right|^{\frac{p}{p-1}}dt\int_{B(x_{0},2^{k+1%
})}\gamma_{k}^{\frac{p}{p-1}}d\nu\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle\left(i^{-\frac{p}{p-1}}\sum\limits_{k=i+1}^{2i}2^{-2k\frac{p}{p-%
1}}2^{2k}\nu(B(x_{0},2^{k+1}))\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle\leq$$
$$\displaystyle C\left(i^{-\frac{p}{p-1}+Q}\sum\limits_{k=i+1}^{2i}2^{k(-2\frac{%
p}{p-1}+2+P)}\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle C\left(i^{-\frac{p}{p-1}+Q+1}\right)^{\frac{p+1/i-1}{p}}$$
$$\displaystyle=$$
$$\displaystyle C<\infty.$$
Combining (3.31), (3.32, (3), (3) with (3.28), we have
$$\displaystyle\int_{0}^{2^{2i}}\int_{B(x_{0},2^{i})}h^{p-1}v^{p}d\mu dt\leq C<\infty.$$
(3.35)
It follows by letting $i\rightarrow\infty$ that
$$\displaystyle\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}d\mu dt\leq C<\infty.$$
Hence, the claim (3.17) is true.
Substituting $\varphi=\varphi_{i}$ and $R=2^{i}$ into (3.15),
combining with (3.31), (3.32, (3), (3) and (3.17),
repeating the same procedures in (3.15), we obtain
$$\displaystyle\int_{0}^{2^{2i}}\int_{B_{2^{i}}}h^{p-1}v^{p}d\mu dt$$
$$\displaystyle\leq$$
$$\displaystyle C\left\{\left(\iint_{D_{2^{i}}^{c}}h^{p-1}v^{p}d\mu dt\right)^{%
\frac{1-\frac{1}{i}}{p}}+\left(\iint_{D_{2^{i}}^{c}}h^{p-1}v^{p}d\mu dt\right)%
^{\frac{1+\frac{1}{i}}{p}}\right.$$
(3.36)
$$\displaystyle\left.+\left(\iint_{D_{2^{i}}^{c}}h^{p-1}v^{p}d\mu dt\right)^{%
\frac{1}{p}}\right\}.$$
Letting $i\rightarrow\infty$, from (3.17), we have
$$\int_{0}^{\infty}\int_{M}h^{p-1}v^{p}d\mu dt=0,$$
which implies
$$v\equiv 0.$$
Noting that $u=hv$, hence $u\equiv 0$. However, the above leads to the contradiction with the positiveness of $u$. Hence, there
exists no global positive solution to problem (1.1).
$\square$
4. Global existence of positive solution
In this section, we show the sharpness of $P,Q$ in Theorem 1.14.
It suffices to show that $Q$ in (1.19) can not be relaxed.
Proof of Theorem 1.14.
Define the operator
$$\displaystyle Tv(x,t)=\int_{M}\tilde{P}_{t}(x,y)v_{0}(y)d\mu(y)+\int_{0}^{t}%
\int_{M}\tilde{P}_{t-s}(x,y)h^{p-1}v^{p}(y,s)d\mu(y)ds.$$
(4.1)
acting on the following space
$$\displaystyle S_{M}=\left\{v\in L^{\infty}(M\times[0,\infty))|\ 0\leq v(x,t)%
\leq\lambda\tilde{P}_{t+\delta}(x,x_{0})\right\}.$$
(4.2)
where $\lambda>0$ is a constant to be chosen later, and $\delta>1$ is a large fixed constant.
It follows that $S_{M}$ is a closed set of $L^{\infty}(M\times[0,\infty),d\mu)$.
Let $v_{0}$ satisfy
$$0\leq v_{0}(x)\leq\frac{\lambda}{2}\tilde{P}_{\delta}(x,x_{0}).$$
(4.3)
Now let us show $TS_{M}\subset S_{M}$.
From (4.3), and applying (1.22), we have
$$\displaystyle\int_{M}\tilde{P}_{t}(x,y)v_{0}(y)d\mu(y)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{\lambda}{2}\int_{M}\tilde{P}_{t}(x,y)\tilde{P}_{\delta}(y,x%
_{0})d\mu(y)$$
(4.4)
$$\displaystyle=$$
$$\displaystyle\frac{\lambda}{2}\tilde{P}_{t+\delta}(x,x_{0}).$$
From $(DUE)$ and (4.2), we have
$$\displaystyle\int_{0}^{t}\int_{M}\tilde{P}_{t-s}(x,y)h^{p-1}v^{p}(y,s)d\mu(y)ds$$
(4.5)
$$\displaystyle\leq$$
$$\displaystyle C_{1}\lambda^{p}\int_{0}^{t}\int_{M}\tilde{P}_{t-s}(x,y)\tilde{P%
}_{s+\delta}^{p}(y,x_{0})d\mu(y)ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{2}\lambda^{p}\int_{0}^{t}\frac{1}{\mu(B(x_{0},\sqrt{s+\delta}%
))^{p-1}}ds\int_{M}\tilde{P}_{t-s}(x,y)\tilde{P}_{s+\delta}(y,x_{0})d\mu(y)$$
$$\displaystyle\leq$$
$$\displaystyle C_{3}\lambda^{p}\tilde{P}_{t+\delta}(x,x_{0})\int_{0}^{t}\frac{1%
}{\mu(B(x_{0},\sqrt{s+\delta}))^{p-1}}ds,$$
where we have used that $h\asymp 1$.
Recalling that for large enough $r$,
$$\mu_{0}(B(x_{0},r))\geq c_{2}r^{P}\ln^{Q+\varepsilon}r,$$
and since $h\asymp 1$, and $d\mu=h^{2}d\mu_{0}$, we have
$$\displaystyle\mu(B(x_{0},r))\geq c_{3}r^{P}\ln^{Q+\varepsilon}r.$$
When $\delta$ is large enough, we obtain
$$\displaystyle\int_{0}^{t}\frac{1}{\mu(B(x_{0},\sqrt{s+\delta}))^{p-1}}ds$$
(4.6)
$$\displaystyle\leq$$
$$\displaystyle\int_{0}^{t}\frac{1}{\left[C_{1}(s+\delta)^{\frac{P}{2}}(\ln\sqrt%
{s+\delta})^{Q+\varepsilon}\right]^{p-1}}ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{4}2^{1+\varepsilon(p-1)}\int_{0}^{t}\frac{1}{(s+\delta)[\ln(s%
+\delta)]^{1+\varepsilon(p-1)}}ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{4}2^{1+\varepsilon(p-1)}\int_{0}^{\infty}\frac{1}{(s+\delta)[%
\ln(s+\delta)]^{1+\varepsilon(p-1)}}ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{5}<\infty,$$
where we have used that $P=\frac{2}{p-1}$, $Q=\frac{1}{p-1}$.
Combining (4.5) with (4.6), we obtain, for small enough $\lambda$,
$$\displaystyle\int_{0}^{t}\int_{M}\tilde{P}_{t-s}(x,y)h^{p-1}v^{p}(y,s)d\mu(y)ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{5}C_{3}\lambda^{p}\tilde{P}_{t+\delta}(x,x_{0})$$
(4.7)
$$\displaystyle\leq$$
$$\displaystyle\frac{\lambda}{2}\tilde{P}_{t+\delta}(x,x_{0}).$$
Combining (4.1),(4.4) with (4.7), we obtain
$$0\leq Tv\leq\lambda\tilde{P}_{t+\delta}(x,x_{0}).$$
Hence
$$TS_{M}\subset S_{M}.$$
Now we show that $T$ is a contraction map. For $v_{1}$, $v_{2}\in S_{M}$, we have
$$\displaystyle\left|Tv_{1}(x,t)-Tv_{2}(x,t)\right|\leq\int_{0}^{t}\int_{M}%
\tilde{P}_{t-s}(x,y)h^{p-1}\left|v_{1}^{p}(y,s)-v_{2}^{p}(y,s)\right|d\mu(y)ds.$$
(4.8)
Noting that
$$\left|v_{1}^{p}(y,s)-v_{2}^{p}(y,s)\right|\leq p\max\{v_{1}^{p-1}(y,s),v_{2}^{%
p-1}(y,s)\}\left|v_{1}(y,s)-v_{2}(y,s)\right|,$$
and combining with $(DUE)$, (4.2) and (4.6),
and using that $h\asymp 1$, we obtain from (4.8)
that
$$\displaystyle\left|Tv_{1}(x,t)-Tv_{2}(x,t)\right|$$
$$\displaystyle\leq$$
$$\displaystyle C_{6}p\lambda^{p-1}\left\|v_{1}-v_{2}\right\|_{L^{\infty}}\int_{%
0}^{t}\int_{M}\tilde{P}_{t-s}(x,y)\tilde{P}_{s+\delta}^{p-1}(y,x_{0})d\mu(y)ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{7}p\lambda^{p-1}\left\|v_{1}-v_{2}\right\|_{L^{\infty}}\int_{%
0}^{t}\frac{1}{\mu(B(x_{0},\sqrt{s+\delta}))^{p-1}}ds\int_{M}\tilde{P}_{t-s}(x%
,y)d\mu(y)$$
$$\displaystyle\leq$$
$$\displaystyle C_{8}p\lambda^{p-1}C_{1}^{p-1}\left\|v_{1}-v_{2}\right\|_{L^{%
\infty}}\int_{0}^{t}\frac{1}{\mu(B(x_{0},\sqrt{s+\delta}))^{p-1}}ds$$
$$\displaystyle\leq$$
$$\displaystyle C_{9}p\lambda^{p-1}\left\|v_{1}-v_{2}\right\|_{L^{\infty}},$$
where we have used that (1.21) and (4.6).
Choosing $\lambda$ small enough so that $C_{9}p\lambda^{p-1}<1$, we obtain that $T$ is a contraction map.
Applying fixed point theorem, we know
there exists a fixed point $v\in S_{M}$ satisfying
$$v(x,t)=\int_{M}\tilde{P}_{t}(x,y)v_{0}(y)d\mu(y)+\int_{0}^{t}\int_{M}\tilde{P}%
_{t-s}(x,y)h^{p-1}(y)v^{p}(y,s)d\mu(y)ds.$$
(4.9)
Since $v_{0}\gneqq 0$, then $v$ is positive on $M$.
Since $v_{0},v\in L^{2}(M,d\mu)$,
by [13, Theorem 7.6 and 7.7],
we know the integrals in (4.9) are both smooth on $M\times(0,\infty)$, hence
we obtain that $v$ is a global positive solution of problem (1.13), Furthermore, $u=hv$ is a global positive solution of problem (1.1).
$\square$
Acknowledgments.
The authors would like to express their deep gratitude to Prof.
Qi S. Zhang from University of California Riverside who initiated the study of the above problems, and bringing our attentions to his paper
[34]. The authors would also like to thank Prof. Verbitsky from University of Missouri for helpful communications in Section 2.
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Abstract
The Standard Model requires the three known leptonic families to
have identical couplings to the gauge bosons.
The present experimental tests on lepton universality are reviewed,
both for the charged and neutral current sectors.
Our knowledge about the Lorentz structure of the
$l^{-}\to\nu_{l}l^{\prime-}\bar{\nu}_{l^{\prime}}$ transition amplitudes is also
discussed.
FTUV/97-02
IFIC/97-02
January 1997
LEPTON UNIVERSALITY***Lectures given at the
Cargèse’96 School –Masses of Fundamental Particles–
(Cargèse, Corsica, 5–17 August 1996)
A. Pich
Departament de Física Teòrica, IFIC
CSIC — Universitat de València
Dr. Moliner 50, E–46100 Burjassot, València, Spain
INTRODUCTION
The Standard Model (SM)
is a gauge theory, based on the group
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$,
which describes strong, weak and electromagnetic interactions,
via the exchange of the corresponding spin–1 gauge fields:
8 massless gluons and 1 massless photon for the strong and
electromagnetic interactions, respectively,
and 3 massive bosons, $W^{\pm}$ and $Z$, for the weak interaction.
The fermionic matter content is given by the known
leptons and quarks, which are organized in a 3–fold
family structure:
$$\left[\begin{array}[]{cc}\nu_{e}&u\\
e^{-}&d\end{array}\right]\,,\qquad\quad\left[\begin{array}[]{cc}\nu_{\mu}&c\\
\mu^{-}&s\end{array}\right]\,,\qquad\quad\left[\begin{array}[]{cc}\nu_{\tau}&t%
\\
\tau^{-}&b\end{array}\right]\,,$$
(1)
where
(each quark appears in 3 different colours)
$$\left[\begin{array}[]{cc}\nu_{l}&q_{u}\\
l^{-}&q_{d}\end{array}\right]\,\,\equiv\,\,\left(\begin{array}[]{c}\nu_{l}\\
l^{-}\end{array}\right)_{\!L},\quad\left(\begin{array}[]{c}q_{u}\\
q_{d}\end{array}\right)_{\!L},\quad l^{-}_{R},\quad(q_{u})_{R},\quad(q_{d})_{R},$$
(2)
plus the corresponding antiparticles.
Thus, the left-handed fields are $SU(2)_{L}$ doublets, while
their right-handed partners transform as $SU(2)_{L}$ singlets.
The 3 fermionic families in (1) appear
to have identical properties (gauge interactions); they only
differ by their mass and their flavour quantum number.
The gauge symmetry is broken by the vacuum,
which triggers the Spontaneous Symmetry Breaking (SSB)
of the electroweak group to the electromagnetic subgroup:
$$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\,\stackrel{{\scriptstyle\mbox{\rm
SSB%
}}}{{\longrightarrow}}\,SU(3)_{C}\otimes U(1)_{QED}\,.$$
(3)
The SSB mechanism generates the masses of the weak gauge bosons,
and gives rise to the appearance of
a physical scalar particle in the model, the so-called Higgs.
The fermion masses and mixings are also generated through the
SSB mechanism.
The SM constitutes one of the most successful achievements
in modern physics. It provides a very elegant theoretical
framework, which is able to describe all known experimental
facts in particle physics.
A detailed description of the SM and its present
phenomenological status can be found in Refs. [1]
and [2], which discuss the electroweak and strong
sectors, respectively.
In spite of its enormous phenomenological success, the SM leaves too many
unanswered questions to be considered as a complete description of the
fundamental forces.
We do not understand yet why fermions are replicated in three
(and only three)
nearly identical copies? Why the pattern of masses and mixings
is what it is? Are the masses the only difference among the three
families? What is the origin of the SM flavour structure?
Which dynamics is responsible for the observed CP violation?
The fermionic flavour is the main source of
arbitrary free parameters in the SM: 9 fermion masses,
3 mixing angles and 1 complex phase (assuming the neutrinos to be
massless).
The problem of fermion–mass
generation is deeply related with the mechanism responsible for the SSB.
Thus, the origin of these parameters lies in the most obscure part of
the SM Lagrangian: the scalar sector.
Clearly, the dynamics of flavour appears to be “terra incognita”
which deserves a careful investigation.
The flavour structure looks richer in the quark sector, where mixing
phenomena among the different families occur (leptons would also mix
if neutrino masses were non-vanishing).
Since quarks are confined within hadrons, an accurate determination of
their mixing parameters requires first a good understanding of
hadronization effects in flavour–changing transitions.
A rather exhaustive description of our present knowledge on the different
quark couplings has been given in Ref. [3].
The leptonic sector is easier to analyze.
The absence of a direct lepton–gluon vertex
provides a much cleaner environment to
study the structure of the weak currents and the universality
of their couplings to the gauge bosons.
In the pure leptonic transitions, strong interactions are only
present through small higher–order corrections
(vacuum polarization, …).
Thus, it is possible to obtain precise theoretical predictions which
can be compared with the available data.
Although hadronization is of course present in
semileptonic decays, such as
$\tau^{-}\to\nu_{\tau}\pi^{-}$, $\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}$, …,
it only involves
gluonic exchanges between the quarks of a single hadronic current.
Taking appropriate ratios of different semileptonic
transitions with identical hadronic components,
the QCD effects cancel to a very good approximation.
Therefore, semileptonic decays also provide
accurate tests of the leptonic couplings.
The measured masses and lifetimes of the known leptons, shown in
table 1, are very different. The mass spectrum indicates a
hierarchy of the original Yukawa couplings, which increase from one
generation to the other. A similar pattern occurs in the quark sector.
The huge lifetime differences can be simply understood as
a kinematic reflection of the different masses
[see Eq. (18)].
How precisely we know
that the underlying interactions are actually identical for the
three lepton generations is the main question we want to address
in the following.
QED COUPLINGS
A general description of the electromagnetic coupling of a
spin–$\frac{1}{2}$ charged lepton to the virtual photon
involves three different form factors:
$$T[l\bar{l}\gamma^{*}]=e\,\varepsilon_{\mu}(q)\,\bar{l}\left[F_{1}(q^{2})\gamma%
^{\mu}+i{F_{2}(q^{2})\over 2m_{l}}\sigma^{\mu\nu}q_{\nu}+{F_{3}(q^{2})\over 2m%
_{l}}\sigma^{\mu\nu}\gamma_{5}q_{\nu}\right]l\ ,$$
(4)
where $q^{\mu}$ is the photon momentum.
Owing to the conservation of the electric charge,
$F_{1}(0)=1$.
At $q^{2}=0$, the other two form factors reduce to the
lepton
magnetic dipole moment,
$\mu_{l}\equiv(e/2m_{l})\,(g_{l}/2)=e(1+F_{2}(0))/2m_{l}$,
and electric dipole moment $d_{l}=eF_{3}(0)/2m_{l}$.
The $F_{i}(q^{2})$ form factors are sensitive quantities to a
possible lepton substructure.
Moreover, $F_{3}(q^{2})$ violates $T$ and $P$ invariance;
thus, the electric dipole moments, which vanish in the SM,
constitute a good probe of CP violation.
Owing to their chiral changing structure, the
magnetic and electric dipole moments
may provide important insights on the mechanism responsible for
mass generation. In general, one expects [6]
that a fermion of mass
$m_{f}$ (generated by physics at some scale $M\gg m_{f}$) will have
induced dipole moments proportional to some power of $m_{f}/M$.
The measurement of the $e^{+}e^{-}\to l^{+}l^{-}$ cross-section has been
used to test the universality of the leptonic QED couplings.
At low energies, where the $Z$ contribution is small, the deviations
from the QED prediction are usually parameterized through†††
A slightly different parameterization is adopted
for $e^{+}e^{-}\to e^{+}e^{-}$, to account for
the $t$–channel contribution [7].
$$\sigma(e^{+}e^{-}\to l^{+}l^{-})\,=\,\sigma_{\mbox{\rm\scriptsize QED}}\,\left%
(1\mp{s\over s-\Lambda_{\pm}^{2}}\right)^{2}.$$
(5)
The cut-off parameters $\Lambda_{\pm}$ characterize the validity of QED
and measure the point-like nature of the leptons.
From PEP and PETRA data, one finds [7]:
$\Lambda_{+}(e)>435$ GeV, $\Lambda_{-}(e)>590$ GeV,
$\Lambda_{+}(\mu)>355$ GeV, $\Lambda_{-}(\mu)>265$ GeV,
$\Lambda_{+}(\tau)>285$ GeV and
$\Lambda_{-}(\tau)>246$ GeV (95% CL),
which correspond to upper limits on the lepton charge radii
of about $10^{-3}$ fm.
The most stringent QED test comes of course from the high–precision
measurements of the $e$ and $\mu$ anomalous magnetic moments
[8, 9, 10, 11, 12, 13]
$a_{l}\equiv(g_{l}-2)/2$:
$$\displaystyle a_{e}$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{cc}(115\,965\,214.0\pm 2.8)\times 10^{-11}%
&(\mbox{\rm Theory})\\
(115\,965\,219.3\pm 1.0)\times 10^{-11}&(\mbox{\rm Experiment})\end{array}\,,\right.$$
(6)
$$\displaystyle a_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{cc}(1\,165\,917.1\pm 1.0)\times 10^{-9}&(%
\mbox{\rm Theory})\\
(1\,165\,923.0\pm 8.4)\times 10^{-9}&(\mbox{\rm Experiment})\end{array}\,.\right.$$
(7)
Experimentally, very little is known about $a_{\tau}$ since the spin
precession method used for the lighter leptons cannot be applied
due to the very short lifetime of the $\tau$.
The effect is however visible in the $e^{+}e^{-}\to\tau^{+}\tau^{-}$
cross-section. The limit $|a_{\tau}|<0.023$ (95% CL) has been derived
[14, 15]
from PEP and PETRA data. This limit actually probes the corresponding
form factor $F_{2}(s)$ at $s\sim 35$ GeV.
A more direct bound at $q^{2}=0$ has been extracted [5]
from the decay $Z\to\tau^{+}\tau^{-}\gamma$:
$$|a_{\tau}|<0.0104\qquad(95\%\,\mbox{\rm CL})\,.$$
(8)
A slightly better, but more model–dependent, limit has been derived
[16] from the $Z\to\tau^{+}\tau^{-}$ decay width:
$-0.004<a_{\tau}<0.006$.
In the SM the overall value of $a_{\tau}$ is dominated by the second order
QED contribution [17],
$a_{\tau}\approx\alpha/2\pi$.
Including QED corrections up to O($\alpha^{3}$),
hadronic vacuum polarization contributions
and the corrections due to the weak interactions
(which are a factor 380
larger than for the muon), the tau anomalous magnetic moment has been
estimated to be [18, 19]
$$a_{\tau}\big{|}_{\mbox{\rm\scriptsize th}}\,=\,(1.1773\pm 0.0003)\times 10^{-3%
}\,.$$
(9)
So far, no evidence has been found for any CP–violation signature
in the lepton sector.
The present limits on the leptonic electric dipole moments are
[4, 5]:
$$d_{e}\,=\,(-0.3\pm 0.8)\times 10^{-26}\,e\,\mbox{\rm cm},$$
$$d_{\mu}\,=\,(3.7\pm 3.4)\times 10^{-19}\,e\,\mbox{\rm cm},$$
(10)
$$|d_{\tau}|\,<\,5.8\times 10^{-17}\,e\,\mbox{\rm cm}.$$
CHARGED CURRENT UNIVERSALITY
In the SM, the charged–current interactions are governed by an
universal coupling $g$:
$${\cal L}_{\mbox{\rm\scriptsize CC}}\,=\,{g\over 2\sqrt{2}}\,\left\{W^{\dagger}%
_{\mu}\,\left[\sum_{ij}\,\bar{u}_{i}\gamma^{\mu}(1-\gamma_{5})V_{ij}d_{j}\,+\,%
\sum_{l}\,\bar{\nu}_{l}\gamma^{\mu}(1-\gamma_{5})l\right]\,+\,\mbox{\rm h.c.}%
\right\}\,.$$
(11)
In the original basis of weak eigenstates quarks and leptons have
identical interactions. The diagonalization of the fermion masses
gives rise to the unitary quark mixing matrix $V_{ij}$, which couples
any up–type quark with all down–type quarks.
For massless neutrinos, the analogous leptonic mixing matrix can
be eliminated by a redefinition of the neutrino fields.
The lepton flavour is then conserved in the minimal SM without
right–handed neutrinos.
$\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}$
The simplest flavour–changing process is the leptonic
decay of the muon, which proceeds through the $W$–exchange
diagram shown in Fig. 2.
The momentum transfer carried by the intermediate $W$ is very small
compared to $M_{W}$. Therefore, the vector–boson propagator reduces
to a contact interaction,
$${-g_{\mu\nu}+q_{\mu}q_{\nu}/M_{W}^{2}\over q^{2}-M_{W}^{2}}\quad\;\stackrel{{%
\scriptstyle q^{2}\ll M_{W}^{2}}}{{\,\longrightarrow\,}}\quad\;{g_{\mu\nu}%
\over M_{W}^{2}}\,.$$
(12)
The decay can then be described through an effective local
4–fermion Hamiltonian,
$${\cal H}_{\mbox{\rm\scriptsize eff}}\,=\,{G_{F}\over\sqrt{2}}\left[\bar{e}%
\gamma^{\alpha}(1-\gamma_{5})\nu_{e}\right]\,\left[\bar{\nu}_{\mu}\gamma_{%
\alpha}(1-\gamma_{5})\mu\right]\,,$$
(13)
where
$${G_{F}\over\sqrt{2}}={g^{2}\over 8M_{W}^{2}}$$
(14)
is called the Fermi coupling constant.
$G_{F}$
is fixed by the total decay width,
$${1\over\tau_{\mu}}\,=\,\Gamma(\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu})\,=\,{G_{%
F}^{2}m_{\mu}^{5}\over 192\pi^{3}}\,\left(1+\delta_{\mbox{\rm\scriptsize RC}}%
\right)\,f\left(m_{e}^{2}/m_{\mu}^{2}\right)\,,$$
(15)
where
$\,f(x)=1-8x+8x^{3}-x^{4}-12x^{2}\ln{x}$,
and
$$(1+\delta_{\mbox{\rm\scriptsize RC}})=\left[1+{\alpha(m_{\mu})\over 2\pi}\left%
({25\over 4}-\pi^{2}\right)\right]\,\left[1+{3\over 5}{m_{\mu}^{2}\over M_{W}^%
{2}}-2{m_{e}^{2}\over M_{W}^{2}}\right]=0.9958\,$$
(16)
takes into account the leading higher-order corrections [20, 21].
The measured lifetime [4],
$\tau_{\mu}=(2.19703\pm 0.00004)\times 10^{-6}$ s,
implies the value
$$G_{F}\,=\,(1.16639\pm 0.00002)\times 10^{-5}\>\mbox{\rm GeV}^{-2}\,\approx\,{1%
\over(293\>\mbox{\rm GeV})^{2}}\,.$$
(17)
$\tau$ Decay
The decays of the $\tau$ lepton proceed through the same
$W$–exchange mechanism as the leptonic $\mu$ decay.
The only difference is that several final states
are kinematically allowed:
$\tau^{-}\to\nu_{\tau}e^{-}\bar{\nu}_{e}$,
$\tau^{-}\to\nu_{\tau}\mu^{-}\bar{\nu}_{\mu}$,
$\tau^{-}\to\nu_{\tau}d\bar{u}$ and $\tau^{-}\to\nu_{\tau}s\bar{u}$.
Owing to the universality of the $W$–couplings, all these
decay modes have equal amplitudes (if final fermion masses and
QCD interactions are neglected), except for an additional
$N_{C}|V_{ui}|^{2}$ factor ($i=d,s$) in the semileptonic
channels, where $N_{C}=3$ is the number of quark colours.
Making trivial kinematical changes in Eq. (15),
one easily gets the lowest–order prediction for the total
$\tau$ decay width:
$${1\over\tau_{\tau}}\equiv\Gamma(\tau)\approx\Gamma(\mu)\left({m_{\tau}\over m_%
{\mu}}\right)^{5}\left\{2+N_{C}\left(|V_{ud}|^{2}+|V_{us}|^{2}\right)\right\}%
\approx{5\over\tau_{\mu}}\left({m_{\tau}\over m_{\mu}}\right)^{5},$$
(18)
where we have used the unitarity relation
$|V_{ud}|^{2}+|V_{us}|^{2}=1-|V_{ub}|^{2}\approx 1$.
From the measured muon lifetime, one has then
$\tau_{\tau}\approx 3.3\times 10^{-13}$ s, to be compared
with the experimental value [5]
$\tau_{\tau}^{\mbox{\rm\scriptsize exp}}=(2.9021\pm 0.0115)\times 10^{-13}$ s.
The branching ratios into the different decay modes are
predicted to be:
$$\displaystyle\mbox{\rm Br}(\tau^{-}\to\nu_{\tau}l^{-}\bar{\nu}_{l})\approx{1%
\over 5}=20\%\ ,$$
$$\displaystyle R_{\tau}\equiv{\Gamma(\tau\to\nu_{\tau}+\mbox{\rm Hadrons})\over%
\Gamma(\tau^{-}\to\nu_{\tau}e^{-}\bar{\nu}_{e})}\approx N_{C}\,,$$
(19)
in good agreement with the measured numbers [5],
given in table 2.
Our naive predictions only deviate
from the experimental results by about 20%. This
is the expected size of the corrections induced by
the strong interactions between the final quarks,
that we have neglected.
Notice that the measured $\tau$ hadronic width provides strong evidence
for the colour degree of freedom.
The pure leptonic decays
$\tau^{-}\to e^{-}\bar{\nu}_{e}\nu_{\tau},\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$
are theoretically understood at the level of the electroweak
radiative corrections [21].
The corresponding decay widths are given by Eqs. (15)
and (16),
making the appropriate changes for the masses of the initial and final
leptons.
Using the value of $G_{F}$
measured in $\mu$ decay, Eq. (15)
provides a relation between the $\tau$ lifetime
and the leptonic branching ratios
$B_{l}\equiv\mbox{\rm Br}(\tau^{-}\to\nu_{\tau}l^{-}\bar{\nu}_{l})$:
$$B_{e}={B_{\mu}\over 0.972564\pm 0.000010}={\tau_{\tau}\over(1.6321\pm 0.0014)%
\times 10^{-12}\,\mbox{\rm s}}\,.$$
(20)
The errors reflect the present uncertainty of $0.3$ MeV
in the value of $m_{\tau}$.
The predicted $B_{\mu}/B_{e}$ ratio is in perfect agreement with the measured
value $B_{\mu}/B_{e}=0.974\pm 0.006$. As shown in
Fig. 3, the relation between $B_{e}$ and
$\tau_{\tau}$ is also well satisfied by the present data. Notice, that this
relation is very sensitive to the value of the $\tau$ mass
[$\Gamma(\tau^{-}\to l^{-}\bar{\nu}_{l}\nu_{\tau})\propto m_{\tau}^{5}$].
The most recent measurements of
$\tau_{\tau}$, $B_{e}$ and $m_{\tau}$ have consistently moved the world averages
in the correct direction, eliminating the previous ($\sim 2\sigma$)
disagreement [22].
The experimental precision (0.4%) is already approaching the
level where a possible non-zero $\nu_{\tau}$ mass could become relevant; the
present bound [5]
$m_{\nu_{\tau}}<18.2$ MeV (95% CL) only guarantees that such
effect is below 0.08%.
Semileptonic Decays
Semileptonic decays such as $\tau^{-}\to\nu_{\tau}P^{-}$ or
$P^{-}\to l^{-}\bar{\nu}_{l}$ [$P=\pi,K$] can be predicted
in a similar way. The effects of the strong interactions
are contained in the so–called decay constants $f_{P}$,
which parameterize the hadronic matrix element of the
corresponding weak current:
$$\begin{array}[]{ccc}\langle\pi^{-}(p)|\bar{d}\gamma^{\mu}\gamma_{5}u|0\rangle&%
\equiv&-i\sqrt{2}f_{\pi}p^{\mu}\,,\\
\langle K^{-}(p)|\bar{s}\gamma^{\mu}\gamma_{5}u|0\rangle&\equiv&-i\sqrt{2}f_{K%
}p^{\mu}\,.\end{array}$$
(21)
Taking appropriate ratios of different semileptonic decay widths
involving the same meson $P$, the dependence on these decay
constants factors out. Therefore, those ratios can be predicted
rather accurately:
$$\displaystyle R_{e/\mu}$$
$$\displaystyle\!\!\!\equiv$$
$$\displaystyle\!\!\!{\Gamma(\pi^{-}\to e^{-}\bar{\nu}_{e})\over\Gamma(\pi^{-}%
\to\mu^{-}\bar{\nu}_{\mu})}\,=\,{m_{e}^{2}(1-m_{e}^{2}/m_{\pi}^{2})^{2}\over m%
_{\mu}^{2}(1-m_{\mu}^{2}/m_{\pi}^{2})^{2}}\,(1+\delta R_{e/\mu})\,=\,(1.2351%
\pm 0.0005)\times 10^{-4},$$
$$\displaystyle R_{\tau/\pi}$$
$$\displaystyle\!\!\!\equiv$$
$$\displaystyle\!\!\!{\Gamma(\tau^{-}\to\nu_{\tau}\pi^{-})\over\Gamma(\pi^{-}\to%
\mu^{-}\bar{\nu}_{\mu})}\,=\,{m_{\tau}^{3}\over 2m_{\pi}m_{\mu}^{2}}{(1-m_{\pi%
}^{2}/m_{\tau}^{2})^{2}\over(1-m_{\mu}^{2}/m_{\pi}^{2})^{2}}\left(1+\delta R_{%
\tau/\pi}\right)\,=\,9774\pm 15\,,$$
(22)
$$\displaystyle R_{\tau/K}$$
$$\displaystyle\!\!\!\equiv$$
$$\displaystyle\!\!\!{\Gamma(\tau^{-}\to\nu_{\tau}K^{-})\over\Gamma(K^{-}\to\mu^%
{-}\bar{\nu}_{\mu})}\,=\,{m_{\tau}^{3}\over 2m_{K}m_{\mu}^{2}}{(1-m_{K}^{2}/m_%
{\tau}^{2})^{2}\over(1-m_{\mu}^{2}/m_{K}^{2})^{2}}\left(1+\delta R_{\tau/K}%
\right)\,=\,480.4\pm 1.1\,,$$
where $\delta R_{e/\mu}=-(3.76\pm 0.04)\%$,
$\delta R_{\tau/\pi}=(0.16\pm 0.14)\%$ and
$\delta R_{\tau/K}=(0.90\pm 0.22)\%$
are the computed [23, 24] radiative corrections.
These predictions are in excellent agreement with the
measured ratios [5, 25, 26]:
$R_{e/\mu}=(1.2310\pm 0.0037)\times 10^{-4}$,
$R_{\tau/\pi}=9878\pm 106$ and
$R_{\tau/K}=465\pm 19$.
Universality Tests
All these measurements can be used to test the universality of
the $W$ couplings to the leptonic charged currents.
Allowing the coupling $g$ in Eq. (11)
to depend on the considered lepton flavour
(i.e., $g_{e}$, $g_{\mu}$, $g_{\tau}$),
the ratios $B_{\mu}/B_{e}$ and $R_{e/\mu}$ constrain $|g_{\mu}/g_{e}|$, while
$B_{e}/\tau_{\tau}$ and $R_{\tau/P}$
provide information on $|g_{\tau}/g_{\mu}|$.
The present results are shown in tables 4 and
4, together with the values obtained from
the comparison of the $\sigma\cdot B$ partial production
cross-sections for the various $W^{-}\to l^{-}\bar{\nu}_{l}$ decay
modes at the $p$-$\bar{p}$ colliders [27, 28, 29].
The present data verify the universality of the leptonic
charged–current couplings to the 0.15% ($\mu/e$) and 0.30%
($\tau/\mu$) level. The precision of the most recent
$\tau$–decay measurements is becoming competitive with the
more accurate $\pi$–decay determination.
It is important to realize the complementarity of the
different universality tests.
The pure leptonic decay modes probe
the charged–current couplings of a transverse $W$. In contrast,
the decays $\pi/K\to l\bar{\nu}$ and $\tau\to\nu_{\tau}\pi/K$ are only
sensitive to the spin–0 piece of the charged current; thus,
they could unveil the presence of possible scalar–exchange
contributions with Yukawa–like couplings proportional to some
power of the charged–lepton mass.
One can easily imagine new physics scenarios which would modify
differently the two types of leptonic couplings [6].
For instance,
in the usual two Higgs doublet model, charged–scalar exchange
generates a correction to the ratio $B_{\mu}/B_{e}$, but
$R_{\pi\to e/\mu}$ remains unaffected.
Similarly, lepton mixing between the $\nu_{\tau}$ and an hypothetical
heavy neutrino would not modify the ratios $B_{\mu}/B_{e}$ and
$R_{\pi\to e/\mu}$, but would certainly correct the relation between
$B_{l}$ and the $\tau$ lifetime.
NEUTRAL CURRENT UNIVERSALITY
In the SM, all leptons with equal electric charge have identical
couplings to the $Z$ boson:
$${\cal L}_{\mbox{\rm\scriptsize NC}}^{Z}\,=\,{g\over 2\cos{\theta_{W}}}\,Z_{\mu%
}\,\sum_{l}\bar{l}\gamma^{\mu}(v_{l}-a_{l}\gamma_{5})l\,,$$
(23)
where
$$v_{l}=T_{3}^{l}(1-4|Q_{l}|\sin^{2}{\theta_{W}})\ ,\qquad\qquad a_{l}=T_{3}^{l}\,.$$
(24)
This has been tested at LEP and SLC [30],
where the effective vector and axial–vector couplings of the three
charged leptons have been determined.
For unpolarized $e^{+}$ and $e^{-}$ beams, the differential
$e^{+}e^{-}\to\gamma,Z\to l^{+}l^{-}$
cross-section can be written, at lowest order, as
$${d\sigma\over d\Omega}\,=\,{\alpha^{2}\over 8s}\,\left\{A\,(1+\cos^{2}{\theta}%
)\,+B\,\cos{\theta}\,-\,h_{l}\left[C\,(1+\cos^{2}{\theta})\,+\,D\cos{\theta}%
\right]\right\},$$
(25)
where $h_{l}$ ($=\pm 1$) is the $l^{-}$ helicity and $\theta$ is
the scattering angle between $e^{-}$ and $l^{-}$.
Here,
$$\begin{array}[]{ccl}A&=&1+2v_{e}v_{l}\,\mbox{\rm Re}(\chi)+\left(v_{e}^{2}+a_{%
e}^{2}\right)\left(v_{l}^{2}+a_{l}^{2}\right)|\chi|^{2},\\
B&=&4a_{e}a_{l}\,\mbox{\rm Re}(\chi)+8v_{e}a_{e}v_{l}a_{l}|\chi|^{2},\\
C&=&2v_{e}a_{l}\,\mbox{\rm Re}(\chi)+2\left(v_{e}^{2}+a_{e}^{2}\right)v_{l}a_{%
l}|\chi|^{2},\\
D&=&4a_{e}v_{l}\,\mbox{\rm Re}(\chi)+4v_{e}a_{e}\left(v_{l}^{2}+a_{l}^{2}%
\right)|\chi|^{2},\\
\end{array}$$
(26)
and $\chi$ contains the $Z$ propagator
$$\chi\,=\,{G_{F}M_{Z}^{2}\over 2\sqrt{2}\pi\alpha}\,\,{s\over s-M_{Z}^{2}+is%
\Gamma_{Z}/M_{Z}}\,.$$
(27)
The coefficients $A$, $B$, $C$ and $D$ can be experimentally determined,
by measuring the total cross-section, the forward–backward asymmetry,
the polarization asymmetry and the forward–backward polarization
asymmetry, respectively:
$$\sigma(s)={4\pi\alpha^{2}\over 3s}\,A\,,$$
$${\cal A}_{\mbox{\rm\scriptsize FB}}(s)\equiv{N_{F}-N_{B}\over N_{F}+N_{B}}={3%
\over 8}{B\over A}\,,$$
$${\cal A}_{\mbox{\rm\scriptsize Pol}}(s)\equiv{\sigma^{(h_{l}=+1)}-\sigma^{(h_{%
l}=-1)}\over\sigma^{(h_{l}=+1)}+\sigma^{(h_{l}=-1)}}\,=\,-{C\over A}\,,$$
(28)
$${\cal A}_{\mbox{\rm\scriptsize FB,Pol}}(s)\equiv{N_{F}^{(h_{l}=+1)}-N_{F}^{(h_%
{l}=-1)}-N_{B}^{(h_{l}=+1)}+N_{B}^{(h_{l}=-1)}\over N_{F}^{(h_{l}=+1)}+N_{F}^{%
(h_{l}=-1)}+N_{B}^{(h_{l}=+1)}+N_{B}^{(h_{l}=-1)}}\,=\,-{3\over 8}{D\over A}\,.$$
Here, $N_{F}$ and $N_{B}$ denote the number of $l^{-}$’s
emerging in the forward and backward hemispheres,
respectively, with respect to the electron direction.
For $s=M_{Z}^{2}$,
the real part of the $Z$ propagator vanishes
and the photon exchange terms can be neglected
in comparison with the $Z$–exchange contributions
($\Gamma_{Z}^{2}/M_{Z}^{2}\ll 1$). Eqs. (28)
become then,
$$\displaystyle\sigma^{0,l}\equiv\sigma(M_{Z}^{2})={12\pi\over M_{Z}^{2}}\,{%
\Gamma_{e}\Gamma_{l}\over\Gamma_{Z}^{2}}\,,$$
$$\displaystyle\qquad\;{\cal A}_{\mbox{\rm\scriptsize FB}}^{0,l}\equiv{\cal A}_{%
FB}(M_{Z}^{2})={3\over 4}{\cal P}_{e}{\cal P}_{l}\,,$$
$$\displaystyle{\cal A}_{\mbox{\rm\scriptsize Pol}}^{0,l}\equiv{\cal A}_{\mbox{%
\rm\scriptsize Pol}}(M_{Z}^{2})={\cal P}_{l}\,,$$
$$\displaystyle\qquad{\cal A}_{\mbox{\rm\scriptsize FB,Pol}}^{0,l}\equiv{\cal A}%
_{\mbox{\rm\scriptsize FB,Pol}}(M_{Z}^{2})={3\over 4}{\cal P}_{e}\,,$$
(29)
where $\Gamma_{l}$
is the $Z$ partial decay width to the $l^{+}l^{-}$
final state, and
$${\cal P}_{l}\,\equiv\,{-2v_{l}a_{l}\over v_{l}^{2}+a_{l}^{2}}$$
(30)
is the average longitudinal polarization of the lepton $l^{-}$,
which only depends on the ratio of the vector and axial–vector couplings.
${\cal P}_{l}$ is a sensitive function of $\sin^{2}{\theta_{W}}$.
The $Z$ partial decay width to the $l^{+}l^{-}$ final state,
$$\Gamma_{l}\equiv\Gamma(Z\to l^{+}l^{-})={G_{F}M_{Z}^{3}\over 6\pi\sqrt{2}}\,(v%
_{l}^{2}+a_{l}^{2})\,\left(1+{3\alpha\over 4\pi}\right),$$
(31)
determines the sum $(v_{l}^{2}+a_{l}^{2})$, while the ratio $v_{l}/a_{l}$
is derived from the asymmetries‡‡‡
The asymmetries determine two possible solutions for $|v_{l}/a_{l}|$.
This ambiguity can be solved with lower–energy data or
through the measurement of the transverse
spin–spin correlation [31]
of the two $\tau$’s in $Z\to\tau^{+}\tau^{-}$,
which requires [32] $|v_{\tau}/a_{\tau}|<<1$..
The signs of $v_{l}$ and $a_{l}$ are fixed by requiring $a_{e}<0$.
The measurement of the final polarization asymmetries can (only) be done for
$l=\tau$, because the spin polarization of the $\tau$’s
is reflected in the distorted distribution of their decay products.
Therefore, ${\cal P}_{\tau}$ and ${\cal P}_{e}$ can be determined from a
measurement of the spectrum of the final charged particles in the
decay of one $\tau$, or by studying the correlated distributions
between the final products of both $\tau^{\prime}s$ [33].
With polarized $e^{+}e^{-}$ beams, one can also study the left–right
asymmetry between the cross-sections for initial left– and right–handed
electrons.
At the $Z$ peak, this asymmetry directly measures
the average initial lepton polarization, ${\cal P}_{e}$,
without any need for final particle identification:
$${\cal A}_{\mbox{\rm\scriptsize LR}}^{0}\,\equiv\,{\cal A}_{\mbox{\rm%
\scriptsize LR}}(M_{Z}^{2})\,=\,{\sigma_{L}(M_{Z}^{2})-\sigma_{R}(M_{Z}^{2})%
\over\sigma_{L}(M_{Z}^{2})+\sigma_{R}(M_{Z}^{2})}\,=\,-{\cal P}_{e}\,.$$
(32)
Tables 6 and 6
show the present experimental results
for the leptonic $Z$--decay widths and asymmetries.
The data are in excellent agreement with the SM predictions
and confirm the universality of the leptonic neutral couplings§§§
A small 0.2% difference between $\Gamma_{\tau}$ and $\Gamma_{e,\mu}$
is generated by the $m_{\tau}$ corrections..
There is however a small ($\sim 2\sigma$) discrepancy between the
${\cal P}_{e}$ values obtained [30] from
${\cal A}^{0,\tau}_{\mbox{\rm\scriptsize FB,Pol}}$
and ${\cal A}_{\mbox{\rm\scriptsize LR}}^{0}$.
Assuming lepton universality,
the combined result from all leptonic asymmetries gives
$${\cal P}_{l}=-0.1500\pm 0.0025\ .$$
(33)
The measurement of ${\cal A}_{\mbox{\rm\scriptsize Pol}}^{0,\tau}$ and
${\cal A}^{0,\tau}_{\mbox{\rm\scriptsize FB,Pol}}$ assumes that the $\tau$ decay
proceeds through the SM charged–current interaction.
A more general analysis should take into account the fact that the
$\tau$–decay width depends on the product $\xi{\cal P}_{\tau}$
(see the next section), where $\xi$
is the corresponding Michel parameter in leptonic decays, or
the equivalent quantity $\xi_{h}$ ($=h_{\nu_{\tau}}$) in the semileptonic
modes.
A separate measurement of $\xi$ and ${\cal P}_{\tau}$ has been performed by
ALEPH [34] (${\cal P}_{\tau}=-0.139\pm 0.040$)
and L3 [35] (${\cal P}_{\tau}=-0.154\pm 0.022$),
using the correlated distribution of the $\tau^{+}\tau^{-}$ decays.
The combined analysis of all leptonic observables from LEP
and SLD (${\cal A}_{\mbox{\rm\scriptsize LR}}^{0}$) results in the
effective vector and axial–vector couplings given in
table 7 [30].
The corresponding 68% probability contours in the $a_{l}$–$v_{l}$ plane
are shown in Fig. 4.
The measured ratios of the $e$, $\mu$ and $\tau$ couplings
provide a test of charged–lepton universality in the neutral–current
sector.
The neutrino couplings can be determined from the invisible
$Z$–decay width, by assuming three identical neutrino generations
with left–handed couplings (i.e., $v_{\nu}=a_{\nu}$),
and fixing the sign from neutrino scattering
data [36].
The resulting experimental value [30],
given in table 7,
is in perfect agreement with the SM.
Alternatively, one can use the SM prediction for
$\Gamma_{\mbox{\rm\scriptsize inv}}/\Gamma_{l}$
to get a determination of the number of (light) neutrino flavours
[30]:
$$N_{\nu}=2.989\pm 0.012\,.$$
(34)
The universality of the neutrino couplings has been tested
with $\nu_{\mu}e$ scattering data, which fixes [37]
the $\nu_{\mu}$ coupling to the $Z$:
$v_{\nu_{\mu}}=a_{\nu_{\mu}}=0.502\pm 0.017$.
The measured leptonic asymmetries can be used to obtain the
effective electroweak mixing angle in the charged–lepton sector:
[30]
$$\sin^{2}{\theta^{\mbox{\rm\scriptsize lept}}_{\mbox{\rm\scriptsize eff}}}%
\equiv{1\over 4}\left(1-{v_{l}\over a_{l}}\right)=0.23114\pm 0.00031\,.$$
(35)
Including also the hadronic asymmetries, one gets [30]
$\sin^{2}{\theta^{\mbox{\rm\scriptsize lept}}_{\mbox{\rm\scriptsize eff}}}=0.23%
165\pm 0.00024$
with a $\chi^{2}/\mbox{\rm d.o.f.}=12.8/6$.
LORENTZ STRUCTURE OF THE CHARGED CURRENTS
Let us consider the decay $l^{-}\to\nu_{l}l^{\prime-}\bar{\nu}_{l^{\prime}}$,
where the lepton pair ($l$, $l^{\prime}$)
may be ($\mu$, $e$), ($\tau$, $e$), or ($\tau$, $\mu$).
The most general, local, derivative–free, lepton–number conserving,
four–lepton interaction Hamiltonian,
consistent with locality and Lorentz invariance
[38, 39, 40, 41, 42, 43],
$${\cal H}=4\frac{G_{l^{\prime}l}}{\sqrt{2}}\sum_{n,\epsilon,\omega}g^{n}_{%
\epsilon\omega}\left[\overline{l^{\prime}_{\epsilon}}\Gamma^{n}{(\nu_{l^{%
\prime}})}_{\sigma}\right]\,\left[\overline{({\nu_{l}})_{\lambda}}\Gamma_{n}l_%
{\omega}\right]\ ,$$
(36)
contains ten complex coupling constants or, since a common phase is
arbitrary, nineteen independent real parameters
which could be different for each leptonic decay.
The subindices
$\epsilon,\omega,\sigma,\lambda$ label the chiralities (left–handed,
right–handed) of the corresponding fermions, and $n$ the
type of interaction:
scalar ($I$), vector ($\gamma^{\mu}$), tensor
($\sigma^{\mu\nu}/\sqrt{2}$).
For given $n,\epsilon,\omega$, the neutrino chiralities $\sigma$ and $\lambda$
are uniquely determined.
Taking out a common factor $G_{l^{\prime}l}$, which is determined by the total
decay rate, the coupling constants $g^{n}_{\epsilon\omega}$
are normalized to [41]
$$\displaystyle 1$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{1\over 4}\,\left(|g^{S}_{RR}|^{2}+|g^{S}_{RL}|^{2}+|g^{S}_%
{LR}|^{2}+|g^{S}_{LL}|^{2}\right)+3\,\left(|g^{T}_{RL}|^{2}+|g^{T}_{LR}|^{2}\right)$$
(37)
$$\displaystyle\!\!\!\mbox{}+\left(|g^{V}_{RR}|^{2}+|g^{V}_{RL}|^{2}+|g^{V}_{LR}%
|^{2}+|g^{V}_{LL}|^{2}\right)\,.$$
In the SM, $g^{V}_{LL}=1$ and all other
$g^{n}_{\epsilon\omega}=0$.
For an initial lepton polarization ${\cal P}_{l}$,
the final charged–lepton distribution in the decaying–lepton
rest frame
is usually parameterized [39] in the form
$${d^{2}\Gamma\over dx\,d\cos\theta}={m_{l}\omega^{4}\over 2\pi^{3}}G_{l^{\prime%
}l}^{2}\sqrt{x^{2}-x_{0}^{2}}\left\{F(x)-{\xi\over 3}\,{\cal P}_{l}\,\sqrt{x^{%
2}-x_{0}^{2}}\,\cos{\theta}\,A(x)\right\},$$
(38)
where $\theta$ is the angle between the $l^{-}$ spin and the
final charged–lepton momentum,
$\,\omega\equiv(m_{l}^{2}+m_{l^{\prime}}^{2})/2m_{l}\,$
is the maximum $l^{\prime-}$ energy for massless neutrinos, $x\equiv E_{l^{\prime-}}/\omega$ is the reduced energy, $x_{0}\equiv m_{l^{\prime}}/\omega$
and
$$\displaystyle F(x)$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!x(1-x)+{2\over 9}\rho\left(4x^{2}-3x-x_{0}^{2}\right)+\eta%
\,x_{0}(1-x)\,,$$
$$\displaystyle A(x)$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!1-x+{2\over 3}\delta\left(4x-4+\sqrt{1-x_{0}^{2}}\right)\,.$$
(39)
For unpolarized $l^{\prime}s$, the distribution is characterized by
the so-called Michel [38] parameter $\rho$
and the low–energy parameter $\eta$. Two more parameters, $\xi$
and $\delta$, can be determined when the initial lepton polarization is known.
If the polarization of the final charged lepton is also measured,
5 additional independent parameters [4]
($\xi^{\prime}$, $\xi^{\prime\prime}$, $\eta^{\prime\prime}$, $\alpha^{\prime}$, $\beta^{\prime}$)
appear.
For massless neutrinos, the total decay rate is given by [43]
$$\Gamma\,=\,{m_{l}^{5}\widehat{G}_{l^{\prime}l}^{2}\over 192\pi^{3}}\,f\!\left(%
{m_{l^{\prime}}^{2}\over m_{l}^{2}}\right)\,(1+\delta_{\mbox{\rm\scriptsize RC%
}})\,,$$
(40)
where
$$\widehat{G}_{l^{\prime}l}\equiv G_{l^{\prime}l}\,\sqrt{1+4\,\eta\,{m_{l^{%
\prime}}\over m_{l}}\,{g\!\left(m_{l^{\prime}}^{2}/m_{l}^{2}\right)\over f\!%
\left(m_{l^{\prime}}^{2}/m_{l}^{2}\right)}}\,,$$
(41)
and $g(z)=1+9z-9z^{2}-z^{3}+6z(1+z)\ln{z}$.
Thus, $\widehat{G}_{e\mu}$ corresponds to the Fermi coupling
$G_{F}$, measured in $\mu$ decay.
The $B_{\mu}/B_{e}$ and $B_{e}\tau_{\mu}/\tau_{\tau}$
universality tests, discussed in the previous section,
actually prove the ratios
$|\widehat{G}_{\mu\tau}/\widehat{G}_{e\tau}|$
and $|\widehat{G}_{e\tau}/\widehat{G}_{e\mu}|$, respectively.
An important point, emphatically stressed by
Fetscher and Gerber [42], concerns the extraction
of $G_{e\mu}$, whose uncertainty is dominated
by the uncertainty in $\eta_{\mu\to e}$.
In terms of the $g_{\epsilon\omega}^{n}$
couplings, the shape parameters in Eqs. (38)
and (LORENTZ STRUCTURE OF THE CHARGED CURRENTS)
are:
$$\displaystyle\rho$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{3\over 4}(\beta^{+}+\beta^{-})+(\gamma^{+}+\gamma^{-})\,,$$
$$\displaystyle\xi$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!3(\alpha^{-}-\alpha^{+})+(\beta^{-}-\beta^{+})+{7\over 3}(%
\gamma^{+}-\gamma^{-})\,,$$
$$\displaystyle\xi\delta$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{3\over 4}(\beta^{-}-\beta^{+})+(\gamma^{+}-\gamma^{-})\,,$$
(42)
$$\displaystyle\eta$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!\frac{1}{2}\mbox{\rm Re}\left[g^{V}_{LL}g^{S\ast}_{RR}+g^{V%
}_{RR}g^{S\ast}_{LL}+g^{V}_{LR}\left(g^{S\ast}_{RL}+6g^{T\ast}_{RL}\right)+g^{%
V}_{RL}\left(g^{S\ast}_{LR}+6g^{T\ast}_{LR}\right)\right],$$
where [44]
$$\displaystyle\alpha^{+}\equiv{|g^{V}_{RL}|}^{2}+{1\over 16}{|g^{S}_{RL}+6g^{T}%
_{RL}|}^{2}\,,$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\alpha^{-}%
\equiv{|g^{V}_{LR}|}^{2}+{1\over 16}{|g^{S}_{LR}+6g^{T}_{LR}|}^{2}\,,$$
$$\displaystyle\beta^{+}\equiv{|g^{V}_{RR}|}^{2}+{1\over 4}{|g^{S}_{RR}|}^{2}\,,$$
(43)
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\beta^{-}%
\equiv{|g^{V}_{LL}|}^{2}+{1\over 4}{|g^{S}_{LL}|}^{2}\,,$$
$$\displaystyle\gamma^{+}\equiv{3\over 16}{|g^{S}_{RL}-2g^{T}_{RL}|}^{2}\,,$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\gamma^{-}%
\equiv{3\over 16}{|g^{S}_{LR}-2g^{T}_{LR}|}^{2}\,,$$
are positive–definite combinations of decay constants, corresponding to
a final right– ($\alpha^{+}$, $\beta^{+}$, $\gamma^{+}$) or
left– ($\alpha^{-}$, $\beta^{-}$, $\gamma^{-}$)
handed lepton.
In the SM, $\rho=\delta=3/4$,
$\eta=\eta^{\prime\prime}=\alpha^{\prime}=\beta^{\prime}=0$ and
$\xi=\xi^{\prime}=\xi^{\prime\prime}=1$.
The normalization constraint (37) is equivalent to
$\alpha^{+}+\alpha^{-}+\beta^{+}+\beta^{-}+\gamma^{+}+\gamma^{-}=1$.
It is convenient to introduce [41] the probabilities
$Q_{\epsilon\omega}$ for the
decay of an $\omega$–handed $l^{-}$
into an $\epsilon$–handed
daughter lepton,
$$\displaystyle Q_{LL}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{1\over 4}|g^{S}_{LL}|^{2}\!+|g^{V}_{LL}|^{2}\phantom{+3|g^%
{T}_{LR}|^{2}}={1\over 4}\left(-3+{16\over 3}\rho-{1\over 3}\xi+{16\over 9}\xi%
\delta+\xi^{\prime}+\xi^{\prime\prime}\right)\!,$$
$$\displaystyle Q_{RR}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{1\over 4}|g^{S}_{RR}|^{2}\!+\!|g^{V}_{RR}|^{2}\phantom{+3|%
g^{T}_{LR}|^{2}}={1\over 4}\left(-3+{16\over 3}\rho+{1\over 3}\xi-{16\over 9}%
\xi\delta-\xi^{\prime}+\xi^{\prime\prime}\right)\!,$$
$$\displaystyle Q_{LR}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{1\over 4}|g^{S}_{LR}|^{2}\!+\!|g^{V}_{LR}|^{2}\!+\!3|g^{T}%
_{LR}|^{2}={1\over 4}\left(5-{16\over 3}\rho+{1\over 3}\xi-{16\over 9}\xi%
\delta+\xi^{\prime}-\xi^{\prime\prime}\right)\!,$$
(44)
$$\displaystyle Q_{RL}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!{1\over 4}|g^{S}_{RL}|^{2}\!+\!|g^{V}_{RL}|^{2}\!+\!3|g^{T}%
_{RL}|^{2}={1\over 4}\left(5-{16\over 3}\rho-{1\over 3}\xi+{16\over 9}\xi%
\delta-\xi^{\prime}-\xi^{\prime\prime}\right)\!.$$
Upper bounds on any of these (positive–semidefinite) probabilities
translate into corresponding limits for all couplings with the
given chiralities.
For $\mu$ decay, where precise measurements of the polarizations of
both $\mu$ and $e$ have been performed, there exist [41]
upper bounds on $Q_{RR}$, $Q_{LR}$ and $Q_{RL}$, and a lower bound
on $Q_{LL}$. They imply corresponding upper bounds on the 8
couplings $|g^{n}_{RR}|$, $|g^{n}_{LR}|$ and $|g^{n}_{RL}|$.
The measurements of the $\mu^{-}$ and the $e^{-}$ do not allow to
determine $|g^{S}_{LL}|$ and $|g^{V}_{LL}|$ separately [41, 45].
Nevertheless, since the helicity of the $\nu_{\mu}$ in pion decay is
experimentally known [46] to be $-1$, a lower limit on $|g^{V}_{LL}|$ is
obtained [41] from the inverse muon decay
$\nu_{\mu}e^{-}\to\mu^{-}\nu_{e}$.
The present (90% CL) bounds [4]
on the $\mu$–decay couplings
are shown in Fig. 6.
These limits show nicely
that the bulk of the $\mu$–decay transition amplitude is indeed of
the predicted V$-$A type.
The experimental analysis of the $\tau$–decay parameters is
necessarily
different from the one applied to the muon, because of the much
shorter $\tau$ lifetime.
The measurement of the $\tau$ polarization and the parameters
$\xi$ and $\delta$
is possible due to the fact that the spins
of the $\tau^{+}\tau^{-}$ pair produced in $e^{+}e^{-}$ annihilation
are strongly correlated
[31, 33, 47, 48, 49, 50, 51, 52].
Another possibility is to use
the beam polarization, as done by SLD.
However,
the polarization of the charged lepton emitted in the $\tau$ decay
has never been measured. In principle, this could be done
for the decay $\tau^{-}\to\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ by stopping the
muons and detecting their decay products [51].
An alternative method would be [53] to use
the radiative decays
$\tau\to l^{-}\bar{\nu}_{l}\nu_{\tau}\gamma$ ($l=e,\mu$),
since the
distribution of the photons emitted by the daughter lepton is
sensitive to the lepton spin.
The measurement of the inverse decay $\nu_{\tau}l^{-}\to\tau^{-}\nu_{l}$
looks far out of reach.
The present experimental status [5]
on the $\tau$–decay Michel parameters
is shown in table 8.
For comparison, the values measured in $\mu$ decay [4]
are also given.
The improved accuracy of the most recent experimental analyses
has brought an enhanced sensitivity to the different shape parameters,
allowing the first measurements [5] of $\eta_{\tau\to\mu}$,
$\xi_{\tau\to e}$, $\xi_{\tau\to\mu}$, $(\xi\delta)_{\tau\to e}$ and
$(\xi\delta)_{\tau\to\mu}$
without any $e/\mu$ universality assumption.
The determination of the $\tau$ polarization parameters
allows us to bound the total probability for the decay of
a right–handed $\tau$ [51],
$$Q_{\tau_{R}}\equiv Q_{RR}+Q_{LR}=\frac{1}{2}\,\left[1+\frac{\xi}{3}-\frac{16}{%
9}(\xi\delta)\right]\;.$$
(45)
One finds (ignoring possible correlations among the measurements):
$$\displaystyle Q_{\tau_{R}}^{\tau\to\mu}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!\phantom{-}0.05\pm 0.10\;<\,0.20\quad(90\%\;\mbox{\rm CL})\,,$$
$$\displaystyle Q_{\tau_{R}}^{\tau\to e}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!-0.03\pm 0.16\;<\,0.25\quad(90\%\;\mbox{\rm CL})\,,$$
(46)
$$\displaystyle Q_{\tau_{R}}^{\tau\to l}$$
$$\displaystyle\!\!\!=$$
$$\displaystyle\!\!\!\phantom{-}0.02\pm 0.06\;<\,0.12\quad(90\%\;\mbox{\rm CL})\,,$$
where the last value refers to the $\tau$ decay into either $l=e$ or $\mu$,
assuming identical $e$/$\mu$ couplings.
Since these probabilities are positive semidefinite quantities, they imply
corresponding limits on all
$|g^{n}_{RR}|$ and $|g^{n}_{LR}|$ couplings.
A measurement of the final lepton polarization could be even more efficient,
since the total probability for the decay into a right–handed lepton
depends on a single Michel parameter:
$$Q_{l^{\prime}_{R}}\equiv Q_{RR}+Q_{RL}={1\over 2}(1-\xi^{\prime})\,.$$
(47)
Thus, a single polarization measurement could bound the five RR and RL
complex couplings.
Another useful positive–definite quantity is [54]
$$\rho-\xi\delta={3\over 2}\beta^{+}+2\gamma^{-}\,,$$
(48)
which provides direct bounds on $|g^{V}_{RR}|$ and $|g^{S}_{RR}|$.
A rather weak upper limit on $\gamma^{+}$ is obtained from the
parameter $\rho$. More stringent is the bound on $\alpha^{+}$ obtained from
$(1-\rho)$, which is also positive–definite; it implies a corresponding
limit on $|g^{V}_{RL}|$.
Table 9 gives the resulting (90% CL) bounds on the
$\tau$–decay couplings.
The relevance of these limits can be better appreciated in
Fig. 6,
where $e$/$\mu$ universality has been assumed.
If lepton universality is assumed,
the leptonic decay ratios $B_{\mu}/B_{e}$ and $B_{e}\tau_{\mu}/\tau_{\tau}$
provide limits on the low–energy parameter $\eta$.
The best sensitivity [55] comes from
$\widehat{G}_{\mu\tau}$,
where the term proportional to $\eta$ is not suppressed by
the small $m_{e}/m_{l}$ factor. The measured $B_{\mu}/B_{e}$ ratio implies
then:
$$\eta_{\tau\to l}\,=\,0.005\pm 0.027\ .$$
(49)
This determination is more accurate that the one in
table 8,
obtained from the shape of the energy distribution,
and is comparable to the value measured in $\mu$ decay.
A non-zero value of $\eta$ would show that there are at least two
different couplings with opposite chiralities for the charged leptons.
Assuming the V$-$A coupling $g_{LL}^{V}$ to be dominant, the
second one would be [51] a Higgs–type coupling $g^{S}_{RR}$.
To first order in new physics contributions,
$\eta\approx\mbox{\rm Re}(g^{S}_{RR})/2$;
Eq. (49) puts then the (90% CL) bound:
$-0.08\,<\mbox{\rm Re}(g^{S}_{RR})<0.10$.
High–precision measurements of the $\tau$ decay parameters have the
potential to find signals for new phenomena. The accuracy
of the present data is still not good enough to provide
strong constraints; nevertheless, it shows
that the SM gives indeed the dominant contribution to the decay
amplitude.
Future experiments should then
look for small deviations of the SM predictions and find out the
possible source of any detected discrepancy.
In a first analysis, it seems natural to assume [43]
that new physics effects would be dominated by the exchange of a single
intermediate boson, coupling to two leptonic currents.
Table 10
summarizes the expected changes
on the measurable shape parameters [43],
in different new physics scenarios.
The four general cases studied correspond to adding a single intermediate
boson exchange, $V^{+}$, $S^{+}$, $V^{0}$, $S^{0}$
(charged/neutral, vector/scalar), to the SM contribution
(a non-standard $W$ would be a particular case of the
SM + $V^{+}$ scenario).
SUMMARY
The flavour structure of the SM is one of the main pending questions
in our understanding of weak interactions. Although we do not know the
reason of the observed family replication, we have learned experimentally
that the number of SM fermion generations is just three (and no more).
Therefore, we must study as precisely as possible the few existing flavours
to get some hints on the dynamics responsible for their observed structure.
The lepton sector provides a clean environment to test the
universality and Lorentz structure of the electroweak couplings.
We want to investigate whether the mass is the only difference
among the three fermion families.
Naïvely, one would expect the $\tau$ to be much more sensitive
than the $e$ or the $\mu$ to new physics related to the flavour and
mass–generation problems.
While many precision measurements of the electron and muon properties
have been done in the past, it is only recently that $\tau$
experiments have achieved a comparable accuracy [5].
Lepton universality has been tested quite precisely,
both in the charged and neutral current sectors.
The leptonic couplings to the charged $W$ have been verified to be
universal at the 0.15% ($g_{\mu}/g_{e}$) and 0.30% ($g_{\tau}/g_{\mu}$) level.
The axial couplings of the $Z$ boson to the charged leptons have been
measured with a comparable accuracy; universality is satisfied to
the 0.17% ($a_{\mu}/a_{e}$) and 0.19% ($a_{\tau}/a_{e}$) level.
The experimental precision is worse for the $Z$ vector couplings,
which are known to be the same for the three charged leptons
to 9% ($v_{\mu}/v_{e}$) and 5% ($v_{\tau}/v_{e}$) accuracy.
The Lorentz structure of the $l^{-}\to\nu_{l}l^{\prime-}\bar{\nu}_{l^{\prime}}$ decay amplitudes
has been investigated by many experiments.
The present data nicely show that the bulk of the
$\mu$–decay transition amplitude in indeed of the predicted
V$-$A type.
The available information on the leptonic $\tau$ decays, is still not
good enough to determine the underlying dynamics;
nevertheless, useful limits on possible new physics contributions
start to emerge.
At present, all experimental results are consistent with the SM.
There is, however, large room for improvements. Future experiments
will probe the SM to a much deeper level of sensitivity and will explore the
frontier of its possible extensions.
ACKNOWLEDGEMENTS
I would like to thank the organizers for
the charming atmosphere of this meeting.
I am indebted to Manel Martinez for keeping me informed about the
most recent LEP averages, and to Wolfgang Lohmann for providing
the PAW files to generate figures 6 and
6.
This work has been supported in part by CICYT (Spain) under grant
No. AEN-96-1718.
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Deterministic Control of Stochastic Reaction-Diffusion Equations
Wilhelm Stannat, Lukas Wessels
Institut für Mathematik
Technische Universität Berlin
Straße des 17. Juni 136
D-10623 Berlin
Germany
[email protected], [email protected]
Abstract.
We consider the control of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise via deterministic controls. Existence of optimal controls and necessary conditions for optimality are derived. Using adjoint calculus, we obtain a representation for the gradient of the cost functional. The restriction to deterministic controls avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schlögl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.
Key words and phrases:Stochastic reaction diffusion equations, variational approach, optimal control, stochastic Schlögl model, stochastic Nagumo Equation, nonlinear conjugate gradient descent
2010 Mathematics Subject Classification: 93E20, 60H15, 65K10, 35K57
1. Introduction
In this paper our objective is to investigate the optimal control of the semilinear SPDE
$$\displaystyle\mathrm{d}u^{g}_{t}$$
$$\displaystyle=\left[\Delta u^{g}_{t}+f\left(u^{g}_{t}\right)+b(t)g(t)\right]%
\mathrm{d}t+\sigma(t,u^{g}_{t})\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(1)
$$\displaystyle u^{g}_{0}(x)$$
$$\displaystyle=u^{0}(x)$$
$$\displaystyle x\in\Lambda$$
on bounded domains $\Lambda\subset\mathbb{R}$ with $f$ satisfying a one-sided Lipschitz condition. Here,
the control $g$ is deterministic. Precise assumptions on the coefficients of (1) will be
stated at the beginning of the following section. In the case where $f(u)=ku(1-u)(u-a)$ for $k>0$ and
$a\in(0,1)$, equation (1) is called the stochastic Schlögl model.
We will be interested in the optimal control of (1) w.r.t. the following quadratic cost functional
$$\displaystyle J(u^{g},g):=$$
$$\displaystyle\mathbb{E}\left[\frac{c_{\overline{\Lambda}}}{2}\int_{0}^{T}\int_%
{\Lambda}\left(u^{g}_{t}(x)-u_{\overline{\Lambda}}(t,x)\right)^{2}\mathrm{d}x%
\mathrm{d}t\right]$$
(2)
$$\displaystyle+\mathbb{E}\left[\frac{c_{T}}{2}\int_{\Lambda}\left(u^{g}_{T}(x)-%
u^{T}(x)\right)^{2}\mathrm{d}x\right]+\frac{\lambda}{2}\int_{0}^{T}\int_{%
\Lambda}g^{2}(t,x)\mathrm{d}x\mathrm{d}t.$$
The optimal control of the deterministic counterpart of (1) (i.e. $\sigma\equiv 0$) has
been well studied in the existing literature (see the monograph [15]). In particular, the
optimal control of the deterministic Schlögl model has been studied in a series of papers by Tröltzsch,
Ryll et al. ([1], [14], [13]).
Recent years have seen a rising interest in the optimal control of SPDEs. Whereas there exists already a
quite substantial literature on the dynamic programming approach to the optimal control of SPDEs (see, e.g.,
the monograph [4] and in particular [2] for the case of stochastic reaction-diffusion systems) direct variational methods have been much less applied.
Results concerning existence of optimal controls of nonlinear SPDEs have first been obtained in
[9] in the case of the stochastic Navier-Stokes equation, see also the recent preprint
[3] for a discussion of existence of optimal controls of semilinear SPDEs. Necessary first
order conditions for optimality are discussed by Fuhrmann et al. in [6] within the mild
approach to SPDEs. The problem of sufficient conditions for optimal controls has been investigated in
[16]. In this paper, the author derives a sufficient maximum principle for a class of
quasilinear SPDEs with a one-dimensional noise term.
In the present paper we will be interested in the optimal control of (1) within the
variational approach to SPDEs. With a view towards the efficient numerical approximation we will restrict to
deterministic controls. The restriction to deterministic controls in our paper allows to avoid the backward
SPDE for the adjoint state and obtain a conceptional much simpler representation in terms of a backward
random PDE. This gives rise to more efficient numerical approximations of optimal controls. In addition it
allows to weaken the regularity assumptions on the coefficients of the state equation. We illustrate our
approach in the case of the stochastic Schlögl model.
The paper is organized as follows. In Section 2 we state precise assumptions for
our analysis, show the well-posedness of the optimal control problem, and prove the existence of an optimal
control. In Section 3 we prove the Gâteaux differentiability of the solution map and the
cost functional and derive a necessary condition for a control to be locally optimal. In Section
4 we derive a representation for the gradient of the cost functional as well as an
equation for the adjoint state that is later on used in the numerical approximation of locally optimal
solutions. Furthermore, we deduce the Stochastic Minimum Principle from the necessary conditions from the
previous section. In Section 5 we present a probabilistic gradient descent method for the
approximation of an optimal control. In Section 6 we are applying our results to two examples
of the stochastic Schlögl model. In the first example, we show how to accelerate traveling waves and change
their direction of travel (cf. Subsection 6.1). The second example is one situation, where the
optimal control for the stochastic equation apparently differs from the optimal control for the deterministic
counterpart (cf. Subsection 6.2). Since we are not able to give a rigorous proof in this case, we
also consider in Subsection 6.3 the simplified setting of a stochastic ordinary differential
equation, where one can rigorously prove that the optimal control for the stochastic case and its
deterministic counterpart are actually different.
2. General Setting and Well-Posedness of the Optimal Control Problem
Consider the stochastic partial differential equation (1) with Neumann boundary conditions, where $\Lambda\subset\mathbb{R}$ is a bounded domain, $T>0$ is fixed, $(W_{t}^{Q})_{t\in[0,T]}$ is a $Q$-Wiener process for some nonnegative, symmetric trace class operator $Q:L^{2}(\Lambda)\to L^{2}(\Lambda)$ on a given filtered probability space $(\Omega,\mathcal{F},\left(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P}\right)$, $u^{0}\in L^{2}(\Omega\times\Lambda)$, $b\in L^{\infty}([0,T]\times\Lambda)$, $\sigma:[0,T]\times L^{2}(\Lambda)\to L(L^{2}(\Lambda))$ is Fréchet differentiable for every fixed $t\in[0,T]$ and satisfies for all $t\in[0,T]$ and $u,v\in L^{2}(\Lambda)$
(3a)
$$\displaystyle\left\|(\sigma(t,u)-\sigma(t,v))\circ\sqrt{Q}\right\|^{2}_{\text{%
HS}(L^{2}(\Lambda))}$$
$$\displaystyle\leq C\|u-v\|^{2}_{L^{2}(\Lambda)},$$
(3b)
$$\displaystyle\left\|\sigma(t,u)\circ\sqrt{Q}\right\|^{2}_{\text{HS}(L^{2}(%
\Lambda))}$$
$$\displaystyle\leq C\left(1+\|u\|^{2}_{L^{2}(\Lambda)}\right),$$
(3c)
$$\displaystyle\left\|\sigma^{\prime}(t,u)v\circ\sqrt{Q}\right\|^{2}_{\text{HS}(%
L^{2}(\Lambda))}$$
$$\displaystyle\leq C\|v\|^{2}_{L^{2}(\Lambda)},$$
for some constant $C\in\mathbb{R}$, where $\|\cdot\|_{\text{HS}(L^{2}(\Lambda))}$ denotes the Hilbert-Schmidt norm on the space of all Hilbert-Schmidt operators on $L^{2}(\Lambda)$. Furthermore, $f:\mathbb{R}\to\mathbb{R}$ is continuously differentiable satisfying $f(0)=0$,
(4)
$$\sup_{x\in\mathbb{R}}f^{\prime}(x)<\infty,$$
and for all $x\in\mathbb{R}$
(5)
$$|f^{\prime}(x)|<C(1+|x|^{2}),$$
for some constant $C\in\mathbb{R}$.
Remark 2.1.
1.
Notice that the upper bound of the derivative implies a one-sided Lipschitz condition, i.e. there exists a constant $\widetilde{\text{Lip}}_{f}\in\mathbb{R}$ such that
(6)
$$(f(u)-f(v))(u-v)\leq\widetilde{\text{Lip}}_{f}(u-v)^{2},$$
for all $u,v\in\mathbb{R}$.
2.
The nonlinearity in the Schlögl equation satisfies these conditions since the leading coefficient of the polynomial is negative and the derivative is a polynomial of degree 2.
Considering the Gelfand triple
(7)
$$H^{1}(\Lambda)\subset L^{2}(\Lambda)\subset\left(H^{1}(\Lambda)\right)^{\ast},$$
the existence of a variational solution to equation (1) in the space
(8)
$$E:=L^{2}([0,T]\times\Omega,\mathrm{d}t\otimes\mathbb{P};H^{1}(\Lambda))\cap L^%
{2}(\Omega;C([0,T];L^{2}(\Lambda)))$$
is assured (see e.g. [10]).
Our objective is to study the optimal control problem associated with the state equation (1). Let $I_{1}:L^{2}([0,T]\times\Omega,\mathrm{d}t\otimes\mathbb{P},L^{2}(\Lambda))\to%
\mathbb{R}$ be given by
(9)
$$I_{1}(v)\;:=\;\mathbb{E}\left[\frac{c_{\overline{\Lambda}}}{2}\int_{0}^{T}\int%
_{\Lambda}\left(v(t,x)-u_{\overline{\Lambda}}(t,x)\right)^{2}\mathrm{d}x%
\mathrm{d}t+\frac{c_{T}}{2}\int_{\Lambda}\left(v(T,x)-u^{T}(x)\right)^{2}%
\mathrm{d}x\right]$$
and $I_{2}:L^{2}([0,T]\times\Lambda)\to\mathbb{R}$
(10)
$$I_{2}(g)\;:=\;\frac{\lambda}{2}\int_{0}^{T}\int_{\Lambda}g^{2}(t,x)\mathrm{d}x%
\mathrm{d}t,$$
where $c_{\overline{\Lambda}}$, $c_{T}$, $\lambda\geq 0$, $u_{\overline{\Lambda}}\in L^{2}\left([0,T]\times\Lambda\right)$, and $u^{T}\in L^{2}(\Lambda)$. We want to minimize the cost functional
(11)
$$J(g)\;:=\;I_{1}(u^{g})+I_{2}(g),$$
subject to the state equation (1), where
(12)
$$g\in G_{\text{ad}}\;:=\;\left\{g\in L^{6}\left([0,T]\times\Lambda\right)|\,\|g%
\|_{L^{6}([0,T]\times\Lambda)}\leq\kappa\right\},$$
for given $\kappa\geq 0$.
Remark 2.2.
The proof of the Gateaux-differentiability of $g\mapsto u^{g}$ (see Proposition 3.1
below), requires a moment bound of the solution in $L^{6}(\Omega\times[0,T]\times\Lambda)$ due to
the upper bound (5) on the derivative $f^{\prime}$ of the nonlinearity. Therefore the
minimal requirement for an admissible control is $g\in L^{6}([0,T]\times\Lambda)$.
In the work by Buchholz et al. ([1]) on the deterministic case, the set of admissible controls
(13)
$$\tilde{G}_{\text{ad}}\;:=\;\left\{g\in L^{\infty}\left([0,T]\times\Lambda%
\right)|\,g_{a}\leq g(t,x)\leq g_{b}\text{ for a.a. }(t,x)\in[0,T]\times%
\Lambda\right\},$$
for some $g_{a}<g_{b}$ is considered. We could use the same set in our analysis as well.
Throughout the whole paper, we are going to work under the aforementioned conditions. First we want to show that the control problem is well-posed. In order to do so, we need the following a priori bound for solutions of the state equation (1).
Proposition 2.3.
There is a constant $C=C(b,f,\sigma,T,Q,u^{0})$ such that for every solution $u^{g}\in E$ of the state equation (1) associated with $g\in G_{\text{ad}}$ on the right hand side we have
(14)
$$\mathbb{E}\left[\sup_{t\in[0,T]}\left\|u^{g}_{t}\right\|_{L^{2}(\Lambda)}^{6}+%
\left(\int_{0}^{T}\left\|u^{g}_{t}\right\|^{2}_{H^{1}(\Lambda)}\mathrm{d}t%
\right)^{3}\right]\leq\;C\left(1+\int_{0}^{T}\left\|g(t)\right\|^{6}_{L^{2}(%
\Lambda)}\mathrm{d}t\right).$$
Proof.
By the Itô formula from [10], Theorem 4.2.5, we have
$$\displaystyle\left\|u^{g}_{t}\right\|_{L^{2}(\Lambda)}^{2}$$
$$\displaystyle=$$
$$\displaystyle\|u^{0}\|_{L^{2}(\Lambda)}^{2}+2\int_{0}^{t}{}_{(H^{1}(\Lambda))^%
{\ast}}\left\langle\Delta u^{g}_{s},u^{g}_{s}\right\rangle_{H^{1}(\Lambda)}%
\mathrm{d}s+2\int_{0}^{t}\left\langle f\left(u^{g}_{s}\right),u^{g}_{s}\right%
\rangle_{L^{2}(\Lambda)}\mathrm{d}s$$
$$\displaystyle+2\int_{0}^{t}\left\langle b(s)g(s),u^{g}_{s}\right\rangle_{L^{2}%
(\Lambda)}\mathrm{d}s+\int_{0}^{t}\|\sigma(s,u^{g}_{s})\circ\sqrt{Q}\|^{2}_{%
\text{HS}(L^{2}(\Lambda))}\mathrm{d}s$$
$$\displaystyle+2\int_{0}^{t}\left\langle u^{g}_{s},\sigma(s,u^{g}_{s})dW^{Q}_{s%
}\right\rangle_{L^{2}(\Lambda)}$$
$$\displaystyle\leq$$
$$\displaystyle\|u^{0}\|_{L^{2}(\Lambda)}^{2}-2\int_{0}^{t}\left\|\nabla u^{g}_{%
s}\right\|^{2}_{L^{2}(\Lambda)}\mathrm{d}s+\left(2\,\widetilde{\text{Lip}}_{f}%
+C+\|b\|_{L^{\infty}([0,T]\times\Lambda)}\right)\int_{0}^{t}\left\|u^{g}_{s}%
\right\|^{2}_{L^{2}(\Lambda)}\mathrm{d}s$$
(15)
$$\displaystyle+\|b\|_{L^{\infty}([0,T]\times\Lambda)}\int_{0}^{t}\left\|g(s)%
\right\|^{2}_{L^{2}(\Lambda)}\mathrm{d}s+T|\Lambda|C+2\left|\int_{0}^{t}\left%
\langle u^{g}_{s},\sigma(s,u^{g}_{s})dW^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}%
\right|,$$
where we used the growth bound (3b) on $\sigma$ and that by the one-sided Lipschitz continuity of $f$ and $f(0)=0$, we have
(16)
$$\int_{0}^{t}\left\langle f(u^{g}_{s}),u^{g}_{s}\right\rangle_{L^{2}(\Lambda)}%
\mathrm{d}s\leq\widetilde{\text{Lip}}_{f}\int_{0}^{t}\left\|u^{g}_{s}\right\|_%
{L^{2}(\Lambda)}^{2}\mathrm{d}s.$$
Taking both sides of equation (2) to the power $3$, taking the supremum with respect to $t\in[0,T]$, and taking expectations yields
$$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\|u^{g}_{t}\right\|_{L^{2}(%
\Lambda)}^{6}\right]\leq C\Bigg{(}$$
$$\displaystyle 1+\int_{0}^{T}\mathbb{E}\left[\sup_{s\in[0,t]}\left\|u^{g}_{s}%
\right\|^{6}_{L^{2}(\Lambda)}\right]\mathrm{d}t+\int_{0}^{T}\left\|g(t)\right%
\|^{6}_{L^{2}(\Lambda)}\mathrm{d}t$$
(17)
$$\displaystyle+\mathbb{E}\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}\left\langle u%
^{g}_{s},\sigma(s,u^{g}_{s})dW^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right|^{3%
}\right]\Bigg{)}.$$
By Burkholder-Davis-Gundy inequality (see e.g. [8]), we get
(18)
$$\mathbb{E}\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}\left\langle u^{g}_{s},%
\sigma(s,u^{g}_{s})\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right|^{3%
}\right]\leq\,C\,\mathbb{E}\left[\left\langle\int_{0}^{\cdot}\langle u^{g}_{s}%
,\sigma(s,u^{g}_{s})\mathrm{d}W^{Q}_{s}\rangle_{L^{2}(\Lambda)}\right\rangle_{%
T}^{\frac{3}{2}}\right].$$
Now, we compute the quadratic variation. To this end, let $(e_{k})_{k\geq 1}$ be an orthonormal basis of $L^{2}(\Lambda)$. Then
$$\displaystyle\left\langle\int_{0}^{\cdot}\langle u^{g}_{s},\sigma(s,u^{g}_{s})%
\mathrm{d}W^{Q}_{s}\rangle_{L^{2}(\Lambda)}\right\rangle_{T}$$
$$\displaystyle=\int_{0}^{T}\sum_{i=1}^{\infty}|\langle u^{g}_{s},(\sigma(s,u^{g%
}_{s})\circ\sqrt{Q})e_{k}\rangle_{L^{2}(\Lambda)}|^{2}\mathrm{d}s$$
$$\displaystyle\leq\int_{0}^{T}\sum_{i=1}^{\infty}\|u^{g}_{s}\|_{L^{2}(\Lambda)}%
^{2}\|(\sigma(s,u^{g}_{s})\circ\sqrt{Q})e_{k}\|_{L^{2}(\Lambda)}^{2}\mathrm{d}s$$
(19)
$$\displaystyle=\int_{0}^{T}\|\sigma(s,u^{g}_{s})\circ\sqrt{Q}\|^{2}_{\text{HS}(%
L^{2}(\Lambda))}\|u^{g}_{s}\|^{2}_{L^{2}(\Lambda)}\mathrm{d}s.$$
Using the linear growth condition (3b) we obtain together with equation (18)
$$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|\int_{0}^{t}\left\langle u^%
{g}_{s},\sigma(s,u^{g}_{s})\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}%
\right|^{3}\right]$$
$$\displaystyle\leq C\,\mathbb{E}\left[\left(\int_{0}^{T}\|u^{g}_{s}\|_{L^{2}(%
\Lambda)}^{2}+\|u^{g}_{s}\|_{L^{2}(\Lambda)}^{4}\mathrm{d}s\right)^{\frac{3}{2%
}}\right]$$
(20)
$$\displaystyle\leq C\left(1+\int_{0}^{T}\mathbb{E}\left[\sup_{s\in[0,t]}\|u^{g}%
_{s}\|_{L^{2}(\Lambda)}^{6}\right]\mathrm{d}t\right)$$
Together with equation (2) and Gronwall’s inequality, this yields
(21)
$$\mathbb{E}\left[\sup_{t\in[0,T]}\left\|u^{g}_{t}\right\|_{L^{2}(\Lambda)}^{6}%
\right]\leq C\left(1+\int_{0}^{T}\left\|g(t)\right\|^{6}_{L^{2}(\Lambda)}%
\mathrm{d}t\right).$$
Furthermore, from (2), we get
(22)
$$\mathbb{E}\left[\left(\int_{0}^{T}\left\|\nabla u^{g}_{t}\right\|^{2}_{L^{2}(%
\Lambda)}\mathrm{d}t\right)^{3}\right]\leq C\left(1+\mathbb{E}\left[\int_{0}^{%
T}\left\|u^{g}_{t}\right\|_{L^{2}(\Lambda)}^{6}\mathrm{d}t\right]+\int_{0}^{T}%
\left\|g(t)\right\|^{6}_{L^{2}(\Lambda)}\mathrm{d}t\right).$$
Putting together equations (21) and (22), we get for some constant
$C=C(b,f,\sigma,T,Q,u^{0})$
(23)
$$\mathbb{E}\left[\left(\int_{0}^{T}\left\|\nabla u^{g}_{t}\right\|^{2}_{L^{2}(%
\Lambda)}\mathrm{d}t\right)^{3}\right]\leq C\left(1+\int_{0}^{T}\left\|g(t)%
\right\|^{6}_{L^{2}(\Lambda)}\mathrm{d}t\right).$$
Together with (21), this completes the proof.
∎
As a consequence, the finiteness of all of the integrals appearing in the cost functional $J$ is assured. Furthermore, we get the following corollary.
Corollary 2.4.
Let $E$ be defined as in (8). Every solution $u^{g}\in E$ of the state equation (1) associated with $g\in G_{ad}$ on the right hand side is in $L^{6}(\Omega\times[0,T]\times\Lambda)$.
Proof.
We apply the Gagliardo-Nirenberg interpolation inequality which can be found in [12]. This yields for almost all $(t,\omega)\in[0,T]\times\Omega$
(24)
$$\left\|u^{g}_{t}\right\|_{L^{6}(\Lambda)}^{6}\leq C\left\|u^{g}_{t}\right\|_{H%
^{1}(\Lambda)}^{2}\left\|u^{g}_{t}\right\|_{L^{2}(\Lambda)}^{4}.$$
Integrating over $[0,T]\times\Omega$ yields
$$\displaystyle\mathbb{E}\left[\int_{0}^{T}\left\|u^{g}_{t}\right\|_{L^{6}(%
\Lambda)}^{6}\mathrm{d}t\right]$$
$$\displaystyle\leq\mathbb{E}\left[\int_{0}^{T}\left\|u^{g}_{t}\right\|_{H^{1}(%
\Lambda)}^{2}\left\|u^{g}_{t}\right\|_{L^{2}(\Lambda)}^{4}\mathrm{d}t\right]$$
$$\displaystyle\leq\mathbb{E}\left[\sup_{t\in[0,T]}\left\|u^{g}_{t}\right\|_{L^{%
2}(\Lambda)}^{4}\int_{0}^{T}\left\|u^{g}_{t}\right\|_{H^{1}(\Lambda)}^{2}%
\mathrm{d}t\right]$$
(25)
$$\displaystyle\leq\mathbb{E}\left[\sup_{t\in[0,T]}\left\|u^{g}_{t}\right\|_{L^{%
2}(\Lambda)}^{6}\right]\mathbb{E}\left[\left(\int_{0}^{T}\left\|u^{g}_{t}%
\right\|_{H^{1}(\Lambda)}^{2}\mathrm{d}t\right)^{3}\right]<\infty,$$
where we used Hölder’s inequality and Proposition 2.3.
∎
Next, we show that the solution map of the state equation (1) is globally Lipschitz continuous.
Proposition 2.5.
Let $E$ be defined as in (8). For the solution map
$$\displaystyle L^{2}\left([0,T]\times\Lambda\right)$$
$$\displaystyle\to E$$
(26)
$$\displaystyle g$$
$$\displaystyle\mapsto u^{g},$$
there exists a constant $C=C(f,b,\sigma,Q,\Lambda,T)\in\mathbb{R}$ such that
(27)
$$\left\|u^{g_{1}}_{t}-u^{g_{2}}_{t}\right\|_{E}^{2}\leq C\int_{0}^{T}\left\|g_{%
1}-g_{2}\right\|_{L^{2}(\Lambda)}^{2}\mathrm{d}s$$
In particular, the solution map is Lipschitz continuous from $L^{2}\left([0,T]\times\Lambda\right)$ to $E$.
Proof.
By the Itô formula from [10], Theorem 4.2.5, we have almost surely
$$\displaystyle\left\|u^{g_{1}}_{t}-u^{g_{2}}_{t}\right\|_{L^{2}(\Lambda)}^{2}=2$$
$$\displaystyle\int_{0}^{t}{}_{(H^{1}(\Lambda))^{\ast}}\left\langle\Delta\left(u%
^{g_{1}}-u^{g_{2}}\right),u^{g_{1}}-u^{g_{2}}\right\rangle_{H^{1}(\Lambda)}%
\mathrm{d}s$$
$$\displaystyle+2\int_{0}^{t}\left\langle f\left(u^{g_{1}}\right)-f\left(u^{g_{2%
}}\right),u^{g_{1}}-u^{g_{2}}\right\rangle_{L^{2}(\Lambda)}\mathrm{d}s$$
$$\displaystyle+2\int_{0}^{t}\left\langle b\left(g_{1}-g_{2}\right),u^{g_{1}}-u^%
{g_{2}}\right\rangle_{L^{2}(\Lambda)}\mathrm{d}s$$
$$\displaystyle+\int_{0}^{t}\|(\sigma(s,u^{g_{1}}_{s})-\sigma(s,u^{g_{2}}_{s}))%
\circ\sqrt{Q}\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s$$
(28)
$$\displaystyle+2\int_{0}^{t}\langle u^{g_{1}}_{s}-u^{g_{2}}_{s},(\sigma(s,u^{g_%
{1}}_{s})-\sigma(s,u^{g_{2}}_{s}))\mathrm{d}W^{Q}_{s}\rangle.$$
Using the Lipschitz condition (3a) and similar arguments as in the proof of Proposition 2.3 yields the claim.
∎
Now we want to prove the existence of an optimal control:
Theorem 2.6.
There is at least one optimal solution $g^{\ast}\in G_{\text{ad}}$ such that
(29)
$$J(g^{\ast})=\inf_{g\in G_{\text{ad}}}J(g).$$
Proof.
First, we notice that $J$ is nonnegative and hence bounded from below. Let $\left(g_{n}\right)_{n\in\mathbb{N}}\subset G_{\text{ad}}$ be a minimizing sequence, i.e.
(30)
$$\lim_{n\to\infty}J(g_{n})=\inf_{g\in G_{\text{ad}}}J(g),$$
and let $u^{g_{n}}\in E$ denote the unique solution of the state equation (1) associated with $g_{n}$ on the right hand side.
Since $\left(g_{n}\right)_{n\in\mathbb{N}}\subset G_{\text{ad}}$, $\left(g_{n}\right)_{n\in\mathbb{N}}$ is in particular bounded in $L^{2}\left([0,T]\times\Lambda\right)$. Hence, we can extract a weakly convergent subsequence - again denoted by $g_{n}$ - such that $g_{n}\rightharpoonup g^{\ast}$ in $L^{2}([0,T]\times\Lambda)$. The point is now to show that $g^{\ast}\in G_{\text{ad}}$, and $g^{\ast}$ minimizes $J$ in $G_{\text{ad}}$.
Since $G_{\text{ad}}$ is convex and strongly closed, it follows that $G_{\text{ad}}$ is also weakly closed, hence $g^{\ast}\in G_{\text{ad}}$.
In order to show that $g^{\ast}$ minimizes $J$, we first show that $u^{g_{n}}$ converges strongly to $u^{g^{\ast}}$. In the deterministic case, the a priori bound in Lemma 2.3 holds pathwise and we can apply a compact embedding theorem in order to show strong convergence of the solutions. Since we only have the a priori bound under the expectation, we cannot use the same technique. Instead we apply the so called compactness method introduced in [5]. Let us sketch this technique here:
From the bound
(31)
$$\sup_{n\in\mathbb{N}}\mathbb{E}\left[\sup_{t\in[0,T]}\left\|u^{g_{n}}_{t}%
\right\|^{2}_{L^{2}(\Lambda)}+\int_{0}^{T}\left\|u^{g_{n}}\right\|_{H^{1}(%
\Lambda)}^{2}\mathrm{d}s\right]<\infty$$
we can conclude tightness of the measures $\mathbb{P}^{n}:=\mathbb{P}\circ(u^{g_{n}})^{-1}$ on $L^{2}([0,T]\times\Lambda)$. Therefore, $(\mathbb{P}^{n})_{n\in\mathbb{N}}$ is relatively compact and we can extract a converging subsequence $\mathbb{P}^{n}\to\mathbb{P}^{\ast}$. It remains to identify the limit $\mathbb{P}^{\ast}$. By the Skorohod embedding theorem there exists a probability space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ and a sequence of random variables $(\tilde{u}^{g_{n}})_{n\in\mathbb{N}}$ and $\tilde{u}^{g^{\ast}}$ defined on $\tilde{\Omega}$ with the same law as $(u^{g_{n}})_{n\in\mathbb{N}}$ and $u^{g^{\ast}}$, respectively, such that $\tilde{u}^{g_{n}}\to\tilde{u}^{g^{\ast}}$ strongly in $L^{2}([0,T]\times\Lambda)$ $\tilde{\mathbb{P}}$-almost surely. Therefore, using the martingale representation theorem, we can identify $\tilde{u}^{g^{\ast}}$ as a solution to our state equation associated with $g^{\ast}$ on the right hand side.
Now, we split the cost functional into one part that depends on $u^{g}$ and into one part that depends on $g$. For the first part, $I_{1}$, we have
$$\displaystyle\lim_{n\to\infty}I_{1}(u^{g_{n}})$$
$$\displaystyle=$$
$$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\frac{c_{\overline{\Lambda}}}{2}%
\int_{0}^{T}\int_{\Lambda}\left(u^{g_{n}}_{t}(x)-u_{\overline{\Lambda}}(t,x)%
\right)^{2}\mathrm{d}x\mathrm{d}t+\frac{c_{T}}{2}\int_{\Lambda}\left(u^{g_{n}}%
_{T}(x)-u^{T}(x)\right)^{2}\mathrm{d}x\right]$$
$$\displaystyle=$$
$$\displaystyle\lim_{n\to\infty}\tilde{\mathbb{E}}\left[\frac{c_{\overline{%
\Lambda}}}{2}\int_{0}^{T}\int_{\Lambda}\left(\tilde{u}^{g_{n}}_{t}(x)-u_{%
\overline{\Lambda}}(t,x)\right)^{2}\mathrm{d}x\mathrm{d}t+\frac{c_{T}}{2}\int_%
{\Lambda}\left(\tilde{u}^{g_{n}}_{T}(x)-u^{T}(x)\right)^{2}\mathrm{d}x\right]$$
$$\displaystyle\geq$$
$$\displaystyle\tilde{\mathbb{E}}\left[\liminf_{n\to\infty}\left(\frac{c_{%
\overline{\Lambda}}}{2}\int_{0}^{T}\int_{\Lambda}\left(\tilde{u}^{g_{n}}_{t}(x%
)-u_{\overline{\Lambda}}(t,x)\right)^{2}\mathrm{d}x\mathrm{d}t+\frac{c_{T}}{2}%
\int_{\Lambda}\left(\tilde{u}^{g_{n}}_{T}(x)-u^{T}(x)\right)^{2}\mathrm{d}x%
\right)\right]$$
$$\displaystyle=$$
$$\displaystyle\tilde{\mathbb{E}}\left[\frac{c_{\overline{\Lambda}}}{2}\int_{0}^%
{T}\int_{\Lambda}\left(\tilde{u}^{g^{\ast}}_{t}(x)-u_{\overline{\Lambda}}(t,x)%
\right)^{2}\mathrm{d}x\mathrm{d}t+\frac{c_{T}}{2}\int_{\Lambda}\left(\tilde{u}%
^{g^{\ast}}_{T}(x)-u^{T}(x)\right)^{2}\mathrm{d}x\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\frac{c_{\overline{\Lambda}}}{2}\int_{0}^{T}\int_%
{\Lambda}\left(u^{g^{\ast}}_{t}(x)-u_{\overline{\Lambda}}(t,x)\right)^{2}%
\mathrm{d}x\mathrm{d}t+\frac{c_{T}}{2}\int_{\Lambda}\left(u^{g^{\ast}}_{T}(x)-%
u^{T}(x)\right)^{2}\mathrm{d}x\right]$$
(32)
$$\displaystyle=$$
$$\displaystyle I_{1}(u^{g^{\ast}}),$$
where we used that uniqueness in law holds for the state equation (1).
Furthermore, since $I_{2}$ is continuous and convex, it is also weakly lower semi continuous, i.e.
(33)
$$g_{n}\rightharpoonup g^{\ast}\qquad\implies\qquad\liminf_{n\to\infty}I_{2}(g_{%
n})\geq I_{2}(g^{\ast}).$$
Therefore, we have
(34)
$$\inf_{g\in G_{\text{ad}}}J(g)=\lim_{n\to\infty}J(g_{n})\geq\lim_{n\to\infty}I_%
{1}(u^{g_{n}})+\liminf_{n\to\infty}I_{2}(g_{n})\geq I_{1}(u^{g^{\ast}})+I_{2}(%
g^{\ast})=J(g^{\ast}),$$
which completes the proof.
∎
3. First Order Condition for Critical Points
In this section, we are first going to derive the Gâteaux derivative of the solution map and the cost functional and then prove a necessary condition for a control to be locally optimal.
Proposition 3.1.
Let $f:\mathbb{R}\to\mathbb{R}$ satisfy the assumptions of Section 2 and $g\in L^{6}([0,T]\times\Lambda)$ be fixed. Then, for every $h\in L^{6}([0,T]\times\Lambda)$, the Gâteaux derivative of the solution map $g\mapsto u^{g}$, $L^{6}([0,T]\times\Lambda)\to E$ in direction $h$ is given by the solution of the linear SPDE
$$\displaystyle\mathrm{d}y_{t}^{h}$$
$$\displaystyle=[\Delta y_{t}^{h}+f^{\prime}(u_{t}^{g})y_{t}^{h}+b(t)h(t)]%
\mathrm{d}t+\sigma^{\prime}(t,u_{t}^{g})y^{h}_{t}\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(35)
$$\displaystyle y^{h}(0,x)$$
$$\displaystyle=0$$
$$\displaystyle x\in\Lambda.$$
Proof.
The idea for this proof stems from [11], Theorem 4.4. Let $y^{h}$ denote the solution of equation (3.1) associated with $h$ on the right hand side. Set
(36)
$$z_{\delta}(t):=\frac{u_{t}^{g+\delta h}-u_{t}^{g}}{\delta}-y_{t}^{h}.$$
We want to show that $z_{\delta}\to 0$ in $L^{2}(\Omega\times[0,T];H^{1}(\Lambda))\cap L^{2}(\Omega;C([0,T];L^{2}(\Lambda%
)))$ as $\delta\to 0$. First notice
$$\displaystyle z_{\delta}(t)=$$
$$\displaystyle\int_{0}^{t}\Delta z_{\delta}(s)+\frac{1}{\delta}\left(f(u_{s}^{g%
+\delta h})-f(u_{s}^{g})\right)-f^{\prime}(u_{s}^{g})y_{s}^{h}\mathrm{d}s$$
(37)
$$\displaystyle+\int_{0}^{t}\frac{1}{\delta}\left(\sigma(s,u^{g+\delta h}_{s})-%
\sigma(s,u^{g}_{s})\right)-\sigma^{\prime}(s,u^{g}_{s})y^{h}_{s}\mathrm{d}W_{s%
}^{Q}.$$
Note that
$$\displaystyle\frac{1}{\delta}\left(f(u_{s}^{g+\delta h})-f(u_{s}^{g})\right)-f%
^{\prime}(u_{s}^{g})y_{s}^{h}$$
(38)
$$\displaystyle=$$
$$\displaystyle\underbrace{\frac{1}{\delta}\left(f(u_{s}^{g}+\delta y_{s}^{h})-f%
(u_{s}^{g})\right)-f^{\prime}(u_{s}^{g})y_{s}^{h}}_{=:R_{\delta}(s)}+%
\underbrace{\frac{1}{\delta}\left(f(u_{s}^{g+\delta h})-f(u_{s}^{g}+\delta y_{%
s}^{h})\right)}_{=:S_{\delta}(s)}.$$
and similarly
$$\displaystyle\frac{1}{\delta}\left(\sigma(s,u_{s}^{g+\delta h})-\sigma(s,u_{s}%
^{g})\right)-\sigma^{\prime}(s,u_{s}^{g})y_{s}^{h}$$
(39)
$$\displaystyle=$$
$$\displaystyle\underbrace{\frac{1}{\delta}\left(\sigma(s,u_{s}^{g}+\delta y_{s}%
^{h})-\sigma(s,u_{s}^{g})\right)-\sigma^{\prime}(s,u_{s}^{g})y_{s}^{h}}_{=:\Xi%
_{\delta}(s)}+\underbrace{\frac{1}{\delta}\left(\sigma(s,u_{s}^{g+\delta h})-%
\sigma(s,u_{s}^{g}+\delta y_{s}^{h})\right)}_{=:\Sigma_{\delta}(s)}.$$
Together with equation (3), Itô’s formula yields
$$\displaystyle\frac{1}{2}\left\|z_{\delta}(t)\right\|_{L^{2}(\Lambda)}^{2}=$$
$$\displaystyle\int_{0}^{t}\!{}_{(H^{1}(\Lambda))^{\ast}}\left\langle\Delta z_{%
\delta}(s),z_{\delta}(s)\right\rangle_{H^{1}(\Lambda)}\mathrm{d}s+\int_{0}^{t}%
\!\left\langle R_{\delta}(s),z_{\delta}(s)\right\rangle_{L^{2}(\Lambda)}%
\mathrm{d}s$$
$$\displaystyle+\int_{0}^{t}\!\left\langle S_{\delta}(s),z_{\delta}(s)\right%
\rangle_{L^{2}(\Lambda)}\mathrm{d}s+\int_{0}^{t}\!\left\langle z_{\delta}(s),%
\Xi_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}$$
$$\displaystyle+\int_{0}^{t}\!\left\langle z_{\delta}(s),\Sigma_{\delta}(s)%
\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}$$
(40)
$$\displaystyle+\frac{1}{2}\int_{0}^{t}\|\left(\Xi_{\delta}(s)+\Sigma_{\delta}(s%
)\right)\circ\sqrt{Q}\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s.$$
First notice that
(41)
$$\int_{0}^{t}{}_{(H^{1}(\Lambda))^{\ast}}\langle\Delta z_{\delta}(s),z_{\delta}%
(s)\rangle_{H^{1}(\Lambda)}\mathrm{d}s=-\int_{0}^{t}\|\nabla z_{\delta}(s)\|^{%
2}_{L^{2}(\Lambda)}\mathrm{d}s.$$
Furthermore, we have $\langle R_{\delta}(s),z_{\delta}(s)\rangle_{L^{2}(\Lambda)}\leq(\|R_{\delta}(s%
)\|_{L^{2}(\Lambda)}^{2}+\|z_{\delta}(s)\|_{L^{2}(\Lambda)}^{2})/2$, and, since $f$ is one-sided Lipschitz continuous, we have
$$\displaystyle\left\langle S_{\delta}(s),z_{\delta}(s)\right\rangle_{L^{2}(%
\Lambda)}$$
$$\displaystyle=\frac{1}{\delta^{2}}\left\langle f(u_{s}^{g+\delta h})-f(u_{s}^{%
g}+\delta y_{s}^{h}),u_{s}^{g+\delta h}-(u_{s}^{g}+\delta y_{s}^{h})\right%
\rangle_{L^{2}(\Lambda)}$$
(42)
$$\displaystyle\leq\widetilde{\text{Lip}}_{f}\|z_{\delta}(s)\|_{L^{2}(\Lambda)}^%
{2}.$$
For the last term in equation (3), we have
$$\displaystyle\frac{1}{2}\int_{0}^{T}\|\left(\Xi_{\delta}(s)+\Sigma_{\delta}(s)%
\right)\circ\sqrt{Q}\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s$$
(43)
$$\displaystyle\leq$$
$$\displaystyle\int_{0}^{T}\|\Xi_{\delta}(s)\circ\sqrt{Q}\|_{\text{HS}(L^{2}(%
\Lambda))}^{2}\mathrm{d}s+\int_{0}^{T}\|\Sigma_{\delta}(s)\circ\sqrt{Q}\|_{%
\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s,$$
where
(44)
$$\|\Xi_{\delta}(s)\circ\sqrt{Q}\|^{2}_{\text{HS}(L^{2}(\Lambda))}\leq\text{tr}Q%
\,\|\Xi_{\delta}(s)\|_{L(L^{2}(\Lambda))}^{2},$$
and, by the Lipschitz condition (3a) on $\sigma$,
$$\displaystyle\left\|\Sigma_{\delta}(s)\circ\sqrt{Q}\right\|^{2}_{\text{HS}(L^{%
2}(\Lambda))}$$
$$\displaystyle=\left\|\left(\frac{1}{\delta}\left(\sigma(s,u^{g+\delta h}_{s})-%
\sigma(s,u^{g}_{s}+\delta y^{h}_{s})\right)\right)\circ\sqrt{Q}\right\|^{2}_{%
\text{HS}(L^{2}(\Lambda))}$$
(45)
$$\displaystyle\leq C\left\|\frac{1}{\delta}\left(u^{g+\delta h}_{s}-u^{g}_{s}%
\right)-y^{h}_{s}\right\|^{2}_{L^{2}(\Lambda)}=C\|z_{\delta}(s)\|^{2}_{L^{2}(%
\Lambda)}.$$
Therefore, taking the supremum with respect to $t\in[0,T]$ in equation (3) and taking expectations, it follows
$$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\|z_{\delta}(t)\right\|_{L^{%
2}(\Lambda)}^{2}\right]+\mathbb{E}\left[\int_{0}^{T}\|\nabla z_{\delta}(s)\|^{%
2}_{L^{2}(\Lambda)}\mathrm{d}s\right]$$
$$\displaystyle\leq C\Bigg{\{}$$
$$\displaystyle\mathbb{E}\left[\int_{0}^{T}\left\|R_{\delta}(s)\right\|^{2}_{L^{%
2}(\Lambda)}\mathrm{d}s\right]+\int_{0}^{T}\mathbb{E}\left[\sup_{s\in[0,t]}\|z%
_{\delta}(s)\|^{2}_{L^{2}(\Lambda)}\right]\mathrm{d}t$$
$$\displaystyle+\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\!\left\langle z_{%
\delta}(s),\Xi_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right]$$
(46)
$$\displaystyle+\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\!\left\langle z_{%
\delta}(s),\Sigma_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}%
\right]\Bigg{\}}.$$
Using Burkholder-Davis-Gundy inequality, we have
(47)
$$\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\!\left\langle z_{\delta}(s),%
\Sigma_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right]\leq
C%
\mathbb{E}\left[\left\langle\int_{0}^{\cdot}\!\left\langle z_{\delta}(s),%
\Sigma_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right%
\rangle^{\frac{1}{2}}_{T}\right].$$
Now we compute the quadratic variation. To this end, let $(e_{k})_{k\geq 1}$ be an orthonormal basis of $L^{2}(\Lambda)$. Then we have
$$\displaystyle\left\langle\int_{0}^{\cdot}\!\left\langle z_{\delta}(s),\Sigma_{%
\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right\rangle^{%
\frac{1}{2}}_{T}$$
$$\displaystyle=$$
$$\displaystyle\left(\int_{0}^{T}\sum_{k=1}^{\infty}\left|\left\langle z_{\delta%
}(s),(\Sigma_{\delta}(s)\circ\sqrt{Q})e_{k}\right\rangle_{L^{2}(\Lambda)}%
\right|^{2}\mathrm{d}s\right)^{\frac{1}{2}}$$
(48)
$$\displaystyle\leq$$
$$\displaystyle\frac{\varepsilon}{2}\sup_{t\in[0,T]}\|z_{\delta}(t)\|_{L^{2}(%
\Lambda)}^{2}+\frac{1}{2\varepsilon}\int_{0}^{T}\left\|\Sigma_{\delta}(s)\circ%
\sqrt{Q}\right\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s,$$
for arbitrary $\varepsilon>0$. With the same estimates as above for $\|\Sigma_{\delta}(s)\circ\sqrt{Q}\|_{\text{HS}(L^{2}(\Lambda))}^{2}$ and with inequality (47) this yields
$$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\!\left\langle z_{%
\delta}(s),\Sigma_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right]$$
(49)
$$\displaystyle\leq$$
$$\displaystyle C\varepsilon\mathbb{E}\left[\sup_{t\in[0,T]}\|z_{\delta}(t)\|_{L%
^{2}(\Lambda)}^{2}\right]+\frac{C}{\varepsilon}\mathbb{E}\left[\int_{0}^{T}\|z%
_{\delta}(s)\|_{L^{2}(\Lambda)}^{2}\mathrm{d}s\right].$$
Furthermore, with similar calculations as above, we get
$$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\!\left\langle z_{%
\delta}(s),\Xi_{\delta}(s)\mathrm{d}W^{Q}_{s}\right\rangle_{L^{2}(\Lambda)}\right]$$
(50)
$$\displaystyle\leq$$
$$\displaystyle C\varepsilon\,\mathbb{E}\left[\int_{0}^{T}\|z_{\delta}(s)\|_{L^{%
2}(\Lambda)}^{2}\mathrm{d}s\right]+\frac{C}{\varepsilon}\,\mathbb{E}\left[\int%
_{0}^{T}\left\|\Xi_{\delta}(s)\circ\sqrt{Q}\right\|_{\text{HS}(L^{2}(\Lambda))%
}^{2}\mathrm{d}s\right],$$
for arbitraty $\varepsilon>0$. Choosing $\varepsilon>0$ in (3) and (3) small enough, we get from (3)
$$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\|z_{\delta}(t)\right\|_{L^{%
2}(\Lambda)}^{2}\right]+\mathbb{E}\left[\int_{0}^{T}\|\nabla z_{\delta}(s)\|^{%
2}_{L^{2}(\Lambda)}\mathrm{d}s\right]$$
$$\displaystyle\leq C\Bigg{\{}$$
$$\displaystyle\int_{0}^{T}\mathbb{E}\left[\sup_{s\in[0,t]}\|z_{\delta}(s)\|^{2}%
_{L^{2}(\Lambda)}\right]\mathrm{d}t+\mathbb{E}\left[\int_{0}^{T}\left\|R_{%
\delta}(s)\right\|^{2}_{L^{2}(\Lambda)}\mathrm{d}s\right]$$
(51)
$$\displaystyle+\mathbb{E}\left[\int_{0}^{T}\left\|\Xi_{\delta}(s)\circ\sqrt{Q}%
\right\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s\right]\Bigg{\}}.$$
By Gronwall inequality, this yields
$$\displaystyle\mathbb{E}\left[\sup_{s\in[0,T]}\|z_{\delta}(s)\|_{L^{2}(\Lambda)%
}^{2}\right]+\mathbb{E}\left[\int_{0}^{T}\|\nabla z_{\delta}(s)\|^{2}_{L^{2}(%
\Lambda)}\mathrm{d}s\right]$$
(52)
$$\displaystyle\leq$$
$$\displaystyle C\left(\mathbb{E}\left[\int_{0}^{T}\|R_{\delta}(s)\|_{L^{2}(%
\Lambda)}^{2}\mathrm{d}s\right]+\mathbb{E}\left[\int_{0}^{T}\left\|\Xi_{\delta%
}(s)\circ\sqrt{Q}\right\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s\right]%
\right).$$
Since $R_{\delta}\to 0$ as $\delta\to 0$ for almost all $(\omega,t,x)\in\Omega\times[0,T]\times\Lambda$, we get by the dominated convergence theorem
(53)
$$\lim_{\delta\to 0}\mathbb{E}\left[\int_{0}^{T}\|R_{\delta}(t)\|^{2}_{L^{2}(%
\Lambda)}\mathrm{d}t\right]=0.$$
Here, we used that $R_{\delta}$ is dominated in the following way: By assumption (5), Taylor’s formula and elementary estimates, we have
(54)
$$|R_{\delta}|\leq C\left(1+\left|u^{g}\right|^{3}+\left|y^{h}\right|^{3}\right).$$
The boundedness of the right hand side in $L^{2}(\Omega\times[0,T]\times\Lambda)$ follows immediately from Corollary 2.4 (notice that we get the boundedness of $y^{h}$ in $L^{6}(\Omega\times[0,T]\times\Lambda)$ by the same arguments as for $u^{g}$). Furthermore, we have
(55)
$$\lim_{\delta\to 0}\mathbb{E}\left[\int_{0}^{T}\left\|\Xi_{\delta}(s)\circ\sqrt%
{Q}\right\|_{\text{HS}(L^{2}(\Lambda))}^{2}\mathrm{d}s\right]=0$$
since by the Lipschitz condition (3a) on $\sigma$ and the bound on the Fréchet derivative (3c) of $\sigma$ we have the following bound:
$$\displaystyle\left\|\Xi_{\delta}(s)\circ\sqrt{Q}\right\|^{2}_{\text{HS}(L^{2}(%
\Lambda))}$$
$$\displaystyle=$$
$$\displaystyle\left\|\left(\frac{1}{\delta}\left(\sigma(s,u_{s}^{g}+\delta y_{s%
}^{h})-\sigma(s,u_{s}^{g})\right)-\sigma^{\prime}(s,u_{s}^{g})y_{s}^{h}\right)%
\circ\sqrt{Q}\right\|^{2}_{\text{HS}(L^{2}(\Lambda))}$$
$$\displaystyle\leq$$
$$\displaystyle 2\left\|\frac{1}{\delta}\left(\sigma(s,u_{s}^{g}+\delta y_{s}^{h%
})-\sigma(s,u_{s}^{g})\right)\circ\sqrt{Q}\right\|^{2}_{\text{HS}(L^{2}(%
\Lambda))}+2\left\|\sigma^{\prime}(s,u_{s}^{g})y_{s}^{h}\circ\sqrt{Q}\right\|^%
{2}_{\text{HS}(L^{2}(\Lambda))}$$
(56)
$$\displaystyle\leq$$
$$\displaystyle C\left(1+\|y^{h}\|^{2}_{L^{2}(\Lambda)}\right).$$
This completes the proof that $z_{\delta}$ converges to $0$ in $L^{2}([0,T]\times\Omega;H^{1}(\Lambda))$ and in
$L^{2}(\Omega;C([0,T];L^{2}(\Lambda)))$. From the definition of $h\mapsto y^{h}$, it follows immediately that this is linear. Thus, for the Gâteaux differentiability it remains to show that $h\mapsto y^{h}$ is continuous. But this follows with the same arguments as in Proposition 2.5.
∎
As a corollary we get the following representation for the Gâteaux derivative of the cost functional.
Corollary 3.2.
For every $h\in L^{6}([0,T]\times\Lambda)$, the cost functional
$J:L^{2}\left([0,T]\times\Lambda\right)\to\mathbb{R}$ is Gâteaux differentiable in the direction $h$ with Gâteaux derivative
$$\displaystyle\frac{\partial J(g)}{\partial h}=\mathbb{E}\Bigg{[}c_{\overline{%
\Lambda}}\int_{0}^{T}\int_{\Lambda}y^{h}_{t}(x)\left(u_{t}^{g}(x)-u_{\overline%
{\Lambda}}(t,x)\right)\mathrm{d}x\mathrm{d}t$$
(57)
$$\displaystyle+c_{T}\int_{\Lambda}y^{h}_{T}(x)\left(u_{T}^{g}(x)-u^{T}(x)\right%
)\mathrm{d}x+\lambda\int_{0}^{T}\int_{\Lambda}g(t,x)h(t,x)\mathrm{d}x\mathrm{d%
}t\Bigg{]},$$
where $y^{h}$ denotes the variational solution of the SPDE (3.1).
Proof.
Recall that the cost functional is given by
(58)
$$J(g)\;:=\;I_{1}(u^{g})+I_{2}(g),$$
where
(59)
$$I_{1}(v)\;:=\;\mathbb{E}\left[\frac{c_{\overline{\Lambda}}}{2}\int_{0}^{T}\int%
_{\Lambda}\left(v(t,x)-u_{\overline{\Lambda}}(t,x)\right)^{2}\mathrm{d}x%
\mathrm{d}t+\frac{c_{T}}{2}\int_{\Lambda}\left(v(T,x)-u^{T}(x)\right)^{2}%
\mathrm{d}x\right]$$
and
(60)
$$I_{2}(g)\;:=\;\frac{\lambda}{2}\int_{0}^{T}\int_{\Lambda}g^{2}(t,x)\mathrm{d}x%
\mathrm{d}t.$$
Hence
(61)
$$\displaystyle\frac{\partial J(g)}{\partial h}$$
$$\displaystyle=\frac{\partial I_{1}\left(u^{g}\right)}{\partial h}+\frac{%
\partial I_{2}(g)}{\partial h}.$$
Let $g\in L^{6}\left([0,T]\times\Lambda\right)$ be fixed. For $h\in L^{6}([0,T]\times\Lambda)$, we get for the Gâteaux derivative of $I_{2}$
(62)
$$\frac{\partial I_{2}(g)}{\partial h}=\lambda\int_{0}^{T}\int_{\Lambda}g(t,x)h(%
t,x)\mathrm{d}x\mathrm{d}t.$$
On the other hand we get for the Gâteaux derivative of $I_{1}$
(63)
$$\frac{\partial I_{1}(v)}{\partial w}=\mathbb{E}\left[c_{\overline{\Lambda}}%
\int_{0}^{T}\int_{\Lambda}w\left(v-u_{\overline{\Lambda}}\right)\mathrm{d}x%
\mathrm{d}t+c_{T}\int_{\Lambda}w\left(v-u^{T}\right)\mathrm{d}x\right].$$
Hence, by the chain rule, we get
(64)
$$\displaystyle\frac{\partial I_{1}\left(u^{g}\right)}{\partial h}=\mathbb{E}%
\left[c_{\overline{\Lambda}}\int_{0}^{T}\int_{\Lambda}\frac{\partial u^{g}}{%
\partial h}\left(u^{g}-u_{\overline{\Lambda}}\right)\mathrm{d}x\mathrm{d}t+c_{%
T}\int_{\Lambda}\frac{\partial u^{g}}{\partial h}\left(u^{g}-u^{T}\right)%
\mathrm{d}x\right],$$
which, together with equation (62) and Proposition 3.1, completes the proof.
∎
Now we can state a necessary condition for $J$ to attain a minimum.
Theorem 3.3.
Let $J$ attain a (local) minimum at $g^{\ast}\in G_{\text{ad}}$. Then, for every $h\in G_{\text{ad}}$ we have
(65)
$$\frac{\partial J(g^{\ast})}{\partial(h-g^{\ast})}\geq 0.$$
Proof.
Let $h\in G_{\text{ad}}$, and set $\delta_{t}:=g^{\ast}+t(h-g^{\ast})\in G_{\text{ad}}$. Since $g^{\ast}$ is a local minimizer, there exists a $t_{0}>0$ such that for all $t\in(0,t_{0})$ we have
(66)
$$J(g^{\ast})\leq J(\delta_{t}).$$
which implies
(67)
$$\frac{1}{t}\left(J(g^{\ast}+t(h-g^{\ast}))-J(g^{\ast})\right)\geq 0.$$
Letting $t$ tend to zero yields the claim.
∎
4. The Gradient of the Cost Functional
In this section, we are going to derive a representation for the gradient of the cost functional via adjoint calculus. Recall the state equation
$$\displaystyle\mathrm{d}u^{g}_{t}$$
$$\displaystyle=\left[\Delta u^{g}_{t}+f\left(u^{g}_{t}\right)+b(t)g(t)\right]%
\mathrm{d}t+\sigma(t,u^{g}_{t})\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(68)
$$\displaystyle u^{g}_{0}(x)$$
$$\displaystyle=u^{0}(x)$$
$$\displaystyle x\in\Lambda$$
In Section 3, we proved the following representation
$$\displaystyle\frac{\partial J(g)}{\partial h}=\mathbb{E}\Bigg{[}c_{\overline{%
\Lambda}}\int_{0}^{T}\int_{\Lambda}y^{h}_{s}(x)\left(u_{t}^{g}(x)-u_{\overline%
{\Lambda}}(t,x)\right)\mathrm{d}x\mathrm{d}t$$
(69)
$$\displaystyle+c_{T}\int_{\Lambda}y^{h}_{T}(x)\left(u_{T}^{g}(x)-u^{T}(x)\right%
)\mathrm{d}x+\lambda\int_{0}^{T}\int_{\Lambda}g(t,x)h(t,x)\mathrm{d}x\mathrm{d%
}t\Bigg{]},$$
where $y^{h}$ is the variational solution of
$$\displaystyle\mathrm{d}y_{t}^{h}$$
$$\displaystyle=[\Delta y_{t}^{h}+f^{\prime}(u_{t}^{g})y_{t}^{h}+b(t)h(t)]%
\mathrm{d}t+\sigma^{\prime}(t,u^{g}_{t})y^{h}_{t}\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(70)
$$\displaystyle y^{h}(0,x)$$
$$\displaystyle=0$$
$$\displaystyle x\in\Lambda.$$
Now, we introduce the following adjoint equation.
$$\displaystyle-\partial_{t}p$$
$$\displaystyle=\Delta p+f^{\prime}(u^{g})p+c_{\overline{\Lambda}}\left(u^{g}-u_%
{\overline{\Lambda}}\right)$$
$$\displaystyle\text{on }[0,T]\times\Lambda$$
(71)
$$\displaystyle p(T,x)$$
$$\displaystyle=c_{T}\left(u^{g}_{T}(x)-u^{T}(x)\right)$$
$$\displaystyle x\in\Lambda.$$
One crucial fact for our algorithm is the fact that the adjoint equation is a random backward PDE. The canonical adjoint equation in this context would be a backward SPDE which makes it difficult to develop efficient algorithms to approximate the optimal control. The following property of the adjoint state is the main ingredient in the derivation of the gradient of the cost functional.
Proposition 4.1.
Let $p$ be the solution of the adjoint equation (4) and let $y^{h}$ be the solution of equation (4) associated with $u^{g}$. Then we have for every $h\in L^{6}([0,T]\times\Lambda)$
(72)
$$\displaystyle\mathbb{E}\left[\int_{0}^{T}\int_{\Lambda}bph\mathrm{d}x\mathrm{d%
}t\right]=\mathbb{E}\left[\int_{0}^{T}\int_{\Lambda}c_{\overline{\Lambda}}(u^{%
g}-u_{\overline{\Lambda}})y^{h}\mathrm{d}x\mathrm{d}t+\int_{\Lambda}c_{T}(u^{g%
}_{T}-u^{T})y^{h}_{T}\mathrm{d}x\right].$$
Proof.
Since $p$ is of zero quadratic variation, we have
(73)
$$y^{h}_{T}p(T,\cdot)-y_{0}^{h}p(0,\cdot)=\int_{0}^{T}y^{h}_{t}\mathrm{d}p(t,%
\cdot)+\int_{0}^{T}p(t,\cdot)\mathrm{d}y_{t}^{h}.$$
Plugging in equations (4) and (4), respectively, this yields
$$\displaystyle y^{h}_{T}c_{T}(u^{g}_{T}-u^{T})=$$
$$\displaystyle-\int_{0}^{T}y^{h}_{t}\left(\Delta p_{t}+f^{\prime}(u^{g}_{t})p_{%
t}+c_{\overline{\Lambda}}(u^{g}_{t}-u_{\overline{\Lambda}}(t,\cdot))\right)%
\mathrm{d}t$$
(74)
$$\displaystyle+\int_{0}^{T}p_{t}\left(\Delta y^{h}_{t}+f^{\prime}(u^{g}_{t})y^{%
h}_{t}+b(t)h(t)\right)\mathrm{d}t+\int_{0}^{T}\sigma^{\prime}(t,u^{g}_{t})y^{h%
}_{t}p_{t}\mathrm{d}W^{Q}_{t}.$$
Integrating over $\Lambda$, integrating the Laplace operator by parts, and taking the expectation, we get
(75)
$$\mathbb{E}\left[\int_{\Lambda}y^{h}_{T}c_{T}(u^{g}_{T}-u^{T})\mathrm{d}x\right%
]=\mathbb{E}\left[\int_{0}^{T}\int_{\Lambda}bhp-c_{\overline{\Lambda}}y^{h}_{t%
}(u^{g}_{t}-u_{\overline{\Lambda}}(t,\cdot))\mathrm{d}x\mathrm{d}t\right],$$
which is the claimed result.
∎
As a corollary, we get the following representation for the gradient of the cost functional.
Theorem 4.2.
The gradient of the cost functional is given by
(76)
$$\nabla J(g)(t,x)=\mathbb{E}\left[b(t)p(t,x)+\lambda g(t,x)\right],$$
where $p$ is the solution of the adjoint equation
$$\displaystyle-\partial_{t}p$$
$$\displaystyle=\Delta p+f^{\prime}(u^{g})p+c_{\overline{\Lambda}}\left(u^{g}-u_%
{\overline{\Lambda}}\right)$$
$$\displaystyle\text{on }[0,T]\times\Lambda$$
(77)
$$\displaystyle p(T,x)$$
$$\displaystyle=c_{T}\left(u^{g}_{T}(x)-u^{T}(x)\right)$$
$$\displaystyle x\in\Lambda.$$
Proof.
By Corollary 3.2, we have
$$\displaystyle\frac{\partial J(g)}{\partial h}=$$
$$\displaystyle\mathbb{E}\Bigg{[}c_{\overline{\Lambda}}\int_{0}^{T}\int_{\Lambda%
}y^{h}_{t}(x)\left(u_{t}^{g}(x)-u_{\overline{\Lambda}}(t,x)\right)\mathrm{d}x%
\mathrm{d}t$$
(78)
$$\displaystyle+c_{T}\int_{\Lambda}y^{h}_{T}(x)\left(u_{T}^{g}(x)-u^{T}(x)\right%
)\mathrm{d}x+\lambda\int_{0}^{T}\int_{\Lambda}g(t,x)h(t,x)\mathrm{d}x\mathrm{d%
}t\Bigg{]},$$
where $y^{h}$ denotes the variational solution of the random PDE (3.1). Now, by Proposition 4.1, this yields
$$\displaystyle\frac{\partial J(g)}{\partial h}=\mathbb{E}\Bigg{[}\int_{0}^{T}%
\int_{\Lambda}bph\mathrm{d}x\mathrm{d}t+\lambda\int_{0}^{T}\int_{\Lambda}gh%
\mathrm{d}x\mathrm{d}t\Bigg{]},$$
which completes the proof.
∎
Furthermore, by plugging this representation into the necessary condition derived in Theorem 3.3, we get the Stochastic Minimum Principle.
Theorem 4.3.
Let $J$ attain a (local) minimum at $g^{\ast}\in G_{\text{ad}}$. Then, for every $h\in G_{\text{ad}}$ we have
(79)
$$\mathbb{E}\left[\int_{0}^{T}\int_{\Lambda}(b(t)p(t,x)+\lambda g^{\ast}(t,x))(h%
(t,x)-g^{\ast}(t,x))\mathrm{d}x\mathrm{d}t\right]\geq 0.$$
5. Nonlinear Conjugate Gradient Descent
Now that we have identified a representation for the gradient, we can apply a probabilistic nonlinear conjugate gradient descent method in order to approximate the optimal control. We’re going to briefly sketch our algorithm here. For a survey of nonlinear conjugate gradient descent methods see [7].
Let the initial control $g_{0}\in L^{6}\left([0,T]\times\Lambda\right)$ be given and fix an initial step size $s_{0}>0$. Then, the next control can be found as follows.
1.
Solve the state equation
$$\displaystyle\mathrm{d}u^{g_{n}}_{t}$$
$$\displaystyle=\left[\Delta u^{g_{n}}_{t}+f\left(u^{g_{n}}_{t}\right)+b(t)g_{n}%
(t)\right]\mathrm{d}t+\sigma(t,u^{g_{n}}_{t})\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on }L^{2}(\Lambda)$$
$$\displaystyle u^{g_{n}}_{0}(x)$$
$$\displaystyle=u^{0}(x)$$
$$\displaystyle x\in\Lambda$$
for one realization of the noise.
2.
Solve the adjoint equation
$$\displaystyle-\partial_{t}p_{n}$$
$$\displaystyle=\Delta p_{n}+f^{\prime}(u^{g_{n}})p_{n}+c_{\overline{\Lambda}}%
\left(u^{g_{n}}-u_{\overline{\Lambda}}\right)$$
$$\displaystyle\text{on }[0,T]\times\Lambda$$
$$\displaystyle p_{n}(T,x)$$
$$\displaystyle=c_{T}\left(u^{g_{n}}_{T}(x)-u^{T}(x)\right)$$
$$\displaystyle x\in\Lambda.$$
with the data given by the sample of the solution of the state equation that was calculated in Step 5.
3.
Repeat Step 5 and Step 5 to approximate
$$\nabla J(g_{n})(t,x)=\mathbb{E}\left[b(t)p_{n}(t,x)+\lambda g_{n}(t,x)\right]$$
via a Monte Carlo method.
4.
The direction of descent is given by $d_{n}=-\nabla J(g_{n})+\beta_{n}d_{n-1}$, where $\beta_{n}=\frac{\|\nabla J(g_{n})\|}{\|\nabla J(g_{n-1})\|}$. (In the first step, $\beta_{1}=0$.)
5.
Compute the new control via $g_{n+1}=g_{n}+s_{n}d_{n}$.
6.
Accept or deny the new control: Again using a Monte Carlo method, we compare the costs under the new control with the costs under the old control. If the new control decreases the costs, we accept the new control and go back to step 5. Otherwise, we decrease the step size $s_{n}=s_{n}/2$ and then go back to step 5. (In our simulations, it has proven useful to accept the new control even if the costs are non-decreasing, once the step size gets too small, e.g. $s_{n}<10^{-4}$.)
7.
Stop if $\|g_{n+1}\|<\eta$, otherwise reset the step size $s_{n}=s_{0}$ and go to step 5.
6. Application to Optimal Control of the Stochastic Schlögl Model
In this section we want to present the application of the algorithm that was introduced in Section 5 to the stochastic Schlögl model. We are going to investigate two examples. The first one is to control the speed and the direction of travel of the wave developing in the Schlögl model with multiplicative noise; the second one is an example, where the optimal control of the deterministic system differs from the optimal control of the stochastic system. Corresponding results for the deterministic model can be found in the work by Buchholz et al. (see [1]).
6.1. Steering of a Wave Front
Let us first recall the Schlögl model. We consider the state equation
$$\displaystyle\mathrm{d}u^{g}_{t}$$
$$\displaystyle=\left[\Delta u^{g}_{t}+f\left(u^{g}_{t}\right)+b(t)g(t)\right]%
\mathrm{d}t+\sigma(t,u^{g}_{t})\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(80)
$$\displaystyle u^{g}_{0}(x)$$
$$\displaystyle=u^{0}(x)$$
$$\displaystyle\text{in}\quad\Lambda$$
with Neumann boundary conditions, where $b\equiv 1$, and the nonlinearities are of the form $f(u)=ku(u-1)(a-u)$ for some $k>0$, $a\in(0,1)$, and $\sigma(t,u)=\overline{\sigma}\min\{0,\max\{-1,u(u-1)\}\}$ for some $\overline{\sigma}\in\mathbb{R}$, i.e. the state equation takes the form
$$\displaystyle\mathrm{d}u^{g}_{t}=$$
$$\displaystyle\left[\Delta u^{g}_{t}+ku^{g}_{t}(u^{g}_{t}-1)(a-u^{g}_{t})+g(t)%
\right]\mathrm{d}t$$
$$\displaystyle+\overline{\sigma}\min\{0,\max\{-1,u^{g}_{t}(u^{g}_{t}-1)\}\}%
\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(81)
$$\displaystyle u^{g}_{0}(x)=$$
$$\displaystyle u^{0}(x)$$
$$\displaystyle\text{in}\quad\Lambda.$$
In our example, we choose the time-horizon $[0,15]$, the space $\Lambda=[0,20]$, $k=1$, and $a=39/40$. These choices lead to two stable steady states, $u=0$ and $u=1$. As initial condition we choose
(82)
$$u^{0}(x)=\begin{cases}0&\text{for }x\in[0,3]\\
1&\text{for }x\in(3,20]\end{cases}.$$
In this case we get a traveling wave. Figure 2 shows the solution in the deterministic case, and Figure 2 shows one realization of the solution in the stochastic case with $\overline{\sigma}=0.5$.
We can see that the traveling wave slowly travels to the left. Our objective is now to first speed up the wave and then change the direction of travel. To this end, we consider the cost functional given by
(83)
$$J(g)=\mathbb{E}\left[\frac{c_{\overline{\Lambda}}}{2}\int_{0}^{T}\int_{\Lambda%
}\left(u^{g}_{t}(x)-u_{\overline{\Lambda}}(t,x)\right)^{2}\mathrm{d}x\mathrm{d%
}t+\frac{c_{T}}{2}\int_{\Lambda}\left(u^{g}_{T}(x)-u^{T}(x)\right)^{2}\mathrm{%
d}x\right],$$
where $c_{\overline{\Lambda}}=1$, $c_{T}=1$, and the reference profile $u_{\overline{\Lambda}}$ is given by
(84)
$$u_{\overline{\Lambda}}(t,x)=\begin{cases}1&\text{for }x>\left(\frac{3}{20}+%
\frac{t}{10}\right)\wedge\left(\frac{33}{20}-\frac{t}{10}\right)\\
0&\text{else}\end{cases},$$
for $(t,x)\in[0,T]\times\Lambda$. The intended terminal profile is given by $u^{T}=u_{\overline{\Lambda}}(T,\cdot)$.
With the algorithm from Section 5 we can approximate the optimal control. Let us apply the algorithm to the stochastic case with $\overline{\sigma}=0.5$. One realization of the solution with applied optimal control is displayed in Figure 4. Figure 4 shows the corresponding optimal control.
6.2. Comparison with the Control of the Deterministic System
Simulations show that the optimal control for the deterministic system in the preceding example does not differ qualitatively from the optimal control for the stochastic system. This is because the fixed points $0$ and $1$ are stable. The situation changes, however, if one of the fixed points becomes unstable from one side, as the following example shows. Consider the state equation
$$\displaystyle\mathrm{d}u^{g}_{t}$$
$$\displaystyle=\left[\Delta u^{g}_{t}-(u^{g}_{t})^{3}+(u^{g}_{t})^{2}+g(t)%
\right]\mathrm{d}t+\overline{\sigma}\mathrm{d}W^{Q}_{t}$$
$$\displaystyle\text{on}\quad L^{2}(\Lambda)$$
(85)
$$\displaystyle u^{g}_{0}(x)$$
$$\displaystyle=u^{0}(x)$$
$$\displaystyle\text{in}\quad\Lambda,$$
where $\Lambda=[0,20]$, $T=30$ and $\overline{\sigma}\in\mathbb{R}$. These choices lead to only one stable steady state, $u=1$ and one unstable steady state $u=0$. Now, as initial condition, we choose $u^{g}_{0}=0$, and consider the cost functional
(86)
$$J(g)=\mathbb{E}\left[\frac{1}{2}\int_{\Lambda}\left(u^{g}_{T}(x)\right)^{2}%
\mathrm{d}x\right],$$
i.e., we want the final state to be unchanged, in the unstable steady state $0$. In the deterministic case, the optimal control is clearly $g^{\ast}=0$, since we start in the steady state $x=0$ and without any forcing, we stay in this state and accomplish the minimal possible costs $J(g^{\ast})=0$. In the stochastic case, however, the noise term pushes the state out of the unstable steady state. Whenever the noise pushes the state above $0$, the dynamics of the state equation force the state towards the stable steady state $x=1$. As an illustration of this effect, Figure 6 displays the potential $F(x)$ of the nonlinearity $f$. Figure 6 shows one realization in the stochastic case without a control function.
When we introduce a control, the control tries to counteract this effect by keeping the state below $0$ for times $t<T$. This effect can be seen in the simulations, as well. As the stopping criterion we used $\eta=0.002$. Figures 8 to 8 display the optimal controls in the stochastic case with $\overline{\sigma}=0.5$ and one realization of the corresponding state.
6.3. Mathematical Analysis in a Simplified Setting
Since we are not able to prove the previous result in that setting rigorously, we consider a simpler similar example in which the optimal control in the deterministic case and the optimal control in the stochastic case differ.
Let us consider the stochastic ordinary differential equation
$$\displaystyle\mathrm{d}u^{g}_{t}$$
$$\displaystyle=\left[-V^{\prime}(u^{g}_{t})+g(t)\right]\mathrm{d}t+\overline{%
\sigma}\mathrm{d}B_{t},\quad t\in[0,T]$$
(87)
$$\displaystyle u^{g}_{0}$$
$$\displaystyle=0,$$
where $(B_{t})_{t\geq 0}$ is a Brownian motion on $\mathbb{R}$, the potential $V:\mathbb{R}\to\mathbb{R}$ is given by
(88)
$$\displaystyle V(x)=\begin{cases}\frac{1}{2}(\arctan(x)-x),&\text{for }x\geq 0%
\\
0,&\text{for }x<0,\end{cases}$$
and hence $-V^{\prime}$ is given by
(89)
$$\displaystyle-V^{\prime}(x)=\begin{cases}\frac{x^{2}}{2(1+x^{2})},&\text{for }%
x\geq 0\\
0,&\text{for }x<0.\end{cases}$$
Notice that this potential qualitatively resembles the potential used in the previous example in the interval $[0,1]$. That is why we observe a similar effect in this example. We consider the cost functional
(90)
$$J(g):=\mathbb{E}\left[\frac{1}{2}\left(u^{g}_{T}\right)^{2}\right].$$
As in the previous example, the initial condition and the desired final state are both the unstable steady state $u=0$. Hence, in the deterministic case ($\overline{\sigma}=0$), the optimal control is given by $g^{\ast}\equiv 0$, since the constant function $u\equiv 0$ solves the deterministic equation without control and the associated costs are zero.
Now, we are going to show that the optimal control in the stochastic case ($\overline{\sigma}>0$), however, is not equal to zero. First, notice that the adjoint equation associated with our control problem is given by
$$\displaystyle-\partial_{t}p$$
$$\displaystyle=-V^{\prime\prime}(u^{g}_{t})p,\quad t\in[0,T]$$
(91)
$$\displaystyle p(T)$$
$$\displaystyle=u^{g}_{T},$$
where $-V^{\prime\prime}$ is given by
(92)
$$\displaystyle-V^{\prime\prime}(x)=\begin{cases}\frac{x}{(1+x^{2})^{2}},&\text{%
for }x\geq 0\\
0,&\text{for }x<0.\end{cases}$$
Hence, the solution of the adjoint equation is given explicitly by
(93)
$$p(t)=u^{g}_{T}\exp\left(\int_{t}^{T}-V^{\prime\prime}(u^{g}_{s})\mathrm{d}s%
\right),$$
and the gradient of the cost functional is given by
(94)
$$\nabla J(g)(t)=\mathbb{E}[p(t)]=\mathbb{E}\left[u^{g}_{T}\exp\left(\int_{t}^{T%
}-V^{\prime\prime}(u^{g}_{s})\mathrm{d}s\right)\right].$$
Now, we are going to show that the gradient for $g\equiv 0$ is not equal to zero and hence, $g\equiv 0$ is not an optimal control. To this end, consider
(95)
$$\displaystyle\partial_{t}(\nabla J(g))(t)$$
$$\displaystyle=\mathbb{E}[\partial_{t}p(t)]=\mathbb{E}\left[V^{\prime\prime}(u^%
{g}_{t})u^{g}_{T}\exp\left(\int_{t}^{T}-V^{\prime\prime}(u^{g}_{s})\mathrm{d}s%
\right)\right].$$
This yields
$$\displaystyle\liminf_{t\to T}\left\{-\partial_{t}(\nabla J(g))(t)\right\}$$
$$\displaystyle=$$
$$\displaystyle\liminf_{t\to T}\mathbb{E}\left[-V^{\prime\prime}(u^{g}_{t})u^{g}%
_{T}\exp\left(\int_{t}^{T}-V^{\prime\prime}(u^{g}_{s})\mathrm{d}s\right)\right]$$
$$\displaystyle\geq$$
$$\displaystyle\mathbb{E}\left[\liminf_{t\to T}\left\{-V^{\prime\prime}(u^{g}_{t%
})u^{g}_{T}\exp\left(\int_{t}^{T}-V^{\prime\prime}(u^{g}_{s})\mathrm{d}s\right%
)\right\}\right]$$
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[-V^{\prime\prime}(u^{g}_{T})u^{g}_{T}\right]$$
(96)
$$\displaystyle=$$
$$\displaystyle\mathbb{E}\left[\frac{(u^{g}_{T})^{2}}{\left(1+\left(u^{g}_{T}%
\right)^{2}\right)^{2}}1_{\{u^{g}_{T}>0\}}\right]>0,$$
where the last part is strictly positive since $u_{T}$ has a strictly positive density with respect to the Lebesgue measure. Therefore, the gradient is not equal to zero and thus, $g\equiv 0$ is not an optimal control.
Remark 6.1.
Notice that we did not use that $g\equiv 0$ in this proof. This shows, that the optimal control in the stochastic case is unbounded.
Figures 10 and 10 illustrate our results in case of the stochastic ordinary differential equation (6.3) as the constraint and the cost functional (90).
Acknowlegdement This work has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 910 “Control of self-organizing nonlinear systems: Theoretical methods and concepts of application,” Project (A10) “Control of stochastic mean-field equations with applications to brain networks.”
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Pointwise
Convergence of the Lloyd algorithm in higher dimension
Gilles Pagès
Jun YU
UPMC, Laboratoire de Probabilités et Modèles aléatoires, UMR 7599, case 188, 4, pl. Jussieu, F-75252 Paris Cedex 5, France. E-mail: [email protected], Laboratoire de Probabilités et Modèles aléatoires, E-mail: [email protected]
Abstract
We establish the pointwise convergence of the iterative Lloyd algorithm, also known as $k$-means algorithm, when the quadratic quantization error of the starting grid (with size $N\geq 2$) is lower than the minimal quantization error with respect to the input distribution is lower at level $N-1$. Such a protocol is known as the splitting method and allows for convergence even when the input distribution has an unbounded support. We also show under very light assumption that the resulting limiting grid still has full size $N$. These results are obtained without continuity assumption on the input distribution. A variant of the procedure taking advantage of the asymptotic of the optimal quantizer radius is proposed which always guarantees the boundedness of the iterated grids.
Keywords: Lloyd algorithm ; $k$-means algorithm ; centroidal Voronoi Tessellation ; optimal vector quantization ; stationary quantizers ; splitting method ; radius of a quantizer.
1 Introduction
A Centroidal Voronoi Tessellation (CVT) with respect to a probability (or mass) distribution is a Voronoi
tessellation of a set of (generating) points in ${\mathbb{R}}^{d}$ (centers of mass) such that each generating point is the centroid of
its corresponding Voronoi region with respect to this density function. This definition can be extended to more general probability measures, typically those assigning no mass to hyperplanes to avoid ambiguity on the boundaries of the Voronoi regions. CVTs enjoy very natural optimization
properties, especially in connection with vector quantization (see further on) which makes them very popular in various scientific and engineering applications including art design, astronomy, clustering, geometric modeling, image and
data analysis, resource optimization, quadrature design, sensor networks, and numerical solution of partial differential equations.
For modern applications of the CVT concept in large-scale scientific and engineering problems, it is important to develop robust and efficient algorithms for constructing CVTs in various settings. Historically, a number of algorithms have been studied and widely used. However, the pioneering contribution is undoubtedly the procedure first developed in the 1960s at Bell Laboratories by S. Lloyd. It remains so far, in its randomized form, one of the most popular methods due to its effectiveness and simplicity.
Let us begin with a more detailed description of the CVT. First assume that the probability distribution, say $\mu$, on $({\mathbb{R}}^{d},{\cal B}or({\mathbb{R}}^{d}))$, has a support included in a closed convex set with non empty interior denoted $\mathbf{U}$ of ${\mathbb{R}}^{d}$. Also, note that, up to a reduction of the dimension $d$, one may always assume that $\mathbf{U}$ has a nonempty interior.
A Voronoi diagram (or partition) of $\mathbf{U}$ refers to a Borel partition $(C_{i}(\Gamma))_{1\leq i\leq N}$ of $\mathbf{U}\subset\mathbb{R}^{d}$ induced by a set $\Gamma=\{x_{i},\,1\leq i\leq N\}\subset\mathbf{U}$ of $N$ given generating points or Generators (the notation $\Gamma$ also refers to the application to numerics where the set of generators is also called a grid). For every $i\!\in\{1,\ldots,N\}$, the Voronoi region (or cell) $C_{i}(\Gamma)$ satisfies
$$C_{i}(\Gamma)\subset\left\{\xi\!\in\mathbf{U}\,:\,|\xi-x_{i}|\leq\min_{1\leq j%
\leq N}|\xi-x_{j}|\right\}$$
where $|\,.\,|$ denotes the canonical Euclidean norm on ${\mathbb{R}}^{d}$. Then
$$\left\{\xi\!\in\mathbf{U}\,:\,|\xi-x_{i}|<\min_{1\leq j\leq N}|\xi-x_{j}|%
\right\}=\stackrel{{\scriptstyle\circ}}{{C}}_{i}(\Gamma)\subset C_{i}(\Gamma)%
\subset\overline{C}_{i}(\Gamma)=\left\{\xi\!\in\mathbf{U}\,:\,|\xi-x_{i}|\leq%
\min_{1\leq j\leq N}|\xi-x_{j}|\right\}$$
so that the $C_{i}(\Gamma)$ have convex interiors and closures. The family of closures is also known as Voronoi tessellation of $\mathbf{U}$ induced by $\Gamma$ and the $\overline{C}_{i}(\Gamma)$, $i=1,\ldots,N$ are called tessels). Furthermore they have a polyhedral structure, in particular their boundaries are contained in $\displaystyle\cup_{i\neq i}H_{ij}$ where $H_{ij}\equiv\frac{x_{i}+x_{j}}{2}+\Big{(}\frac{x_{i}-x_{j}}{|x_{i}-x_{j}|}\Big%
{)}^{\perp}$ is the median hyperplane of $x_{i}$ and $x_{j}$. Of course, a notion of Voronoi regions can be defined with respect to any norm $N$ on ${\mathbb{R}}^{d}$ but the above (polyhedral) convexity properties fail (see $e.g.$ [7], chapter 1) for non Euclidean norms.
We will often assume that $\mu$ is strongly continuous in the sense that it assigns no mass to hyperplanes (so is the case if $\mu$ is absolutely continuous $i.e.$ $\mu(d\xi)=\rho(\xi)d\xi$ where $\rho$ is a probability density function defined on ${\mathbb{R}}^{d}$ whose support is contained in $\mathbf{U}$). Then the boundaries of the Voronoi regions are $\mu$-negligible so that we can define in a unique way the centroids $x^{*}_{i}$, $i=1,\ldots,N$ of the Voronoi regions by setting
$$x_{i}^{*}=\left\{\begin{array}[]{ll}\displaystyle\frac{\int_{C_{i}}\xi\mu(d\xi%
)}{\mu(C_{i})}&\mbox{if }\mu(C_{i})>0,\\
x_{i}&\mbox{if }\mu(C_{i})=0,\end{array}\right.\;i=1,\ldots,N.$$
(1.1)
Note that, owing to the convexity of the Voronoi cells $C_{i}$ and the finiteness of the measure $\mu$, one has $x^{*}_{i}\!\in\overline{C}_{i}$ (closure in $\mathbf{U}$) for every $i\!\in\{1,\ldots,N\}$. From a more probabilistic point of view, if $X$ denotes an ${\mathbb{R}}^{d}$-valued random vector with distribution ${\mathbb{P}}_{{}_{X}}=\mu$, then (with an obvious convention when ${\mathbb{P}}(X\!\in C_{i})=0$)
$$x^{*}_{i}={\mathbb{E}}\big{(}X\,|\,X\in C_{i}\big{)},\;i=1,\ldots,N.$$
This naturally leads to the definition of a CVT which is but a Voronoi tessellation whose generators $x_{i}$ are the centroids of their respective Voronoi regions. With the notation given above, the Lloyd algorithm for constructing CVTs can be described more precisely by the following procedure.
The paradigm of Lloyd’s algorithm is to consider the definition of CVT as a fixed point equality for the so-called Lloyd map $T^{\mu}_{{}_{N}}$ defined on the set of $\mathbf{U}$-valued grids $\Gamma$ with at most $N$ values by (1.1), $i.e.$
$$T^{\mu}_{{}_{N}}(\Gamma)=\{x^{*}_{i},\;i=1,\ldots,N\}\quad\mbox{if}\quad\Gamma%
=\{x_{i},\;i=1,\ldots,N\}\subset\mathbf{U}.$$
As mentioned above, $T(\Gamma)_{i}\!\in\overline{C}_{i}(\Gamma)$ since the Voronoi tessels are convex and $\mu$ is probability distribution.
Note that, furthermore, if ${\rm supp}(\mu)=\mathbf{U}$ or if $\mu$ is contain,ious (assigns no mass to hyperplanes) then, a supporting hyperplane argument shows that $T(\Gamma)_{i}\in\,\stackrel{{\scriptstyle\circ}}{{C}}_{i}(\Gamma)$ (interior in $\mathbf{U}$) for every $i=1,\ldots,N$ (see further on Lemma 2.2, see also [7], p.22). In particular, $T(\Gamma)$ and $\Gamma$ have the same size $N$.
The Lloyd algorithm is simply the formal fixed point search procedure for the Lloyd map $T_{{}_{N}}$ starting from a given grid $\Gamma^{(0}=\{x^{(0)}_{i},\,i=1,\ldots,N\}$ of full size $N$ $i.e.$
$$\Gamma^{(k+1)}=T^{\mu}_{{}_{N}}(\Gamma^{(k)}),\;k\geq 0.$$
Algorithm 1 (Lloyd’s algorithm for computing CVTs):
$\rhd$ Inputs:
•
$\mathbf{U}$, the domain of interest;
•
$\mu$ a probability distribution supported by $\mathbf{U}$;
•
$\Gamma^{(0)}=\{x_{i}^{(0)},\,i=1,\ldots,N\}\subset\mathbf{U}$, the initial set of $N$ generators.
$\rhd$ Pseudo-script:
Formally, at the $k$th iteration, one has to proceed as follows:
1.
Construct the Voronoi tessellation $\{C_{i}(\Gamma^{(k)}),\;i=1,\ldots,N\}$ of $\mathbf{U}$ with the grid of generators $\Gamma^{(k)}=\{x_{i}^{(k)},\;i=1,\ldots,N\}$.
2.
Compute the $\mu$-centroids of $\{C_{i}(\Gamma^{(k)}),\;i=1,\ldots,N\}$ as the new grid of generators $\Gamma^{(k+1)}=\{x_{i}^{(k+1)},\;i=1,\ldots,N\}$.
$\rhd$ Repeat the iteration above until some stopping criterion is met to provide a grid of generators as close as possible of a $\mu$-centroid.
$\rhd$ end.
In $1$-dimension, Kieffer has proved in [9] that $T_{{}_{N}}$ is contracting if $\mu$ has a $\log$-concave density over a compact interval so that only one $\mu$-centroid with $N$ points exists for such distribution and the above procedure converges exponentially fast toward it. See also, more recently a convergence result in [4].
In practice, these two steps become intractable in higher dimension by analytic or even deterministic approximation methods, say when $d\geq 3$ or $4$ (see however the website QHull: www.qhull.org). So this theoretical procedure has to be replaced for numerical purpose by a randomized version in which:
– Step 1 is replaced by a systematic nearest neighbour search of simulated random $\mu$-distributed vectors.
– Step 2 is replaced by a Monte Carlo estimation of both terms of the ratio which define the Lloyd map.
In the community of data analysis, note that when $\mu=\frac{1}{M}\sum_{m=1}^{M}\delta_{\xi_{m}}$ is the empirical measure of a $\mathbf{U}$-valued data set $(\xi_{m})_{m=1,\ldots,M}$, it is still possible to define and compute the Lloyd map (using appropriate conventions like $e.g.$ random allocation of points lying on the boundary of (closed) Voronoi tessels). In such a case the Lloyd procedure is known as the Forgy algorithm or the batch-$k$-means procedure. When the data set is so huge that a uniform sampling (of size $M$) of the dataset is necessary at each iteration, the procedure is known as the $k$-means procedure.
In this paper we will focus on the converging properties of the theoretical (or batch in the data-mining community) Lloyd procedure, prior to any randomization or approximation, although we are aware that in higher dimension for continuous distributions $\mu$, it is a pseudo-algorithm. So far we have presented the Lloyd procedure in an intrinsic manner. In fact Lloyd’s algorithm is deeply connected with the theory of Optimal Vector Quantization of probability distribution. This connection turns out to provide very powerful tools to investigate its convergence properties. It is also a major field of application when trying to compute with a sharp accuracy optimal quantizers of simulatable distribution arising in the design of numerical schemes for solving nonlinear problems (optimal stopping problems, (possibly Reflected) Backward Stochastic Differential Equations, Stochastic Control, etc, see [1, 16]).
Quantization is a way to discretize the path space of a random phenomenon: a random vector in finite dimension (but also stochastic process in infinite dimension viewed as a random variable taking values in its path space which we will not investigate in this paper). We consider here a random vector $X$ defined on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ taking its values in ${\mathbb{R}}^{d}$ equipped with its Borel $\sigma$-field ${\cal B}or({\mathbb{R}}^{d})$.
It is convenient for what follows to introduce a few notions and results about vector quantization and its (mean quadratic) optimization. It makes a connection between CVTs and stochastic optimization, gives a rigorous meaning to the notion of “goodness” of a CVT. Optimal vector quantization goes back to the early 1950’s in the Bell laboratories and have been developed for the optimization of signal transmission.
Let $X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow{\mathbb{R}}^{d}$ be a square integrable random vector ($i.e.$ ${\mathbb{E}}|X|^{2}<+\infty$) or equivalently $X\!\in L_{{\mathbb{R}}^{d}}^{2}(\mathbb{P})$. Assume that its distribution $\mu={\mathbb{P}}_{{}_{X}}$ is included in $\mathbf{U}$ (defined as above). The terminology $N$-quantizer (or a quantizer at level $N$) is assigned to any $\mathbf{U}$-valued subset with cardinality $N$. When used in a numerical framework, it is also known as quantization grid.
$$\Gamma:=\{x_{1},x_{2},\cdots,x_{N}\}\subset\mathbf{U}.$$
The cardinality of $\Gamma$ is $N$. In numerical applications, $\Gamma$ is also called a (quantization) grid. It is the set of genrators of its (borel) Voroni regions $(C_{i}(\Gamma))_{1\leq i\leq N}$. Then can discretize $X$ in pointwise way by $q(X)$ where $q$: ${\mathbb{R}}^{d}\rightarrow\Gamma$ is a Borel function. Then we get
$$\forall\omega\!\in\Omega,\;|X(\omega)-q(X(\omega))|\geq{\rm dist}(X(\omega),%
\Gamma)=\min_{1\leq i\leq N}|X(\omega)-x_{i}|$$
so that the best pointwise approximation of $X$ is provided by considering any (Borel) nearest neighbour projection $q={\rm Proj}_{\Gamma}$ associated with the Voronoi tessellation $(C_{i}(\Gamma))_{1\leq i\leq N}$ by setting
$$\operatorname{Proj}_{\Gamma}(\xi)=\sum_{i=1}^{N}x_{i}\mathbf{1}_{C_{i}(\Gamma)%
}(\xi),\;\xi\!\in{\mathbb{R}}^{d}.$$
It is clear that such a projection is in one-to-one correspondence with the Voronoi partitions (or diagrams) of ${\mathbb{R}}^{d}$ induced by $\Gamma$.
These projections only differ on the boundaries of the Voronoi cells $C_{i}(\Gamma)$ so that, as soon as $\mu={\mathbb{P}}_{{}_{X}}$ is strongly contoinuous, these neratest neighbour projections are all ${\mathbb{P}}_{{}_{X}}$ -$a.s.$ equal. We define a Voronoi $N$-quantization of $X$ (or at level $N$) by setting for every $\omega\!\in\Omega$,
$$\widehat{X}^{\Gamma}(\omega):=\operatorname{Proj}_{\Gamma}(X(\omega))=\sum_{i=%
1}^{N}x_{i}\mathbf{1}_{C_{i}(\Gamma)}(X(\omega)).$$
Thus for all $\omega\in\Omega$,
$$|X(\omega)-\widehat{X}^{\Gamma}(\omega)|={\rm dist}(X(\omega),\Gamma)=\min_{1%
\leq i\leq N}|X(\omega)-x_{i}|.$$
(1.2)
We will call $\widehat{X}^{\Gamma}$ a Voronoi $\Gamma$-quantization of $X$ or, in short, a quantization of $X$.
The mean quadratic quantization error is then defined by
$$e(\Gamma,X)=\|X-\widehat{X}^{\Gamma}\|_{2}=\sqrt{\mathbb{E}\left(\min_{1\leq i%
\leq N}|X-x_{i}|^{2}\right)}$$
where $\|\cdot\|_{2}$ is the norm in $L_{{\mathbb{R}}^{d}}^{2}(\mathbb{P})$. The distribution of $\widehat{X}^{\Gamma}$ as a random vector is given by the $N$-tuple $\left(\mathbb{P}(X\in C_{i}(\Gamma))\right)_{1\leq i\leq N}$. This distribution clearly depends on the choice of the Voronoi partition.
We naturally wonder whether it is possible to design some optimally fitted grids to a given distribution $\mu=\mathbb{P}_{X}$ i.e. which induces the lowest possible mean quadratic quantization error among all
grids of size at most $N$. This optimization problem, known as the optimal quantization problem at level $N$, reads as follows:
$$e_{N}(X):=\inf_{\Gamma\subset\mathbb{R}^{d},\operatorname{Card}(\Gamma)\leq N}%
e(\Gamma,X)$$
By introducing the energy function or distortion value function
$$\mathcal{G}:\,\big{(}\mathbb{R}^{d}\big{)}^{N}\longrightarrow\mathbb{R}_{+}$$
$$x=(x_{1},x_{2},\cdots,x_{N})\longmapsto\mathbb{E}\left(\min_{1\leq i\leq N}|X-%
x_{i}|^{2}\right)$$
the optimization problem also reads
$$e_{N}(X)=\inf_{x\in(\mathbb{R}^{d})^{N}}\sqrt{\mathcal{G}(x)}$$
since the value of $\mathcal{G}$ at an $N$-tuple $x=(x_{1},x_{2},\cdots,x_{N})\!\in({\mathbb{R}}^{d})^{N}$ only depends on its value grid $\Gamma=\Gamma_{x}=\{x_{1},\ldots,x_{N}\}$ of size at most $N$ of the $N$-tuple (in particular $\mathcal{G}$ is a symmetric function). We will make occasionally the abuse of notation consisting in denoting $\mathcal{G}(\Gamma)$ instead of $\mathcal{G}(x)$.
One proves (see $e.g.$ [2, 7, 12]) that there always exists at least one optimal $N$-point grid $\Gamma_{{}_{N}}^{*}=\{x_{1}^{*},x_{2}^{*},\cdots,x_{N}^{*}\}\subset\mathbb{R}^%
{d}$ with cardinal $N$ such that $e_{N}(X)=\sqrt{\mathcal{G}(\Gamma_{{}_{N}}^{*})}$. If the support of ${\mathbb{P}}_{{}_{X}}$ has at least $N+1$ elements ($e.g.$ because it is infinite), then $\Gamma_{{}_{N}}^{*}$ has full size $N$. Furthermore, $\Gamma^{*}_{{}_{N}}\subset\mathbf{U}$; this last claim strongly relies on the Euclidean feature of the norm on ${\mathbb{R}}^{d}$: if $\Gamma^{*}_{{}_{N}}\,/\hskip-11.381102pt\subset\mathbf{U}$, then the projection of the elements of $\Gamma^{*}_{{}_{N}}$ on the closed (nonempty) convex $\mathbf{U}$ strictly reduces the mean quadratic quantization error (see $e.g.$ [12, 7, 10]).
Note that this existence result does not require $\mu={\mathbb{P}}_{X}$ to be strongly continuous. In fact, even if $\mu$ has an atomic component, it is shown in [7] (see Theorem 4.2, p.38) that $\mu\big{(}\cup_{i}\partial C_{i}(\Gamma^{*}_{{}_{N}})\big{)}=0$.
Furthermore, the function $\mathcal{G}$ is differentiable on $({\mathbb{R}}^{d})^{N}$ at every $N$-tuple $x=(x_{1},\ldots,x_{{}_{N}})$ such that $x_{i}\neq x_{j}$, $i\neq j$ and $\displaystyle{\mathbb{P}}\big{(}X\!\in\cup_{1\leq i\leq N}\partial C_{i}(x)%
\big{)}=0$ and its gradient is given by
$$\nabla\mathcal{G}(x)=\frac{1}{2}{\mathbb{E}}\left(\mbox{\bf 1}_{\{X\!\in C_{i}%
(x)\}}(x_{i}-X)\right)$$
(1.3)
where $(C_{i}(x))_{1\leq i\leq N}$ denotes any Voronoi diagram of $\{x_{1},\ldots,x_{{}_{N}}\}$ (or $x$ with, once again, an obvious abuse of notation).
In particular, if $x^{*}=(x^{*}_{1},\ldots,x^{*}_{{}_{N}})$ is an optimal quadratic quantizer, one shows (see [7], Theorem 4.2 p.38) that ${\mathbb{P}}\big{(}X\!\in\cup_{i}\partial C_{i}(x^{*})\big{)}=0$ even if ${\mathbb{P}}_{{}_{X}}$ assigns mass to (at most countably many) hyperplanes. As a consequence,
$$\nabla\mathcal{G}(x^{*})=0\quad\mbox{ or equivalently}\quad T^{{\mathbb{P}}_{{%
}_{X}}}_{N}(x^{*})=x^{*}$$
where $T^{{\mathbb{P}}_{{}_{X}}}_{N}$ stands for the Lloyd map related to the distribution $\mu={\mathbb{P}}_{{}_{X}}$ of the random vector $X$. The equivalence is a straightforward consequence of (1.3)). Hence any optimal quantizer induces a CVT for the distribution of $X$. In reference to the fact that such an $N$-quantizer is a zero of a gradient, this property is also known as stationarity for the $N$-tuple $x^{*}$ itself.
Unfortunately, the converse is not true since $\mathcal{G}$ may have many local minima, various types of saddle points (and a “pin” behaviour on affine manifolds induced by clusters of stuck components). This phenomenon becomes more and more intense as $d$ grows. However, it makes a strong connection between search for optimal quantizers and Lloyd’s algorithm as described above. And there is no doubt that what practitioners are interested in are the optimal quantizers rather than any “saddle” stationary quantizers. One can also derive a stochastic gradient algorithm from the representation of $\nabla\mathcal{G}$ as an expectation of a computable function of the quantizer $x$ and the random vector $X$. This second approach leads to a stochastic optimization procedure, a stochastic gradient descent to be more precise, known as the Competitive learning vector Quantization algorithm ($CLVQ$) which has also been extensively investigated (see among others [12]).
The paper is organized as follows: in Section 2 we establish the convergence of the Lloyd procedure at level $N$ under some natural assumptions on the probability distribution (at least for numerical probability purpose) but assuming that the starting quantizer (or generators) induces a lower quantization error than the lowest quantization error at level $N-1$. In Section 3, we propose a modified Lloyd’s procedure, inspired by recent results on the asymptotics of the “radius” of optimal quantizers at level $N$ as $N\to+\infty$, to overcome partially this constraint on the starting grid.
Notations: $\bullet$ ${\rm supp}(\mu)$ denotes the support of the Borel probability measure $\mu$ on ${\mathbb{R}}^{d}$. $\lambda_{d}$ denotes the Lebesgue measure on ${\mathbb{R}}^{d}$.
$\bullet$ $(.\,,.)$ denotes the canonical inner product on ${\mathbb{R}}^{d}$. $B(x,r)$, $r\!\in{\mathbb{R}}_{+}$, denotes the canonical Euclidean ball centered at $x\!\in{\mathbb{R}}^{d}$ with radius $r>0$.
$\bullet$ $|A|$ denotes the cardinality of set $A$.
2 Convergence analysis of Lloyd’s algorithm with unbounded inputs
2.1 The main result
Owing to both simplicity and efficiency of of practical implementations of Lloyd’s algorithm in various fields of applications, it is important to study its convergence as it has been carried out, at least partially, in [12] for its “counterpart” in the world of Stochastic Approximation, the recursive stochastic gradient descent attached to the above gradient of the distortion function $\mathcal{G}$. This procedure is also known as $CLVQ$ (for Competitive Learning Vector Quantization algorithm). In fact, as concerns the convergence properties of Lloyd’s algorithm, many investigations have already been carried out . Thus, as mentioned in the introduction, true convergence for $\log$-concave densities has been established in [9] whereas global “weak” convergence has been proved in a one dimensional setting (see [4]). However, there are not many general mathematical results on the convergence analysis for
distributions on multi-dimensional spaces, especially when the support of the distribution $\mu$ of interest is not bounded.
It is convenient to rewrite the iterations of the Lloyd algorithm in a more probabilistic form, using quantization formalism (with a generic notation for the grids: $\Gamma^{(k)}=\big{\{}x^{(k)}_{1},\ldots,x^{(k)}_{{}_{N}}\big{\}}$, $k\geq 0$). Let $\Gamma^{(0)}\subset{\mathbb{R}}^{d}$. For every $k\geq 0$,
$$\left\{\begin{array}[]{lll}(I)\hbox{\em Centroid updating}:&\hskip-28.452756pt%
\widetilde{X}^{(k+1)}={\mathbb{E}}\big{(}X\,|\,\widehat{X}^{\Gamma^{(k)}}\big{%
)},\;\Gamma^{(k+1)}&=\widetilde{X}^{(k+1)}(\Omega)\\
&&=\left\{\frac{{\mathbb{E}}\big{(}\mbox{\bf 1}_{\{\widehat{X}^{\Gamma^{(k)}}=%
x^{(k)}_{i}\}}X\big{)}}{{\mathbb{P}}(\widehat{X}^{\Gamma^{(k)}}=x^{(k)}_{i})},%
\,i=1,\ldots,N\right\}\\
(II)\hbox{\em Voronoi cells re-allocation}:&\widehat{X}^{\Gamma^{(k+1)}}%
\leftarrow\widetilde{X}^{(k+1)}.&\end{array}\right.$$
with the following degeneracy convention:
if ${\mathbb{P}}(\widehat{X}^{\Gamma^{(k)}}=x^{(k)}_{i})={\mathbb{P}}(X\!\in C_{i}%
(\Gamma^{(k)}))=0$ then $\widehat{X}^{\Gamma^{(k+1)}}=x^{(k)}_{i}$.
We need now to formalize in a more precise way what can be a consistent connection between the sequence of iterated grids $(\Gamma^{(k)})_{k\geq 0}$ in the Lloyd procedure and the $N$-tuples that can be associated to their values.
Definition 2.1.
Let $(\Gamma^{(k)})_{k\geq 0}$ be a sequence of iterates of the Lloyd procedure where $|\Gamma^{(0)}|=N$. A sequence of $N$-tuples
$(x^{(k)})_{k\leq 0}$ is a consistent representation of the sequence $\Gamma^{(k)}$ if
(i)
$\Gamma^{(k)}=\{x^{(k)}_{i},\,i=1,\ldots,N\}$,
(ii)
For every integer $k\geq 0$ and every $i\!\in\{1,\ldots,N\}$, $x^{(k+1)}_{i}$ is the centroid of the cell of $x^{(k)}_{i}$ $i.e.$
$$x^{(k+1)}_{i}={\mathbb{E}}\big{(}X\,|\,X\!\in C_{i}(\Gamma^{(k)})\big{)}=\frac%
{{\mathbb{E}}\big{(}\mbox{\bf 1}_{\{\widehat{X}^{\Gamma^{(k)}}=x^{(k)}_{i}\}}X%
\big{)}}{{\mathbb{P}}(\widehat{X}^{\Gamma^{(k)}}=x^{(k)}_{i})}$$
(still with the above degeneracy convention).
There are clearly $N!$ consistent representations of a sequence of Lloyd iterates, corresponding to the possible numbering of $\Gamma^{(0)}$. But then this numbering is frozen as $k$ increases. It is also clear that roundedness, convergence (in $({\mathbb{R}}^{d})^{N}$) of such consistent representations does not depend on the selected representation. So is true for a possible limit $x^{(\infty)}$ of such sequences since $\Gamma^{(\infty)}$ will not depend on the selected representation. However on may have $|\Gamma^{(\infty)}|<N$ in case of an asymptotic merging of some of the components of the representation. One checks that under the assumptions we make ($\mu$ continuous, or convex support or splitting assumption on $\Gamma^{(0)}$) no merging occurs at finite range.
Throughout the paper $(x^{(k)})_{k\geq 0}$ will always denote a consistent representation of the sequence of iterates $\Gamma^{(k)})_{k\geq 0}$. These remark lead naturally to the following definition
Definition 2.2 (Convergence of iterated grids).
We will say that $\Gamma^{(k)}\to\Gamma^{(\infty)}$ (converges) as $k\to+\infty$ if there exists a constant representation $(x^{(k)})_{k\geq 0}$ converging in $({\mathbb{R}}^{d})^{N}$ toward $x^{(\infty)}$ such that $\Gamma^{(\infty)}=\{x^{(k)}_{i},\,i=1,\ldots,N\}$.
Several specific results established in what follows are known in the literature, but we chose to provide all proofs for self-completeness of the paper and reader’s convenience.
Let us first recall a basic fact which is at the origin of the efficiency of the Lloyd algorithm.
Lemma 2.1.
The iteration of Lloyd’s algorithm makes the quantization error decrease.
$$e(\Gamma^{(k+1)},X)\leq e(\Gamma^{(k)},X).$$
Furthermore, $e(\Gamma^{(k+1)},X)<e(\Gamma^{(k)},X)$ as long as $\widehat{X}^{\Gamma^{(k)}}\neq\mathbb{E}(X|\widehat{X}^{\Gamma^{(k)}})$ with positive ${\mathbb{P}}$-probability. Conversely, if $\widehat{X}^{\Gamma^{(k_{0})}}=\mathbb{E}(X|\widehat{X}^{\Gamma^{(k_{0})}})$ ${\mathbb{P}}$-$a.s.$ for an integer $k_{0}\!\in{\mathbb{N}}$, then $\widehat{X}^{\Gamma^{(k)}}=\widehat{X}^{\Gamma^{(k_{0})}}$ ${\mathbb{P}}$-$a.s.$ for every $k\geq k_{0}$.
Proof:
By its very definition, $\widetilde{X}^{(k+1)}$ is the best approximation of $X$ among the $\sigma(\widehat{X}^{\Gamma^{(k)}})$-measurable functions, including $\widehat{X}^{\Gamma^{(k)}}$ itself. Thus
$$\|X-\widetilde{X}^{(k+1)}\|_{2}=\|X-\mathbb{E}(X|\widehat{X}^{\Gamma^{(k)}})\|%
_{2}\leq\|X-\widehat{X}^{\Gamma^{(k)}}\|_{2}=e(\Gamma^{(k)},X)$$
(2.4)
with equality if and only if $\widehat{X}^{\Gamma^{(k)}}=\mathbb{E}(X|\widehat{X}^{\Gamma^{(k)}})$. Note that, if so is the case, $\Gamma^{(k+1)}=\Gamma^{(k)}$ since $\widetilde{X}^{(k+1)}=\widehat{X}^{\Gamma^{(k)}}$. On the other hand, by (1.2), $\widehat{X}^{\Gamma^{(k+1)}}$ is the best approximation of $X$ among all $\Gamma^{(k+1)}$-valued random vectors since
$$e(\Gamma^{(k+1)},X)=\|X-\widehat{X}^{\Gamma^{(k+1)}}\|_{2}=\|{\rm dist}(X,%
\Gamma^{(k+1)})\|_{2}\leq\|X-\widetilde{X}^{(k+1)}\|_{2}$$
and if $\Gamma^{(k+1)}=\Gamma^{(k)}$, this inequality holds as an equality.
$\qquad\Box$
This seminal property of Lloyd’s algorithm is striking not only by its simplicity. It is also at the origin of its success. Morally speaking, it suggests a convergence toward a stationary – and hopefully optimal or at least locally optimal – quantizer of the distribution $\mu$ of $X$. In fact, things are less straightforward, at least from a theoretical point of view since this property provides absolutely no information on the boundedness of the sequence of grids $(\Gamma^{(k)})_{k\geq 0}$ generated by the procedure, although it is a crucial property the way toward convergence.
The main result of this paper is the following theorem.
Theorem 2.1.
Let $X:(\Omega,{\cal A},{\mathbb{P}})\to{\mathbb{R}}^{d}$ be a square integrable random vector with a distribution $\mu$ having a convex support $\mathbf{U}$ assigning no mass to hyperplanes. Let $\Gamma^{(0)}\subset\mathbf{U}$ with size $|\Gamma^{(0)}|=N$. Then, all the iterates $\Gamma^{(k)}=\{x^{(k)}_{i},\,i=1,\ldots,N\}$ have full size $N$ and are $\mathbf{U}$-valued.
$(a)$ If the sequence $\big{(}\Gamma^{(k)}\big{)}_{k\geq 1}$ is bounded (or equivalently kits consistent representations $(x^{(k)})_{k\geq 0}$), then
$$\liminf_{k}\min_{1\leq i,j\leq N}|x^{(k)}_{i}-x^{(k)}_{j}|>0$$
and there exists $\ell_{\infty}\!\in\big{[}0,e_{N}(\Gamma^{(0)}(X))\big{)}$ and a connected component ${\cal C}_{\ell_{\infty},\mathbf{U}}$ of $\displaystyle\Lambda_{\infty}:=\big{\{}x\!\in\mathbf{U}^{N},\,e_{N}\big{(}\{x_%
{i},i=1,\ldots,N\},X\big{)}=\ell_{\infty},\;\mathcal{G}(x)=0\big{\}}$ such that
$${\rm dist}\big{(}x_{i}^{(k)},{\cal C}_{\ell_{\infty},\mathbf{U}}\big{)}\to 0%
\quad\mbox{as}\quad k\to+\infty.$$
In particular, if $\Lambda_{\infty}$ is locally finite ($i.e$ is reduced to finitely many points on each compact set), then $\Gamma^{(k)}\to\Gamma^{(\infty)}\!\in\Lambda_{\infty}$.
$(b)$ Splitting method: If furthermore $e(\Gamma^{(0)},X)\!\in(e_{N}(X),e_{N-1}(X)]$, then the sequence $\big{(}\Gamma^{(k)}\big{)}_{k\geq 1}$ is always ${\mathbb{P}}$-$a.s.$ bounded and $\ell_{\infty}\!\in[e_{N}(X),e_{N-1}(X))$.
Remarks. $\bullet$ The result in claim $(a)$ does not depend on the original numbering of $\Gamma^{(0)}$ $i.e.$ on the selected order (among $N!$) selected to define $x^{(0)}$ and the then the sequence $(x^{(k)})$. In particular in case of true convergence of the sequence $x^{(k)}$, all its permutations do converge as well toward the corresponding permutations of $x^{(\infty)}$ so that one can, by an abuse of notion write that $\Gamma^{(k)}\to\Gamma^{(\infty)}$. A direct approach based on a formal notion of set convergence is also possible but would be of no help in practice.
$\bullet$ Nothing ensures that the limiting grid $\Gamma^{(\infty)}$ is optimal or even a local minimum. We refer to the Appendix for a brief discussion and an closed formula for the Hessian of $\mathcal{G}$.
The first claim of this theorem relies on a boundedness assumption for the sequence $\big{(}\Gamma^{(k)}\big{)}_{k\geq 0}$. This condition is of course satisfied if the support of the distribution $\mu$ of $X$ is compactly supported and $\mathbf{U}=\overline{{\cal H}({\rm supp}(\mu))}$ (closed convex hull of the support of $\mu$) since we assume that $\Gamma^{(0)}\subset\mathbf{U}$: then, for every $k\!\in{\mathbb{N}}$, so will be the case for $\Gamma^{(k)}$ as emphasized in the description of the procedure.
Claim $(b)$ emphasizes that, by an appropriate choice of (the quadratic quantization error) $\Gamma^{(0)}$
may imply the boundedness of the whole sequence of iterates $(\Gamma^{(k)})_{k\geq 0}$.
This approach known by practitioners as the splitting method is investigated in the next subsection.
We will prove this theorem step by step, establishing intermediary results, often under less stringent assumptions than the above theorem, which may have their own interest.
2.2 Possibly unbounded inputs: the splitting method
In practical computations, if one aims at computing (hopefully) optimal quantization grids (or $CVT$) of $X$ on a wide range of levels $N$ (see $e.g.$ the website www.quantize.maths-fi.com), the so-called splitting method appears as an extremely efficient “level-by-level” procedure. The principle is to compute the grids in a telescopic manner based on their size $N$: assume we have access to an optimal grid of size $N$. Imagine we add to this grid an $(N+1)^{th}$ $\mathbf{U}$-valued component $e.g.$ sampled randomly from the distribution of $X$ (or any $\mathbf{U}$-supported distribution). Doing so, we make up a grid which has for sure a lower quadratic mean quantization error than any grid with $N$ points. This grid of size $N+1$ is likely to lie in the attracting basin of the optimal quadratic quantizer (or $CVT$) at level $N+1$. This is often observed in practice and, even if not optimal, it provides at least very good quantizers. Its main the trade-off is that it requires a systematic, hence heavily time consuming, simulation. Many variants or potential improvements can be implemented (like adding an optimal quantizer of size $N_{0}\geq 2$ at each new initialization to directly obtain (hopefully) optimal quantizers of size $N+N_{0}$ (see $e.g.$ [13]).
Splitting Assumption (on the starting grid): Let $N\geq 1$ be such that card$({\rm supp}\mu)>N$ (where $\mu={\mathbb{P}}_{{}_{X}}$). Let $\Gamma_{N-1}^{*}=\{x^{*}_{i},\,i=1,\ldots,n\}\subset\mathbf{U}$ be an optimal grid of size $N-1$ for ${\mathbb{P}}_{{}_{X}}$ and let $x_{N}^{(0)}\!\in{\rm supp}(\mu)\setminus\Gamma_{N-1}^{*}$. The Lloyd algorithm is initialized as follows:
$$\Gamma^{(0)}=\Gamma_{N-1}^{*}\cup\{x_{N}^{(0)}\}=\Big{\{}x_{1}^{*},x_{2}^{*},%
\cdots,x_{N-1}^{*},x_{N}^{(0)}\Big{\}}.$$
(2.5)
One natural way to generate the additional element $x^{(0)}_{{}_{N}}$ is in practice to simulate randomly a copy of $X$ (in fact one can also simulate a copy of a random vector whose distribution is equivalent to that of $X$, see the remark below).
Remark. If $\mu$ has a density $\rho$, it is more efficient to simulate according to the distribution whose probability density is proportional to $\rho^{\frac{d}{d+2}}$ (one checks that this function is integrable if $X\!\in L^{2+\eta}({\mathbb{P}})$ for an $\eta>0$, see $e.g.$ [7]). The reason is that the resulting distribution $\mu_{d}=\kappa_{d}\rho^{\frac{d}{d+2}}.\lambda_{d}$ provides the best random $N$-quantizers of $\mu=\rho.\lambda_{d}$ for every $N\geq 1$ in the following sense: the asymptotic minimization problem
$$\inf\Big{\{}\limsup_{N\to+\infty}N^{\frac{2}{d}}{\mathbb{E}}\big{(}\min_{1\leq
i%
\leq N}|X-Y_{i}|^{2}\big{)},\;Y_{1},\ldots,Y_{{}_{N}}\;i.i.d.,\;\perp\!\!\!%
\perp X,\;Y_{1}\sim\nu\Big{\}}$$
has $\nu=\mu_{d}$ as a solution ($\perp\!\!\!\perp$ denotes here independence).
If the Splitting Assumption on the initialization of the procedure is satisfied, the grids of the iterations in the Lloyd algorithm share an interesting property (which implies their global boundedness). The arguments developed in the proof below are close to those used in the proof of the existence of an optimal quantizer at level $N$ (see $e.g.$ [12, 7]).
Proposition 2.1.
Assume the Splitting Assumption (2.5).
$(a)$ The quantization error induced by the grid $\Gamma^{(0)}$ defined by (2.5) is strictly smaller than $e_{N-1}(X)$ ($i.e.$ that of the optimal $(N-1)$-quantizer $\Gamma^{*}_{{}_{N-1}}$) so that $|\Gamma^{(k)}|=N$ and
$$e(\Gamma^{(k)},X)=\big{\|}X-\widehat{X}^{\Gamma^{(k)}}\big{\|}_{2}\leq e(%
\Gamma^{(0)},X)<e_{N-1}(X).$$
$(b)$ The sequence of iterated grids $(\Gamma^{(k)})_{k\geq 0}$ is bounded in ${\mathbb{R}}^{d}$ $i.e.$ there exists a compact set $K\subset{\mathbb{R}}^{d}$ such that
$$\forall\,k\!\in{\mathbb{N}},\quad\Gamma^{(k)}\subset K.$$
$(c)$ All the limiting values of the sequence of consistent representations $(x^{(k)})_{k\geq 0}$ have $N$ pairwise distinct components ($i.e.$ the the sequence of grids $(\Gamma^{(k)})_{k\geq 0}$ has asymptotically full size full size $N$).
Proof: $(a)$ The (squared) quadratic means quantization error (or “distortion value”) induced by $\Gamma^{(0)}$ satisfies:
$$e(\Gamma^{(0)},X)^{2}=\mathcal{G}(x_{1}^{*},x_{2}^{*},\cdots,x_{(N-1)}^{*},x_{%
N}^{(0)})=\mathbb{E}\Big{(}\min_{1\leq i\leq N}|X-x_{i}^{(0)}|^{2}\Big{)}.$$
where $\,x_{j}^{(0)}=x_{j}^{*}$, $1\leq j\leq N-1$.
Let $\varepsilon=\frac{1}{3}\min_{1\leq i\leq N-1}|x_{i}^{*}-x_{N}^{(0)}|$. For every $\xi\!\in B(x_{N}^{(0)},\varepsilon)$,
$$|\xi-x_{i}^{(0)}|\geq|x^{*}_{i}-x^{(0)}_{{}_{N}}|-|\xi-x^{(0)}_{{}_{N}}|\geq 3%
\varepsilon-\varepsilon=2\varepsilon>|\xi-x^{(0)}_{{}_{N}}|^{2}.$$
Consequently, on the event $A_{\varepsilon}=\{X\!\in B(x_{N}^{(0)},\varepsilon)\}$, we have
$$\min_{1\leq i\leq N}|X(\omega)-x_{i}^{(0)}|_{2}^{2}=|X(\omega)-x_{N}^{(0)}|_{2%
}^{2}>\min_{1\leq i\leq N-1}|X(\omega)-x_{i}^{(0)}|_{2}$$
whereas we always have that $\min_{1\leq i\leq N}|X(\omega)-x_{i}^{(0)}|^{2}\leq\min_{1\leq i\leq N-1}|X(%
\omega)-x_{i}^{(0)}|^{2}$. On the other hand we know that $x^{(0)}_{{}_{N}}\!\in{\rm supp}({\mathbb{P}}_{X})$ so that ${\mathbb{P}}(A_{\varepsilon})>0$ which implies in turn that
$$e(\Gamma^{(0)},X)^{2}<e_{N-1}(X)^{2}.$$
One concludes by noting that Lloyd’s algorithm makes the sequence of (squared) quadratic quantization error induced by the iterated grids non-increasing.
$(b)$
Suppose that there exists a subsequence $(\varphi(k))$ and a component $i_{0}$ such that $|x_{i_{0}}^{(\varphi(k))}|\to+\infty$ as $k\to+\infty$.
Then, by re-extracting finitely many subsequences, we can split the set $\{1,\cdots,N\}$ into two disjoint non empty subsets $I$ and $I^{c}$ and find a subsequence (still denoted $(\varphi(k))_{k\geq 0}$ for convenience) such that
$$I=\Big{\{}j\!\in\{1\ldots,N\}\mbox{ such that }\,\big{|}x_{j}^{(\varphi(k))}%
\big{|}\rightarrow+\infty\Big{\}}\neq\emptyset$$
and
$$I^{c}=\Big{\{}j\!\in\{1\ldots,N\}\mbox{ such that }\,x_{j}^{(\varphi(k))}%
\rightarrow x_{j}^{\infty}\in\mathbb{R}^{d}\Big{\}}.$$
It is clear that
$$\forall\,\xi\!\in{\mathbb{R}}^{d},\quad\lim_{k\to+\infty}\min_{1\leq i\leq N}%
\big{|}\xi-x_{i}^{(\varphi(k))}\big{|}^{2}=\lim_{k\to\infty}\min_{j\in I^{c}}%
\big{|}\xi-x_{j}^{(\varphi(k))}\big{|}^{2}=\min_{j\in I^{c}}|\xi-x_{j}^{\infty%
}|^{2}.$$
Therefore, it follows from Fatou’s Lemma that
$$\displaystyle\liminf_{k\to\infty}\mathcal{G}(\Gamma^{(\varphi(k))})$$
$$\displaystyle=\liminf_{k\to\infty}\mathbb{E}\left(\min_{1\leq i\leq N}\big{|}x%
-x_{i}^{(\varphi(k))}\big{|}^{2}\right)$$
$$\displaystyle\geq\mathbb{E}\left(\liminf_{k\to\infty}\min_{1\leq i\leq N}\big{%
|}x-x_{i}^{(\varphi(k))}\big{|}^{2}\right)$$
$$\displaystyle=\mathbb{E}\left(\liminf_{k\to\infty}\min_{j\in I^{c}}\big{|}x-x_%
{i}^{(\varphi(k))}\big{|}^{2}\right)$$
$$\displaystyle=\mathbb{E}\left(\min_{j\in I^{c}}\big{|}x-x_{j}^{\infty}\big{|}^%
{2}\right)$$
$$\displaystyle\geq e_{|I^{c}|}(X)^{2}.$$
Finally $e_{|I^{c}|}(X)^{2}\geq e_{N-1}(X)^{2}$ since $I$ contains at least $i_{0}$.
On the other hand, we know from $(a)$ that $\mathcal{G}(\Gamma^{(\varphi(k))})\leq\mathcal{G}(\Gamma^{(0)})$, so that we get the following contradictory inequality
$$e_{N-1}(X)^{2}\leq\liminf_{k\to\infty}\mathcal{G}(\Gamma^{(\varphi(k))})\leq%
\mathcal{G}(\Gamma^{(0)})<e_{N-1}(X)^{2}.$$
Consequently $I$ is empty which completes the proof.
$(c)$ The proof is similar to that of the above item $(b)$. Assume $x^{(\varphi(k))}\to x^{(\infty)}$. If there exists two components $x_{i}^{(\varphi(k))}$ and $x_{j}^{(\varphi(k))}$ converging to $x^{\infty}_{i}=x^{\infty}_{j}$. Then, owing to Fatou’s Lemma
$$e(\Gamma^{(\infty)},X)^{2}=\mathbb{E}\left(\min_{1\leq i\leq N}|X-x_{i}^{%
\infty}|_{2}^{2}\right)\leq\liminf_{k}\mathbb{E}\left(\min_{1\leq i\leq N}|X-x%
_{i}^{(\varphi(k))}|_{2}^{2}\right)\leq e(\Gamma^{(\varphi(0))},X)<e_{N-1}(X)$$
which yields a contradiction. $\qquad\Box$
2.3 Convergence of the Lloyd procedure under a boundedness assumption
From now on, our aim is to investigate the structure of the set $\Upsilon^{\infty}$ of limiting grids of the sequence $(\Gamma^{(k)})_{k\geq 0}$ $i.e.$ the set of grids $\Gamma^{(\infty)}$ such that there exists a subsequence $(\varphi(k))_{k\geq 0}$ for which $\displaystyle\lim_{k\to+\infty}\Gamma^{(\varphi(k))}=\Gamma^{(\infty)}$.
The two lemmas and the proposition below establish several properties of the iterated grids $(\Gamma^{(k)})_{k\geq 0}$ which are the basic “bricks” of the proof of Claim $(a)$ of Theorem 2.1.
Lemma 2.2.
Let $C$ be a closed convex set of ${\mathbb{R}}^{d}$ with non-empty interior and let $\nu$ be a Borel probability distribution such that $\int_{{\mathbb{R}}^{d}}|\xi|^{2}\nu(d\xi)<+\infty$ and $\nu(C)>0$. Furthermore assume that, either $\nu$ satisfies $\nu(\stackrel{{\scriptstyle\circ}}{{C}})>0$, or $\nu$ assigns no mass to hyperplanes. Then the function defined on $C$ by
$$I_{C}:y\longmapsto\int_{C}|y-\xi|^{2}\nu(d\xi)$$
is continuous, strictly convex and atteins its unique minimum at $y^{*}_{{}_{C}}=\displaystyle\frac{\int_{C}\xi\nu(d\xi)}{\nu(C)}\in\,\stackrel{%
{\scriptstyle\circ}}{{C}}$.
Proof.
Elementary computations show that
$$\displaystyle I_{C}(y)$$
$$\displaystyle=$$
$$\displaystyle\nu(C)|y|^{2}+\int_{C}|\xi|^{2}\nu(d\xi)-2\left(y|\int_{C}\xi\nu(%
d\xi)\right)$$
$$\displaystyle=$$
$$\displaystyle\nu(C)\left(|y-y^{*}_{{}_{C}}|^{2}+\int_{C}|\xi-y^{*}_{{}_{C}}|^{%
2}\frac{\nu(d\xi)}{\nu(C)}\right)$$
so that $\displaystyle y^{*}_{{}_{C}}={\rm argmin}_{C}I_{C}$.
Now assume $y^{*}_{{}_{C}}\!\in\partial C$. There exists a supporting hyperplane $H^{*}$ to $C$ at $y^{*}_{{}_{C}}$ defined by $H^{*}=y^{*}_{{}_{C}}+\vec{u}^{\perp}$, $|\vec{u}|=1$. For every $\xi\!\in C$, $(\xi-y^{*}_{{}_{C}}|\vec{u})\geq 0$ so that
$$0=(0|\vec{u})=\frac{1}{\nu(C)}\int_{C}\underbrace{(\xi-y^{*}_{{}_{C}}|\vec{u})%
}_{\geq 0}\nu(d\xi)$$
so that $(\xi-y^{*}_{{}_{C}}|\vec{u})=0$ $\nu(d\xi)$-$a.s.$ $i.e.$ $\xi\!\in H^{*}$ $\nu(d\xi)$-$a.s.$. This leads to a contradiction with the assumptions made on $\nu$. $\quad\Box$
Lemma 2.3.
Assume that a subsequence $\Gamma^{(\varphi(k))}\to\Gamma^{(\infty)}$ as $k\to+\infty$ and that the boundary of the Voronoi tessellation of $\Gamma^{(\infty)}$ are ${\mathbb{P}}$-negligible. Let $Y\in L^{1}(\Omega,{\cal A},\mathbb{P})$. Then
$$\mathbb{E}(Y|\widehat{X}^{\Gamma^{(\varphi(k))}})\rightarrow\mathbb{E}(Y|%
\widehat{X}^{\Gamma^{(\infty)}})\qquad a.\,s.\;\mbox{ as }\;k\to+\infty.$$
Proof of Lemma 2.3: By definition,
$$\displaystyle\mathbb{E}(Y|\widehat{X}^{\Gamma^{(\varphi(k))}})$$
$$\displaystyle=\sum_{i=1}^{N}\frac{{\mathbb{E}}\,Y\mathbf{1}_{X\in C_{i}(\Gamma%
^{(\varphi(k))})}}{\mathbb{P}(X\in C_{i}(\Gamma^{(\varphi(k))}))}\mbox{\bf 1}_%
{\{X\in C_{i}(\Gamma^{(k)})\}}.$$
We know from what precedes that, since $\Gamma^{(\varphi(k))}\rightarrow\Gamma^{(\infty)}$, if $X(\omega)\!\in\mathbb{R}^{d}\setminus\bigcup_{i=_{1}}^{N}\partial C_{i}(\Gamma%
^{(\infty)})$, the functions
$$\mathbf{1}_{X\in C_{i}(\Gamma^{(\varphi(k))})}(\omega)Y(\omega)\rightarrow%
\mathbf{1}_{X\in C_{i}(\Gamma^{(\infty)})}(\omega)Y(\omega)\quad\mbox{ as }\;k%
\to+\infty,\;i=1,\ldots,N,$$
Our assumption on $\Gamma^{(\infty)}$ implies that these convergences hold ${\mathbb{P}}$-$a.s.$.
We conclude by Lebesgue’s dominated convergence theorem that for every $i\!\in\{1,\ldots,N\}$
$${\mathbb{E}}\big{(}Y(\xi)^{a}\mathbf{1}_{X\in C_{i}(\Gamma^{(\varphi(k))})}%
\big{)}\longrightarrow{\mathbb{E}}\big{(}Y(\xi)^{a}\mathbf{1}_{X\in C_{i}(%
\Gamma^{(\infty)})}\big{)}$$
Therefore, with our convention for the index $i$ such that $\mathbb{P}(X\in C_{i}(\Gamma^{(\infty)}))=0$ (if any), we get by applying the above convergence to $Y$ and ${\bf 1}$
$$\displaystyle\mathbb{E}(Y|\widehat{X}^{\Gamma^{(\varphi(k))}})$$
$$\displaystyle=\sum_{i=1}^{N}\frac{{\mathbb{E}}\big{(}Y\mathbf{1}_{X\in C_{i}(%
\Gamma^{(\varphi(k))})}\big{)}}{\mathbb{P}(X\in C_{i}(\Gamma^{(\varphi(k))}))}%
\mbox{\bf 1}_{\{X\in C_{i}(\Gamma^{(k)})\}}$$
$$\displaystyle\xrightarrow[k\to\infty]{\,}\sum_{i=1}^{N}\frac{{\mathbb{E}}\big{%
(}Y\mathbf{1}_{X\in C_{i}(\Gamma^{(\infty)})}\big{)}}{\mathbb{P}(X\in C_{i}(%
\Gamma^{(\infty)})}\mbox{\bf 1}_{\{X\in C_{i}(\Gamma^{(\infty)})\}}=\mathbb{E}%
(Y|\widehat{X}^{\Gamma^{(\infty)}})\qquad{\mathbb{P}}\mbox{-}a.s.\qquad\qquad\hfill\Box$$
Proposition 2.2 (Grid convergence I).
Assume that $\Gamma^{(0)}$ is a $\mathbf{U}$-valued grid and that the iterates $(\Gamma^{(k)})_{k\geq 0}$ of the Lloyd algorithm are bounded.
$(a)$ Assume that $\mu$ assigns no mass to hyperplanes and ${\rm supp}(\mu)=\mathbf{U}$ ($i.e.$ is convex).
If the sequence $(\Gamma^{(k)})_{k\geq 0}$ of iterations of the Lloyd procedure is bounded ($e.g.$ because $\mathbf{U}$ is itself bounded) then
$$\liminf_{k}\min_{i\neq j}|x^{(k)}_{i}-x^{(k)}_{j}|>0$$
$i.e.$ no components of the grids get asymptotically stuck as $k$ goes to infinity.
$(b)$ Assume $X\!\in L^{2}({\mathbb{P}})$. Let $\Gamma^{(\infty)}$ be a limiting grid of $(\Gamma^{(k)})_{k\geq 0}$. If the boundary of the Voronoi cells of $\Gamma^{(\infty)}$ are ${\mathbb{P}}_{{}_{X}}$-negligible $i.e.$ ${\mathbb{P}}\big{(}X\!\in\cup_{i}\partial C_{i}(\Gamma^{(\infty)})\big{)}=0$, then the grid $\Gamma^{(\infty)}$ is stationary $i.e.$ it is a fixed point of the Lloyd map $T_{{}_{N}}$ or equivalently that (any of) its induced Voronoi tessellation(s) is a CVT. Moreover if $\Gamma^{(\varphi(k))}\to\Gamma^{(\infty)}$, then
$$\widehat{X}^{\Gamma^{(\varphi(k))}}\stackrel{{\scriptstyle a.s.\,\&\,L^{2}}}{{%
\longrightarrow}}\widehat{X}^{\infty}\quad\mbox{ as }\;k\to+\infty.$$
$(c)$ If the distribution of $X$ assigns no mass to hyperplanes, then $\nabla\mathcal{G}(\Gamma^{(k)})\to 0$ as $k\to+\infty$.
$(d)$ If the distribution of $X$
has a convex support $\mathbf{U}={\rm supp}(\mu)$, then the consistent representations of $(\Gamma^{(k)})_{k\geq 0}$ satisfy
$$\sum_{k\geq 0}|x^{(k+1)}-x^{(k)}|^{2}_{({\mathbb{R}}^{d})^{N}}<+\infty.$$
In particular, $x^{(k+1)}-x^{(k)}\to 0$ as $k\to+\infty$. Hence, the set $\mathcal{X}_{\infty}$ of (consistent representations of) limiting grids of $(\Gamma^{(k)})_{k\geq 0}$ is a (compact) connected subset of $\mathbf{U}^{N}$.
Remarks. $\bullet$ In the literature the convergence of the gradient of the iterated grids as established in $(c)$ is sometimes known as “weak convergence” of the Lloyd procedure.
$\bullet$ Lemma 2.2 improves a result obtained in [5] which show that component do not asymptotically merge in Lloyd’s procedure even when the Splitting Assumption is not satisfied. In [5], the distribution $\mu$ is supposed to be absolutely continuous with a density a continuous $\rho$, everywhere strictly positive on its support. Our method of proof allows for relaxing this absolute continuity assumption.
Proof of Proposition 2.2: $(a)$
For every $i,\,j\!\in\{1,\ldots,N\}$, $i\neq j$, we define the median hyperplane of $x^{(k)}_{i}$ and $x^{(k)}_{j}$ by
$$\vec{u}^{(k)}_{ij}=\frac{x^{(k)}_{i}-x^{(k)}_{j}}{|x^{(k)}_{i}-x^{(k)}_{j}|},%
\;H^{(k)}_{ij}=\frac{x^{(k)}_{i}+x^{(k)}_{j}}{2}+\vec{u}_{ij}^{\perp}\quad%
\mbox{ and }\quad$$
We define together the affine form
$$\varphi_{ij}(\xi)=\Big{(}\xi-\frac{x^{(k)}_{i}+x^{(k)}_{j}}{2}\Big{|}\vec{u}_{%
ij}\Big{)},\quad\xi\!\in{\mathbb{R}}^{d}$$
which satisfies $\varphi^{(k)}_{ij}\geq 0$ on $C_{i}(\Gamma^{(k)})$ and $\varphi^{(k)}_{ij}(x^{(k)}_{i})=\frac{1}{2}|x^{(k)}_{i}-x^{(k)}_{j}|>0$. Moreover, note that $\varphi^{(k)}_{ji}=-\varphi^{(k)}_{ij}$ and that
$$\bigcap_{j\neq i}\Big{\{}\varphi_{ij}^{\infty}>0\Big{\}}=\stackrel{{%
\scriptstyle\circ}}{{C}}_{i}(\Gamma^{(k)})\subset C_{i}(\Gamma^{(k)})\subset%
\overline{C}_{i}(\Gamma^{(k)})\subset\bigcap_{j\neq i}\{\varphi_{ij}^{\infty}%
\geq 0\}.$$
The sequence of iterated grids $(\Gamma^{(k)})_{k\geq 0}$ being bounded by assumption, we may assume without loss of generality, up to an extraction $(\varphi(k))_{k\geq 0}$,
$$x^{(\varphi(k))}_{i}\to x^{\infty}_{i},\;i\!\in\{1,\ldots,N\},\quad\vec{u}^{(%
\varphi(k))}_{ij}\to u^{\infty}_{ij},\;i,\,j\!\in\{1,\ldots,N\}$$
and
$$x^{(\varphi(k)+1)}_{i}\to\tilde{x}^{\infty}_{i},\;i\!\in\{1,\ldots,N\}\quad%
\mbox{ as }k\to+\infty.$$
Set for every $i,\,j\!\in\{1,\ldots,N\}$
$$\varphi_{ij}^{\infty}=\lim_{\varphi(k)\to\infty}\varphi^{(\varphi(k))}_{ij}%
\quad\mbox{and}\quad C^{\infty}_{i}=\bigcap_{j\neq i}\{\varphi_{ij}^{\infty}%
\geq 0\}.$$
Hence, for every $i\!\in\{1,\ldots,N\}$, $C^{\infty}_{i}$ is a closed polyhedral convex set containing $x_{i}$. It also contains $\tilde{x}_{i}$ since $x^{(\varphi(k)+1)}_{i}\!\in C_{i}(\Gamma^{(\varphi(k))})$ owing to the stationarity property. Then, for every $j\!\in\{1,\ldots,N\}$,
$$\varphi_{ij}^{\infty}(\tilde{x}^{\infty}_{i})=\lim_{\varphi(k)\to\infty}%
\varphi_{ij}^{(\varphi(k))}(x^{(\varphi(k)+1)}_{i})\geq 0$$
since $\varphi^{(\varphi(k))}_{ij}$ uniformly on compact sets toward $\varphi^{\infty}_{i}$ (simple convergence of affine forms in finite dimension implies locally uniform convergence).
Assume there exists $i_{0}\!\in\{1,\ldots,N\}$ such that $I_{0}=\{i\,|\,x^{\infty}_{i}=x^{\infty}_{i_{0}}\}$ is not reduces to $\{i_{0}\}$ $i.e.$ contains at least two indices.
It is clear that
$$\emptyset\neq\Big{\{}\xi\!\in{\mathbb{R}}^{d}\,|\,|\xi-x^{\infty}_{i_{0}}|<d%
\big{(}\xi,\Gamma^{(\infty)}\setminus\{x_{i_{0}}\}\big{)}\Big{\}}\subset%
\bigcup_{i\in I_{0}}C^{\infty}_{i}.$$
The above nonempty open set has non zero $\mu$-mass since $x_{i_{0}}\!\in\mathbf{U}$. Hence, there exists two indices $i_{1},i_{2}\!\in I_{0}$ such that $\mu(C^{\infty}_{i_{1}})$ and $\mu(C^{\infty}_{i_{2}})>0$. First note that $\varphi^{\infty}_{i_{1}i_{2}}(x^{\infty}_{i_{1}})\geq 0$ and $\varphi_{i_{2}i_{1}}(x^{\infty}_{i_{2}})\geq 0$ but both quantities being opposite since $x^{\infty}_{i_{1}}=x^{\infty}_{i_{2}}$ they are equal to $0$ which means that
$$x^{\infty}_{i_{0}}=x^{\infty}_{i_{1}}=x^{\infty}_{i_{2}}\!\in\partial C^{%
\infty}_{i_{1}}\cap\partial C^{\infty}_{i_{2}}.$$
Since these sets are polyhedral and $\mu$ assigns no mass to hyperplanes, $\mu(\stackrel{{\scriptstyle\circ}}{{C}}_{i_{1}})$ and $\mu(\stackrel{{\scriptstyle\circ}}{{C}}_{i_{2}})>0$. Then one checks that
$$\mbox{\bf 1}_{\stackrel{{\scriptstyle\hskip-7.113189pt\circ}}{{C_{i_{\ell}}^{%
\infty}}}}=\lim_{k}\mbox{\bf 1}_{\stackrel{{\hskip-21.339567pt{}_{\circ}}}{{C_%
{i_{\ell}}^{(\varphi(k))}}}},\;\ell=1,2$$
so that, still using that $\mu$ assigns no mass to the boundaries of these polyhedral convex sets, we get
$$0<\mu(\stackrel{{\hskip-7.113189pt{}_{\circ}}}{{C_{i_{\ell}}^{\infty}}})=\lim_%
{k\to+\infty}\mu\big{(}\stackrel{{\hskip-21.339567pt{}_{\circ}}}{{C_{i_{\ell}}%
^{(\varphi(k))}}}\big{)},\;\ell=1,2.$$
Set $\varepsilon_{0}=\min_{\ell=1,2}\mu(\stackrel{{\scriptstyle\hskip-7.113189pt%
\circ}}{{C_{i_{\ell}}^{\infty}}})>0$. It follows form (2.7) that
$$\|X-\widehat{X}^{\Gamma{(\varphi(k))}}\|^{2}_{2}-\|X-\widehat{X}^{\Gamma{(%
\varphi(k)+1)}}\|^{2}_{2}\geq\min_{\ell=1,2}\mu\big{(}\stackrel{{\scriptstyle%
\!\!\!\!\circ}}{{C_{i_{\ell}}}}^{(\varphi(k))}\big{)}\big{(}\big{|}x^{(\varphi%
(k)+1)}_{i_{1}}-x^{(\varphi(k))}_{i_{1}}\big{|}^{2}+\big{|}x^{(\varphi(k)+1)}_%
{i_{2}}-x^{(\varphi(k))}_{i_{2}}\big{|}^{2}\big{)}.$$
Letting $\varphi(k)$ go to infinity implies that
$$\varepsilon_{0}\big{(}\big{|}\tilde{x}^{\infty}_{i_{1}}-x^{\infty}_{i_{1}}\big%
{|}^{2}+\big{|}\tilde{x}^{\infty}_{i_{2}}-x^{\infty}_{i_{2}}\big{|}^{2}\big{)}\leq
0$$
since $\|X-\widehat{X}^{\Gamma{(k)}}\|_{2}$ is a converging sequence. Hence $\tilde{x}^{\infty}_{i_{\ell}}=x^{\infty}_{i_{\ell}}$, $\ell=1,2$, which in turn implies that $\tilde{x}^{\infty}_{i_{1}}=\tilde{x}^{\infty}_{i_{1}}=x_{i_{0}}$.
One shows likewise, still taking advantage of the $\mu$-negligibility of the boundaries of the polyhedral sets $C^{\infty}_{i}$, that
$$\lim_{k\to+\infty}\int_{C_{i_{\ell}}(\Gamma^{(\varphi(k))})}\xi\mu(d\xi)=\int_%
{C_{i_{\ell}}^{\infty}}\xi\mu(d\xi),\;\ell=1,2.$$
Passing to the limit in the stationary equation satisfies by $x_{{}_{\ell}}^{(\varphi(k))}$, $\ell=1,2$, finally implies that
$$\tilde{x}^{\infty}_{i_{\ell}}=\frac{\int_{C_{i_{\ell}}^{\infty}}\xi\mu(d\xi)}{%
\mu(C_{i_{\ell}}^{\infty})}\!\in\,\stackrel{{\scriptstyle\hskip-7.113189pt%
\circ}}{{C_{i_{\ell}}^{\infty}}},\quad\ell=1,2.$$
But, owing to Lemma 2.2 (see also [7], p.22), this implies that $\tilde{x}^{\infty}_{i_{\ell}}\!\in\,\stackrel{{\scriptstyle\hskip-7.113189pt%
\circ}}{{C_{i_{\ell}}^{\infty}}}$, $\ell=1,2$. This yields a contradiction to the fact that both $x_{\ell}$ are equal $x_{i_{0}}$.
$(b)$ Let $\xi\!\in\mathbb{R}^{d}\setminus\bigcup_{i=_{1}}^{N}\partial C_{i}(\Gamma^{(%
\infty)})$.
Then $\xi$ belong to the interior of one of the tessels $C_{i}(\Gamma^{(\infty)})$, say $\mathring{C}_{i_{0}}(\Gamma^{(\infty)})$. Hence, $|\xi-x_{i_{0}}^{\infty}|$ is strictly smaller than $\min_{i\neq i_{0}}|\xi-x_{i}^{\infty}|$. Consequently, there exists an $n(\xi)\in\mathbb{N}^{*}$ such that for all $n\geq n(\xi)$,
$$|\xi-x_{i_{0}}^{(\varphi(k))}|<\min_{i\neq i_{0}}|\xi-x_{i}^{(\varphi(k))}|$$
or equivalently $\xi\!\in\mathring{C}_{i_{0}}(\Gamma^{(\varphi(k))})$.
Thus, for every $\xi\!\in\mathbb{R}^{d}\setminus\bigcup_{i=_{1}}^{N}\partial C_{i}(\Gamma^{(%
\infty)})$,
$$\operatorname{Proj}_{\Gamma^{(\varphi(k))}}(\xi)=\sum_{i=1}^{N}x_{i}^{(\varphi%
(k))}\mathbf{1}_{C_{i}(\Gamma^{(\varphi(k))})}(\xi)\xrightarrow[k\to\infty]{\,%
}\sum_{i=1}^{N}x_{i}^{\infty}\mathbf{1}_{C_{i}(\Gamma^{(\infty)})}(\xi)=%
\operatorname{Proj}_{\Gamma^{(\infty)}}(\xi).$$
This clearly implies that ${\mathbb{P}}(d\omega)$-$a.s.$, $\widehat{X}^{\Gamma^{(\varphi(k))}}(\omega)\to\widehat{X}^{\Gamma^{(\infty)}}(\omega)$ as $k\to+\infty$
since $\mathbb{P}(X\!\in\bigcup_{i=_{1}}^{N}\partial C_{i}(\Gamma^{(\infty)}))=0$. To carry on the proof, we need the following lemma.
If we set $Y=X$ in Lemma 2.3, then
$$\mathbb{E}(X|\widehat{X}^{\Gamma^{(\varphi(k))}})\longrightarrow\mathbb{E}(X|%
\widehat{X}^{\Gamma^{(\infty)}})\qquad a.s.$$
Moreover, the sequence $\big{(}{\mathbb{E}}(X\,|\,\widehat{X}^{\Gamma^{(k)}})\big{)}_{k\geq 0}$ being $L^{2}$-uniformly integrable since $X\!\in L^{2}({\mathbb{P}})$, the above convergence also holds in $L^{2}({\mathbb{P}})$.
Now, since $\varphi(k+1)\geq\varphi(k)+1$ and $\widetilde{X}^{\Gamma^{(\varphi(k)+1)}}={\mathbb{E}}\big{(}X\,|\,\widehat{X}^{%
\Gamma^{(\varphi(k))}}\big{)}$ is $\Gamma^{(\varphi(k)+1)}$-valued, we derive from Lemma 2.1 that
$$\forall k\in\mathbb{N}^{*},\quad\|X-\widehat{X}^{\Gamma^{(\varphi(k+1))}}\|_{2%
}\leq\|X-\widehat{X}^{\Gamma^{(\varphi(k)+1)}}\|_{2}\leq\|X-\widetilde{X}^{%
\Gamma^{(\varphi(k)+1)}}\|_{{}_{2}}=\|X-{\mathbb{E}}\big{(}X\,|\,\widehat{X}^{%
\Gamma^{(\varphi(k))}}\big{)}\|_{2}.$$
Since we know that $\widehat{X}^{\Gamma^{(\varphi(k))}}\longrightarrow\widehat{X}^{\Gamma^{(\infty%
)}}$ ${\mathbb{P}}$-$a.s.$, it follows from from Fatou’s Lemma that
$$\|X-\widehat{X}^{\Gamma^{(\infty)}}\|_{2}\leq\liminf_{k\to\infty}\|X-\widehat{%
X}^{\Gamma^{(\varphi(k+1))}}\|_{2}.$$
On the other hand, we derive from the convergence $\mathbb{E}(X|\widehat{X}^{\Gamma^{(\varphi(k))}})\longrightarrow\mathbb{E}(X|%
\widehat{X}^{\Gamma^{(\infty)}})$ in $L^{2}({\mathbb{P}})$ that
$$\lim_{k\to\infty}\|X-{\mathbb{E}}\big{(}X\,|\,\widehat{X}^{\Gamma^{(\varphi(k)%
)}}\big{)}\|_{2}=\|X-\mathbb{E}(X|\widehat{X}^{\Gamma^{(\infty)}})\|_{2}.$$
so that
$$\|X-\widehat{X}^{\Gamma^{(\infty)}}\|_{2}\leq\|X-\mathbb{E}(X|\widehat{X}^{%
\Gamma^{(\infty)}})\|_{2}$$
which in turn implies by the very definition of conditional expectation as an orthogonal projection on $L^{2}(\widehat{X}^{\Gamma^{(\infty)}})$ that
$$\widehat{X}^{\Gamma^{(\infty)}}=\mathbb{E}(X|\widehat{X}^{\Gamma^{(\infty)}})%
\qquad{\mathbb{P}}\mbox{-}a.s.$$
$(c)$ First we note that, for any grid $\Gamma=\{x_{1},\ldots,x_{{}_{N}}\}$, Schwarz’s Inequality implies
$$\displaystyle|\nabla\mathcal{G}(\Gamma)|_{2}^{2}$$
$$\displaystyle=\sum_{i=1}^{N}\left|\frac{\partial\mathcal{G}}{\partial x_{i}}%
\left(\Gamma\right)\right|^{2}$$
$$\displaystyle=4\sum_{i=1}^{N}\left|\int_{C_{i}(\Gamma}\left(x_{i}-\xi\right)%
\mathbb{P}(\mathrm{d}\xi)\right|^{2}\leq 4\sum_{i=1}^{N}\int_{C_{i}(\Gamma}%
\left|x_{i}-\xi\right|^{2}\mathbb{P}(\mathrm{d}\xi)=4\mathcal{G}(\Gamma)$$
so that the sequence $\big{(}\nabla\mathcal{G}(\Gamma^{(k)})\big{)}_{k\geq 0}$ is bounded. Now, as ${\mathbb{P}}_{X}$ assigns no mass to the boundary of any Voronoi tessellations, we derive from what precedes that any limiting grid of $(\Gamma^{(k)})_{k\geq 0}$ is stationary $i.e.$ $\nabla\mathcal{G}(\Gamma^{(\infty)})=0$. It is clear by an extraction procedure that $0$ is the only limiting value for the bounded sequence $\big{(}\nabla\mathcal{G}(\Gamma^{(k)})\big{)}_{k\geq 0}$ which consequently converges toward $0$.
$(d)$ The set of limiting grids of the sequence $(\Gamma^{(k)})_{k\geq 0}$ is compact by construction. Its connectedness will classically follow from
$$|\Gamma^{(k+1)}-\Gamma^{(k)}|\xrightarrow[]{k\to\infty}0$$
(see $e.g.$ [15]).
It is clear from item $(a)$ that $K_{\Gamma}=\{\Gamma^{(k)},\,k\geq 0\}\cup\Upsilon_{\infty}$ is a compact whose intersection with the closed set $\{(x_{1},\ldots,x_{{}_{N}})\!\in({\mathbb{R}}^{d})^{N},\,x_{i}\neq x_{j},\,i%
\neq j\}^{c}$. Hence, $K_{\Gamma}$ stands at a positive distance of this set or, equivalently, there exists $\delta>0$, such that
$$\forall\,k\geq 0,\quad\forall\,i,\,j\in\{1,\ldots,N\},\,i\neq j,\quad|x^{(k)}_%
{i}-x^{(k)}_{j}|\geq\delta.$$
As a consequence, every Voronoi cell satisfies
$$B^{\hskip-5.121496pt{}^{{}^{\circ}}}(x^{(k)}_{i},\delta/2)\subset C_{i}(\Gamma%
^{(k)})$$
where $B^{\hskip-5.121496pt{}^{{}^{\circ}}}(\xi,r)$ denotes the open ball with center $\xi$ and radius $r$. Since $\mathbf{U}={\rm supp}(\mu)$ is a (closed) convex set and $\Gamma^{(0)}\subset{\rm supp}(\mu)$, then $\Gamma{(k)}\subset\mathbf{U}$ for every $k\!\in{\mathbb{N}}$. As a consequence, $K_{\Gamma}\subset\mathbf{U}$ which in turn implies that the function $\xi\mapsto{\mathbb{P}}\big{(}X\!\in B^{\hskip-5.121496pt{}^{{}^{\circ}}}(\xi,%
\delta/2)\big{)}$ is (strictly) positive on the compact set $K_{\Gamma}$, so that we can define
$$m_{*}:=\inf_{\xi\in\mathbf{K_{\Gamma}}}{\mathbb{P}}\big{(}X\!\in B^{\hskip-5.1%
21496pt{}^{{}^{\circ}}}(\xi,\delta/2)\big{)}>0$$
(2.6)
which implies in particular that
$$M^{(k)}_{i}=\mathbb{P}(X\!\in C_{i}(\Gamma^{(k)}))\geqslant m_{*},\;1\leq i%
\leq N,\;k\geq 0.$$
Now we introduce the energy gap
$$\Delta(k)=\big{\|}X-\widehat{X}^{\Gamma^{(k)}}\big{\|}^{2}_{2}-\big{\|}X-%
\widetilde{X}^{\Gamma^{(k+1)}}\big{\|}^{2}_{2}$$
known to be non-negative by (2.4) in Lemma 2.1. Then
$$\displaystyle\Delta(k)$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{N}\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{(k)}|^{2}%
\mathbb{P}_{{}_{X}}(\mathrm{d}\xi)-\sum_{j=1}^{N}\int_{C_{j}(\Gamma^{(k)})}|%
\xi-x_{j}^{(k+1)}|^{2}\mathbb{P}_{{}_{X}}(\mathrm{d}\xi).$$
(2.7)
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{N}\int_{C_{j}(\Gamma^{(k)})}(|x_{j}^{(k)}|^{2}-|x_{j}%
^{(k+1)}|^{2})+2\left(x_{j}^{(k+1)}-x_{j}^{(k)}|\xi\right)\mathbb{P}_{{}_{X}}(%
\mathrm{d}\xi)$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{N}M^{(k)}_{j}(|x_{j}^{(k)}|^{2}-|x_{j}^{(k+1)}|^{2})+%
2\Big{(}x_{j}^{(k+1)}-x_{j}^{(k)}|\int_{C_{j}(\Gamma^{(k)})}\xi\,\mathbb{P}_{{%
}_{X}}(\mathrm{d}\xi)\Big{)}$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{N}M^{(k)}_{j}(|x_{j}^{(k)}|^{2}-|x_{j}^{(k+1)}|^{2})+%
2M^{(k)}_{j}\left(x_{j}^{(k+1)}-x_{j}^{(k)}|x_{j}^{(k+1)}\right)$$
$$\displaystyle=$$
$$\displaystyle\sum_{j=1}^{N}M^{(k)}_{j}|x_{j}^{(k)}-x_{j}^{(k+1)}|^{2}$$
$$\displaystyle\geq$$
$$\displaystyle m_{*}|\Gamma^{(k+1)}-\Gamma^{(k)}|^{2}_{({\mathbb{R}}^{d})^{N}}.$$
On the other hand $\Delta(k)\leq\big{\|}X-\widehat{X}^{\Gamma^{(k)}}\big{\|}^{2}_{2}-\big{\|}X-%
\widehat{X}^{\Gamma^{(k+1)}}\big{\|}^{2}_{2}$ so that, finally,
$$\qquad\qquad\qquad\sum_{k\geq 0}|\Gamma^{(k+1)}-\Gamma^{(k)}|^{2}_{({\mathbb{R%
}}^{d})^{N}}\leq\frac{1}{m_{*}}\big{\|}X-\widehat{X}^{\Gamma^{(0)}}\big{\|}^{2%
}_{2}<+\infty.\qquad\qquad\qquad\qquad\qquad\qquad\hfill\Box$$
Comments. $\bullet$ The restrictions on the possible limiting grids in Theorem 2.1 do not imply uniqueness in general in higher dimensions: the symmetry properties shared by the distribution itself already induces multiple limiting grids as emphasized by the case of the multivariate normal distribution $\mathcal{N}(0,I_{d})$.
In fact, for an orthogonal matrix $P\!\in\mathcal{O}_{d}$ that is $PP^{*}=I_{d}$ ($P^{*}$ stands for the transpose of $P$) and any optimal grid $\Gamma=\{x_{1},\ldots,x_{{}_{N}}\}$,
$$\|X-\widehat{X}^{P\Gamma}\|^{2}_{2}={\mathbb{E}}\min_{1\leq i\leq N}|X-Px_{i}|%
^{2}={\mathbb{E}}\min_{1\leq i\leq N}|P^{*}X-x_{i}|^{2}={\mathbb{E}}\min_{1%
\leq i\leq N}|X-x_{i}|^{2}=\|X-\widehat{X}^{\Gamma}\|^{2}_{2}.$$
$\bullet$ For distribution with less symmetries like $X\sim\mathcal{N}(0,\Sigma_{d})$ with $\Sigma=\operatorname{diag}(\lambda_{1},\lambda_{2},\cdots,\lambda_{d}),\,0<%
\lambda_{1}<\lambda_{2},\cdots<\lambda_{d}$, one can reasonably hope that at least local uniqueness of stationary grids (CVT) holds true. One way to check that is to establish that the Hessian $D^{2}\mathcal{G}(\Gamma)$ is invertible at each stationary grid $\Gamma$. A closed form is available for this Hessian (see $e.g.$ [6]).
Proof of Theorem 2.1: The preliminary claim follows by induction from the structural stationary properties of the iterates and Lemma 2.2.
$(a)$ Combining the results obtained in the above proposition and the convergence $\big{\|}X-\widehat{X}^{\Gamma^{(k)}}\big{\|}_{2}$ toward a non-negative real number $\ell_{\infty}$ completes the proof.
$(b)$ follows from Proposition 2.1 in Section 2.2 which implies that the sequence $(\Gamma^{(k)})_{k\geq 0}$ is bounded. Then one concludes by $(a)$. $\quad\Box$
3 A bounded variant of Lloyd’s procedure based on spatial estimation of the optimal quantizers
So far our results are based on the hypothesis that we initialize the Lloyd algorithm using a grid of size $N$ whose induced quadratic quantization error is lower than the minimal quantization error achievable with a grid of size at most $N-1$. This is clearly the key point to ensure that the iterates of the procedure remain bounded. From a practical point of view, this choice for the initial grid is not very realistic, in particular if we are processing a “splitting method”: nothing ensures, even if the Lloyd procedure converges at a level $N-1$, that the limiting grid will be optimal with the consequence that the initialization at level $N$ “below” $e_{N-1}(X)$ becomes impossible.
However, we know from theoretical results on optimal vector quantization where the optimal quantizers are located a priori.
3.1 A priori bounds for optimal quantizers
The following proposition can be found in [7].
Proposition 3.1.
Let $L_{N,X}(c)=\{\Gamma\,:\,|\Gamma|=N\mbox{ and }e(\Gamma,X)\leqslant c\}$.
Let $c\!\in(0,e_{N-1}(X)]$. There exists $R\!\in(0,+\infty)$ such that
$$L_{N,X}(c)\subset B(m_{{}_{X}},R)\mbox{ with }m_{{}_{X}}={\mathbb{E}}\,X.$$
An upper bound $S$ satisfies the following conditions:
(i)
$\exists\,r>0$ such that $\mathbb{P}\big{(}X\!\in B(m_{{}_{X}},r)\big{)}>0$ and $(\frac{R}{5}-r)^{2}\mathbb{P}\big{(}X\!\in B(m_{{}_{X}},r)\big{)}>c$,
(ii)
$4\displaystyle\int_{B(0,\frac{2R}{5})^{c}}|\xi-m_{{}_{X}}|_{2}^{2}\mathbb{P}_{%
X}(\mathrm{d}\xi)<e_{N-1}(X)-c$.
If we specify $c=e_{N}(X)$, then the set $L_{N,X}(c)$ will be the set of grids corresponding to optimal $N$-quantizers, and $R$ will be an upper bound of the optimal $N$-quantizers.
Consequently, in order to given a numerical estimation of $R$, we have to estimate the asymptotic behaviour of $e_{N-1}(X)-e_{N}(X)$. We know from ([14], see also [10]) that
$$e_{N-1}(X)-e_{N}(X)\propto N^{-\frac{d+2}{d}}$$
If $X$ has a standard Gaussian distribution ${\cal N}(m;I_{d})$, then
$$\int_{B(m_{{}_{X}},\frac{2R}{5})^{c}}|\xi-m_{{}_{X}}|^{2}\mathbb{P}_{X}(%
\mathrm{d}\xi)\int_{B(0,\frac{2R}{5})^{c}}|\xi|^{2}\mathbb{P}_{{}_{X}}(\mathrm%
{d}\xi)\propto R^{d}e^{-\frac{R^{2}}{2}}.$$
Thus
$$R\propto\sqrt{\ln(N)}.$$
More precise results can be found in [14] and [8] for various families of distributions with exponential or polynomial tails at infinity. Under certain condition on $X$, the asymptotic behaviour of $R=R(N)$ can be analyzed sharply as $N$ goes to infinity. Typically if $X\sim{\cal N}(m;I_{d})$,
$$\lim_{N}\frac{R(N)}{\sqrt{\log(N)}}=\frac{1}{\sqrt{2}}\Big{(}1+\frac{2}{d}\Big%
{)}^{\frac{1}{2}}.$$
3.2 A variant of Lloyd’s algorithm
Since we know that all optimal $N$-quantizers are constrained in a bounded domain (depending on $N$ in a more or less controlled way), a natural idea is to constrain the exploration of the Lloyd iterates inside it to take advantage of this information.
We can fix an area that we are sure that the optimal quantizer have its grids in it. Once an iteration of the algorithm includes some points that go beyond the area, we will be sure that it is not the optimal grid. So we can do something to pull the points back into the area while keeping the error non-increasing.
To this end, we now compute the difference made on the second phase of the Lloyd iteration in condition that we pick a point other than the mass center point. If we take $x_{j}^{\prime}$ instead of $x_{j}^{(k+1)}$ for a certain $j$, the resulting difference in the $j$th Voronoi region created by the second phase will be:
$$\displaystyle\Delta_{j}^{\prime}(k):=$$
$$\displaystyle\;{\mathbb{E}}|X-x^{(k+1)}_{j}|^{2}\mbox{\bf 1}_{\{X\in C_{j}(%
\Gamma^{(k)})\}}-{\mathbb{E}}|X-x^{\prime}_{j}|^{2}\mbox{\bf 1}_{\{X\in C_{j}(%
\Gamma^{(k)})\}}$$
$$\displaystyle=$$
$$\displaystyle\;\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{(k)}|^{2}\mathbb{P}(%
\mathrm{d}\xi)-\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{\prime}|^{2}\mathbb{P}(%
\mathrm{d}\xi)$$
$$\displaystyle=$$
$$\displaystyle\;\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{(k)}|^{2}\mathbb{P}(%
\mathrm{d}\xi)-\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{(k+1)}|^{2}\mathbb{P}(%
\mathrm{d}\xi)$$
$$\displaystyle+\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{(k+1)}|^{2}\mathbb{P}(%
\mathrm{d}\xi)-\int_{C_{j}(\Gamma^{(k)})}|\xi-x_{j}^{\prime}|^{2}\mathbb{P}(%
\mathrm{d}\xi)$$
$$\displaystyle=$$
$$\displaystyle\;M_{j}(k)\Big{(}|x_{j}^{(k)}-x_{j}^{(k+1)}|^{2}-|x_{j}^{\prime}-%
x_{j}^{(k+1)}|^{2}\Big{)}.$$
This shows that if we pick a point which lies at the same distance to $x^{(k+1)}_{j}$ as $x^{(k)}_{j}$, then the above difference becomes zero. The idea is then to choose the point inside the prescribed domain if $x^{(k+1)}_{j}$ is outside. It is always possible, $e.g.$ by keeping $x^{(k)}_{j}$ still (although this is probably not the optimal way to proceed).
With this idea we present a modified version of Lloyd’s algorithm, which does not need the Splitting Assumption, to be run successfully. We set a $R>0$ that all optimal quantizers are in the ball $B(0,R)$. The algorithm is as follows (assuming that $X$ is centered for convenience):
Algorithm 2 (Modified Lloyd’s algorithm):
Inputs:
•
$B(0,R)$, the domain of interest (with $R$ close to $R(N)$) hopefully.
•
$\mu={\mathbb{P}}_{{}_{X}}$ a simulatable probability distribution (with a convex support $\mathbf{U}$) and assigning no mass to hyperplanes.
•
$\Gamma^{(0)}=\{x_{i}^{(0)},\,i=1,\ldots,N\}$, the initial set of $N$ generators (starting grid).
Pseudo-script:
$\rhd$ At the $k$th iteration:
1.
Compute the position of $\Gamma^{(k+1)}=\{x_{i}^{(k)+1},\,i=1,\ldots,N\}$, the mass centroid of $\{C_{i}^{(k)},\,i=1,\ldots,N\}$. If there is an index $j$ such that $x_{j}^{(k+1)}$ lies outside $B(0,R)$ (and for every such point), replace the current value $x_{j}^{(k+1)}$ by a point $x_{j}^{(k+1)\prime}$ defined $e.g.$ by $x_{j}^{(k+1)\prime}=\partial B(x_{j}^{(k+1)},|x_{j}^{(k)}-x_{j}^{(k+1)}|)\cap[%
x^{(k)}_{j},x^{(k+1)}_{j}]$ (other choices are possible like choosing it randomly on $\partial B(x_{j}^{(k+1)},|x_{j}^{(k)}-x_{j}^{(k+1)}|)\cap B(0,R)$.
2.
If every point lies in the ball $B(0,R)$, then take these mass centroids of $\{C_{i}^{(k)},\,i=1,\ldots,N\}$ as the new set of generators $\Gamma^{(k+1)}=\{x_{i}^{(k+1)},\,i=1,\ldots,N\}$,
3.
Construct the Voronoi tessellation $\mathcal{C}(\Gamma^{(k+1)})=\{C_{i}^{(k+1)},\,i=1,\ldots,N\}$ of $\mathbf{U}$ with the grid of generators $\{x_{i}^{(k+)},\,i=1,\ldots,N\}$.
$\rhd$ Repeat the above iteration until a stopping criterion is met. And the output is the CVT $\{C_{i}^{(n)},\,i=1,\ldots,N\}$ with generators $\{x_{i}^{(n)},\,i=1,\ldots,N\}$ in $\mathbf{U}$.
$\rhd$ end.
In this new version we modify Phase I (2.4) of the Lloyd procedure in such a way that the quadratic approximation error $\|X-\widetilde{X}^{\Gamma^{(k+1)}}\|_{2}$ still decreases.
The second phase being unchanged, so this modified Lloyd algorithm is still energy descending and furthermore it lives in the ball $B(0,R)$. The trade-off is that with the $R$ fixed at the beginning of the algorithm, we loose the (theoretical) possibility that the iterated sequence cruises very far during the iterations to finally come back with a lower energy. Therefore the energy level of the new limit points will be higher than setting $R=+\infty$. We also see that the larger the radius $R$ we take, the lower limit energy level we can get.
Another trade-off of the modified procedure is that it does not guarantee Lemma 3 because we do not use the Splitting Assumption. In this case we cannot prove the non-degeneracy of the limit grid by this global energy reasoning. However, we can now rely on Proposition 2.2 to ensure that no merging occurs.
Provisional remarks. One verifies on numerical implementation of Lloyd algorithm, that the main default that slows down the procedure is more the freezing of one component of the grid which is“ too far from the core” of the support of the distribution $\mu$ than the explosion of the grid with components going to infinity. but in some sense these seeming radically different behavior are the two sides of the same coin and the above procedure is an efficient way to prevent these parasitic effects. Though, in practice we proceed in a less formal way: using the theoretical estimates on the radius of the distribution allows for an adequate choice of the initial grid $\Gamma^{(0)}$ as confirmed by various numerical experiments carried $e.g.$ in [18].
References
[1]
V. Bally and G. Pagès (2003): A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems. Bernoulli, 9(6):1003-1049.
[2]
J. A. Cuesta, C. Matrán (1988): The strong law of large numbers for k-means and best possible nets of Banach valued random variables. Probab. Theory Related Fields, 78(4):523-534.
[3]
Q. Du, V. Faber and M. Gunzburger (1999): Centroidal Voronoi tessellations: Applications and algorithms, SIAM Review, 41:637-676.
[4]
Q. Du, M. Emelianenko and L. Ju (2006): Convergence of the Lloyd algorithm for computing centroidal Voronoi tessellations, SIAM Journal on Numerical Analysis, 44:102-119.
[5]
M. Emelianenko, L. Ju and A. Rand (2008): Nondegeneracy and Weak Global Convergence of the Lloyd Algorithm in $\mathbb{R}^{d}$, SIAM Journal on Numerical Analysis, 46(3):1423 - 1441.
[6]
J.C. Fort and G. Pagès (1995): On the a.s. convergence of the Kohonen algorithm with a general neighborhood function. Ann. Appl. Probab. 5(4):1177-1216.
[7]
S. Graf and H. Luschgy (2000): Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics 1730, Berlin: Springer.
[8]
S. Junglen (2011): Quantization Balls and Asymptotics of Quantization Radii for Probability Distributions with Radial Exponential Tails, EJP, 16:283-295.
[9]
J.C. Kieffer (1982): Exponential rate of convergence for Lloyd’s method I, IEEE Trans. on Inform. Theory, Special issue on quantization, 28(2):205-210.
[10]
S. Graf, H. Luschgy and G. Pagès (2012): The local quantization behaviour of absolutely continuous probabilities. Annals of Probability, 40(4):1795-1828, 2012
[11]
D.
G. Luenberger and Y. Ye (2008): Linear and Nonlinear Programming, 3rd ed. Springer.
[12]
G. Pagès (1998): A space vector quantization method for numerical
integration, J. Computational and Applied Mathematics, 89,
1-38 (Extended version of “Voronoi Tessellation, space quantization algorithms and numerical integration” (1993), in: M. Verleysen (Ed.), Proceedings of the ESANN’ 93, Bruxelles, Quorum Editions, 221-228).
[13]
G. Pagès and J.. Printems (2003): Optimal quadratic quantization for numerics: the Gaussian case, Monte Carlo Methods & Applications Journal, 9(2):135-166.
[14]
G. Pagès and A. Sagna (2008): Asymptotics of the maximal radius of an $L^{r}$-optimal sequence of quantizers, Bernoulli 18(1):360-389.
[15]
M.D. Ašić and D.D. Adamović (1970): Limit points of sequences in metric spaces, The American Mathematical Monthly, 77(6): 613-616.
[16]
G. Pagès, H. Pham and J. Printems (2004): Optimal quantization methods and applications to numerical problems in finance, in Handbook on Numerical Methods in Finance (S. Rachev, ed.), Birkhauser, Boston, 253-298.
[17]
D. Pollard (1982): A central limit theorem for $k$-means clustering, Ann. Probab., 10, 919-926.
[18]
A. Sagna (2008): Méthodes de quantification optimale avec applications à la finance, PhD thesis, UPMC.
4 Appendix: Numerical detection of the nature of a limiting grid
We provide here a formula for the Hessian of the distortion value function $\mathcal{G}$ when the distribution $\mu$ of $X$ is absolutely continuous with density function $\rho$. From such a formula it is possible, at least numerically in low dimensions, to detect the status of a stationary grid/quantizer in terms of stability: local minimum, saddle point, etc.
As a first step we need a Lemma which provides a formula for the differentiation of integrals overs teasels of the Voronoi partition of an $N$-tuple $x=(x_{1},\ldots,x_{{}_{N}})$.
Lemma 4.1.
Let $\varphi\!\in{\cal C}({\mathbb{R}}^{d},{\mathbb{R}})$. Set for every $x\!\in{\mathbb{R}}^{d}$ with pairwise distinct components,
$$\Phi_{i}(x)=\int_{C_{i}(x)}\varphi(\xi)\lambda_{d}(d\xi),\;i=1,\ldots,N,$$
where $\lambda_{d}$ denotes the Lebesque measure on $({\mathbb{R}}^{d},{\cal B}or({\mathbb{R}}^{d}))$. Then $\Phi_{i}$ is continuously differentiable on the open set of $N$-tuples with pairwise distinct components and
$$\forall\,j\!\in\{1,\ldots,N\},\;j\neq i,\;\frac{\partial\Phi_{i}}{\partial x_{%
j}}(x)=\int_{\overline{C}_{i}(x)\cap\overline{C}_{j}(x)}\varphi(\xi)\left(%
\frac{1}{2}n_{x}^{ij}+\frac{1}{|x_{i}-x_{j}|}\left(\frac{x_{i}+x_{j}}{2}-\xi%
\right)\right)\lambda_{x}^{ij}(d\xi)$$
where $\lambda_{x}^{ij}(d\xi)$ denotes the Lebesque measure on the median hyperplane $H_{ij}^{x}$ of $x_{i}$ and $x_{j}$ and $n^{ij}_{x}=\frac{x_{j}-x_{i}}{|x_{j}-x_{i}|}$. Furthermore,
$$\frac{\partial\Phi_{i}}{\partial x_{i}}(x)=-\sum_{j\neq i}\frac{\partial\Phi_{%
j}}{\partial x_{i}}(x).$$
We refer to [6] or [17] for a proof. This leads to the announced general result concerning the Hessian of the distortion function $\mathcal{G}$. We set for every $u=(u^{1},\ldots,u^{d}),\,v=(v^{1},\ldots,v^{d})\!\in{\mathbb{R}}^{d}$, $u\otimes v=[u^{i}v^{j}]_{1\leq i,j\leq d}$ and $I_{d}=[\delta_{ij}]_{1\leq i,j\leq d}$ ($\delta_{ij}$ denotes the Kronecker symbol).
Proposition 4.1.
Let $\mu={\mathbb{P}}_{{}_{X}}=\rho.\lambda_{d}$ with $\rho$ continuous. Then, for every $i,\,j\!\in\{1,\ldots,N\}$, $i\neq j$,
$$\frac{\partial^{2}\mathcal{G}}{\partial x_{i}\partial x_{j}}(x)=\int_{%
\overline{C}_{i}(x)\cap\overline{C}_{j}(x)}(x_{i}-\xi)\otimes\left(\frac{1}{2}%
n^{ij}_{x}+\frac{1}{|x_{i}-x_{j}|}\Big{(}\frac{x_{i}+x_{j}}{2}-\xi\Big{)}%
\right)\rho(\xi)\lambda^{ij}_{x}(d\xi)\quad\mbox{if }i\neq j$$
and
$$\frac{\partial^{2}\mathcal{G}^{\ell}}{\partial x_{i}\partial x_{j}}(x)=\mu(C_{%
i}(x))I_{d}+\sum_{j\neq i}\int_{\overline{C}_{i}(x)\cap\overline{C}_{j}(x)}(x_%
{j}-\xi)\otimes\left(\frac{1}{2}n^{ij}_{x}-\frac{1}{|x_{i}-x_{j}|}\Big{(}\frac%
{x_{i}+x_{j}}{2}-\xi\Big{)}\right)\rho(\xi)\lambda^{ij}_{x}(d\xi)\;\mbox{if }%
\,i=j.$$
This formula is used $e.g.$ in [6] to show the instability of “square”, ”hyper”-rectangular stationary grids for the uniform distribution over the unit hypercube for Kohonen’s Self-Organizing Maps ($SOM$). When the neighborhood function of the $SOM$ is degenerated (no true neighbor) the $SOM$ amounts to the $CLVQ$ and its equilibrium points are those of the Lloyd procedure, with the same (un-)stability properties. These quantities also appear in the asymptotic variance of the $CLT$ established in [17]. |
The Inverse Weighted Lindley Distribution: Properties, Estimation and an Application on a Failure Time Data
Pedro L. Ramos${}^{1}$111Corresponding author: Pedro Luiz Ramos, Email: [email protected] Francisco Louzada${}^{1}$ Taciana K.O. Shimizu${}^{1}$ Aline O. Luiz${}^{1}$
${}^{1}$Institute of Mathematical Science and Computing
University of São Paulo
São Carlos
Brazil
Abstract
In this paper a new distribution is proposed. This new model provides more flexibility to modeling data with upside-down bathtub hazard rate function. A significant account of mathematical properties of the new distribution is presented. The maximum likelihood estimators for the parameters in the presence of complete and censored data are presented. Two corrective approaches are considered to derive modified estimators that are bias-free to second order. A numerical simulation is carried out to examine the efficiency of the bias correction. Finally, an application using a real data set is presented in order to illustrate our proposed distribution.
keywords:
Inverse weighted Lindley distribution; Maximum Likelihood Estimation; Bias correction; Random censoring.
††articletype:
1 Introduction
In recent years, several new distributions have been introduced
in literature for describing real problems. An important distribution was presented by Lindley [16] in the context of fiducial statistics and Bayes’ theorem. Ghitany et al. [11] argued that the Lindley distribution provides flexible mathematical properties and outlined that in many cases this distribution outperforms the exponential distribution. Since then, new generalizations of Lindley distribution have been proposed such as the generalized Lindley [26], extended Lindley [3], and Power Lindley [9] distribution.
The study of weight distributions provide new comprehension of standard distributions and contributes in adding more flexibility for fitting data [18]. Ghitany et al. [10] presented a two-parameter weighted Lindley (WL) distribution which has bathtub and increasing hazard rate. The WL distribution has probability density function (PDF) given by
$$f(t|\phi,\lambda)=\frac{\lambda^{\phi+1}}{(\phi+\lambda)\Gamma(\phi)}t^{\phi-1%
}(1+t)e^{{}^{-}\lambda t},$$
(1)
for all $t>0$, $\phi>0$ and $\lambda>0$ where $\Gamma(\phi)=\int_{0}^{\infty}{e^{-x}x^{\phi-1}dx}$ is the gamma function. Mazucheli et al. [17] compared the finite sample properties of the parameters of the WL distribution numerical simulations using four methods. Wang and Wang [25] presented bias-corrected MLEs and argued that the proposed estimators are strongly recommended over other estimators without bias-correction. Ali [2] considered a Bayesian approach and derived several informative and noninformative priors under different loss functions. Ramos and Louzada [20] introduced three parameters generalized weighted Lindley distribution.
In this study, a new two-parameter distribution with upside-down bathtub hazard rate is proposed, hereafter, inverse weighted Lindley (IWL) distribution. This new model can be rewritten as the inverse of the WL distribution. A significant account of mathematical properties for the IWD distribution is presented such as moments, survival properties and entropy functions. The maximum likelihood estimators of the parameters and its asymptotic properties are obtained. Further, two corrective approaches are discussed to derive modified MLEs that are bias-free to second order. The first has an analytical expression derived by Cox and Snell (12) and the second is based on the bootstrap resampling method (see Efron [8] for more details), which can be used to reduce bias. Similar corrective approaches has been considered by many authors for other distributions, e.g., Cordeiro et al. [5], Lemonte [15], Teimouri and Nadarajah [24], Giles et al. [12], Ramos et al. [19], Schwartz et al. [22] and Reath et al. [21]. In addition, the MLEs in the presence of randomly censored data is presented. Approximated bias-corrected MLEs for censored data are also discussed. A numerical simulation is performed to examine the effect of the bias corrections in the MLEs for complete and censored data.
The new distribution is a useful generalization of the inverse Lindley distribution [23] and can be represented by a two-component mixture model. Mixture models play an important role in statistics for describing heterogeneity (see, Aalen [1]). Therefore, the IWL distribution can be used to describe data sets in the presence of heterogeneity. For instance, we can be interested in describing the lifetime of components that are composed of new and repaired products, however, only the failure time is observed and the groups are latent variables. In this case, the proposed distribution, as a mixture distribution, can express the heterogeneity in the data. In reliability, this model may be used to describe the lifetime of components associated with a high failure rate
after short repair time. In studies involving the lifetime of patients this model can be useful to describe the course of a disease, where their mortality rate reaches a peak and then declines as the time increase, i.e., problems where their hazard function has upside-down bathtub shape.
In order to illustrate our proposed methodology, we considered a real data set related to failure time of devices of an airline company. Such study is important in order to prevent customer dissatisfaction and customer attrition, and consequently to avoid customer loss. In this context, the choice of the distribution that fits better this data is fundamental for the company reduces its costs. We showed that the inverse weighted Lindley distribution fits better than other well-known distributions for this data set.
The paper is organized as follows. Section 2 introduces the inverse weighted Lindley distribution. Section 3 presents the properties of the IWL distribution such as moments, survival properties and entropy. Section 4 discusses the inferential procedure based on MLEs for complete and censored data. A bias correction approach is also presented for complete and censored data. Section 5 describes two corrective approaches to reduce the bias in the MLEs for complete and censored data. Section 6 presents a simulation study to verify the performance of the proposed estimators. Section 7 illustrates the relevance of our proposed methodology in a real lifetime data. Section 8 summarizes the present study.
2 Inverse Weighted Lindley distribution
A non-negative random variable T follows the IWL distribution with parameters $\phi>0$ and $\lambda>0$ if its PDF is given by
$$f(t|\phi,\lambda)=\frac{\lambda^{\phi+1}}{(\phi+\lambda)\Gamma(\phi)}t^{-\phi-%
1}\left(1+\frac{1}{t}\right)e^{-\lambda t^{-1}}.$$
(2)
Note that if $\phi=1$, the IWL distribution reduces to the inverse Lindley distribution [23]. The IWL distribution can be expressed as a two-component mixture
$$f(t|\phi,\lambda)=pf_{1}(t|\phi,\lambda)+(1-p)f_{2}(t|\phi,\lambda),$$
where $p=\lambda/(\lambda+\phi)$ and $T_{j}\sim\operatorname{IG}(\phi+j-1,\lambda)$, for $j=1,2$, i.e., $f_{j}(t|\lambda,\phi)$ is Inverse Gamma distribution, given by
$$f_{j}(t|\phi,\lambda)=\frac{\lambda^{\phi+j-1}}{\Gamma(\phi+j-1)}t^{-\phi-j}e^%
{-\lambda t^{-1}}.$$
Therefore, the IWL distribution is a mixture distribution and can express the heterogeneity in the data.
Proposition 2.1.
Let $T\sim\operatorname{IWL}(\phi,\lambda)$ then $X=1/T$ follows a weighted Lindley distribution [10].
Proof.
Define the transformation $X=g(T)=\frac{1}{T}$ then the resulting transformation is
$$\displaystyle f_{X}(x)$$
$$\displaystyle=f_{T}\left(g^{-1}(x)\right)\left|\frac{d}{dx}g^{-1}(x)\right|=%
\frac{\lambda^{\phi+1}}{(\phi+\lambda)\Gamma(\phi)}{x}^{\phi+1}\left(1+x\right%
)e^{-\lambda x}\frac{1}{x^{2}}$$
$$\displaystyle=\frac{\lambda^{\phi+1}}{(\phi+\lambda)\Gamma(\phi)}{x}^{\phi-1}%
\left(1+x\right)e^{-\lambda x}.$$
∎
Figure 1 gives examples from the shapes of the density function for different values of $\phi$ and $\lambda$.
The cumulative distribution function from the IWL distribution is given by
$$F(t|\phi,\lambda)=\frac{\Gamma\left(\phi,\lambda t^{-1}\right)(\lambda+\phi)+(%
\lambda t^{-1})^{\phi}e^{-\lambda t^{-1}}}{(\lambda+\phi)\Gamma(\phi)},\ $$
where $\Gamma(x,y)=\int_{x}^{\infty}{w^{y-1}e^{-x}dw}$ is the upper incomplete gamma.
3 Properties of IWL Distribution
In this section, we provide a significant account of mathematical properties of the new distribution.
3.1 Moments
Moments play an important role in statistics. They can be used in many applications, for instance the first moment of the PDF is the well know mean, while the second moment is used to obtain the variance, skewness and kurtosis are also obtained from the moments. In the following, we will derive the moments for the IWL distribution.
Proposition 3.1.
For the random variable $T$ with $\operatorname{IWL}$ distribution,
the r-th moment is given by
$$\mu_{r}=E[T^{r}]=\frac{\lambda^{r}(\phi+\lambda-r)}{(\lambda+\phi)(\phi-1)(%
\phi-2)\cdots(\phi-r)}\,,\quad\mbox{where}\quad\phi>r.$$
(3)
Proof.
Note that if $W\sim\operatorname{IG}(\phi,\lambda)$ distribution then the r-th moment from the random variable $W$ is given by
$$E_{(\phi,\lambda)}[W^{r}]=\frac{\lambda^{r}\Gamma(\phi-r)}{\Gamma(\phi)}=\frac%
{\lambda^{r}}{(\phi-1)(\phi-2)\ldots(\phi-r)}\,,\quad\mbox{where}\quad\phi>r.$$
Since the IWL distribution can be expressed as a two-component mixture, we have
$$\displaystyle\mu_{r}=E[T^{r}]$$
$$\displaystyle=\int_{0}^{\infty}t^{r}f(t|\phi,\lambda)dt=pE_{(\phi,\lambda)}[W^%
{r}]+(1-p)E_{(\phi+1,\lambda)}[W^{r}]$$
$$\displaystyle=\frac{\lambda}{(\lambda+\phi)}\frac{\Gamma(\phi-r)}{\Gamma(\phi)%
}+\frac{\phi}{(\lambda+\phi)}\frac{\Gamma(\phi+1-r)}{\Gamma(\phi+1)}=\frac{%
\lambda^{r}(\lambda+\phi-r)\Gamma(\phi-r)}{(\lambda+\phi)\Gamma(\phi)}$$
$$\displaystyle=\frac{\lambda^{r}(\phi+\lambda-r)}{(\lambda+\phi)(\phi-1)(\phi-2%
)\cdots(\phi-r)}\,,\quad\mbox{where}\quad\phi>r.$$
∎
Proposition 3.2.
The r-th central moment for the random variable $T$ is given by
$$\displaystyle M_{r}$$
$$\displaystyle=E[T-\mu]^{r}=\sum_{i=0}^{r}\binom{r}{i}(-\mu)^{r-i}E[T^{i}]$$
(4)
$$\displaystyle=\sum_{i=0}^{r}\binom{r}{i}\left(-\frac{\lambda(\phi+\lambda-1)}{%
(\lambda+\phi)(\phi-1)}\right)^{r-i}\left(\frac{\lambda^{i}(\phi+\lambda-i)}{(%
\lambda+\phi)(\phi-1)(\phi-2)\cdots(\phi-i)}\right).$$
Proof.
The result follows directly from the proposition 3.1.∎
Proposition 3.3.
A random variable $T$ with $\operatorname{IWL}$ distribution,
has the mean and variance given by
$$\mu=\frac{\lambda(\phi+\lambda-1)}{(\lambda+\phi)(\phi-1)},$$
$$\sigma^{2}=\frac{\lambda^{2}\left((\phi+\lambda-2)(\phi-1)-{(\phi+\lambda-1)}^%
{2}(\phi-2)\right)}{(\lambda+\phi)(\phi-2){(\phi-1)}^{2}}.$$
Proof.
From (3) and considering $r=1$, it follows that $\mu_{1}=\mu$. The second result follows from (4) considering $r=2$ and with some algebraic operation the proof is completed.
∎
3.2 Survival Properties
Survival analysis has become a popular branch of statistics with wide range of applications. Although many functions related to survival analysis can be derived for this model, in this section we will present the most common functions. The survival function of IWL distribution representing the probability of an observation does not fail until a specified time $t$ is given by
$$S(t|\phi,\lambda)=\frac{\gamma\left(\phi,\lambda t^{-1}\right)(\lambda+\phi)-(%
\lambda t^{-1})^{\phi}e^{-\lambda t^{-1}}}{(\lambda+\phi)\Gamma(\phi)},$$
where $\gamma(y,x)=\int_{0}^{x}{w^{y-1}e^{-w}}dw$ is the lower incomplete gamma function. The hazard function of $T$ is given by
$$h(t|\phi,\lambda)=\frac{\lambda^{\phi+1}t^{-\phi-1}\left(1+t^{-1}\right)e^{-%
\lambda t^{-1}}}{\gamma\left(\phi,\lambda t^{-1}\right)(\lambda+\phi)-(\lambda
t%
^{-1})^{\phi}e^{-\lambda t^{-1}}}\,.$$
(5)
This model has upside-down bathtub hazard rate. The following Lemma is useful to prove such result.
Lemma 3.4.
Glaser [13]: Let T be a non-negative continuous random variable with twice differentiable PDF $f(t)$, hazard rate function $h(t)$ and $\eta(t)=-\frac{\partial}{\partial t}\log f(t)$. Then if $\eta(t)$ has an upside-down bathtub shape, $h(t)$ has an upside-down bathtub shape.
Theorem 3.5.
The hazard function (5) is upside-down bathtub for all $\phi>0$ and $\lambda>0$.
Proof.
For IWL distribution we have
$$\eta(t)=\frac{\phi}{t}+\frac{2}{t}-\frac{1}{(t+1)}-\frac{\lambda}{t^{2}}\,,$$
it follows that
$$\eta^{\prime}(t)=-\frac{\phi}{t^{2}}-\frac{2}{t^{2}}+\frac{1}{{(t+1)}^{2}}+%
\frac{2\lambda}{t^{3}}.$$
The study of the behaviour of $\eta^{\prime}(t)$ is not simple. However using the Wolfram$|$Alpha software, we can check that for all $\phi>0$ and $\lambda>0$, $\eta^{\prime}(t)$ is increasing in $(0,\xi(t|\phi,\lambda))$ and decreasing in $(\xi(t|\phi,\lambda),\infty)$, i.e., $\eta^{\prime}(t)=0$ at $\xi(t|\phi,\lambda)$, where $\xi(t|\phi,\lambda)$ is a very large function computed to the Wolfram$|$Alpha (available upon request). Therefore, $\eta(t)$ and consequently $h(t)$ has upside-down bathtub shape.
∎
This properties make the IWL distribution an useful model for reliability data. Figure 2 gives examples of different shapes for the hazard function.
Proposition 3.6.
The mean residual life function $r(t|\phi,\lambda)$ of the $\operatorname{IWL}$ distribution is given by
$$\displaystyle r(t|\phi,\lambda)$$
$$\displaystyle=\frac{1}{S(t)}\int_{t}^{\infty}yf(y|\lambda,\phi)dy-t=\frac{%
\lambda\gamma\left(\phi,\lambda t^{-1}\right)+\lambda^{2}\gamma\left(\phi,%
\lambda t^{-1}\right)}{\gamma\left(\phi,\lambda t^{-1}\right)(\lambda+\phi)-(%
\lambda t^{-1})^{\phi}e^{-\lambda t^{-1}}}-t.$$
Proof.
Note that, for the Inverse Gamma distribution we have that
$$\int_{t}^{\infty}yf_{j}(y|\phi,\lambda)dy=\frac{\lambda}{\Gamma(\phi+j-1)}%
\gamma\left[\phi+j-2,\lambda t^{-1}\right],\ \ j=1,2.$$
Using the following relationship
$$r(t|\phi,\lambda)=\frac{1}{S(t)}\left[p\int_{t}^{\infty}yf_{1}(y|\lambda,\phi)%
dy+(1-p)\int_{t}^{\infty}yf_{2}(y|\lambda,\phi)dy\right]-t\,,$$
and after some algebraic manipulations, the proof is completed.
∎
3.3 Entropy
In information theory, entropy has played a central role as a measure of the uncertainty associated with a random variable. Shannon’s entropy is one of the most important metrics in information theory. The Shannon’s Entropy from IWL distribution is given by solving the following equation
$$H_{S}(\phi,\lambda)=-\int_{0}^{\infty}\log\left(\frac{\lambda^{\phi+1}}{(\phi+%
\lambda)\Gamma(\phi)}t^{-\phi-1}\left(1+\frac{1}{t}\right)e^{-\lambda t^{-1}}%
\right)f(t|\phi,\lambda)dt.$$
(6)
Proposition 3.7.
A random variable $T$ with $\operatorname{IWL}$ distribution,
has the Shannon’s Entropy given by
$$\displaystyle H_{S}(\phi,\lambda)=$$
$$\displaystyle\log(\lambda+\phi)+\log\Gamma(\phi)+\frac{\phi(\lambda+\phi+1)}{(%
\lambda+\phi)}-(\phi+1)\left(\frac{1}{\lambda+\phi}+\psi(\phi)\right)$$
$$\displaystyle-\frac{\lambda^{\phi+1}\Omega(\phi,\lambda)}{(\lambda+\phi)\Gamma%
(\phi)}.$$
where $\Omega(\phi,\lambda)=\int_{0}^{\infty}(x+1)\log(x+1)x^{(}\phi-1)e^{-\lambda x}dx$.
Proof.
From the equation (6) we have
$$H_{S}(\phi,\lambda)=\ (\phi+1)\log\lambda-\log(\lambda+\phi)-\log\Gamma(\phi)-%
\lambda E\left[t_{i}^{-1}\right]-(\phi+1)E[\log(t_{i})]+E\left[\log(1+t_{i}^{-%
1})\right].$$
Since
$$E[\log(t)]=\log(\lambda)-\frac{1}{\lambda+\phi}-\psi(\phi),\quad\mbox{and }$$
$$E\left[t_{i}^{-1}\right]=\frac{(\phi+1)}{\lambda}-\frac{1}{\lambda+\phi}=\frac%
{\phi(\lambda+\phi+1)}{\lambda(\lambda+\phi)}.$$
$$E\left[\log(1+t_{i}^{-1})\right]=\frac{\lambda^{\phi+1}}{(\lambda+\phi)\Gamma(%
\phi)}\int_{0}^{\infty}(x+1)\log(x+1)x^{(}\phi-1)e^{-\lambda x}dx.$$
Then
$$\displaystyle H_{S}(\phi,\lambda)=$$
$$\displaystyle\ (\phi+1)\left(\frac{1}{\lambda+\phi}+\psi(\phi)\right)-\log(%
\lambda+\phi)-\log\Gamma(\phi)-\frac{\phi(\lambda+\phi+1)}{(\lambda+\phi)}$$
$$\displaystyle-\frac{\lambda^{\phi+1}\Omega(\phi,\lambda)}{(\lambda+\phi)\Gamma%
(\phi)}.$$
∎
4 Inference
In this section, we present the maximum likelihood estimator of the parameters $\phi$ and $\lambda$ of the IWL distribution. Additionally, MLEs considering randomly censored data are also discussed.
4.1 Maximum Likelihood Estimation
Among the statistical inference methods, the maximum likelihood method is widely used due to its better asymptotic properties. Under the maximum likelihood method, the estimators are obtained from maximizing the likelihood function. Let $T_{1},\ldots,T_{n}$ be a random sample such that $T\sim\operatorname{IWL}(\phi,\mu)$. In this case, the likelihood function from (2) is given by
$$L(\boldsymbol{\theta};\boldsymbol{t})=\frac{\lambda^{n(\phi+1)}}{{(\phi+%
\lambda)}^{n}{\Gamma(\phi)}^{n}}\left\{\prod_{i=1}^{n}{t_{i}^{-\phi-1}}\right%
\}\prod_{i=1}^{n}\left(1+\frac{1}{t_{i}}\right)\exp\left\{-\lambda\sum_{i=1}^{%
n}\frac{1}{t_{i}}\right\}.$$
The log-likelihood function $l(\boldsymbol{\theta};\boldsymbol{t})=\log{L(\boldsymbol{\theta};\boldsymbol{t%
})}$ is given by
$$l(\boldsymbol{\theta};\boldsymbol{t})=\ n(\phi+1)\log\lambda-n\log(\lambda+%
\phi)-n\log\Gamma(\phi)-\lambda\sum_{i=1}^{n}\frac{1}{t_{i}}-(\phi+1)\sum_{i=1%
}^{n}\log(t_{i}).$$
(7)
From the expressions $\frac{\partial}{\partial\phi}l(\boldsymbol{\theta};\boldsymbol{t})=0$, $\frac{\partial}{\partial\lambda}l(\boldsymbol{\theta};\boldsymbol{t})=0$, we get the likelihood equations
$$n\log(\lambda)-\sum_{i=1}^{n}\log(t_{i})-\frac{n}{\lambda+\phi}-n\psi(\phi)=0\,,$$
$$\frac{n(\phi+1)}{\lambda}-\sum_{i=1}^{n}\frac{1}{t_{i}}-\frac{n}{\lambda+\phi}%
=0\,,$$
where $\psi(k)=\frac{\partial}{\partial k}\log\Gamma(k)=\frac{\Gamma^{\prime}(k)}{%
\Gamma(k)}$ is the digamma function. After some algebraic manipulation the solution of $\lambda_{MLE}$ is given by
$$\hat{\lambda}_{MLE}=\frac{-\hat{\phi}_{MLE}\left(\xi(\boldsymbol{t})-1\right)+%
\sqrt{\left(\hat{\phi}_{MLE}\left(\xi(\boldsymbol{t})-1\right)\right)^{2}+4\,%
\xi(\boldsymbol{t})\left(\hat{\phi}_{MLE}^{2}+\hat{\phi}_{MLE}\right)}}{2\xi(%
\boldsymbol{t})}\,,$$
where $\xi(\boldsymbol{t})=\sum_{i=1}^{n}(nt_{i})^{-1}$
and $\hat{\phi}_{MLE}$ can be obtained solving the nonlinear system
$$n\log(\hat{\lambda}_{MLE})-\sum_{i=1}^{n}\log(t_{i})-\frac{n}{\hat{\lambda}_{%
MLE}+\hat{\phi}_{MLE}}-n\psi(\hat{\phi}_{MLE})=0.$$
(8)
These results are a simple modification of the results obtained for Ghitany et al. [11] for the WL distribution. Under mild conditions the ML estimates are asymptotically normal distributed with a bivariate normal distribution given by
$$(\hat{\phi},\hat{\lambda})\sim N_{2}[(\phi,\lambda),I^{-1}(\phi,\lambda)]\mbox%
{ for }n\to\infty,$$
where the elements of the Fisher information matrix I$(\phi,\lambda)$ are given by
$$h_{11}(\phi,\lambda)=-\frac{n}{{(\lambda+\phi)}^{2}}+n\psi^{\prime}(\phi)\,,$$
$$h_{12}(\phi,\lambda)=h_{21}(\phi,\lambda)=-\frac{n}{\lambda}-\frac{n}{{(%
\lambda+\phi)}^{2}}\,,$$
$$h_{22}(\phi,\lambda)=\frac{n(\phi+1)}{\lambda^{2}}-\frac{n}{{(\lambda+\phi)}^{%
2}}\,,$$
and $\psi^{\prime}(k)=\frac{\partial}{\partial^{2}k}\log\Gamma(k)$ is the trigamma function. An interesting property of the IWL distribution is that the observed matrix information is equal to the expected information matrix.
4.2 Random Censoring
In survival analysis and industrial lifetime testing, random censoring schemes have been received special attention. Suppose that the $i$th individual has a lifetime $T_{i}$ and a censoring time $C_{i}$, moreover the random censoring times $C_{i}$s are independent of $T_{i}$s and their distribution does not depend on the parameters, then the data set is $(t_{i},\delta_{i})$, where $t_{i}=\min(T_{i},C_{i})$ and $\delta_{i}=I(T_{i}\leq C_{i})$. This type of censoring have as special case the type I and II censoring mechanism. The likelihood function for $\boldsymbol{\theta}$ is given by
$$L(\boldsymbol{\theta,t})=\prod_{i=1}^{n}f(t_{i}|\boldsymbol{\theta})^{\delta_{%
i}}S(t_{i}|\boldsymbol{\theta})^{1-\delta_{i}}.$$
Let $T_{1},\cdots,T_{n}$ be a random sample of IWL distribution, the likelihood function considering data with random censoring is given by
$$\displaystyle L(\lambda,\phi|\boldsymbol{t})=$$
$$\displaystyle\frac{\lambda^{d(\phi+1)}}{(\lambda+\phi)^{n}\Gamma(\phi)^{n}}%
\prod_{i=1}^{n}\left((\lambda+\phi)\gamma(\phi,\lambda t_{i}^{-1})-\left(%
\lambda t_{i}^{-1}\right)^{\phi}e^{-\lambda t_{i}^{-1}}\right)^{1-\delta_{i}}$$
(9)
$$\displaystyle\times\left(t_{i}^{-\phi-1}(1+t_{i}^{-1})e^{-\lambda t_{i}^{-1}}%
\right)^{\delta_{i}}.$$
The logarithm of the likelihood function (9) is given by
$$\displaystyle l(\lambda,\phi|\boldsymbol{t})=$$
$$\displaystyle\,-(\phi+1)\sum_{i=1}^{n}\delta_{i}\log(t_{i})-\lambda\sum_{i=1}^%
{n}\delta_{i}t_{i}^{-1}+d(\phi+1)\log(\lambda)-n\log(\phi+\lambda)$$
$$\displaystyle+\sum_{i=1}^{n}(1-\delta_{i})\log\left((\lambda+\phi)\gamma(\phi,%
\lambda t_{i}^{-1})-{(\lambda t_{i}^{-1})}^{\phi}e^{-\lambda t_{i}^{-1}}\right%
)-n\log\left(\Gamma(\phi)\right)$$
$$\displaystyle+\sum_{i=1}^{n}\delta_{i}\log(1+t_{i}^{-1}).$$
From ${\partial}l(\lambda,\phi|\boldsymbol{t})/{\partial\lambda}=0$ and ${\partial}l(\lambda,\phi|\boldsymbol{t})/{\partial\phi}=0$, the likelihood equations are given as follows
$$\sum_{i=1}^{n}\frac{(1-\delta_{i})\left(\gamma(\phi,\lambda t_{i}^{-1})+(%
\lambda+\phi)\left(\lambda t_{i}^{-1}\right)^{\phi-1}e^{-\lambda t_{i}^{-1}}-%
\phi\lambda^{\phi-1}t_{i}^{-\phi}e^{-\lambda t_{i}^{-1}}-\left(\lambda t_{i}^{%
-1}\right)^{\phi+1}e^{-\lambda t_{i}^{-1}}\right)}{\left((\lambda+\phi)\gamma(%
\phi,\lambda t_{i}^{-1})\right)-\left(\lambda t_{i}^{-1}\right)^{\phi}e^{-%
\lambda t_{i}^{-1}}}=$$
$$\frac{n}{\lambda+\phi}-\frac{d(\phi+1)}{\lambda}+\sum_{i=1}^{n}\delta_{i}t_{i}%
^{-1},$$
(10)
$$\sum_{i=1}^{n}\frac{(1-\delta_{i})\left(\gamma(\phi,\lambda t_{i}^{-1})+(%
\lambda+\phi)\Psi(\phi,\lambda t_{i}^{-1})-\left(\lambda t_{i}^{-1}\right)^{%
\phi}\log(\lambda t_{i}^{-1})e^{-\lambda t_{i}^{-1}}\right)}{\left((\lambda+%
\phi)\gamma(\phi,\lambda t_{i}^{-1})\right)-\left(\lambda t_{i}^{-1}\right)^{%
\phi}e^{-\lambda t_{i}^{-1}}}=-d\log(\lambda)$$
$$+\frac{n}{\lambda+\phi}+n\psi(\phi)+\sum_{i=1}^{n}\delta_{i}\log(t_{i}^{-1})\,,$$
(11)
where $\Psi(k,x)={\partial}\,\gamma(k,x)/{\partial k}$ can be computed numerically. Numerical methods are required in order to find the solution of these non-linear equations.
5 Bias correction for the maximum likelihood estimators
In this section, we discuss modified MLEs based on two corrective approaches that are bias-free to second order. Firstly a corrective analytical approach is presented than the bootstrap resampling method is presented.
5.1 A corrective approach
Consider the likelihood function $L(\boldsymbol{\theta};\boldsymbol{t})$ with a $p$-dimensional vector of parameters $\boldsymbol{\theta}$. Thus, the joint cumulants of the derivatives of $l(\boldsymbol{\theta};\boldsymbol{t})$ can be written by
$$\displaystyle h_{ij}(\boldsymbol{\theta})$$
$$\displaystyle=E\left(\frac{\partial^{2}l(\boldsymbol{\theta};\boldsymbol{t})}{%
\partial\theta_{i}\theta_{j}}\right),\quad h_{ijl}(\boldsymbol{\theta})=E\left%
(\frac{\partial^{3}l(\boldsymbol{\theta};\boldsymbol{t})}{\partial\theta_{i}%
\partial\theta_{j}\partial\theta_{l}}\right)\mbox{and}$$
$$\displaystyle h_{ij,l}(\boldsymbol{\theta})$$
$$\displaystyle=E\left(\frac{\partial^{2}l(\boldsymbol{\theta};\boldsymbol{t})}{%
\partial\theta_{i}\partial\theta_{j}}.\frac{\partial l(\boldsymbol{\theta};%
\boldsymbol{t})}{\partial\theta_{l}}\right),\quad\mbox{for}\quad i,j,l=1,%
\ldots,p.$$
Consequently, the derivatives of such cumulants are given by
$$\displaystyle h_{ij}^{(l)}(\boldsymbol{\theta})$$
$$\displaystyle=\dfrac{\partial h_{ij}(\boldsymbol{\theta})}{\partial\theta_{l}}%
,\quad\mbox{for}\quad i,j,l=1,\ldots,p.$$
The bias of $\theta_{m}$ studied by Cox and Snell [7] for independent sample without necessarily be identically distributed can be written by
$$Bias(\hat{\theta}_{m})=\sum_{i=1}^{p}\sum_{j=1}^{p}\sum_{k=1}^{p}s_{mi}(%
\boldsymbol{\theta})s_{jl}(\boldsymbol{\theta})\left(h_{ij,l}(\boldsymbol{%
\theta})+0.5h_{ijl}(\boldsymbol{\theta})\right)+O(n^{-2})\,,$$
(12)
where $s^{ij}$ is the $(i,j)$-th element of the inverse of Fisher’s information matrix of $\boldsymbol{\hat{\theta}}$, $K=\{-h_{ij}\}$.
Cordeiro and Klein [6] proved that even if the data are dependent the expression (12) can be re-written as
$$Bias(\hat{\theta}_{m})=\sum_{i=1}^{p}s_{mi}(\boldsymbol{\theta})\sum_{j=1}^{p}%
\sum_{k=1}^{p}s_{jl}(\boldsymbol{\theta})\left(h_{ij}^{(l)}(\boldsymbol{\theta%
})-0.5h_{ijl}(\boldsymbol{\theta})\right)+O(n^{-2}).$$
(13)
Let $a_{ij}^{l}=h_{ij}^{(l)}-\frac{1}{2}h_{ij}^{(l)}$ and define the matrix $A=[A^{(1)}|A^{(2)}|\ldots|A^{(p)}]$ with $A^{(l)}=\{a_{ij}^{(l)}\}$, for $i,j,l=1,\ldots,p$. Thus, the expression for the bias of $\boldsymbol{\hat{\theta}}$ can be expressed as
$$Bias(\hat{\theta}_{m})=K^{-1}A.\mbox{vec}{(K^{-1})}+O(n^{-2}).$$
(14)
A bias corrected MLE for $\hat{\boldsymbol{\theta}}$ is obtained as
$$\hat{\boldsymbol{\theta}}_{CMLE}=\hat{\boldsymbol{\theta}}-K^{-1}A.\mbox{vec}{%
(K^{-1})}\,,$$
(15)
where $\hat{\boldsymbol{\theta}}$ is the MLE of the parameter $\boldsymbol{\theta}$, $\hat{K}=K|_{\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}}$ and $\hat{A}=A|_{\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}}$. The bias of $\hat{\boldsymbol{\theta}}_{CMLE}$ is unbiased $O(n^{-2})$.
For the IWL distribution the higher-order derivatives can be easily obtained since they do not involve $\boldsymbol{t}$, thus, we have
$$\displaystyle h_{111}(\boldsymbol{\theta})$$
$$\displaystyle=h_{11}^{(1)}(\boldsymbol{\theta})=-\frac{2n}{{(\lambda+\phi)}^{3%
}}-n\psi^{\prime\prime}(\phi)\,,$$
$$\displaystyle h_{122}(\boldsymbol{\theta})$$
$$\displaystyle=h_{221}(\boldsymbol{\theta})=h_{212}(\boldsymbol{\theta})=h_{12}%
^{(2)}(\boldsymbol{\theta})=h_{22}^{(1)}(\boldsymbol{\theta})=-\frac{2n}{{(%
\lambda+\phi)}^{3}}-\frac{n}{\lambda^{2}},$$
$$\displaystyle h_{222}(\boldsymbol{\theta})$$
$$\displaystyle=h_{22}^{(2)}(\boldsymbol{\theta})=-\frac{2n}{{(\lambda+\phi)}^{3%
}}-\frac{2n(\phi+1)}{\lambda^{3}}\ \ \mbox{ and }$$
$$\displaystyle h_{211}(\boldsymbol{\theta})$$
$$\displaystyle=h_{112}(\boldsymbol{\theta})=h_{121}(\boldsymbol{\theta})=h_{12}%
^{(1)}(\boldsymbol{\theta})=h_{11}^{(2)}(\boldsymbol{\theta})=-\frac{2n}{{(%
\lambda+\phi)}^{3}},$$
where $\psi^{\prime\prime}(k)=\frac{\partial}{\partial^{3}k}\log\Gamma(k)$. The matrix $K$ is given by
$$K=\begin{bmatrix}\frac{n}{(\lambda+\phi)^{2}}-n\psi^{\prime}(\phi)&\frac{n}{%
\lambda}+\frac{n}{(\lambda+\phi)^{2}}\\
\frac{n}{\lambda}+\frac{n}{(\lambda+\phi)^{2}}&-\frac{n(\phi+1)}{\lambda^{2}}+%
\frac{n}{(\lambda+\phi)^{2}}\end{bmatrix}.$$
To obtain the matrix $A$ of (14), we present the elements of $A^{(1)}$
$$\displaystyle a_{11}^{(1)}$$
$$\displaystyle=h_{11}^{(1)}-\frac{1}{2}h_{111}=-\frac{n}{(\lambda+\phi)^{3}}-%
\frac{n\psi^{\prime\prime}(\phi)}{2}\,,$$
$$\displaystyle a_{12}^{(1)}$$
$$\displaystyle=a_{21}^{(1)}=h_{12}^{(1)}-\frac{1}{2}h_{112}=-\frac{n}{(\lambda+%
\phi)^{3}}\,,$$
$$\displaystyle a_{22}^{(1)}$$
$$\displaystyle=h_{22}^{(1)}-\frac{1}{2}h_{221}=-\frac{n}{(\lambda+\phi)^{3}}-%
\frac{n}{2\lambda^{2}}\,,$$
and the elements of $A^{(2)}$ are
$$\displaystyle a_{11}^{(2)}$$
$$\displaystyle=h_{11}^{(2)}-\frac{1}{2}h_{112}=-\frac{n}{(\lambda+\phi)^{3}}\,,$$
$$\displaystyle a_{12}^{(2)}$$
$$\displaystyle=a_{21}^{(2)}=h_{12}^{(2)}-\frac{1}{2}h_{122}=-\frac{n}{(\lambda+%
\phi)^{3}}-\frac{n}{2\lambda^{2}}\,,$$
$$\displaystyle a_{22}^{(2)}$$
$$\displaystyle=h_{22}^{(2)}-\frac{1}{2}h_{222}=-\frac{n}{(\lambda+\phi)^{3}}-%
\frac{n(\phi+1)}{\lambda^{3}}.$$
Thus, the matrix $A=\ [A^{(1)}|A^{(2)}]$ is expressed by
$$\displaystyle A=\ n\begin{pmatrix}-\frac{1}{(\lambda+\phi)^{3}}-\frac{\psi^{%
\prime\prime}(\phi)}{2}&-\frac{1}{(\lambda+\phi)^{3}}&-\frac{1}{(\lambda+\phi)%
^{3}}&-\frac{1}{(\lambda+\phi)^{3}}-\frac{1}{2\lambda^{2}}\\
-\frac{1}{(\lambda+\phi)^{3}}&-\frac{1}{(\lambda+\phi)^{3}}-\frac{1}{2\lambda^%
{2}}&-\frac{1}{(\lambda+\phi)^{3}}-\frac{1}{2\lambda^{2}}&-\frac{1}{(\lambda+%
\phi)^{3}}-\frac{(\phi+1)}{\lambda^{3}}\end{pmatrix}.$$
Finally, the bias-corrected maximum likelihood estimators are given by
$$\begin{pmatrix}\hat{\phi}_{CMLE}\\
\hat{\lambda}_{CMLE}\end{pmatrix}=\begin{pmatrix}\hat{\phi}_{MLE}\\
\hat{\lambda}_{MLE}\end{pmatrix}-\hat{K}^{-1}\hat{A}.vec(\hat{K}^{-1})\,,$$
(16)
where $\hat{K}=K|_{\phi=\hat{\phi},\lambda=\hat{\lambda}}$ and $\hat{A}=A|_{\phi=\hat{\phi},\lambda=\hat{\lambda}}$. It is important to point out that, since the higher-order do not involve $\boldsymbol{t}$, they are the same of the WL distribution [25].
A bias corrected approach can be considered for censored data. Although the Fisher information matrix related to the MLEs (9) does not present closed-form expressions, we can consider the bias corrected presented in (5.1). In this case, approximated bias-corrected maximum likelihood estimates (ACMLE) are archived by
$$\begin{pmatrix}\hat{\phi}_{ACMLE}\\
\hat{\lambda}_{ACMLE}\end{pmatrix}=\begin{pmatrix}\hat{\phi}_{MLE}\\
\hat{\lambda}_{MLE}\end{pmatrix}-\hat{K}^{-1}\hat{A}.vec(\hat{K}^{-1})\,,$$
where $\hat{K}=K|_{\phi=\hat{\phi}_{MLE},\lambda=\hat{\lambda}_{MLE}}$, $\hat{A}=A|_{\phi=\hat{\phi}_{MLE},\lambda=\hat{\lambda}_{MLE}}$ and $\hat{\phi}_{MLE}$ and $\hat{\lambda}_{MLE}$ are the solutions of $(\ref{verowl21})$ and $(\ref{verowl22})$. However, the bias of $\hat{\theta}_{ACMLE}$ is not an unbiased estimator with $O(n^{-2})$.
5.2 Bootstrap resampling method
In what follows we consider the bootstrap resampling method proposed by Efron [8] to reduce the bias of the MLEs. Such method consists in generating pseudo-samples from the original sample to estimate the bias of the MLEs. Thus, the bias-corrected MLEs is given by subtraction of the estimated bias with the original MLEs.
Let $\mathbf{y}=(y_{1},\ldots,y_{n})^{\top}$ be a sample with $n$ observations randomly selected from the random variable $Y$ in which has the distribution function $F=F_{\nu}(y)$. Thus, let the parameter $\nu$ be a function of $F$ given by $\nu=t(F)$. Finally, let $\hat{\nu}$ be an estimator of $\nu$ based on $\mathbf{y}$, i.e., $\hat{\nu}=s(\mathbf{y})$. The pseudo-samples $\mathbf{y^{*}}=(y^{*}_{1},\ldots,y^{*}_{n})^{\top}$ is obtained from the original sample $\mathbf{y}$ through resampling with replacement. The bootstrap replicates of $\hat{\nu}$ is calculated, where $\hat{\nu}^{*}=s(\mathbf{y}^{*})$ and the empirical cdf (ecdf) of $\hat{\nu}^{*}$ is used to estimate $F_{\hat{\nu}}$ (cdf of $\hat{\nu}$). Let $B_{F}(\hat{\nu},\nu)$ be the bias of the estimator $\hat{\nu}=s(\mathbf{y})$ given by
$$B_{F}(\hat{\nu},\nu)=E_{F}[\hat{\nu},\nu]=E_{F}[s(\mathbf{y})]-\nu(F).$$
Note that the subscript of the expectation $F$ indicates that is taken with respect to $F$. The bootstrap estimators of the bias were obtained by replacing $F$ with $F_{\hat{\nu}}$, where $F$ generated the original sample. Therefore, the bootstrap bias estimate is given by
$$\hat{B}_{F_{\hat{\nu}}}(\hat{\nu},\nu)=E_{F_{\hat{\nu}}}[\hat{\nu^{*}}]-\hat{%
\nu}.$$
If we have $B$ bootstrap samples $(\mathbf{y}^{*(1)},\mathbf{y}^{*(2)},\ldots,\mathbf{y}^{*(B)})$ which are generated independently from the original sample $\mathbf{y}$ and the respective bootstrap estimates $(\hat{\nu}^{*(1)},\hat{\nu}^{*(2)},\ldots,\hat{\nu}^{*(B)})$ are calculated, then it is achievable to determine the bootstrap expectations $E_{F_{\hat{\nu}}}[\hat{\nu^{*}}]$ approximately by
$$\hat{\nu}^{*(.)}=\frac{1}{B}\sum_{i=1}^{B}\hat{\nu}^{*(i)}.$$
Therefore, the bootstrap bias estimate based on $B$ replications of $\hat{\nu}$ is $\hat{B}_{F}(\hat{\nu},\nu)=\hat{\nu}^{*(.)}-\hat{\nu}$, which results in the bias corrected estimators obtained through by bootstrap resampling method that is given by
$$\nu^{B}=\hat{\nu}-\hat{B}_{F}(\hat{\nu},\nu)=2\hat{\nu}-\hat{\nu}^{*(.)}.$$
In our case, we have $\nu^{B}$ denoted by $\hat{\theta}_{BOOT}=(\hat{\phi}_{BOOT},\hat{\lambda}_{BOOT})^{\top}$.
6 Simulation Analysis
In this section a simulation study is presented to compare the efficiency of the maximum likelihood method and the bias correction approaches in the presence of complete and censored data. The proposed comparisons are performed by computing the mean relative errors (MRE) and the relative mean square errors (RMSE) given by
$$\operatorname{MRE}_{i}=\frac{1}{N}\sum_{j=1}^{N}\frac{\hat{\theta}_{i,j}}{%
\theta_{i}}\ ,\quad\operatorname{RMSE}_{i}=\frac{1}{N}\sum_{j=1}^{N}\frac{(%
\hat{\theta}_{i,j}-\theta_{i})^{2}}{\theta_{i}^{2}},\quad\operatorname{for}\ %
\ i=1,2,$$
where $N$ is the number of estimates obtained through the MLE, CMLE and the bootstrap approach. The $95\%$ coverage probability of the asymptotic confidence intervals are also evaluated. Considering this approach, we expected that the most efficient estimation method returns the MREs closer to one with smaller RMSEs. Moreover, for a large number of experiments, using a $95\%$ confidence level, the frequencies of intervals that covered the true values of $\boldsymbol{\theta}$ should be closer to $95\%$. Following Reath et al. [21] we used B=1,000 for the bootstrap method. The programs can be obtained, upon request. The random sample of the IWL were generated considering the following algorithm:
1.
Generate $U_{i}\sim\operatorname{Uniform}(0,1),i=1,\ldots,n$;
2.
Generate $X_{i}\sim\operatorname{IG}(\phi,\lambda),i=1,\ldots,n$;
3.
Generate $Y_{i}\sim\operatorname{IG}(\phi+1,\lambda),i=1,\ldots,n$;
4.
If $U_{i}\leq p=\lambda/(\lambda+\phi)$, then set $T_{i}=X_{i}$, otherwise, set $T_{i}=Y_{i},i=1,\ldots,n$.
6.1 Complete Data
The simulation study is performed considering the values: $\boldsymbol{\theta}=((0.5,2)$,$(2,4))$, $N=30,000$ and $n=(20,25,\ldots$, $130)$. It is important to point out that, similar results were achieved for different choices of $\phi$ and $\lambda$. The uniroot procedure available in R is considered to find the solution of the non-linear equation (8). The bias correction is computed directly from (16). Figures 3 and 4 present the MRE, RMSE and the coverage probability with a $95\%$ confidence level related to the MLE, CMLE and the bootstrap under different values of $n$.
From Figures 3 and 4, we observed that the estimates of $\phi$ and $\lambda$ are asymptotically unbiased, i.e., the MREs tend to one when $n$ increases and the RMSEs decrease to zero for $n$ large. The CMLE present superior performance than the bootstrap approach for both parameters for any sample sizes. Taking into account the results of the simulation studies, the maximum likelihood estimators combined with the corrective bias approach discussed in Section 5.1 should be considered for estimating the parameters of the IWL distribution.
6.2 Censored Data
In this section, we considered the MLES in the presence of random censored data. The censored data is generated following the same procedure presented by Goodman et al. [14]. In our case, we presented two scenarios where we obtained approximately $0.3$ and $0.5$ proportions of censored data, i.e., $30\%$ and $50\%$ of censorship. The simulation study is performed considering $\boldsymbol{\theta}=(2,4)$, $N=2,000$ and $n=(10,15,\ldots$, $130)$ The maximum likelihood estimates were computed using the log-likelihood functions (4.2) with the maxLik package available in R. The solution for the maximum was unique for all initial values.
Figures 5 and 6 present the MRE, the RMSE and the coverage probability with a $95\%$ confidence level related to the MLE, CMLE and the bootstrap under different values of $n$.
As shown in Figures 5 and 6 the proposed ACMLE returned more accurate estimates for both parameters when compared with the bootstrap approach or the MLEs. Taking into account the results of the simulation studies, the approximated corrected bias approach combined with the maximum likelihood estimators should be consider for estimating the parameters of the IWL distribution in the presence of censorship.
7 Application
In this section, recall the real data set briefly presented in Section 1. The analyze of the distribution that better fit the proposed data is relevant to avoid higher costs for the company. Table 1 presents the data related to failure time of (in days) of $194$ devices in an aircraft (+ indicates the presence of censorship).
The results obtained from the IWL distribution were compared to the Weibull, Gamma, Lognormal, Logistic, Inverse Weibull and Inverse Lindley distribution and the nonparametric survival curve adjusted using the Kaplan-Meier estimator. Initially, in order to verify the behavior of the empirical hazard function it will be considered the TTT-plot (total time on test) proposed by Barlow and Campo [4]. The TTT-plot is achieved through the consecutive plot of the values $[r/n,G(r/n)]$ where
$G(r/n)=\left(\sum_{i=1}^{r}t_{i}+(n-r)t_{(r)}\right)/{\sum_{i=1}^{n}t_{i}},%
\quad r=1,\ldots,n,\ i=1,\ldots,n,$
and $t_{i}$ is the order statistics. If the curve is concave (convex), the hazard function is increasing (decreasing), when it starts convex and then concave (concave and then convex) the hazard function will have a bathtub (inverse bathtub) shape.
Different discrimination criterion methods based on log-likelihood function evaluated at the MLEs were also considered. The discrimination criterion methods are respectively: Akaike information criterion (AIC) computed through $\operatorname{AIC}=-2l(\boldsymbol{\hat{\theta}};\boldsymbol{x})+2k$, Corrected Akaike information
criterion $\operatorname{AICC}=\operatorname{AIC}+[{2\,k\,(k+1)}/{(n-k-1)}]$, Hannan-Quinn information
criterion $\operatorname{HQIC}=-2\,l(\boldsymbol{\hat{\theta}};\boldsymbol{x})+2\,k\,\log%
\left(\log(n)\right)$ and the consistent Akaike information criterion $\operatorname{CAIC=AIC}+k\log(n)-k$, where $k$ is the number of
parameters to be fitted and $\boldsymbol{\hat{\theta}}$ the estimates of $\boldsymbol{\theta}$. The best model is the one which provides the minimum values
of those criteria.
Since the data has random censoring mechanism, consequently the equations (10) and (11) were used to compute the MLEs. Table 2 displays the MLEs, standard-error and $95\%$ confidence intervals for $\phi$ and $\lambda$. Table 3 presents the results of AIC, AICC, HQIC, CAIC criteria, for different probability distributions.
Figure 7 presents the TTT-plot, the survival function adjusted by different distributions and the Kaplan-Meier estimator and the hazard function adjusted by the IWL distribution.
Comparing the empirical survival function with the adjusted models we observed a goodness of the fit for the inverse weighted Lindley distribution. This result is also confirmed by the different discrimination criterion methods considered since IWL distribution has the minimum value. Based on the TTT-plot there is an indication that the hazard function has upside-down bathtub failure rate this result is confirmed by the adjusted hazard function. Therefore, from the proposed methodology the data related to the failure time of $194$ devices in an aircraft can be described by the inverse weighted Lindley distribution.
8 Concluding Remarks
In this paper, a new distribution called inverse weighted Lindley is proposed and its mathematical properties were studied in detail. The maximum likelihood estimators of the parameters and their asymptotic properties were obtained, we also presented two corrective approaches to derive a modified MLEs that are bias-free to second order, as well as the MLEs in the presence of randomly censored data. The simulation study showed that the CMLE and ACLME present extremely efficient estimators for both parameters for any sample sizes. The practical importance of the IWL distribution was reported in a real application, in which our new distribution returned better fitting in comparison with other well-known distributions.
Acknowledgements
The authors are very grateful to the reviewers for their helpful and useful comments that improved the manuscript. The authors’ researchers are partially supported by the Brazilian Institutions: CNPq, CAPES and FAPESP.
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The ISM Properties and Gas Kinematics of a Redshift 3 Massive Dusty Star-forming Galaxy
T. K. Daisy Leung11affiliation: Department of Astronomy, Space Sciences Building, Cornell University, Ithaca, NY 14853, USA 2 2affiliationmark:
Dominik A. Riechers11affiliation: Department of Astronomy, Space Sciences Building, Cornell University, Ithaca, NY 14853, USA
Andrew J. Baker33affiliation: Department of Physics and Astronomy, Rutgers, the State University of New Jersey,
136 Frelinghuysen Road, Piscataway, NJ, 08854-8019
Dave L. Clements44affiliation: Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK
Asantha Cooray55affiliation: Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA
Christopher C. Hayward22affiliation: Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
R. J. Ivison 66affiliation: European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany 7 7affiliationmark:
Roberto Neri88affiliation: Institut de Radioastronomie Millimétrique (IRAM), 300 Rue de la Piscine, Domaine Universitaire de Grenoble,
38406 St. Martin d’Hères, France
Alain Omont99affiliation: Institut d’Astrophysique de Paris, Centre national de la recherche scientifique (CNRS) & Université Pierre et Marie Curie (UPMC),
98 bis boulevard Arago, 75014 Paris, France
Ismael Pérez-Fournon1010affiliation: Instituto de Astrofisica de Canarias, E-38200 La Laguna, Tenerife, Spain 11 11affiliationmark:
Douglas Scott1212affiliation: Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
Julie L. Wardlow1313affiliation: Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK 14 14affiliationmark:
[email protected]
Abstract
We present CO ($J$ = 1$\rightarrow$0; 3$\rightarrow$2; 5$\rightarrow$4; 10$\rightarrow$9) and 1.2-kpc resolution [CII] line observations of the dusty star-forming galaxy (SFG) HXMM05 —
carried out with the Karl G. Jansky Very Large Array, the Combined Array for Research in Millimeter-wave Astronomy,
the Plateau de Bure Interferometer, and the Atacama Large Millimeter/submillimeter Array, measuring an unambiguous redshift of $z$ = 2.9850 $\pm$ 0.0009.
We find that HXMM05 is a hyper-luminous infrared galaxy ($L_{\rm IR}$ $=$ (4 $\pm$ 1)$\times 10^{13}$ $L_{\odot}$)
with a total molecular gas mass of (2.1 $\pm$ 0.7)$\times 10^{11}$($\alpha_{\rm CO}$/0.8) $M_{\odot}$.
The CO ($J$ = 1 $\rightarrow$ 0) and [CII] emission are extended over $\sim$9 kpc in diameter, and the CO line FWHM exceeds 1100 km s${}^{-1}$.
The [CII] emission shows a monotonic velocity gradient consistent with a disk,
with a maximum rotation velocity of $v_{\rm c}$ = 616 $\pm$ 100 km s${}^{-1}$ and a dynamical
mass of (7.7 $\pm$ 3.1)$\times 10^{11}$ $M_{\odot}$.
We find a star formation rate (SFR) of 2900${}^{+750}_{-595}$ $M_{\odot}$ yr${}^{-1}$.
HXMM05 is thus among the most intensely star-forming galaxies known at high redshift.
Photo-dissociation region modeling suggests physical conditions similar to nearby SFGs, showing
extended star formation, which is consistent with our finding that the gas and dust emission are co-spatial.
Its molecular gas excitation resembles the local major merger Arp 220.
The broad CO and [CII] lines and a pair of compact dust nuclei
suggest the presence of a late-stage major merger at the center of the extended disk, again reminiscent of Arp 220.
The observed gas kinematics and conditions together with the presence of a companion and the pair of nuclei
suggest that HXMM05 is experiencing multiple mergers as a part of the evolution.
Subject headings:infrared: galaxies –
galaxies: high-redshift –
galaxies: ISM –
galaxies: evolution –
galaxies: starburst –
radio lines: ISM
††slugcomment: Accepted to the ApJ77affiliationtext: Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK1111affiliationtext: Departamento de Astrofisica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, Spain1414affiliationtext: Physics Department, Lancaster University, Lancaster, LA1 4YB, UK
1. Introduction
Most of the stellar mass in the Universe is assembled in the first few billion years of cosmic time, in the redshift range
1$\lesssim$ $z$ $\lesssim$3
(see e.g., review by Madau & Dickinson, 2014).
Galaxies at this epoch typically have higher star formation rates (SFRs) compared to the present day.
Among the high-$z$ galaxy populations discovered, dusty star-forming galaxies (DSFGs) represent
the most IR-luminous
systems at this peak epoch.
They are typically gas-rich, with molecular gas masses exceeding $M_{\rm gas}$ $=$10${}^{10}$ $M_{\odot}$ and IR luminosities exceeding those of nearby (ultra-)luminous infrared galaxies
(U/LIRG; $L_{\rm IR}$ $>$10${}^{11-13}$ $L_{\odot}$; see reviews by Carilli & Walter 2013; Casey et al. 2014).
Given the differences found between nearby ULIRGs and high-$z$ DSFGs (e.g., Younger et al., 2010; Rujopakarn et al., 2011, 2013), studying their interstellar medium (ISM) properties, gas dynamics, and star-forming environments directly are
essential to understanding how galaxies are initially assembled at early epochs.
In the classical model of disk galaxy formation (Fall & Efstathiou, 1980),
disk galaxies form out of the gas that is cooling off from the hot halos associated with
dark matter (DM) potential wells while maintaining the specific angular momentum as the gas
settles into rotationally supported disks (Mo et al., 1998).
The structure and dynamics of disk galaxies are therefore closely correlated
with the properties of their parent DM halos. Probing the structure and dynamics of disk galaxies at high redshift
can thus inform us about the processes driving the assembly history of galaxies at early cosmic times.
For instance, by tracing the gas dynamics, the Tully-Fisher relation (Tully & Fisher, 1977),
which links the angular momentum of the parent DM halo of a disk
galaxy with the luminosity/mass of its stellar populations, can be studied out to earlier epochs.
Past observations have led to two physical pictures for the nature and origin of DSFGs: compact
irregular starbursts resulting from major mergers (of two or more disks) and extended disk-like galaxies with high SFRs
(e.g., Tacconi et al., 2006, 2008; Shapiro et al., 2008; Engel et al., 2010; Riechers et al., 2010; Ivison et al., 2010a, 2011; Riechers et al., 2011a, c; Hodge et al., 2012; Riechers et al., 2013; Ivison et al., 2013; Bothwell et al., 2013; Riechers et al., 2014b; Hodge et al., 2015; Oteo et al., 2016a; Riechers et al., 2017)
resulting from minor mergers and/or cold gas accreted
from the intergalactic medium (IGM; also known as cold mode accretion; CMA; e.g., Kereš et al. 2005; Dekel et al. 2009a; Davé et al. 2010).
However, as individual DSFGs can fall into either physical picture, a third interpretation is
that DSFGs are a heterogeneous population composed of both compact
starbursts and extended disks (e.g., Hayward et al. 2013), presumably observed at different stages of evolution.
Determining their gas kinematics is therefore key to better understanding their formation mechanisms and shedding light on
whether major mergers or continuous accretion dominate and sustain their intense star formation.
However, such studies require high spatial resolution and sensitivity in order to
image their gas reservoirs, and thus,
are relatively expensive to carry out.
To date, only a handful of high-$z$ galaxies have been mapped in their molecular gas at high resolution,
revealing a mixture of rotating disks and galaxy mergers (e.g., Swinbank et al., 2011; Hodge et al., 2012; Ivison et al., 2013; Oteo et al., 2016b, 2017a, 2018).
With the goal to better understand the star-forming conditions and the gas dynamics of high-$z$ DSFGs,
we observed multi-$J$ CO and
[CII] line emission in HerMES J022547-041750 (HXMM05; RA, Dec = 02${}^{\rm h}$25${}^{\rm m}$47${}^{\rm s}$, $-$04°17′50″; J2000),
one of the brightest DSFGs known, at $\lesssim$ 0$\farcs$15 resolution.
Line emission from different rotational transitions of CO is useful for determining molecular gas mass and physical properties
of the ISM. The [CII] (${}^{2}$P${}_{3/2}$ $\rightarrow$ ${}^{2}$P${}_{1/2}$) fine-structure line at rest-frame157.7 $\micron$ is one of the brightest emission lines in star-forming galaxies, and can contribute up to
1% of the FIR luminosity of galaxies (Malhotra et al., 1997; Nikola et al., 1998; Colbert et al., 1999).
In addition, [CII] and CO ($J$ = 1 $\rightarrow$ 0) line emission trace similar gas kinematics in nearby star-forming galaxies
(e.g., Mittal et al. 2011; Braine et al. 2012; Kramer et al. 2013; Pineda et al. 2013),
making the former a powerful probe of high-$z$ gas kinematics,
especially when paired with the exceptional capabilities of the Atacama Large Millimeter/submillimeter Array (ALMA).
The target HXMM05 was discovered in the
Herschel Multi-tiered Extragalactic Survey (HerMES; Oliver et al. 2012)
as one of 29 high-$z$ strongly-lensed galaxy candidates identified (Wardlow et al., 2013; Bussmann et al., 2015).
The parent sample was selected based on a flux density threshold of
$S_{\rm 500}\geq$80 mJy at 500 $\micron$.
The surface density of such bright DSFGs is (0.31 $\pm$ 0.06) deg${}^{-2}$ (Wardlow et al. 2013).
Previous high-resolution imaging obtained with the Hubble Space Telescope (HST) and ALMA
and lens modeling of $0\farcs 4$ resolution dust continuum data at 870 $\micron$
show that HXMM05 is at most weakly lensed, with magnification factor
$\mu_{870}\lesssim 1.4$
(Bussmann et al., 2015)111The orientation of the HST image of HXMM05 shown in Figure 3 of Calanog et al. (2014); Bussmann et al. (2015)
is incorrect (i.e., North is down instead of up), but the correct locations of all galaxies were used in the lens modeling..
HXMM05 is therefore intrinsically extremely IR-luminous, unlike other typically strongly-lensed DFSGs in the parent sample with
similar sub-millimeter flux densities.
Bussmann et al. (2015) find a total of three unlensed, intrinsically-bright DSFGs out of the parent sample of 29.
This yields a surface density of $\sim$0.03 deg${}^{-2}$ for such sources, which makes them even rarer than
strongly-lensed DSFGs.
HXMM05 therefore belongs to a rare and understudied luminous/massive high-$z$ galaxy population.
Currently, the general consensus is these unlensed DSFG with $S_{\rm 500}\gtrsim$ 100 mJy appear to be
predominantly
major galaxy mergers (e.g., HXMM01 and G09v124; Fu et al. 2013; Ivison et al. 2013).
In this work, we investigate the nature of HXMM05 — to examine whether it is a dispersion-dominated merger, or an isolated HyLIRG.
We securely determine its redshift to be $z$ = 2.9850 through multi-$J$ CO and [CII] line observations,
indicating that HXMM05 is near the peak epoch of cosmic star formation.
This paper is structured as follows.
In §2, we summarize the observations and procedures used to reduce the data.
We also briefly describe the ancillary data used in our analysis.
In §3, we present the main results.
In §4, we present the results from spectral energy distribution (SED) modeling and
dynamical modeling of the [CII] line data using the tilted-ring and “envelope”-tracing methods.
In §5, we discuss the properties of HXMM05 and
compare them to those of other galaxy populations.
We discuss the key implications of our findings in §6, and
summarize the main results and present our conclusions in §7.
Throughout this paper, we use a concordance
$\Lambda$CDM cosmology, with
parameters from the WMAP9 results:
$H_{0}$ = $69.32$ km s${}^{-1}$ Mpc${}^{-1}$, $\Omega_{\rm M}$ = $0.29$, and
$\Omega_{\Lambda}$ = $0.71$ (Hinshaw et al., 2013).
2. Observations and Ancillary Data
{turnpage}
2.1. Combined Array for Research in Millimeter-wave Astronomy (CARMA) CO ($J$ = 3 $\rightarrow$ 2)
Based on the Herschel/SPIRE multi-band colors of $S_{\rm 500}<S_{\rm 250}<S_{\rm 350}$, we expected
the redshift of HXMM05 to be 2 $\lesssim z\lesssim$ 3.5, and its CO ($J$ = 3 $\rightarrow$ 2) line — at rest-frame frequency $\nu_{\rm rest}$ = 345.79599 GHz —
to be redshifted into the 3 mm receiver window of CARMA.
We therefore performed a blind CO line search in HXMM05 with CARMA
in the D array configuration.
Five tracks were executed under excellent
weather conditions between 2010 September 02 and 21 (Program ID: cx310; PI: D. Riechers).
A total of 10.1 hours of on-source time was obtained after combining all data.
We scanned the 3 mm window using four distinct frequency setups, covering a frequency range of $\nu_{\rm obs}$ = 84.98$-$111.97 GHz.
For each setup, the correlator provided sixteen spectral windows, each with a bandwidth of 494.792 MHz
and 95 channels, resulting in an effective bandwidth of 3.75 GHz per sideband after accounting for overlapping edge channels.
This correlator setup provides a
spectral resolution of $\Delta\nu$ = 5.208 MHz (i.e., $\Delta v$ = 18 km s${}^{-1}$ at $\nu_{\rm obs}$ = 86.8 GHz).
All tracks used the same calibrators, as summarized in Table 1.
We estimate a flux calibration accuracy of $\sim$15%.
The miriad package was used to calibrate the visibility data.
The calibrated visibility data were
imaged and deconvolved using the CLEAN algorithm with natural weighting, yielding a synthesized
beam size of 7$\farcs$68 $\times$ 5$\farcs$00, at a position angle (PA) of $-$53°.
The final rms noise is typically $\sigma_{\rm ch}$ = 2.26 mJy beam${}^{-1}$ over a
channel width of 90 km s${}^{-1}$.
We form four continuum images at $\nu_{\rm cont}$ = 90, 93.4, 103, and 107 GHz,
by averaging across the line-free channels in each setup (i.e., one per spectral tuning).
The final rms of the continuum images are $\sigma_{\rm cont}$ = 0.17, 0.37, 0.33, and 0.43 mJy beam${}^{-1}$, respectively.
2.2. Plateau de Bure Interferometer (PdBI) CO ($J$ = 5 $\rightarrow$ 4) and 131 GHz Continuum
We detected a single line in the CARMA data (see §3.1).
Based on the SPIRE colors, the line is most likely CO($J$ = 3$\rightarrow$2),
suggesting a redshift of $z\approx$ 2.985 for HXMM05.
This redshift was spectroscopically confirmed through the detection of a second CO line, which was observed with IRAM
PdBI (Program ID: U–3; PI: N. Fiolet).
Based on the redshift suggested by the CARMA data, we expected the CO ($J$ = 5 $\rightarrow$ 4) line
($\nu_{\rm rest}$ = 576.26793 GHz) to be redshifted to an observed frequency of $\nu_{\rm obs}$ = 144.6093 GHz.
Observations were carried out in good weather conditions in the
D array configuration with six antennas on
2010 September 23 and 26.
A total on-source time of 1.4 hours was obtained in the combined tracks.
The 2 mm receivers were used to cover the expected frequency of the CO ($J$ = 5 $\rightarrow$ 4) line and the underlying continuum.
The WideX correlator was used,
providing a spectral resolution of 1.95 MHz (about 4 km s${}^{-1}$ at $\nu_{\rm obs}$) over an effective
bandwidth of 3.6 GHz, in dual polarization mode.
Calibrators used for bandpass, flux, and complex gain calibrations are listed in Table 1.
We estimate a flux calibration accuracy of 15%.
The gildas package was used to calibrate and analyze the visibility data.
The calibrated visibility data were imaged and deconvolved using the CLEAN algorithm with natural
weighting, yielding a synthesized beam of 7$\farcs$43 $\times$ 4$\farcs$08 at PA = 111$\arcdeg$.
The final rms noise is 5.53 mJy beam${}^{-1}$ over 20 MHz (41.3 km s${}^{-1}$).
A continuum image at an average frequency of $\nu_{\rm cont}$ = 145.4 GHz was produced by averaging over the line-free channels ($\Delta\nu$ = 3.12 GHz), yielding
an rms noise of 0.44 mJy beam${}^{-1}$.
We also observed the $\nu_{\rm obs}$ = 131.1 GHz continuum emission in HXMM05 with the PdBI (Program ID: U–3; PI: N. Fiolet) to rule out an alternative redshift option.
Observations were carried out on 2010 September 21 under good weather conditions in the D array configuration
for 0.6 hours of on-source time (Table 1).
The visibility data were calibrated using gildas. Imaging and deconvolution were performed using
the CLEAN algorithm with natural weighting.
We formed a continuum image by averaging across all channels within an effective bandwidth of 3.6 GHz,
reaching an rms of $\sigma_{\rm cont}$ = 0.21 mJy beam${}^{-1}$ and a beam size of 14$\farcs$85 $\times$ 2$\farcs$59 at PA = $-$36$\arcdeg$.
2.3. NSF’s Karl G. Jansky Very Large Array (VLA) CO ($J$ = 1 $\rightarrow$ 0)
Based on the redshift determined from the CO($J$ = 3$\rightarrow$2) and CO ($J$ = 5 $\rightarrow$ 4) lines,
we targeted the CO ($J$ = 1 $\rightarrow$ 0) line ($\nu_{\rm rest}$ = 115.27120 GHz) in HXMM05 using the
the VLA, for a total of ten observing sessions (Program ID: 14B-302;
PI: S. Bussmann).
One session was carried out on 2014 September 20 in the DnC array configuration and the remaining
nine sessions were carried out between
2014 November 17 and December 11 in the C array configuration,
A total of 10.5 hours of on-source time was obtained in the combined ten sessions.
The Ka-band receivers were used to cover the redshifted CO ($J$ = 1 $\rightarrow$ 0) line.
The WIDAR correlator was used in full polarization mode, providing a total bandwidth of
2 GHz covered by sixteen sub-bands, each with a bandwidth of 128 MHz
and a channel spacing of 2 MHz (29 km s${}^{-1}$).
Calibrators are listed in Table 1.
We estimate a flux calibration accuracy of $\lesssim$15%.
Visibility data were calibrated and analyzed using version 4.7.1 of
the casa package. We combined all calibrated data and imaged
the visibilities using the CLEAN algorithm with natural weighting to maximize sensitivity, yielding
a synthesized beam size of 1$\farcs$21$\times$ 0$\farcs$80 at PA = 36$\arcdeg$.
The final rms noise is 0.041 mJy beam${}^{-1}$ over 6 MHz (62 km s${}^{-1}$), or 0.028 mJy beam${}^{-1}$ per $\Delta v$ = 145 km s${}^{-1}$ velocity bin. A continuum image at $\nu_{\rm cont}$ = 31.27 GHz
was produced by averaging over all the line-free channels, yielding an rms
noise of $\sigma_{\rm cont}$ = 3.19 $\mu$Jy beam${}^{-1}$. To examine the kinematics of the CO ($J$ = 1 $\rightarrow$ 0) line emission at higher resolution,
we made an additional line cube using Briggs weighting with robustness $R$ = 0.5.
An rms noise of $\sigma_{\rm ch}$ = 0.031 mJy beam${}^{-1}$ per velocity bin ($\Delta v$ = 145 km s${}^{-1}$) is reached in the resulting line cube, with a beam size of 0$\farcs$94 $\times$ 0$\farcs$71 at PA = 31$\arcdeg$.
2.4. ALMA [CII]
We observed the [CII] fine-structure line ($\nu_{\rm rest}$ = 1900.536900 GHz) in
HXMM05 with ALMA on 2015 June 15 and August 27 during Cycle 2
(ID: 2013.1.00749.S, PI: D. Riechers).
The [CII] line is redshifted to Band 8 at the redshift of HXMM05 determined from our CO data ($z$ = 2.9850).
We employed the frequency division mode (FDM) correlator setup with dual polarization, providing
an effective bandwidth of 7.5 GHz and a spectral resolution of 1.95 MHz (1.2 km s${}^{-1}$).
The on-source time, baseline coverage, and calibrators used in each track are listed in Table 1.
All data were calibrated manually due to the uncertain
flux scale of Ceres, which was used as the flux calibrator in one
of the two tracks. The calibrated amplitudes of both the phase and bandpass calibrators are consistent with those
found in the ALMA Calibrator Source Catalogue.
The flux scale was also verified by comparing the calibrated
amplitudes of the same phase calibrator across the two tracks.
We estimate a flux calibration accuracy of 15%.
All data were calibrated using casa version 4.5.0 and were then combined,
imaged, and deconvolved using the CLEAN algorithm with
natural weighting,
yielding a synthesized beam of 0$\farcs$18$\times$0$\farcs$14 at PA = 61.3°.
To obtain an optimal balance between sensitivity and spectral resolution,
we binned the data cubes to spectral resolutions of $\Delta v$ = 25 km s${}^{-1}$ and 300 km s${}^{-1}$,
reaching typical rms noise values of $\sigma_{\rm ch}$ = 2.36 and 0.75 mJy beam${}^{-1}$ per channel, respectively.
A continuum image was obtained by averaging across the line-free channels and
excluding any channels that were affected by atmospheric features.
The bandwidth used to form the continuum images is 5.47 GHz, yielding
an rms noise level of $\sigma_{\rm cont}$ = 0.22 mJy beam${}^{-1}$.
We also imaged the visibilities
with $uv$-tapering applied at 500 k$\lambda$ (311.5 m) to recover potential diffuse low surface brightness
emission and structure on larger spatial scales.
After tapering, a line cube binned to a spectral resolution of
$\Delta v$ = 150 km s${}^{-1}$ was imaged and deconvolved using the CLEAN algorithm and natural weighting.
We used the tapered data cube and image to define the apertures used for extracting the line
and underlying continuum fluxes, and the line spectrum (see §3).
The beam size for the tapered data is 0$\farcs$31 $\times$ 0$\farcs$26 at PA = 69.5$\arcdeg$, which is roughly twice
the untapered beam size.
The final rms noise is
$\sigma_{\rm cont}$ = 0.33 mJy beam${}^{-1}$ for the tapered continuum map,
and $\sigma_{\rm ch}$ = 1.25 mJy beam ${}^{-1}$ per 150 km s${}^{-1}$ bin for the data cube.
2.5. ALMA CO($J$ = 10$\rightarrow$9)
In ALMA Cycle 4, we observed the CO($J$ = 10$\rightarrow$9) line ($\nu_{\rm rest}$ = 1151.98545200 GHz) in
HXMM05 on 2017 September 11 and 16 (ID: 2016.2.00105.S, PI: D. Riechers) using the 7 m Atacama Compact Array (ACA).
The CO ($J$ = 10 $\rightarrow$ 9) line is redshifted to Band 7 for HXMM05.
We employed the time division mode (TDM) correlator setup with dual polarization, providing
an effective bandwidth of 7.5 GHz
and a spectral resolution of 15.6 MHz (16.2 km s${}^{-1}$).
The on-source time, baseline coverage, and calibrators of each track are listed in Table 1.
We conservatively estimate a flux calibration accuracy of 15%.
All data were calibrated using version 5.1.1 of casa, and were then combined,
imaged, and deconvolved using the CLEAN algorithm with natural weighting.
This yields a clean beam of 5$\farcs$35$\times$3$\farcs$65 at PA = $-$85°.
We binned the data cube to a spectral resolution of $\Delta v$ = 49 km s${}^{-1}$, reaching a typical rms noise
of $\sigma_{\rm ch}$ = 1.20 mJy beam${}^{-1}$ per channel.
A continuum image was obtained by averaging across the line-free channels
over a bandwidth of 5.61 GHz, yielding an rms noise of $\sigma_{\rm cont}$ = 0.37 mJy beam${}^{-1}$.
2.6. Ancillary Data
2.6.1 Herschel/SPIRE and PACS, and MAMBO 1.2 mm
HXMM05 was observed with Herschel/PACS and SPIRE at 100, 160, 250, 350, and 500 $\micron$
as part of the HerMES project (Oliver et al. 2012).
HXMM05 remains undetected at 100 $\micron$ down to a 5$\sigma$ limit of
$S_{\rm 100}$ $<$ 28.8 mJy,
but is detected at 160 $\micron$.
The 160 $\micron$ photometry was
extracted from the Level 5 XMM-VIDEO3 data using a positional prior from the Spitzer/MIPS 24 $\micron$ catalog
with aperture photometry, and with appropriate aperture corrections applied (PACS DR4).
For the SPIRE photometry, we adopted the fluxes reported by
Wardlow et al. (2013), which were extracted using
StarFinder (Diolaiti et al., 2000). We also include the 1.2 mm photometry obtained
with the IRAM 30-m telescope/MAMBO in modeling the SED of HXMM05 (Wardlow et al. 2013; Table 2; see §4.1).
2.6.2 SMA 870 $\micron$
We also make use of 870 $\micron$ continuum data obtained with
the Submillimeter Array (SMA; IDs: 2010A-S091 and 2011A-S068, PIs: A. Cooray and S. Bussmann; Wardlow et al. 2013).
Observations were carried out in the extended and subcompact array configurations on
2010 August 16 and September 25, and 2011 August 05,
with local oscillator frequencies of
342.224 GHz and 342.003 GHz (extended),
and 340.017 GHz (subcompact), respectively.
The on-source time of each track is listed in Table 1.
Uranus was used as the primary flux calibrator,
and the quasars J0238$+$166 and J0217$+$017 were used as
complex gain calibrators for all three tracks.
Quasars 3C454.3 and 3C84 were used for bandpass calibration.
MWC349A and Callisto were observed as secondary flux calibrators in the extended configuration tracks.
All visibility data were calibrated using the IDL-based mir package and imaged using miriad.
We combined all tracks to form a continuum image using the CLEAN algorithm with natural weighting, yielding
a synthesized beam of 0$\farcs$99 $\times$ 0$\farcs$78 at PA = $-$68.2$\arcdeg$ and
an rms noise of 0.92 mJy beam${}^{-1}$ over the full bandwidth of 7.5 GHz.
2.6.3 ALMA Cycle-0 870 $\micron$
We previously observed the 870 $\micron$ continuum emission in HXMM05 with ALMA in Band 7 (ID: 2011.0.00539.S; PI: D. Riechers; also see Bussmann et al. 2015).
Visibilities were imaged using the CLEAN algorithm with Briggs weighting (robustness $R$ = 0.5), yielding
a synthesized beam of 0$\farcs$50 $\times$ 0$\farcs$40 (PA = 76.4$\arcdeg$) and
an rms noise of $\sigma_{\rm cont}$ = 0.28 mJy beam${}^{-1}$.
2.6.4 Spitzer/IRAC and MIPS Near- and Mid-IR
HXMM05 was observed with Spitzer/IRAC and MIPS as part of the Spitzer
Wide-area InfraRed Extragalactic Survey (SWIRE; Lonsdale et al. 2003)
in the XMM-LSS field.
The survey depths (5$\sigma$) for point sources are
$S_{\nu}$ $<$ 3.7, 5.4, 48, and 37.8 $\mu$Jy for the IRAC channels at 3.6, 4.5, 5.8 and 8.0 $\micron$, respectively,
and 230 $\mu$Jy, 18 mJy, and 150 mJy for the MIPS bands at 24, 70, and 160 $\micron$,
respectively222http://swire.ipac.caltech.edu/swire/astronomers/program.html.
In the MIPS bands, HXMM05 is detected at 24 $\micron$
(SWIRE catalog DR2)333https://irsa.ipac.caltech.edu/data/SPITZER/SWIRE/docs/delivery_doc_r2_v2.pdf.
The 24 $\micron$ photometry was extracted using aperture photometry and SExtractor (Savage & Oliver, 2007).
Appropriate aperture corrections have been applied.
HXMM05 remains undetected at 70 and 160 $\micron$;
we adopt 3$\sigma$ levels as the upper limits for the non-detections (see Table 2).
In the post-cryogenic period of Spitzer, more sensitive continuum images at 3.6 and 4.5 $\micron$ were obtained
in the deeper Spitzer Extragalactic Representative Volume Survey (SERVS), which reaches 5$\sigma$ limits
of 1.25 $\mu$Jy (Mauduit et al., 2012; Nyland et al., 2017).
For the two SWIRE images observed at longer wavelengths (IRAC 5.8 and 8.0 $\micron$),
we perform aperture photometry to extract the fluxes of HXMM05 at the centroid position determined from the SMA 870 $\micron$ map.
Final flux densities are reported in Table 2.
2.6.5 Wide-field Infrared Survey Explorer (WISE) Near- and Mid-IR
HXMM05 was observed with WISE as part of the ALLWISE program.
Its flux density limits are reported in the ALLWISE source catalog available on the NASA/IRAC Infrared Science Archive (IRSA)
and were extracted through profile-fitting.
In Vega magnitude units, we find
15.460 $\pm$ 0.040, 14.905 $\pm$ 0.065, $<$12.457, and $<$8.817
for the four WISE bands (at 3.4, 4.6, 12, and 22 $\micron$, respectively). The latter two are 3$\sigma$ upper limits. Since a few sources with IR emission
near HXMM05 are detected in the Spitzer images, we expect emission toward HXMM05 to be unresolved and blended within the WISE beam.
As such, we adopt all the WISE fluxes as upper limits only, yielding 3$\sigma$ limits of 0.20, 0.19,
0.52, and 3.24 mJy, respectively (Table 2).
2.6.6 Visible and Infrared Survey Telescope for Astronomy (VISTA) Near-IR
The XMM-LSS field was imaged with VISTA in the $Z$-, $Y$-, $J$-, $H$-, and $Ks$-bands
as part of the VISTA Deep Extragalactic Observations (VIDEO) Survey
(Jarvis et al., 2013), reaching 5$\sigma$ limits of 25.7, 24.6, 24.5, 24.0, and 23.5 AB mag
for a point source in a 2${}^{\prime\prime}$ diameter aperture.
HXMM05 is undetected in all bands.
In Table 2, we report the corresponding 3$\sigma$ levels as upper limits.
2.6.7 CFHT UV-optical-IR
HXMM05 was imaged with the CFHT/MegaCam
in $u^{*}$, $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, $z^{\prime}$ bands as part of the CFHT Legacy Survey Deep-1 field (CFHTLS-D1).
In the final CFHTLS release (version T0007),
the sensitivity limits corresponding to 80% completeness for a point source
are 26.3, 26.0, 25.6, 25.4, and 25.0 AB mag for the five bands, respectively,
or 3$\sigma$ point-source sensitivities of 0.19, 0.14, 0.20, 0.24, 0.35 $\mu$Jy.
We show the $\sim$0.8${}^{\prime\prime}$ resolution CFHT deep field images retrieved from
the CFHT Science Archive from the Canadian Astronomy Data Centre (CADC) in the Appendix.
HXMM05 remains undetected in all bands according to the T0007 CFHTLS-Deep catalog (Hudelot et al. 2012; Table 2).
2.6.8 Galaxy Evolution Explorer (GALEX) Near and Far-UV
UV emission in the HXMM05 field was observed with GALEX in the FUV-1500 and NUV-2300 bands
as part of the XMM-LSS Deep Imaging Survey (DIS).
HXMM05 was covered in the XMMLSS_00 tile, which was observed for
75262 and 60087 seconds in the NUV and the FUV bands, respectively444Based on the
images and catalog released in GR6.,
reaching 3$\sigma$ limits of 25.5 in AB mag (Pierre et al., 2004; Martin et al., 2005).
2.6.9 XMM-Newton X-ray
HXMM05 is located in the CFHTLS-D1 field, which was observed with the European Photon Imaging Camera (EPIC)
onboard XMM-Newton for an integration time of around 20 ks in the XMM Medium Deep Survey (XMDS;
Chiappetti et al. 2005), reaching 3$\sigma$ point source limits
of 3.7$\times 10^{-15}$ erg s${}^{-1}$ cm${}^{-2}$ and 1.2$\times 10^{-14}$ erg s${}^{-1}$ cm${}^{-2}$ in the soft (0.5$-$2 keV)
and hard (2$-$10 keV) X-ray bands, respectively.
These limits correspond to
$L_{X{\rm,0.5-2\,keV}}$ $<$ 7.4$\times 10^{43}$ erg s${}^{-1}$ (soft) and
$L_{X{\rm,2-10\,keV}}$ $<$ 9.5$\times 10^{44}$ erg s${}^{-1}$ (hard) at $z$ = 2.9850,
which reach the levels
of
powerful Seyfert galaxies (Elvis et al., 1978).
HXMM05 remains undetected in these observations.
3. Results
3.1. CO Line Emission and Redshift Identification
From the first two CO lines we detected — CO ($J$ = 3 $\rightarrow$ 2) and CO ($J$ = 5 $\rightarrow$ 4) with CARMA and the PdBI — we spectroscopically determine
the redshift of HXMM05 to be $z$ = 2.9850 $\pm$ 0.0009.
The CO($J$ = 3$\rightarrow$2; 5$\rightarrow$4; 10$\rightarrow$9) lines remain spatially unresolved, and are
detected at $>$ 8$\sigma$, $>$ 6$\sigma$, and $>$5$\sigma$ significance, respectively (Figures 1 and 2).
Due to the near-equatorial declination of HXMM05 and the sparse $uv$-sampling of the data,
the PdBI synthesized beam is highly elongated,
and the image fidelity is heavily affected by strong sidelobes.
We fit single Gaussian profiles to the line spectra, as shown in Figure 1.
The resulting best-fit parameters are summarized in Table 3.
We note that given the broad linewidths observed up to the $J$ = 10$\rightarrow$9 transition, the lack of emission at $v>0$ km s${}^{-1}$ in the CO ($J$ = 5 $\rightarrow$ 4) line
may be attributed to the limited S/N of the data. The true CO ($J$ = 5 $\rightarrow$ 4) flux may be a factor of two higher.
Upon determining the redshift of HXMM05, we observed the CO ($J$ = 1 $\rightarrow$ 0) line with the VLA.
We detect marginally spatially-resolved CO ($J$ = 1 $\rightarrow$ 0) line emission at $>$14 $\sigma$ peak significance (Figure 2).
The emission centroid is centered at the position of HXMM05, but shifts from NW to SE with increasing velocity.
A second peak is detected at 2$\farcs$6 NE of HXMM05,
at 6$\sigma$ significance in the blueshifted channels (see Figure 2), corresponding to
a projected separation of 20 kpc.
In the subsequent sections of this paper, this NE component is referred to as X-NE, and the main component is referred to as “X-Main”.
We extract a spectrum using an aperture defined by the
2$\sigma$ contours centered at the coordinates of HXMM05 (middle left panel of Figure 3),
and a spectrum for just X-NE (bottom left panel of Figure 3).
The centroid of X-NE is blueshifted by $-$535 $\pm$ 55 km s${}^{-1}$ with respect to X-Main.
Assuming that the line detected is CO ($J$ = 1 $\rightarrow$ 0) , the redshift of X-NE would be $z$ = 2.9779 $\pm$ 0.0007.
We also extract a spectrum for the HXMM05 system as a whole, including emission from both X-Main and X-NE
(top panel of Figure 1 and top left panel of Figure 3).
The best-fit linewidths and intensities are listed in Table 3.
The CO ($J$ = 1 $\rightarrow$ 0) line is remarkably broad ($>$1100 km s${}^{-1}$ FWHM) and shows a hint of a double-horned profile, which
likely results from contributions from both X-Main and X-NE (see Figure 3).
We fit 2D Gaussians to the two components detected in the velocity-integrated line intensity map, finding
a deconvolved source size of (1$\farcs$12 $\pm$ 0$\farcs$37) $\times$ (0$\farcs$81 $\pm$ 0$\farcs$45) at PA = 173 $\pm$ 49$\arcdeg$
for HXMM05.
This corresponds to a physical diameter of 8.8 kpc $\times$ 6.4 kpc at $z$ = 2.9850.
For the NE component, we find a deconvolved source
size of (1$\farcs$12 $\pm$ 0$\farcs$41) $\times$ (0$\farcs$26 $\pm$ 0$\farcs$42) at PA = 72 $\pm$ 37$\arcdeg$, which corresponds to a physical size of 8.8 kpc $\times$ 2.0 kpc at $z$ = 2.9779.
The extent of the cold molecular gas in both HXMM05 and the NE component are consistent
with those observed in other DSFGs (e.g., Ivison et al., 2011; Riechers et al., 2011a).
3.2. [CII] Line Emission
We detect spatially resolved [CII] line emission toward HXMM05 at a peak significance of $>$13$\sigma$
(in a tapered intensity map).
At the full spatial resolution of the data (0$\farcs$15), HXMM05 is resolved over $>$25 beams.
To better determine the line profile shape, we
create two [CII] line cubes — with and without $uv$-tapering (see §2).
The 635 $\micron$ continuum emission has been subtracted from both line cubes in the $uv$-plane.
We collapse them to form velocity-integrated line intensity (i.e., zeroth moment)
maps
as shown in Figure 4.
We show the [CII] line spectrum of HXMM05 in the last panel of Figure 1 and the top right panel of Figure 3.
The best-fit parameters obtained from fitting a single-Gaussian are
listed in Table 3, together with those derived for the CO lines.
We extract separate spectra for X-Main and X-NE from the high resolution data cube using an aperture
defined by the 1$\sigma$ contours of the tapered intensity map.
The resulting spectrum of X-Main is shown in the middle right panel of Figure 3.
Fitting a single Gaussian yields a peak flux density of $S_{\rm peak}$ = 172 $\pm$ 8 mJy, a line FWHM of $\Delta v$ = 667 $\pm$ 46 km s${}^{-1}$,
and a line intensity of $I$ = 122 $\pm$ 10 Jy km s${}^{-1}$.
We also fit a double-Gaussian profile, yielding
best-fit peak fluxes of $S_{\rm peak}$ = 53 $\pm$ 30 and 164 $\pm$ 10 mJy, and line FWHMs of
$\Delta v$ = 167 $\pm$ 85 and 659 $\pm$ 101 km s${}^{-1}$, respectively.
The peaks are separated by $\Delta v_{\rm sep}$ = 346 $\pm$ 124 km s${}^{-1}$.
X-NE is detected at $\sim$6$\sigma$ significance (see bottom right panel of Figure 3 and also Figure 4).
We fit a 2D Gaussian to the tapered intensity map of X-Main, which yields a deconvolved FWHM source size of
(0$\farcs$91 $\pm$ 0$\farcs$08)$\times$ (0$\farcs$75 $\pm$ 0$\farcs$07),
or a physical size of (7.2 $\pm$ 0.6) $\times$ (5.9 $\pm$ 0.6) kpc, consistent with the extent seen in the higher resolution image.
The first and second moment maps of the [CII] emission representing the
velocity
and the velocity dispersion of X-Main along the line-of-sight (LOS)
are shown in Figure 5.
Moment maps are created from the line cube after clipping at 3$\sigma_{\rm ch}$ per channel.
Structures on the scale of the angular resolution ($\lesssim 1.2$ kpc) are seen
in the channel maps (see Appendix §A).
A velocity gradient along the NW-to-SE direction, varying over a velocity range of $\Delta v\simeq$ 600 km s${}^{-1}$,
is seen in the velocity field (Figure 5).
The dispersion map is remarkably uniform across the whole galaxy, with $\sigma_{v}\simeq$ 75 km s${}^{-1}$,
except in the central $\lesssim$0$\farcs$2 region,
where the dispersion reaches its peak at $\sigma_{v}\simeq$ 200 km s${}^{-1}$.
A position-velocity (PV) diagram extracted along the major kinematic axis of X-Main
(see §4.2.1) is shown in Figure 6.
The rising part of a rotation curve and the outer envelope are both detected. The latter is
usually more pronounced in more inclined disks (as seen in nearby galaxies; see review by Sofue & Rubin 2001).
The PV diagram is consistent with broad [CII] line emission, which varies by $>$ 700 km s${}^{-1}$ within about 9 kpc.
We find comparable deconvolved source sizes for CO ($J$ = 1 $\rightarrow$ 0) and [CII] emission (see Table 3), as
confirmed by the comparable extents found after convolving the high resolution [CII] data to the CO ($J$ = 1 $\rightarrow$ 0) line resolution (Figure 7).
At the resolution of the VLA data, the velocity gradient seen in the CO ($J$ = 1 $\rightarrow$ 0) line emission is
consistent with that of the [CII] line, but more sensitive and
higher angular resolution data are required to match the detailed velocity structures of both lines.
3.3. H${}_{2}$O Line Emission
The H${}_{2}$O ($1_{11}$$\rightarrow$$0_{00}$; 3${}_{12}$$\rightarrow$$2_{21}$)
lines at redshifted frequencies of 279.383 and 289.367 GHz are covered by the ALMA CO ($J$ = 10 $\rightarrow$ 9) line observations.
We do not detect the ground state H${}_{2}$O line in emission or absorption down to
a 3$\sigma$ limit of $<$ 0.80 Jy km s${}^{-1}$ beam${}^{-1}$, assuming the same linewidth as the CO ($J$ = 10 $\rightarrow$ 9) line (760 km s${}^{-1}$).
The H${}_{2}$O(3${}_{12}$$\rightarrow$$2_{21}$) line is next to the CO ($J$ = 10 $\rightarrow$ 9) line and is at most weakly detected; we
conservatively report a 3$\sigma$ upper limit of $<$ 0.87 Jy km s${}^{-1}$ beam${}^{-1}$,
assuming the same linewidth as for the CO ($J$ = 10 $\rightarrow$ 9) line.
3.4. Continuum
We show the Spitzer/IRAC images in Figure 8.
Sources near HXMM05 are detected in the IRAC IR and CFHT NUV bands (see also Appendix §B), but HXMM05 remains undetected.
Among the four 3 mm spectral setups of the CARMA observations, we do not detect continuum emission in the individual tunings.
A final continuum image created by averaging across all the tunings yields a weak
detection at 4$\sigma$ significance (see Table 2).
In the PdBI 2 mm setups, continuum emission remains undetected.
On the other hand, we detect Ka-band continuum emission underlying the CO ($J$ = 1 $\rightarrow$ 0) line at 31.3 GHz
at $\gtrsim$ 5$\sigma$ significance, which remains unresolved at the resolution and sensitivity of the VLA data (Figure 7, see Table 2).
The centroid of the 31.3 GHz continuum emission coincides with that of the CO ($J$ = 1 $\rightarrow$ 0) line emission, and its
flux density is consistent with that obtained from fitting a four-parameter model
(Gaussian plus a first order polynomial) to the CO ($J$ = 1 $\rightarrow$ 0) line spectrum extracted at the peak pixel.
We also detect unresolved continuum emission at observed-frame $\sim$1 mm (rest-frame 260 $\micron$) underlying the CO ($J$ = 10 $\rightarrow$ 9) line
at $\sim$15 $\sigma$ significance (Table 2).
Continuum emission underlying the [CII] line at observed-frame 635 $\micron$ is detected at a peak significance of $>$31$\sigma$
(see Table 2).
Two dust peaks, separated by 2.4 kpc, are detected at high significance.
One peak coincides with the 870 $\micron$ emission centroid (Figure 8)
and with the CO ($J$ = 1 $\rightarrow$ 0) emission centroid of X-Main (see Figure 7), whereas the other dust peak
is offset to the SW (we denote these as XD1 and XD2, respectively, hereafter).
We measure the total continuum flux density using an aperture defined by the 1$\sigma$ contours.
We fit a two-component 2D Gaussian to the continuum image and find deconvolved source sizes of
(0$\farcs$39 $\pm$ 0$\farcs$05) $\times$ (0$\farcs$36 $\pm$ 0$\farcs$05) for
XD1 and (0$\farcs$39 $\pm$ 0$\farcs$06) $\times$ (0$\farcs$35 $\pm$ 0$\farcs$06) for XD2,
corresponding to physical sizes of about 3 kpc for both components.
Since the deconvolved source sizes are larger than the beam size, the size measurements
are not limited by the resolution of the observations.
The peak flux densities are 4.95 $\pm$ 0.38 mJy beam${}^{-1}$ for XD1
and 2.75 $\pm$ 0.28 mJy beam${}^{-1}$ for XD2 (Table 4).
Based on their total flux densities and sizes, their brightness temperatures
in the Rayleigh-Jeans limit are 1.12 and 0.63 K, respectively,
corresponding to $T_{\rm B,RJ}$ = 4.5 and 2.5 K in the rest-frame.
We overplot the 635 $\micron$ continuum emission with
the SMA and ALMA data observed at 870 $\micron$ in Figure 8.
We fit a single-component elliptical Gaussian model to each of the 870 $\micron$ images.
Only XD1 is detected at 870 $\micron$.
We also convolve the 635 $\micron$ data to the native resolution of the ALMA 870 $\micron$ data, and
find a spatial offset between two peaks emission centroids.
The emission centroids are determined by fitting a two-component Gaussian model to the 635 $\micron$ data
and a single component Gaussian model to the 870 $\micron$ data.
We thus conclude that XD2 is likely to be much fainter than XD1 at 870 $\micron$, in order for it to remain undetected
down to a 3$\sigma$ limit of 0.84 mJy beam${}^{-1}$.
While the [CII] emission shows a monotonic velocity gradient (Figure 5), which suggests that HXMM05 is a
rotating disk with ordered motions, the dust continuum is almost
exclusively produced at the two peaks embedded within the kpc-scale [CII] disk (Figure 8).
Likely due to the limited surface brightness sensitivity of our observations, the [CII] line emission appears more irregular compared to the continuum.
We detect low surface brightness emission in the outer region of the 635 $\micron$ dust continuum map,
which is consistent with the overall extent of the [CII] and CO ($J$ = 1 $\rightarrow$ 0) emission (Figures 7 and 8).
This diffuse component is likely to be more optically thin compared to XD1 and XD2,
which likely dominate the dust optical depth estimated at 635 $\micron$ based on the integrated SED model (see §4.1), given that
they contribute $>$80% to the total continuum flux at this wavelength.
X-NE (which is detected in CO and [CII] line emission) is also weakly detected in the continuum at 635 $\micron$
at $>$3$\sigma$ significance, and in the UV, optical, and NIR wavebands
(see the last two panels of Figure 7, Figure 8, and Figure 18 in Appendix §B).
4. Analysis
4.1. Spectral Energy Distribution Modeling
We use the extensive multi-wavelength photometric data
available in the XMM-LSS field to determine the IR, dust, and stellar properties of HXMM05 via SED modeling.
Previously, Wardlow et al. (2013) modeled the dust SED of HXMM05 by fitting a simple modified blackbody
to the photometry measured at (sub-)mm wavebands (Herschel-SMA-MAMBO),
assuming a dust emissivity index of $\beta$ = 1.5.
This model suggests an IR luminosity (rest-frame
$\lambda_{\rm rest}$ = 8$-$1000 $\micron$) of $L_{\rm IR}$ $=$ (3.2 $\pm$ 0.4) $\times$ 10${}^{13}$ $L_{\odot}$ and a dust
temperature of $T_{d}$ = (45 $\pm$ 1) K.
Here, we update the SED with more photometric data obtained since,
covering UV through radio wavelengths (see Table 2).
We model the observed dust SED using a modified blackbody (MBB)
and the full SED using the magphys code (da Cunha et al., 2015)
to derive a stellar mass in a self-consistent way from the dust and stellar emission.
4.1.1 Modified Blackbody Model
We model the dust SED of HXMM05 by assuming a single-temperature modified blackbody,
which is parameterized by the characteristic dust temperature $T_{d}$.
We fit MBB-based SED models to 16 photometric points covering rest-frame IR-to-mm wavelengths (observed-frame
24 $\micron$$-$3 mm; see Table 2)
using the code mbb_emcee (e.g., Riechers et al., 2013; Dowell et al., 2014).
To account for the absolute flux-scale uncertainties associated with the photometry
obtained with ALMA, SMA, PdBI, and CARMA,
we add in quadrature an additional 15% uncertainty.
The model consists of a MBB component that accounts for the FIR emission
and a power-law component blue-ward
thereof to describe the warmer dust emission at mid-IR wavelengths.
The dust optical depth (as a function of wavelength) is
taken into account via the parameter $\lambda_{0}$, where dust emission at $\lambda<\lambda_{0}$ (rest-frame) is
optically thick ($\tau_{\nu}>1$).
The dust mass is calculated using
$$M_{d}=S_{\nu}~{}D_{L}^{2}~{}[(1+z)~{}\kappa~{}B_{\nu}(T)]^{-1}~{}\tau_{\nu}~{}%
[1~{}-~{}\exp{(-\tau_{\nu})}]^{-1},$$
(1)
where $D_{L}$ is the luminosity distance and $B_{\nu}$ is the Planck function.
In estimating the dust mass, we assume an absorption mass coefficient of
$\kappa$ = 2.64 m${}^{2}$ kg${}^{-1}$ at $\lambda$ = 125.0 $\micron$ (Dunne et al., 2003).
This (general) model is therefore parameterized by five free parameters:
a characteristic dust temperature ($T_{d}$); emissivity index ($\beta$); power-law index ($\alpha$); normalization
factor ($f_{\rm norm}$); and $\lambda_{0}$.
We impose uniform priors such that $T_{d}$ $>$ 1 K,
$\beta$ $\in$ [0.1, 20.0],
$\lambda_{0}$ $\in$ [1.0, 400.0] $\micron$, and
$\alpha$ $\in$ [0.1, 20.0].
We adopt the statistical means and 68${}^{\rm th}$ percentiles of the
resulting posterior probability distributions as the “best-fit” parameters.
For comparison with literature values, we also fit MBB+power-law models
without the wavelength-dependent optical depth parameter (i.e., assuming optically thin dust emission).
All the best-fit parameters are listed in Table 5.
We note that the 160 $\micron$ photometry data is poorly fitted, which may suggest the
presence of a warmer dust component in HXMM05.
However, with the data at hand, this dust component cannot be constrained.
Fitting models to photometry excluding the 160 $\micron$ data
yields physical parameters that are consistent with those listed in Table 5 within the
uncertainties.
4.1.2 MAGPHYS model
To determine the stellar mass of HXMM05, we fit models to its full SED, sampled by the
FUV-to-radio wavelength photometry using
the high-$z$ extension of magphys (da Cunha et al., 2008, 2015).
This code exploits a large library of optical and IR templates that are linked together physically through
energy balance, such that the UV-to-optical starlight is absorbed by dust and re-radiated in the FIR.
A detailed explanation of the magphys code and the model priors are given by da Cunha et al. (2015).
Following da Cunha et al. (2015), upper limits are taken into
account by setting the input flux densities to zero and uncertainties to upper limits.
The best-fit SED is shown in Figure 9 and
the resulting best-fit parameters
are listed in Table 6.
Since in the best-fit model, the Herschel/PACS 160 $\micron$ measurement forces the
dust peak to shorter wavelengths and worsens the fit at long wavelengths
(similar to the MBB fit), we re-model the SED excluding this outlier.
The resulting best-fit parameters are listed in Table 6.
The dust peak in this fit is in good agreement with the (sub-)mm and radio photometry.
The best-fit parameters determined with and without the
PACS 160 $\micron$ photometry are consistent within the uncertainties.
We thus adopt the parameters from the latter fit (i.e., excluding the 160 $\micron$ outlier)
in the following sections.
4.2. Dynamical Modeling
We fit dynamical models to the 1-kpc resolution [CII] data obtained with ALMA to study the gas dynamics of HXMM05 (more specifically, X-Main).
The monotonic velocity gradient observed in [CII] suggests that HXMM05 is a rotating disk galaxy,
an interpretation further supported by the analysis of §4.2.1 below.
Assuming that the disk is circular and infinitesimally thin,
we use the kinematic major and minor axes to estimate the inclination angle, which yields $i$ = 46 $\pm$ 8$\arcdeg$.
This is slightly different from the value estimated using the morphological axes, which yields
$i$ = 35 $\pm$ 5$\arcdeg$, but the two are
consistent within the error bars. We initialize the inclination angle in the following analyses based on these estimates.
4.2.1 Harmonic Decomposition and Tilted-ring Model
To assess whether the velocity field observed towards HXMM05 is consistent with its gas being distributed in a disk rather than
effects caused by e.g., merging clumps, tidal debris, or inflows, we apply
harmonic decomposition analysis (Schoenmakers et al., 1997).
Briefly,
this method describes higher order moments, $K$ (e.g., LOS velocity) as a Fourier series:
$$\displaystyle K\left(\psi\right)$$
$$\displaystyle=A_{0}+A_{1}\sin(\psi)+B_{1}\cos{(\psi)}~{}+$$
$$\displaystyle A_{2}\sin(2\psi)+B_{2}\cos(2\psi)+\dotsb,$$
where $\psi$ is the azimuthal angle measured from the major axis.
The above can be recast into the following form:
$$K\left(r,\psi\right)=A_{0}(r)+\sum_{m}K_{m}(r)~{}\cos{\{m[\psi-\psi_{m}(r)]\}},$$
(2)
where the amplitude and phase of the $m$-th order term are defined as
$$K_{m}\equiv\sqrt{A_{m}^{2}+B_{m}^{2}}\quad\mathrm{and}\quad\psi_{m}\equiv%
\arctan{\frac{A_{m}}{B_{m}}}.$$
(3)
Since the velocity field is expected to be dominated by the cosine term
in the case of an ideal rotating disk;
in this scenario, $B_{1}$ should dominate the harmonic terms,
with higher order terms $K_{m}$ measuring deviations from the ideal case.
Following Krajnović et al. (2006), we compare the fifth-order amplitude term to the first-order cosine term ($K_{5}/B_{1}$) to
quantify deviations in the [CII] velocity map of HXMM05 from a rotating disk,
and thus, differentiate between a rotation-dominated disk and a dispersion-dominated merger.
As shown in Figure 10, the higher order term is insignificant compared to $B_{1}$ across the majority of the disk, especially towards the
center, where the data have higher S/N.
We thus interpret HXMM05 to be a rotating disk for the remainder of this paper.
Given the modest inclination of HXMM05, we fit tilted-ring models (Begeman, 1989)
to the observed velocity field using the task rotcur provided in the gipsy software package to
analyze the gas dynamics of HXMM05 due to bulk motions (i.e., driven by the gravitational potential).
The tilted-ring model
assumes that the gas is in a circular, rotating thin disk, and describes the disk using a series of
concentric rings, where each ring can have an independent inclination angle ($i$),
major axis PA, rotation velocity ($v_{\rm rot}$), and
expansion velocity ($v_{\rm exp}$).
The rotation velocity is related to the projected LOS velocity via
$$v_{\rm LOS}=v_{\rm sys}+v_{\rm rot}\,\cos{(\psi)}\sin{(i)}+v_{\rm exp}\,\sin{(%
\psi)}\sin{(i)}.$$
(4)
Here, we assume that
the observed LOS velocity is due entirely to
disk rotation and ignore any radial motions (e.g., due to inflow/outflow) by
setting the expansion velocity to 0 km s${}^{-1}$ (i.e., the higher order $K_{m}$ terms).
We fit the model iteratively with different sets of parameters held fixed, while varying others freely.
We adopted this approach because each ring
would have six free parameters otherwise ($x_{\rm cen}$, $y_{\rm cen}$, $v_{\rm sys}$,
$i$, PA, $v_{\rm rot}$), which our data do not allow us to fix simultaneously, especially
because $v_{\rm rot}$ and $i$ are highly degenerate. Without doing so, models
struggle to converge to a solution555We have tested this by allowing $i$ and PA
also to vary across rings.. The fact that this approach is also adopted
in modeling the kinematics of local galaxies, where the data obtained
have much higher S/N and spatial resolution, shows that our data do not
offer such constraining power (e.g., Swaters et al., 2009; van Eymeren et al., 2009; Elson, 2014; Hallenbeck et al., 2014; Di Teodoro & Fraternali, 2015; Jovanović, 2017).
This approach is also adopted in fitting
low-S/N and coarser spatial resolution data obtained at high redshift
(see e.g., Shapiro et al. 2008).
Here, we minimize the set of freely varying parameters via least-squares fitting.
Except in the last iteration, the width of each ring is set to the beam size.
In the first iteration, the dynamical center ($x_{\rm cen},y_{\rm cen}$) and
systemic velocity ($v_{\rm sys}$) vary freely, whereas the inclination angle is fixed to
the average value found from the kinematic and morphological axes,
and the PA is fixed to the photometric/morphological PA.
We then constrain $i$, PA, and $v_{\rm rot}$ while fixing
$x_{\rm cen},y_{\rm cen}$, and $v_{\rm sys}$ to their weighted-average values
found in the previous iteration.
To better determine the inclination angle, we further fix the PA and fit for $i$ and $v_{\rm rot}$ only.
In the final run, we fix all parameters to the weighted averages found in the previous iterations
and only fit for $v_{\rm rot}$, and the width of each ring is set to half the beam size to sample the rotation curve.
From the model, we find a best-fit PA of 133.6$\arcdeg$ $\pm$ 0.6$\arcdeg$
(east of north).
and an inclination of $i$ = 41.3$\arcdeg$ $\pm$ 3.9$\arcdeg$.
After this determination, the best-fit parameters are used to form the model velocity field using the velfi task.
A residual image (Figure 11) is obtained by subtracting the model (after convolving with the beam) from the data.
The residual is largely uniform across the entire disk, with velocities varying by less than 100 km s${}^{-1}$,
consistent with the velocity dispersion map observed in Figure 5.
The relatively low residuals indicate
that the best-fit model is a reasonable description of the observed velocity field, and
that non-circular motions (e.g., streaming motions along unseen
spiral arms or bars, or large-scale tidal torquing from galaxy interactions)
are unlikely to be detected in the kpc-scale resolution data.
We note that beam smearing means that velocity information within the inner kpc region will be
largely lost in the data.
We fit an arctangent model (e.g., Courteau, 1997) to the rotcur rotation curve (RC).
The model is parameterized as:
$$V_{\rm rot}=V_{0}+\frac{2}{\pi}V_{a}\arctan\left(\frac{R}{R_{t}}\right),$$
(5)
where $V_{\rm rot}$ is the rotation velocity found with rotcur,
$V_{0}$ is the systemic velocity,
$V_{a}$ is the asymptotic velocity, and $R_{t}$ is the “turnover”
radius at which the rising part of the rotation curve begins to flatten.
We perform non-linear least-squares fitting using the Levenberg-Marquardt
algorithm to find the best-fit parameters.
We limit the turnover radius to 0$<$ $R_{t}$ $<$25 kpc
in order to keep this parameter within a physically meaningful range.
Using this model, we find $V_{a}$ = 503 $\pm$ 83 km s${}^{-1}$,
$R_{t}$ = 0.8 $\pm$ 0.3 kpc,
and $V_{0}$ = 0 $\pm$ 28 km s${}^{-1}$ (relative to the systemic redshift).
We thus find an inclination-corrected rotation velocity of
$v_{\rm rot}$ = 474 $\pm$ 78 km s${}^{-1}$ at a spatial offset of 8.8 kpc
(the extent of the ground state CO line emission; Table 3).
We note that the model underestimates the velocities at $R\gtrsim 4$ kpc because of the
outermost three data points at $>$6 kpc, which deviate from the trend of increasing velocity with radius.
Such a trend — a declining rotation curve with increasing galactocentric radius — has been reported in some
studies of high-$z$ galaxies (e.g., Genzel et al. 2017; Lang et al. 2017, cf. e.g., Tiley et al. 2018).
In our data, this trend is likely an artifact due to the limited S/N at those PV-positions
(i.e., low number of pixels fitted; see Figures 6 and 16 in Appendix §A).
In other words, the decreasing velocities seen at increasing radius in our target could easily be mimicked by
a lack of sensitivity to low surface brightness emission in the outer regions.
If we instead fit the arctangent model excluding these three data points,
we find an inclination-corrected rotation velocity of $v_{\rm rot}$ = 537 $\pm$ 83 km s${}^{-1}$ at 8.8 kpc
and an asymptotic velocity of $V_{a}$ = 617 $\pm$ 97 km s${}^{-1}$. Both models are consistent within the uncertainties.
Rotation curves from both arctangent models do not reach the
terminal velocity666Terminal velocity is not the same as asymptotic velocity, which
the arctangent model does constrain. (i.e., the flat part of the rotation curve).
Therefore, the rotation velocities inferred here may be lower limits only.
On the other hand, part of the rotation curve that is flattening is clearly detected in the PV-diagram (Figure 6).
This discrepancy is related to the fact that fitting models to velocity fields can underestimate
true rotation velocities,777This underestimation occurs because velocity fields are intensity-weighted and the tilted-ring model assumes that all the
gas in a ring is at a unique position along the LOS; however,
gas emission from other velocities along the LOS is blended within the beam.
Thus, the lower the resolution, the more likely the true velocities are underestimated by fitting models to the velocity fields.
and that the decreasing velocities seen in the outermost three data points of the rotation curve are of limited S/N.
This flattening part of the rotation curve detected in HXMM05 is likely to be mainly driven by
the dynamics of the parent dark matter halo, as in nearby galaxies;
we see no evidence indicating that HXMM05 is dominated by baryons from the data at hand.
Adopting the inclination-corrected $V_{a}$ as the maximum rotation velocity, we find that HXMM05 is consistent with
the gas Tully-Fisher relation found for nearby galaxies,
given its gas mass (see Table 7; McGaugh & Schombert 2015).
4.2.2 Envelope-Tracing Method
As an alternative approach to estimate the rotation velocity of HXMM05, we also use the
envelope-tracing (ET) method, where we fit models to
the PV diagram extracted along the kinematic major axis
(Figure 6; see review by Sofue & Rubin 2001).
The ET method attempts to trace out the material
that has the maximum tangential motion along each LOS
(see Figure 5 of Chemin et al. 2009 for a schematic depiction of this geometric effect).
We fit a third order ($h_{3}$) Gauss-Hermite
polynomial to a (Hanning-smoothed) spectrum extracted at each position along the PV cut (Figure 6)
to account for any asymmetries in the spectra.
The rotation curve (traced by the “envelope”) is derived from the terminal velocity ($v_{t}^{\rm obs}$) at which
8% of the total flux under the fitted curve is outside $v_{t}^{\rm obs}$.
In essence, this approach traces the
isophotes at each position along the kinematic major axis.
The innermost 1.5 kpc region of the PV diagram is steeply rising (Figure 6), which is due in part to
the facts that the velocity gradient in this region is changing rapidly
from positive to negative, and that contributions from multiple radii overlap in the inner roughly 1 kpc
(which remains unresolved at the $\sim$1.2 kpc resolution of the data).
Structures within the “envelope” modulo inclination and beam smearing effects may result from
the presence of spiral- or ring-like structures, or a clumpy gas distribution in HXMM05.
Based on the terminal velocity, we derive the
rotation velocity of HXMM05 using the following equation:
$$v_{\rm rot}=(v_{t}^{\rm obs}-v_{\rm sys})/\sin{(i)}-\sqrt{(\sigma_{\rm PSF}^{2%
}+\sigma_{\rm ISM}^{2})},$$
(6)
where $v_{\rm sys}$ is the systemic velocity determined from fitting a double-Gaussian to the [CII] spectrum (Figure 1),
$i$ is the inclination angle from rotcur,
$\sigma_{\rm PSF}$ is the spectral resolution, and $\sigma_{\rm ISM}$ is the velocity dispersion of the
gas (see e.g., Vollmer et al. 2016 and Sofue 2017).
Here, we adopt
the observed velocity dispersion of $\sigma_{v}$ = 75 km s${}^{-1}$ as the
subtracted term.
We then re-sample the rotation profile every half beam (instead of every pixel) and show the output
rotation curve in Figure 6.
We find an inclination-corrected rotation velocity of $v_{\rm rot}$ = 616 $\pm$ 100 km s${}^{-1}$ at
the last measured radius of the rotation curve ($R$ = 8.0 kpc), which is consistent with the rotation
velocity of $v_{\rm rot}$ = 537 $\pm$ 83 km s${}^{-1}$ derived from the arctangent model within the uncertainties.
4.3. PDR modeling
Photo-dissociation regions (PDRs) are the warm and dense surfaces of molecular clouds exposed to
FUV photons with energies 6 $<$ $h\nu$ $<$ 13.6 eV escaping from HII regions.
In PDRs, gas temperatures and densities are typically
$T$ = 100 $-$ 500 K and $n$ = $10^{2-5}$ cm${}^{-3}$. Since the [CII] 158 $\micron$ line is the primary coolant in PDR conditions satisfying
$n$ $\lesssim$ 10${}^{3}$ cm${}^{-3}$ and $T\lesssim 100$ K, [CII] and other ISM lines near or in PDRs
are sensitive probes of the physical conditions of the PDR gas and
the intensity of the ambient interstellar radiation field
($G_{0}$; conventionally expressed in units of 1.6$\times 10^{-3}$ erg cm${}^{-2}$ s${}^{-1}$, the Habing flux; e.g., Hollenbach & Tielens, 1999).
Using the [CII] and CO line luminosities
and the PDR model grids from Tielens & Hollenbach (1985) and
Kaufman et al. (1999, 2006)999Available through the PDR Toolbox888http://dustem.astro.umd.edu described by Pound & Wolfire (2008) and Wolfire et al. (2010)
, we constrain the globally-averaged $G_{0}$, hydrogen density ($n$), and surface temperature
for the PDRs in HXMM05 101010While it is physically unrealistic to model an entire galaxy as a single PDR,
we infer the
$G_{0}$ and $n$ values of HXMM05 in a globally-averaged sense to facilitate
comparison with other studies in the literature..
We adopt CO grids from an updated version of the code
(Hollenbach et al. 2012; M. Wolfire 2017, private communication) that is
merged with the Meudon code (Le Petit et al., 2006)
for a more detailed treatment of H${}_{2}$ chemistry and thermal balance.
The observed line ratios are shown in Figure 12 as functions of $G_{0}$ and $n$.
Since a fraction of the [CII] emission in the ISM arises outside of PDRs, and we lack other diagnostic lines to
determine this fraction in HXMM05, we adopt a canonical value of 30% to account for this non-PDR contribution
(Carral et al. 1994; Colbert et al. 1999; Malhotra et al. 2001; Oberst et al. 2006; see also Pavesi et al. 2016, 2018; Zhang et al. 2018b).
CO line emission is typically optically thick (especially the low-$J$ lines), and so we
multiply their line intensities by a factor of $2$ to account for the emission on the other side of the surface.
Corrections are incorporated into the line ratios as uncertainties (filled regions in Figure 12).
The best-fit model is determined based on the global minimum $\chi^{2}$,
corresponding to $\log\,n$ = 4.5 cm${}^{-3}$ and $\log\,G_{0}$ = 2.25.
Based on the $\chi^{2}$ surface, the uncertainties in both $n$ and $G_{0}$
are approximately an order of magnitude.
As discussed by Röllig et al. (2007), physical parameters inferred from any PDR models should not be taken too literally,
since they are subject to differences depending on the assumptions adopted and the implementation of microphysics in the code.
Nevertheless, we use the best-fit parameters as simple approximations to compare HXMM05 with other galaxies.
The lower $G_{0}$ solution ($G_{0}<$1) implied by the mid-$J$ CO and
[CII]-to-FIR ($L_{\rm[CII]}$/$L_{\rm FIR}$) luminosity ratios disagrees with that implied by the CO (10$\rightarrow$9)-to-(1$\rightarrow$0)
and [CII]-to-CO ($J$ = 1 $\rightarrow$ 0) luminosity ratios.
We reject this low $G_{0}$ solution since it would require a physically enormous emitting
region to account for the observed high $L_{\rm FIR}$ in HXMM05 ($G_{0}\propto\mbox{$L_{\rm FIR}$}/D^{2}$; Wolfire et al. 1990).
Assuming the values for M82 ($D\simeq$ 300 pc, $G_{0}\simeq$ 1000, $L_{\rm FIR}$ $\simeq$ 2.8$\times 10^{10}$ $L_{\odot}$),
the solution with $G_{0}\simeq$ 0.2 would require an emitting region $D$ = 600 kpc in size, contrary to what is observed. On the other hand, the best-fit $G_{0}$ $\sim$ 200
corresponds to an emitting region of $D\simeq$ 20 kpc,
which is more consistent with the sizes observed in [CII] and CO ($J$ = 1 $\rightarrow$ 0) line emission (Table 3).
The FUV radiation field intensity of HXMM05 is thus stronger than the local Galactic interstellar
radiation field intensity by a factor of around 200, comparable to the values found in
nearby normal star-forming galaxies and those found in some other DSFGs (e.g., Malhotra et al., 2001; Wardlow et al., 2017).
The best-fit $G_{0}$ and $n$ together suggest a surface temperature of $T_{\rm surf}$ = 290 K for the PDR.
We approximate the PDR pressure using $P\propto nT$, yielding $P/k_{B}$ = 9.0$\times 10^{6}$ cm${}^{-3}$ K.
We note that an offset is found between the CO ($J$ = 10$\rightarrow$9)/CO($J$ = 1$\rightarrow$0) line and the other luminosity ratios
in the $\log n-\log G_{0}$ plane. This
offset likely results from the fact that CO ($J$ = 10 $\rightarrow$ 9) emission preferentially traces a more highly-excited phase of the ISM
than the other lines (e.g.,
due to mechanical heating or X-ray heating; see also §5.6).
However, with the data at hand, the presence and properties of an X-ray dominated region (and/or a second PDR component,
and/or shock excitation) are unconstrained and indistinguishable from a simple single PDR.
The PDR properties thus suggest that the
high far-IR luminosity of HXMM05 ($>$10${}^{13}$ $L_{\odot}$) may result from extended star formation, with only a modest FUV radiation field intensity.
This is in stark contrast with the compact starbursts seen in the cores of many nearby ULIRGs (less than
a few hundred parsecs),
which are found to have stronger FUV radiation fields compared to HXMM05 (e.g., Stacey et al., 2010).
The inferred PDR conditions also suggest that HXMM05 is unlikely to host an AGN or a powerful quasar,
consistent with §5.1.
5. Discussion
Since X-Main and X-NE remain spatially unresolved from each other in the IR photometry and most of the
spectral line data (except in CO $J$ = 1$\rightarrow$0 and [CII] emission), we discuss the
properties of HXMM05 as a combined system in the following sections.
5.1. No Evidence of an AGN in HXMM05
Given the upper limits imposed on the X-ray luminosity of HXMM05, we
find no evidence for the presence of a powerful AGN,
but we cannot rule out the possibility of a heavily dust-obscured AGN in HXMM05 or
a Seyfert galaxy nucleus with modest X-ray emission.
To assess the reliability of the stellar mass derived from SED fitting,
we examine if the mid-IR spectral slope of HXMM05 ($S_{\nu}$$\propto\nu^{\alpha}$) may be consistent with a low X-ray luminosity AGN
(see e.g., Stern et al., 2005; Donley et al., 2007).
We fit a power-law to the IRAC 5.8- and MIPS 24-$\micron$ photometry, which correspond to
rest-frame 1.5$-$6.0 $\micron$.
We find a spectral index $\alpha_{\rm 1.5-6,rest}$ = 1.46 $\pm$ 0.58,
which is much flatter than those observed in AGN host galaxies111111
Spectral indices reported in the literature are based on photometry taken at 3.6$-$8.0 $\micron$, which
correspond to the closest wavelength range used here for HXMM05 in the rest-frame.
(Stern et al., 2005; Donley et al., 2007, 2008), suggesting that the NIR emission in HXMM05 may be dominated by stellar emission.
Thus, we assume in the following that all the NIR emission
detected in the IRAC channels 3 and 4 bands arises solely from the starlight in HXMM05.
That is, the accuracy of the stellar mass estimated is dominated by the uncertainty on the IMF adopted (see e.g., Zhang et al., 2018a).
If HXMM05 were to host an AGN, however, its stellar mass and SFR would be overestimated.
5.2. ISM Properties
5.2.1 Stellar Mass and Specific Star Formation Rate
We find an unusually high stellar mass of 10${}^{12}$ $M_{\odot}$ for HXMM05 from SED modeling.
The stellar mass estimate relies heavily on IRAC channels 3 and 4 (i.e., rest-frame 1.4 and 2.0 $\micron$)
photometry.
Previous studies have shown that rest-frame $K$-band (2.2 $\micron$) photometry appears to be a
reliable proxy121212Since the dust optical depth of HXMM05 at rest-frame 158 $\micron$ is $\tau_{\nu}$ $\sim$ 1, its $K$-band emission
could be highly attenuated, unless most of the
starlight is less attenuated than the dust (e.g., if the latter is dominated by compact star-forming knots
and the former is much more extended), which
remains possible given its dust morphology, gas excitation, and $G_{0}$. for the
stellar mass of galaxies, since photometry in this band is relatively insensitive to the past star-formation histories of galaxies
(e.g., Kauffmann & Charlot 1998; Lacey et al. 2008; cf. Kannappan & Gawiser 2007),
and because NIR emission is less affected by dust extinction compared to optical light.
In particular, the difference in the $K$-band luminosity between initial burst and constant
star formation models is less than a factor of 3 (e.g., Pérez-González et al. 2008).
The main systematic uncertainties associated with $M_{*}$ are therefore
the star-formation histories assumed, the IMF and stellar population synthesis model adopted,
and the fact that differential dust extinction is not captured in simple energy balance models (e.g., magphys)131313Alternatively, hot dust emission due to a deeply buried AGN could contribute to the observed IR luminosity, and thus
lead to an overestimate of $M_{*}$ (but see Michałowski et al. 2014, who find insignificant effects of AGN on the SED-derived
$M_{*}$)..
Nevertheless, the stellar masses inferred from magphys are found to match the true masses of mock galaxies in
simulations fairly well
(e.g., Michałowski et al., 2014; Hayward & Smith, 2015; Smith & Hayward, 2015), unless the dust attenuation in HXMM05 is underestimated by magphys.
Taken at face value141414Note, however, that even assuming no AGN is present in HXMM05,
the $M_{*}$ estimate is accurate to only $\lesssim$0.5 dex (see also Michałowski et al. 2014)
on top of the large statistical error bars reported in Table 6., the high stellar mass suggests that a substantial fraction of
stars may have already formed in some massive galaxies by $z$ = 3 (approximately 2 Gyr after the Big Bang).
The relatively tight “correlation” found between SFR and $M_{*}$ for
star-forming galaxies at low- and high-$z$ suggests that the majority of galaxies are
forming stars over a long duty cycle in a secular mode (e.g., Rodighiero et al., 2011; Lehnert et al., 2015).
The specific SFR of sSFR = 2.37${}^{+4.31}_{-1.43}$ Gyr${}^{-1}$ of HXMM05 is consistent with the star-forming “main-sequence” (SFMS) within the
scatter of the MS relations derived by
Tacconi et al. (2013), Lilly et al. (2013), Speagle et al. (2014), and Schreiber et al. (2015), if we extrapolate them to higher masses
and include the uncertainties associated with the SFR and stellar mass inferred for HXMM05.
One possible caveat is the applicability of the SFMS relation, and whether
our current knowledge of
the MS is meaningful at high stellar mass (10${}^{12}$ $M_{\odot}$).
In this paper, we only consider HXMM05 as a MS galaxy for the sake of comparing its ISM properties
with other high-$z$ MS and starburst systems.
5.2.2 Gas Mass, Gas-to-Dust Ratio, and Metallicity
Using the CO ($J$ = 1 $\rightarrow$ 0) line intensities (Table 3) and
assuming a CO luminosity-to-H${}_{2}$ mass conversion factor of $\alpha_{\rm CO}$ = 0.8 $M_{\odot}~{}($K km pc${}^{2})^{-1}$ (e.g., Downes & Solomon, 1998),
we derive molecular gas masses of
$M_{\rm gas}^{\rm X-Main}$ = (1.68 $\pm$ 0.43)$\times 10^{11}$ $M_{\odot}$ for X-Main,
$M_{\rm gas}^{\rm NE}$ = (6.52 $\pm$ 1.63)$\times 10^{10}$ $M_{\odot}$ for X-NE,
and $M_{\rm gas}^{\rm total}$ = (2.48 $\pm$ 0.65)$\times 10^{11}$ $M_{\odot}$ for the entire system (Figure 2).
Using the molecular gas mass of the system, we find a gas-to-dust mass ratio of
GDR ($\alpha_{\rm CO}$ $/$ 0.8)${}^{-1}$ = 50$-$145, which is
consistent with those measured in the Milky Way, local spiral galaxies,
ULIRGs, and DSFGs (Draine & Li, 2007; Wilson et al., 2008; Combes et al., 2013; Bothwell et al., 2013).
Based on the fundamental metallicity relation (FMR)
determined by Mannucci et al. (2010), we infer a gas-phase metallicity of $Z$ = 12$+$log(O/H) = 9.07 for HXMM05 151515This assumes that the FMR relation holds up to $z$ = 3 and a
stellar mass of 10${}^{12}$ $M_{\odot}$ (see e.g., Steidel et al. 2014).,
which is comparable to the that of the $z$ = 4 SMG GN20 (Magdis et al., 2011).
We express the metallicity on the Pettini & Pagel (2004; PP04) scale using the calibration proposed by
Kewley & Dopita (2002) and Kewley & Ellison (2008), yielding $Z_{\rm PP04}$ = 8.74.
The range of GDR derived for HXMM05 is consistent with the best-fit GDR $-$ $Z_{\rm PP04}$ relation presented by Magdis et al. (2011), which was determined for
a sample of local galaxies studied by Leroy et al. (2011). If the CO-to-H${}_{2}$ conversion
factor were $\alpha_{\rm CO}$$>$0.8 $M_{\odot}~{}($K km pc${}^{2})^{-1}$, then HXMM05 would lie above this relation.
By applying the $\alpha_{\rm CO}$ $-$ $Z$ relations found by Leroy et al. (2011) and Genzel et al. (2012),
we find a range of $\alpha_{\rm CO}$ of 1.4$-$1.9 $M_{\odot}~{}($K km pc${}^{2})^{-1}$, which would increase
the molecular gas mass by a factor of 1.7$-$2.4 compared to the value assumed here.
5.2.3 Dust, Gas, and Stellar Mass Ratios
The dust-to-stellar mass ratio (DSR) measures the amount of dust
per unit stellar mass that survives all dust destruction processes in a galaxy (e.g., type II SN explosions).
The DSR of HXMM05 is 2.3${}^{+2.7}_{-1.8}$$\times 10^{-3}$, which is within the range measured in local
star-forming galaxies and ULIRGs, but is among the lowest measured in
intermediate-$z$ ULIRGs and quasars (e.g., Dunne et al., 2011; Combes et al., 2013; Leung et al., 2017).
This ratio is also lower than those measured in
DSFGs at similar redshifts (e.g., Magdis et al., 2011; Calura et al., 2017).
The molecular gas-to-stellar mass ratio of HXMM05 is $M_{\rm gas}$/$M_{*}$ = 0.2${}^{+0.2}_{-0.1}$, which is
higher than those observed in local SFGs and early-type galaxies (e.g., Leroy et al., 2008; Saintonge et al., 2011; Young et al., 2014).
Previous studies
report a positive evolution in this ratio with redshift (e.g., Tacconi et al., 2010; Davé et al., 2012).
The $M_{\rm gas}$/$M_{*}$ ratio of HXMM05 is lower than those typically measured in
other high-$z$ SFGs and DSFGs at $z$ $>$ 1.2, and is
the lowest161616The $M_{\rm gas}$/$M_{*}$ ratio is susceptible to
uncertainties in the $\alpha_{\rm CO}$ conversion factor and in stellar mass.
If we were to adopt a conversion factor of 4.6 $M_{\odot}~{}($K km pc${}^{2})^{-1}$, the gas-to-stellar mass ratio of HXMM05 would be consistent with the
expected redshift evolution of the molecular gas mass content in galaxies (Geach et al., 2011).
found among massive galaxies at
$z$ $\simeq$ 3 to date (Leroy et al., 2008; Tacconi et al., 2010; Daddi et al., 2010; Geach et al., 2011; Decarli et al., 2016).
The low DSR and gas-to-stellar ratio of HXMM05 may indicate that it is a relatively evolved system, in which
a large fraction of its gas has been converted into
stars and a large fraction of dust has been locked up in stars.
That said, as discussed in §5.1 and §5.2.1, it is possible that the stellar mass maybe overestimated.
5.3. Star Formation Efficiency and Gas Depletion Timescale
To first order, the star formation efficiency (SFE) measures the star formation rate per unit mass of molecular gas available in a galaxy.
The SFE of HXMM05 is $L_{\rm FIR}$/$L^{\prime}_{\textrm{CO(1-0)}}$ = 91 $\pm$ 25 $L_{\odot}$ (K km s${}^{-1}$ pc${}^{2}$)${}^{-1}$ (or 13 $\pm$ 4 Gyr${}^{-1}$), which is
slightly higher than but consistent with the range found in
nearby active star-forming spiral galaxies
($z$ $<$ 0.1; Gao & Solomon, 2004; Solomon & Vanden Bout, 2005; Stevens et al., 2005; Leroy et al., 2008; Wilson et al., 2009; Leroy et al., 2013) and
high-$z$ massive disk-like galaxies (Daddi et al., 2008, 2010; Aravena et al., 2014).
Assuming that the star formation in HXMM05 continues at its current rate without gas replenishment,
the gas will be depleted in $\tau_{\rm depl}$ = 72 $\pm$ 27 Myr171717However, the gas reservoir would last 6 times longer if we
had instead adopted $\alpha_{\rm CO}$ = 4.6 $M_{\odot}~{}($K km pc${}^{2})^{-1}$ in deriving $M_{\rm gas}$.,
comparable to the depletion times in starburst systems.
HXMM05 thus lies between SFMS and starburst galaxies in the so-called “transition region”
on the integrated version of the “star-formation law” (i.e., $L_{\rm FIR}$ $-$ $L^{\prime}_{\textrm{CO(1-0)}}$ relation; Daddi et al. 2010; Magdis et al. 2012; Sargent et al. 2014).
We conclude that the gas depletion timescale in HXMM05 is short compared to those of SFMS galaxies at high redshift.
5.4. Dynamical Mass
The rotation curve of a galaxy reflects
its dynamics due to the total (i.e., baryonic and dark matter) enclosed mass.
We estimate the total dynamical mass enclosed within 8 kpc
using $M_{\rm dyn}$ $=v_{\rm rot}^{2}~{}R/G$. We find an inclination-corrected dynamical mass of $M_{\rm dyn}$ = (7.7 $\pm$ 3.1)$\times 10^{11}$ $M_{\odot}$.
Taken at face value181818The dominant systematic uncertainties in $M_{\rm dyn}$ are the
uncertainties in the rotation velocity due to
the potential presence of inflows or outflows, in the
velocity dispersion, and in our assumption that HXMM05 is a thin disk with negligible scale height., we find a molecular gas mass fraction of
$f_{\rm gas}^{\rm dyn}$ = $M_{\rm mol}$/$M_{\rm dyn}$ = 18 $\pm$ 8%
using the gas mass of
the main component of HXMM05 only ($M_{\rm gas}^{\rm X-Main}$) and
33 $\pm$ 15% using the total molecular gas mass of the system ($M_{\rm gas}^{\rm total}$).
The dynamical mass is consistent with the stellar mass within the considerable uncertainties.
Since the dynamical masses derived for most other high-$z$ galaxies in the literature are based on marginally
resolved or unresolved observations,
we also estimate the dynamical mass of HXMM05 using the isotropic estimator
$M_{\rm dyn}^{\rm iso}$ = 2.8$\times 10^{5}$ $\Delta v_{\rm FWHM}^{2}$ $R_{\rm FWHM}$ (e.g., Engel et al., 2010),
where $\Delta v_{\rm FWHM}$ is the CO ($J$ = 1 $\rightarrow$ 0) line FWHM measured by fitting a single-Gaussian to the line profile
in units of km s${}^{-1}$, and $R_{\rm FWHM}$ is the FWHM extent of the galaxy measured from CO ($J$ = 1 $\rightarrow$ 0) line emission in units of kpc.
Here, we adopt the linewidth of HXMM05 excluding X-NE as $\Delta v_{\rm FWHM}$
and the average between the major and minor axes
as the extent ($R_{\rm FWHM}$ = 7.6 kpc).
We thus find an inclination-corrected dynamical mass of $M_{\rm dyn}^{\rm iso}$ = (3.89 $\pm$ 1.09)$\times 10^{12}$ $M_{\odot}$,
yielding a molecular gas mass fraction of
$f_{\rm gas}^{\rm dyn,iso}$ = 4.3 $\pm$ 2.9% using $M_{\rm gas}^{\rm HXMM05}$
for X-Main only and 5.3 $\pm$ 2.4% for the HXMM05 system.
However, given the evidence of disk-like rotation for HXMM05, we consider
the first dynamical mass estimate (i.e., $M_{\rm dyn}$) to be more reliable.
5.5. Star Formation Rate and Gas Surface Densities and the Spatially-resolved Star-formation Law
The Schmidt-Kennicutt relation (i.e., the star-formation law) is an empirical
relation relating SFR and gas surface densities as
$\Sigma_{\rm SFR}\propto\Sigma_{\rm gas}^{N}$, where $N\simeq$1.4 is established
from measurements of different nearby galaxy populations
(e.g., Schmidt, 1959; Kennicutt, 1998a, 2008).
Based on the SFR of 2900${}^{+750}_{-595}$ $M_{\odot}$ yr${}^{-1}$ and the sizes and flux ratio of XD1 and XD2 at 635 $\micron$,
we find star formation rates of SFR = 1500 and 860 $M_{\odot}$ yr${}^{-1}$ and
SFR surface densities of $\Sigma_{\rm SFR}^{\rm XD1}$ = 210 $M_{\odot}$ yr${}^{-1}$ kpc${}^{-2}$ and
$\Sigma_{\rm SFR}^{\rm XD2}$ = 120 $M_{\odot}$ yr${}^{-1}$ kpc${}^{-2}$
for XD1 and XD2, respectively.
These SFR surface densities are elevated compared to those measured in the circumnuclear starburst regions
of nearby galaxies (Kennicutt, 1998b), consistent with the overall somewhat shorter gas depletion timescale
but are much lower than those observed in high-$z$ “maximal starburst”-like galaxies, such as
the $z$ = 5.3 SMG AzTEC-3, the $z$ = 5.7 HyLIRG ADFS 27, and the $z$ = 6.3
HFLS3 (Riechers et al., 2013, 2014b, 2017; Oteo et al., 2017b).
For the low surface brightness diffuse dust component, which is
almost as extended as the [CII] line emission (Figure 8), we find
a source-averaged SFR surface density of
$\Sigma_{\rm SFR}\,=\,$10 $M_{\odot}$ yr${}^{-1}$ kpc${}^{-2}$
(or 60 $M_{\odot}$ yr ${}^{-1}$ kpc${}^{-2}$ including the pair of nuclei).
Based on the CO($J$ = 1$\rightarrow$0) line-emitting source size of (8.8 $\pm$ 2.9) $\times$ (6.4 $\pm$ 3.5) kpc and the
total molecular gas mass measured in the HXMM05 system,
the molecular gas surface density
is $\Sigma_{\rm gas}$ $=$ (590 $\pm$ 410) $\times$ ($\alpha_{\rm CO}$/0.8) $M_{\odot}$ pc${}^{-2}$.
We thus find that HXMM05 lies along the “starburst sequence” of the Schmidt-Kennicutt relation reported
by Bouché et al. (2007).
Accounting for uncertainties in SFR and gas surface densities,
we find that HXMM05 lies in the same region of parameter space as the sub-regions of GN20 and the
$z$ = 2.6 SMG SMM J14011$+$0252 (Sharon et al., 2013; Hodge et al., 2015).
5.6. CO Gas Excitation
Due to the different physical conditions
required to excite the various rotational transitions of CO, flux ratios between the low- and high-$J$ CO
lines are sensitive to the molecular gas volume densities and kinetic temperatures.
With the data at hand (i.e., only four CO lines spanning the CO “ladder” up to $J$ = 10$\rightarrow$9),
we do not attempt to fit radiative transfer models to the observed line fluxes. Instead,
we compare the line ratios measured in HXMM05 (Table 7) with those of other galaxy populations to
study the connection between the SFR surface density and the gas excitation in HXMM05 (since SFR surface density is tightly linked to gas density, temperature, and line optical depths).
The global SFR of the HXMM05 system is comparable to those of the most luminous DSFGs known, but
their different CO spectral line energy distribution (SLED)
shapes indicate that the underlying physical conditions in their ISM may be different.
As shown in Figure 13, the gas excitation of the HXMM05 system probed by transitions up to $J_{\rm upper}$ = 5
is lower than those typically observed in nuclear starbursts, SMGs, and quasars,
but is comparable to those observed in the outer disk of the Milky Way (despite HXMM05’s much higher SFR), and
those observed in high-$z$ BzK disks. Such a relatively modest gas excitation
is in accord with the modest source-averaged SFR surface density of HXMM05 and
its PDR gas conditions (§4.3) — i.e.,
its total SFR of 2900 $M_{\odot}$ yr${}^{-1}$ is spread across the entire disk (as seen in the co-spatial gas and FIR dust distribution),
and its extended star formation is embedded within a medium with only moderate radiation flux and pressure.
Including the highest-$J$ line probed with the data at hand ($J$ = 10$\rightarrow$9), we find that the overall
SLED shape (and thus gas excitation) of HXMM05 resembles that of the local merger-driven ULIRG Arp 220.
This may suggest that the molecular ISM of the HXMM05 system is composed of (at least) two gas-phase components —
a diffuse extended cold component and a dense warm component.
If we exclude X-NE, we find that the molecular gas in X-Main (i.e., grey symbols in Figure 13) is more highly excited
than the system overall, which is comparable to other high-$z$ DSFGs (e.g., Riechers, 2011; Sharon et al., 2016).
As noted in §3.1, the true CO ($J$ = 5 $\rightarrow$ 4) flux may be a factor of two higher.
In this case, the excitation conditions of HXMM05 and X-Main would be consistent with
(and possibly more excited than) those of other DSFGs.
However, higher fidelity data are needed to confirm this scenario.
5.7. Morphology and Kinematics of the [CII] and CO Emission
The cold molecular gas reservoir of HXMM05 is approximately 9 kpc $\times$6 kpc in diameter, which is comparable
to the mean size of nearby disk-like U/LIRGs
(Ueda et al., 2014),
and of ULIRGs in general (Gao & Solomon 1999).
Similarly extended gas reservoirs have also been observed
in some other high-$z$ galaxies (Daddi et al., 2010; Ivison et al., 2010a; Riechers et al., 2011a; Ivison et al., 2011; Hodge et al., 2012).
The CO ($J$ = 1 $\rightarrow$ 0) FWHM linewidth of HXMM05 is much broader than those typically observed in
“normal” star-forming galaxies at low- and high-redshifts,
ULIRGs,
and high-$z$ gas-rich galaxies (Solomon et al., 1997; Solomon & Vanden Bout, 2005; Daddi et al., 2010; Danielson et al., 2011; Ivison et al., 2011; Riechers et al., 2011a, b; Carilli & Walter, 2013; Combes et al., 2013; Sharon et al., 2016),
although galaxies with similarly broad line do exist (e.g., J13120$+$4242; Hainline et al., 2006; Riechers et al., 2011a).
Similarly, the [CII] line of HXMM05 ($\Delta v$ = 667 $\pm$ 46 km s${}^{-1}$)
is also broader than those seen in many other high-$z$ galaxies apart from
major mergers (e.g., Ivison et al., 2010b; Walter et al., 2012; Ivison et al., 2013; Riechers et al., 2013, 2014a; Neri et al., 2014; Rhoads et al., 2014).
The velocity dispersion traced by the [CII] line emission in HXMM05 is the highest in the central 0$\farcs$2 region,
as seen in Figure 5.
The higher dispersion at the center may indicate gas dynamics
affected by late-stage merger activity, intense cold gas accretion/inflows, or
enhanced turbulence caused by an undetected AGN,
or the fact that systemic motions/radial velocities change abruptly in this region,
where the velocity curve is also the steepest.
We therefore estimate the
gas dispersion in the extended part of HXMM05 based on the velocity dispersion observed in its outskirts,
yielding $\sigma_{v}\simeq$75 km s${}^{-1}$.
We estimate the $v/\sigma$ stability parameter for HXMM05 using the maximum rotation velocity and
the observed velocity dispersion of $\sigma$ = 75 km s${}^{-1}$ (see §3.2), yielding $v/\sigma$ = 7 $\pm$ 3.
This ratio is closer to those measured in nearby disk galaxies ($\simeq$10) than in
other high-$z$ galaxies (e.g., Genzel et al. 2006; Förster Schreiber et al. 2006; Cresci et al. 2009; Gnerucci et al. 2011; Schreiber et al. 2017).
This distinction may suggest that the ISM of HXMM05 is not as turbulent as other high-$z$ galaxies studied to date (e.g., Law et al., 2009; Jones et al., 2010; Swinbank et al., 2011) ---
perhaps a result of its lower gas mass fraction191919Compared based on $f_{\rm gas}^{\rm dyn,iso}$. compared to other high-$z$ galaxies.
However, in most high-$z$ studies with reliable $v/\sigma$ estimates (requiring spatially resolved information),
the ratio is typically derived from stellar kinematics
(examples based on CO line emission are still limited in number; e.g., Swinbank et al. 2011).
Determining the gas stability of galaxies by imaging their molecular gas reservoirs
is more meaningful for characterizing their prospects for star formation, since molecular gas is the raw fuel for star formation.
In other words, the stability of gas against gravitational collapse is more closely linked to star formation than the velocity structure of the existing stellar component, which may (re-)settle on a different timescale
from the gas after perturbations.
5.8. Dust
5.8.1 Morphology and Optical Depth
HXMM05 remains undetected in deep UV and optical images,
indicating that it is highly dust obscured, consistent with its rest-frame 158$\micron$ optical depth of $\tau_{\nu}\simeq$1
(determined from SED modeling; §4.1).
This optical depth exceeds those of most “normal” star-forming galaxies and nearby
disk galaxies, but is similar to that seen in
Arp 220 and high-$z$ starburst galaxies — e.g., HFLS3, AzTEC-3, and ADFS 27 (Riechers et al., 2013, 2014b, 2017).
The dust emission morphologies at 635 and 870 $\micron$ appear different (Figure 8).
While two compact dust components are found to be embedded within
an extended component at 635 $\micron$, only one compact component
coincides with XD1 at 870 $\micron$.
The second 635 $\micron$ dust peak, XD2, is 1.8 times fainter than XD1 at its peak flux (see Table 4).
If XD1 and XD2 were to have the same peak flux ratio at 635 and 870 $\micron$, we
would expect
a peak flux density of 6.2 mJy beam${}^{-1}$ for XD2 at 870 $\micron$, which we would have detected at $>$20 $\sigma$
significance.
Hence, the non-detection of XD2 at 870 $\micron$ may be a result of
the lower dust column density at 870 $\micron$, where the emission is optically thin on average based on202020The
optical depth derived from
a galaxy-averaged SED model is luminosity-weighted, and thus, biased toward compact dust components.
The true optical depth is likely to be even lower in the outskirts of a galaxy.
the best-fit dust SED model ($\tau_{\nu}$ = 0.54).
5.8.2 Interpretation of the compact dust components
The compact dust components, XD1 and XD2, detected at 635 $\micron$
could be two regions of intense star formation, or the remnant cores from
a previous merger (e.g., Johansson et al., 2009).
At the positions of the double nuclei,
the velocity field of HXMM05 is the steepest (see markers in Figure 5), but we find no obvious signs of a misaligned velocity gradient at their positions,
which would be expected for the latter scenario.
However, the velocity field is only a first-order representation of the kinematics of a galaxy,
since it is calculated based on intensity-weighted LOS velocities and is affected by the limited
spatial resolution of the data (similarly for the velocity dispersion map).
Thus, it will not capture the full kinematics in the system.
As such, we cannot rule out the possibility that the
double nuclei may be the cores of a pair of progenitor galaxies,
where the gas disk may have reconfigured itself into rotation already (e.g., Springel & Hernquist, 2005; Robertson et al., 2006; Narayanan et al., 2008; Robertson & Bullock, 2008; Hopkins et al., 2009).
Such a scenario would be reminiscent of the nearby ULIRG Arp 220 (e.g., Sakamoto et al., 2008; Scoville et al., 2017),
but with a greater separation between the pair in HXMM05.
Alternatively, if the dust peaks were truly giant star-forming “clumps” that are virialized, we would expect their velocity
dispersions to be $\sigma_{v}\simeq$ 40 km s${}^{-1}$ or $\simeq$ 400 km s${}^{-1}$ based on the size-linewidth relations found for
local GMCs in a quiescent environment or the Galactic center, respectively (Larson, 1981).
As shown in Figure 5, the observed velocity dispersion in the nuclei of HXMM05 is 160$-$200 km s${}^{-1}$. Thus, the
dust peaks are unlikely to be virialized clumps.
Similarly, a scenario in which XD1 and XD2
correspond to the “twin peaks” produced in response to an
$m=2$ (i.e., bar or oval)
perturbation (see e.g., Kenney et al., 1992)
is disfavored for two main reasons:
the lack of obvious non-circular motions (§4.2.1), and
the pronounced asymmetry in the 635/870 $\micron$ flux ratio of XD1 and XD2 (i.e., implying differences in
their optical depths and dust column densities).
5.9. [CII] and FIR Luminosity Ratio
The $L_{\rm[CII]}$/$L_{\rm FIR}$ ratio measures the fraction of far-UV photons
that is heating up the gas versus that deposited onto dust grains.
We find a $L_{\rm[CII]}$/$L_{\rm FIR}$ ratio of 0.20 $\pm$ 0.03% for HXMM05.
Thus, HXMM05 lies in the same region of parameter space as nearby star-forming galaxies and LIRGs,
despite its two orders of magnitude higher $L_{\rm FIR}$ (e.g., Stacey et al., 2010).
This ratio is consistent with those measured in other high-$z$ star formation-dominated
galaxies with similar far-IR luminosities in the $L_{\rm[CII]}$/$L_{\rm FIR}$ $-$ $L_{\rm FIR}$ plane
(cf. nearby ULIRGs and high-$z$ quasars; Malhotra et al., 2001; Hailey-Dunsheath et al., 2010; Stacey et al., 2010; Wang et al., 2013; Díaz-Santos et al., 2013; Zhang et al., 2018b),
suggesting that HXMM05 is dominated by extended star formation rather than
a compact starbursts or AGN (see also §2.6.9).
This evidence is consistent with the extent observed in its gas and dust emission.
5.10. Spatially Resolved $L_{\rm[CII]}$/$L_{\rm FIR}$ Map and Star Formation
We investigate the spatially resolved [CII]-to-FIR luminosity ratio in HXMM05 on 1-kpc scale to examine the connection
between SFR and [CII] surface densities.
To create the surface density plots in Figures 14 and 15, we clipped
both the [CII] and the 635 $\micron$ continuum maps at 3$\sigma$212121
We also test our results with less clipping (at 1$\sigma$) to
confirm that the trends and relationships found are not artificial or biased because of “excessive” clipping..
Essentially, the notion of using the [CII] luminosity as a proxy for SFR relies on the assumption
that [CII] dominates the cooling budget of the neutral ISM,
in which heating is dominated by the photoelectric effect of UV photons from young and massive stars.
We show the $\Sigma_{\rm SFR}-\Sigma_{\rm[C{\scriptsize II}]}$ relation for HXMM05 in the left panel of Figure 15.
The large scatter suggests that [CII] emission traces both
star-forming regions and “diffuse” gas reservoirs.
The trend of decreasing $L_{\rm[CII]}$/$L_{\rm FIR}$ at high $\Sigma_{\rm SFR}$ suggests that
the former is suppressed in compact, high-SFR surface density regions (Figure 14).
On the other hand, we find a tighter relation between
$\Sigma_{\rm[C{\scriptsize II}]}/\Sigma_{\rm FIR}$ and $\Sigma_{\rm FIR}$, as shown in the right panel of Figure 15.
We fit power-laws of the forms $\Sigma_{\rm SFR}$ = A$\Sigma_{\rm[C{\scriptsize II}]}^{N}$ and
$L_{\rm[C{\scriptsize II}]}/L_{\rm FIR}$ = A$\Sigma_{\rm FIR}^{N}$ to our data for HXMM05, and
find the following best-fit relations:
$$\begin{split}\displaystyle\log{\left(\frac{\Sigma_{\rm SFR}}{M_{\odot}~{}%
\textrm{yr}^{-1}~{}\textrm{pc}^{-2}}\right)}&\displaystyle=-3.0~{}(\pm 0.3)~{}%
+\\
&\displaystyle 1.4~{}(\pm 0.1)\times\log{\left(\frac{\Sigma_{\rm[C{\scriptsize
II%
}]}}{L_{\odot}~{}\textrm{pc}^{-2}}\right)};\end{split}$$
(7)
and
$$\log\left({\frac{L_{\rm[C{\scriptsize II}]}}{L_{\rm FIR}}}\right)=5.91~{}(\pm 0%
.14)-0.81~{}(\pm 0.01)\times\log{\left(\frac{\Sigma_{\rm FIR}}{L_{\odot}~{}%
\textrm{kpc}^{-2}}\right)}.$$
(8)
The slope of the former relation coincidentally resembles the slope of the
Schmidt-Kennicutt relation for the CO ($J$ = 1 $\rightarrow$ 0) line on kpc scales, albeit with a large scatter (Kennicutt, 1998b).
The slope of the latter relation
is steeper than those reported by Díaz-Santos et al. (2013, 2017) for nearby ULIRGs, perhaps
due to the different tracers used.
The latter authors use $L_{\rm IR}$ and source size measured at 24 $\micron$ to derive the FIR surface density,
whereas we use $L_{\rm FIR}$ and FIR size (or pixels) measured at rest-frame 158 $\micron$ for HXMM05.
The steeper relation found in HXMM05 can also be understood if we were to assume that its outer region,
where the $L_{\rm[CII]}$/$L_{\rm FIR}$ ratio is the highest, has a
lower metallicity (and thus, a higher photoelectric heating efficiency).
In any case, the tight $L_{\rm[C{\scriptsize II}]}/L_{\rm FIR}-\Sigma_{\rm FIR}$ relationship
is consistent with the notion that the two quantities are connected through the the local FUV radiation field intensity.
We find that the $L_{\rm[CII]}$/$L_{\rm FIR}$ ratio of HXMM05 decreases toward the center, as shown in the
spatially resolved map in
Figure 14. Such a negative gradient has been observed in nearby star-forming
galaxies (e.g., Kramer et al., 2013; Smith et al., 2017)
and U/LIRGs (Díaz-Santos et al. 2014).
The deficit at the center may be explained by a higher dust temperature (see §3.4) and a
more intense radiation field at the center (given that $G_{0}\propto R^{-2}$).
In addition, the gas density at the center is likely to exceed the critical density of [CII],
where collisional de-excitation dominates and saturates the
[CII] emission
(Luhman et al. 1998; Malhotra et al. 1997; Goldsmith et al. 2012).
This effect also explains the decreasing $L_{\rm[C{\scriptsize II}]}/L_{\rm FIR}$ ratio found with increasing FIR surface densities
in the $L_{\rm[C{\scriptsize II}]}/L_{\rm FIR}-\Sigma_{\rm FIR}$ relation (Figure 15).
5.11. HXMM05 in the Context of High-$z$ Galaxy Populations
HXMM05 is one of the most IR-luminous galaxies known at high redshift.
Given its IR luminosity of $L_{\rm IR}$ = 4$\times 10^{13}$ $L_{\odot}$, it can be classified as a HyLIRG.
However, we find that its ISM properties differ from those observed in some other unlensed HyLIRGs studied to date.
For instance, both the gas and SFR surface densities of HXMM05 are much lower than those observed in
GN20 and the $z$ = 5.7 binary HyLIRG ADFS 27 (Hodge et al., 2012; Riechers et al., 2017),
but are comparable to those of the $z$ = 2.4 HyLIRG merger HATLAS J084933
and the sub-regions of GN20 (Ivison et al., 2013; Hodge et al., 2015), suggesting that the star formation in HXMM05 is relatively modest compared to “maximum”-starburst-like HyLIRGs.
Given the dynamical mass, stellar mass, and sSFR of HXMM05, it is among the most massive
galaxies known at $z$ = 3.
However, as discussed in §3.4,
HXMM05 was discovered with Herschel/SPIRE observations at submillimeter wavelengths
and remains undetected in deep UV and optical observations.
It therefore differs from other high-$z$ massive disk galaxy populations, such as those typically selected in the UV, optical, and NIR wavebands
by applying the $U$-, $B$-, $G$-, $R$-, $z$-, $K$-band color-selection and the Lyman Break “dropout” technique
(i.e., BzK, BM/BX, and LBG; Steidel et al. 1996; Adelberger et al. 2004; Steidel et al. 2004; Daddi et al. 2004), in that it has a larger dust content, which may
suggest different evolutionary histories for these high-$z$ populations.
The molecular gas extent, kinematics, gas excitation, SFR,
dust mass, SFE, SFR surface density, and metallicity of HXMM05 are similar to those of GN20.
This agreement suggests that HXMM05 and GN20 may belong to the same class of DSFG.
The finding of such rare massive disk galaxies at $z$ $\sim$ 3 could be consistent with model predictions that disk-wide
star formation plays an important role for some of the
most massive DSFGs at early epochs, whether as a phase
in a merger event or (Hayward et al., 2013) independent of
a major merger altogether222222This statement does not explicitly address the relevance of merger activity in
the overall evolution of massive disks..
6. Implications on the Formation Scenarios of HXMM05 — Major Merger and Cold Mode Accretion
With a SFR of 2900 $M_{\odot}$ yr${}^{-1}$ and a stellar mass of $10^{12}$ $M_{\odot}$,
distributed across a rotating disk 9 kpc in diameter,
HXMM05 is a massive rotation-dominated star-forming galaxy.
One of the main goals in studying high-$z$ star-forming galaxies
is to examine and understand what drives their high SFRs.
A critical question concerns whether an interaction is required
to drive the high SFRs observed in high-$z$ starbursting DSFGs — which would put them in a transient phase —
or whether DSFGs are just a massive galaxy population
undergoing “quiescent” star formation, but at higher rates due simply to their higher masses and/or gas mass fractions
compared to nearby and low-mass galaxies.
Previous theoretical and observational studies have suggested that star formation in the most massive
starburst-dominated DSFGs is likely triggered by major mergers, whereas less massive systems
could be triggered by
gravitational instability as a result of their high gas mass fractions
(e.g., Chapman et al., 2003; Engel et al., 2010; Narayanan et al., 2010; Hayward et al., 2011, 2013; Riechers et al., 2017).
In cosmological N-body zoom-in and hydrodynamic simulations,
massive galaxies with stellar masses similar to that of HXMM05,
albeit rare, can be formed quickly by $z$ = 6
via multiple gas-rich major mergers
(Li et al., 2007; Davé et al., 2010, see also Ruszkowski & Springel 2009).
From a theoretical point of view, it is thus conceivable that HXMM05 may have recently experienced a
major merger that would explain its broad CO lines, high SFR, large molecular gas mass,
and its 3 kpc size double nuclei observed at 635 $\micron$.
In this scenario, the double nuclei may correspond to two compact obscured starburst regions
triggered by massive gas inflows, or to the remnant cores of two similar mass progenitor galaxies (Johansson et al., 2009).
On the other hand, the observed spatial extent, velocity gradient, $G_{0}$, and gas surface density
observed in HXMM05 are more consistent with a rotation-dominated “normal” star-forming galaxy, suggesting that additional mechanisms
may be at work to form a system like HXMM05.
In the standard model of dissipational disk formation,
infant disk galaxies form from the gas that is infalling into hierarchically growing dark matter halos.
However, since a substantial fraction of the angular momentum of gas is lost to the surrounding
halo through dynamical friction (up to 90%)
while it configures itself into a rotationally supported disk in the inner
portion of the dark matter halo, disks are an order of magnitude
smaller than those observed (also known as the angular momentum “catastrophe”; e.g., Steinmetz & Navarro, 1999).
In this formation paradigm, a massive extended disk like HXMM05 is quite unexpected
at $z$ = 3 (only about 2 Gyr after the Big Bang).
While feedback and the continuation of tidal torquing and
accretion of satellite galaxies/minor mergers
have been proposed to resolve the disagreement between models and observations, as they
can prevent the gas from over-cooling and losing its angular momentum
(e.g., Sommer-Larsen & Dolgov, 2001; Robertson et al., 2004; Scannapieco et al., 2008; Zavala et al., 2008), it remains
unclear whether minor mergers232323Major mergers would take a few Gyr to form
an extended disk again from two progenitor disks, if ever (e.g., Governato et al., 2009).
alone could increase the angular momentum sufficiently to explain the properties and number density
of disk galaxies observed
(e.g., Vitvitska et al. 2002).
In recent years, the cold mode accretion formation model has been put forward as an alternative mechanism capable of driving the
high SFRs seen in high-$z$ gas-rich star-forming galaxies,
which may also explain the discrepancy with major mergers
(i.e., there are not enough major mergers in models to explain all DSFGs as merger-driven starbursts; Kereš et al. 2005; Dekel et al. 2009a, b; Davé et al. 2010; see also
Narayanan et al. 2015 and Lacey et al. 2016).
Since cold streams
can provide additional angular momentum, extended gas-rich disk galaxies with kpc-scale star formation can be explained naturally
(Kereš et al., 2005; Dekel et al., 2009a).
Given that some properties of HXMM05 are consistent with the major merger scenario and others are
consistent with
the cold mode accretion scenario, it is conceivable that both
mechanisms together are important to give rise to a galaxy like HXMM05, which perhaps is similar to the
case of GN20 (see Carilli et al. 2010).
7. Summary and Conclusions
We determine the redshift and gas excitation of the
Herschel-selected DSFG HXMM05 at $z$ = 2.9850 $\pm$ 0.0009
by observing its CO($J$ = 1$\rightarrow$0; 3$\rightarrow$2; 5$\rightarrow$4; 10$\rightarrow$9) line emission.
We image its gas reservoir and dust-obscured star formation on 1.2 kpc scales using
[CII] line and dust continuum emission.
We detect a companion galaxy (hereafter X-NE) about 20 kpc NE of the main component of HXMM05 (hereafter X-Main) in CO ($J$ = 1 $\rightarrow$ 0) and [CII] line emission at a redshift close to X-Main ($\Delta v$ = $-$535 $\pm$ 55 km s${}^{-1}$).
X-NE is also detected in the UV, optical, and NIR continuum emission.
Based on the CO ($J$ = 1 $\rightarrow$ 0) line flux, we infer a
total molecular gas mass of $M_{\rm gas}^{\rm total}$ = (2.12 $\pm$ 0.71) $\times$ ($\alpha_{\rm CO}$/0.8)$\times 10^{11}$ $M_{\odot}$ residing in the HXMM05 system (composed of X-NE and X-Main),
yielding a gas mass fraction of $f_{\rm gas}^{\rm dyn}$ = 33 $\pm$ 15%.
Based on the CO ($J$ = 1 $\rightarrow$ 0) and [CII] line data, the velocity structure of X-Main is consistent with a rotating disk, with a diameter of $\sim$9 kpc.
Thus, the gas reservoir of HXMM05 is more extended than those typically observed in high-$z$ DSFGs and quasars, but comparable to those observed in high-$z$ “main-sequence” galaxies and the $z$ = 4 starburst galaxy GN20 (Carilli et al., 2010; Hodge et al., 2015).
We find that the widths of its CO($J$ = 1$\rightarrow$0; 10$\rightarrow$9) and [CII] lines are broader
than those typically observed in “normal” star-forming galaxies, ULIRGs, and high-$z$ SMGs, but comparable to
those observed in the more extreme systems (e.g., J13120$+$4242 and G09v124; Riechers et al. 2011a; Ivison et al. 2013).
We find that the overall
gas excitation of HXMM05 resembles that of the nearby galaxy merger Arp 220.
The shape of the CO excitation ladder (i.e., SLED) suggests that the molecular ISM of HXMM05 may consist of (at least) two
gas phases — a diffuse extended cold component and a dense compact warm component.
The X-Main component of the HXMM05 system
remains undetected in deep UV and optical observations, indicating that it is highly dust obscured.
We find a pair of compact dust components (XD1 and XD2) in the dust continuum emission at 635 $\micron$,
which are about 3 kpc across each and are separated by 2 kpc.
The pair is embedded within an extended dust component, which also appears to be as extended
as the CO ($J$ = 1 $\rightarrow$ 0) and [CII] line emission.
The brightness temperatures of the nuclei suggest that they may be
warmer, more optically thick, and/or
with higher beam filling factors than the extended dust component.
We find that the source-averaged FUV radiation field intensity of HXMM05 is around 200 times stronger than
that of the local Galactic interstellar medium,
but is comparable to those observed in nearby star-forming galaxies and other DSFGs.
The PDR properties of HXMM05 together with its gas properties and excitation are indicative of galaxy-wide star formation,
consistent with its extended gas and dust emission observed
(as opposed to those typically observed in compact starburst galaxies).
We find a stellar mass of $M_{*}$ $\simeq$ 10${}^{12}$ $M_{\odot}$ and an SFR of $\simeq$ 2900 $M_{\odot}$ yr${}^{-1}$ for HXMM05 from SED modeling, consistent with it being one of the most massive star-forming galaxies at $z$ = 3.
We also find source-averaged SFR and molecular gas surface densities of
$\Sigma_{\rm SFR}$ = 10 $-$ 60 $M_{\odot}$ yr${}^{-1}$ kpc${}^{-2}$ and
$\Sigma_{\rm gas}$ = 590 $\times$ $($$\alpha_{\rm CO}$$/0.8)$ $M_{\odot}$ pc${}^{-2}$.
Thus, HXMM05 lies along the “starburst sequence” of the Schmidt-Kennicutt relation (e.g., Bouché et al., 2007),
similar to the
sub-regions of GN20 and the $z$ $\sim$ 2.6 SMG SMM J14011$+$0252
(Sharon et al. 2013; Hodge et al. 2015). This locus corresponds to an elevated SFE
compared to
other SFMS galaxies.
The SFR surface densities for the double nuclei are elevated compared to those observed in
the circumnuclear starburst regions of nearby galaxies (Kennicutt, 1998b), but are much lower than those observed in other
(not strongly lensed) high-$z$ HyLIRGs (“maximum starbursts”; e.g., Riechers et al., 2013, 2014b, 2017; Oteo et al., 2017b).
A large scatter seen in the $\Sigma_{\rm SFR}-\Sigma_{\rm[C{\scriptsize II}]}$ relation for HXMM05 on 1 kpc scale
suggests that its [CII] emission traces both star-forming regions and “diffuse” gas reservoirs.
We find a tighter relation between $L_{\rm[CII]}$/$L_{\rm FIR}$ and $\Sigma_{\rm SFR}$ across HXMM05, which
is consistent with our understanding that the two quantities are connected through the local FUV radiation field intensity (e.g., Tielens & Hollenbach, 1985, and references therein).
We find that the $L_{\rm[CII]}$/$L_{\rm FIR}$ ratio is “suppressed” at high SFR surface densities
(e.g., near the center of HXMM05), which is suggestive of a stronger UV radiation field and warmer dust emission there.
On the other hand, the source-averaged $L_{\rm[CII]}$/$L_{\rm FIR}$ ratio of HXMM05 is comparable to those of nearby star-forming galaxies and
LIRGs rather than nearby ULIRGs and quasars, despite its two orders of magnitude higher $L_{\rm FIR}$.
The scatter observed in the spatially resolved and galaxy-integrated [CII] and FIR luminosity relations for HXMM05 are consistent with our understanding that $L_{\rm[CII]}$ and SFR are not related linearly.
The spatially resolved data presented in this paper thus confirm the speculation put forward by
Stacey et al. (2010) based on unresolved observations: that high-$z$ DSFGs are not simple scaled-up ULIRGs and that
starburst-dominated DSFGs can be much more extended than ULIRGs,
which is also consistent with previous findings of spatially extended CO emission (e.g., Riechers et al., 2011a; Ivison et al., 2011).
While rotationally-supported clumps may yield velocity
gradients (§5.8.2), we find no evidence of such with the data at hand, in spite of the
pair of dust peaks identified. Even in
the merging clump scenario (e.g., in late stage merger), it is unlikely for the clumps
to have a huge impact on the global scale across the entire galaxy as to cause a
monotonic velocity gradient over $\sim$9 kpc across,
especially given the observed centrally peaked velocity
dispersion observed in the [CII] data, which is relatively uniform outside the central $\sim$1.2 kpc242424
Approximately the beam size..
Another piece of evidence disfavoring HXMM05 from being strictly a dispersion-dominated merger
system comes from the fact that the potential merger candidates
(the pair of dust peaks) are oriented almost-perpendicular to the
velocity gradient. We further quantified the disk-like kinematics of
HXMM05 based on the higher order Fourier coefficients of the harmonic
decomposition (§4.2.1), which are found to be insignificant compared
to the $m$ = 0 term. We thus interpret HXMM05 to be a rotating disk252525Note that
this does not rule out the possibility that the disk is part of a merger..
The disk-like kinematics, extended star formation, high SFR and $M_{*}$, and
gas and SFR surface densities of HXMM05 are quite similar to those of GN20 (Hodge et al., 2012, 2015), suggesting that
the two may correspond to the same class of DSFG — massive extended rotating disks with highly dust-obscured star formation.
HXMM05 can be classified as a HyLIRG, making it one of the most IR-luminous galaxies known.
In a sample of the brightest high-$z$ DSFGs discovered in the 95 deg${}^{2}$ surveyed by HerMES,
only around 10% appear to be intrinsically comparably luminous, corresponding to a surface density of only 0.03 deg${}^{-2}$ (Bussmann et al., 2015).
In fact, the stellar mass function also suggests that massive galaxies like HXMM05 are very rare at $z$ = 3 (Davé et al. 2010; Schreiber et al. 2015).
In the framework of the hierarchical formation model, one would expect a
massive galaxy like HXMM05 to form via major mergers, given its high SFR and $M_{*}$.
The two compact dust nuclei and enhanced central velocity dispersion as well as the detection of a companion galaxy at only 20 kpc away may be consistent
with such a scenario.
However, its extended massive gas disk, monotonic velocity gradient, $G_{0}$, and gas and SFR surface densities at $z$ = 3
suggest additional mechanisms such as proposed in the cold mode accretion model may also play an important role
in shaping the existence and subsequent evolution of massive DSFGs.
HXMM05 could thus be a rare example of such system showing both mechanisms at play.
We thank the referee for providing detailed and constructive comments that have significantly improved the clarify of this manuscript.
We thank Mark Lacy and the data analysts at the North American ALMA Science Center (NAASC)
for assistance with the ALMA data reduction.
T.K.D.L. thanks Amit Vishwas and Drew Brisbin for helpful discussions,
Carlos Gómez Guijarro for assistance with the IRAC flux extraction and setting up the galfit software,
and Gregory Hallenback and Luke Leisman for helpful discussions on dynamical modeling.
We thank Shane Bussmann for leading a proposal to obtain some of the data presented in this paper.
T.K.D.L. acknowledges support from the NSF through award SOSPA4-009
from the NRAO and support from the Simons Foundation.
D.R. acknowledges support from the NSF under grant
number AST-1614213 to Cornell University.
A.J.B acknowledges support from NSF grant AST-0955810, to Rutgers, The State University of New Jersey.
I.P.-F. acknowledges support from the Spanish research grants ESP2015-65597-C4-4-R and ESP2017-86852-C4-2-R
J.L.W acknowledges support from an STFC Ernest Rutherford Fellowship (ST/P004784/1 and ST/P004784/2),
and additional support from STFC (ST/P000541/1).
The Flatiron Institute is supported by the Simons Foundation.
This work is based on observations carried out
with the IRAM PdBI Interferometer. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).
Support for CARMA construction was derived from the Gordon and Betty Moore
Foundation, the Kenneth T. and Eileen L. Norris Foundation, the James S.
McDonnell Foundation, the Associates of the California Institute of
Technology, the University of Chicago, the states of Illinois, California, and
Maryland, and the National Science Foundation.
CARMA development and operations were supported by the National Science Foundation under a
cooperative agreement and by the CARMA consortium universities.
The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated
Universities, Inc.
This publication makes use of data products from the Wide-field Infrared Survey Explorer,
which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space Administration.
This paper makes use of the following ALMA data:
ADS/JAO.ALMA# 2016.2.00105.S;
ADS/JAO.ALMA# 2013.1.00749.S; and
ADS/JAO.ALMA# 2011.0.00539.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
This research has made use of NASA’s Astrophysics Data System Bibliographic
Services.
The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.
This work is based in part on observations
made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble
Legacy Archive, which is a collaboration between the Space Telescope Science
Institute (STScI/NASA), the Space Telescope European Coordinating Facility
(ST-ECF/ESA), and the Canadian Astronomy Data Centre (CADC/NRC/CSA).
This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al., 2013).
This research made use of APLpy, an open-source plotting package for Python hosted at http://aplpy.github.com.
This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.
This work is also based on observations obtained with the MegaPrime/MegaCam instrument, a joint project of CFHT and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This study is also based in part on data products produced at Terapix available at the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.
Based on observations made with the NASA Galaxy Evolution Explorer.
GALEX is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034.
This work uses data based on observations obtained with
the XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
Facilities: IRAM PdBI, CARMA, VLA, HST(WFC3), SMA, ALMA, Spitzer (IRAC, MIPS), WISE, Herschel(PACS, SPIRE), VISTA, CFHT(MegaCam), XMM-Newton(EPIC), GALEX
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Appendix A CO and [CII] Channel Maps
Since we are investigating the gas kinematics, it is essential to show and acknowledge
the limited significance of the detected signal per velocity bin. We show the channel maps for the
CO ($J$ = 1 $\rightarrow$ 0) and [CII] lines in Figures 16 and 17.
In the [CII] maps, structures on the scale of the angular resolution ($\lesssim 1.2$ kpc) are seen, but at low S/N significance.
We therefore do not discuss the properties of potential star-forming “clumps”/structures in this paper.
Exploring such direction with higher resolution and better sensitivity data would be useful to better understand the physics behind
the high SFR of HXMM05.
As noted in §4.2.1,
a drop off is seen in the rotation velocity beyond a radius of $R$ = 6 kpc (in Figure 11).
This is most likely a result of the limited S/N in the reddest velocity channels, as illustrated in Figure 16.
Appendix B Non-detection of X-Main at UV/optical Wavebands
As shown in the RGB image created from Spitzer/IRAC 4.5 (blue), 5.8 (green), and 8 $\micron$ (red) data (Figure 8),
emission detected at 4.5 $\micron$ is dominated by foreground sources (see also Figure 19), but emission at 5.8 and 8 $\micron$ is dominated by HXMM05.
We therefore model the surface brightness profiles of the sources near HXMM05 based on their morphologies seen in the CFHT and VISTA images in order to de-blend the emission observed at
3.6 and 4.5 $\micron$ (see Appendix §C).
On the other hand, X-NE is detected in the UV, optical, and NIR wavebands (as shown in Figures 8 and 18).
As discussed in §3, this component is also detected in CO ($J$ = 1 $\rightarrow$ 0) and [CII] line emission (see Figures 3 and 4).
With the available data, we cannot discriminate and obtain reliable constraints on the stellar
masses and SFRs for X-NE and X-Main separately. We thus infer the properties of the system as
a whole in §5 and subsequent sections.
That said, optically-selected high-$z$ sources (e.g,
BzKs, LBGs) appear to be different populations from these highly dusty
starburst galaxies (possibly due to different evolutionary stages), and
surveys done at only one wavelength are likely to miss other high-z
candidates in the field. Given that X-NE is optically visible, and
thus may have less dust than X-Main, it may be a young nearby galaxy
soon to be engulfed by X-Main. We report the pair’s gas mass ratios in
Table 7.
More observations will be useful to better understand the physical properties of
X-NE, and thus, its nature in relation to X-Main in the
HXMM05 system.
Appendix C De-blending Spitzer/IRAC Emission
Multiple sources are detected near HXMM05 at 3.6 and 4.5 $\micron$ (channels 1 and 2; Figures 18 and 19).
We examine whether part of the emission detected at 3.6 and 4.5 $\micron$ may arise from HXMM05 by
using the publicly available software galfit (Peng et al., 2002) to de-blend the emission.
We initialize the fitting parameters based on the positions, brightnesses, and morphologies
of the sources near HXMM05 as observed in the higher resolution NIR images
(HST/WFC3 F110W, VISTA, and CFHT; see e.g., Figure 18).
We use a total of six components and a sky background
to account for all the emission detected in the high resolution NIR images.
We model the surface brightness distributions of the two brightest components using Sersic profiles, each with seven free
parameters: $x$, $y$, $I$, $R_{e}$, $n$, $b/a$, and PA, where $x$ and $y$ describe the position of the component,
$I$ is the integrated flux,
$R_{e}$ is the effective radius, $n$ is the Sersic index, $b/a$ is the axial ratio, and PA is the position angle.
We model the remaining four components as point sources, for which
we adopt the point response functions (PRF), described by three free parameters $x$, $y$, and $I$ per source.
We allow all parameters to vary without imposing any priors in order to avoid biasing the best-fit parameters.
The PRFs
are adopted from the IRAC calibration routines262626http://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/psfprf/.
We do not detect any statistically significant emission at the position of HXMM05 in the residual maps (Figure 19).
We thus adopt the SWIRE survey depths at the two IRAC wavebands
as 3$\sigma$ upper limits (Table 2). |
Fusarium Damaged Kernels Detection Using Transfer Learning on Deep Neural Network Architecture
Márcio Nicolau${}^{1,2}$, Márcia Barrocas Moreira Pimentel${}^{1}$, Casiane Salete Tibola${}^{1}$
José Mauricio Cunha Fernandes${}^{1,2}$, Willingthon Pavan${}^{2}$
Abstract
The present work shows the application of transfer learning for a pre-trained deep neural network (DNN), using a small image dataset ($\approx$ 12,000) on a single workstation with enabled NVIDIA GPU card that takes up to 1 hour to complete the training task and archive an overall average accuracy of 94.7%. The DNN presents a 20% score of misclassification for an external test dataset. The accuracy of the proposed methodology is equivalent to ones using HSI methodology (81%-91%) used for the same task, but with the advantage of being independent on special equipment to classify wheat kernel for FHB symptoms.
Keynotes. Deep Learning, Fusarium Damaged Kernels, ImageNet, TensorFlow, Transfer Learning.
\sidecaptionvpos
figuret
1 Introduction
The wheat is the main source of nutrients to the world population, most of its production is converted into flour for human consumption. In Southern Brazil, where 90% of the domestic wheat is produced, Fusarium head blight (FHB), a fungal disease, is a major concern. Apart from yield loss, the causal agent Fusarium graminearum may cause mycotoxin contamination of wheat products, creating health problems.
Therefore, to avoid potential health risks, Fusarium affected grains must be identified and segregated, before their processing, to avoid its incorporation into food for humans and animal feed.
Usually, the detection of Fusarium head blight (FHB) is carried out manually by human experts using a process that may be both lengthy and tiresome. Moreover, the effectiveness of this kind of detection may drop with factors such as fatigue, external distractions and optical illusions [2]. Thus, improving the detection of Fusarium Head Blight (FHB) in wheat kernels has been a major goal, due to the health risks associated with the ingestion of the mycotoxin, mainly deoxynivalenol (DON).
Methods capable of performing this disease detection in an automatic way are highly demanded in the productive wheat chain, to segregate lots. Most automatic methods proposed to date rely on image processing to perform their tasks [2, 3, 8].
Barbedo et al. ] used Hyperspectral imaging (HSI) for detecting Fusarium head blight (FHB) in wheat kernels using an algorithm. The outcome was a Fusarium index (FI), ranging from 0 to 1, that can be interpreted as the likelihood of the kernel to be damaged by FHB. According to the authors, hyperspectral imagery is currently not sensitive enough to estimate DON content directly. However, an indirect estimation from the Fusarium damaged kernels was successfully achieved, with an accuracy of approximately 91% [2].
Other study investigated the use of hyperspectral imaging (HSI) for deoxynivalenol (DON) screening in wheat kernels. The developed algorithm achieved accuracies of 72% and 81% for the three- and two-class classification schemes, respectively. The results, although not accurate enough to provide conclusive screening, indicating that the algorithm could be used for initial screening to detect wheat batches that warrant further analysis regarding their DON content [3].
Min and Cho ] presented a review about nondestructive detection of fungi and mycotoxins in grains, focusing on spectroscopic techniques and chemometrics. The spectroscopy has advantages over conventional methods including the rapidness and nondestructive nature of this approach. However, some limitations as expensive and complex setup equipment’s and low accuracy due to external interferents exist, which must be overcome before widespread adoption of these techniques.
The application of computer vision on digital images offers a high-throughput and non-invasive alternative to analytical and immunological methods. This paper presents an automated method to detect Fusarium Damaged Kernels, which uses the application of computer vision to digital images.
The main goal of this work is the use of machine learning algorithms and computer vision techniques to detect Fusarium Damaged Kernels in wheat, based on digital images.
2 Material and Methods
Digital images of Fusarium Head Blight symptomatic and non-symptomatic wheat kernels were available at the National Research Centre for Wheat (Embrapa Wheat), located in Passo Fundo, Rio Grande do Sul State, Brazil.
The images were obtained by recording a video of 06:25 minutes using an Olympus SP-810UZ digital camera with 36x optical zoom, 24mm wide-angle view, and 14-megapixel resolution, 720p HD video and Olympus Lens 36x Wide Optical Zoom ED 4.3-154.8mm 1:2.9-5.
All tasks run on an iMac workstation configured with 32GB of RAM DDR3 1600MHz, a 3.5GHz quad-core Intel Core i7 processor, and a NVIDIA GeForce GTX 780M GPU with 4GB of GDDR5 memory. The TensorFlow 1.0.1 built from source with CUDA Toolkit 8.0 and cuDNN v5.1 to enable GPU support. All scripts were developed using Python 2.7.
2.1 Methodology
To classify digital images in predefined classes, we could use one of the several methods developed in recent two decades [10, 6]. Methodologies to solve this kind of problem was developed both in Artificial Intelligence (AI), a research area in Computer Science, and in Statistics.
The main differences between Statistics and AI approach are the size of the task, for statistics point of view, algorithms and techniques are limited when then input size of image dataset are greater than tens of thousand pictures and for a large number of classification sets.
During the training stage of a system to classify images and objects based only on information content embedded in a single digital file, it is necessary that this system would detect all possible contexts where the object could occur. Small differences in color, luminosity, angle and other could be misinterpreted by the proposed system and results in the wrong or low-level prediction classification.
The key advantage of using AI strategy in the image classification task are related to knowing how to combine layers and manage the relationship between levels of information, without needing any strong assumption related to the type of dependency or relational structure among input information.
In some cases, the improvements in the accuracy and precision gain are archive using these fine tune setting, but the most of the effort is made only with the input information, in other words, the processes are designed to take the most of the self-learning way.
2.2 Deep Convolution Neural Network (DNN)
The most used architecture of the neural network for image classification task is called convolutional neural network (CNN). The convolution operation or, sometimes called convolution layer is related to the operation to process or respond to “stimuli” in a limited region known as the receptive field.
The receptive field from each neuron contains a partial overlap of information from input layer (raw image) and, in this way, the preprocessing or further operations occur with a minimal amount of effort. Other advantages of this technique are related to the possibility of using distributed algorithms (even GPU versions) to calculate, filters and processing small pieces of information, one each time and aggregated the results when necessary [7].
The deep portion of CNN come from the stacking or combination of several layers where the output of preceding layer is used as an input for the next one. The most common layers employed in DNN are convolution, ReLU (Rectified Linear Units), tanh (Hyperbolic Tangent Function), max pooling, average pooling, fully connected, concat, dropout and softmax. To better understand this relationship see a CNN example in Figure 1.
2.3 Transfer Learning
Transfer learning is a machine learning method which utilizes a pre-trained neural network, this technique allows the detachment of the lasts outer layer (classification layer) and uses the remains structure to retraining and get new weights corresponding the classes of interest – damaged kernel in our case (Figure 2).
In this work, we use as a pre-trained neural network the output from [11]. Szegedy et al. ] developed a 22 layer deep convolution neural network on top of ImageNet for classifying 1000 leaf-node categories, using 1.2 million images for training, 50,000 for validation and 100,000 images for testing.
In brief, transfer learning makes it possible to classify new classes based on a new set of images, reusing the feature extraction part and re-train the classification part of the new picture set. Since feature extraction part of the network it was already trained (which is the most complex part), the new neural network could be trained with less computational resources and time.
2.4 Machine Learning Framework
Nowadays we have some options to build and analyze deep neural networks using machine learning algorithms. The final choice would base on the computational infrastructure available to run this task, either the number of classes, the intended purpose and where the final Net will be deployed to handle.
For this work, it is necessary a framework for machine learning that could run on a distributed system with both CPU and GPU, with the possibility to deploy the final network to servers, desktops, mobile applications and embedded systems in an easy way.
Alongside these needs, it is necessary that the chosen framework could easily implement the Transfer Learning techniques, described before. Based on these requirements, the natural choice is TensorFlow [1].
Abadi et al. ] presented TensorFlow as an interface for expressing and executing machine learning algorithms that can be performed with little or no change in a broad range of heterogeneous systems, ranging from mobile devices such as phones and tablets up to large-scale distributed systems of hundreds of machines and thousands of computational devices such as GPU cards.
2.5 Image Pre-processing
To generate the normal kernel images was used the FFMPEG library111FFMPEG library, see more information on http://www.ffmpeg.org. to split the 6:25 minutes movie into 11,555 individual files (1280x720). Both normal and damaged kernel images were arranged in the separate folder for later use.
Before using the images for generating the neural network, they as randomly allocated in two distinct image sets: 80% and 20% for training and validation set, respectively. In the process of training the Net, other parts of wheat plant structure like spikes and leaves were used too in the composition of the DNN intending to classify better the wheat damaged kernels.
3 Results and Rationale
Comparing the effort described in [11] to training a whole DNN from scratch with the number of pictures necessary to get reasonable results. In our case, the number of pictures available at the moment ($\approx$ 12,000) probably will not archive this scores and definitely, the time and computational infrastructure necessary to training and evaluate the resultant DNN must be larger than installed capacity.
A broad range of applications are using transfer learning, Devikar ] describe the use in image classification of various dog breeds with an overall accuracy of 96% from 11 dog breeds. Wang et al. ] describes the application for remote scene classification and attempt to form a baseline for transferring pre-trained DNN to remote sensing images with various spatial and spectral information.
Esteva et al. ] describes the use of DNN for dermatologist-level classification of skin cancer, trained end-to-end from images directly, using only pixels and disease labels as inputs for a dataset of 129,450 clinical images.
Tkáč and Verner ] presents many business applications, using artificial neural networks, related to financial distress and bankruptcy problems, stock price forecasting, and decision support, with particular attention to classification tasks. For other uses of CNN and DNN, see [10, 6].
The training procedure was carried out on the described workstation and took up to 1 hour to finish, and the output retrained neural network archive an overall average accuracy of 94.7%. The main structure of the final neural network is presented in Figure 3.
For validation purposes and to check the correctness in classifying new images of wheat kernels (with and without Fusarium damage) we choose a new dataset to validate against DNN. These images were from National Research Centre for Wheat articles published over last decade about this content, alongside with other open source images of Fusarium damaged kernel found over the Internet.
At this point, we could share our positive experience in using the transfer learning techniques in relation to time to training a DNN with a new set of images and classes and the overall accuracy achieved with this initial image dataset.
The results from validation dataset present a total of 20% of misclassification, and in half (10%) the new images were classified as damaged leaves class. This results could be related to two main reasons: (a) the small number of pictures with Fusarium Damaged Kernels and (b) the prevalence of the normal wheat kernels present in the initial dataset ($\approx$ 80%).
4 Conclusions
The associated use of Transfer Learning, TensorFlow, and Inception-v3 cut the time necessary to training and the necessity to have a large image dataset ($\approx$ 120,000) to start the classification procedure with a good accuracy level, compared to training a DNN from scratch.
The misclassification for damaged leaves class could be associated with the characteristics of damage both in kernel and leaves (in the most cases) where the region color of lesions was more blight that the standard wheat kernel.
Unfortunately, the symptoms on leaves present in this initial dataset was not separated by diseases.Thus, it was not possible to claim that some visual characteristics of FHB in leaves could be transferred to kernel evaluations in our context. This hypothesys needs to be investigated by adding new images to the present dataset related to Fusarium Damaged Kernel and wheat leaves with FHB symptoms.
Beside this misclassification for a new dataset, the overall average accuracy archived (94.7%) for this Fusarium Damaged Kernels Deep Neural Network (FDK-DNN). Therefore, there is a potential of using this methodology for classifying Fusarium Damaged Kernel by means of smartphone camera.
The accuracy of the proposed methodology is equivalent to ones using HSI methodology presented by [3].
An interesting future work could be related to using a mixed of RGB pictures, and layers from HSI operational spectra for Fusarium Damaged Kernel proposed by [2].
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Anisotropic Exchange in ${\bf LiCu_{2}O_{2}}$
Z. Seidov
Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, D-86135 Augsburg, Germany
Institute of Physics, Azerbaijan National Academy of Sciences, H. Cavid pr. 33, AZ-1143 Baku, Azerbaijan
T. P. Gavrilova
Kazan E. K. Zavoisky Physical-Technical Institute RAS, 420029 Kazan, Russia
Kazan (Volga region) Federal University, 420008 Kazan, Russia
R. M. Eremina
Kazan E. K. Zavoisky Physical-Technical Institute RAS, 420029 Kazan, Russia
Kazan (Volga region) Federal University, 420008 Kazan, Russia
L. E. Svistov
P. L. Kapitza Institute for Physical Problems RAS, 119334 Moscow, Russia
A. A. Bush
Moscow Institute of Radiotechnics, Electronics, and Automation, RU-117464 Moscow, Russia
A. Loidl
Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, D-86135 Augsburg, Germany
H.-A. Krug von Nidda
[email protected]
Experimental Physics V, Center for Electronic Correlations and Magnetism, University of Augsburg, D-86135 Augsburg, Germany
(November 19, 2020)
Abstract
We investigate the magnetic properties of the multiferroic quantum-spin system LiCu${}_{2}$O${}_{2}$ by electron spin resonance (ESR) measurements at $X$- and $Q$-band frequencies in a wide temperature range $(T_{\rm N1}\leq T\leq 300$ K). The observed anisotropies of the $g$ tensor and the ESR linewidth in untwinned single crystals result from the crystal-electric field and from local exchange geometries acting on the magnetic Cu${}^{2+}$ ions in the zigzag-ladder like structure of LiCu${}_{2}$O${}_{2}$. Supported by a microscopic analysis of the exchange paths involved, we show that both the symmetric anisotropic exchange interaction and the antisymmetric Dzyaloshinskii-Moriya interaction provide the dominant spin-spin relaxation channels in this material.
PACS numbers
76.30.Fc, 75.30.Et, 75.47.Lx
Keywords
electron spin resonance, anisotropic exchange interactions, low-dimensional magnetism, multiferroics
††preprint: APS/123-QED
I Introduction
Unconventional magnetic ground states and excitations of frustrated quantum-spin chains represent attractive issues in solid-state physics during the last decades.Mikeska_2004 They appear under a fine balance and partly compensation of competing dominant exchange interactions and are often caused by much weaker interactions
or fluctuations.Chubukov_1991 ; Kolezhuk_2000 ; Kolezhuk_2005 ; Dmitriev_2008 ; Amiri_2015
Typically, frustration in quasi-one-dimensional (1D) chain magnets is provided by competing interactions, if the nearest-neighbor (NN) exchange is ferromagnetic and the next-nearest neighbor (NNN) exchange is antiferromagnetic. Numerical investigations of frustrated chain magnets within different models
Hikihara_2008 ; Sudan_2009 ; Heidrich_2009
have predicted a number of exotic magnetic phases, such as planar, spiral, or different multipolar phases. Moreover, theoretical studies show that the magnetic phases are very sensitive to inter-chain interactions and anisotropic interactions in the system.Furukawa_2008 ; Zhitomirsky_2010 ; Nishimoto_2010
There is a number of magnets which are attractive objects for experimental investigations as realizations of 1D frustrated systems like LiCuVO${}_{4}$, Rb${}_{2}$Cu${}_{2}$Mo${}_{3}$O${}_{12}$, NaCu${}_{2}$O${}_{2}$, Li${}_{2}$CuO${}_{2}$, Li${}_{2}$ZrCuO${}_{4}$, and CuCl${}_{2}$,
(see for example, Refs. Enderle_2005, ; Hase_2004, ; Drechsler_2007, ; Boehm_1998, ; Drechsler_2007_2, ; Banks_2009, ). Here we concentrate on LiCu${}_{2}$O${}_{2}$ with its fascinating interplay of competing exchange interactions both within the Cu${}^{2+}$ chains and the zigzag-ladders formed by neighboring two chains.Masuda_2005
LiCu${}_{2}$O${}_{2}$ was first discovered in 1990 during the study of Li/CuO electrochemical cells Hibble90a and in turn synthesized on purpose for search of new candidates for high-temperature superconductivity.Hibble90b After it was characterized as a low-dimensional quantum antiferromagnet in the late nineties,Fritschij98 ; Vorotynov its exotic ground-state properties received large interest and triggered further experimental efforts, revealing a complex phase diagram,Roessli2001 suggesting a dimer-liquid state,Zvyagin2002 coexistence of dimerization and long-range order,Choi2004 as well as helimagnetism.Masuda2004 ; Gippius ; Mihaly2006 The interest in LiCu${}_{2}$O${}_{2}$ was even stronger intensified by the discovery of its ferroelectric properties in 2007,Park2007 i.e. it turned out to be a paramount example of a multiferroic due to the correlation between spin helicity and electric polarization.Seki2008 ; Yasui2009 ; Kobayashi2009 Detailed investigations to resolve the phase diagram of LiCu${}_{2}$O${}_{2}$ have been performed by means of magnetization and dielectric measurements Bush as well as neutron scattering,Masuda_2005 electron spin resonance (ESR), and nuclear magnetic resonance (NMR) studies.Bush2013 Basically, a susceptibility maximum at a temperature $T_{\rm max}=38$ K typical for a low-dimensional antiferromagnet and two subsequent phase transitions at $T_{\rm N1}=24.5$ K and $T_{\rm N2}=23$ K into the spin-spiral structure, where the latter is accompanied by the occurrence of ferroelectricity, characterize the magnetic and electric properties of LiCu${}_{2}$O${}_{2}$ at low magnetic fields.
In this paper, we report the results of an ESR study of LiCu${}_{2}$O${}_{2}$ single crystals in the paramagnetic regime. This study is performed in order to obtain information on the anisotropic exchange interactions in this material. The knowledge of the anisotropic exchange parameters is important for the interpretation of the magnetic and magneto-electric properties of LiCu${}_{2}$O${}_{2}$ in the magnetically ordered phase.
Previous ESR experiments revealed a single exchange-narrowed Lorentz-shaped absorption line with $g$ values $g_{c}\approx 2.22$ and $g_{a}\approx g_{b}\approx 2.0$ at a microwave frequency of 9 GHz and $T\gg T_{\rm max}$ as well as $g_{c}\approx 2.29$ at 227 GHz.Vorotynov ; Zvyagin2002 The ESR linewidth $\Delta H$ was found to amount more than 1 kOe at room temperature and to diverge to low temperature on approaching magnetic order with a critical behavior $\Delta H\propto(T-T_{\rm crit})^{(-p)}$ with $T_{\rm crit}=30$ K and $p=1.28$ or 1.35 for $H||c$ or $H\perp c$, respectively, at 9 GHz and $T_{\rm crit}=23$ K and $p=0.58$ for $H||c$ at 227 GHz.Vorotynov ; Zvyagin2002 So far the discussion and analysis of the paramagnetic resonance remained on a qualitative level.
Here we present a quantitative analysis of the angular dependence of the paramagnetic resonance linewidth in LiCu${}_{2}$O${}_{2}$ to determine the anisotropic exchange parameters. For this purpose ESR is the method of choice, because the anisotropy of the line broadening is extremely sensitive to anisotropic interactions, while the isotropic exchange contributions only result in a general isotropic narrowing of the ESR signal.
While previous ESR studies have been limited by twinning of the crystals, our present investigations are performed on high-quality untwinned single crystals, which is an essential precondition to determine the anisotropy unequivocally. Based on our ESR data we will show that besides the symmetric anisotropic exchange interaction, the antisymmetric Dzyaloshinskii-Moriya (DM) interaction substantially contributes to the linewidth, and we will suggest a possible DM exchange path.
II Crystal structure and exchange interactions
LiCu${}_{2}$O${}_{2}$ crystallizes within an orthorhombic structure (space group $Pnma$). The lattice constants at room
temperature are given by $a=5.726$ Å, $b=2.8587$ Å, and $c=12.4137$ Å.Berger Besides four nonmagnetic Li${}^{+}$ cations its unit cell contains four monovalent nonmagnetic cations Cu${}^{+}$ (electronic configuration $3d^{10}$) and four divalent cations Cu${}^{2+}$ ($3d^{9}$) with spin $S=1/2$. Each magnetic Cu${}^{2+}$ ion is surrounded by five oxygen ions forming slightly distorted pyramids. Thus, all Cu${}^{2+}$ ions are structurally and magnetically equivalent, because the corresponding oxygen pyramids differ from each other by a $180^{\circ}$ rotation, only. There are two linear Cu${}^{2+}$ chains in the crystal structure of LiCu${}_{2}$O${}_{2}$, which propagate along the $b$ axis and form a zigzag-ladder like structure as indicated in Fig. 1. The ladders are isolated from each other by both Li${}^{+}$ ions in the ab plane and layers of nonmagnetic Cu${}^{+}$ ions along the c direction. The distance between the magnetic nearest-neighbor Cu${}^{2+}$ ions along the chains amounts 2.869 Å, and the spacing between the next-nearest neighbor Cu${}^{2+}$ ions (between the chains in one ladder) is about 3.10 Å.
The unit-cell parameter $a$ is approximately twice the unit-cell parameter $b$. Consequently, LiCu${}_{2}$O${}_{2}$ crystals, as a rule, are characterized by twinning due to formation of crystallographic domains
rotated by 90${}^{\circ}$ around their common crystallographic $c$ axis.Bush
The exchange constants of the quasi-one-dimensional helimagnet LiCu${}_{2}$O${}_{2}$ were determined by T. Masuda et al.Masuda_2005 in a single-crystal inelastic neutron-scattering study. Based on these experiments the authors investigated the validity of three different exchange models and concluded that in LiCu${}_{2}$O${}_{2}$ the frustration mechanism is rather complex and involves a competition between a combination of antiferromagnetic intra-ladder $J_{1}$ and ferromagnetic intra-chain $J_{2}$ exchange interactions against an additional antiferromagnetic long-range intra-chain $J_{4}$ coupling as illustrated in Fig. 2. The three corresponding exchange constants turned out to be of comparable strength, i.e. $J_{1}=3.2$ meV, $J_{2}=-5.95$ meV, and $J_{4}=3.7$ meV. Moreover, a sizable antiferromagnetic inter-ladder exchange $J_{\perp}=0.9$ meV was obtained. Masuda_2005
Note that the key features of this exchange model, namely, a ferromagnetic $J_{2}$ bond and a substantial antiferromagnetic $J_{4}$ coupling constant, are similar to those of theoretical LDA calculations.Gippius
As a further corroboration of this model, it can be inferred that recently Y. Qi and A. Du Qi adopted the suggestion of Masuda et al.Masuda_2005 about a strong antiferromagnetic ’rung’ interaction $J_{1}$ and a weak inter-ladder coupling $J_{\perp}$, to explain the fascinating magnetoelectric coupling effects observed in LiCu${}_{2}$O${}_{2}$.
Thus, the present analysis of our ESR results will be based on Masuda’s exchange model and on our earlier workKrug von Nidda ; Eremina1 in the related compounds LiCuVO${}_{4}$ and CuGeO${}_{3}$.
III Theoretical Background
Electron spin resonance (ESR) measures the resonant microwave-power absorption at a given frequency $\omega$ due to induced magnetic dipolar transitions between the Zeeman levels of magnetic ions split by an external magnetic field $H$. The resonance condition $\hbar\omega=g\mu_{\rm B}H$ yields the $g$ value, which contains information on crystal-electric field and spin-orbit coupling. Here $\hbar$ denotes the Planck constant $h$ divided by $2\pi$, and $\mu_{\rm B}$ the Bohr magneton. The resonance linewidth $\Delta H$ provides microscopic access to the anisotropic interactions acting on the electron spins.
In general, in the case of sufficiently strong exchange interaction the ESR linewidth can be analyzed in terms of the high-temperature approach $(k_{\rm B}T\gg J)$: Abragam ; Anderson1953 ; Kubo1954
$$\Delta H=\frac{\hbar}{g\mu_{\rm B}}\frac{M_{2}}{\omega_{\rm ex}}$$
(1)
where the second moment $M_{2}$ is defined by:
$$M_{2}=-\frac{1}{\hbar^{2}}\,\frac{Tr\,[{\cal H}_{\rm int},S_{x}]^{2}}{Tr\,S_{x%
}^{2}},$$
(2)
The second moment $M_{2}$ and the exchange frequency $\omega_{\rm ex}$ can be expressed via microscopic Hamiltonian parameters ${\cal H}_{\rm int}$. The second moment shows an orientation dependence with respect to the external magnetic field, which is characteristic for the anisotropic interaction responsible for the line broadening. The exchange frequency is defined by the dominating exchange interactions shown in Fig. 2 as
$$\omega_{\rm ex}=\sqrt{2J_{1}^{2}+2J_{2}^{2}}.$$
(3)
In LiCu${}_{2}$O${}_{2}$ the second moment is defined by anisotropic interactions of relativistic nature.
Due to the fact that in the case of spin $S=\frac{1}{2}$ the usually dominating single-ion anisotropy is absent, relativistic interactions of neighboring spins, i.e., anisotropic exchange interactions have to be considered.
Note that the magnetic resonance properties of several spin $S=\frac{1}{2}$ chain compounds,Zakharov e.g., LiCuVO${}_{4}$ (Ref. Krug von Nidda, ) and CuGeO${}_{3}$ (Ref. Eremina1, ) as well as CuTe${}_{2}$O${}_{5}$ (Refs. Eremina, ; Gavrilova, ) have been well explained taking into account the anisotropic exchange interactions.
Thus, the results of our present paramagnetic resonance experiments in LiCu${}_{2}$O${}_{2}$ will be discussed in the frame of the following model Hamiltonian:
$$\displaystyle{\cal H}_{\rm int}$$
$$\displaystyle=$$
$$\displaystyle J_{2}^{n}\,(S_{i,j,k}\cdot S_{i,j+1,k})+S_{i,j,k}\textbf{J}_{%
\textbf{2}}^{\textbf{n}}S_{i,j+1,k}$$
(4)
$$\displaystyle+$$
$$\displaystyle J_{1}^{\alpha\delta}\,(S_{i,j,k}\cdot S_{i+1,j,k-1})+S_{i,j,k}%
\textbf{J}_{\textbf{1}}^{\alpha\delta}S_{i+1,j,k-1}$$
$$\displaystyle+$$
$$\displaystyle J_{1}^{\alpha\delta}\,(S_{i,j,k}\cdot S_{i+1,j-1,k-1})+S_{i,j,k}%
\textbf{J}_{\textbf{1}}^{\alpha\delta}S_{i+1,j-1,k-1}$$
$$\displaystyle+$$
$$\displaystyle J_{1}^{\beta\gamma}\,(S_{i,j,k}\cdot S_{i-1,j,k-1})+S_{i,j,k}%
\textbf{J}_{\textbf{1}}^{\beta\gamma}S_{i-1,j,k-1}$$
$$\displaystyle+$$
$$\displaystyle J_{1}^{\beta\gamma}\,(S_{i,j,k}\cdot S_{i-1,j-1,k-1})+S_{i,j,k}%
\textbf{J}_{\textbf{1}}^{\beta\gamma}S_{i-1,j-1,k-1}$$
$$\displaystyle+$$
$$\displaystyle J_{\bot}\,(S_{i,j,k}\cdot S_{i+1,j,k})+J_{4}^{n}\,(S_{i,j,k}%
\cdot S_{i,j+2,k})$$
$$\displaystyle+$$
$$\displaystyle\textbf{D}_{\textbf{2}}^{\textbf{n}}\cdot[S_{i,j,k}\times S_{i,j+%
1,k}]+\mu_{B}H\cdot\textbf{g}_{\textbf{i,j,k}}\cdot S_{i,j,k}$$
where $n=\alpha,\beta,\gamma,\delta$ denotes the chains corresponding to Fig. 2. The summation over all $i,j,k$ is dropped for brevity.
In this model spin Hamiltonian, we included isotropic and symmetric anisotropic exchange interactions between a few types of ions (see Fig. 2): ferromagnetic isotropic intra-chain exchange $J_{2}$ between nearest Cu${}^{2+}$ ions in the chains with the tensor of the anisotropic contribution $\textbf{J}_{\textbf{2}}$, antiferromagnetic isotropic intra-ladder exchange $J_{1}$ along the rungs with the tensor of anisotropic contribution $\textbf{J}_{\textbf{1}}$, long-range antiferromagnetic isotropic intra-chain exchange $J_{4}$ and antiferromagnetic isotropic exchange $J_{\bot}$ between neighboring ladders without anisotropic contributions. The anisotropic contribution to $J_{4}$ can be expected to be very small compared to $\textbf{J}_{\textbf{1}}$ because of the longer Cu–O–O–Cu super-super exchange path. A similar argument holds for $J_{\bot}$, which is not indicated in Fig. 2, because its direction is oriented along the crystallographic a axis and so $J_{\bot}$ is perpendicular to the plane of Fig. 2. The first term of the last line of Eq. 4 introduces a possible antisymmetric anisotropic exchange interaction, i.e. a Dzyaloshinskii-Moriya (DM) interaction, within the chains, which in this way has not been considered so far, but will appear to be essential to explain the experimentally observed anisotropy of the linewidth. The last term in Eq. 4 denotes the Zeeman interaction of all spins with the magnetic field.
To evaluate the anisotropic exchange contributions in ${\cal H}_{\rm int}$, one has to consider the respective bond geometries. For each anisotropic exchange contribution a local coordinate system has to be defined such that the corresponding tensor of anisotropic interaction is diagonal and the sum of diagonal elements equals zero. One of the local axes is directed along the exchange bond. The directions of the two other axes are defined by the symmetry of the local environment. As indicated in Fig. 1, for the intra-chain anisotropic exchange interaction $\textbf{J}_{\textbf{2}}$ the local axes are defined as: $x^{\prime}$ - along the O-O direction within the chain, $y^{\prime}$ - along the Cu-Cu direction within the chain, and $z^{\prime}$ - perpendicular to the plane spanned by the Cu-O${}_{2}$ ribbons within the chain. The local axes of the intra-ladder anisotropic exchange between neighboring chains $\textbf{J}_{\textbf{1}}$ are chosen as: $x^{\prime\prime}$ - along the Cu-Cu direction between neighboring chains, $y^{\prime\prime}$ - perpendicular to the plane spanned by the Cu-O-Cu bridge between neighboring chains, and $z^{\prime\prime}$ - perpendicular to $x^{\prime\prime}$ and $y^{\prime\prime}$ axes. The unit vectors of the local coordinates in the crystallographic system are given in the Appendix.
For details of second-moment calculations for anisotropic exchange interactions we refer to Ref. Rushana, . The intra-chain anisotropic contribution $\textbf{J}_{\textbf{2}}$ can be adopted from the identical ionic configuration in the related compound LiCuVO${}_{4}$, where we considered the same so called ring-exchange geometry of the Cu-O${}_{2}$ ribbons yielding $J_{2}^{cc}/k_{\rm B}=-2$ K.Krug von Nidda The remaining intra-ladder anisotropic contribution $\textbf{J}_{\textbf{1}}$ needs a deeper analysis which will be discussed in the following.
The schematic pathway of the relevant anisotropic spin-spin coupling $\textbf{J}_{\textbf{1}}$ between two neighboring chains within the same ladder is illustrated in Fig. 3.
Here we use local coordinates $x,y,z$ adapted to the conventional rotation of the unperturbed $d$-orbitals neglecting distortion of lattice and any mixing of the wave functions.
We consider the case where the hole ground state $d_{x^{2}-y^{2}}$ is coupled with the excited $d_{xz}$ state by spin-orbit interaction (Fig. 3). Following this scheme, we estimate the intra-ladder anisotropic exchange parameter $A_{yy}$ according to Ref. Bleaney, :
$$A_{yy}=\frac{1}{2}\frac{\lambda^{2}}{\Delta_{x^{2}-y^{2},xz}^{2}}(\langle x^{2%
}-y^{2}|l_{y}|xz\rangle)^{2}J_{x^{2}-y^{2},xz}$$
(5)
$J_{x^{2}-y^{2},xz}$ denotes the corresponding isotropic exchange integral, which is significantly larger than $J_{1}$. A similar expression was obtained earlier in Refs. Eremina1, ; Zakharov, .
To estimate the value of $J_{x^{2}-y^{2},xz}$ we used the formula:
$$J_{x^{2}-y^{2},xz}\approx 4\frac{t^{2}_{\sigma}t^{2}_{\pi}}{\Delta_{12}\Delta_%
{\pi}\Delta_{\sigma}}$$
(6)
Here we insert $\lambda/k_{\rm B}\approx 913$ K for the spin-orbit coupling and $\Delta_{x^{2}-y^{2},xz}/k_{\rm B}=(\varepsilon_{5,6}-\varepsilon_{1,2})/k_{\rm
B%
}\approx 13600$ K for the crystal-field splitting between the respective $d$ states of Cu${}^{2+}$ as derived in the Appendix of this paper.
For $\sigma$ and $\pi$ bonds between copper and oxygen, $t_{\sigma}$ and $t_{\pi}$ denote the transfer integrals, $\Delta_{\sigma}$ and $\Delta_{\pi}$ denote the charge-transfer energies. $\Delta_{12}$ corresponds to the charge-transfer energy from one Cu site to the other excited Cu site, which amounts to $\Delta_{12}\approx 7$ eV.Papagno The ratio of the oxygen-copper transfer parameters $t_{\sigma}$ and $t_{\pi}$ to the charge-transfer energy $\Delta_{\sigma}\approx\Delta_{\pi}$ is known for oxides from studies of the transferred hyperfine interactions as $t^{2}_{\pi}/\Delta^{2}_{\pi}\approx t^{2}_{\sigma}/\Delta^{2}_{\sigma}\approx 0%
.077$. Walstedt The oxygen-copper transfer integrals are approximately equal ($t_{\sigma}\approx t_{\pi}$), and according to different estimations their value is about $1.3\leq t_{\sigma}\leq 2.5$ eV. Eskes ; Hybertsen Thus, we obtain $74\leq J_{x^{2}-y^{2},xz}\leq 275$ meV, which is significantly larger than $J_{1}\simeq 3.2$ meV.
Our estimation of the isotropic exchange interaction parameter between the ground and excited states is quite rough and probably strongly overestimated because of the idealized geometry. Therefore,
to obtain a more realistic value of the anisotropic exchange interaction, we refer to experimental values found for such an exchange geometry in other compounds. In literature the values range from $J=15$ meV (=174 K) in Sr${}_{2}$VO${}_{4}$,Eremin2011 , where the $d_{xy}$ and $p_{x}$ orbitals exhibit $\pi$-overlapping, to $J=112$ meV (=1298 K) in La${}_{2}$CuO${}_{4}$,Coldea2001 where the overlapping orbitals form $\sigma$ bonds. Using the minimum value $J_{xz,x^{2}-y^{2}}=15$ meV in Eq. 5, we get $A_{yy}/k_{\rm B}\approx 3.5$ K, which is still significant and cannot be neglected compared to the isotropic exchange.
Finally, we indicate a possible exchange path allowing the existence of the DM interaction between neighboring Cu${}^{2+}$ ions within the chains. Recently the DM interaction was suggested to be important for stabilization of the spin spiral order,Furukawa2010 ; Chen2014 but there is no consensus about its microscopic origin, yet. Furukawa et al.Furukawa2010 make the inter-layer exchange responsible for a nonzero DM interaction, while Chen amd Hu Chen2014 suppose that the DM interaction arises within the ladders resulting in a DM vector oriented preferably along the $b$ direction, but at least within the $ab$ plane.
The analysis of our ESR results described below demands a DM vector along the $a$ axis. This could be realized as follows: as the Cu${}^{2+}$ ions are built in O${}^{2-}$ square pyramids, which in $c$ direction are separated by Cu${}^{+}$ planes, we find a Cu${}^{2+}$ – O${}^{2-}$ – O${}^{2-}$ – Cu${}^{2+}$ exchange path via the apical oxygen ions giving rise to a DM vector pointing along the $a$ axis, if we neglect the distortions. This exchange path does not have any symmetric counterpart, which would compensate the DM vector. The neighboring chain within the same ladder exhibits the analogue geometry rotated by $180^{\circ}$ giving rise to a DM vector in opposite direction. But as these DM vectors belong to different pairs of Cu${}^{2+}$ ions, they do not compensate each other. Due to the admixture of excited orbitals the DM interaction exists, but an estimation of its magnitude is very difficult and demands a deeper theoretical analysis. Hence, we confine ourselves to the experimental determination of the DM contribution in LiCu${}_{2}$O${}_{2}$.
IV Experimental Results and Discussion
The untwinned single crystals under investgation were taken from the series of samples grown by solution in the melt described in Ref. Svistov2009, . For the ESR measurements they were fixed in Suprasil quartz tubes with paraffin to provide a well defined rotation axis for angular dependent investigations. The ESR measurements were performed in a Bruker ELEXSYS E500-CW spectrometer equipped with continuous-flow He cryostats (Oxford Instruments) at X- (9.47 GHz) and Q-band (34 GHz) frequencies in the temperature range $4.2\leq T\leq 300$ K. Like in earlier reports Vorotynov ; Zvyagin2002 and as shown in Fig. 5, the observed ESR absorption is well described by a single Lorentzian line with resonance field $H_{\rm res}$ and half-width at half maximum linewidth $\Delta H$ within the whole paramagnetic range above $T>35$ K. Note that the lines with the large linewidth $\Delta H\approx H_{res}$ were fitted including the counter resonance at $-H_{\rm res}$ as described in Ref. Joshi2004, .
Fig. 6 shows the angular dependence of the $g$ value at room temperature at $X$-band frequency and partially at $Q$-band frequency for the three principal crystallographic planes.
The $g$ values are independent from temperature for $T\geq 35$ K, with $g_{c}=2.28(1)$ and $g_{a}=g_{b}=2.05(1)$ as shown in the inset of Fig. 6. The observed anisotropy of the $g$ tensor is in agreement with the crystal-field analysis described in the Appendix. Note that due to the point-charge model the calculated $g$ values slightly overestimate the experimental ones.
On decreasing temperature the ESR line strongly broadens and disappears close to the ordering temperature.
This increase of the linewidth towards low temperatures is depicted in Fig. 7 for the field applied along all three principal axes. Fitting the data in terms of a critical law yields different exponents for the three orientations. This results from the competition of different relaxation processes with different temperature dependence and anisotropy. As shown by Oshikawa and Affleck,Oshikawa2002 the symmetric anisotropic exchange interaction gives rise to a monotonously increasing linewidth with increasing temperature which finally saturates at high temperature. In contrast the DM interaction provokes a divergence of the linewidth on decreasing temperature. In addition, critical behavior may arise because of critical fluctuations close to the Néel temperature. Due to different anisotropies of these relaxation processes, we cannot scale the temperature dependences of the three main directions on each other.
Now we focus on the angular dependence of the linewidth depicted in Fig. 8 for $T=150$, 200 and 300 K. For all temperatures the maximum of the linewidth is found for $H\parallel a$, the minimum for $H\parallel b$ and an intermediate value for $H\parallel c$ indicating the leading anisotropic exchange contribution to be connected to the $a$ direction. To fit the angular dependencies of the ESR linewidth in LiCu${}_{2}$O${}_{2}$ we used Eqs. 1-4. The isotropic exchange parameters were taken from Ref. Masuda_2005, . Hence, as fitting parameters we used the components of the symmetric anisotropic exchange interactions and of the antisymmetric DM interaction $\mathbf{D_{2}}$ (See Eqs. 4).
Taking into account the geometry of the exchange bonds, we reduced the number of relevant components to three: in their respective local coordinates the intra-chain interaction $\mathbf{J_{2}}$ is axial with respect to the $z^{\prime}$ axis, i.e. $J_{2}^{z^{\prime}z^{\prime}}=-2J_{2}^{x^{\prime}x^{\prime}}=-2J_{2}^{y^{\prime%
}y^{\prime}}$, the inter-chain interaction $\mathbf{J_{1}}$ is axial with respect to the $y^{\prime\prime}$ axis, i.e. $J_{1}^{y^{\prime\prime}y^{\prime\prime}}=-2J_{1}^{x^{\prime\prime}x^{\prime%
\prime}}=-2J_{1}^{z^{\prime\prime}z^{\prime\prime}}$, and only the DM vector component $D_{2}^{a}$ does not vanish.
As one can see, the model provides a good description of the experimental data. The resulting fitting parameters are given in Tab. 1 using the local coordinate systems of the symmetric anisotropic exchange interactions in LiCu${}_{2}$O${}_{2}$ and the crystallographic coordinate system for the DM vector. Note that from the analysis of the angular dependence of the linewidth one obtains the absolute value of the anisotropic exchange parameters. The sign of the anisotropic exchange interaction was derived from the theoretical analysis of the exchange bonds.
Fig. 9 shows the angular dependence of the three contributions separately. While the intra-chain symmetric anisotropic exchange $\mathbf{J_{2}}$ results in a maximum linewidth for $H\parallel c$ and a nearly constant contribution for $H\perp c$, the inter-chain symmetric anisotropic exchange $\mathbf{J_{1}}$ leads to a minimum linewidth for $H\parallel c$ and nearly isotropic behavior in the plane $H\perp c$, which can be understood by the superposition of the two inter-chain bonds in the zig-zag ladder. Thus, the symmetric anisotropic exchange contributions $\mathbf{J_{1}}$ and $\mathbf{J_{2}}$ together can only result in an effective axial anisotropy of the linewidth with respect to the crystallographic $c$ axis. Therefore, the antisymmetric DM interaction has to be introduced with the only relevant component $D_{2}^{a}\neq 0$ which allows describing the observed maximum linewidth for $H\parallel a$.
The ferromagnetic intra-chain $J_{2}^{z^{\prime}z^{\prime}}$ contribution is of comparable magnitude like in LiCuVO${}_{4}$, where the symmetric anisotropic exchange resulting from the ring geometry in the CuO${}_{2}$ ribbons dominated the spin-spin relaxation. In addition, the intra-ladder contribution is of high importance for the line broadening in LiCu${}_{2}$O${}_{2}$ as predicted by our estimation given above. Interestingly, the DM interaction yields the leading contribution. Here further theoretical efforts will be necessary to understand its origin and possible impact on the still unresolved problem, how to explain the multiferroicity in LiCu${}_{2}$O${}_{2}$.Sadykov2012
V Summary
We investigated the spin-spin relaxation of the antiferromagnetic $S=1/2$ quantum spin-ladder compound LiCu${}_{2}$O${}_{2}$ in the paramagnetic regime by means of electron spin resonance. From the anisotropy of the ESR linewidth obtained on untwinned single crystals we were able to extract the symmetric anisotropic exchange contributions resulting from nearest-neighbor super exchange $|J_{2}^{z^{\prime}z^{\prime}}|\sim 1$ K within the chains and next-nearest neighbor super exchange $|J_{1}^{y^{\prime\prime}y^{\prime\prime}}|\sim 2$ K within the rungs of the ladders formed by every two neighboring chains. In addition we discovered a sizable intra-chain antisymmetric anisotropic DM contribution $|D_{a}|\sim 5$ K which is necessary to describe the observed anisotropy of the linewidth accurately.
Concluding the discussion, let us recall the main microscopic interactions in LiCu${}_{2}$O${}_{2}$ suggested for explanation of the experimental data. The dominant interactions are of isotropic nature: intra-ladder ($J_{1}$), intra-chain ($J_{2}$, $J_{4}$), and inter-ladder exchange interactions ($J_{\perp}$) between the zig-zag ladders located within the $ab$ plane. The inter-plane exchange interactions are at least one order of magnitude smaller.Masuda_2005 The relativistic interactions in small fields are approximately 5-10 times smaller than the isotropic exchange interactions. The analysis of our ESR data suggests that the strongest of them is the intra-chain antisymmetric anisotropic DM interaction with the DM vector $\mathbf{D_{a}}$ directed parallel to the crystallographic $a$ direction. The symmetric anisotropic interactions for different exchange paths are found to be 3-5 times smaller. The values of these contributions have the values close to the related chain antiferromagnet LiCuVO${}_{4}$.
The suggested essential intra-chain DM interaction can be important for modeling of the magnetic structure of LiCu${}_{2}$O${}_{2}$ in the magnetically ordered state. In the limit of strong intra-chain DM interaction the chirality vectors of two spiral chains of every zig-zag ladder tend to be antiparallel, because the vectors $\mathbf{D_{a}}$ for these chains have different signs. Probably, this interaction providing the alternation of chirality vectors explains the absence of spontaneous electrical polarization in the structurally similar magnet NaCu${}_{2}$O${}_{2}$.Leininger2010
VI Acknowledgments
We thank M. V. Eremin for useful discussions concerning the crystal-field analysis.
This work was financially supported by the German Research Foundation (DFG) within the Transregional Collaborative Research Center TRR 80 ”From Electronic Correlations to Functionality” (Augsburg, Munich, Stuttgart). L. E. S., R. M. E., and T. P. G. gratefully acknowledge support by the Program of the Steering Committee of the Russian Academy of Sciences.
Appendix A Local Coordinate Systems
In the unit cell there are two different ladders which both consist of two chains. In the following expressions the upper and lower signs correspond to the directions of the individual vectors for different ladders. In the crystallographic coordinate system $(a,b,c)$ the unit vectors of the local coordinate systems of the intra-chain anisotropic exchange $\textbf{J}_{\textbf{2}}$ read
$$\Biggl{(}\begin{array}[]{cccc}x^{\prime}&0.982&0&\mp 0.187\\
y^{\prime}&0&1&0\\
z^{\prime}&\pm 0.187&0&0.982\end{array}\Biggr{)}$$
(7)
for first and second ladder, respectively. This means that only the local $y^{\prime}$ axis coincides with the crystallographic $b$ axis parallel to the Cu${}^{2+}$ chains, whereas $x^{\prime}$ and $z^{\prime}$ axes are slightly rotated from $a$ and $c$ axes, respectively. In the first ladder the unit vectors of the local coordinate system of the intra-ladder anisotropic exchange $\textbf{J}_{\textbf{1}}$ are given by
$$\Biggl{(}\begin{array}[]{cccc}x^{\prime\prime}&0.460&\pm 0.463&0.758\\
y^{\prime\prime}&\mp 0.725&0.689&\pm 0.018\\
z^{\prime\prime}&-0.513&\mp 0.557&0.653\end{array}\Biggr{)}$$
(8)
for first and second super-exchange bond, respectively. Analogously, for the second ladder the unit vectors of the local coordinate systems of $\textbf{J}_{\textbf{1}}$ are given by
$$\Biggl{(}\begin{array}[]{cccc}x^{\prime\prime}&-0.460&\pm 0.463&0.758\\
y^{\prime\prime}&\mp 0.725&-0.689&\mp 0.018\\
z^{\prime\prime}&0.513&\mp 0.557&0.653\end{array}\Biggr{)}$$
(9)
Appendix B Crystal-Field Analysis
In LiCu${}_{2}$O${}_{2}$ the Cu${}^{2+}$ ions (electronic configuration $3d^{9}$, spin $S=1/2$) are surrounded by five O${}^{2-}$ ions and four Cu${}^{+}$ ions. The nearest-neighbor environment of the magnetic Cu${}^{2+}$
ion is depicted in Fig. 10. To calculate the energy-level scheme of Cu${}^{2+}$ in LiCu${}_{2}$O${}_{2}$, we start from the following Hamiltonian:
$${\cal H}_{0}=\lambda(\mathbf{L}\cdot\mathbf{S})+\sum_{k\,=\,2;\,4}\,\sum_{q\,=%
\,-k}^{k}B_{q}^{(k)}\,C_{q}^{(k)}(\vartheta,\,\varphi)$$
(10)
The first term corresponds to the spin-orbit coupling. S and L are total spin and orbital moment, respectively. For Cu${}^{2+}$ the spin-orbit coupling parameter amounts $\lambda\approx 830$ cm${}^{-1}$.Abragam The second term represents the crystal-field operator, where $C_{q}^{(k)}$ denote the components of the spherical tensor. We use a coordinate system with the Cartesian axes $x$, $y$, and $z$ chosen along the crystallographic axes $a$, $b$, and $c$, respectively. The crystal-field parameters $B_{q}^{(k)}$ (in eV) were calculated using a superposition model with exchange contributions.Malkin ; Eremin The relevant overlap integrals were calculated using Hartree-Fock wave functions for Cu${}^{2+}$ and O${}^{2-}$.Eremin11 The exchange-charge parameter $G=9.9$ was chosen in accordance with the optical excitation energy $\Delta=1.95$ eV.Papagno
Using the crystal-field parameters listed in Table 2 for the position Cu${}^{2+}$ (0.124; 1/4; 0.906) we obtain the following set of Kramers doublets: $\varepsilon_{1,2}=0$, $\varepsilon_{3,4}=1.077$ eV, $\varepsilon_{5,6}=1.192$ eV, $\varepsilon_{7,8}=1.213$ eV, and $\varepsilon_{9,10}=1.963$ eV. In the local coordinate system the wave functions read:
$$|\varepsilon_{n}\rangle=\sum_{m_{l}\,=\,-2}^{+2}\,\sum_{m_{S}\,=\,\uparrow,\,%
\downarrow}a_{m_{l},\,m_{S}}^{(n)}\,|m_{l},\,m_{S}\rangle$$
(11)
The values of the coefficients are given in Tab. 4. Using these wave functions we calculated the $g$-tensor components $g_{z}=2\langle k_{z}l_{z}+2s_{z}\rangle$, $g_{x}=2\langle k_{x}l_{x}+2s_{x}\rangle$, and $g_{y}=2\langle k_{y}l_{y}+2s_{y}\rangle$, which are equal for all four copper positions: assuming the reduction factors of the orbital momentum due to covalency effects as $k_{x}=k_{y}=k_{z}=0.8$ we obtained $g_{z}=2.41$, $g_{x}=2.09$, and $g_{y}=2.09$.
Note that the energy level scheme derived here, differs from that reportedHuang2011 for the CASSCF/MRCI $d-d$ excitation energies for edge sharing chains of CuO${}_{4}$ plaquettes in LiCu${}_{2}$O${}_{2}$; $0$ $(d_{xy});$ $1.13/1.43$ eV $(d_{x^{2}-y^{2}});$ $1.58/1.88$ eV $(d_{xz});$ $1.64/1.94$ eV $(d_{yz});$ $1.67/1.98$ eV $(d_{z^{2}})$, since we have taken into account the contributions to the crystal field from long-distant ligands, which are not negligible.
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Projective connections associated with second order ODEs
Ezra T Newman
Department of Physics and Astronomy
University of Pittsburgh
Pittsburgh PA
USA
[email protected]
Pawel Nurowski
Instytut Fizyki Teoretycznej
Uniwersytet Warszawski
ul. Hoza 69, Warszawa
Poland
[email protected]
Abstract
We show that every 2nd order ODE defines a 4-parameter
family of projective connections on its 2-dimensional
solution space. In a special case of ODEs, for which a certain point
transformation invariant vanishes, we find that this family of
connections always has a preferred representative. This preferred
representative turns out to
be identical to the projective connection described in Cartan’s classic paper
Sur les Varietes a Connection
Projective.
1 Introduction
In recent years there has been a return of interest in the two
related classical issues associated with differential equations: (1) the
equivalence problem (under a variety of transformation types) for the
equations and (2) the natural geometric structures induced by the equations
on their solution spaces. The original studies began, among others, with the
work of Lie [10] and his student, Tresse [11, 12]. This was
soon followed by Wünschmann’s contribution
[13] and reached its peak with the work of
Cartan [2] and Chern[3]. Cartan devised an extremely
powerful but difficult scheme for the analysis of the equivalence problem
under the three classes of transformation: fiber preserving, point and
contact. Though equivalence relations were established for a variety of
equations and transformation classes, the calculations were extraordinarily
complicated and long, and as a consequence, many problems were only
partially completed. (The modern advent of algebraic computers has allowed
the completion of many of these problems and opened the door to a variety of
new problems [8, 5, 4, 7].) Early in these studies - then confined
to general 2nd and 3rd order odes - it was realized that the equations
themselves defined on the (finite dimensional) solution spaces certain
geometric structures. For example, Wünschmann discovered that a (large)
class of 3rd order ode’s define a conformal (Lorentzian) metric on the
3-dimensional solution space. This class was defined by the vanishing of a
certain function of the 3rd order equation. Later, in the context of
Cartan’s and Chern’s work, this function was understood as a (relative)
invariant of the equation under contact transformations and became known as
the Wünschmann invariant. (As an aside we mention that in the modern context
of general relativity, this work was generalized to pairs of 2nd order odes
whose solution space is 4-dimensional. The vanishing of a generalized
Wünschmann invariant for these equations leads to a conformal Lorentzian
metric on the solution space. All four dimensional Lorentzian metrics are
obtainable in this manner[4, 5].)
Cartan, following Lie and Tresse, using his scheme for the
analysis of 2nd order odes under point transformations, realized[1]
that a large class of 2nd order odes induced a natural projective structure
on their 2-dimensional solution space. This class was defined (analogously
to the 3rd order ode case) by the vanishing of a certain Wünschmann-like
function of the 2nd order equation.
In the present work we return to the problem of the
geometry associated with any 2nd order ode. Without recourse to Cartan’s
equivalence technique, we find that any 2nd order ode defines, via the
torsion-free 1st Cartan structure equation, a 4-parameter family
of projective connections on the solution space.
In the second section we review the general theory of normal
projective connections on $n$-manifolds from the point of view of Cartan
connections. We also define projective structures as equivalence
classes of certain sets of one-forms on these manifolds.
As an example of projective connections, in the third section, we
consider the geometry associated with a second order ode.
We find a natural 4-parameter family of projective connections living
on its two dimensional space of solutions. In general these
connections are quite complicated. They are parametrized by the
solutions of a certain linear ode of fourth order,
which is naturally associated with our ode.
We find that among all the odes $y^{\prime\prime}=Q(x,y,y^{\prime})$ there is a large class
for which the associated 4th order ode is homogeneous. This
class of equations is characterized in terms of the
vanishing af a certain function constructed solely from $Q$ and its
derivatives, which is directly analogous to the Wünschman function.
It turns out that the trivial solution of the homegeneous 4th
order ode singles out a preferred connection from the 4-parameter
family. Then this
class of second order odes together with this preferred connection
turns out to be identical to the class that Cartan obtained from a
study of the equivalence problem. In the last section we discuss the
relationship between our and Cartan’s method of obtaining this class.
The work described here is part of a larger project, namely the
study of natural geometric structures induced on the finite dimensional
solution spaces of both odes and certain overdetermined pdes. In earlier work
[6] we saw how all 4-dimensional conformal metrics and Cartan normal
conformal connections were contained in the space of pairs of pdes
satisfying generalized Wünschmann equations. (Similar results hold for all
3rd order odes satisfying the Wünschmann equation.) In the present work we
have extended these results to unique Cartan normal projective connections
associated with 2nd order odes satisfying a Wünschmann-like equation.
2 Projective connection
2.1 Cartan connection
In this subsection we will first define a Cartan connection and then
specialize it to a Cartan projective connection (see [9] for
more details).
Consider a structure $(P,H,M,G)$ such that
•
$(P,H,M)$ is the principal fibre bundle, over an $n$-dimensional
manifold, with a structure Lie group $H$
•
$G$ is a Lie group, of dimension dim$G=$dim$P$, for which $H$ is a
closed subgroup.
Denote by $B^{*}$ the fundamental vector field associated with an element
$B$ of the Lie algebra $H^{\prime}$ of $H$.
Let $\omega$ be a $G^{\prime}$-valued 1-form on $P$ such that
–
$\omega(B^{*})=B$ for each $B\in H^{\prime}$
–
$R_{b}^{*}\omega=b^{-1}\omega b$ for each $b\in H$
–
$\omega(X)=0$ if and only if the vector field $X$ vanishes
identically on $P$.
Then $\omega$ is called Cartan’s connection on $(P,H,M,G)$.
The Cartan projective connection is a Cartan connection for which
$$G={\bf SL}(n+1,{\bf R})/({\rm center}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
~{}~{}~{}~{}$$
$$H=\{\begin{pmatrix}{\bf A}&0\\
A^{T}&(\det{\bf A})^{-1}\end{pmatrix},~{}{\bf A}\in{\bf GL}(n,{\bf R}),~{}A\in%
{\bf R}^{n}\}/({\rm center})$$
In next two subsections we present a convenient way of defining a projective connection on a local
trivialization $U\times H$ of the bundle $P$.
2.2 Normal projective connection on $U\in M$
Here working on the base space $M$ we define a normal projective
connection on $U\subset M$.
Consider a coframe $(\omega^{i})$, $i=1,2,...,n$ on an open
neighbourhood $U$ of $M$. Suppose that in addition you have $n^{2}$
1-forms $\omega^{i}_{~{}j}$, $i,j=1,2,...,n$ on $M$ such that
$${\rm d}\omega^{i}+\omega^{i}_{~{}j}\wedge\omega^{j}=0,~{}~{}~{}\forall i=1,2,.%
..n.$$
(1)
Then, the system of forms $(\omega^{i},\omega^{i}_{~{}j})$ defines a torsion-free
connection on $U$.
Take $n$ arbitrary 1-forms $(\omega_{i})$, $i=1,2,...n$ on
$U$. The forms $(\omega^{i},\omega^{i}_{~{}j},\omega_{j})$ define the $n^{2}$ 2-forms
$\Omega^{i}_{~{}j}$ and $n$ 2-forms $\Psi_{j}$ on $U$ by
$$\Omega^{i}_{~{}j}={\rm d}\omega^{i}_{~{}j}+\omega^{i}_{~{}k}\wedge\omega^{k}_{%
~{}j}+\omega^{i}\wedge\omega_{j}+\delta^{i}_{~{}j}\omega^{k}\wedge\omega_{k},$$
(2)
$$\Psi_{i}={\rm d}\omega_{i}+\omega_{k}\wedge\omega^{k}_{~{}j}.$$
(3)
Decompose $\Omega^{i}_{~{}j}$ onto the basis $(\omega^{i})$,
$$\Omega^{i}_{~{}j}=\frac{1}{2}\Omega^{i}_{~{}jkl}\omega^{k}\wedge\omega^{l}.$$
Find all $(\omega_{i})$ for which the so called normal
condition
$$\Omega^{i}_{~{}jil}=0,~{}~{}~{}~{}~{}\forall j,l=1,2,...n$$
(4)
is satisfied. It turns out that if $n\geq 2$ the forms $\omega_{i}$ are determined
uniquely by the equations (4). Indeed, by using the Riemann 2-forms
$$R^{i}_{~{}j}=\frac{1}{2}R^{i}_{~{}jkl}\omega^{k}\wedge\omega^{l}={\rm d}\omega%
^{i}_{~{}j}+\omega^{i}_{~{}k}\wedge\omega^{k}_{~{}j}$$
(5)
and the Ricci tensor
$$R_{jl}=R^{i}_{jil}$$
(6)
of the connection $\omega^{i}_{~{}j}$ one finds that
$$\omega_{i}=~{}[~{}\frac{1}{1-n}R_{(ij)}-\frac{1}{1+n}R_{[ij]}~{}]~{}\omega^{j}.$$
(7)
Having determined the forms $\omega_{i}$, collect the system of 1-forms
$(\omega^{i},\omega^{i}_{~{}j},\omega_{j})$ into a matrix
$$\omega_{u}=\begin{pmatrix}\omega^{i}_{~{}k}-\frac{1}{n+1}\omega^{l}_{~{}l}%
\delta^{i}_{~{}k}&\omega^{i}\\
&\\
\omega_{k}&-\frac{1}{n+1}\omega^{l}_{~{}l}\end{pmatrix}.$$
(8)
Note that $\omega_{u}$ is a 1-form on $U$ which has values in the Lie
algebra $G^{\prime}={\bf SL}^{\prime}(n+1,{\bf R})$. It is called
a normal projective connection on $U$.
2.3 Normal projective connection on $U\times H$
Earlier we defined a Cartan projective connection on the principal
$H$-bundle $(P,M,H,G)$. Here we show how the normal projective
connection on $U\subset M$ can be lifted to $(P,M,H,G)$.
Choose a generic element of $H$ in the form
$$b=\begin{pmatrix}A^{i}_{~{}k}&0\\
&\\
A_{k}&a^{-1}\end{pmatrix},$$
(9)
where $(A^{i}_{~{}j})$ is a real-valued
$n\times n$ matrix with nonvanishing determinant $a=\det(A^{i}_{~{}j})$, and $(A_{i})$ is a
real row $n$-vector.
Define a $G^{\prime}$-valued 1-form $\omega$ on $U\times H$ by
$$\omega=b^{-1}\omega_{u}b+b^{-1}{\rm d}b.$$
(10)
The 1-form $\omega$ defines a projective connection on $U\times H$.
This projective connection
on $U\times H$ is called the normal projective connection. The
term normal referes to the condition (4), which this
connection satisfies.
The explicit formulae for the normal projective connection
(10) are written below.
$$\omega=\begin{pmatrix}\omega^{\prime i}_{~{}k}-\frac{1}{n+1}\omega^{\prime l}_%
{~{}l}\delta^{i}_{~{}k}&\omega^{\prime i}\\
&\\
\omega^{\prime}_{k}&-\frac{1}{n+1}\omega^{\prime l}_{~{}l}\end{pmatrix},$$
(11)
where
$$\omega^{\prime i}=a^{-1}A^{-1i}_{~{}~{}~{}~{}j}\omega^{j},$$
(12)
$$\omega^{\prime i}_{~{}j}=A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}_{~{}l}A^{l}_{~{}j}+%
A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}A_{j}+\delta^{i}_{~{}j}A_{l}A^{-1l}_{~{}~{}~{%
}~{}k}\omega^{k}+A^{-1i}_{~{}~{}~{}~{}k}{\rm d}A^{k}_{~{}j}+\delta^{i}_{~{}j}a%
^{-1}{\rm d}a,$$
(13)
$$\omega^{\prime}_{i}=a~{}(~{}\omega_{k}A^{k}_{~{}i}-A_{l}A^{-1l}_{~{}~{}~{}~{}j%
}\omega^{j}_{~{}k}A^{k}_{~{}i}-A_{l}A^{-1l}_{~{}~{}~{}~{}j}\omega^{j}A_{i}+{%
\rm d}A_{i}-A_{l}A^{-1l}_{~{}~{}~{}~{}j}{\rm d}A^{j}_{~{}i}~{})~{},$$
(14)
and we have used the fact that
$${\rm d}a=aA^{-1l}_{~{}~{}~{}~{}k}{\rm d}A^{k}_{~{}l}.$$
(15)
The curvature
$$\Omega={\rm d}\omega+\omega\wedge\omega$$
(16)
of $\omega$ has the form
$$\Omega=b^{-1}\Omega_{u}b,~{}~{}~{}{\rm where}~{}~{}~{}\Omega_{u}={\rm d}\omega%
_{u}+\omega_{u}\wedge\omega_{u}=\begin{pmatrix}\Omega^{i}_{~{}j}-\frac{1}{n+1}%
\delta^{i}_{~{}j}\Omega^{l}_{~{}l}&0\\
&\\
\Psi_{j}&-\frac{1}{n+1}\Omega^{l}_{~{}l}\end{pmatrix}$$
(17)
It is worthwhile to note that if $n\geq 3$ then the vanishing of $\Omega^{i}_{~{}j}$ implies
the vanishing of $\Psi_{i}$. This follows from the Bianchi identity
${\rm d}\Omega-\Omega\wedge\omega+\omega\wedge\Omega=0$.
It is known that in dimension $n=2$ the forms $\Omega^{i}_{~{}j}$
are identically equal to zero. In this dimension all the information
about the curvature of the normal projective connection is encoded in the forms $\Psi_{i}$.
Remark
To globalize the local trivialization construction of the normal projective
connection described above one needs assumptions about topology of
$M$. In the local treatment we use in this paper these assumptions are not neccessary.
2.4 Projective structure on $M$
An alternative view of the formulae
(12)-(14) is to consider them as an
equivalence class of connections on $U$. This motivates the following definition.
A projective structure on an $n$-dimensional manifold $M$ is an
equivalence class $[(\omega^{i},\omega^{i}_{~{}j})]$ of sets of 1-forms $(\omega^{i},\omega^{i}_{~{}j})$ on $M$ such that
•
$(\omega^{i})$, $i=1,2,...,n$ is a coframe on $M$ such that
$${\rm d}\omega^{i}+\omega^{i}_{~{}j}\wedge\omega^{j}=0,~{}~{}~{}\forall i=1,2,.%
..,n$$
•
two sets $(\omega^{i},\omega^{i}_{~{}j})$ and $(\omega^{\prime i},\omega^{\prime i}_{~{}j})$
are in the same equivalence class iff there exists functions
$A^{i}_{~{}j}$ and $A_{i}$ on $M$ such that
$$\omega^{\prime i}=a^{-1}A^{-1i}_{~{}~{}~{}~{}j}\omega^{j}$$
(18)
and
$$\omega^{\prime i}_{~{}j}=A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}_{~{}l}A^{l}_{~{}j}+%
A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}A_{j}+\delta^{i}_{~{}j}A_{l}A^{-1l}_{~{}~{}~{%
}~{}k}\omega^{k}+A^{-1i}_{~{}~{}~{}~{}k}{\rm d}A^{k}_{~{}j}+\delta^{i}_{~{}j}a%
^{-1}{\rm d}a,$$
(19)
with $a=\det(A^{i}_{~{}j})\neq 0$ at every point of $M$.
It turns out that all the torsion-free
connections from the equivalence class of a given projective structure
have the same set of geodesics on $M$. To see this consider a representative $(\omega^{i},\omega^{i}_{~{}j})$ of a
projective structure on $M$. Let $(e_{i})$ be the set of $n$-vector fields dual to the coframe
$(\omega^{i})$, i.e. $\omega^{i}(e_{j})=\delta^{i}_{~{}j}$. Let $\gamma(t)$ be a geodesic curve for the
connection 1-forms $\omega^{i}_{~{}j}=\omega^{i}_{~{}jk}\omega^{k}$. This means that if
$V=\frac{{\rm d}}{{\rm d}t}=V^{i}e_{i}$ is a
vector tangent to this curve then
$$\frac{{\rm d}V^{i}}{{\rm d}t}+\omega^{i}_{~{}jk}V^{j}V^{k}=fV^{i},$$
(20)
with a certain function $f$ on $M$. If
$(\omega^{\prime i},\omega^{\prime i}_{~{}j})$ belongs to the same projective structure as
$(\omega^{i},\omega^{i}_{~{}j})$ then the equation
(20) for $V^{i}$ and the relations between $(\omega^{i},\omega^{i}_{~{}j})$
and $(\omega^{\prime i},\omega^{\prime i}_{~{}j})$ imply that in the coframe $(\omega^{\prime i})$ the
$V^{\prime i}$ component of the vector $V=V^{\prime i}e^{\prime}_{i}$ satisfies geodesic equation
$$\frac{{\rm d}V^{\prime i}}{{\rm d}t}+\omega^{\prime i}_{~{}jk}V^{\prime j}V^{%
\prime k}=f^{\prime}V^{\prime i},$$
(21)
with merely new function $f^{\prime}=f+2aA_{j}V^{\prime j}$. Thus the curve $\gamma(t)$ is also
a geodesic in connection $\omega^{\prime i}_{~{}j}$.
Note that if $A^{i}_{~{}j}=\delta^{i}_{~{}j}$ then
$$\omega^{\prime i}=\omega^{i}$$
(22)
and
$$\omega^{\prime i}_{~{}j}=\omega^{i}_{~{}j}+\omega^{i}A_{j}+\delta^{i}_{~{}j}A,$$
(23)
with $A=A_{i}\omega^{i}$. Thus, for a given projective structure
$(\omega^{i},\omega^{i}_{~{}j})$, fixing the coframe does not fix
the gauge in the choice of $\omega^{\prime i}_{~{}j}$. There exists an entire
class (23) of connections
that, together with the fixed coframe $(\omega^{i})$, represents the
same projective structure.
2.5 Equivalence of projective structures
We say that two projective structures $(\omega^{i},\omega^{i}_{~{}j})$ and
$(\bar{\omega}^{i},\bar{\omega}^{i}_{~{}j})$ on two respective $n$-dimensional
manifolds $M$ and $\bar{M}$ are (locally) equivalent iff there
exists a (local) diffeomorphism $\phi:M\to\bar{M}$ and functions
$A^{i}_{~{}j}$ and $A_{j}$ on $M$ such that
$$\phi^{*}(\bar{\omega}^{i})=a^{-1}A^{-1i}_{~{}~{}~{}~{}j}\omega^{j}$$
and
$$\phi^{*}(\bar{\omega})^{i}_{~{}j}=A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}_{~{}l}A^{l%
}_{~{}j}+A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}A_{j}+\delta^{i}_{~{}j}A_{l}A^{-1l}_%
{~{}~{}~{}~{}k}\omega^{k}+A^{-1i}_{~{}~{}~{}~{}k}{\rm d}A^{k}_{~{}j}+\delta^{i%
}_{~{}j}a^{-1}{\rm d}a,$$
with $a=\det(A^{i}_{~{}j})\neq 0$.
If, given a projective structure $(\omega^{i},\omega^{i}_{~{}j})$ on
$M$, we have a diffeomorphism $\phi:M\to M$ with $A^{i}_{~{}j}$
and $A_{j}$ as above, such that
$$\phi^{*}(\omega^{i})=a^{-1}A^{-1i}_{~{}~{}~{}~{}j}\omega^{j}$$
(24)
and
$$\phi^{*}(\omega)^{i}_{~{}j}=A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}_{~{}l}A^{l}_{~{}%
j}+A^{-1i}_{~{}~{}~{}~{}k}\omega^{k}A_{j}+\delta^{i}_{~{}j}A_{l}A^{-1l}_{~{}~{%
}~{}~{}k}\omega^{k}+A^{-1i}_{~{}~{}~{}~{}k}{\rm d}A^{k}_{~{}j}+\delta^{i}_{~{}%
j}a^{-1}{\rm d}a,$$
(25)
then we call $\phi$ a symmetry of
$(\omega^{i},\omega^{i}_{~{}j})$. Locally, a 1-parameter group of
symmetries $\phi_{t}:M\to M$ of $(\omega^{i},\omega^{i}_{~{}j})$ is expressible in terms of the
corresponding vector field $X$, called an infinitesimal
symmetry.
Taking the Lie derivative with respect to $X$ of equations
(24)-(25) one obtains the following characterization
of infinitesimal symetries.
A vector field $X$ is an infinitesimal symmetry of
a projective structure $(\omega^{i},\omega^{i}_{~{}j})$ iff there
exist functions
$B^{i}_{~{}j}$ and $B_{j}$ on $M$ such that
$${\cal L}_{X}\omega^{i}=-(B^{i}_{~{}j}+B^{k}_{~{}k}\delta^{i}_{~{}j})\omega^{j}$$
(26)
$${\cal L}_{X}\omega^{i}_{~{}j}=\omega^{i}_{~{}j}B^{l}_{~{}j}-B^{i}_{~{}l}\omega%
^{l}_{~{}j}+\omega^{i}B_{j}+\delta^{i}_{~{}j}B_{l}\omega^{l}+{\rm d}B^{i}_{~{}%
j}+\delta^{i}_{~{}j}{\rm d}B^{k}_{~{}k}.$$
(27)
It is easy to check that a Lie bracket $[X_{1},X_{2}]$ of two
infinitesimal symmetries is an infinitesimal symmetry, hence the
infinitesimal symmetries generate a Lie algebra. This is the Lie
algebra of infinitesimal symmetries of the structure $(\omega^{i},\omega^{i}_{~{}j})$.
3 Projective structures of second order ODEs
3.1 Contact forms associated with a second order ODE
We now show that a second order ODE defines a projective structure on
the space of its solutions.
A second order ODE
$$\frac{{\rm d}^{2}y}{{\rm d}x^{2}}~{}=~{}Q(~{}x,~{}y,~{}\frac{{\rm d}y}{{\rm d}%
x}~{})$$
(28)
for a function ${\bf R}\ni x\to y=y(x)\in{\bf R}$, can be
alternatively written as a system of the two first order ODEs
$$\frac{{\rm d}y}{{\rm d}x}~{}=~{}p,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
\frac{{\rm d}p}{{\rm d}x}~{}=~{}Q(x,y,p)$$
(29)
for two functions ${\bf R}\ni x\to y=y(x)\in{\bf R}$ and ${\bf R}\ni x\to p=p(x)\in{\bf R}$. This system defines two (contact) 1-forms
$$\omega^{1}={\rm d}y-p{\rm d}x,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\omega%
^{2}={\rm d}p-Q{\rm d}x,$$
(30)
which live on a 3-dimensional manifold $J^{1}$, the first jet space, parametrized by
coordinates $(x,y,p)$. All the information about the ODE (28)
is encoded in these two forms. For example any solution to
(28) is a curve $\gamma(x)=(~{}x,y(x),p(x)~{})\subset J^{1}$ on
which the forms (30) vanish.
Given an ODE (28), we look for a set
$(\omega^{1}_{~{}1},\omega^{1}_{~{}2},\omega^{2}_{~{}1},\omega^{2}_{~{}2})$ of 1-forms on $J^{1}$ such that
$${\rm d}\omega^{1}+\omega^{1}_{~{}1}\wedge\omega^{1}+\omega^{1}_{~{}2}\wedge%
\omega^{2}=0,~{}~{}~{}~{}~{}~{}{\rm d}\omega^{2}+\omega^{2}_{~{}1}\wedge\omega%
^{1}+\omega^{2}_{~{}2}\wedge\omega^{2}=0.$$
(31)
Introducing the third 1-form
$$\omega^{3}={\rm d}x,$$
(32)
which together with
$\omega^{1}$ and $\omega^{2}$ constitutes a basis of 1-forms on $J^{1}$, we
find that
$${\rm d}\omega^{1}=-\omega^{2}\wedge\omega^{3},~{}~{}~{}~{}~{}~{}~{}{\rm d}%
\omega^{2}=-(Q_{y}\omega^{1}+Q_{p}\omega^{2})\wedge\omega^{3},$$
(33)
and that the general solution to the ‘vanishing torsion’ equations
(31) is
$$\omega^{1}_{~{}1}=\omega^{1}_{~{}11}\omega^{1}+\omega^{1}_{~{}12}\omega^{2},~{%
}~{}~{}~{}~{}~{}\omega^{1}_{~{}2}=\omega^{1}_{~{}12}\omega^{1}+\omega^{1}_{~{}%
22}\omega^{2}-\omega^{3},$$
(34)
$$\omega^{2}_{~{}1}=\omega^{2}_{~{}11}\omega^{1}+\omega^{2}_{~{}12}\omega^{2}-Q_%
{y}\omega^{3},~{}~{}~{}~{}~{}~{}\omega^{2}_{~{}2}=\omega^{2}_{~{}12}\omega^{1}%
+\omega^{2}_{~{}22}\omega^{2}-Q_{p}\omega^{3},$$
(35)
with some unspecified functions
$(\omega^{1}_{~{}11},\omega^{1}_{~{}12},\omega^{1}_{~{}22},\omega^{2}_{~{}11},%
\omega^{2}_{~{}12},\omega^{2}_{~{}22})$
on $J^{1}$. Here, and in the following, we denoted the partial
derivatives with respect to a variable, as a subscript on the function
whose partial derivative is evaluated, e.g.
$Q_{y}:=\frac{\partial Q}{\partial y}$.
The anihilator of the contact forms $\omega^{1}$ and $\omega^{2}$ is
spanned by the vector field
$$D=\partial_{x}+p\partial_{y}+Q\partial_{p},$$
(36)
which is defined
up to a multiplicative
factor. Its integral curves, which coincide with the
solutions $\gamma(x)$ of the
original equation, are intrinsically
defined. Also the notion of surfaces $S$, transversal
to $D$ is unambigous.
Any choice of 1-forms $(\omega^{1}_{~{}1},\omega^{1}_{~{}2},\omega^{2}_{~{}1},\omega^{2}_{~{}2})$ of
the form given by equations
(34)-(35) on the jet space $J^{1}$ determines projective
structures $[(\omega^{k};\omega^{i}_{~{}j})_{|S}]$
on each 2-dimensional surface $S$ transversal to $D$. These projective
structures are defined on each
$S$ by transformations (18)-(19) applied to the 1-forms
$(\omega^{k};\omega^{i}_{~{}j})_{|S}$.
They, in turn, were defined as the restrictions of the 1-forms
$(\omega^{1},\omega^{2};\omega^{1}_{~{}1},\omega^{1}_{~{}2},\omega^{2}_{~{}1},%
\omega^{2}_{~{}2})$
from $J^{1}$ to $S$. Given a particular choice of functions $\omega^{i}_{~{}jk}$ in
(34)-(35) and a pair of transversal to $D$ surfaces $S$ and
$S^{\prime}$, the projective structures
$[(\omega^{k};\omega^{i}_{~{}j})_{|S}]$ and $[(\omega^{k};\omega^{i}_{j})_{|S^{\prime}}]$ will
be in general inequivalent. It is therefore interesting to ask as to
whether there exist a choice of forms (34)-(35) which, on
all transversal surfaces $S$, defines the same (modulo
equivalence) projective structure. Locally, this requirement is
equivalent to the existence of a choice of forms
(34)-(35) on $J^{1}$ such that
the Lie derivative of the forms $(\omega^{i};\omega^{k}_{~{}j})$ along $D$ is
simply the infinitesimal version of the transformations
(24)-(25). Explicitly, we ask for the existence of
$\omega^{i}_{~{}jk}$ of (34)-(35) and the existence of
functions $B^{i}_{~{}j}$ and $B_{k}$ on $J^{1}$ such that
$${\cal L}_{D}\omega^{i}=-(B^{i}_{~{}j}+B^{k}_{~{}k}\delta^{i}_{~{}j})\omega^{j}$$
(37)
$${\cal L}_{D}\omega^{i}_{~{}j}=\omega^{i}_{~{}j}B^{l}_{~{}j}-B^{i}_{~{}l}\omega%
^{l}_{~{}j}+\omega^{i}B_{j}+\delta^{i}_{~{}j}B_{l}\omega^{l}+{\rm d}B^{i}_{~{}%
j}+\delta^{i}_{~{}j}{\rm d}B^{k}_{~{}k}~{}~{}~{}~{}~{}~{}i,j=1,2.$$
(38)
If we were able to find a solution $\omega^{i}_{~{}jk}$ to the above equations,
then it would generate the same projective structure on all
surfaces transversal to $D$. This structure would therefore descend to
the 2-dimensional space of integral lines of $D$ endowing it, or what
is the same, endowing the parameter space of
solutions to the original ODE, with a projective structure.
To solve equations (37)-(38) we take the most
general forms $(\omega^{1}_{~{}1},\omega^{1}_{~{}2},\omega^{2}_{~{}1},\omega^{2}_{~{}2})$ (from
(34)-(35)) that are associated with the ODE. We then use the
gauge freedom (22)-(23) preserving
$$\omega^{1}={\rm d}y-p{\rm d}x,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\omega%
^{2}={\rm d}p-Q{\rm d}x$$
to achieve
$$\omega^{1}_{~{}1}=0$$
everywhere on $J^{1}$. The forms
$(\omega^{1},\omega^{2};\omega^{1}_{~{}1},\omega^{1}_{~{}2},\omega^{2}_{~{}1},%
\omega^{2}_{~{}2})$ with
$\omega^{1}_{~{}1}=0$, when restricted to each $S$, will therefore represent the same
projective structure on $S$ as the original general forms we started with. Thus, without loss of
generality, we solve equations
(37)-(38) for forms
$$\omega^{1}={\rm d}y-p{\rm d}x,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\omega%
^{2}={\rm d}p-Q{\rm d}x$$
(39)
and
$$\omega^{1}_{~{}1}=0,~{}~{}~{}~{}~{}~{}\omega^{1}_{~{}2}=\omega^{1}_{~{}22}%
\omega^{2}-\omega^{3},$$
(40)
$$\omega^{2}_{~{}1}=\omega^{2}_{~{}11}\omega^{1}+\omega^{2}_{~{}12}\omega^{2}-Q_%
{y}\omega^{3},~{}~{}~{}~{}~{}~{}\omega^{2}_{~{}2}=\omega^{2}_{~{}12}\omega^{1}%
+\omega^{2}_{~{}22}\omega^{2}-Q_{p}\omega^{3}.$$
It is a matter of straigthforward calculation to achieve the following
proposition.
Proposition 1
The forms (39)-(40) satisfy equations
(37)-(38)
if and only if
$$\displaystyle\omega^{2}_{~{}22}=$$
$$\displaystyle D\omega^{1}_{~{}22}+2Q_{p}\omega^{1}_{~{}22},$$
$$\displaystyle\omega^{2}_{~{}12}=$$
$$\displaystyle\frac{1}{4}~{}[-D^{2}\omega^{1}_{~{}22}-3Q_{p}D\omega^{1}_{~{}22}%
+(3Q_{y}-2Q_{p}^{2}-2DQ_{p})\omega^{1}_{~{}22}-Q_{pp}~{}]$$
(41)
$$\displaystyle\omega^{2}_{~{}11}=$$
$$\displaystyle\frac{1}{6}~{}[D^{3}\omega^{1}_{~{}22}+3D^{2}\omega^{1}_{~{}22}+(%
5DQ_{p}+2Q_{p}^{2}-7Q_{y})D\omega^{1}_{~{}22}+$$
$$\displaystyle~{}~{}~{}~{}(2D^{2}Q_{p}-3DQ_{y}+4Q_{p}DQ_{p}-8Q_{p}Q_{y})\omega^%
{1}_{~{}22}+DQ_{pp}-4Q_{py}~{}]$$
and $\omega^{1}_{~{}22}$ fulfills the differential equation
$$D^{4}\omega^{1}_{~{}22}+a_{4}D^{3}\omega^{1}_{~{}22}+a_{3}D^{2}\omega^{1}_{~{}%
22}+a_{2}D\omega^{1}_{~{}22}+a_{1}\omega^{1}_{~{}22}+a_{0}=0$$
(42)
with coefficients $a_{0},a_{1},a_{2},a_{3},a_{4}$ given by
$$a_{4}=2Q_{p},$$
$$a_{3}=(8DQ_{p}-Q_{p}^{2}-10Q_{y}),$$
$$a_{2}=(7D^{2}Q_{p}-10DQ_{y}+3Q_{p}DQ_{p}-2Q_{p}^{3}-10Q_{p}Q_{y}),$$
(43)
$$a_{1}=(2D^{3}Q_{p}-3D^{2}Q_{y}+4(DQ_{p})^{2}+2Q_{p}D^{2}Q_{p}-5Q_{p}DQ_{y}-4Q_%
{p}^{2}DQ_{p}-14Q_{y}DQ_{p}+2Q_{p}^{2}Q_{y}+9Q_{y}^{2}),$$
$$a_{0}=D^{2}Q_{pp}-4DQ_{py}-Q_{p}DQ_{pp}+4Q_{p}Q_{py}-3Q_{pp}Q_{y}+6Q_{yy}.$$
Thus, modulo equivalence, the only forms (30)-(35) that
generate the same projective structure on all surfaces transversal to
$D$ are given by (40)-(41)
with the coefficient $\omega^{1}_{~{}22}$ satisfying differential equation
(42)-(43). Now, recalling that the space of solutions of
the second order ODE can be identified with the 2-dimensional space of
integral lines of $D$ in $J^{1}$, we obtain the following theorem.
Theorem 1
Every solution $\omega^{1}_{~{}22}$ to the fourth order
differential equation (42)-(43) defines a natural
projective structure on the space of solutions $J^{1}/D$ of the second
order ODE $y^{\prime\prime}=Q(x,y,y^{\prime})$. The structure is
given by the projection from $J^{1}$ to $J^{1}/D$ of forms
$$\omega^{1}={\rm d}y-p{\rm d}x,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}%
\omega^{2}={\rm d}p-Q{\rm d}x$$
with
$$\omega^{1}_{~{}1}=0,$$
$$\omega^{1}_{~{}2}=\omega^{1}_{~{}22}\omega^{2}-\omega^{3},~{}~{}~{}~{}~{}~{}~{%
}\omega^{3}={\rm d}x,$$
$$\displaystyle\omega^{2}_{~{}1}=$$
$$\displaystyle\frac{1}{6}~{}[D^{3}\omega^{1}_{~{}22}+3D^{2}\omega^{1}_{~{}22}+(%
5DQ_{p}+2Q_{p}^{2}-7Q_{y})D\omega^{1}_{~{}22}+$$
$$\displaystyle~{}~{}~{}~{}(2D^{2}Q_{p}-3DQ_{y}+4Q_{p}DQ_{p}-8Q_{p}Q_{y})\omega^%
{1}_{~{}22}+DQ_{pp}-4Q_{py}~{}]~{}~{}\omega^{1}$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{4}~{}[-D^{2}\omega^{1}_{~{}22}-3Q_{p}D\omega^{1}_{~{}22}%
+(3Q_{y}-2Q_{p}^{2}-2DQ_{p})\omega^{1}_{~{}22}-Q_{pp}~{}]~{}~{}\omega^{2}$$
$$\displaystyle-$$
$$\displaystyle Q_{y}~{}~{}\omega^{3},$$
$$\displaystyle\omega^{2}_{~{}2}=$$
$$\displaystyle\frac{1}{4}~{}[-D^{2}\omega^{1}_{~{}22}-3Q_{p}D\omega^{1}_{~{}22}%
+(3Q_{y}-2Q_{p}^{2}-2DQ_{p})\omega^{1}_{~{}22}-Q_{pp}~{}]~{}~{}\omega^{1}$$
$$\displaystyle+$$
$$\displaystyle[~{}D\omega^{1}_{~{}22}+2Q_{p}\omega^{1}_{~{}22}~{}]~{}~{}\omega^%
{2}~{}~{}-~{}~{}Q_{p}~{}~{}\omega^{3}.$$
Since the equation (42)-(43) is of 4th order it has four
independent solutions. Thus, all the corresponding
projective structures on $J^{1}/D$ should be treated on equal
footing. However, in the case of second order ODEs satisfying some
additional conditions, some of these structures may be more
distinguished. In particular, Sophus Lie [10] and Elie Cartan
[1] considered
2nd order ODEs satisfying the additional condition
$$a_{0}=D^{2}Q_{pp}-4DQ_{py}-Q_{p}DQ_{pp}+4Q_{p}Q_{py}-3Q_{pp}Q_{y}+6Q_{yy}%
\equiv 0.$$
(44)
For such ODEs equation (42)-(43) is homogeneous and as such
has a preferred solution $\omega^{1}_{~{}22}=0$. Thus, for this class of
second order ODEs there exists a distinguished, natural projective
structure on $J^{1}/D$ associated with the solution $\omega^{1}_{~{}22}=0$ of
(42)-(43). Explicitely, for any second order ode
satisfying $a_{0}\equiv 0$, this structure is given by
$$\omega^{1}={\rm d}y-p{\rm d}x,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\omega%
^{2}={\rm d}p-Q{\rm d}x$$
(45)
with
$$\omega^{1}_{~{}1}=0,~{}~{}~{}~{}~{}\omega^{1}_{~{}2}=-\omega^{3},~{}~{}~{}~{}~%
{}\omega^{3}={\rm d}x,$$
(46)
$$\omega^{2}_{~{}1}=\frac{1}{6}(DQ_{pp}-4Q_{py})\omega^{1}-\frac{1}{4}~{}Q_{pp}%
\omega^{2}-Q_{y}\omega^{3},$$
(47)
$$\omega^{2}_{~{}2}=-\frac{1}{4}Q_{pp}\omega^{1}-Q_{p}\omega^{3}.$$
(48)
In general, any projective structure described by Theorem 1
leads to a projective ${\bf SL}(3,{\bf R})$ connection on an
8-dimensional bundle
$P\to J^{1}/D$. One of the features of the projective
structures which via $\omega^{1}_{~{}22}=0$ are associated with $a_{0}\equiv 0$, is that each of them
leads to a normal projective ${\bf SL}(3,{\bf R})$ connection on
$P$. Using the local parameters
$(x,y,p,\alpha,\beta,\gamma,\nu,\mu)$ for $P$ and equations
(11)-(14), (45)-(48) we find that this ${\bf SL}(3,{\bf R})$ connection reads
$$\omega=\begin{pmatrix}\frac{1}{3}(\Omega_{2}-2\Omega_{1})&-\theta^{3}&\theta^{%
1}\\
&&\\
-\Omega_{3}&\frac{1}{3}(\Omega_{1}-2\Omega_{2})&\theta^{2}\\
&&\\
\Omega_{5}&-\Omega_{4}&\frac{1}{3}(\Omega_{1}+\Omega_{2})\end{pmatrix}$$
(49)
where
$(\theta^{1},\theta^{2},\theta^{3},\Omega_{1},\Omega_{2},\Omega_{3},\Omega_{4},%
\Omega_{5})$
are given by
$$\theta^{1}=\alpha\omega^{1},~{}~{}~{}~{}\theta^{2}=\beta(\omega^{2}+\gamma%
\omega^{1}),~{}~{}~{}~{}\theta^{3}=\frac{\alpha}{\beta}(\omega^{3}+\nu\omega^{%
1}),$$
$$\displaystyle\Omega_{1}={\rm d}\log\alpha-\mu~{}\theta^{1}+\frac{\nu}{\beta}~{%
}\theta^{2}-\frac{\beta\gamma}{\alpha}~{}\theta^{3}$$
$$\displaystyle\Omega_{2}={\rm d}\log\beta-\frac{1}{4\alpha}~{}[~{}6\gamma\nu+4%
\nu Q_{p}-Q_{pp}+2\alpha\mu~{}]~{}\theta^{1}+2\frac{\nu}{\beta}~{}\theta^{2}+%
\frac{\beta}{\alpha}~{}[~{}\gamma+Q_{p}~{}]~{}\theta^{3}$$
$$\displaystyle\Omega_{3}=\frac{\beta}{\alpha}{\rm d}\gamma-\frac{\beta}{6\alpha%
^{2}}~{}[~{}DQ_{pp}-6\gamma^{2}\nu-6\gamma\nu Q_{p}+3\gamma Q_{pp}-4Q_{py}+6%
\nu Q_{y}~{}]~{}\theta^{1}-\frac{1}{4\alpha}~{}[~{}2\gamma\nu-Q_{pp}+2\alpha%
\mu~{}]~{}\theta^{2}$$
$$\displaystyle-\frac{\beta^{2}}{\alpha^{2}}~{}[~{}\gamma^{2}+\gamma Q_{p}-Q_{y}%
~{}]~{}\theta^{3}$$
$$\displaystyle\Omega_{4}=\frac{1}{\beta}{\rm d}\nu-\frac{1}{6\alpha\beta}~{}[~{%
}6\gamma\nu^{2}+6\nu^{2}Q_{p}-3\nu Q_{pp}+Q_{ppp}~{}]~{}\theta^{1}+\frac{\nu^{%
2}}{\beta^{2}}~{}\theta^{2}-\frac{1}{4\alpha}~{}[~{}-2\gamma\nu-4\nu Q_{p}+Q_{%
pp}+2\alpha\mu~{}]~{}\theta^{3}$$
$$\displaystyle 2\Omega_{5}={\rm d}\mu+\mu{\rm d}\log\alpha-\frac{\nu}{\alpha}{%
\rm d}\gamma+\frac{\gamma}{\alpha}{\rm d}\nu$$
$$\displaystyle-\frac{1}{24\alpha^{2}}~{}[~{}12\alpha^{2}\mu^{2}+48\nu Q_{py}-48%
\nu^{2}Q_{y}-12\nu DQ_{pp}+36\gamma^{2}\nu^{2}+48\gamma\nu^{2}Q_{p}-36\gamma%
\nu Q_{pp}+12\gamma Q_{ppp}$$
$$\displaystyle+8DQ_{ppp}+8Q_{p}Q_{ppp}-12Q_{ppy}-3Q_{pp}^{2}~{}]~{}\theta^{1}+%
\frac{1}{6\alpha\beta}~{}[~{}6\gamma\nu^{2}-3\nu Q_{pp}+Q_{ppp}+6\alpha\nu\mu~%
{}]~{}\theta^{2}$$
$$\displaystyle-\frac{\beta}{6\alpha^{2}}~{}[~{}DQ_{pp}-6\gamma^{2}\nu-12\gamma%
\nu Q_{p}+3\gamma Q_{pp}-4Q_{py}+12\nu Q_{y}+6\alpha\gamma\mu~{}]~{}\theta^{3}.$$
The curvature of this connection reads
$$\Omega=\begin{pmatrix}0&0&0\\
&&\\
0&0&0\\
&&\\
\frac{1}{6\alpha^{2}\beta}~{}b_{01}~{}\theta^{1}\wedge\theta^{2}&-\frac{1}{6%
\alpha\beta^{2}}~{}b_{0}~{}\theta^{1}\wedge\theta^{2}&0\end{pmatrix},$$
where we have introduced
$$b_{0}=Q_{pppp},~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}b_{01}=Db_{0}+(\gamma+2Q_{p})b%
_{0}.$$
The relatively simple form of this curvature agrees with the general theory of normal projective connections
for $n=2$ (compare with the note at the end of section 2.3).
The next section is devoted to explaining the Lie/Cartan motivation for
considering the class of ODEs leading to the structure defined above.
3.2 Equivalence classes of 2nd order ODEs modulo point
transformations
A point transformation of variables
$$(~{}x,~{}y~{})=(~{}x(\bar{x},\bar{y}),~{}y(\bar{x},\bar{y})~{})$$
(50)
applied to the second order ODE
$$y^{\prime\prime}=Q(x,y,y^{\prime})$$
(51)
changes it to the new form
$$\bar{y}^{\prime\prime}=\bar{Q}(\bar{x},\bar{y},\bar{y}^{\prime}).$$
(52)
The function $Q=Q(x,y,y^{\prime})$ transforms in a rather complicated way into a new function
$\bar{Q}=\bar{Q}(\bar{x},\bar{y},\bar{y}^{\prime})$. But, using appropriate
derivatives of $Q$ one can construct functions which have
nice transformation properties under transformations (50). In
particular, the relative invariants
of the equation (51)
are such functions which, under transformations (50), scale by a factor.
Their vanishing is the point
invariant property of the equation. One of such relative invariants is
$$a_{0}=D^{2}Q_{pp}-4DQ_{py}-Q_{p}DQ_{pp}+4Q_{p}Q_{py}-3Q_{pp}Q_{y}+6Q_{yy},$$
the same function that appears in equations (43).
This fact was
already known to Sophus Lie [10]. Elie Cartan [1]
considered the
problem of finding all point invariants of (51). He
used his equivalence method which, enabled him to determine another
relative invariant
$$b_{0}=Q_{pppp}.$$
Both $a_{0}$ and $b_{0}$ are of the same order and, it follows from the
Cartan analysis, that the equation (51) has no more point
invariants of order less than or equal to 4. Thus, according to Cartan, the second order ODEs modulo point
transformations split into four major classes which are
i)
$a_{0}=b_{0}=0$
ii)
$a_{0}=0$ and $b_{0}\neq 0$
iii)
$a_{0}\neq 0$ and $b_{0}=0$
iv)
$a_{0}\neq 0$ and $b_{0}\neq 0$.
Cases i)-ii) were analyzed by Cartan completely. In particular, he showed that
if $a_{0}=0$ then with each point equivalence class of second order ODEs
is associated a natural normal projective connection, whose curvature provides
all the point invariants of the class. This connection equips the
space of solutions of each of the equations from the
equivalence class with a projective structure. It follows that the projective structures
originating in this way from different equations from the same point
equivalence class are equivalent. This distinguished projective
structure associated with the class of equation $y^{\prime\prime}=Q(x,y,y^{\prime})$
coincides with the structure (47) defined in the previous
section.
4 Acknowledgments
We acknowledge support from NSF Grant No PHY-0088951 and the Polish
KBN Grant No 2 P03B 12724.
References
[1]
Cartan, E. (1924) “Sur les Varietes a Connection
Projective” Bull. Soc. Math. France 52 205-241; Oeuvres III, 1, N70, 825-862, Paris, (1955)
[2]
Cartan, E. (1941) “La Geometria de las Ecuaciones
Diferenciales der Tercen orden” Rev. Hispano-Amer. 4 1-31
[3]
Chern, S-S. (1978) “The Geometry of the
Differential Equation $y^{\prime\prime\prime}=F(x,y,y^{\prime},y^{\prime\prime})$” in Selected Papers, Springer, Berlin (original 1942)
[4]
Frittelli, S., Kamran, N., Newman, E T. (2002) “Differential
Equations and Conformal Geometry” J. Geom. Phys. 43, 133-145
[5]
Frittelli S., Kozameh C., Newman E T. (2001) “Differential
Geometry from Differential Equations”, Commun. Math. Phys. 223, 383-408
[6]
Frittelli, S., Kozameh, C., Newman, E T., Nurowski
P., (2002) “Cartan Normal Conformal Connections from Differential Equations”,
Class. Quantum Grav. 19, 5235-5247
[7]
Godlinski, M., Nurowski, P., “Cartan Connections
Associated with the Third Order ODEs” in preparation
[8]
Hurtubise, J.C., Kamran, N., (1992) “Differential
Invariants, Double Fibrations and Painleve Equations”, in Painleve
Transcendents, eds., Levi, D., Winternitz, P. Plenum Press, NY,
271-298
[9]
Kobayashi, S. (1970) Transformation Groups in
Differential Geometry, Springer, Berlin
[10]
Lie, S. (1924) “Klassifikation und Integration von
gewohnlichen Differentialgleichungen zwischen $x$, $y$, die eine Gruppe von
Transformationen gestatten III”, in Gesammelte Abhandlungen, Vol. 5, Teubner, Leipzig
[11]
Tresse, M.A., (1894) “Sur les Invariants
Differentiels des Groupes Continus de Transformations”, Acta Math
18 1-88
[12]
Tresse, M.A., (1896) “Determination des Invariantes
Ponctuels de l’Equation Differentielle du Second Ordre $y^{\prime\prime}=\omega(x,y,y^{\prime})$”, Hirzel, Leipzig
[13]
Wünschmann, K., (1905) “Über Beruhrungsbedingungen bei
Integralcurven von Differentialgleichungen, Inaug. Dissert.,
Teubner, Leipzig |
A large deviation principle for
empirical measures
on Polish spaces: Application
to singular Gibbs measures on manifolds
David García Zelada
Abstract
We prove a large deviation principle
for a sequence of point processes defined
by Gibbs probability measures
on a Polish space. This is obtained as
a consequence of a more general
Laplace principle for
the nonnormalized Gibbs measures.
We consider three main applications:
Conditional Gibbs measures on compact spaces, Coulomb
gases on compact Riemannian manifolds and
the usual Gibbs measures in the Euclidean space.
Finally, we study
the generalization of Fekete points
and prove the convergence
of the empirical measures
to the minimizer of the energy if it is unique.
The approach is partly inspired by
the works of Dupuis and co-authors.
It is remarkably natural and general
compared to the usual strategies
for singular Gibbs measures.
Keywords: Gibbs measure;
Coulomb gas; empirical measure; large deviation principle;
interacting particle system;
singular potential;
two-dimensional Einstein manifold;
relative entropy;
Fekete points.
2010 Mathematics Subject Classification:
60F10; 60K35; 82C22.
1 Introduction
The present article
is inspired by part of the work of
Dupuis, Laschos and Ramanan on large deviations
for a sequence of point processes
given by Gibbs measures associated to
very general singular two-body interactions
[12]. In fact, we follow
the philosophy of
Dupuis and Ellis
[11] about
the use of variational formulas
to make plausible and
sometimes easier to find a Laplace principle.
We are interested in
proving the Laplace principle
and the large deviation principle
for a very general sequence of energies
in a not necessarily compact space.
Part of our work has an overlap with
the article of Berman
[6] and it was developed independently.
As in [6] the peculiarity of this result
is the generality of the sequence of energies:
they do not need to be made of a two-body
interaction potential but they may still be
very singular.
The proof given here is very simple compared
to the ad hoc methods
used in the usual proofs of the large deviation
principles for a sequence
of Gibbs measures such as in [3],
[14], [15]
[9] and [13].
It can be seen as a generalization
of the Laplace-Varadhan lemma
for the Sanov’s theorem to the case
of more singular functions that may
depend on $n$ (Section 6).
Among the applications we can give
we are particularly interested
in explaining a simple case inspired by
[5]. This is the
case of a Coulomb gas on a two-dimensional
Riemannian manifold.
As a second application
we study
a large deviation principle for a
conditional Gibbs measure, i.e. we fix the position of some
of the particles and leave the rest of them random.
The last application we discuss is
a different proof of already known results
such as the one-dimensional log-gas of
[1, Section 2.6],
the special two-dimensional log-gas [3]
related to the Ginibre ensemble of random matrices
and its generalization
to an $n$-dimensional Coulomb gas
in [9] and [12],
and the note in
[13] about two-dimensional log-gases
with a weakly confining potential.
We now explain the contents of each section.
The rest of Section 1 will be dedicated to
the main definitions and assumptions we will
need to state the theorems. The two main theorems
will be stated and we will discuss some properties about the
assumptions. Those properties are used in the
examples and applications mainly for simplification.
Section 2 includes the most famous example,
the $k$-body interaction. We give sufficient
conditions to satisfy the hypothesis and these
will be important when we treat the Euclidean space case.
In Section 3 we
start giving an idea of the proofs.
This includes mainly
the variational formula we talked about above
and some properties
of the entropy.
Then
we give the proofs
of the two main theorems and of their corollaries.
We discuss some particular
examples in Section 4.
More precisely, the conditional Gibbs measure,
the Coulomb gas on a Riemannian manifold
and
a new
way to obtain already known results in the Euclidean space.
Section 5
is about the nonrandom case, also known sometimes
as Fekete points. We give an easy proof
for the convergence of the empirical measure
of minimizers of the energy
by using the same hypothesis as in the random case.
The relation to the generalized Laplace-Varadhan lemma is given in Section 6.
Finally, we have four appendices where
we prove some known results for
the reader’s convenience.
1.1 Model
Let $M$ be a Polish space, i.e. a
separable topological space
such that there exists
a complete metric that induces its topology.
Choose a probability measure $\pi$ on $M$. We say
that $(M,\pi)$ is a Polish probability space.
Denote by $\mathcal{P}(M)$
the space of probability measures in $M$ and
endow it with the topology of weak convergence
of probability measures, also known as vague
convergence
(see Appendix A
for definitions).
Suppose we have a sequence
$\{W_{n}\}_{n\in\mathbb{N}}$
of symmetric measurable functions
$$W_{n}:M^{n}\to(-\infty,\infty]$$
and a
measurable function
$$W:\mathcal{P}(M)\to(-\infty,\infty].$$
Along the article our sequences will
be indexed by $\mathbb{N}$. Nevertheless they
will make sense for $n$ large enough.
In fact, we may consider
$W_{n}$ not defined for every $n$
but just for an
increasing sequence of positive integers
and
the results will always be
true with the obvious changes.
Stable sequence.
We shall say that the sequence
$\{W_{n}\}_{n\in\mathbb{N}}$ is a
stable sequence if it is
uniformly bounded from below, i.e. if
there exists $C\in\mathbb{R}$ such that
$$W_{n}\geq C\mbox{ for all }n\in\mathbb{N}.$$
Take a sequence of nonnegative numbers
$\{\beta_{n}\}_{n\in\mathbb{N}}$ that converges
to some $\beta\in(0,\infty]$.
We define the Gibbs measures (not normalized)
$$d{\gamma}_{n}=e^{-n\beta_{n}W_{n}}d\pi^{\otimes_{n}}$$
(1.1)
and the free energy with parameter $\beta$
as
$$F=W+\frac{1}{\beta}D(\cdot\|\pi),$$
(1.2)
(we suppose $0\times\infty=0$)
where $D(\mu\|\nu)$ denotes the
relative entropy of $\mu$ with
respect to $\nu$, also known as the
Kullback–Leibler divergence i.e.
$$D(\mu\|\nu)=\int_{M}\frac{d\mu}{d\nu}\log\left(\frac{d\mu}{d\nu}\right)d\nu$$
(1.3)
if $\mu$ is absolutely continuous with respect to $\nu$
and $D(\mu\|\nu)=\infty$ otherwise.
We shall state the main assumptions
we will make in the theorems.
Lower limit assumption (H1).
Define the ‘extension’ of $W_{n}$ to $\mathcal{P}(M)$
as
$$\begin{split}\displaystyle\tilde{W}_{n}:\mathcal{P}(M)\to(-\infty,\infty]\\
\displaystyle\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}\mapsto W_{n}(x_{1},...,x_%
{n})\end{split}$$
(1.4)
and $\infty$ otherwise.
We say that the pair
$(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower lower limit assumption (H1) if
for every sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures on $M$ that
converges to some
probability measure $\mu$ we have
$$\liminf_{n\to\infty}\tilde{W}_{n}(\mu_{n})\geq W(\mu).$$
Upper limit assumption (H2).
We say that the pair
$(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the upper limit assumption (H2) if
for each $\mu\in\mathcal{P}(M)$ we have that
$$\limsup_{n\to\infty}\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]\leq W(\mu).$$
Regularity assumption (H3).
Define the set of ‘nice’ probability measures
$$\mathcal{N}=\left\{\mu\in\mathcal{P}(M):\ D(\mu\|\pi)<\infty\mbox{ and }%
\limsup_{n\to\infty}\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]\leq W(\mu)\right\}.$$
(1.5)
We shall say that the pair
$(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the regularity assumption (H3) if
for every $\mu\in\mathcal{P}(M)$ such
that $W(\mu)<\infty$ we can find a sequence
of probability measures
$\{\mu_{n}\}_{n\in\mathbb{N}}$ in $\mathcal{N}$
such that
$\mu_{n}\to\mu$
and $\limsup_{n\to\infty}W(\mu_{n})\leq W(\mu)$.
Notice that if the upper limit assumption (H2)
is satisfied then the regularity assumption (H3)
depends
only on $W$ and not on the sequence
$\{W_{n}\}_{n\in\mathbb{N}}$.
Confining assumption (H4).
We shall
say that $\{W_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4)
if the following is true. Let
$\{n_{j}\}_{j\in\mathbb{N}}$ be any
increasing sequence of natural numbers
and let $\{\mu_{j}\}_{j\in\mathbb{N}}$ be any sequence
of probability measures on $M$. If there exists
a real constant $A$ such that
$$\tilde{W}_{n_{j}}(\mu_{j})\leq A$$
for every $j\in\mathbb{N}$, where $\tilde{W}_{n}$
is defined in 1.4, then
$\{\mu_{j}\}_{j\in\mathbb{N}}$ is relatively
compact in $\mathcal{P}(M)$.
The previous condition is
always true in
a compact space.
Now we can state the Laplace
principles and the large deviation principles.
1.2 Main results
Denote by
$$\begin{split}\displaystyle i_{n}:M^{n}\to\mathcal{P}(M)\\
\displaystyle(x_{1},...,x_{n})\mapsto\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}%
\end{split}$$
(1.6)
the usual continuous ‘inclusion’ of $M^{n}$ in $\mathcal{P}(M)$.
The following is the first main theorem. This
treats the case where $\beta$ is finite.
Theorem 1.1 (Laplace principle for finite $\beta$).
Let $\{W_{n}\}_{n\in\mathbb{N}}$ be a stable
sequence and
$W:\mathcal{P}(M)\to(-\infty,\infty]$
a measurable function
such that $(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower and upper limit assumptions,
(H1) and (H2).
Take a sequence
of positive numbers $\{\beta_{n}\}_{n\in\mathbb{N}}$
that converges to some $\beta\in(0,\infty)$.
Define the Gibbs measures $\gamma_{n}$ by 1.1
and the free energy $F$ by 1.2.
Then, the following Laplace’s principle is satisfied.
For every bounded continuous
function $f:\mathcal{P}(M)\to\mathbb{R}$
$$\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-n\beta_{n}f\circ i_{n}}d\gamma_{n}\,%
\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right%
)+F\left(\mu\right)\}.$$
This Laplace principle implies the following
large deviation principle.
Corollary 1.2 (Large deviation principle for finite $\beta$).
Define $Z_{n}=\gamma_{n}(M^{n})$.
Suppose $Z_{n}>0$ for every $n$.
As the sequence $\{W_{n}\}_{n\in\mathbb{N}}$ is
a stable sequence we also have
$Z_{n}<\infty$ for every $n$.
Take the sequence of probability measures
$\{\mathbb{P}_{n}\}_{n\in\mathbb{N}}$ defined by
$$d\mathbb{P}_{n}=\frac{1}{Z_{n}}d\gamma_{n}.$$
(1.7)
Suppose the same conditions as in Theorem
1.1.
Additionally, suppose $W$ is lower
semicontinuous, i.e.
$\{\mu\in\mathcal{P}(M):\ W(\mu)\leq a\}$
is closed for every $a<\infty$.
For each $n\in\mathbb{N}$, let $i_{n}(\mathbb{P}_{n})$
be the pushforward measure of $\mathbb{P}_{n}$
by $i_{n}$.
Then the sequence
$\{i_{n}(\mathbb{P}_{n})\}_{n\in\mathbb{N}}$
satisfies a large deviation
principle with speed $n\beta_{n}$ and with rate function
$$I=F-\inf F,$$
i.e.
for every open set $A\subset\mathcal{P}(M)$
we have
$$\liminf_{n\to\infty}\frac{1}{n\beta_{n}}\log\mathbb{P}_{n}(i_{n}^{-1}(A))\geq-%
\inf_{\mu\in A}I(\mu)$$
and for every closed set $C\subset\mathcal{P}(M)$
we have
$$\limsup_{n\to\infty}\frac{1}{n\beta_{n}}\log\mathbb{P}_{n}(i_{n}^{-1}(C))\leq-%
\inf_{\mu\in C}I(\mu).$$
The second main theorem is the following. It is
the case of infinite $\beta$.
Theorem 1.3 (Laplace principle for infinite $\beta$).
Let $\{W_{n}\}_{n\in\mathbb{N}}$ be a stable
sequence and
$W:\mathcal{P}(M)\to(-\infty,\infty]$
a measurable function.
Suppose $(\{W_{n}\}_{n\in\mathbb{N}},W)$
satisfies the lower limit assumption (H1)
and the regularity assumption (H3) and
suppose $\{W_{n}\}_{n\in\mathbb{N}}$ satisfies
the confining assumption (H4).
Take a sequence
of positive numbers $\{\beta_{n}\}_{n\in\mathbb{N}}$
that converges to infinity, i.e. $\beta=\infty$.
Define the Gibbs measures $\gamma_{n}$ by 1.1
and the free energy $F$ by 1.2.
Then, the following Laplace’s principle is satisfied.
For every continuous
function $f:\mathcal{P}(M)\to\mathbb{R}$
$$\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-n\beta_{n}f\circ i_{n}}d\gamma_{n}\,%
\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right%
)+F\left(\mu\right)\}.$$
Corollary 1.4 (Large deviation principle for infinite $\beta$).
Define $Z_{n}=\gamma_{n}(M^{n})$.
Suppose $Z_{n}>0$ for every $n$.
As the sequence $\{W_{n}\}_{n\in\mathbb{N}}$ is
a stable sequence we also have
$Z_{n}<\infty$ for every $n$.
Take the sequence of probability measures
$\{\mathbb{P}_{n}\}_{n\in\mathbb{N}}$ defined by
$$d\mathbb{P}_{n}=\frac{1}{Z_{n}}d\gamma_{n}.$$
(1.8)
Suppose the same conditions as in Theorem
1.3.
Additionally, suppose $W$ has compact level sets, i.e.
$\{\mu\in\mathcal{P}(M):\ W(\mu)\leq a\}$
is compact for every $a<\infty$.
For each $n\in\mathbb{N}$, let $i_{n}(\mathbb{P}_{n})$
be the pushforward measure of $\mathbb{P}_{n}$
by $i_{n}$.
Then, the sequence $\{i_{n}(\mathbb{P}_{n})\}_{n\in\mathbb{N}}$
satisfies a large deviation
principle with speed $n\beta_{n}$ and with rate function
$$I=F-\inf F,$$
i.e.
for every open set $A\subset\mathcal{P}(M)$
we have
$$\liminf_{n\to\infty}\frac{1}{n\beta_{n}}\log\mathbb{P}_{n}(i_{n}^{-1}(A))\geq-%
\inf_{\mu\in A}I(\mu)$$
and for every closed set $C\subset\mathcal{P}(M)$
we have
$$\limsup_{n\to\infty}\frac{1}{n\beta_{n}}\log\mathbb{P}_{n}(i_{n}^{-1}(C))\leq-%
\inf_{\mu\in C}I(\mu).$$
In Section 2 we shall study the
usual case of $k$-body interaction. We will see
when the associated sequence satisfy
the assumptions. Section 4
will be about
some more specific examples,
such as
the conditional Gibbs measure,
the Coulomb gas on a compact Riemannian manifold
and the usual Gibbs measures on a
noncompact space such as the Euclidean space.
But first we need to make some remarks
about the assumptions.
1.3 Comments about the assumptions
In this section we are going to mention some
properties of the assumptions.
These will help us simplify
many proofs in the applications.
The first one is just a reformulation
of the lower limit assumption (H1)
using $W$ instead of $\tilde{W}$.
Remark 1.5 (Equivalent lower limit assumption).
The lower limit assumption (H1)
is equivalent to the following.
For every increasing sequence of
natural numbers $\{n_{j}\}_{j\in\mathbb{N}}$
and a sequence $\{a_{j}\}_{j\in\mathbb{N}}$
where
$a_{j}\in M^{n_{j}}$ such that
$$i_{n_{j}}(a_{j})\xrightarrow[\>j\to\infty\>]{}\mu$$
for some $\mu\in\mathcal{P}(M)$
we have
$$\liminf_{j\to\infty}W_{n_{j}}(a_{j})\geq W(\mu).$$
The following remarks
tell us that any positive
linear combination of stable sequences
that satisfy the
lower or upper limit assumption,
(H1) or (H2), satisfies
the lower or upper limit assumption,
(H1) or (H2).
Remark 1.6 (Closure by scalar multiplication).
If $\{a_{n}\}_{n\in\mathbb{N}}$
is a sequence of nonnegative numbers
that converge to some $a\in(0,\infty)$ and
$(\{W_{n}\}_{n\in\mathbb{N}},W)$
satisfies the lower limit assumption (H1) then
$(\{a_{n}W_{n}\}_{n\in\mathbb{N}},aW)$
also satisfies the
lower limit assumption (H1). The
same is true if we change
lower limit assumption (H1)
by
upper limit assumption (H2).
Remark 1.7 (Closure by sum).
If $(\{W^{1}_{n}\}_{n\in\mathbb{N}},W^{1})$ and
$(\{W^{2}_{n}\}_{n\in\mathbb{N}},W^{2})$
satisfy the lower limit assumption (H1)
then
$(\{W^{1}_{n}+W^{2}_{n}\}_{n\in\mathbb{N}},W^{1}+W^{2})$
satisfy the lower limit assumption (H1).
The
same is true if we change
lower limit assumption (H1)
by
upper limit assumption (H2).
The following two remarks are related
to the confining assumption (H4).
The first one may be used to simplify proofs
and the second remark is important
when we treat the usual cases in $\mathbb{R}^{n}$.
Remark 1.8 (Confining assumption
is preserved under affine transformations).
If $\{W_{n}\}_{n\in\mathbb{N}}$ satisfies
the confining assumption (H4),
$\{b_{n}\}_{n\in\mathbb{N}}$
is a sequence of real numbers
such that $b_{n}\to b\in\mathbb{R}$
and $\{a_{n}\}_{n\in\mathbb{N}}$ is a sequence of
nonnegative numbers such that $a_{n}\to a>0$ then
the sequence
$\{a_{n}W_{n}+b_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4).
Remark 1.9 (Confining assumption of a perturbed sequence).
Let $\{W^{1}_{n}\}_{n\in\mathbb{N}}$ and
$\{W^{2}_{n}\}_{n\in\mathbb{N}}$
be two stable sequences.
Let $\{a_{n}\}_{n\in\mathbb{N}}$ be a sequence
of nonnegative numbers such that
$a_{n}\to 1$.
If $\{W^{1}_{n}+W^{2}_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4) then
$\{W^{1}_{n}+a_{n}W^{2}_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4).
Proof.
By Remark 1.8
and because
the sequences are uniformly bounded from below
we can assume $W^{1}_{n}$ and $W^{2}_{n}$ positive.
Let $\{n_{j}\}_{j\in\mathbb{N}}$
be an increasing sequence of natural numbers and
$\{\mu_{j}\}_{j\in\mathbb{N}}$ a sequence of probability
measures
such that
$$\tilde{W}^{1}_{n_{j}}(\mu_{j})+a_{n_{j}}\tilde{W}^{2}_{n_{j}}(\mu_{j})\leq A$$
for every $j\in\mathbb{N}$ and some real number $A$,
where $\tilde{W}_{n}$ is defined in 1.4.
Then
$$\tilde{W}^{1}_{n_{j}}(\mu_{j})\leq A$$
and
$$a_{n_{j}}\tilde{W}^{2}_{n_{j}}(\mu_{j})\leq A.$$
But, if $j$ is large enough,
we have $a_{n_{j}}\geq 1/2$.
So,
$$\tilde{W}^{2}_{n_{j}}(\mu_{j})\leq 2A$$
and we get
$$\tilde{W}^{1}_{n_{j}}(\mu_{j})+\tilde{W}^{2}_{n_{j}}(\mu_{j})\leq 3A$$
for $j$ large enough. Then
$\{\mu_{j}\}_{j\in\mathbb{N}}$ is tight.
∎
2 Example of a stable sequence: k-body interaction
We will give
the most basic nontrivial example of a
stable sequence. We start with
a simple case, the more general case
being a straightforward generalization.
Lemma 2.1 (Two-body interaction).
Take a symmetric lower semicontinuous function
bounded from below
$G:M\times M\to(-\infty,\infty].$
Define
$$W_{n}:M^{n}\to(-\infty,\infty]$$
$$(x_{1},...,x_{n})\mapsto\frac{1}{n^{2}}\sum_{i<j}^{n}G(x_{i},x_{j})$$
and
$$W:\mathcal{P}(M)\to(-\infty,\infty]$$
$$\mu\mapsto\frac{1}{2}\int_{M\times M}G(x,y)\,d\mu(x)d\mu(y).$$
Then $\{W_{n}\}_{n\in\mathbb{N}}$ is a stable sequence,
i.e. measurable and uniformly bounded from below,
$W$ is lower semicontinuous and
the pair $(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower and upper limit assumption,
(H1) and (H2).
Proof.
To see that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence
we notice that
if $C\leq G$ then $\frac{n(n-1)}{2n^{2}}C\leq W_{n}$.
The lower semicontinuity of $W$ is
a consequence of the lower semicontinuity of
$G$ and the fact
that it is bounded from below.
Now,
let us prove that $(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower and upper limit assumption,
(H1) and (H2).
$\bullet$ Lower limit assumption (H1).
Let $\mu\in\mathcal{P}(M)$. Take
$N>0$ and define $G_{N}=G\wedge N$. Then,
we can prove that
$$\tilde{W}_{n}(\mu)+\frac{1}{2}\frac{N}{n}\geq\frac{1}{2}\int_{M\times M}G_{N}(%
x,y)\,d\mu(x)d\mu(y)$$
(2.1)
where $\tilde{W}_{n}$ is the extension defined
in 1.4.
If $\tilde{W}_{n}(\mu)=\infty$ there is nothing to prove.
If $\tilde{W}_{n}(\mu)<\infty$ then
$\mu=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$
for some $(x_{1},...,x_{n})\in M^{n}$. Then we have
$$\frac{1}{n^{2}}\sum_{i<j}^{n}G_{N}(x_{i},x_{j})+\frac{1}{2}\frac{N}{n}\geq%
\frac{1}{2}\int_{M\times M}G_{N}(x,y)\,d\mu(x)d\mu(y),$$
but
$$\frac{1}{n^{2}}\sum_{i<j}^{n}G(x_{i},x_{j})\geq\frac{1}{n^{2}}\sum_{i<j}^{n}G_%
{N}(x_{i},x_{j}),$$
and we get the inequality 2.1.
Let $\mu_{n}\to\mu\in\mathcal{P}(M)$. Then, taking
the lower limit
as $n$ goes to infinity in the inequality
$$\tilde{W}_{n}(\mu_{n})+\frac{1}{2}\frac{N}{n}\geq\frac{1}{2}\int_{M\times M}G_%
{N}(x,y)\,d\mu_{n}(x)d\mu_{n}(y),$$
we get
$$\displaystyle\liminf_{n\to\infty}\tilde{W}_{n}(\mu_{n})$$
$$\displaystyle\geq\liminf_{n\to\infty}\frac{1}{2}\int_{M\times M}G_{N}(x,y)d\mu%
_{n}(x)d\mu_{n}(y)$$
$$\displaystyle\geq\frac{1}{2}\int_{M\times M}G_{N}(x,y)d\mu(x)d\mu(y),$$
where the last inequality is due to the lower semicontinuity
of $G_{N}$ and the fact that it is
bounded from below.
Finally, as $G$ is bounded from below we can take
$N$ to infinity and use
the monotone convergence theorem to get
$$\liminf_{n\to\infty}\tilde{W}_{n}(\mu_{n})\geq\frac{1}{2}\int_{M\times M}G(x,y%
)d\mu(x)d\mu(y).$$
$\bullet$ Upper limit assumption (H2).
Take $\mu\in\mathcal{P}(M)$
and notice that
$$\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]=\frac{n(n-1)}{2n^{2}}\int_{M\times M}G(x%
,y)d\mu(x)d\mu(y).$$
Then,
$$\lim_{n\to\infty}E_{\mu^{\otimes_{n}}}[W_{n}]=\frac{1}{2}\int_{M\times M}G(x,y%
)d\mu(x)d\mu(y).$$
∎
Lemma 2.2 ($k$-body interaction).
A generalization of the previous example is to take
an integer $k>0$ and
a symmetric lower semicontinuous function
bounded from below
$G:M^{k}\to(-\infty,\infty].$
Define
$$W_{n}:M^{n}\to(-\infty,\infty]$$
$$(x_{1},...,x_{n})\mapsto\frac{1}{n^{k}}\sum_{\stackrel{{\scriptstyle\{i_{1},..%
.,i_{k}\}\subset\{1,...,n\}}}{{\#\{i_{1},...,i_{k}\}=k}}}G(x_{i_{1}},...,x_{i_%
{k}}).$$
and
$$W:\mathcal{P}(M)\to(-\infty,\infty]$$
$$\mu\mapsto\frac{1}{k!}\int_{M^{k}}G(x_{1},...,x_{k})d\mu(x_{1})...d\mu(x_{k}).$$
Then $\{W_{n}\}_{n\in\mathbb{N}}$ is a stable sequence,
i.e. measurable and uniformly bounded from below,
$W$ is lower semicontinuous and
the pair $(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower and upper limit assumption,
(H1) and (H2).
Proof.
(We shall almost copy the same proof as the previous
example.)
To see that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence
we notice that if $C\leq G$ then
${n\choose k}\frac{1}{n^{k}}C\leq W_{n}$.
The lower semicontinuity of $W$ is
a consequence of the lower semicontinuity of
$G$ and the fact
that it is bounded from below. Now,
let us prove that $(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower and upper limit assumption,
(H1) and (H2).
$\bullet$ Lower limit assumption (H1).
Let $\mu\in\mathcal{P}(M)$. As the previous example, take
$N>0$ and define $G_{N}=G\wedge N$. Then,
we can prove that
$$\tilde{W}_{n}(\mu)+\frac{N}{k!\,n^{k}}\left(n^{k}-\frac{n!}{(n-k)!}\right)\geq%
\frac{1}{k!}\int_{M^{k}}G_{N}(x_{1},...,x_{k})d\mu(x_{1})...d\mu(x_{k})$$
(2.2)
where $\tilde{W}_{n}$ is the extension defined
in 1.4.
If $\tilde{W}_{n}(\mu)=\infty$ there is nothing to prove.
If $\tilde{W}_{n}(\mu)<\infty$ then
$\mu=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$
for some $(x_{1},...,x_{n})\in M^{n}$. Then we have
$$\frac{1}{n^{k}}\sum_{\stackrel{{\scriptstyle\{i_{1},...,i_{k}\}\subset\{1,...,%
n\}}}{{\#\{i_{1},...,i_{k}\}=k}}}G_{N}(x_{i_{1}},...,x_{i_{k}})+\frac{N}{k!\,n%
^{k}}\left(n^{k}-\frac{n!}{(n-k)!}\right)\geq\frac{1}{k!}\int_{M^{k}}G_{N}(x_{%
1},...,x_{k})d\mu(x_{1})...d\mu(x_{k}),$$
but
$$\frac{1}{n^{k}}\sum_{\stackrel{{\scriptstyle\{i_{1},...,i_{k}\}\subset\{1,...,%
n\}}}{{\#\{i_{1},...,i_{k}\}=k}}}G(x_{i_{1}},...,x_{i_{k}})\geq\frac{1}{n^{k}}%
\sum_{\stackrel{{\scriptstyle\{i_{1},...,i_{k}\}\subset\{1,...,n\}}}{{\#\{i_{1%
},...,i_{k}\}=k}}}G_{N}(x_{i_{1}},...,x_{i_{k}})$$
and we get the inequality 2.2.
Let $\mu_{n}\to\mu\in\mathcal{P}(M)$.
Then, taking
the lower limit
as $n$ goes to infinity in the inequality
$$\tilde{W}_{n}(\mu_{n})+\frac{N}{k!\,n^{k}}\left(n^{k}-\frac{n!}{(n-k)!}\right)%
\geq\frac{1}{k!}\int_{M^{k}}G_{N}(x_{1},...,x_{k})d\mu_{n}(x_{1})...d\mu_{n}(x%
_{k}),$$
we get
$$\displaystyle\liminf_{n\to\infty}\tilde{W}_{n}(\mu_{n})$$
$$\displaystyle\geq\liminf_{n\to\infty}\frac{1}{k!}\int_{M^{k}}G_{N}(x_{1},...,x%
_{k})d\mu_{n}(x_{1})...d\mu_{n}(x_{k})$$
$$\displaystyle\geq\frac{1}{k!}\int_{M^{k}}G_{N}(x_{1},...,x_{k})d\mu(x_{1})...d%
\mu(x_{k}),$$
where the last inequality is due to the lower semicontinuity
of $G_{N}$ and the fact that it is
bounded from below.
Finally, as $G$ is bounded from below we can take
$N$ to infinity and use
the monotone convergence theorem to get
$$\liminf_{n\to\infty}\tilde{W}_{n}(\mu_{n})\geq\frac{1}{k!}\int_{M^{k}}G(x_{1},%
...,x_{k})d\mu(x_{1})...d\mu(x_{k}).$$
$\bullet$ Upper limit assumption (H2).
Take $\mu\in\mathcal{P}(M)$
and notice that
$$\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]=\frac{1}{n^{k}}{n\choose k}\int_{M^{k}}G%
(x_{1},...,x_{k})d\mu(x_{1})...d\mu(x_{k}).$$
Then,
$$\lim_{n\to\infty}E_{\mu^{\otimes_{n}}}[W_{n}]=\frac{1}{k!}\int_{M^{k}}G(x_{1},%
...,x_{k})d\mu(x_{1})...d\mu(x_{k}).$$
∎
Now we give a sufficient condition
for a $k$-body interaction
to satisfy the confining assumption (H4).
Lemma 2.3 ($k$-body interaction and confining
assumption).
Take $k>0$ and
a symmetric function
bounded from below
$G:M^{k}\to(-\infty,\infty]$.
Define
$$W_{n}:M^{n}\to(-\infty,\infty]$$
$$(x_{1},...,x_{n})\mapsto\frac{1}{n^{k}}\sum_{\stackrel{{\scriptstyle\{i_{1},..%
.,i_{k}\}\subset\{1,...,n\}}}{{\#\{i_{1},...,i_{k}\}=k}}}G(x_{i_{1}},...,x_{i_%
{k}}).$$
Suppose $G(x_{1},...,x_{k})$ tends to infinity
when $x_{i}\to\infty$ for all $i\in\{1,...,k\}$,
i.e. suppose that for every $C\in\mathbb{R}$ there exists
a compact set $K$ such that
$G|_{K^{c}\times...\times K^{c}}\geq C$. Then
the sequence $\{W_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4).
Proof.
By Remark 1.8 we can suppose $G$ positive. Remember the definition of $\tilde{W}_{n}$
in 1.4. All we need
is the following result.
Lemma 2.4.
Suppose that $G$ is positive.
Take $n\in\mathbb{N}$,
$A\in\mathbb{R}$ and $\mu\in\mathcal{P}(M)$
that satisfies
$$\tilde{W}_{n}(\mu)\leq A.$$
If $K$ is a compact set such that
$G|_{K^{c}\times...\times K^{c}}\geq C$
with $C>0$, then
$$\mu(K^{c})\leq\left(\frac{A}{C}\,k!\right)^{1/k}+\frac{k}{n}.$$
Proof.
We first
notice that
$\mu=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$
for some $(x_{1},...,x_{n})\in M^{n}$.
By the hypotheses we can see that
$$\frac{1}{n^{k}}[\mbox{number of $k$-combinations outside K}]\,C\leq\frac{1}{n^%
{k}}\sum_{\stackrel{{\scriptstyle\{i_{1},...,i_{k}\}\subset\{1,...,n\}}}{{\#\{%
i_{1},...,i_{k}\}=k}}}G(x_{i_{1}},...,x_{i_{k}})\leq A.$$
(2.3)
where, more precisely,
$[\mbox{number of $k$-combinations outside K}]$
denotes
the cardinal of the following set,
$\{S\subset\{1,...,n\}:\ \#S=k\mbox{ and }\forall i\in S,\,x_{i}\notin K\}$.
But, if $m$ denotes the
number of points among $x_{1},...,x_{n}$ outside $K$
and if $k\leq m$, we have
$$\frac{(m-k)^{k}}{k!}\leq\frac{m!}{(m-k)!\,k!}=[\mbox{number of $k$-%
combinations outside K}]$$
which, along with the inequality
2.3,
implies
$$\frac{1}{n^{k}}\frac{(m-k)^{k}}{k!}C\leq A,$$
or, equivalently
$$\frac{m}{n}\leq\left(\frac{A}{C}\,k!\right)^{1/k}+\frac{k}{n}.$$
As $\mu=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$ then
$$\mu(K^{c})\leq\left(\frac{A}{C}\,k!\right)^{1/k}+\frac{k}{n}.$$
∎
Take an
increasing sequence of natural numbers
$\{n_{j}\}_{j\in\mathbb{N}}$
and a sequence
of probability measures $\{\mu_{j}\}_{j\in\mathbb{N}}$
on M.
Suppose there exists a real constant $A$
such that
$$\tilde{W}_{n_{j}}(\mu_{j})\leq A$$
for every $j\in\mathbb{N}$.
We will prove that
$\{\mu_{j}\}_{j\in\mathbb{N}}$
is tight and, by Prokhorov’s theorem
7.3,
relatively compact. Take $\epsilon>0$. We
are looking for a compact set
$K$ such that
$$\forall j\in\mathbb{N},\ \mu_{j}(K^{c})\leq\epsilon.$$
Choose $C>0$ that satisfies
$$\left(\frac{A}{C}\,k!\right)^{1/k}\leq\frac{\epsilon}{2}$$
and $N\in\mathbb{N}$ that satisfies
$$\frac{k}{N}\leq\frac{\epsilon}{2}.$$
Take a compact set $K_{0}$ such that
$G|_{K_{0}^{c}\times...\times K_{0}^{c}}\geq C$.
Then, if $j\geq N$ we have $n_{j}\geq N$ and
by Lemma 2.4 we get
$$\forall j\geq N,\ \mu_{j}(K_{0}^{c})\leq\epsilon.$$
As every probability measure is tight we can find
a bigger compact set $K\supset K_{0}$ such that
$$\forall j\in\mathbb{N},\ \mu_{j}(K^{c})\leq\epsilon.$$
∎
Finally we notice that in the
regularity assumption (H3)
we can replace finite entropy by
absolute continuity with respect to $\pi$.
Lemma 2.5 ($k$-body interaction and regularity assumption).
Take $k>0$ and
a measurable symmetric function
bounded from below
$G:M^{k}\to(-\infty,\infty]$.
Define
$$W:\mathcal{P}(M)\to(-\infty,\infty]$$
$$\mu\mapsto\int_{M^{k}}G(x_{1},...,x_{k})\,d\mu(x_{1})...d\mu(x_{k}).$$
Let
$$\mathcal{N}_{1}=\left\{\mu\in\mathcal{P}(M):\ D(\mu\|\pi)<\infty\right\},$$
and
$$\mathcal{N}_{2}=\left\{\mu\in\mathcal{P}(M):\ \mu\mbox{ is absolutely %
continuous with respect to }\pi\right\}.$$
Suppose that for
every $\mu$ with $W(\mu)<\infty$, there exists
a sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
in $\mathcal{N}_{2}$ such that
$\mu_{n}\to\mu$ and $W(\mu_{n})\to W(\mu)$
then the same is true if we replace $\mathcal{N}_{2}$ by
$\mathcal{N}_{1}$.
Proof.
Notice that what we want to prove
is equivalent to the following.
If the closure of
$\{(\mu,W(\mu)):\ \mu\in\mathcal{N}_{2}\}$
contains
$\{(\mu,W(\mu)):\ W(\mu)<\infty\}$ then
the closure of
$\{(\mu,W(\mu)):\ \mu\in\mathcal{N}_{1}\}$
contains
$\{(\mu,W(\mu)):\ W(\mu)<\infty\}$.
So, it is enough to see that
the closure of
$\{(\mu,W(\mu)):\ \mu\in\mathcal{N}_{1}\}$
contains $\{(\mu,W(\mu)):\ \mu\in\mathcal{N}_{2}\}$,
i.e. for
every $\mu\in\mathcal{N}_{2}$ there exists a sequence
$\{\mu_{n}\}_{n\in\mathbb{N}}$
in $\mathcal{N}_{1}$ such that $\mu_{n}\to\mu$
and $W(\mu_{n})\to W(\mu)$.
Let $\rho$ be the density of $\mu$
with respect to $\pi$, i.e. $d\mu=\rho\,d\pi$.
For each $n>0$ define $\mu_{n}\in\mathcal{N}_{1}$ by
$d\mu_{n}=\frac{\rho\wedge n\,d\pi}{\int_{M}\rho\wedge n\,d\pi}$.
Then, by the monotone
convergence theorem we can see that
$\mu_{n}\to\mu$. And, again, by the monotone convergence
theorem, by supposing $G\geq 0$, we can see
that $W(\mu_{n})\to W(\mu)$.
∎
3 Proof of the theorems
This section is dedicated to the proof of the two main
theorems, i.e.
Theorem 1.1
and 1.3.
We start giving a proof sketch along
with some properties of the entropy that we will
need.
3.1 Idea of the proofs
To prove both Laplace principles we shall use the
following very known result that
tells us the Legendre transform of $D(\cdot\|\mu)$,
defined in 1.3.
See [11, Proposition 4.5.1] or
Appendix B for a proof.
Lemma 3.1.
Let $(E,\mu)$ be a Polish probability space and $g:E\to(-\infty,\infty]$
a measurable function bounded from below.
Then
$$\log\mathbb{E}_{\mu}\left[e^{-g}\right]=-\inf_{\tau\in\mathcal{P}(E)}\left\{%
\mathbb{E}_{\tau}\left[g\right]+D(\tau\|\mu)\right\}.$$
Remember the definition
of $\gamma_{n}$ in 1.1 and $F$ in 1.2.
With the help of Lemma 3.1
we can write
$$\displaystyle\frac{1}{n\beta_{n}}\log$$
$$\displaystyle\int_{M^{n}}e^{-n\beta_{n}f\circ i_{n}}d\gamma_{n}$$
$$\displaystyle=\frac{1}{n\beta_{n}}\log\,\mathbb{E}_{\pi^{\otimes_{n}}}\left[e^%
{-n\beta_{n}\left(f\circ\,i_{n}+W_{n}\right)}\right]$$
$$\displaystyle=-\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}%
\left[f\right]+\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau%
\|\pi^{\otimes_{n}})\right\}.$$
So, we need to prove that
$$\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}\left[f\right]+%
\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau\|\pi^{\otimes_{%
n}})\right\}\xrightarrow[\>n\to\infty\>]{}\inf_{\mu\in\mathcal{P}(M)}\{f\left(%
\mu\right)+F\left(\mu\right)\}.$$
(3.1)
To prove this we need the following properties
of the entropy proven in Appendix
D. The first
one is analogous to the
lower limit assumption (H1).
Lemma 3.2 (Lower limit property of the entropy).
Let $\{n_{j}\}_{j\in\mathbb{N}}$
be an increasing sequence in $\mathbb{N}$.
For each $j\in\mathbb{N}$
take $\tau_{j}\in\mathcal{P}(M^{n_{j}})$.
If $i_{n_{j}}(\tau_{j})\to\zeta\in\mathcal{P}(\mathcal{P}(M))$,
then
$$\mathbb{E}_{\zeta}\left[D\left(\cdot|\pi\right)\right]\leq\liminf_{j\to\infty}%
\frac{1}{n_{j}}D(\tau_{j}\|\pi^{\otimes_{n_{j}}}).$$
And the second one is analogous to the confining assumption (H4).
Lemma 3.3 (Confining property of the entropy).
Let $\{n_{j}\}_{j\in\mathbb{N}}$
be an increasing sequence in $\mathbb{N}$.
For each $j\in\mathbb{N}$
take $\tau_{j}\in\mathcal{P}(M^{n_{j}})$. If
there exists a real constant $C$ such that
$$\frac{1}{n_{j}}D(\tau_{j}\|\pi^{\otimes_{n_{j}}})\leq C$$
for every $j\in\mathbb{N}$, then the sequence
$\{i_{n_{j}}(\tau_{j})\}_{j\in\mathbb{N}}$ is tight.
3.2 Proof of
Theorem 1.1:
Case of finite
$\beta$
In this subsection we shall prove
the Laplace principle and the
large deviation principle
for the case of finite $\beta$, i.e.
Theorem 1.1 and
Corollary 1.2.
By Remark 1.6,
we can suppose $\beta_{n}=1$
for every $n$ by redefinition of $W_{n}$
and $W$. Then
the Gibbs measure 1.1 and
the free energy 1.2 are
$$d\gamma_{n}=e^{-nW_{n}}d\pi^{\otimes_{n}}\quad\text{and}\quad F=W+D(\cdot\|\pi).$$
As explained in subsection 3.1 we need to prove
3.1.
We shall repeat the argument.
Proof of Theorem
1.1.
By definition of $\gamma_{n}$ we have
$$\frac{1}{n}\log\,\int_{M^{n}}e^{-nf\circ i_{n}}d\gamma_{n}=\frac{1}{n}\log\,%
\mathbb{E}_{\pi^{\otimes_{n}}}\left[e^{-nf\circ i_{n}}e^{-nW_{n}}\right],$$
By Lemma 3.1, we can write
$$\displaystyle\frac{1}{n}\log\,\mathbb{E}_{\pi^{\otimes_{n}}}\left[e^{-n\left(f%
\circ\,i_{n}+W_{n}\right)}\right]$$
$$\displaystyle=-\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{\tau}\left[f%
\circ i_{n}+W_{n}\right]+\frac{1}{n}D(\tau\|\pi^{\otimes_{n}})\right\}$$
$$\displaystyle=-\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}%
\left[f\right]+\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n}D(\tau\|\pi^{%
\otimes_{n}})\right\}.$$
First, we will proof the
lower limit bound
$$\begin{split}\displaystyle\liminf_{n\to\infty}&\displaystyle\inf_{\tau\in%
\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}\left[f\right]+\mathbb{E}_{%
\tau}\left[W_{n}\right]+\frac{1}{n}D(\tau\|\pi^{\otimes_{n}})\right\}\\
\displaystyle\geq&\displaystyle\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)+%
F\left(\mu\right)\}.\end{split}$$
(3.2)
We shall proceed by contradiction.
Suppose 3.2 is not true. Then,
$$\liminf_{n\to\infty}\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(%
\tau)}\left[f\right]+\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n}D(\tau\|%
\pi^{\otimes_{n}})\right\}$$
$$<\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)+F\left(\mu\right)\}.$$
So, there exists an increasing sequence
of natural numbers $\{n_{j}\}_{j\in\mathbb{N}}$
and a constant $C\in\mathbb{R}$ such that
$$\inf_{\tau\in\mathcal{P}(M^{n_{j}})}\left\{\mathbb{E}_{i_{n_{j}}(\tau)}\left[f%
\right]+\mathbb{E}_{\tau}\left[W_{n_{j}}\right]+\frac{1}{{n_{j}}}D(\tau\|\pi^{%
\otimes_{n_{j}}})\right\}<\,C\,<\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)%
+F\left(\mu\right)\}$$
for every $j\in\mathbb{N}$. For each $j\in\mathbb{N}$
we can find $\tau_{j}\in\mathcal{P}(M^{n_{j}})$ such that
$$\mathbb{E}_{i_{n_{j}}(\tau_{j})}\left[f\right]+\mathbb{E}_{\tau_{j}}\left[W_{n%
_{j}}\right]+\frac{1}{n_{j}}D(\tau_{j}\|\pi^{\otimes_{n_{j}}})<\,C\,<\inf_{\mu%
\in\mathcal{P}(M)}\{f\left(\mu\right)+F\left(\mu\right)\}.$$
(3.3)
Using that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence,
i.e. it is uniformly bounded from below,
we get that
$\frac{1}{n_{j}}D(\tau_{j}\|\pi^{\otimes_{n_{j}}})$
is uniformly bounded from above.
By the confining property of the entropy,
Lemma 3.3, we get that
$i_{n_{j}}(\tau_{j})$ is tight. By taking a subsequence
using Prokhorov’s theorem,
we shall
assume it converges to some
$\zeta\in\mathcal{P}(\mathcal{P}(M))$.
Then, by the lower limit property of the entropy,
Lemma 3.2,
we get
$$\mathbb{E}_{\zeta}\left[D\left(\cdot|\pi\right)\right]\leq\liminf_{j\to\infty}%
\frac{1}{n_{j}}D(\tau_{n_{j}}\|\pi^{\otimes_{n_{j}}}).$$
By Skorokhod’s representation theorem
we can realize the weak convergence
$i_{n_{j}}(\tau_{n_{j}})\to\zeta$
as an almost sure convergence
$X_{j}\to X$ where $\{X_{j}\}_{j\in\mathbb{N}}$
is a sequence of random variables
taking values in $\mathcal{P}(M)$
such that for every $j\in\mathbb{N}$ the variable
$X_{j}$ has law $i_{n_{j}}(\tau_{n_{j}})$.
Then, by
the lower limit assumption (H1),
$$\liminf_{j\to\infty}\tilde{W}_{n_{j}}(X_{j})\geq W(X).$$
We note that
$\tilde{W}_{n}$ is measurable (see Appendix
C)
and that, by the upper
limit assumption (H2)
and the fact that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence,
$W$ is bounded from below.
So, as
$\{W_{n}\}_{n\in\mathbb{N}}$
is uniformly bounded from below, we can
use Fatou’s lemma to get
$$\mathbb{E}[W(X)]\leq\liminf_{j\to\infty}\mathbb{E}[\tilde{W}_{n_{j}}(X_{j})]$$
or, what is the same,
$$\mathbb{E}_{\zeta}[W]\leq\liminf_{j\to\infty}\mathbb{E}_{\tau_{n_{j}}}\left[W_%
{n_{j}}\right].$$
Then, by taking the lower limit
when $j$ tends to infinity
in the inequality 3.3
we get
$$\mathbb{E}_{\zeta}\left[f+W+D(\cdot\|\pi)\right]\leq\,C\,<\inf_{\mu\in\mathcal%
{P}(M)}\{f\left(\mu\right)+F\left(\mu\right)\},$$
or, equivalently,
$$\mathbb{E}_{\zeta}\left[f+F\right]\leq\,C\,<\inf_{\mu\in\mathcal{P}(M)}\{f%
\left(\mu\right)+F\left(\mu\right)\},$$
which is impossible.
Now let us prove the upper limit bound
$$\begin{split}\displaystyle\limsup_{n\to\infty}&\displaystyle\inf_{\tau\in%
\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}\left[f\right]+\mathbb{E}_{%
\tau}\left[W_{n}\right]+\frac{1}{n}D(\tau\|\pi^{\otimes_{n}})\right\}\\
\displaystyle\leq&\displaystyle\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)+%
F\left(\mu\right)\}.\end{split}$$
(3.4)
We need to prove that for every probability
measure
$\mu\in\mathcal{P}(M)$
$$\limsup_{n\to\infty}\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(%
\tau)}\left[f\right]+\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n}D(\tau\|%
\pi^{\otimes_{n}})\right\}$$
$$\leq f\left(\mu\right)+F\left(\mu\right).$$
It is enough to find a sequence
$\tau_{n}\in\mathcal{P}(M^{n})$ such that
$$\limsup_{n\to\infty}\left\{\mathbb{E}_{i_{n}(\tau_{n})}\left[f\right]+\mathbb{%
E}_{\tau_{n}}\left[W_{n}\right]+\frac{1}{n}D(\tau_{n}\|\pi^{\otimes_{n}})%
\right\}\leq f\left(\mu\right)+F\left(\mu\right).$$
We shall choose $\tau_{n}=\mu^{\otimes_{n}}$.
Then, we know that, by the
law of large numbers and Proposition 7.1
in Appendix A,
we have the weak convergence
$i_{n}(\tau_{n})\to\delta_{\mu},$ so
$$\lim_{n\to\infty}\mathbb{E}_{i_{n}\left(\tau_{n}\right)}\left[f\right]=f(\mu).$$
By the upper
limit assumption (H2), we get that
$$\limsup_{n\to\infty}\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]\leq W(\mu).$$
Finally, we use that
$$D(\tau_{n}\|\pi^{\otimes_{n}})=n\,D(\mu\|\pi)$$
to obtain
$$\lim_{n\to\infty}\left\{\mathbb{E}_{i_{n}(\tau_{n})}\left[f+\tilde{W}_{n}%
\right]+\frac{1}{n}D(\tau_{n}\|\pi^{\otimes_{n}})\right\}=f(\mu)+W(\mu)+D(\mu%
\|\pi)$$
completing the proof.
∎
Proof of Corollary
1.2.
We know that the large deviation principle is
equivalent to the Laplace principle for the sequence
$i_{n}(\mathbb{P}_{n})$ if
the rate function has compact level sets
(see
[11, Theorem 1.2.1]
and
[11, Theorem 1.2.3])
This is the case because
the entropy has compact level sets
(see [11, Lemma 1.4.3 (c)]) and
$W$ is bounded from below,
due to the upper
limit assumption (H2)
and the fact that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence, and lower semicontinuous.
Then we have to prove that,
for every bounded continuous
function $f:\mathcal{P}(M)\to\mathbb{R}$,
$$\frac{1}{n\beta_{n}}\log\,\mathbb{E}_{i_{n}\left(\mathbb{P}_{n}\right)}\left[e%
^{-n\beta_{n}f}\right]\,\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{%
P}(M)}\{f\left(\mu\right)+F\left(\mu\right)-\inf F\}$$
or, using the measures $\gamma_{n}$,
$$\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-n\beta_{n}f\circ i_{n}}\frac{d\gamma%
_{n}}{Z_{n}}\,\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{f%
\left(\mu\right)+F\left(\mu\right)-\inf F\}.$$
But, by Theorem 1.1
applied to the function
$f=0$ we get
$$\frac{1}{n\beta_{n}}\log\,Z_{n}\,\xrightarrow[\>n\to\infty\>]{}\,-\inf F$$
and combining it with
$$\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-n\beta_{n}f\circ i_{n}}d\gamma_{n}\,%
\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right%
)+F\left(\mu\right)\}$$
we complete the proof.
∎
3.3 Proof of Theorem
1.3:
Case of infinite $\beta$
In this subsection we provide a proof for Theorem
1.3
by modifying the proof of Theorem
1.1.
Recall that from the definition
of Gibbs measure 1.1 and
free energy 1.2
now we have
$$d\gamma_{n}=e^{-n\beta_{n}W_{n}}d\pi^{\otimes_{n}},\quad\text{and}\quad F=W.$$
where $\beta_{n}\to\infty$.
We first notice that the
confining assumption (H4)
implies an
a priori stronger property.
This is a direct consequence of
Proposition 7.4
in Appendix A.
Proposition 3.4.
Assume the sequence $\{W_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4)
and take
a sequence of
probability measures $\{\chi_{j}\}_{j\in\mathbb{N}}$ on
$\mathcal{P}(M)$, i.e. $\chi_{j}\in\mathcal{P}(\mathcal{P}(M))$.
Suppose there exists an increasing sequence
$\{n_{j}\}_{j\in\mathbb{N}}$
of natural numbers and a constant $C<\infty$
such that $\mathbb{E}_{\chi_{j}}\left[\tilde{W}_{n_{j}}\right]\leq C$
for every $j\in\mathbb{N}$ . Then
$\{\chi_{j}\}_{n\in\mathbb{N}}$ is
relatively compact in
$\mathcal{P}(\mathcal{P}(M))$.
Now we shall prove Theorem
1.3 .
Proof of Theorem
1.3.
Take $f:\mathcal{P}(M)\to\mathbb{R}$ bounded continuous.
By Lemma 3.1 and following
the argument in Subsection 3.1
about the idea of the proof we need to obtain
3.1, i.e.
$$\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}\left[f\right]+%
\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau\|\pi^{\otimes_{%
n}})\right\}\xrightarrow[\>n\to\infty\>]{}\inf_{\mu\in\mathcal{P}(M)}\{f\left(%
\mu\right)+F\left(\mu\right)\}.$$
To prove the lower limit bound
$$\displaystyle\liminf_{n\to\infty}$$
$$\displaystyle\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}%
\left[f\right]+\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau%
\|\pi^{\otimes_{n}})\right\}$$
$$\displaystyle\geq$$
$$\displaystyle\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)+F\left(\mu\right)\}$$
we proceed by contradiction as in Theorem
1.1
saying that
if this were not the case
there would exist an increasing sequence
of natural numbers $\{n_{j}\}_{j\in\mathbb{N}}$,
a constant $C\in\mathbb{R}$ and for each $j\in\mathbb{N}$
a probability measure
$\tau_{j}\in\mathcal{P}(M^{n_{j}})$ such that
$$\mathbb{E}_{i_{n_{j}}(\tau_{j})}\left[f\right]+\mathbb{E}_{\tau_{j}}\left[W_{n%
_{j}}\right]+\frac{1}{n_{j}\beta_{n_{j}}}D(\tau_{j}\|\pi^{\otimes_{n_{j}}})<\,%
C\,<\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)+F\left(\mu\right)\}.$$
Then, as the entropy is always positive we get
$$\mathbb{E}_{i_{n_{j}}(\tau_{j})}\left[f\right]+\mathbb{E}_{\tau_{j}}\left[W_{n%
_{j}}\right]<\,C\,<\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right)+F\left(\mu%
\right)\}.$$
We see that
$\mathbb{E}_{\tau_{j}}\left[W_{n_{j}}\right]=\mathbb{E}_{i_{n_{j}}(\tau_{j})}%
\left[\tilde{W}_{n_{j}}\right]$
is a bounded sequence and, by the
confining assumption (H4)
and Proposition 3.4,
we get that $i_{n_{j}}(\tau_{j})$ is
relatively compact in
$\mathcal{P}(\mathcal{P}(M))$. We proceed as
in the proof of Theorem
1.1
where now $W$ is bounded from below
by the regularity assumption (H3) and
because $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence.
The proof of the upper limit bound
follows the same reasoning as in Theorem
1.1. Take $\mu\in\mathcal{P}(M)$.
To obtain
$$\begin{split}\displaystyle\limsup_{n\to\infty}&\displaystyle\inf_{\tau\in%
\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}\left[f\right]+\mathbb{E}_{%
\tau}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau\|\pi^{\otimes_{n}})\right\}%
\\
\displaystyle\leq&\displaystyle\ f\left(\mu\right)+F\left(\mu\right),\end{split}$$
(3.5)
we choose $\tau_{n}=\mu^{\otimes_{n}}$ and
try to prove
$$\limsup_{n\to\infty}\left\{\mathbb{E}_{i_{n}(\tau_{n})}\left[f\right]+\mathbb{%
E}_{\tau_{n}}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau_{n}\|\pi^{\otimes_{%
n}})\right\}\leq f\left(\mu\right)+F\left(\mu\right).$$
By the law of large numbers,
as in the proof of Theorem 1.1,
this is true for
$\mu$ in $\mathcal{N}$ defined in 1.5.
For general $\mu\in\mathcal{P}(M)$ take a sequence
$\{\mu_{k}\}_{k\in\mathbb{N}}$ in $\mathcal{N}$
such that
$\mu_{k}\to\mu$ and
$\limsup_{k\to\infty}W(\mu_{k})\leq W(\mu)$. This
exists due to the regularity assumption (H3).
Finally, take the upper limit when $k$
goes to infinity in
$$\displaystyle\ \ \limsup_{n\to\infty}$$
$$\displaystyle\inf_{\tau\in\mathcal{P}(M^{n})}\left\{\mathbb{E}_{i_{n}(\tau)}%
\left[f\right]+\mathbb{E}_{\tau}\left[W_{n}\right]+\frac{1}{n\beta_{n}}D(\tau%
\|\pi^{\otimes_{n}})\right\}$$
$$\displaystyle\leq$$
$$\displaystyle\ f\left(\mu_{k}\right)+F\left(\mu_{k}\right),$$
to obtain
3.5.
∎
Proof of Corollary
1.4.
It is the same proof as that of Corollary
1.2
with the additional assumption that
we demand
$W$ to have compact level sets because $F$ does not
contain an entropy term.
∎
4 Applications
In this section we shall give the three main applications
we are thinking of: Conditional Gibbs measure,
Coulomb gas on a Riemannian manifold and
the known results in $\mathbb{R}^{n}$.
4.1 Conditional Gibbs measure
In this subsection we shall treat the case
of the Gibbs measure associated to a two-body interaction
but with some of the points conditioned to
be deterministic. We begin
by a more general result.
We shall suppose $M$ compact.
Proposition 4.1 (Varying environment).
Let $\{\nu_{n}\}_{n\in\mathbb{N}}$ be a
sequence of probability measures on $M$ such that
$\nu_{n}\to\nu$. Suppose we have two lower
semicontinuous functions
$G^{E},G^{I}:M\times M\to(-\infty,\infty]$
(and thus bounded from below)
such that
$x\mapsto\int_{M}G^{E}(x,y)d\nu(y)$
is continuous.
Internal potential energy.
For each $n$ we shall think of $n$ particles
interacting with the two particle potential $G^{I}$.
This would give rise to
an internal energy
$W^{I}_{n}:M^{n}\to(-\infty,\infty]$
$$W^{I}_{n}(x_{1},...,x_{n})=\frac{1}{n^{2}}\sum_{i<j}^{n}G^{I}(x_{i},x_{j})$$
and a macroscopic internal energy
$W^{I}:\mathcal{P}(M)\to(-\infty,\infty]$
$$W^{I}(\mu)=\frac{1}{2}\int_{M\times M}G^{I}(x,y)\,d\mu(x)d\mu(y).$$
External potential energy.
The probability measure $\nu_{n}$ will interact with
the $n$ particles via the external potential
$V_{n}:M\to\mathbb{R}$ defined by
$V_{n}(x)=\int_{M}G^{E}(x,y)d\nu_{n}(y)$.
This gives rise to the external energy
$W^{E}_{n}:M^{n}\to(-\infty,\infty]$
$$W^{E}_{n}(x_{1},...,x_{n})=\frac{1}{n}\sum_{i=1}^{n}V_{n}(x_{i})$$
with a macroscopic external energy
$W^{E}:\mathcal{P}(M)\to(-\infty,\infty]$
$$W^{E}(\mu)=\int_{M\times M}G^{E}(x,y)\,d\nu(x)d\mu(y).$$
Total potential energy.
For each $n$ we define
$$W_{n}=W^{E}_{n}+W^{I}_{n}\ \ \ \mbox{ and }\ \ \ W=W^{E}+W^{I}.$$
Let
$$\tilde{\mathcal{N}}=\left\{\mu\in\mathcal{P}(M):\ D(\mu\|\pi)<\infty\mbox{ and%
}x\mapsto\int_{M}G^{E}(x,y)d\mu(y)\mbox{ is continuous}\right\}$$
and suppose that for every
$\mu\in\mathcal{P}(M)$ such that $W^{I}(\mu)<\infty$
there exists a sequence
$\{\mu_{n}\}_{n\in\mathbb{N}}$ of probability
measures
in $\tilde{\mathcal{N}}$ such that $\mu_{n}\to\mu$
and $W^{I}(\mu_{n})\to W^{I}(\mu)$.
Then, $\{W_{n}\}_{n\in\mathbb{N}}$ is a stable sequence,
$W$ is a lower semicontinuous function and
$(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower limit assumption (H1),
and the regularity assumption (H3).
In particular, if
we choose $\beta_{n}\to\infty$,
Theorem 1.3
and Corollary 1.4 may
be applied for $(\{W_{n}\}_{n\in\mathbb{N}},W)$.
Proof.
It is not hard to prove
the stability and the lower semicontinuity.
For instance, we can imitate
the proof of Lemma 2.1.
Let us prove
the lower limit assumption (H1).
Lower limit assumption (H1).
By Lemma 2.1, we already know that
$(\{W^{I}_{n}\}_{n\in\mathbb{N}},W^{I})$
satisfies the lower limit assumption (H1).
By Remark 1.7,
we need to check this for $(\{W^{E}_{n}\}_{n\in\mathbb{N}},W^{E})$.
If $\mu_{n}=i_{n}(x_{1},...,x_{n})$ then
$$\tilde{W}^{E}_{n}(\mu_{n})=\int_{M}V_{n}d\mu_{n}=\int_{M\times M}G^{E}(x,y)d%
\mu_{n}(x)d\nu_{n}(y),$$
where $\tilde{W}^{E}_{n}$ is defined in
1.4.
So, the lower limit assumption (H1)
is a consequence of the lower semicontinuity of $G^{E}$.
Regularity assumption (H3).
To prove the regularity assumption (H3)
we take
$\mu\in\mathcal{P}(M)$ such that
$W(\mu)<\infty$.
Then $W^{I}(\mu)<\infty$.
By hypothesis,
we know that there exists
a sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$ of probability
measures
in $\tilde{\mathcal{N}}$ such that $\mu_{n}\to\mu$
and $W^{I}(\mu_{n})\to W^{I}(\mu)$.
As $x\mapsto\int_{M}G^{E}(x,y)d\nu(y)$ is continuous
we also have that
$W^{E}(\mu_{n})\to W^{E}(\mu)$.
So, $W(\mu_{n})\to W(\mu)$.
We have to prove that the sequence we chose is in the set
$\mathcal{N}$ defined in 1.5 by
$$\mathcal{N}=\left\{\mu\in\mathcal{P}(M):\ D(\mu\|\pi)<\infty\mbox{ and }%
\limsup_{n\to\infty}\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]\leq W(\mu)\right\},$$
i.e. we need to see that
$\tilde{\mathcal{N}}\subset\mathcal{N}$.
Let $\mu\in\tilde{\mathcal{N}}$. Then
$$\displaystyle\mathbb{E}_{\mu^{\otimes_{n}}}[W^{E}_{n}]$$
$$\displaystyle=\mathbb{E}_{\mu}[V_{n}]$$
$$\displaystyle=\int_{M}\left(\int_{M}G^{E}(x,y)d\nu_{n}(y)\right)d\mu(x)$$
$$\displaystyle=\int_{M}\left(\int_{M}G^{E}(x,y)d\mu(x)\right)d\nu_{n}(y).$$
So, as $y\mapsto\int_{M}G^{E}(x,y)d\mu(x)$ is continuous,
we get
$$\mathbb{E}_{\mu^{\otimes_{n}}}[W^{E}_{n}]\,\xrightarrow[\>n\to\infty\>]{}\,%
\int_{M\times M}G^{E}(x,y)d\nu(y)d\mu(x)=W^{E}(\mu).$$
By Lemma 2.1 we already know that
$\lim_{n\to\infty}\mathbb{E}_{\mu^{\otimes_{n}}}[W^{I}_{n}]=W^{I}(\mu)$
and then
$\mu\in\mathcal{N}$.
∎
Now, we take the particular case where $\nu_{n}$
is an empirical measure and interpret it
as a conditioning.
Example 4.2 (Conditional Gibbs measure).
Take a lower semicontinuous function
$G:M\times M\to(-\infty,\infty]$.
Define $G^{I}=G^{E}=G$.
For each $n\in\mathbb{N}$ choose
$(y_{1},...,y_{n})\in M^{n}$ and define
$\nu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{y_{i}}$.
Suppose $\nu_{n}\to\nu\in\mathcal{P}(M)$
such that $x\mapsto\int_{M}G(x,y)d\nu(y)$ is
continuous.
With the notation of Proposition 4.1
we have that $W_{n}^{E}:M^{n}\to(-\infty,\infty]$
is given by $W_{n}^{E}(x_{1},...,x_{n})=\frac{1}{n^{2}}\sum_{i,j=1}^{n}G(x_{i},y_{j})$
and then that
$$W_{n}(x_{1},...,x_{n})=\frac{1}{n^{2}}\sum_{i<j}^{n}G(x_{i},x_{j})+\frac{1}{n^%
{2}}\sum_{i,j=1}^{n}G(x_{i},y_{j}).$$
This suggests that we can think on the Gibbs
probability measure (normalized Gibbs measure)
associated to $W_{n}$ and to some parameter $\beta_{n}$
as the Gibbs probability measure associated
to the energy
$$\mathcal{W}_{n}(x_{1},...,x_{2n})=\frac{1}{n^{2}}\sum_{i<j}^{2n}G(x_{i},x_{j}).$$
conditioned to
$(x_{n+1},...,x_{2n})=(y_{1},...,y_{n})$.
Proposition 4.1 treats this case
if
$\beta_{n}\to\infty$
and if for every $\mu\in\mathcal{P}(M)$
such that
$\int_{M\times M}G(x,y)d\mu(x)d\mu(y)<\infty$
there exists a sequence
$\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures on $M$ such that
$D(\mu_{n}\|\pi)<\infty$ and
$x\mapsto\int_{M}G(x,y)d\mu_{n}(y)$ is continuous
for every $n\in\mathbb{N}$,
$\mu_{n}\to\mu$ and
$\int_{M\times M}G(x,y)d\mu_{n}(x)d\mu_{n}(y)\to\int_{M\times M}G(x,y)d\mu(x)d%
\mu(y)$.
Proposition 4.3 (One particle conditional Gibbs measure).
Take again a lower semicontinuous function
$G:M\times M\to(-\infty,\infty]$.
For each $n\in\mathbb{N}$ choose
$(y_{1},...,y_{n})\in M^{n}$ and suppose that
$\frac{1}{n}\sum_{i=1}^{n}\delta_{y_{i}}\to\nu$
for some $\nu\in\mathcal{P}(M)$ and
that $x\mapsto\int_{M}G(x,y)d\nu(y)$
is continuous.
Define $V_{n}:M\to(-\infty,\infty]$
by $V_{n}(x)=\frac{1}{n}\sum_{j=1}^{n}G(x,y_{j})$.
Take $\beta_{n}\to\infty$
and define the measures $\gamma^{c}_{n}$ by
$$d\gamma^{c}_{n}=e^{-\beta_{n}V_{n}}d\pi.$$
Then, we have
the following Laplace principle.
For every (bounded) continuous function
$f:M\to\mathbb{R}$ we
have that
$$\frac{1}{\beta_{n}}\log\,\int_{M}e^{-\beta_{n}f}d\gamma^{c}_{n}\,\xrightarrow[%
\>n\to\infty\>]{}\,-\inf_{x\in M}\left\{f(x)+\int_{M}G(x,y)d\nu(y)\right\}.$$
The normalization
of $\gamma^{c}_{n}$ can be thought of as the
Gibbs probability measure associated to
the energy
$\mathcal{W}_{n}:M^{n+1}\to(-\infty,\infty]$
$$\mathcal{W}_{n}(x_{1},...,x_{n+1})=\frac{1}{n^{2}}\sum_{i<j}^{n+1}G(x_{i},x_{j})$$
conditioned to $(x_{1},...,x_{n})=(y_{1},...,y_{n})$.
Proof.
To apply Proposition 4.1
define $G^{I}=0$ and $G^{E}=G$.
With the notation of Proposition 4.1
we have that $V_{n}:M\to(-\infty,\infty]$
is given by $V_{n}(x)=\frac{1}{n}\sum_{j=1}^{n}G(x,y_{j})$
and $W_{n}^{E}:M^{n}\to(-\infty,\infty]$
is given by $W_{n}^{E}(x_{1},...,x_{n})=\frac{1}{n}\sum_{i=1}^{n}V_{n}(x_{i})$.
Then, $W_{n}=W_{n}^{E}$.
As $G^{I}=0$, there is no regularity assumption to
prove.
From Proposition 4.1 we know that
if $\beta_{n}\to\infty$ then
for every (bounded) continuous
function $\hat{f}:\mathcal{P}(M)\to\mathbb{R}$
$$\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-n\beta_{n}\hat{f}\circ i_{n}}d\gamma%
_{n}\,\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{\hat{f}%
\left(\mu\right)+F\left(\mu\right)\}$$
where
$$d\gamma_{n}=e^{-n\beta_{n}W_{n}^{E}}d\pi^{\otimes_{n}}.$$
In particular, if $\hat{f}$ is given by integration against
some (bounded) continuous function $f:M\to\mathbb{R}$, i.e.
$\hat{f}(\mu)=\int_{M}fd\mu$, then
$$\displaystyle\frac{1}{n\beta_{n}}\log$$
$$\displaystyle\,\int_{M^{n}}e^{-n\beta_{n}\hat{f}\circ i_{n}}d\gamma_{n}$$
$$\displaystyle=\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-\beta_{n}\sum_{i=1}^{n%
}f(x_{i})}e^{-\beta_{n}\sum_{i=1}^{n}V_{n}(x_{i})}d\pi(x_{1})...d\pi(x_{n})$$
$$\displaystyle=\frac{1}{\beta_{n}}\log\,\int_{M}e^{-\beta_{n}f(x)}e^{-\beta_{n}%
V_{n}(x)}d\pi(x)$$
$$\displaystyle=\frac{1}{\beta_{n}}\log\,\int_{M}e^{-\beta_{n}f(x)}d\gamma_{n}^{%
c}(x)$$
converges to
$$\displaystyle\inf_{\mu\in\mathcal{P}(M)}\{\hat{f}\left(\mu\right)+F\left(\mu%
\right)\}$$
$$\displaystyle=\inf_{\mu\in\mathcal{P}(M)}\left\{\int_{M}fd\mu+\int_{M}\left(%
\int_{M}G(x,y)d\nu(y)\right)d\mu(x)\right\}$$
$$\displaystyle=\inf_{x\in M}\left\{f(x)+\int_{M}G(x,y)d\nu(y)\right\}.$$
∎
4.2 A Coulomb gas on a Riemannian manifold
Let $(M,g)$ be a compact oriented $n$-dimensional
Riemannian manifold without boundary where
$g$ denotes the Riemannian metric.
Denote by $\pi$ the normalized volume form of $(M,g)$.
Notice that integrability with
respect to $\pi$ is equivalent to integrability
with respect to any differentiable volume form
and we may say
that some function is integrable in this case
without saying explicitly
that it is with respect to $\pi$.
A
signed measure $\Lambda$ will be called a
differentiable signed measure
if it is given by an $n$-form or
equivalently if it has a differentiable
density with respect to $\pi$.
From now on we shall identify $\Omega^{n}(M)$ with the
space of differentiable signed measures.
Let $\Delta:C^{\infty}(M)\to\Omega^{n}(M)$
be the Laplacian
operator, i.e. $\Delta=d*d$ where $*$
is the Hodge star operator or, equivalently,
$\Delta f=\nabla^{2}f\,d\pi$ where $\nabla^{2}$
is the Laplace-Beltrami operator.
Proposition 4.4 (Green function).
Take any differentiable signed measure $\Lambda$.
Then, there exists
a symmetric continuous function
$G:M\times M\to(-\infty,\infty]$ such that
for every $x\in M$ the function
$G_{x}:M\to(-\infty,\infty]$ defined
by $G_{x}(y)=G(x,y)$ is integrable and
$$\Delta G_{x}=-\delta_{x}+\Lambda.$$
More explicitly, the previous equality can be written
as follows. For every $f\in C^{\infty}(M)$
we have
$$\int_{M}G_{x}\,\Delta f=-f(x)+\int_{M}fd\Lambda.$$
Such a function will be called a Green
function associated to $\Lambda$.
Furthermore $G$ is integrable with respect
to $\pi\otimes\pi$. If $\mu$ is a
differentiable signed measure
then $\psi:M\to\mathbb{R}$
defined by $\psi(x)=\int_{M}G(x,y)d\mu(y)$ belongs
to $C^{\infty}(M)$ and
$$\Delta\psi=-\mu+\mu(M)\Lambda.$$
In particular, we can get that
$G$ is bounded from below,
$\int_{M}G_{x}d\Lambda$ does not depend on $x\in M$
and the Green function
associated to $\Lambda$
is unique up to an additive constant.
Proof.
This result is well known if $\Lambda=\pi$.
See for instance [19, Chapter 4].
Then we can take
a symmetric continuous function
$H:M\times M\to(-\infty,\infty]$ such that
for every $x\in M$ the function
$H_{x}:M\to(-\infty,\infty]$ defined
by $H_{x}(y)=H(x,y)$ is integrable and
$$\Delta H_{x}=-\delta_{x}+\pi.$$
We also know that $H$ is integrable
with respect to $\pi\otimes\pi$.
For the case of general
$\Lambda$ choose
$\phi\in C^{\infty}(M)$ such that
$$\Delta\phi=-\pi+\Lambda.$$
See [19, Theorem 4.7] for the existence
of such $\phi$.
Define $G:M\times M\to(-\infty,\infty]$
by $G(x,y)=H(x,y)+\phi(x)+\phi(y)$.
This is a symmetric continuous function
such that for every $x\in M$ the function
$G_{x}:M\to(-\infty,\infty]$ defined
by $G_{x}(y)=G(x,y)$ is integrable and
$$\Delta G_{x}=\Delta H_{x}+\Delta\phi=-\delta_{x}+\pi+\left(-\pi+\Lambda\right)%
=-\delta_{x}+\Lambda.$$
The integrability
of $G$ with respect to $\pi\otimes\pi$
is a consequence of the integrability
of $H$ with respect to $\pi\otimes\pi$.
We can see that $G$ is bounded from below
by the continuity of $G$ so we shall suppose that
$G$ is positive for simplicity.
Let $\mu$ be a differentiable signed measure
and take $\rho\in C^{\infty}(M)$ such that
$d\mu=\rho\,d\pi$.
Then,
$\int_{M\times M}G(x,y)d\pi(x)|\rho(y)|d\pi(y)<\infty$ and
$\psi(x)=\int_{M\times M}G(x,y)\rho(y)d\pi(y)$
is defined for $\pi$ - almost every $x\in M$, it is integrable with respect to $\pi$
and, for every $f\in C^{\infty}(M)$,
$$\displaystyle\int_{M}\psi\,\Delta f$$
$$\displaystyle=\int_{M\times M}G(x,y)\Delta f(x)\rho(y)d\pi(y)$$
$$\displaystyle=\int_{M}\left(\int_{M}G(x,y)\Delta f(x)\right)\rho(y)d\pi(y)$$
$$\displaystyle=\int_{M}\left(-f(y)+\int_{M}f\,d\Lambda\right)\rho(y)d\pi(y)$$
$$\displaystyle=-\int_{M}f\,d\mu+\mu(M)\int_{M}f\,d\Lambda.$$
This is precisely the
relation
$$\Delta\psi=-\mu+\mu(M)\Lambda$$
in the sense of distributions.
Then, by the ellipticity of $\Delta$,
we obtain that $\psi$ is differentiable and
$$\Delta\psi=-\mu+\mu(M)\Lambda$$
in the usual pointwise sense.
In particular, we get that
if we define $\psi:M\to\mathbb{R}$ by
$\psi(x)=\int_{M}G_{x}d\Lambda$ then
$\Delta\psi=0$ and so $\psi$ is a constant.
To prove that $G$ is unique up to an additive constant,
take another Green function $\tilde{G}$
associated to $\Lambda$. Then,
for every $x\in M$,
$\Delta\tilde{G}_{x}=\Delta G_{x}$,
i.e.
$\int_{M}\tilde{G}_{x}\,\Delta f=\int_{M}G_{x}\,\Delta f$
for every $f\in C^{\infty}(M)$.
This is equivalent to say that
$\int_{M}\tilde{G}_{x}\,d\mu,=\int_{M}G_{x}\,d\mu$
for every differentiable signed measure $\mu$
such that $\mu(M)=0$ because
in this case $\mu=\Delta f$ for some
$f\in C^{\infty}(M)$ by
[19, Theorem 4.7].
Take any signed differentiable measure $\mu$.
Define the signed measure
$\mu_{0}=\mu-\mu(M)\Lambda$.
Then, for every $x\in M$,
$$\displaystyle\int_{M}\tilde{G}_{x}\,d\mu,$$
$$\displaystyle=\int_{M}\tilde{G}_{x}\,d\mu_{0}+\mu(M)\int_{M}\tilde{G}_{x}\,d\Lambda,$$
$$\displaystyle=\int_{M}G_{x}\,d\mu_{0}+\mu(M)\left(\int_{M}\tilde{G}_{x}\,d%
\Lambda-\int_{M}G_{x}\,d\Lambda\right)+\mu(M)\int_{M}G_{x}\,d\Lambda$$
$$\displaystyle=\int_{M}G_{x}\,d\mu+\mu(M)\left(\int_{M}\tilde{G}_{x}\,d\Lambda-%
\int_{M}G_{x}\,d\Lambda\right)$$
where we have already seen that
$C=\int_{M}\tilde{G}_{x}\,d\Lambda-\int_{M}G_{x}\,d\Lambda$
does not depend on $x\in M$. Then,
for every $x\in M$ we have
the equality in distributions
$$\tilde{G}_{x}=G_{x}+C$$
that, by the continuity of $\tilde{G}_{x}$ and
$G_{x}$, becomes a pointwise equality
$$\forall x,y\in M,\,\tilde{G}(x,y)=G(x,y)+C.$$
∎
We fix
a differentiable signed measure $\Lambda$.
For simplicity we choose the Green function $G$
associated to $\Lambda$ that satisfies
$\int_{M}G_{x}d\Lambda=0$ for every $x\in M$.
By Proposition 4.4 we know that
$G$
satisfies the conditions of
Lemma 2.1 about the two-body interaction.
Define $W:\mathcal{P}(M)\to(-\infty,\infty]$
by $W(\mu)=\int_{M\times M}G(x,y)d\mu(x)d\mu(y)$.
Then
we can prove a strong form of the regularity assumption
for $W$.
Proposition 4.5 (Regularity property
of the Green energy).
Let $\mu\in\mathcal{P}(M)$. There exists
a sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of differentiable probability measures such that
$\mu_{n}\to\mu$ and $W(\mu_{n})\to W(\mu)$.
Proof.
We can assume $W(\mu)<\infty$, otherwise
any sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of differentiable probability measures such that
$\mu_{n}\to\mu$
will satisfy $W(\mu_{n})\to W(\mu)$
due to the lower semicontinuity of $W$.
Using the proof of [2, Lemma 3.13]
for the case of probability measures we know
that the result is true for
the Green function $H$ associated to $\pi$.
For general $\Lambda$,
take $\phi\in C^{\infty}(M)$ such that
$\Delta\phi=-\pi+\Lambda$ as in
the proof of Proposition 4.4.
Then $G:M\times M\to(-\infty,\infty]$
defined by $G(x,y)=H(x,y)+\phi(x)+\phi(y)$
is a Green function for $\Lambda$ and
for every $\mu\in\mathcal{P}(M)$ we have
$$\int_{M\times M}G(x,y)d\mu(x)d\mu(y)=\int_{M\times M}H(x,y)d\mu(x)d\mu(y)+2%
\int_{M}\phi\,d\mu.$$
From this relation and the result for $H$ we get
the result for $G$.
∎
As a corollary we obtain the following result.
Corollary 4.6.
$G$ satisfies the conditions
of Theorem 1.1,
Theorem
1.3
and
Proposition
4.1
with $G^{I}=G$ and $G^{E}=G$.
Remember the definitions of $i_{n}$,
1.6,
and $\mathbb{P}_{n}$,
1.7.
Let $\{X_{n}\}_{n\in\mathbb{N}}$
be a sequence of random variables
taking values in $\mathcal{P}(M)$ such that, for every
$n\in\mathbb{N}$,
$X_{n}$ has law $i_{n}(\mathbb{P}_{n})$.
By studying the minimizers of the
free energy $F$ defined in 1.2
we can understand
the possible limit points of
$\{X_{n}\}_{n\in\mathbb{N}}$.
In particular, if $F$ attains its minimum
at a unique probability measure
$\mu_{eq}$, we get
$$X_{n}\xrightarrow[\>n\to\infty\>]{a.s.}\mu_{eq}.$$
This is a consequence of Borel-Cantelli lemma
and the large deviation principle
in Corollary 1.2.
We shall now specialize
to the case of dimension two
and finite $\beta$
because the minimizer of $F$
has
a nice geometric meaning in this case.
Proposition 4.7 (Convexity of $W$).
$W$ is convex.
Proof.
To prove the convexity it is enough to show that
for every $\mu,\nu\in\mathcal{P}(M)$
$$\frac{1}{2}W(\mu)+\frac{1}{2}W(\nu)\geq W\left(\frac{1}{2}\mu+\frac{1}{2}\nu%
\right).$$
due to the lower semicontinuity of $W$.
Equivalently, we need to verify that,
for every $\mu,\nu\in\mathcal{P}(M)$,
$$\frac{1}{2}W(\mu)+\frac{1}{2}W(\nu)\geq\int_{M\times M}G(x,y)\,d\mu(x)\,d\nu(y).$$
(4.1)
If $\mu$ and $\nu$ are
differentiable probability measures,
i.e. differentiable signed measures
that are probability measures,
we can notice that
the previous inequality
4.1
is equivalent to
the inequality
$$\int_{M}\|\nabla(f-g)\|^{2}\,d\pi\geq 0.$$
where
$$f(x)=\int_{M}G(x,y)\,d\mu(y)\ \ \ \ \mbox{and}\ \ \ \ g(x)=\int_{M}G(x,y)\,d%
\nu(y).$$
This is due to the properties
$$\Delta f=-\mu+\Lambda\ \ \ \ \mbox{and}\ \ \ \ \Delta g=-\nu+\Lambda$$
given in Proposition 4.4.
In general, take $\mu,\nu$ arbitrary. Choose
two sequences
$\{\mu_{n}\}_{n\in\mathbb{N}}$,
$\{\nu_{n}\}_{n\in\mathbb{N}}$
of differentiable probability measures
such that $\mu_{n}\to\mu$, $\nu_{n}\to\nu$
and $W(\mu_{n})\to W(\mu)$, $W(\nu_{n})\to W(\nu)$,
possible by Proposition
4.5.
By the lower semicontinuity and lower boundedness
of $G$ we know that
$$\liminf_{n\to\infty}\int_{M\times M}G(x,y)d\mu_{n}(x)d\nu_{n}(y)\geq\int_{M%
\times M}G(x,y)d\mu(x)d\nu(x).$$
So, taking the lower limit
when $n$ tends to infinity in the inequalities
$$\frac{1}{2}W(\mu_{n})+\frac{1}{2}W(\nu_{n})\geq\int_{M\times M}G(x,y)\,d\mu_{n%
}(x)\,d\nu_{n}(y)$$
we get
$$\frac{1}{2}W(\mu)+\frac{1}{2}W(\nu)\geq\int_{M\times M}G(x,y)\,d\mu(x)\,d\nu(y).$$
∎
As $D(\cdot\|\pi)$ is strictly convex
(see [11, Lemma 1.4.3]) we know that
the free energy $F$
of parameter $\beta<\infty$ is strictly convex.
Let $\rho\in C^{\infty}(M)$ be a
differentiable positive solution of the following equation
(see [8])
$$\Delta\log\rho=\beta\,\mu_{eq}-\beta\Lambda.$$
where $d\mu_{eq}=\rho\,d\pi$.
We shall prove that
the functional $F$ achieves its minimum at
$\mu_{eq}$.
For this, we should calculate
its derivative at $\mu_{eq}$.
Lemma 4.8 (Derivative of $W$ and of the
entropy).
Let $\mu$ be
any probability measure different from $\mu_{eq}$
such that $F(\mu)<\infty$.
Define
$$\mu_{t}=t\mu+(1-t)\mu_{eq},\ \ t\in[0,1].$$
Then,
$W(\mu_{t})$ and $D(\mu_{t}\|\pi)$
are differentiable at $t=0$, and
$$\hskip-113.811024pt\frac{d}{dt}W(\mu_{t})|_{t=0}=2\int_{M\times M}G(x,y)\,d\mu%
_{eq}(x)\,\left(d\mu(y)-d\mu_{eq}(y)\right)\,,$$
(4.2)
$$\hskip-184.942913pt\frac{d}{dt}D(\mu_{t}\|\pi)|_{t=0}=\int_{M}\log\rho(y)\left%
(d\mu(y)-d\mu_{eq}(y)\right).$$
(4.3)
Proof.
As $W(\mu)$ and $W(\mu_{eq})$ are finite
and due to the convexity of $W$
$$\begin{split}\displaystyle W(\mu_{t})&\displaystyle=t^{2}\int_{M\times M}G(x,y%
)d\mu(x)\,d\mu(y)\\
\displaystyle+\,2t(1-t)\int_{M\times M}G(x,y)&\displaystyle d\mu_{eq}(x)\,d\mu%
(y)\ \ +\ \ (1-t)^{2}\int_{M\times M}G(x,y)d\mu_{eq}(x)\,d\mu_{eq}(y)\end{split}$$
(4.4)
is finite.
The linear term (in the variable $t$) in
4.4 is
$$2\int_{M\times M}G(x,y)\,d\mu_{eq}(x)\,\left(d\mu(y)-d\mu_{eq}(y)\right),$$
and we have proven the first equality
4.2.
To prove the second equality
4.3 we write
$$D(\mu_{t}\|\pi)=\int_{M}\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,\right)%
\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\right)\,d\pi(x)$$
(4.5)
and due to the finiteness of $D(\mu_{t}\|\pi)$
we can notice that
$\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,\right)$
is integrable with respect to $\mu$ and $\mu_{eq}$
for $0<t<1$. Developing the expression
4.5 we get
$$\displaystyle D(\mu_{t}\|\pi)$$
$$\displaystyle=t\,\int_{M}\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,\right)%
\left(\frac{d\mu}{d\pi}(x)-\rho(x)\right)\,d\pi(x)$$
$$\displaystyle\ \ \ +\int_{M}\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,%
\right)\rho(x)\,d\pi(x)$$
and we also have
$$D(\mu_{0}\|\pi)=\int_{M}\log(\rho(x))\,\rho(x)\,d\pi(x).$$
Then
$$\begin{split}\displaystyle\frac{1}{t}&\displaystyle\left(D(\mu_{t}\|\pi)-D(\mu%
_{0}\|\pi)\right)\\
&\displaystyle=\ \ \ \int_{M}\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,%
\right)d\mu(x)\\
&\displaystyle\ \ \ -\int_{M}\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,%
\right)d\mu_{eq}(x)\\
&\displaystyle\ \ \ +\int_{M}\frac{1}{t}\left[\log\left(t\frac{d\mu}{d\pi}(x)+%
(1-t)\rho(x)\,\right)-\log\rho(x)\right]\rho(x)\,d\pi(x).\end{split}$$
(4.6)
For the first two terms
on the right-hand side of
4.6
we notice that, when $t\searrow 0$,
$$\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,\right)\to\log\rho(x).$$
Since each term is
integrable with respect to $\mu$ and $\mu_{eq}$
for $0<t<1$
we can use
the increasing monotone convergence theorem
where $\rho(x)\geq\frac{d\mu}{d\pi}(x)$ and
the decreasing monotone convergence theorem where
$\rho(x)<\frac{d\mu}{d\pi}(x)$.
We could have also said that
$\left|\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,\right)\right|$ is bounded by
$\left|\log\left(\frac{1}{2}\frac{d\mu}{d\pi}(x)+\frac{1}{2}\rho(x)\,\right)%
\right|+\left|\log\left(\rho(x)\right)\right|$
for $0<t\leq\frac{1}{2}$ and use
the dominated convergence theorem.
For the last term in
4.6
we see that
$$\frac{1}{t}\left[\log\left(t\frac{d\mu}{d\pi}(x)+(1-t)\rho(x)\,\right)-\log%
\rho(x)\right]\rho(x)\nearrow\left[\frac{d\mu}{d\pi}(x)-\rho(x)\right].$$
as $t\searrow 0$ and since each term is integrable
with respect to $\pi$
we can use the monotone
convergence theorem again to get a zero limit.
∎
Theorem 4.9 (Minimizer of F).
Let $\rho$ be a positive differentiable
solution to
$$\begin{split}\displaystyle\Delta\log\rho&\displaystyle=\beta\,\mu_{eq}-\beta%
\Lambda\\
\displaystyle d\mu_{eq}&\displaystyle=\rho\,d\pi.\end{split}$$
(4.7)
Then $F(\mu_{eq})<F(\mu)$
for every $\mu\in\mathcal{P}(M)$ different
from $\mu_{eq}$.
In particular, the solution to 4.7
is unique.
Proof.
As in Lemma 4.8 let $\mu$ be
any probability measure different from $\mu_{eq}$
such that $F(\mu)<\infty$.
Define
$$\mu_{t}=t\mu+(1-t)\mu_{eq},\ \ t\in[0,1].$$
Take the equality
$$\Delta\log\rho=\beta\,\mu_{eq}-\beta\Lambda,$$
multiply it by $G(x,y)$ and integrate in one variable
to get
$$-\log\rho(y)+\int_{M}\log\rho(x)\,d\Lambda(x)=\beta\int_{M}G(x,y)\rho(x)\,d\pi%
(x).$$
Then, we have that
$$\displaystyle\frac{d}{dt}$$
$$\displaystyle F(\mu_{t})|_{t=0}$$
$$\displaystyle=\int_{M\times M}G(x,y)d\mu_{eq}(x)\,\left(d\mu(y)-d\mu_{eq}(y)%
\right)+\frac{1}{\beta}\int_{M\times M}\log\rho(y)\left(d\mu(y)-d\mu_{eq}(y)\right)$$
$$\displaystyle=\int_{M}\left(\int_{M}G(x,y)\rho(x)\,d\pi(x)+\frac{1}{\beta}\log%
\rho(y)\right)\left(d\mu(y)-d\mu_{eq}(y)\right)$$
$$\displaystyle=\frac{1}{\beta}\int_{M}\left(\int_{M}\log\rho(x)\,d\Lambda(x)%
\right)\left(d\mu(y)-d\mu_{eq}(y)\right)$$
$$\displaystyle=\frac{1}{\beta}\left(\int_{M}\log\rho(x)\,d\Lambda(x)\right)%
\left(\int_{M}\left(d\mu(y)-d\mu_{eq}(y)\right)\right)=0.$$
This implies, due to the strict convexity of $F(\mu_{t})$
in $t$, that
$$F(\mu_{eq})<F(\mu).$$
∎
Remark 4.10 (Scalar curvature relation).
We shall suppose that $\chi(M)$,
the Euler characteristic of $M$, is different from
zero.
If $\bar{g}$ is any metric, we denote by $R_{\bar{g}}$
the scalar curvature of $\bar{g}$.
Choose
$$d\Lambda=\frac{R_{g}\,d\pi}{4\pi\chi(M)}.$$
It can be seen that if $\bar{g}=\rho g$,
where $\int_{M}\rho\,d\pi=1$, then
$$\Delta\log\rho=R_{g}d\pi-R_{\bar{g}}\rho\,d\pi.$$
With this identity we can prove that
$\rho$ is a solution to
$$\begin{split}\displaystyle R_{\bar{g}}&\displaystyle=\left(4\pi\chi(M)+\beta%
\right)R_{g}\rho^{-1}-\beta\\
\displaystyle\bar{g}&\displaystyle=\rho\,g\end{split}$$
if and only if
$\rho$ is a solution to
$$\begin{split}\displaystyle\Delta\log\rho&\displaystyle=\beta\,\mu_{eq}-\beta%
\Lambda\\
\displaystyle d\mu_{eq}&\displaystyle=\rho\,d\pi.\end{split}$$
In particular, if $\chi(M)<0$ and
$\beta=-4\pi\chi(M)$
then
$\bar{g}$ satisfies
$$R_{\bar{g}}=4\pi\chi(M),$$
i.e. $\bar{g}$ is a metric with constant curvature.
4.3 Relation to other articles
Suppose that $l$ is a not necessarily finite measure
on the Polish space $M$.
Let $V:M\to(-\infty,\infty]$
and
$G:M\times M\to(-\infty,\infty]$
be lower semicontinuous functions with $G$ symmetric
and such that $(x,y)\mapsto G(x,y)+V(x)+V(y)$
is bounded from below.
Define $H_{n}:M^{n}\to(-\infty,\infty]$ by
$$H_{n}(x_{1},...,x_{n})=\sum_{i<j}^{n}G(x_{i},x_{j})+n\sum_{i=1}^{n}V(x_{i})$$
and $W:\mathcal{P}(M)\to(-\infty,\infty]$
by
$$W(\mu)=\frac{1}{2}\int_{M\times M}\left(G(x,y)+V(x)+V(y)\right)d\mu(x)d\mu(y).$$
Take a sequence $\{\beta_{n}\}_{n\in\mathbb{N}}$
such that $\beta_{n}\to\infty$ and
let $\gamma_{n}$ be the Gibbs measure defined by
$$d\gamma_{n}=e^{-\frac{\beta_{n}}{n}H_{n}}dl^{\otimes_{n}}.$$
We shall give some hypothesis that imply
that $\gamma_{n}$ satisfies
a Laplace principle.
The first example is the most straightforward.
Proposition 4.11 (Weakly confining case).
Take $\beta_{n}=n$. Suppose that
$\bullet$ $\int_{M}e^{-V}dl<\infty$,
$\bullet$ the function
$(x,y)\mapsto G(x,y)+V(x)+V(y)$
is bounded from below,
$\bullet$ $G(x,y)+V(x)+V(y)\to\infty$ when
$x,y\to\infty$ at the same time, and
$\bullet$
for every $\mu\in\mathcal{P}(M)$
such that $W(\mu)<\infty$,
there exists a sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures absolutely continuous
with respect to $l$ such that
$\mu_{n}\to\mu$ and $W(\mu_{n})\to W(\mu)$.
Then, for every bounded continuous function
$f:\mathcal{P}(M)\to\mathbb{R}$
we have
$$\frac{1}{n^{2}}\log\,\int_{M^{n}}e^{-n^{2}f\circ i_{n}}d\gamma_{n}\,%
\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right%
)+W\left(\mu\right)\}.$$
Proof.
Assume $\int_{M}e^{-V}dl=1$
for simplicity. We notice that
$$d\gamma_{n}=e^{-\left(\sum_{i<j}^{n}G(x_{i},x_{j})+(n-1)\sum_{i=1}^{n}V(x_{i})%
\right)}d(e^{-V}l)^{\otimes_{n}}.$$
If we define
$$\tilde{G}(x,y)=G(x,y)+V(x)+V(y)$$
and
$$W_{n}(x_{1},...,x_{n})=\frac{1}{n^{2}}\sum_{i<j}^{n}\tilde{G}(x_{i},x_{j})$$
we have
$$d\gamma_{n}=e^{-n^{2}W_{n}}d(e^{-V}l)^{\otimes_{n}}.$$
We now prove that $\{W_{n}\}_{n\in\mathbb{N}}$
satisfies
the conditions necessary to apply
Theorem 1.3.
Lower and upper limit assumption, (H1) and (H2).
By hypotheses, $\tilde{G}$ is lower semicontinuous
and bounded from below.
We can apply Lemma 2.1 to get
that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence and
that $(\{W_{n}\}_{n\in\mathbb{N}},W)$ satisfies
the lower limit assumption (H1)
and the upper limit assumption (H2).
Regularity assumption (H3).
Take $\mu\in\mathcal{P}(M)$ such that
$W(\mu)<\infty$. Then, by hypothesis,
there exists a sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures absolutely continuous
with respect to $l$ such that $\mu_{n}\to\mu$ and
$W(\mu_{n})\to W(\mu)$. As $W(\mu)<\infty$
we can assume $W(\mu_{n})<\infty$ for every
$n\in\mathbb{N}$. Fix $n\in\mathbb{N}$.
We want to prove that $\mu_{n}$
is absolutely continuous with respect to
the measure defined by $e^{-V}dl$.
For this it is enough
to notice that
$\mu_{n}(\{x\in M:\ V(x)=\infty\})=0$.
We can see that the set
$\{(x,y)\in M\times M:\ V(x)=\infty\mbox{ and }V(y)=\infty\}$
is included in the set
$\{(x,y)\in M\times M:\ G(x,y)+V(x)+V(y)=\infty\}$. The latter
has zero measure because $W(\mu)<\infty$ and
we conclude by definition of product measure.
Confining assumption (H4).
Using that
$\tilde{G}(x,y)\to\infty$ when
$x,y\to\infty$ at the same time
and Lemma 2.3
we get that $\{W_{n}\}_{n\in\mathbb{N}}$ satisfies
the confining assumption (H4).
We can finally
apply Theorem 1.3.
∎
The second example is
related to the article this
work is inspired on, i.e. [12].
Proposition 4.12 (Strongly confining case).
Suppose that
$\bullet$
There exists $\xi>0$
such that $\int_{M}e^{-\xi V}dl<\infty$,
$\bullet$
$V$ is bounded from below,
$\bullet$
there exists $\epsilon\in[0,1)$ such that
$(x,y)\mapsto G(x,y)+\epsilon V(x)+\epsilon V(y)$
is
bounded from below,
$\bullet$
the function
$G(x,y)+V(x)+V(y)$
tends to infinity when $x,y\to\infty$
at the same time, and
$\bullet$
for every $\mu\in\mathcal{P}(M)$
such that $W(\mu)<\infty$,
there exists a sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures absolutely continuous
with respect to $l$ such that
$\mu_{n}\to\mu$ and $W(\mu_{n})\to W(\mu)$.
Then, for every bounded continuous function
$f:\mathcal{P}(M)\to\mathbb{R}$
we have
$$\frac{1}{n\beta_{n}}\log\,\int_{M^{n}}e^{-n\beta_{n}f\circ i_{n}}d\gamma_{n}\,%
\xrightarrow[\>n\to\infty\>]{}\,-\inf_{\mu\in\mathcal{P}(M)}\{f\left(\mu\right%
)+W\left(\mu\right)\}.$$
Proof.
We can assume $\int_{M}e^{-\xi V}dl=1$ for
simplicity.
Then we can write
$$d\gamma_{n}=e^{-\frac{\beta_{n}}{n}\left(\sum_{i<j}^{n}G(x_{i},x_{j})+\left(n-%
\frac{n}{\beta_{n}}\xi\right)\sum_{i=1}^{n}V(x_{i})\right)}d(e^{-\xi V}l)^{%
\otimes_{n}}.$$
which may only make sense
for $n$ large enough
due to some positive and negative
infinities. If we define
$$G^{n}(x,y)=G(x,y)+\frac{1}{n-1}\left(n-\frac{n}{\beta_{n}}\xi\right)V(x)+\frac%
{1}{n-1}\left(n-\frac{n}{\beta_{n}}\xi\right)V(y)$$
and
$$W_{n}(x_{1},...,x_{n})=\frac{1}{n^{2}}\sum_{i<j}^{n}G^{n}(x_{i},x_{j})$$
we have
$$d\gamma_{n}=e^{-n\beta_{n}W_{n}}d(e^{-\xi V}l)^{\otimes_{n}}.$$
Now we can try to apply Theorem
1.3
to get the Laplace principle.
Define
$$G_{1}(x,y)=G(x,y)+\epsilon V(x)+\epsilon V(y),\ \ \ W^{1}_{n}(x_{1},...,x_{n})%
=\frac{1}{n^{2}}\sum_{i<j}^{n}G_{1}(x_{i},x_{j}),$$
$$G_{2}(x,y)=(1-\epsilon)V(x)+(1-\epsilon)V(y),\ \ \ W^{2}_{n}(x_{1},...,x_{n})=%
\frac{1}{n^{2}}\sum_{i<j}^{n}G_{2}(x_{i},x_{j})$$
and
$$a_{n}=\frac{1}{1-\epsilon}\left(\frac{1}{n-1}\left(n-\frac{n}{\beta_{n}}\xi%
\right)-\epsilon\right)\to 1.$$
This definitions allow us to write
$$W_{n}=W^{1}_{n}+a_{n}W^{2}_{n}.$$
We start by proving the
lower limit assumption (H1)
and the upper limit assumption (H2).
Lower and upper limit assumption, (H1) and (H2).
By the hypotheses,
we can see that $G_{1}$ and $G_{2}$ are lower
semicontinuous functions bounded from below.
Then, we can apply Lemma 2.1 about
the two-body interaction to get that
$\{W^{1}_{n}\}_{n\in\mathbb{N}}$
and $\{W^{2}_{n}\}_{n\in\mathbb{N}}$
are stable sequences and if we define
the lower semicontinuous functions
$W^{1}(\mu)=\frac{1}{2}\int_{M\times M}G_{1}(x,y)d\mu(x)d\mu(y)$
and
$W^{2}(\mu)=\frac{1}{2}\int_{M\times M}G_{2}(x,y)d\mu(x)d\mu(y)$,
then
$(\{W^{1}_{n}\}_{n\in\mathbb{N}},W^{1})$
and $(\{W^{2}_{n}\}_{n\in\mathbb{N}},W^{2})$
satisfy the lower limit assumption (H1)
and the upper limit assumption (H2).
Then, as $a_{n}>0$ for $n$ large enough,
we get that $\{W_{n}\}_{n\in\mathbb{N}}$
is a stable sequence for $n$ large enough.
Noticing that
$$W^{1}(\mu)+W^{2}(\mu)=W(\mu)=\frac{1}{2}\int_{M\times M}\left(G(x,y)+V(x)+V(y)%
\right)d\mu(x)d\mu(y).$$
we obtain,
by Remark 1.6 and 1.7
about the linear combination of stable sequences,
that $(\{W_{n}\}_{n\in\mathbb{N}},W)$
satisfies the lower limit assumption (H1)
and the upper limit assumption (H2).
Confining assumption (H4).
By Lemma 2.3 about
the confining
assumption in the k-body interaction
and by the
fact that $G(x,y)+V(x)+V(y)\to\infty$
when $x,y\to\infty$ at the same time,
we get that
$\{W^{1}_{n}+W^{2}_{n}\}_{n\in\mathbb{N}}$ satisfies
the confining assumption (H4), and by
Remark 1.9,
$\{W_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4).
Regularity assumption (H3).
By an argument similar to the one given
in the proof of Proposition 4.11 we can prove
the regularity assumption (H3) for $W$.
We have proven the conditions to apply Theorem
1.3.
∎
5 About Fekete points:
case of infinite $\beta_{n}$
Instead of defining the sequence of probability
measures
$$d\mathbb{P}_{n}=\frac{1}{Z_{n}}e^{-n\beta_{n}W_{n}}d\pi^{\otimes_{n}},$$
we will take $\beta_{n}=\infty$. Intuitively
the important points
in this case will be the points that minimize
$W_{n}$. The empirical measures associated
to those points should converge to
the minimizer of $W$.
More precisely, we have the following result.
Theorem 5.1 (Almost minimizers converge
to minimizers).
Let $\{W_{n}\}_{n\in\mathbb{N}}$ be a stable
sequence and let $W:\mathcal{P}(M)\to\mathbb{R}$
be some
function such that
$(\{W_{n}\}_{n\in\mathbb{N}},W)$
satisfies the lower and upper limit assumptions
(H1) and (H2).
For each $n$ take $x^{n}\in M^{n}$ such that
$$\epsilon_{n}:=|W_{n}(x^{n})-\inf(W_{n})|\xrightarrow[\>n\to\infty\>]{}0.$$
Then, if $\mu\in\mathcal{P}(M)$ is a limit point of
$\{i_{n}(x^{n})\}_{n\in\mathbb{N}}$,
i.e. if there exists an increasing
sequence of natural numbers
$\{n_{j}\}_{j\in\mathbb{N}}$ such that
$i_{n_{j}}(x^{n_{j}})\to\mu$,
we get
$$W(\mu)=\inf(W)\ \ \ \mbox{ and }\ \ \ W_{n_{j}}(x^{n_{j}})\xrightarrow[\>j\to%
\infty\>]{}W(\mu).$$
In particular, if $\{W_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4)
we get that
$$\inf(W_{n})\xrightarrow[\>n\to\infty\>]{}\inf(W)$$
and if additionally $W$ has a unique minimizer $\mu$
we get
$$i_{n}(x^{n})\xrightarrow[\>n\to\infty\>]{}\mu.$$
Proof.
By the upper limit assumption (H2)
we have that,
for every probability measure $\alpha$ on $M$,
$$\limsup_{n\to\infty}\mathbb{E}_{\alpha^{\otimes_{n}}}[W_{n}]\leq W(\alpha).$$
Then,
as $\inf(W_{n})\leq\mathbb{E}_{\alpha^{\otimes_{n}}}[W_{n}]$,
we get
$$\limsup_{n\to\infty}\left\{\inf(W_{n})\right\}\leq W(\alpha)$$
for every $\alpha\in\mathcal{P}(M)$. This implies that
$$\limsup_{n\to\infty}\left\{\inf(W_{n})\right\}\leq\inf(W).$$
But,
$$W_{n}(x^{n})=\inf(W_{n})+\epsilon_{n},$$
so we also have
$$\limsup_{n\to\infty}W_{n}(x^{n})\leq\inf(W).$$
Now, let $\{x^{n_{j}}\}_{j\in\mathbb{N}}$
be the subsequence such that
$$i_{n_{j}}(x^{n_{j}})\xrightarrow[\>j\to\infty\>]{}\mu.$$
By the lower limit assumption (H1)
and Remark
1.5
we know that
$$W(\mu)\leq\liminf_{j\to\infty}W_{n_{j}}(x^{n_{j}}),$$
and we get
$$W(\mu)\leq\liminf_{j\to\infty}W_{n_{j}}(x^{n_{j}})\leq\limsup_{n\to\infty}W_{n%
}(x^{n})\leq\inf(W).$$
∎
Notice that we have proved that even if
$W$ were not
measurable and
$\{W_{n}\}_{n\in\mathbb{N}}$
satisfies the confining assumption (H4)
then $W$ attains its infimum.
6 Final comments: About Laplace-Varadhan lemma
As mentioned in the introduction, Theorem
1.1 has a nice
corollary as some kind of Laplace-Varadhan lemma
for Sanov’s theorem.
Corollary 6.1 (Generalized Laplace-Varadhan lemma
for Sanov’s theorem).
Define
$\mathbb{P}_{n}=i_{n}(\pi^{\otimes_{n}})$
and denote by $\mathbb{E}_{n}$ the expected
value with respect to $\mathbb{P}_{n}$.
Let $\{f_{n}:\mathcal{P}(M)\to(-\infty,\infty]\}_{n\in\mathbb{N}}$ be a sequence
of measurable functions uniformly bounded
from below and suppose
we have a
measurable function
$f:\mathcal{P}(M)\to(-\infty,\infty]$ such that
$\bullet$ for every $\mu\in\mathcal{P}(M)$,
$\limsup_{n\to\infty}\mathbb{E}_{i_{n}(\mu^{\otimes_{n}})}[f_{n}]\leq f(\mu),\ and$
$\bullet$ for every sequence of probability
measures
$\{\mu_{n}\}_{n\in\mathbb{N}}$ that converges
to some $\mu\in\mathcal{P}(M)$,
$$\liminf_{n\to\infty}f_{n}(\mu_{n})\geq f(\mu).$$
Then,
$$\frac{1}{n}\log\mathbb{E}_{n}[e^{-nf_{n}}]\xrightarrow[\>n\to\infty\>]{}-\inf_%
{\mu\in\mathcal{P}(M)}\{f(\mu)+D(\mu\|\pi)\}$$
Proof.
To see it as a corollary of Theorem
1.1 we define
$W_{n}=f_{n}\circ i_{n}$
and
$W=f$. We need to prove the
lower limit assumption (H1) and the
upper limit assumption (H2).
Lower limit assumption (H1).
Notice that $\tilde{W}_{n}\geq f_{n}$,
where $\tilde{W}_{n}$ is defined as
in 1.4.
Take a sequence of probability measures
$\{\mu_{n}\}_{n\in\mathbb{N}}$ that converges
to some $\mu\in\mathcal{P}(M)$. We get
$$\liminf_{n\to\infty}\tilde{W}_{n}(\mu_{n})\geq\liminf_{n\to\infty}f_{n}(\mu_{n%
})\geq f(\mu)$$
and this proves the lower limit assumption (H1).
Upper limit assumption (H2). Take
$\mu\in\mathcal{P}(M)$. Then,
$\mathbb{E}_{i_{n}(\mu^{\otimes_{n}})}[f_{n}]=\mathbb{E}_{\mu^{\otimes_{n}}}[f_%
{n}\circ i_{n}]=\mathbb{E}_{\mu^{\otimes_{n}}}[W_{n}]$ and the
hypothesis
$\limsup_{n\to\infty}\mathbb{E}_{i_{n}(\mu^{\otimes_{n}})}[f_{n}]\leq f(\mu)$ implies the
upper limit assumption (H2).
∎
In fact, it is interesting to notice
that we could have proven this corollary first
and use it to prove Theorem
1.1 by making
$f_{n}=\tilde{W}_{n}$ and $f=W$.
Acknowledgments.
I would like to thank Raphaël Butez,
Djalil Chafaï and Adrien Hardy for their useful remarks.
7 Appendices
A Convergence of
probability measures
Let $M$ be a Polish space,
i.e. a separable topological space
metrizable by a complete metric.
Endow it with the Borel $\sigma$-algebra associated
to this topology, i.e. the $\sigma$-algebra
generated by the topology.
We will endow $\mathcal{P}(M)$, the space of probability
measures on $M$, with a topology such that a
sequence
$\{\mu_{n}\}_{n\in\mathbb{N}}$
in $\mathcal{P}(M)$ converge to
$\mu\in\mathcal{P}(M)$ if and only if
$\int_{M}fd\mu_{n}\to\int_{M}fd\mu$ for every
bounded continuous function $f:M\to\mathbb{R}$.
To define this topology,
for any bounded measurable function
$f:M\to\mathbb{R}$
we define the function
$$\hat{f}:\mathcal{P}(M)\to\mathbb{R},\ \ \ \ \mu\mapsto\int_{M}fd\mu.$$
(7.1)
The topology on $\mathcal{P}(M)$ we will consider
is the smallest such that
$\hat{f}$ is continuous for every
bounded continuous function $f:M\to\mathbb{R}$.
With this topology, $\mathcal{P}(M)$ is also a Polish space
(see [16, Section 1.4]
and [16, Section 1.6] for a proof).
This is called weak or vague topology.
We need
the following characterization of the
convergence.
Proposition 7.1 (Convergence in $\mathcal{P}(M)$).
There exists a countable family
of bounded
continuous functions
$\{f_{i}:M\to\mathbb{R}\}_{i\in\mathbb{N}}$ such that
given any sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures on $M$
and $\mu\in\mathcal{P}(M)$
we have that
$$\mu_{n}\xrightarrow[\>n\to\infty\>]{}\mu\ \mbox{ if and only if }\ \forall i%
\in\mathbb{N},\int_{M}f_{i}d\mu_{n}\xrightarrow[\>n\to\infty\>]{}\int_{M}f_{i}%
d\mu.$$
Proof.
As $M$ is a separable metrizable
topological space, there exists
a metric $d$ that induces its topology and such that
the space of bounded uniformly continuous
functions is separable
with respect to the topology of uniform convergence
associated with $d$.
See [18, Lemma 9.1.4.]
or [16, Lemme 1.2 and 1.3] for
a proof. Then we take $\{f_{i}\}_{i\in\mathbb{N}}$
as a dense sequence
of bounded
uniformly continuous functions
to finish the proof using Portmanteau theorem.
∎
Corollary 7.2 (Equality of probability
measures).
There exists a countable family
of bounded
continuous functions
$\{f_{i}:M\to\mathbb{R}\}_{i\in\mathbb{N}}$
that determines probability measures, i.e. such that
given $\mu$ and $\nu$ any two probability
measures on $M$
$$\mu=\nu\ \mbox{ if and only if }\ \forall i\in\mathbb{N},\int_{M}f_{i}d\mu=%
\int_{M}f_{i}d\nu.$$
A very important concept in analysis is
the one of relatively compact.
A subset $A$ of a topological space $X$
is said to be relatively compact (in $X$)
if its closure is compact.
If $X$ is metrizable this is equivalent to say that
every sequence in $A$ has a subsequence that converges
(to some point in $X$). If $\{x_{n}\}_{n\in\mathbb{N}}$
is a sequence in $X$ then the set
$A=\{x_{n}:\ n\in\mathbb{N}\}$ is relatively compact
if and only if for every subsequence of
$\{x_{n}\}_{n\in\mathbb{N}}$
there exists a further subsubsequence that converges
(to some point in $X$).
This tells us that if we can characterize
the relatively compact sets then
we are able to know when we can extract subsequences.
Theorem 7.3 (Prokhorov’s theorem).
Let $M$
be a Polish space.
A subset $A\subset\mathcal{P}(M)$ is relatively compact
if and only if it is tight, i.e. for every $\epsilon>0$
there exists a compact set $K\subset M$ such that
$$\forall\mu\in A,\ \mu(K)\geq 1-\epsilon.$$
Proof.
See [7, Section 5] or
[16, Theorem 1.7].
∎
The following two propositions are related to the
notion
of tightness and are important for
taking subsequences on the proofs of the Laplace
principles.
Proposition 7.4 (Tightness and the confining
assumption).
Let $M$ be a Polish space and
$\{f_{n}:M\to(-\infty,\infty]\}_{n\in\mathbb{N}}$ a sequence of measurable functions
uniformly
bounded from below such
that,
for all $C\in\mathbb{R}$, every sequence $\{x_{n}\}_{n\in\mathbb{N}}$
of points in $M$
with $f_{n}(x_{n})\leq C$ for every $n\in\mathbb{N}$
is relatively compact. Then, for all
$C\in\mathbb{R}$, every sequence
$\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures on $M$
such that $\int_{M}f_{n}d\mu_{n}\leq C$ for every
$n\in\mathbb{N}$
is tight.
Proof.
By adding a constant we
can suppose that $f_{n}$ is positive
for every $n\in\mathbb{N}$.
Take $C>0$ and a
sequence $\{\mu_{n}\}_{n\in\mathbb{N}}$
of probability measures on $M$
such that $\int_{M}f_{n}d\mu_{n}\leq C$ for every
$n\in\mathbb{N}$.
Let $\epsilon\in(0,1)$.
We look for a compact set $K\subset M$ such that
$$\mu_{n}(K)\geq 1-\epsilon$$
for every $n\in\mathbb{N}$.
Fix $n\in\mathbb{N}$ for a moment.
Take $A_{n}=\{x\in M:\ f_{n}(x)\leq\frac{2C}{\epsilon}\}$. Then,
by the positivity of $f_{n}$ we have
$$\frac{2C}{\epsilon}\mu_{n}(A_{n}^{c})\leq\int_{A_{n}^{c}}f_{n}d\mu_{n}\leq C.$$
So,
$$\mu_{n}(A_{n})\geq 1-\frac{\epsilon}{2}.$$
Take a compact set $K_{n}$ such that
$$\mu_{n}(K_{n})\geq 1-\frac{\epsilon}{2}.$$
Then, consider the set
$$A=\bigcup_{n\in\mathbb{N}}\left(K_{n}\cap A_{n}\right).$$
We affirm that $A$ is relatively compact. For this,
take a sequence $\{x_{k}\}_{k\in\mathbb{N}}$ in $A$.
We want to prove that it has a convergent
subsequence.
If there exists $N\in\mathbb{N}$ such that
$x_{k}\in\bigcup_{n=0}^{N}\left(K_{n}\cap A_{n}\right)$
for every $k\in\mathbb{N}$,
then there exists $P\in\mathbb{N}$
such that
$x_{k}\in K_{P}\cap A_{P}$ for an infinite number of $k$.
As $K_{P}$ is compact we can find a convergent subsequence.
If there does not exists such $N\in\mathbb{N}$ then
we can find a subsequence
$\{x_{k_{i}}\}_{i\in\mathbb{N}}$
such that there exists an increasing sequence of
natural numbers $\{n_{i}\}_{i\in\mathbb{N}}$ with
$x_{k_{i}}\in A_{n_{i}}$ for every
$i\in\mathbb{N}$.
We can think of the subsequence
$\{x_{k_{i}}\}_{i\in\mathbb{N}}$ as a subsequence
of some other sequence
$\{y_{n}\}_{n\in\mathbb{N}}$
that satisfies $y_{n_{i}}=x_{k_{i}}$
for every $i\in\mathbb{N}$ and
$f_{n}(y_{n})\leq C$ for the rest of $n$.
So, as
$f_{n}(y_{n})\leq 2C/\epsilon$ for every
$n\in\mathbb{N}$,
we know by
hypothesis that
$\{y_{n}\}_{n\in\mathbb{N}}$ is relatively compact
and then
that $\{x_{k_{i}}\}_{i\in\mathbb{N}}$
is relatively compact and
we can find a convergent subsequence.
We take $K=\bar{A}$ the closure of $A$ and notice that
$$\mu_{n}(K)\geq 1-\epsilon$$
for every $n\in\mathbb{N}$.
∎
Let $\chi\in\mathcal{P}(\mathcal{P}(M))$.
We shall say that $\mu\in\mathcal{P}(M)$
is the expected value
of $\chi$ if one of the following equivalent conditions
is satisfied.
$\bullet$ For every
measurable set $A\subset M$,
$\mathbb{E}_{\chi}\left[\hat{1}_{A}\right]=\mu(A)$.
$\bullet$
For every bounded continuous function
$f:M\to\mathbb{R}$, $\mathbb{E}_{\chi}\left[\hat{f}\right]=\int_{M}fd\mu$ .
We are using the notation introduced in
7.1.
Proposition 7.5 (Tightness criterion using expected values).
Let $\{\chi_{\lambda}\}_{\lambda\in\Lambda}$
be a family of probability measures on the space of
probability measures,
i.e. $\chi_{\lambda}\in\mathcal{P}(\mathcal{P}(M))$
for every $\lambda\in\Lambda$,
and let
$\{\mu_{\lambda}\}_{\lambda\in\Lambda}$
be the family of expected values.
Then, $\{\chi_{\lambda}\}_{\lambda\in\Lambda}$
is tight if and only if
$\{\mu_{\lambda}\}_{\lambda\in\Lambda}$ is tight.
Proof.
Suppose
$\{\chi_{\lambda}\}_{\lambda\in\Lambda}$
is tight. That means that
for every $\epsilon>0$ there exists a compact
set $B\subset\mathcal{P}(M)$ such that
$\chi_{\lambda}(B)\geq 1-\frac{\epsilon}{2}$ for every
$\lambda\in\Lambda$. Then, as $B$ is compact,
it is tight. There exists a compact set $K\subset M$
such that $\mu(K)\geq 1-\frac{\epsilon}{2}$ for every
$\mu\in B$. We are going to prove
that $\mu_{\lambda}(K)\geq 1-\epsilon.$
for every $\lambda\in\Lambda$.
In the following we will write explicitly
the variable $\mu$ in the functions
we will take expected value.
Take $\lambda\in\Lambda$. We have
$$\mathbb{E}_{\chi_{\lambda}}\left[1_{B^{c}}(\mu)\,\mu(K^{c})\right]+\mathbb{E}_%
{\chi_{\lambda}}\left[1_{B}(\mu)\,\mu(K^{c})\right]=\mu_{\lambda}(K^{c}).$$
But
$$\mathbb{E}_{\chi_{\lambda}}\left[1_{B^{c}}(\mu)\,\mu(K^{c})\right]\leq\mathbb{%
E}_{\chi_{\lambda}}\left[1_{B^{c}}(\mu)\right]\leq\frac{\epsilon}{2}$$
and
$$\mathbb{E}_{\chi_{\lambda}}\left[1_{B}(\mu)\,\mu(K^{c})\right]\leq\mathbb{E}_{%
\chi_{\lambda}}\left[1_{B}(\mu)\,\frac{\epsilon}{2}\right]\leq\frac{\epsilon}{%
2}.$$
So,
$$\mu_{\lambda}(K^{c})\leq\epsilon.$$
Now suppose
$\{\mu_{\lambda}\}_{\lambda\in\Lambda}$ is tight
and take $\epsilon>0$.
Take a sequence of positive numbers
$\{\epsilon_{n}\}_{n\in\mathbb{N}}$ such that
$\sum_{n\in\mathbb{N}}\epsilon_{n}=\epsilon$. For
each $n\in\mathbb{N}$
take $K_{n}\subset M$ compact such that
$\mu_{\lambda}(K_{n})\geq 1-\epsilon_{n}^{2}$ for every
$\lambda\in\Lambda$. Then, by Chebyshev’s inequality,
$$\chi_{\lambda}\left(\{\mu\in\mathcal{P}(M):\ \mu(K_{n}^{c})\geq\epsilon_{n}\}%
\right)\leq\frac{1}{\epsilon_{n}}\mathbb{E}_{\chi_{\lambda}}[\mu(K_{n}^{c})]=%
\frac{1}{\epsilon_{n}}\mu_{\lambda}(K_{n}^{c})\leq\epsilon_{n}.$$
So,
$$\chi_{\lambda}\left(\bigcup_{n\in\mathbb{N}}\{\mu\in\mathcal{P}(M):\ \mu(K_{n}%
^{c})\geq\epsilon_{n}\}\right)\leq\epsilon.$$
Then
$$B=\bigcap_{n\in\mathbb{N}}\{\mu\in\mathcal{P}(M):\ \mu(K_{n})>1-\epsilon_{n}\}$$
satisfies
$$\chi_{\lambda}(B)\geq 1-\epsilon$$
for every $\lambda\in\Lambda$.
By definition $B$ is tight and, by Prokhorov’s theorem
7.3, it is relatively
compact. Then its closure $\bar{B}$ is a compact
subset of $\mathcal{P}(M)$ that satisfies
$$\chi_{\lambda}(\bar{B})\geq 1-\epsilon$$
for every $\lambda\in\Lambda$.
∎
B Variational representation
We shall prove Lemma 3.1.
Let $(E,\mu)$ be a Polish probability space and $g:E\to(-\infty,\infty]$
a measurable function bounded from below.
We want to prove that
$$\log\mathbb{E}_{\mu}\left[e^{-g}\right]=-\inf_{\tau\in\mathcal{P}(E)}\left\{%
\mathbb{E}_{\tau}\left[g\right]+D(\tau\|\mu)\right\}.$$
(7.2)
The easiest case is when $g$ is infinite
$\mu$ - almost surely or, equivalently,
$E_{\mu}\left[e^{-g}\right]=0$. In particular,
if
$\tau\in\mathcal{P}(E)$
is absolutely continuous with respect to
$\mu$ then $\mathbb{E}_{\tau}\left[g\right]$ is
infinite. So,
$\mathbb{E}_{\tau}\left[g\right]+D(\tau\|\mu)$
is infinite for every $\tau\in\mathcal{P}(E)$
and, as
$\log\mathbb{E}_{\mu}\left[e^{-g}\right]$
is minus infinity, we have proven
7.2.
Now suppose
$\mathbb{E}_{\mu}\left[e^{-g}\right]>0$.
First, let us prove that, for every $\tau\in\mathcal{P}(E)$,
$$-\log\mathbb{E}_{\mu}\left[e^{-g}\right]\leq\mathbb{E}_{\tau}\left[g\right]+D(%
\tau\|\mu).$$
(7.3)
If $D(\tau\|\mu)=\infty$
or $\mathbb{E}_{\tau}\left[g\right]=\infty$,
the inequality is clearly true.
Suppose $D(\tau\|\mu)<\infty$ and
$\mathbb{E}_{\tau}\left[g\right]<\infty$.
Then $d\tau=\rho\,d\mu$ for some
measurable function
$\rho:E\to[0,\infty)$ and $g$ is finite
$\tau$ - almost surely.
As the support of $\tau$ is contained in
the support of $\mu$ we have
$$-\log\mathbb{E}_{\mu}\left[e^{-g}\right]\leq-\log\mathbb{E}_{\tau}\left[e^{-g}%
e^{-\log\rho}\right]$$
where $\log\rho$ is defined as $0$ if
$\rho=0$ for simplicity.
By Jensen’s inequality,
$$-\log\mathbb{E}_{\tau}\left[e^{-g}e^{-\log\rho}\right]\leq\mathbb{E}_{\tau}%
\left[g+\log\rho\right]=\mathbb{E}_{\tau}\left[g\right]+D(\tau\|\mu),$$
and we get 7.3.
Now, as
$\mathbb{E}_{\mu}\left[e^{-g}\right]>0$
we can define $\tau\in\mathcal{P}(E)$ by
$d\tau=\frac{e^{-g}}{\mathbb{E}_{\mu}\left[e^{-g}\right]}\,d\mu$
and see that
$$-\log\mathbb{E}_{\mu}\left[e^{-g}\right]=\mathbb{E}_{\tau}\left[g\right]+D(%
\tau\|\mu)$$
which proves that the infimum in
7.2 is attained.
C Measurability of $\tilde{W}_{n}$
We will prove that the applications $\tilde{W}_{n}$ defined
in 1.4 are measurable.
Proposition 7.6 (Measurability of $\tilde{W}_{n}$).
Take a symmetric measurable function
$$W_{n}:M^{n}\to\mathbb{R}.$$
Then $\tilde{W}_{n}$ defined by
$$\tilde{W}_{n}:\mathcal{P}(M)\to(-\infty,\infty]$$
$$\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}\to W_{n}(x_{1},...,x_{n})$$
and $\infty$ otherwise,
is measurable.
Proof.
The application
$$i_{n}:M^{n}\to\mathcal{P}(M)$$
$$(x_{1},...,x_{n})\mapsto\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$$
is continuous
and closed. To see why it is closed take a closed
subset $F\subset M^{n}$ and a sequence
$\{z_{i}\}_{i\in\mathbb{N}}$ in $F$ such that
$i_{n}(z_{i})$ converge to some
$\mu\in\mathcal{P}(M)$ as $i\to\infty$.
We have to prove that $\mu\in i_{n}(F)$.
As the sequence is tight
we can get a compact subset $K\subset M$ such that
$i_{n}(z_{i})(K)>1-\frac{1}{n}$ for every
$i\in\mathbb{N}$. This
implies, by the nature of $i_{n}(z_{i})$, that $i_{n}(z_{i})(K)=1$ for every $i\in\mathbb{N}$
or, what is the same, $z_{i}\in K^{n}$
for every $i\in\mathbb{N}$.
So, $\{z_{i}\}_{i\in\mathbb{N}}$
has a subsequence that converge to
some $z\in F$ and, by continuity,
$i_{n}(z)=\mu$. In particular, $\mu\in i_{n}(F)$.
Take $S(M^{n})=M^{n}/S_{n}$ where
$S_{n}$ is the symmetric group of $n$ symbols that acts
continuously on $M^{n}$. Let $p:M^{n}\to S(M^{n})$
be the canonical projection map. We can see that the
unique application
$$\tilde{i}_{n}:S(M^{n})\to\mathcal{P}(M),$$
such that $\tilde{i}_{n}\circ p=i_{n}$, is continuous. In fact,
it is an homeomorphism onto its image by the
continuity and the
closedness of $i_{n}$, and its image
$\tilde{i}_{n}(S(M^{n}))=i_{n}(M^{n})$ is closed.
So, what we need to prove is that the
unique function
$$w_{n}:S(M^{n})\to\mathbb{R},$$
such that $w_{n}\circ p=W_{n}$, is measurable
(with respect to the Borel $\sigma$-algebra on $S(M^{n})$).
For this we shall use the following result.
Lemma 7.7 (Invariant open sets generate
invariant measurable sets).
Let $(X,\tau)$ be a metrizable topological space
and $G$ a finite group
that acts continuously on $X$. Let $\mathcal{B}=\sigma(\tau)$ (the $\sigma$-algebra generated by $\tau$).
Take
$$S_{\tau}=\{A\in\tau:\ \forall g\in G,\ g(A)=A\}$$
and
$$S_{\mathcal{B}}=\{A\in\mathcal{B}:\ \forall g\in G,\ g(A)=A\}.$$
Then
$$\sigma(S_{\tau})=S_{\mathcal{B}}.$$
Proof of the lemma.
As $S_{\mathcal{B}}$ is a $\sigma$-algebra
and $S_{\tau}\subset S_{\mathcal{B}}$,
it is clear that
$\sigma(S_{\tau})\subset S_{\mathcal{B}}$.
We want to prove that
$S_{\mathcal{B}}\subset\sigma(S_{\tau})$.
Define
$$F=\left\{A\subset X:\ \bigcup_{g\in G}g(A)\in\sigma(S_{\tau})\right\}.$$
To get $S_{\mathcal{B}}\subset\sigma(S_{\tau})$
it is enough to prove that $\mathcal{B}\subset F$.
Then we can complete the proof by saying that if
$A\in S_{\mathcal{B}}$ then
$A\in F$ (because $\mathcal{B}\subset F$) and
$A=\bigcup_{g\in G}\,g(A)$ which implies
$A\in\sigma(S_{\tau})$ .
By the continuity of the action we know that
$$\tau\subset F.$$
We shall prove that $F$ is a monotone class and that
the $\sigma$-algebra generated by $\tau$ is
the same as the monotone class generated by $\tau$
to conclude that
$\mathcal{B}\subset F$.
To prove that
$F$ is a monotone class we notice that
if $A_{n}\nearrow A$ we have
$\bigcup_{g\in G}g(A_{n})\nearrow\bigcup_{g\in G}g(A)$
by definition of increasing limit and
if $A_{n}\searrow A$ then
$\bigcup_{g\in G}g(A_{n})\searrow\bigcup_{g\in G}g(A)$.
The last fact follows from the general following fact.
$$A_{n}\searrow A\mbox{ and }B_{n}\searrow B\mbox{ implies }A_{n}\cup B_{n}%
\searrow A\cup B$$
(7.4)
This is a consequence
of the following reasoning. If $A_{n}\searrow A$
and $B_{n}\searrow B$
we have that $A_{n}\cup B_{n}$ is decreasing.
To see that
$\bigcap_{n\in\mathbb{N}}\left(A_{n}\cup B_{n}\right)=A\cup B$ we take
$x\in\bigcap_{n\in\mathbb{N}}\left(A_{n}\cup B_{n}\right)$.
So, by definition
$x\in A_{n}\cup B_{n}$ for all $n$ and then
$x\in A_{n}$ for all $n$ or $x\in B_{n}$ for all $n$.
If not there would exist $N_{1}$ and $N_{2}$ such that
$x\notin A_{N_{1}}$ and $x\notin B_{N_{2}}$, so
$x\notin A_{\max(N_{1},N_{2})}\cup B_{\max(N_{1},N_{2})}$.
We conclude that $x\in A$ or $x\in B$.
To see that the monotone class
generated by $\tau$ equals the
$\sigma$-algebra generated by $\tau$
we use the monotone class theorem.
By this theorem it suffices to show that the algebra generated by
$\tau$ is contained in the monotone class generated by $\tau$.
We know that every closed set is a decreasing limit
of open sets
(if $F$ is closed then
$\{x\in X:\ d(x,F)<\frac{1}{n}\}\searrow F$ as $n\to\infty$).
But the algebra generated by $\tau$ is
the smallest family of sets
closed under finite union and under finite intersection
that contains the open and closed sets. We know
that if $A_{n}\searrow A$ and $B_{n}\searrow B$
then $A_{n}\cap B_{n}\searrow A\cap B$ by
definition of limit and we saw before in
7.4 that
$A_{n}\cup B_{n}\searrow A\cup B$. So,
the sets that can be written as a decreasing limit
of open sets is closed under finite union and
finite intersection.
This proves that every element of the algebra
generated by $\tau$ is a decreasing limit of open sets
and thus belongs to the monotone class
generated by $\tau$.
∎
Now, to prove that $w_{n}$ is measurable we can notice
that $w_{n}$ is measurable with respect to
the quotient $\sigma$-algebra, i.e. the largest
$\sigma$-algebra such that $p$ is measurable.
What we can get is that this $\sigma$-algebra
is the Borel $\sigma$-algebra associated to
the quotient topology.
Lemma 7.8 (Relation between quotient topology
and quotient $\sigma$-algebra).
As in Lemma 7.7,
let $(X,\tau)$ be a metrizable topological space
and $G$ a finite group
that acts continuously on $X$.
Denote by $p:X\to X/G$ the canonical projection
map and
let $\bar{\tau}$ be the
quotient topology on $X/G$.
Denote by $\sigma(\bar{\tau})$
the Borel $\sigma$-algebra on $(X/G,\bar{\tau})$,
i.e. the $\sigma$-algebra generated by $\bar{\tau}$.
Consider also
the quotient $\sigma$-algebra $Q$ on $X/G$, i.e.
$A\in Q$ if and only if
$p^{-1}(A)\subset X$ is measurable. Then,
$$\sigma(\bar{\tau})=Q.$$
Proof of the lemma.
We note that the application
$$P:2^{X/G}\to 2^{X}$$
$$A\mapsto p^{-1}(A)$$
is injective due to the surjectivity of $p$.
It is enough to prove that
$P(\sigma(\bar{\tau}))=P(Q)$.
We have that
$$P(\sigma(\bar{\tau}))=\sigma(P(\bar{\tau}))=\sigma(\{A\in\tau:\ \forall g\in G%
,\ g(A)=A\})$$
and
$$P(Q)=\{A\in\mathcal{B}:\ \forall g\in G,\ g(A)=A\}.$$
By Lemma 7.7,
$$\sigma(\{A\in\tau:\ \forall g\in G,\ g(A)=A\})=\{A\in\mathcal{B}:\ \forall g%
\in G,\ g(A)=A\}$$
and we finish the proof.
∎
Applying this lemma to $X=M^{n}$ and $G=S_{n}$
we complete the proof.
∎
D Confining and lower limit properties of the entropy
In this appendix, instead of writing $d\mu(x)$
we shall write
$\mu(dx)$.
Given a probability measure
$\tau\in\mathcal{P}(M^{n})$
we consider a random element
$(X_{1},...,X_{n})$ in $M^{n}$
with law $\tau$ and
an $n$-tuple of
random probability measures
$(\tau^{1},\tau^{2},...,\tau^{n})$
in $\mathcal{P}(M)^{n}$
such that
$$\mathbb{E}_{\tau^{i}}[f]=\mathbb{E}[f(X_{i})|X_{1},...,X_{i-1}]$$
for every bounded measurable $f:M\to\mathbb{R}$.
A construction of
$(\tau^{1},\tau^{2},...,\tau^{n})$
can be made using transition
kernels as in
[18, Theorem 9.9.2].
To understand the behavior of
$\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$
we will study the behavior of
$\frac{1}{n}\sum_{i=1}^{n}\tau^{i}$.
Proposition 7.9 (Chain rule).
$$D(\tau\|\pi^{\otimes_{n}})=\mathbb{E}\left[\sum_{k=1}^{n}D(\tau^{k}\|\pi)%
\right].$$
Sketch of the proof.
We will give an idea of the proof ignoring issues
of measurability and finiteness of the entropy.
For more details we refer to
[11, Theorem C.3.1].
We shall define explicitly the random elements
$(\tau^{1},\tau^{2},...,\tau^{n})$
in $\mathcal{P}(M)^{n}$ and
$(X_{1},...,X_{n})$ in $M^{n}$.
The domain of our random variables will be
$M^{n}$ endowed with the probability measure
$\tau$.
We define $X_{k}:M^{n}\to M$ as
the projection onto the $k$-th coordinate.
By definition the law of $(X_{1},...,X_{n})$ is $\tau$.
Suppose
$\tilde{\tau}^{k}:M^{k-1}\to\mathcal{P}(M)$
is a transition kernel from $(X_{1},...,X_{k-1})$
to $X_{k}$ and define
$\tau^{k}=\tilde{\tau}^{k}\circ p_{k-1}$,
where $p_{k-1}:M^{n}\to M^{k-1}$ is the projection
onto the first $k-1$ coordinates.
We may notice that the case
$k=1$ is already treated in the previous
definitions if we consider $M^{0}=\{\emptyset\}$.
For instance, $\tau^{1}$ is constant equal to
$p_{1}(\tau)$.
We can see that
$(\tau^{1},...,\tau^{n})$ satisfies the
desired properties. If we assume all entropies
are finite,
$$\displaystyle\mathbb{E}\left[D(\tau^{k}\|\pi)\right]$$
$$\displaystyle=\mathbb{E}\left[D(\tilde{\tau}^{k}\circ p_{k-1}\|\pi)\right]$$
$$\displaystyle=\int_{M^{n}}D(\tilde{\tau}^{k}\circ p_{k-1}(x)\|\pi)\,\tau(dx)$$
$$\displaystyle=\int_{M^{k-1}}D(\tilde{\tau}^{k}(x)\|\pi)\,\,\,p_{k-1}(\tau)(dx)$$
$$\displaystyle=\int_{M^{k-1}}\left(\int_{M}\log\left(\frac{\tilde{\tau}^{k}(x,%
dy)}{\pi(dy)}\right)\tilde{\tau}^{k}(x,dy)\right)\,\,\,p_{k-1}(\tau)(dx)$$
$$\displaystyle=\int_{M^{k-1}\times M}\log\left(\frac{\tilde{\tau}^{k}(x,dy)}{%
\pi(dy)}\right)\,\,\,p_{k}(\tau)(dx,dy)$$
where we have used that $p_{k}(\tau)(dx,dy)=\tilde{\tau}^{k}(x,dy)\,p_{k-1}(\tau)(dx)$. Define
$\tilde{\rho}_{k}:M^{k-1}\times M\to[0,\infty]$
by $\tilde{\rho}_{k}(x,y)=\frac{\tilde{\tau}^{k}(x,dy)}{\pi(dy)}$
and $\rho_{k}=\tilde{\rho}_{k}\circ p_{k}$. We get
$$\mathbb{E}\left[D(\tau^{k}\|\pi)\right]=\int_{M^{n}}\log\left(\rho_{k}(z)%
\right)\tau(dz).$$
Then, summing over $k$ we get
$$\mathbb{E}\left[\sum_{k=1}^{n}D(\tau^{k}\|\pi)\right]=\sum_{k=1}^{n}\mathbb{E}%
\left[D(\tau^{k}\|\pi)\right]=\sum_{k=1}^{n}\int_{M^{n}}\log\left(\rho_{k}(z)%
\right)\tau(dz).$$
Finally we notice that
$$\Pi_{k=1}^{n}\rho_{k}(z)=\frac{\tau(dz)}{\pi^{\otimes_{n}}(dz)}$$
by definition of transition kernel
and conclude the proof.
∎
Proof of Lemma
3.3,
the confining property of the entropy.
We use the notation introduced
at the beginning of this appendix
and we shall not write the dependence on $j$
for simplicity.
By Proposition 7.9 and
by the convexity of $D(\cdot\|\pi)$
(see [11, Lemma 1.4.3]) we have
$$\mathbb{E}\left[D\left(\left.\frac{1}{n}\sum_{i=1}^{n}\tau^{i}\right\|\pi%
\right)\right]\leq\frac{1}{n}\mathbb{E}\left[\sum_{i=1}^{n}D(\tau^{i}\|\pi)%
\right]=\frac{1}{n}D(\tau\|\pi^{\otimes_{n}}).$$
Then, as the sequence
$\frac{1}{n}D(\tau\|\pi^{\otimes_{n}})$
is bounded from above we get that
$\mathbb{E}\left[D\left(\left.\frac{1}{n}\sum_{i=1}^{n}\tau^{i}\right\|\pi%
\right)\right]$
is also bounded from above. So, the sequence of laws
of the random variables
$\frac{1}{n}\sum_{i=1}^{n}\tau^{i}$
is tight due to Proposition
7.4
in Appendix A
and the fact that $D(\cdot\|\pi)$ has
compact level sets (for the latter property see [11, Lemma 1.4.3 (c)]).
Now, to see that the sequence of laws of
$\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$ is tight
we notice that
$$\mathbb{E}\left[\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}\right]=\mathbb{E}\left%
[\frac{1}{n}\sum_{i=1}^{n}\tau^{i}\right].$$
because
$$\mathbb{E}\left[\delta_{X_{i}}\right]=\mathbb{E}\left[\tau^{i}\right]$$
by definition of $\tau^{i}$.
Then we use the fact that the sequence of
laws of random probability measures is tight
if and only if the sequence of expected values
is tight (see Appendix A
, proposition 7.5 or
[10, Lemma 10]).
∎
Now we want to prove Lemma 3.2,
the lower limit property of the entropy.
For this we will compare $\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$
and
$\frac{1}{n}\sum_{i=1}^{n}\tau^{i}$ and define
a metric such that the distance between them
converges to zero in probability. But
first we take a bounded measurable function
$f:M\to\mathbb{R}$ and compare
its integral with respect to both
probability measures.
Lemma 7.10 (Concentration
inequality for a test function $f$).
Let $(X_{1},...,X_{n})\in M^{n}$ and $(\tau^{1},...,\tau^{n})\in\mathcal{P}(M)^{n}$ be the random elements
defined at the beginning of this appendix.
Let $f:M\to\mathbb{R}$ be
a bounded measurable function.
Consider the random measures
$$\hat{\mu}=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$$
and
$$\hat{\tau}=\frac{1}{n}\sum_{i=1}^{n}\tau^{i}.$$
Then
$$\mathbb{P}\left(\left|\int_{M}f(x)\hat{\mu}(dx)-\int_{M}f(y)\hat{\tau}(dy)%
\right|>\epsilon\right)\leq 4\frac{\|f\|_{\infty}^{2}}{n\epsilon^{2}}.$$
Proof.
By Chebyshev’s inequality, we need to understand
$$\mbox{ var }\left(\int_{M}f(x)\hat{\mu}(dx)-\int_{M}f(y)\hat{\tau}(dy)\right).$$
The first term is
$$\int_{M}f(x)\hat{\mu}(dx)=\frac{1}{n}\sum_{k=1}^{n}f(X_{k})$$
and the second is
$$\int_{M}f(y)\hat{\tau}(dy)=\frac{1}{n}\sum_{k=1}^{n}\int_{M}f(y)\,\tau^{k}(dy)%
=\frac{1}{n}\sum_{k=1}^{n}\mathbb{E}[f(X_{k})|X_{1},...,X_{k-1}].$$
We can see that both have the same expected value
and if $i<j$ we have
$$\begin{split}\displaystyle\mathbb{E}\Big{[}&\displaystyle\Big{(}f(X_{i})-%
\mathbb{E}[f(X_{i})|X_{1},...,X_{i-1}]\Big{)}\mathbb{E}[f(X_{j})|X_{1},...,X_{%
j-1}]\Big{]}\\
&\displaystyle\qquad=\mathbb{E}\Big{[}\Big{(}f(X_{i})-\mathbb{E}[f(X_{i})|X_{1%
},...,X_{i-1}]\Big{)}f(X_{j})\Big{]}\end{split}$$
because
$\left(f(X_{i})-\mathbb{E}[f(X_{i})|X_{1},...,X_{i-1}]\right)$
is $(X_{1},...,X_{j-1})$ measurable. Then
$$\mathbb{E}\Big{[}\Big{(}f(X_{i})-\mathbb{E}[f(X_{i})|X_{1},...,X_{i-1}]\Big{)}%
\Big{(}f(X_{j})-\mathbb{E}[f(X_{j})|X_{1},...,X_{j-1}]\Big{)}\Big{]}=0.$$
We have
$$\mbox{ var }\left(\int_{M}f(x)\hat{\mu}(dx)-\int_{M}f(y)\hat{\tau}(dy)\right)=%
\frac{1}{n^{2}}\sum_{i=1}^{n}\mathbb{E}\left[\left(f(X_{i})-\mathbb{E}[f(X_{i}%
)|X_{1},...,X_{i-1}]\right)^{2}\right]$$
$$\leq\frac{1}{n^{2}}\sum_{i=1}^{n}4\|f\|_{\infty}^{2}=\frac{1}{n}4\|f\|_{\infty%
}^{2}$$
and by Chebyshev’s inequality we can
complete the proof.
∎
Choose any sequence
of bounded continuous functions
$\{f_{k}\}_{k\in\mathbb{N}}$
that determines probability measures
(see Corollary 7.2 in Appendix
A) and such that
$$\sum_{k\in\mathbb{N}}\|f_{k}\|_{\infty}^{2}=1.$$
Take a sequence of strictly positive numbers
$\{a_{k}\}_{k\in\mathbb{N}}$
such that $\sum_{k\in\mathbb{N}}a_{k}=1$.
Define the metric
$$d:\mathcal{P}(M)\times\mathcal{P}(M)\to\mathbb{R}$$
$$(\mu,\nu)\mapsto\sum_{k\in\mathbb{N}}a_{k}\left|\int_{M}f_{k}(x)\mu(dx)-\int_{%
M}f_{k}(y)\nu(dy)\right|$$
This metric induces the weak topology.
We have the following corollary of Lemma
7.10.
Corollary 7.11 (Concentration inequality).
With the same notation as Lemma
7.10
we have
$$\mathbb{P}(d(\hat{\mu},\hat{\tau})>\epsilon)\leq\frac{4}{n\epsilon^{2}}$$
Proof.
We can notice that
$$\displaystyle\mathbb{P}(d(\hat{\mu},\hat{\tau})>\epsilon)$$
$$\displaystyle=\mathbb{P}\left(\sum_{k\in\mathbb{N}}a_{n}\left|\int_{M}f_{k}(x)%
\hat{\mu}(dx)-\int_{M}f_{k}(y)\hat{\tau}(dy)\right|>\epsilon\right)$$
$$\displaystyle\leq\mathbb{P}\left(\bigcup_{k\in\mathbb{N}}\left\{\left|\int_{M}%
f_{k}(x)\hat{\mu}(dx)-\int_{M}f_{k}(y)\hat{\tau}(dy)\right|>\epsilon\right\}\right)$$
$$\displaystyle\leq\sum_{k\in\mathbb{N}}\mathbb{P}\left(\left|\int_{M}f_{k}(x)%
\hat{\mu}(dx)-\int_{M}f_{k}(y)\hat{\tau}(dy)\right|>\epsilon\right)$$
$$\displaystyle\leq 4\sum_{k\in\mathbb{N}}\frac{\|f_{k}\|_{\infty}^{2}}{n%
\epsilon^{2}}\leq\frac{4}{n\epsilon^{2}}$$
∎
The previous corollary gives us a convergence in probability
when $n$ tends to infinity. This implies that
if the law of $\hat{\mu}$ converges
then the law of $\hat{\tau}$ converges to the same
limit. With this idea we prove Lemma
3.2.
Proof of Lemma
3.2,
the lower limit property of the entropy.
As in the proof of
Lemma 3.3
we do not write the dependence on $j$.
By the convexity of $D(\cdot\|\pi)$
(see [11, Lemma 1.4.3]),
$$\mathbb{E}\left[D\left(\left.\frac{1}{n}\sum_{i=1}^{n}\tau^{i}\right\|\pi%
\right)\right]\leq\frac{1}{n}\mathbb{E}\left[\sum_{i=1}^{n}D(\tau^{i}\|\pi)%
\right]=\frac{1}{n}D(\tau\|\pi^{\otimes_{n}}),$$
(7.5)
where we have used the
chain rule, Proposition
7.9.
As the law of $\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$
converges
to $\zeta$ and as
the distance
between $d\left(\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}},\frac{1}{n}\sum_{i=1}^{n}\tau^{%
i}\right)$
converges to zero in probability
we get that the law of $\frac{1}{n}\sum_{i=1}^{n}\tau^{i}$
also converges
to $\zeta$.
By the semicontinuity of
$D\left(\cdot\|\pi\right)$ we
have
$$\mathbb{E}_{\zeta}\left[D\left(\cdot\|\pi\right)\right]\leq\liminf\mathbb{E}%
\left[D\left(\left.\frac{1}{n}\sum_{i=1}^{n}\tau^{i}\right\|\pi\right)\right]$$
and we take the lower limit
in 7.5 to complete the proof.
∎
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E-mail address: [email protected] |
MOST††thanks: Based on data from the Most satellite, a Canadian Space Agency mission, jointly operated by Dynacon Inc., the University of Toronto Institute of
Aerospace Studies and the University of British Columbia with the assistance of the University of Vienna. photometry and modeling of the rapidly oscillating
(roAp) star $\gamma$ Equ
Michael Gruberbauer
1
Institute for Astronomy (IfA), University of Vienna,
Türkenschanzstrasse 17, A-1180 Vienna,
1last name @ astro.univie.ac.at
Hideyuki Saio
2Astronomical Institute, Graduate School of Science, Tohoku University, Sendai, 980-8578, Japan,
2saio @ astr.tohoku.ac.jp
Daniel Huber
1
Institute for Astronomy (IfA), University of Vienna,
Türkenschanzstrasse 17, A-1180 Vienna,
1last name @ astro.univie.ac.at
Thomas Kallinger
1
Institute for Astronomy (IfA), University of Vienna,
Türkenschanzstrasse 17, A-1180 Vienna,
1last name @ astro.univie.ac.at
Werner W. Weiss
1
Institute for Astronomy (IfA), University of Vienna,
Türkenschanzstrasse 17, A-1180 Vienna,
1last name @ astro.univie.ac.at
David B. Guenther
3Department of Astronomy and Physics, St. Mary’s University Halifax, NS B3H 3C3, Canada
[email protected]
Rainer Kuschnig
4Dept. Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
[email protected], [email protected], [email protected]
Jaymie M. Matthews
4Dept. Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
[email protected], [email protected], [email protected]
Anthony F.J. Moffat
5Dépt. de physique, Univ. de Montréal C.P. 6128, Succ. Centre-Ville, Montréal, QC H3C 3J7, Canada
[email protected]
Slavek Rucinski
6Dept. Astronomy & Astrophysics, David Dunlop Obs., Univ. Toronto P.O. Box 360, Richmond Hill, ON L4C 4Y6, Canada
[email protected]
Dimitar Sasselov
7Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
[email protected]
Gordon A.H. Walker
4Dept. Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
[email protected], [email protected], [email protected]
[email protected]
(Received / Accepted)
Key Words.:
stars: chemically peculiar – stars: oscillations – stars: individual: $\gamma$ Equ – stars: magnetic fields – methods: data analysis
††offprints: M. Gruberbauer :
Abstract
Context:
Aims:Despite photometry and spectroscopy of its oscillations
obtained over the past 25 years, the pulsation frequency spectrum
of the rapidly oscillating Ap (roAp) star $\gamma$ Equ has remained poorly understood.
Better time-series photometry, combined with recent advances to incorporate interior magnetic
field geometry into pulsational models, enable us to perform improved asteroseismology of this roAp star.
Methods:We obtained 19 days of continuous high-precision photometry
of $γ$ Equ with the Most (Microvariability & Oscillations of
STars) satellite. The data were reduced with two different reduction
techniques and significant frequencies were identified. Those
frequencies were fitted by interpolating a grid of pulsation models
that include dipole magnetic fields of various polar strengths.
Results:We identify 7 frequencies in $\gamma$ Equ that we associate
with 5 high-overtone p-modes and 1st and 2nd harmonics of the dominant p-mode.
One of the modes and both harmonics are new discoveries for this star. Our best model
solution (1.8 $M_{\sun}$, log $T_{\rm eff}\sim 3.882$; polar field
strength $\sim$ 8.1 kG) leads to unique mode identifications for
these frequencies ($\ell=$ 0, 1, 2 and 4). This is the first
purely asteroseismic fit to a grid of magnetic models. We measure amplitude
and phase modulation of the primary frequency due to beating with a
closely spaced frequency that had never been resolved. This casts
doubts on theories that such modulation – unrelated to the rotation
of the star – is due to a stochastic excitation mechanism.
Conclusions:
1 Introduction - the history of $\gamma$ Equ
Rapidly oscillating Ap (roAp) stars form a class of variables consisting of cool magnetic Ap stars with spectral types ranging from A-F, luminosity class V, and were discovered by Kurtz (1982). Photometric and spectroscopic observations during the last 2 decades characterize roAp stars with low amplitude pulsations ($<$ 13 mmag) and periods between 5 to 21 minutes. It is now widely accepted that the oscillations are due to high-overtone ($n>15$) non-radial p-mode pulsations. The observed frequencies can be described in a first-order approximation by the oblique pulsator model as described by Kurtz (1982). In this model the pulsation axis is aligned with the magnetic axis, which itself is oblique to the rotation axis of the star. It has since been refined and improved to include additional effects like the coriolis force (Bigot & Dziembowski 2002).
The oscillations in roAp stars are most probably excited by the $\kappa$ mechanism acting in the hydrogen ionization zone (Dziembowski & Goode 1996), while the prime candidate for the selection mechanism of the observed pulsation modes is the magnetic field (e.g. Bigot et al. 2000; Cunha & Gough 2000; Balmforth et al. 2001; Bigot & Dziembowski 2002, 2003; Saio 2005; Cunha 2006). Similar to non-oscillating Ap (noAp) stars, the group of roAp stars also shows a distinctive chemical pattern in their atmospheres, in particular an overabundance of rare earth elements, which points to stratification and vertical abundance gradients. Tackling the question of what drives the roAp pulsation, it is important to note that apart from the similar chemical composition, noAp stars show comparable rotation periods and magnetic fields but higher effective temperatures (Ryabchikova et al. 2004). Thus, their hydrogen ionization zone lies further out in the stellar envelope, which prevents efficient driving of pulsation. All together, roAp stars have given new impulse to asteroseismology and 2D mapping of pulsation (Kochukhov 2004), as well as detailed 3D mapping of (magnetic) stellar atmospheres, has become possible.
For reviews on primarily observational aspects of roAp stars we refer to Kurtz & Martinez (2000) and Kurtz et al. (2004) and on theoretical aspects to Shibahashi (2003) and Cunha (2005).
$\gamma$ Equ, (HD 201601, HR 8097, A9p, $m_{\rm{V}}=4.7$) was the 6th discovery as a roAp star.
It was long known to have a significant magnetic field (Babcock 1958) which is variable with a period of $\sim$ 72 years (Bonsack & Pilachowski 1974; Scholz 1979; Leroy et al. 1994). Rotation, a solar-like magnetic cycle or precession of the star’s rotation axis due to its binary companion are candidate mechanisms for this cyclic variation. The binary hypothesis was supported by Scholz et al. (1997), who observed a temporary drop in radial velocity for $\gamma$ Equ during 4 consecutive nights. Their results are however contradicted by Mkrtichian et al. (1998, 1999). Magnetic field data spanning more than 58 years indicate that the variation is most likely due to rotation and therefore $P_{\rm{mag}}=P_{\rm{rot}}$ (Bychkov et al. 2006). This explanation is also supported by many spectroscopic observations showing very sharp absorption lines (e.g. Kanaan & Hatzes 1998) which are typical for a slow rotator (or a star seen pole-on).
Kurtz (1983) was the first to detect a pulsation period of 12.5 min (1.339 mHz) with an amplitude varying between 0.32 and 1.43 mmag and speculated about a rotation period of 38 days, inconsistent with the magnetic field measurements mentioned above. Aside from rotation, beating with a closely spaced frequency has been proposed as a cause, for which the present paper gives further evidence. The pulsation was confirmed shortly after Kurtz’s study by Weiss (1983). Bychkov (1987) reported first evidence for radial velocity (RV) variations, but it took two more years for a clear detection (Libbrecht 1988). Although unable to detect the 1.339 mHz oscillations reported by Kurtz (1983), Libbrecht discovered three frequencies at 1.365 mHz, 1.369 mHz and 1.427 mHz. He suggested that the amplitude modulation observed in the spectra of roAp stars may not be due to closely spaced frequencies, but rather caused by short mode lifetimes in the order of $\sim$ 1 d. He concluded that both peaks therefore belong to a single p-mode oscillation. The follow-up study by Weiss & Schneider (1989) aimed at performing simultaneous spectroscopic and photometric studies but failed to confirm $\gamma$ Equ’s pulsation. The latter authors applied a correlation technique using more than 100 Å wide spectral ranges in order to improve the S/N ratio relative to previous techniques, based on individual lines. This approach turned out to be inapplicable, considering the peculiar abundance structure of roAp star atmospheres which is now well known. Matthews & Scott (1995) then claimed to have detected variability with rather large radial velocity amplitudes but their results were inconsistent with previous findings.
Detailed photometric observations were conducted by Martinez et al. (1996) using a multi-site campaign in 1992, spanning a total of 26 nights. Their results also suggested limited life times of pulsation modes, because additional frequencies appeared in their analysis of individual nights. However, all three frequencies detected so far were confirmed, and prewhitening of 1.366 mHz revealed a fourth eigenfrequency at 1.397 mHz. Using these four frequencies they concluded a spacing of consecutive radial overtones of $\Delta\nu\sim 30\,\mu$Hz based on the asymptotic theory for low-degree, high-overtone p-mode pulsations (Tassoul 1990). They did not detect any evidence for non-linear behavior of the eigenfrequencies (i.e. harmonics) which, as will be shown in this paper, is indeed present in the Most data. So far their observing campaign was the most recent photometric investigation of $\gamma$ Equ. A period of spectroscopic studies focusing on pulsation, abundance and stratification analyses followed.
Kanaan & Hatzes (1998) reported on RV amplitudes for chromium and titanium and speculated that this is due to their concentration close to the magnetic – hence also pulsation – poles, while other elements (e.g. iron) showing no RV variations may be concentrated at the magnetic equator. Malanushenko et al. (1998) independently arrived at a similar conclusion and they are the first who identified the rare earth elements (REE) Pr iii and Nd iii to show the largest variations (up to 800 $\rm ms^{-1}$). Most other atomic species had very low or non-measurable RV variations and the authors concluded that previous spectroscopic observations failed to detect significant radial velocity variations due to the chosen spectral regions and/or low spectral resolution. Savanov et al. (1999) further clarified the situation by drawing attention to a possible incorrect identification of spectral lines in Kanaan & Hatzes (1998) and concluded that Pr iii and Nd iii were responsible for the large RV amplitudes discussed in their analysis.
Further investigations by Kochukhov & Ryabchikova (2001) and Ryabchikova et al. (2002) impressively explained amplitude modulations of different elements and even ions of the same element by lines being formed at different atmospheric depths. The authors also tried a first mode identification based on rather short data sets of line profile variations (LPVs) and they argue for $\ell=2$ or $3$, and $m=-\ell$ or $-\ell+1$ modes. Shibahashi et al. (2004) disagreed with this analysis and suggested a shock wave causing the observed LPVs which in turn was questioned by Kochukhov et al. (2007) who argued that the pulsational velocity should not exceed the local sound speed. The latter authors propose a modified oblique pulsator model where LPVs are caused by pulsation velocity fields superposed by sinusoidal line width changes due to convection, but which previously was thought to be suppressed. In any case, they agree with Shibahashi et al. that the identification of the primary frequency as an $\ell=1,m=0$ mode is still the most likely explanation, which is also supported by this paper. The issue is still being debated though (Shibahashi et al. 2007).
A still unsolved issue are magnetic field variations synchronized with pulsation. Leone & Kurtz (2003) showed evidence for this, but subsequent investigations (Kochukhov et al. 2004; Bychkov et al. 2005; Savanov et al. 2006) could not confirm their findings. Hubrig et al. (2004) corroborate this null detection for $\gamma$ Equ with ESO-FORS1 data.
Despite a history of more than 25 years of observations, the pulsation frequency spectrum of $\gamma$ Equ remains poorly understood. Several frequencies have been published, but up to now these have never been observed simultaneously. This is where Most steps in, providing the most complete high precision photometric campaign ever conducted for $\gamma$ Equ and covering continuously a timespan of 19 days.
2 Observations and data reduction
2.1 Most observations
Most is a Canadian space satellite designed for the detection of stellar variability with amplitudes down to several ppm on time scales up to several days (Walker et al. 2003). The satellite was launched in June 2003 and has since proven to deliver photometric time series of unprecedented precision. It carries a Maksutov telescope with an aperture of 15 cm, uses a broadband filter (350–700 nm) and operates in a low Earth orbit following the terminator in an almost polar orbit. Most can observe stars in the continuous viewing zone for up to 6 weeks with a pointing precision of $\sim\,1$”. An array of Fabry lenses is used to suppress effects of satellite jitter in the data of primary targets of $V<6$ mag. $\gamma$ Equ was observed for 19 days from 07/28/2004 to 08/16/2004 with exposures (integration time $=11$ sec) taken continuously once every $30$ sec. As $\gamma$ Equ was one of the first primary targets observed by Most, stray light, produced by earthshine, and other instrumental effects were still being investigated at this time with the aim of improving the observing procedures for future runs. In fact, this data set was the basis for the development of the C reduction pipeline for Most Fabry Imaging targets (Reegen et al. 2006).
2.2 Data reduction using decorrelation
Stray light effects are corrected by computing a correlation between target and background pixels distinguished by a fixed aperture, where target denotes pixels inside the aperture and background characterizes pixels outside the aperture, which are assumed to contain no stellar signal at all. A “neutral” area flag, for pixels not used for either purpose, is meant to exclude pixels at the transition zone close to the edge of the aperture.
As it turned out throughout the Most mission, each target star’s photometry suffered specific problems which needed to be addressed individually by optimizing the reduction parameters. Due to bad data quality at the beginning of the $\gamma$ Equ-run about 1060 frames had to be rejected. Additional frames needed to be eliminated due to pointing problems in the early days of Most operations which resulted in distorted Fabry image geometries identified by comparison to a mean normalized Fabry image.
If the pixel values in an individual (normalized) frame deviate by more than $g\,\sigma$, where $g$ is an integer, from the pixel values of the mean normalized reference image, the frame is rejected. Since the mean reference image changes after the elimination process, this step is repeated iteratively until no more frames need to be rejected. Statistics of the reduction process are shown in Table 1.
Tests showed that the number of images in which pixels are deviating by more than 5 $\sigma$ first decreased as expected, but increased after about ten iterations (see Fig. 1, upper panel) and later decreased again, indicating two separate causes for deviating Fabry image geometry. Fig. 2 shows the raw light curve after correcting for cosmic rays, phased with the orbital period, clearly indicating which phases are responsible for the image geometry deviations. Not surprisingly, the majority of these rejections concern exposures taken during high stray light phases and therefore cause regular gaps in the final data set (see Fig. 3). To ensure that this reduced duty cycle does not impair the identification of intrinsic frequencies, the analysis was repeated with the data set obtained after the first iteration step in the image geometry evaluation, which eliminated only 807 frames. In the lower panel of Fig. 1 we compare the spectral windows of both reductions, showing that the rejections mainly affected orbit-induced artifacts ($f_{\rm{orbit}}=14.19$ d${}^{-1}$). The final analysis did not yield a different set of intrinsic, but considerably fewer instrumental frequencies.
2.3 Data reduction using “doughnut” fitting
Another (less invasive) reduction method was developed to verify the consistency of our frequency analysis. In a first step an average Fabry frame is determined from all frames obtained during orbital phases with very low stray light signal. The mean intensity, derived from pixels which are defined as background (“sky”) pixels, is subtracted from all pixels. The resulting frame is scaled to the mean intensity of the $N$ pixels with the highest intensity values, where $N$ is of the order of 100. The resulting frame serves as sort of normalized point spread function (PSF) for a Fabry image (Most’s Fabry lenses produce images of the entrance aperture of the optics which resemble a doughnut - hence the name “doughnut fitting”). As a next step a linear regression between pixel intensities of an image and the corresponding pixel intensities of the mean PSF frame is performed. For each of these images the slope $k$ of the linear fit gives a scaling factor, while the offset $d$ corresponds to the image mean background intensity. Each frame is replaced by the PSF frame multiplied with the corresponding scaling factor $k$ and the mean background intensity $d$ is added. The average intensities of all target and background pixels result in the target and background light curve, respectively. Finally, a linear fit between the background and target light curve is determined and subtracted from the target light curve resulting in the final light curve. In the case of $\gamma$ Equ the latter consists of 48955 datapoints.
The advantage of this method is the insensitivity of the linear regression to extreme pixel values, for instance due to cosmic ray hits or local stray light effects. Also, contrary to the method described in section 2.2 (see also Reegen et al. 2006), the data set is not split into subsets and processed individually, which could distort the low frequency signal. The entire data set is reduced en-block, no subsets are created which qualifies this procedure as a perfect cross-check for detecting artifacts due to data reduction. The disadvantage of this method is the stray light correction which is less efficient as for the decorrelation method.
3 Frequency analysis
Time series resulting from both reductions outlined in section 2 were analyzed individually. SigSpec (Reegen 2007) was used to obtain frequencies by means of the spectral significance between 0 and 360 d${}^{-1}$ in a prewhitening sequence down to a significance level of 5.46, corresponding roughly to an amplitude S/N ratio of 4. Only frequencies without a counterpart in the sky background were considered, following the procedure described by Reegen et al. (2008).
Most background light curves typically contain several dozen frequencies between 0 and 360 d${}^{-1}$, produced by stray light and instrumental effects. These are, in most cases well confined to the orbital frequency ($f_{\rm{orbit}}=14.19$ d${}^{-1}$) and harmonics of it, to 1d${}^{-1}$ side lobes due to the passages of Most above nearly the same ground pattern after 14 orbits, or are caused by temperature drifts of electronic boards. In the case of $\gamma$ Equ we used a method similar to what is described in Gruberbauer et al. (2007) to identify non-intrinsic frequencies. Finally, the remaining frequencies deduced from both reduction methods were compared and, again, only matching frequencies were taken into account for our final analysis (Fig. 4).
For the early Most runs the low-frequency region is known to be affected by an instrumental effect producing spectral features centered on 3.16 d${}^{-1}$ and multiples of it (Reegen et al. 2006). This instrumental effect is much weaker for the background pixels which explains why the corresponding frequencies remain even after comparisons with the background frequency spectrum and what justifies the rejection of all frequencies below 10 d${}^{-1}$. Some of the power excess in the low frequency region might be still of stellar origin, but the current data is inconclusive. We also found the frequencies between 10 and 35 d${}^{-1}$ to be multiples of 3.16 d${}^{-1}$, but no other Most data set shows these instrumental effects at such high frequencies. Because we prefer a critical approach, in the end, only the six higher frequencies with $f>100$ d${}^{-1}$ were considered to be intrinsic to $\gamma$ Equ beyond doubt.
Table 2 lists the result of our conservative analysis. Two additional significant frequencies, not included in the table, can be found very close to $f_{\rm{1}}$ with amplitudes of $\sim 40$ ppm, but including them in a Period04 multi-sine fit (Lenz & Breger 2005) fails, because the solution does not converge. Their nature is discussed in section 6.2.
No obvious spacing, as predicted by the asymptotic theory of non-radial pulsation, can be found in our final set of frequencies. This suggests that the regular spacing of modes with the same degree $\ell$ is disturbed, e.g. by the magnetic field, or that modes of different degree are excited. We therefore turned to fit our observed frequencies to the latest generation of roAp-pulsation models, but using only $f_{\rm{1}}$ to $f_{\rm{5}}$ for the fitting process, as $f_{\rm{6}}$ and $f_{\rm{7}}$ correspond to the harmonics of $f_{\rm{1}}$.
4 Pulsation models
To compare theoretical frequencies with observed ones of $\gamma$ Equ, we have computed main-sequence evolutionary models for a mass range of $1.75\,M_{\rm{\odot}}-1.85\,M_{\rm{\odot}}$ with heavy element abundance of $0.015\leq Z\leq 0.025$ and OPAL opacities (Iglesias & Rogers 1996). A list of the individual models, and how we organised them into grids, is presented in Table 3. For standard models, envelope convection is suppressed ($\alpha=0.0$), and to mimic depletion of helium abundance in the outermost layers of a star, the helium abundance $Y$ is given as $Y=0.01+0.27(x_{2}+x_{3})$, where $x_{2}$ and $x_{3}$ are fractions of singly and doubly ionized helium, respectively (cf Balmforth et al. 2001). Nonetheless, the effects of convection were also tested by computing all grids except Grid 4 with $\alpha=1.5$ and homogeneous He abundance in the outer layers, but otherwise the same parameters. Grid 4 included a homogeneous distribution (mixing) of He in the envelope but no convection.
Nonadiabatic frequencies of axisymmetric ($m=0$) high order p-modes under the presence of a dipole magnetic field were calculated using the method described in Saio (2005). Outer boundary conditions are imposed at an optical depth of $10^{-3}$. A reflective mechanical condition ($\delta p/p\rightarrow$ constant) is adopted. All frequencies presented in this paper are less than the acoustic critical frequencies of corresponding models.
Since the latitudinal dependence of amplitude under the presence of a magnetic field cannot be expressed by a single Legendre function, we expand it into a truncated series of components proportional to Legendre functions $P_{l_{j}}$ with $l_{j}=2j-1$ for odd modes and $l_{j}=2j$ for even modes, where we have included twelve terms; i.e., $j=1,2,\ldots,12$. The latitudinal degree $\ell$ is not a definite quantity for a pulsation mode any more, because pulsation energy is distributed among twelve components associated with $P_{l_{j}}$. For convenience we still use $\ell$ representing the $l_{j}$ value of the component associated with the largest kinetic energy. Sometimes the distribution of the kinetic energy among the components is broad so that the identification of $\ell$ is ambiguous, and the value of $\ell$ may change as the strength of the magnetic field changes. Also, for a mode associated with $\ell$, the latitudinal dependence of amplitude on the stellar surface may be considerably different from that of $P_{\ell}(\cos\theta)$ as shown in Saio & Gautschy (2004); Saio (2005) (see also below).
5 Model fitting
Pulsation models were calculated taking into account a magnetic dipole with polar field strength, $B_{\rm{P}}$, up to 12 kG. These models were tried to match the currently estimated parameter space of $\gamma$ Equ, which is $\log T_{\rm{eff}}=3.882\pm 0.011$ K, $\log L/L_{\rm{\odot}}=1.10\pm 0.03$, and $M\sim 1.74\pm 0.03$ M${}_{\rm{\odot}}$ (Kochukhov & Bagnulo 2006). Fig. 5 shows the position and parameters of all models involved. For the corresponding values and further details, we again refer to Table 3.
To increase the resolution of our model grid, the mode frequencies were interpolated linearly in $B_{\rm{P}}$ and ($\log T_{\rm{eff}}$, $\log L/L_{\rm{\odot}}$) for a fixed mass and chemical composition. The step width for the interpolation was chosen to be 0.02 kG in $B_{\rm{P}}$ and 0.00002 in $\log T_{\rm{eff}}$. Since not all modes with the same degree $\ell$ and radial order $n$ converge in the model calculations, the sequence of frequencies for a specific mode may not cover the entire chosen stellar fundamental parameter space ($\log T_{\rm{eff}},\log L/L_{\rm{\odot}}$, and $M$). Finally we used a single coordinate, $\Delta\nu$, according to
$$\Delta\nu=0.1349\cdot\left(\frac{\rho}{\rho_{\odot}}\right)^{0.5},$$
(1)
where $\rho$ is the mean density of the stellar model. All model fitting was carried out for a given mass in the ($B_{\rm{P}}$,$\Delta\nu$)–space with a $\chi^{2}$–test similar to Guenther & Brown (2004). Since an a priori mode identification was not a reasonable option, each model’s eigenfrequencies of spherical degrees $\ell=0$ to $\ell=4$ were compared to the five observed frequencies. The 1 $\sigma$–uncertainties of the observed frequencies were estimated to be roughly 0.25 of the upper frequency error derived according to Kallinger et al. (2007). The corresponding 1 $\sigma$–errors assigned to the individual model frequencies were assumed to be 0.2 $\mu$Hz.
The best fitted models, represented by a minimum $\chi^{2}$, are listed in Table 4. Grid 3, as defined in Table 3, outperforms the other models by far. It is the only model grid that manages to produce frequencies that fit all five observed frequencies on average to within the uncertainties ($\chi^{2}\leq 1$), as it is illustrated by the upper panel of Fig. 6. As expected, including convection for Grid 3 destroys the fit, while for grids with large $\chi^{2}$ it has the opposite effect. The best fit is located at the center of an extended patch. Its proximity to a genuine calculated model ensures that this good fit is not an artifact produced by the interpolation (also see Section 6). The mean parameter values for the $\chi^{2}\leq 1$ region with the mentioned interpolation step size are $\log T_{\rm{eff}}=3.8818$, $\log L/L_{\rm{\odot}}=1.0871$ and $B_{\rm{P}}=8.1\,\rm{kG}$.
Fig. 7 shows an échelle diagram of the best fitted model together with the observations. $f_{\rm{1}}$, the primary frequency, is identified as an $\ell=1$ mode. $f_{\rm{2}}$ and $f_{\rm{3}}$ are matched by consecutive $\ell=4$ modes. The remaining two frequencies, $f_{\rm{4}}$ and $f_{\rm{5}}$, are fitted by modes of degree $\ell=2$ and $\ell=0$. As a comparison, the lower panel of Fig. 6 shows the fitting results to Grid 6, a grid with considerably larger $\chi^{2}$ values. Interestingly, while all grids except Grid 3 fail at delivering $\chi^{2}<1$–results, the mode identification remains stable for grids with $\chi^{2}<4$. In particular, the two closely spaced model frequencies in the vicinity of $f_{\rm{1}}$ and $f_{\rm{2}}$ are found to fit the same degrees $\ell$ = 1 and 4 for Grids 3, 7$\alpha$, 5$\alpha$, and 6. This is not the case for 6$\alpha$ and 1$\alpha$. Thus, in our grids there seems to be a tendency towards the mode identification of Grid 3 for better fitted models. Also, convection is shown to radically influence the model frequencies.
In Fig. 6 obvious discontinuities can be seen in the fitting results. They are produced when the interpolation routine cannot find two modes of equal spherical degree $\ell$ and radial order $n$
in both models that are acting as sampling points. This can happen, as already mentioned in Section 4, when the $\ell$-value of a mode formally changes, because the former associated spherical harmonic does no longer supply most of the kinetic energy. Also, the cyclic variation of the damping rate as a function of $B_{\rm{P}}$, as shown in Saio & Gautschy (2004), has an effect on the “availability” of certain modes. Around the maximum of the damping rate, the kinetic energy is so broadly distributed among the different $\ell$-components that convergence of the model calculations starts to fail. This effect is clearly visible as the two large discontinuities in each panel of Fig. 6, which bears resemblance to what is presented in Saio & Gautschy (2004).
6 Discussion
6.1 The frequencies of $\gamma$ Equ
As mentioned in the introduction, roAp stars are characterized by an interaction of a strong global magnetic field with velocity fields due to pulsation of still poorly known origin, all leading to a complex eigenfrequency spectrum. The latter provide the only directly accessible information concerning the internal structure of these stars.
Data obtained by Most have allowed us to unambiguously identify 7 frequencies (Table 2) above 1.3 mHz, including the first two harmonics of the primary frequency, which significantly exceed the mean noise level of $\sim 10$ ppm between 1.15 and 1.75 mHz. The comparison with Martinez et al. yields a match for $f_{\rm{1}}$, $f_{\rm{3}}$, and $f_{\rm{4}}$. Our $f_{\rm{2}}$ has never been detected before, probably because $f_{\rm{1}}$ and $f_{\rm{2}}$ could not be resolved as individual frequencies. This is also important for the discussion on amplitude modulation of the primary frequency, for which we refer to the next section. Our value for $f_{\rm{5}}$ is comparable to their $\nu_{1}=1.321$ mHz (taken from Weiss 1983), but differs by $\sim 0.01$ mHz. We assume that they have misidentified a 1-day alias of the real frequency but this is difficult to asses, since no frequency uncertainties are known for this data set. Given that the data set from Weiss (1983) spans only three nights, the poor frequency resolution $1/T_{\rm{obs}}\simeq 0.33\,\rm{d}^{-1}=0.004$ mHz supports our assumption. Thus, we can for the first time confirm and expand the previous set of frequencies unambiguously.
When discussing published amplitudes (including those in the present paper) one has to keep in mind that two different properties prevent a direct comparison. First, amplitude modulations are present which lead to different amplitudes for observations obtained at different beating phases. Second, it has been well known since the eighties that amplitudes (Weiss & Schneider 1984) and phases (Weiss 1986) depend on wavelength and the former are rapidly decreasing towards the red. The passband of Most is broad compared to the Strömgren system, frequently used for roAp star photometry, hence the observed Most amplitudes are intrinsically smaller.
6.2 Amplitude modulation
Amplitude modulation of $\gamma$ Equ’s primary frequency has often been mentioned in the literature and rotation or closely spaced frequencies have been proposed as possible explanations. Because of the relatively small effect and the limited accuracy of the data a specific modulation period could never be accurately determined. This situation, however, changed with Most. To distinguish between amplitude modulation of a single frequency and beating of two closely spaced frequencies one needs to discuss simultaneous phase $and$ amplitude changes (Breger & Pamyatnykh 2006). Figure 8 illustrates the correlation of amplitude and phase variations for $f_{\rm{1}}$ with a relative phase shift of $\frac{\pi}{2}$, which is indicative for beating of a close pair of frequencies spaced by $f_{\rm{beat}}\sim 0.07$ d${}^{-1}$ ($\sim 0.0008$ mHz).
We find $f_{\rm{1}}+f_{\rm{beat}}\simeq f_{\rm{2}}$. The best fitting pulsation model predicts both $f_{\rm{1}}$ and $f_{\rm{2}}$ to be eigenfrequencies of the star, hence we conclude that the “closely spaced frequencies”-hypothesis is correct. It comes as no surprise that previous observations, especially time-resolved spectroscopy, could not explain the modulation effect, since it is impossible to resolve $f_{\rm{2}}$ with short time bases. The possibility of unresolved modes in such data should be taken into account when studying pulsation via residual spectra, produced by subtraction of a mean spectrum phased by only a single pulsation frequency. For ground-based photometry, the aliasing problem and the higher noise level might also have contributed to the difficulty of detecting $f_{\rm{2}}$.
We have found no evidence for modulation of the other frequencies, but we can comment on two other significant frequencies with amplitudes $\sim 40$ ppm close to $f_{\rm{1}}$ which are not instrumental. These frequencies appear to be Fourier artifacts produced by irregularities in the beating of $f_{\rm{1}}$ and $f_{\rm{2}}$. As mentioned in Section 3, a multi-sine fit fails to converge if they are included in the solution. We have to mention here again that the broad filter band pass used by Most may be disadvantageous for such investigations.
6.3 On the validity of interpolated frequencies
To increase the resolution of our grid, we used linear interpolation in temperature, luminosity, and polar magnetic field strength, respectively. It may be questioned if such an interpolation is a sensible approach. For a test we compared interpolated values with genuine models. For all genuine models $m_{\rm{g}}$ enclosed by 2 additional genuine models $m_{\rm{g,1}}$ and $m_{\rm{g,2}}$ , we produced frequencies at the position of $m_{\rm{g}}$ by interpolation between $m_{\rm{g,1}}$ and $m_{\rm{g,2}}$. The deviation of the interpolated frequencies from their genuine values, normalized to the gap in $\log T_{\rm{eff}}$ between the enclosing and the enclosed models, was calculated and over 16000 frequencies were compared. Figure 9 shows a histogram of the results. It illustrates that our interpolation delivers a reasonable approximation to genuine models, as long as the gaps in between the reference models remain reasonably small.
In the case of Grid 3 (see Table 3) we also repeated the frequency fitting procedure.
We interpolated between 3a and 3c and omitted the 3b–models of Grid 3 . Figure 10 shows the $\chi^{2}$–statistics for our observed frequencies based on an interpolation in this much coarser grid (best fit with $\chi^{2}=2.642$). A comparison of Figure 10 with the upper panel of Fig. 6 again shows that interpolation is applicable for small gaps in between genuine models, which is also reflected in the results of the fitting procedure. Since not all modes can be approximated sufficiently through linear interpolation, as indicated by the outliers in Figure 9, it is necessary to check whether the modeled frequencies that fit the observations satisfy the linear assumption. Figure 11 presents the absolute (rather than the normalized) deviation of the interpolated frequencies from the genuine frequencies for the 3b–models. It shows that the interpolated values of all 5 modes that fit the observations are hardly deviating from their genuine values - the deviations lie well within the model uncertainties.
It is important to note that all of the modes which fit the observed frequencies are actually not expected to be excited according to current model physics. This hints at remaining problems concerning the stability analysis or the implemented driving mechanism.
6.4 Latitudinal amplitude dependence
As discussed in §5 the observed frequencies of $\gamma$ Equ are identified as modes of $\ell=$ 0, 1, 2, and 4. Those modes, however, have amplitude distribution on the stellar surface considerably deviating from that of a single Legendre function $P_{\ell}(\cos\theta)$ due to the strong magnetic effect (Fig.12). The amplitude for $f_{\rm{1}}$ ($\ell=1$) and $f_{\rm{4}}$ ($\ell=2$) is more concentrated toward the polar regions than $P_{1}(\cos\theta)$ and $P_{2}(\cos\theta)$. It is interesting that the amplitude distribution of $f_{\rm{4}}$ ($\ell=2$) on a hemisphere is not very different from that of $f_{\rm{1}}$ ($\ell=1$) having very small amplitude near the equator, although $f_{\rm{1}}$ is antisymmetric and $f_{\rm{4}}$ symmetric to the equator.
For $f_{\rm{2}}$, $f_{\rm{3}}$ ($\ell=4$) and $f_{\rm{5}}$ ($\ell=0$) the amplitude distributions are strongly concentrated around $\cos\theta\approx 0.65$, drastically different from those for non-magnetic stars. Although the light variation from an $\ell=4$ mode of a non-magnetic star is expected to suffer from a strong cancellation on the stellar disk, Fig. 12 indicates that our cases $f_{\rm{2}}$ and $f_{\rm{3}}$ seem hardly affected by this effect.
7 Conclusion
We have demonstrated that our models of roAp stars reproduce the observations in great detail. The temperature and luminosity parameters of our best fit are in good agreement with those expected from previous observations (Kochukhov & Bagnulo 2006), although we find a slightly higher mass. According to Ryabchikova et al. (1997) the mean magnetic field modulus of $\gamma$ Equ is $\rm\simeq 4$ kG, which is half of the polar magnetic field strength of 8 kG as suggested by our best model. At present we can only speculate about a difference of internal magnetic field strengths, to which pulsation is sensitive, relative to surface magnetic fields, which are accessible to spectroscopy. Furthermore, our models assume a simple, axisymmetric magnetic dipole, which may not sufficiently reflect a more complicated reality. The factor of 2 between (surface) magnetic field modulus and best fitting magnetic pulsation model appears again for the roAp star 10 Aql (Huber et al. 2008), another Most primary target. Certainly, more investigations of this sort have to be conducted in order to solve this problem.
Pulsation amplitude changes for roAp stars were discussed in the literature as a consequence of limited mode life time or beating frequencies. For $\gamma$ Equ we can clearly identify the beating frequencies, which seriously questions excitation mechanisms for roAp stars based on stochastic processes.
Finally, we could identify the modes of all 7 frequencies detected so far in $\gamma$ Equ as $\ell=0$, $1$, $2$ and $4$, with $f_{\rm{6}}$ and $f_{\rm{7}}$ being the harmonics of $f_{\rm{1}}$ produced by non-linear pulsation.
Since we are not able to calculate non-axisymmetric modes, all of these frequencies are assumed to be ($m=0$) p-modes. While we cannot rule out the possibility of excitation of frequencies with ($m\neq 0$), axisymmetric modes are most probable, because of the extremely long rotation period of $\gamma$ Equ and the assumption of a magnetic dipole.
The assignment of any $\ell$-value to these modes, however, has to be understood as a convenient simplification. In the presence of strong magnetic fields a mode’s oscillation behavior cannot be described by a single spherical harmonic. Consequently, we alert the reader that analysis techniques using this assumption, e.g. mode identification based on LPVs, should be adjusted for magnetic effects on the pulsation geometry.
Acknowledgements.
We would like to thank the referee for all the useful comments and suggestions, which greatly helped to improve the quality of this paper. It is also a pleasure to thank M. Cunha, O. Kochukhov, P. Reegen, T. Ryabchikova and H. Shibahashi for valuable discussions. MG, TK and WW have received financial support by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (P17890-N02) and by the Austrian Research Promotion Agency, FFG-ALR. JMM, DBG, AFJM, SR, DS, and GAHW acknowledge funding from the Natural Sciences & Engineering Research Council (NSERC) Canada. RK is supported by the Canadian Space Agency (CSA).
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Restrictions on symplectic fibrations
Jarosław Kȩdra
University of Szczecin
(with an appendix written jointly with Kaoru Ono)
Partially supported by
State Committee for Scientific Research grant 2PO3A 035 14. The paper was
written during the author’s stay at MPI in Bonn
and at IMPAN in Warsaw.
Keywords: symplectic fibration, spectral sequence, flux
AMS classification(2000): Primary 57R17;
Secondary 53D45
(November 23, 2020)
1 Introduction
This paper is devoted to restrictions on symplectic fibrations
coming from Gromov-Witten invariants. They might have two types
of nature. First is reflected in the properties of the Leray-Serre
spectral sequence. The behavior of its differentials
gives an information about the topology of symplectomorphism
groups. There arises two conjectures:
1.
The flux conjecture which says that the group
$Ham(M,\omega)$ of Hamiltonian symplectomorphisms is $C^{1}$-closed
in the group $Symp(M,\omega)$ of all symplectomorphisms, where
$(M,\omega)$ is a compact symplectic manifold. This conjecture
can be expressed equivalently in terms of so called
flux groups, which are defined with the use of
the differentials in spectral sequences associated to
symplectic fibrations over two dimensional sphere [LMP1, MS1].
2.
The c-splitting conjecture which states that
the spectral sequence associated to any Hamiltonian fibration
degenerates at $E_{2}$ term [LM]. This means that the
rational cohomology of the total space is additively isomorphic
to the tensor product of the cohomology of the base and the
cohomology of the fiber.
The second nature of restrictions arises under assumption
that the spectral sequence collapses at $E_{2}$, e.g. for
symplectic manifolds satisfying the c-splitting conjecture.
Namely, one may ask how much the cohomology ring of the total
space differs from the tensor product of the cohomology ring
of the base and of the fiber. This was already taken up by Seidel
[Se] in case when $(M,\omega)$ is a product of complex
projective spaces.
1.1 Sketch of the argument
We are mainly interested in the properties of
the Leray-Serre spectral sequences associated to symplectic
and Hamiltonian fibrations. The main idea comes from the
fact that Gromov-Witten invariants may be defined parametrically.
Here we give the rough sketch of the main points. We need to
slightly restrict the form of fibrations we are interested in.
Definition 1.1
Let $M\to P\to B$ be a fibration whose local coefficient
system is trivial and the base is an $m$-dimensional finite CW-complex
with one dimensional top (co-) homology.
We call such a fibration simple. For example,
$B$ might be a simply connected compact manifold.
Let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ be a symplectic fibration which is simple.
(We will indicate where the simplicity of the fibration is
needed.)
Then
each fiber $M_{b}$, $b\in B$, admits a tamed almost
complex structure $J_{b}$. These
structures form a smooth family parameterized by the base
of the fibration (see Proposition 2.2).
Over each fiber, one has a moduli space
${\cal M}_{g,k}(A,\mathbb{J}_{b})$
of
$\mathbb{J}_{b}$-holomorphic stable maps representing a homology
class $A\in H_{2}(M;\mathbb{Z})$ such that $i_{*}(A)\neq 0$.
Here we use the assumption that the local
coefficient system of the fibration is trivial
in order to make sense that a map to a fiber $M_{b}$
represents the class $A\in H_{2}(M;\mathbb{Z})$.
These moduli spaces match up to a bigger moduli space denoted
by ${\cal M}^{P}_{g,k}(A,\mathbb{J})$.
Let $ev_{k}^{P}:{\cal M}_{g,k}^{P}(A,\mathbb{J})\to P\oplus\dots\oplus P$
be the evaluation of stable maps at $k$ marked points.
Out of the image of
this mapping we obtain a cycle whose homology class
$[ev_{k}({\cal M}_{g,k}(A,\mathbb{J}))]$
serves for defining a parametrized Gromov-Witten invariant.
Observe that the intersection of this image with
the fiber $M_{b}$ is the image of the evaluation map
$ev_{k}:{\cal M}_{g,k}(A,\mathbb{J}_{b})\to M_{b}\times\dots\times M_{b}$
from the moduli space associated to this fiber
(see Section 3).
Here we touch the main point (Theorem 4.2).
Let us assume that $E_{2}=E_{3}=\dots=E_{m}$ in the spectral
sequence associated to the fibration. It means that
all the differentials are trivial except possibly those
on $E_{m}$.
It follows from
the above discussion that the homology class
$[ev_{k}({\cal M}_{g,k}(A,\mathbb{J}_{b}))]$
made of the image of the evaluation map for
a generic fiber is the homological intersection
of the homology class coming from the parametrized
moduli space with the fiber of the appropriate
Whitney sum. This implies that the image of
$[ev_{k}({\cal M}_{g,k}(A,\mathbb{J}_{b}))]$
under the differential in the Leray-Serre
spectral sequence is zero.
A cohomological
version of this argument gives a formula which connects
Gromov-Witten invariant $\Phi_{A}$ and the differential
$\partial$ of the spectral sequence
(Theorem 4.4):
$$\sum_{i=1}^{k}(-1)^{\sum_{1\leq j<i}\deg({\alpha}_{j})}\Phi_{A}({\alpha}_{1},%
\dots,\partial{\alpha}_{i},\dots,{\alpha}_{k})=0$$
(see Section 4
for more precise statements and details).
1.2 A consequence for flux groups
We present two main applications of the above argument. The first
consists of an estimate of the rank of flux groups. This is
a completion of Theorem A from [K]. We need to introduce
some notions before giving the statement of the result.
Let $ev:Symp(M,\omega)\to M$ denotes the evaluation map,
111 Warning: Notice that we use two completely
different evaluation maps.
that is
$ev(\phi)=\phi(pt)$, where $pt\in M$ is a base point.
Definition 1.2
The image of
$ev_{*}:\pi_{1}(Symp(M,\omega))\to\pi_{1}(M)$
is called
symplectic Gottlieb group and denote
by $G(M,\omega)$. Similarly, the image of
$ev_{*}:H_{1}(Symp(M,\omega);\mathbb{Q})\to H_{1}(M;\mathbb{Q})$
we call a
rational symplectic Gottlieb group and denote
by $G_{Q}(M,\omega)$.
Recall that the definition of the classical Gottlieb
group uses the space of homotopy equivalences instead of
the group of symplectomorphisms [Got].
Of course symplectic Gottlieb group is a subgroup of
the classical one.
It is also
worth noticing that the nontriviality of
$G_{Q}(M,{\omega})$ is quite restrictive. Indeed, if this holds
then the rational cohomology of $M$ splits as
$H^{*}(M/S^{1})\otimes H^{*}(S^{1})$ [LM]. Moreover,
elements of Gottlieb group acts trivially on homotopy
groups.
Our method relies on the examination of the $J$-holomorphic
curves in total spaces symplectic fibrations.
That’s why the homology classes
represented by these curves have to belong to the image
of the map induced by the inclusion of fibers.
This is the motivation for the following notion.
Definition 1.3
A homology class $A\in H_{2}(M;\mathbb{Z})$ is said to be
flux free if for every $\xi\in\pi_{1}(Symp(M,\omega))$,
$i_{*}A\neq 0$. Here $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P_{\xi}\to S^{2}$ is the
symplectic fibration
associated to $\xi$ by the clutching construction.
Theorem 1.4
Let $(M,\omega)$ be a compact symplectic manifold.
Then the
rank of its flux group ${\Gamma}_{{\omega}}$ satisfies the following
estimate:
$${\hbox{\em rank }}G(M,\omega)\leq{\hbox{\em rank }}{\Gamma}_{{\omega}}\leq\dim
G%
_{Q}(M,\omega)+\dim\big{[}\ker(\cup[{\omega}]^{n-1})\cap\ker(\cup PDA)\big{]},$$
for any flux free class $A\in H_{2}(M;\mathbb{Z})$ such that
$\dim{\cal M}_{g,1}(A,J)=2$.
Usually, the above estimate
reduces to a simpler one, which does not involve the
Gottlieb group because it is trivial.
Moreover, in this case (cf. Proposition 5.2)
$${\Gamma}_{{\omega}}\subset\ker(\cup[{\omega}]^{n-1})\cap\ker(\cup PDA).$$
Recall that the flux conjecture depends only on the behavior
of the flux homomorphism on the loops with trivial
(in $\pi_{1}(M)$) evaluation [LMP1] (Proposition 1.2).
Namely, if the flux group is not discrete then
its nondicreteness occur in the above intersection of
the kernels. This argument easy shows that the conjecture
holds for manifolds with trivial $\ker\cup[{\omega}]^{n-1}$,
e.g. Kähler manifolds or more generally Lefschetz.
1.3 A consequence for Hamiltonian fibrations
The second application establishes the c-splitting conjecture
for manifolds whose Gromov-Witten invariants satisfy some
restrictions.
Theorem 1.5
Let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ be a compact Hamiltonian fibration.
Suppose that for any
${\alpha}\in H^{i}(M)$, where $i\leq 2n-4$,
there exist ${\beta}_{0},{\beta}_{1},\dots,{\beta}_{k}\in\text{im \!}i^{*}$ such
that
$$\Phi_{A}({\alpha}\cup{\beta}_{0},{\beta}_{1},\dots,{\beta}_{k})\neq 0$$
for some Gromov-Witten invariant $\Phi_{A}$.
Then the spectral sequence associated
to $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ collapses at $E_{2}$.
Remark 1.6
1.
One can choose the elements ${\beta}_{i}$ from the subalgebra
of $H^{*}(M)$ generated by the class of symplectic form,
Chern classes and elements of degree at most 2 (or 3 provided
that $B$ is a compact manifold). Indeed, they
are contained in $\text{im \!}i^{*}$ as follows. The class
of symplectic form is in this image because the fibration is
Hamiltonian. The Chern classes of $(M,\omega)$ are images of the Chern
classes of the vector bundle tangent to the fibers [K].
An argument for the elements of small degree is explained in
the proof of the theorem (Section 5.3).
2.
If $B$ is a manifold then the Poincare duality for $P$
allows us to consider elements ${\alpha}$ such that
$1\leq\deg({\alpha})<\frac{2n-b}{2}+1$, where $2n=\dim M$
and $b=\dim B$. The second inequality is an immediate consequence
of Poincaré duality for $P$ and the first one follows from the
fact that $[{\omega}]^{n}$ survives since the fibration is Hamiltonian.
3.
By the above observations,
the theorem seems to be relevant in dimension four.
But then c-splitting conjecture easy follows from the theorem
which establishes the conjecture for Hamiltonian fibrations
over 3-dimensional CW-complexes [LM] (Lemma 4.1.8).
The same argument
works also for simply connected 6-dimensional $(M,\omega)$.
A more explicit application is the
establishing c-splitting conjecture for projective space
blown-up along 4-dimensional submanifold (Section
5.4).
Acknowledgments. I am deeply grateful to Kaoru Ono
who patiently answered my questions and pointed out some
gaps in my arguments. I thank Dusa McDuff and Tomek Maszczyk
for valuable comments and remarks.
Also I thank the Max-Planck-Institute,
for providing a wonderful research atmosphere.
2 Symplectic fibrations
Definition 2.1
Let $(M,\omega)$ be a symplectic manifold. A fibration
$(M,\omega)\to P\to B$ is said to be symplectic if
its structure group is contained in the group $Symp(M,\omega)$
of symplectomorphisms of $(M,\omega)$.
Recall that an almost complex structure $J$ on $M$ is called
${\omega}$-tamed if ${\omega}(J\cdot,\cdot)$ defines
a Riemannian metric. Every symplectic manifold admits
a contractible nonempty set of ${\omega}$-tamed almost complex
structures. This fact can be extended to fibrations
as follows.
Proposition 2.2
Let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ be a symplectic fibration.
Then there exists a complex bundle $(Vert,\mathbb{J})\to P$ such
that $i_{b}^{*}\mathbb{J}$ is ${\omega}_{b}$-tamed for every $b\in B$.
The set of such almost complex structures $\mathbb{J}$ is
contractible.
Proof: We construct a bundle $TM\to Vert\to B$ using the differentials
of the transition functions of the fibration $P\to B$. The
the natural projection $Vert\to P$ defines a
symplectic vector bundle. This means that
the structure group of the bundle is $Sp(2n,\mathbb{R})$.
Reducing the structure group to the maximal compact
subgroup $U(n)$ we obtain required almost complex
structure. Since such a structure is a section of
the bundle whose fibers are contractible
($Sp(2n,\mathbb{R})/U(n)$), then the set of
of such structures is also contractible.
$\Box$
3 Parametrized Gromov-Witten invariants
Let $J$ be an ${\omega}$-tamed almost complex structure on $M$ and
${\Sigma}$ be a closed Riemann surface with complex structure $j$.
Let ${\cal M}_{g,k}(A,J)$ be the moduli space of
stable $J$-holomorphic maps of genus $g$ with $k$ marked
points representing the class $A$. This space is
a compactification of the moduli space of $J$-holomorphic maps
$u:{\Sigma}\to M$ with $k$ marked points, where ${\Sigma}$ is a
Riemann surface of genus $g$. The fundamental result of
Fukaya and Ono [FO](Theorem 1.3) states that the above
moduli space carries a kind of fundamental class of
${\cal M}_{g,k}(A,J)$ over $\mathbb{Q}$ in the following sense.
The $k$-point evaluation map
$$ev_{k}:{\cal M}_{g,k}(A,J)\to M\times...\times M$$
is defined by
$$ev_{k}(u,z_{1},...,z_{k})=(u(z_{1}),...,u(z_{k})).$$
Then the result of Fukaya and Ono says that
$[ev_{k}({\cal M}_{g,k}(A,J))]$ is well defined as a homology class
in $H^{*}(M\times\dots\times M;\mathbb{Q})$. This allows them to
define a Gromov-Witten
invariant as a map
$\Phi_{A}:H^{*}(M)\times\dots\times H^{*}(M)\to\mathbb{Q}$ by
$$\Phi_{A}({\alpha}_{1},...,{\alpha}_{k})=\left<a_{1}\times...\times a_{k},[ev_{%
k}({\cal M}_{g,k}(A,J))]\right>.$$
The concept of the Gromov-Witten invariants
can be generalized to the parametrized situation.
Namely, let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ be a symplectic fibration over
a finite simply connected CW-complex.
Then every fiber is, up to
symplectic isotopy, identified with $(M,\omega)$.
Suppose that $i_{*}(A)\neq 0$
for some $A\in H_{2}(M;\mathbb{Z})$. Then
for an almost complex structure $\mathbb{J}$ in
the vertical bundle of a symplectic fibration,
we consider the space ${\cal M}^{P}_{g,k}(A,\mathbb{J})$
of pairs $(u,b)$ where
$b\in B$ and $u:C_{g}\to P_{b}$
is a genus $g$, $\mathbb{J}_{b}$-holomorphic
map representing the class $A$. It does make sense to say that
some map ${\Sigma}\to P_{b}$ represents class $A$ since every
fiber is identified with $M$ up to symplectic isotopy.
A parametrized $k$-point evaluation map is defined
as follows.
$$\begin{CD}ev^{P}_{k}:{\cal M}^{P}_{g,k}(A,\mathbb{J})\to P\oplus\dots\oplus P%
\\
ev^{P}_{k}(u,b,z_{1},...,z_{k})=(u(z_{1}),...,u(z_{k})),\end{CD}$$
where $P\oplus\dots\oplus P$ is $k$-fold Whitney sum of the
fibration $P$:
$$\begin{CD}P\oplus\dots\oplus P:={\Delta}^{*}(P\times\dots\times P)@>{%
\widetilde{{\Delta}}}>{}>P\times\dots\times P\\
\qquad\qquad\qquad\qquad @V{}V{}V@V{}V{}V\\
\qquad\qquad\qquad\qquad B@>{{\Delta}}>{}>B\times\dots\times B\end{CD}$$
To define the parametrized Gromov-Witten invariant
we need to construct a homology class
$[ev^{P}_{k}({\cal M}_{g,k}^{P}(A,\mathbb{J}))]\in H_{*}(P\oplus\dots\oplus P)$. To do this one has
to observe that the construction of Fukaya and Ono
of the cycle $ev_{k}({\cal M}_{g,k}(A,J))$ can be done parametrically.
Finally the parametrized Gromov-Witten invariant
is a map
$\Phi^{P}_{A}:H^{*}(P)\times\dots\times H^{*}(P)\to\mathbb{Q}$ defined by
$$\Phi^{P}_{A}({\alpha}_{1},...,{\alpha}_{k})=\left<\widetilde{{\Delta}}^{*}({%
\alpha}_{1}\times\dots\times{\alpha}_{k}),\left[ev_{k}^{P}({\cal M}^{P}_{g,k}(%
A,\mathbb{J}))\right]\right>.$$
It is obvious from the definition that
$ev_{k}^{P}({\cal M}_{g,k}^{P}(A,\mathbb{J}))\cap(P\oplus\dots\oplus P)_{b}=ev_%
{k}({\cal M}_{g,k}(A,\mathbb{J}_{b}))$ for any symplectic fibration over
simply connected base. We need an analogous statement in
homology. In more detail, let
$$i_{!}:H_{k+m}(P\oplus\dots\oplus P)\to H_{k}(M\times\dots\times M)$$
be the homology transfer, where
$i:M\times\dots\times M\to P\oplus\dots\oplus P$ is an inclusion
of the fiber and $\dim B=m$. Recall that the transfer can be
expressed as follows.
$$i_{!}(a)=[C_{a}\cap i(M\times\dots\times M)],$$
where $C_{a}$ is a cycle representing the class $a$.
Lemma 3.1
Let $\mathbb{J}$ be an almost complex structure in $Vert$
compatible with the fibration.
Then
$$i_{!}\left[ev^{P}_{k}({\cal M}^{P}_{g,k}(A,\mathbb{J})\right]=[ev_{k}({\cal M}%
_{g,k}(A,\mathbb{J}_{b})],$$
for any $b\in B$.
Proof: This follows from the above observation that the construction of
the cycle $ev_{k}({\cal M}_{g,k}(A,J)$ in [FO] can
be done parametrically.
We proceed using the framework of [FO].
Fix a point $b\in B$ and construct Kuranishi structure
on the moduli space
${\cal M}_{g,k}(A,\mathbb{J}_{b})$.
Next choose
multivalued perturbation in order to get a virtual
fundamental cycle. Then extend the Kuranishi structure to
a Kuranishi structure on the parametrized moduli space
and also extend the multivalued pertubation (for
${\cal M}_{g,k}(A,\mathbb{J}_{b})$) to a multivalued pertubation
of the Kuranishi map for the parametrized moduli space.
For the above choice of the Kuranishi structure and
the multivalued perturbation, the intersection
$(P\oplus\dots\oplus P)_{b}\cap ev^{P}_{k}({\cal M}^{P}_{g,k}(A,\mathbb{J}))=ev%
_{k}({\cal M}_{g,k}(A,\mathbb{J}_{b})$ is transversal, which
completes the proof.
$\Box$
Remark 3.2
A similar argument should also work for the other approaches
to Gromov-Witten invariants, such as those presented by
Li and Tian [LT] or Siebert [Si].
The case of curves of genus zero was
already done by Seidel [Se]. Notice also that in the
weakly monotone case the above lemma might be proved using
genericity of almost complex structure involved.
Example 3.3
Let $(M,\omega):=(S^{2}\times S^{2},{\omega})$ be equipped with symplectic structure
such that $[\{x\}\times S^{2}]$ has area 1 and $[S^{2}\times\{x\}]$
has area 2. Let $A:=[S^{2}\times\{x\}]+[\{x\}\times S^{2}]$
There exist a symplectic fibration
$$(M,\omega)\to P\to S^{2}$$
which admits a compatible almost complex structure $\mathbb{J}$ in $Vert$
such that ${\cal M}_{0,3}(A,\mathbb{J}_{z})$ is empty for any $z\in S^{2}$ except
one point.
This phenomenon can be explained as follows (cf. [AM]).
The space of
${\omega}$-tamed almost complex structures contains an open
stratum consisting almost complex structures $J$ for which
class $A$ cannot be represented by any $J$-holomorphic
sphere and codimension 2 stratum consisting of almost
complex structures for which $A$ can be represented.
The complex structure $\mathbb{J}$ can be seen as a map
from $S^{2}$ to the space of ${\omega}$-tamed almost complex
structures which intersects transversely this codimension
2 stratum at one point. $\Box$
4 The main theorem
Our main results gives the restrictions to the form of the
differentials in the
Leray-Serre spectral sequences associated to symplectic
fibrations. We consider the spectral sequences for
fibration $P\to B$ as well as for $P\oplus\dots\oplus P\to B$.
Corresponding differentials in the homology spectral
sequences we denote by $\partial^{r}:E^{r}_{p,q}\to E^{r}_{p-r,q+r-1}$
and $\overline{\partial}^{r}:\overline{E}^{r}_{p,q}\to\overline{E}^{r}_{p-r,q+r-1}$, respectively. The analogous
differential in the cohomology spectral sequence we
will denote with subscripts.
4.1 Generalized Wang homomorphisms
Now we define a generalizations of the Wang homomorphisms,
which originally were associated to fibrations over homology
spheres (cf. [Sp] Section 9.5). Suppose
that the differentials in the Leray-Serre cohomology spectral
sequence associated to a simple fibration over
$m$-dimensional base
are trivial up to the $m^{\text{th}}$ term.
In other words $\partial^{m}$ (respectively $\partial_{m}$)
is the only nontrivial differential in the homology
(resp. cohomology) spectral sequence.
For the differential
$\partial_{m}:H^{0}(B)\otimes H^{k}(M)\to H^{m}(B)\otimes H^{k-m+1}(M)$
we define a generalized Wang homomorphism
$\partial:H^{k}(M)\to H^{k-m+1}(M)$ by
$$\partial_{m}(1\otimes{\alpha})={\beta}\otimes\partial{\alpha},$$
where ${\beta}\in H^{m}(B)$ is a generator dual to the fundamental
class $[B]$.
Lemma 4.1
The generalized Wang homomorphism:
1.
is natural with respect to bundle maps which induce
the identity on the top cohomology of the base and
2.
satisfies the Leibniz rule.
Proof: We use the fact that the differential in the spectral
sequence has required properties.
1.
Naturality:
Let $f:P_{1}\to P_{2}$ be a bundle map.
$$\displaystyle{\beta}\otimes\partial(f|M)^{*}{\alpha}=$$
$$\displaystyle\partial_{m}(1\otimes(f|_{M})^{*}{\alpha})=$$
$$\displaystyle\partial_{m}(f^{*}(1\otimes{\alpha}))=$$
$$\displaystyle f^{*}(\partial_{m}(1\otimes{\alpha}))=$$
$$\displaystyle f^{*}({\beta}\otimes\partial{\alpha})=$$
$$\displaystyle{\beta}\otimes(f|M)^{*}\partial{\alpha})$$
2.
The Leibniz rule:
$$\displaystyle{\beta}\otimes\partial({\alpha}_{1}\cup{\alpha}_{2})=$$
$$\displaystyle\partial_{m}(1\otimes({\alpha}_{1}\cup{\alpha}_{2})=$$
$$\displaystyle\partial_{m}((1\otimes{\alpha}_{1})\cup(1\otimes{\alpha}_{2}))=$$
$$\displaystyle\partial_{m}(1\otimes{\alpha}_{1})\cup(1\otimes{\alpha}_{2})\pm(1%
\otimes{\alpha}_{1})\cup\partial_{m}(1\otimes{\alpha}_{2})=$$
$$\displaystyle{\beta}\otimes\partial({\alpha}_{1})\cup(1\otimes{\alpha}_{2})\pm%
\cup(1\otimes{\alpha}_{1})\cup{\beta}\otimes\partial({\alpha}_{2})=$$
$$\displaystyle{\beta}\otimes(\partial({\alpha}_{1})\cup{\alpha}_{2}\pm{\alpha}_%
{1}\cup\partial({\alpha}_{2})).$$
$\Box$
The homology case, denoted by the same symbol,
$\partial:H_{k-m+1}(M)\to H_{k}(M)$ (that should not be confusing)
is defined analogously by
$$\partial^{m}([B]\otimes a)=[pt]\otimes\partial a.$$
It is also natural with respect to bundle maps which
induce the identity on the top homology of the base.
Moreover, it straightforward that homology and cohomology Wang homomorphisms
are dual to each other in the sense that
$$\left<\partial a,{\alpha}\right>=\left<a,\partial{\alpha}\right>.$$
4.2 The Wang homomorphism vanishes on Gromov-Witten invariants
Let $\overline{\partial}$ denote
the generalized Wang homomorphisms associated to the Whitney
sum of the fibration.
Theorem 4.2
Let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ be a simple (in the sense of Definition
1.1)
symplectic fibration over $m$-dimensional
base and
$A\in H_{2}(M;\mathbb{Z})$ be such that $i_{*}A\neq 0$.
Suppose that the differential
$\partial^{r}:E^{r}_{*,*}\to E^{r}_{*-r,*+r-1}$
is zero for $r\neq m$.
Then
$\overline{\partial}[ev_{k}({\cal M}_{g,k}(A,J))]=0$
for any moduli space ${\cal M}_{g,k}(A,J)$.
Proof: Since $E^{m}_{*,*}$ is the only term with possibly
nontrivial differential then so does $\overline{E}^{m}_{*,*}$.
Thus
there is the following commutative diagram in which the
upper row is exact
$$\!\!\!\!\!\!\!\begin{CD}{\scriptstyle\!\!0\to\overline{E}^{\infty}_{m,k}}@>{}>%
{}>{\scriptstyle\overline{E}^{m}_{m,k}}@>{{\overline{\partial}^{m}}}>{}>{%
\scriptstyle\overline{E}^{m}_{0,k+m-1}}\to{\scriptstyle\overline{E}^{\infty}_{%
0,k+m-1}\to 0}\\
@A{}A{}A\Big{\|}\Big{\|}\Big{\|}\\
{\scriptstyle H_{m+k}(P\oplus\dots\oplus P)}@>{i_{!}}>{}>{\scriptstyle H_{k}(M%
\times\dots\times M)}@>{\overline{\partial}}>{}>{\scriptstyle H_{k+m-1}(M%
\times\dots\times M)}\to{\scriptstyle i_{*}H_{k+m-1}(M\times\dots\times M)\to 0%
}\end{CD}$$
Here $H_{m+k}(P\oplus\dots\oplus P)\to\overline{E}^{\infty}_{m,k}=H_{m+k}(P\oplus%
\dots\oplus P)/F_{m-1}(H_{m+k}(P\oplus\dots\oplus P)$ is the projection
and $F_{s}$ denotes the filtration to which is associated
$\overline{E}^{\infty}$ (see [Sp]).
It follows from Lemma 3.1, that
$$i_{!}[ev^{P}_{k}({\cal M}^{P}_{g,k}(A,\mathbb{J}))]=[ev_{k}({\cal M}_{g,k}(A,J%
))].$$
Then, accordingly to the exactness of the above sequence,
we get that
$$\overline{\partial}[ev_{k}({\cal M}_{g,k}(A,J))]=0$$
as required. $\Box$
Remark 4.3
Of course, we can relax the assumption that there is the only
one possibly nontrivial differential. Indeed,
since the class $[{\cal M}_{g,k}(A,J)]=i_{!}[{\cal M}^{P}_{g,k}(A,\mathbb{J})]$
then it is in the kernel of any differential in the spectral
sequence.
The motivation for this assumption is the following. In case of
Hamiltonian fibrations, it is already proven that the first
nontrivial term in the spectral sequence is $E_{4}$. By taking
appropriate restrictions we can prove c-splitting in some
cases by induction.
On the other hand, in non Hamiltonian case we are mostly
interested in fibrations over $S^{2}$.
4.3 The Wang homomorphism is compatible with Gromov-Witten
invariants
The following theorem is a cohomological incarnation of
the previous one.
Theorem 4.4
Under the assumption of Theorem 4.2, the following
symmetry holds
$$\sum_{i=1}^{k}(-1)^{\sum_{1\leq j<i}\deg({\alpha}_{j})}\Phi_{A}({\alpha}_{1},%
\dots,\partial{\alpha}_{i},\dots,{\alpha}_{k})=0,$$
where ${\alpha}_{i}\in H^{*}(M)$. In particular,
$$\Phi_{A}(\partial{\alpha},{\alpha}_{1},\dots,{\alpha}_{k})=0$$
for ${\alpha}_{i}\in\ker\partial$.
Proof: This is the following computation, which uses the
naturality of the generalized Wang homomorphism and the fact that
it satisfies the Leibniz rule.
$$\displaystyle\sum_{i=1}^{k}(-1)^{\sum_{1\leq j<i}\deg({\alpha}_{j})}\Phi_{A}({%
\alpha}_{1},\dots,\partial{\alpha}_{i},\dots,{\alpha}_{k})=$$
$$\displaystyle\sum_{i=1}^{k}(-1)^{\sum_{1\leq j<i}\deg({\alpha}_{j})}\left<{%
\alpha}_{1}\times\dots\times\partial{\alpha}_{i}\times\dots\times{\alpha}_{k},%
[ev({\cal M}_{g,k}(A,J))]\right>=$$
$$\displaystyle\sum_{i=1}^{k}(-1)^{\sum_{1\leq j<i}\deg({\alpha}_{j})}\left<\pi_%
{1}^{*}{\alpha}_{1}\cup\dots\cup\pi_{i}^{*}\partial{\alpha}_{i}\cup\dots\cup%
\pi_{k}^{*}{\alpha}_{k},[ev({\cal M}_{g,k}(A,J))]\right>=$$
$$\displaystyle\sum_{i=1}^{k}(-1)^{\sum_{1\leq j<i}\deg({\alpha}_{j})}\left<\pi_%
{1}^{*}{\alpha}_{1}\cup\dots\cup\overline{\partial}\pi_{i}^{*}{\alpha}_{i}\cup%
\dots\cup\pi_{k}^{*}{\alpha}_{k},[ev({\cal M}_{g,k}(A,J))]\right>=$$
$$\displaystyle\left<\overline{\partial}({\alpha}_{1}\times\dots\times{\alpha}_{%
k}),[ev({\cal M}_{g,k}(A,J))]\right>=$$
$$\displaystyle\left<{\alpha}_{1}\times\dots\times{\alpha}_{k},\overline{%
\partial}[ev({\cal M}_{g,k}(A,J))]\right>=0$$
$\Box$
5 Consequences
It follows from the work of Lalonde, McDuff and Polterovich
[LMP1, M2] that the spectral sequence for Hamiltonian
fibration over $S^{2}$ degenerates at $E_{2}$ term. Moreover,
the same holds for Hamiltonian fibrations over any 3-dimensional
CW-complex and it is conjectured by Lalonde and McDuff
to be true in general [LM].
We divide the consequences of the main theorem into two parts.
First, symplectic fibrations over $S^{2}$ which are not
Hamiltonian. Second, Hamiltonian fibrations over arbitrary
bases.
5.1 Fibrations over $S^{2}$ and flux groups
Every symplectic fibration over $S^{2}$ is naturally
associated to an element $\xi\in\pi_{1}(Symp(M,\omega))$
by the clutching construction. We denote this fibration
by $P_{\xi}$. Recall that the evaluation map
$ev:Symp(M,\omega)\to M$ is defined by $ev(\phi)=\phi(pt)$,
where $pt\in M$ is a point.
Proposition 5.1
Let $(M,\omega)\to P_{\xi}\to S^{2}$ be a symplectic fibration
and $A\in H_{2}(M;\mathbb{Z})$ is such that $i_{*}A\neq 0$.
Suppose that $[ev_{*}(\xi)]\neq 0$ in homology.
Then
$$\Phi_{A}(\partial{\alpha}_{1},\dots,\partial{\alpha}_{k})=0,$$
for any Gromov-Witten invariant.
Proof: It is a result of Lalonde and McDuff [LM], that
$[ev_{*}(\xi)]\neq 0$ in homology iff
$\ker\partial=\text{im \!}\partial$. Thus the statement easily
follows from Theorem 4.4.
$\Box$
The next result gives the restriction for the rank of flux
groups. We recall the definition of these groups.
Flux homomorphism $F:\pi_{1}(Symp(M,\omega))\to H^{1}(M;\mathbb{R})$
is defined by
$$F(\xi)=\partial_{\xi}[{\omega}],$$
where $\partial_{\xi}$ is the Wang homomorphism
associated to the fibration $P_{\xi}$. By definition
the flux group ${\Gamma}_{{\omega}}$ is the image
of the flux homomorphism. The importance of flux groups
comes from the fact that they, in some sense, measure
the difference between the group of symplectomorphisms
$Symp(M,\omega)$
and the group of Hamiltonian symplectomorphisms
$Ham(M,\omega)$. Moreover,
the discreteness of flux groups in equivalent to closeness
of $Ham(M,\omega)$ in $Symp(M,\omega)$ (see [LMP1, MS1] for details).
Proposition 5.2
Let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P_{\xi}\to S^{2}$ be a symplectic fibration.
Suppose that $[ev_{*}(\xi)]=0$ in homology and $F(\xi)\neq 0$.
Then
$$F(\xi)\in\ker(\cup[{\omega}]^{n-1})\cap\ker(\cup PDmA)$$
for any $A\in H_{2}(M;\mathbb{Z})$ for which
$\dim{\cal M}_{g,1}(A,J)=2$ and $i_{*}A\neq 0$.
Here $m\in\mathbb{Q}$ denotes the virtual number of stable
curves representing the class $A$, that is
$mA=[ev_{1}({\cal M}_{g,1}(A,J))]$.
Proof: The statement that $F(\xi)\in\ker(\cup[{\omega}]^{n-1})$
was already proved in [K]. The proof relies on the fact that
the assumption that $[ev_{*}(\xi)]=0$ in homology implies
that $\partial_{\xi}[{\omega}]^{n}=0$. The latter is equal
to $n\partial_{\xi}[{\omega}]\cup[{\omega}]^{n-1}=F(\xi)\cup[{\omega}]^{n-1}$ and the statement follows.
The inclusion
$F(\xi)\in\ker(\cup PDmA)$
follows from Theorem 4.2 and the observation
that $mA=[ev_{1}({\cal M}_{g,1}(A,J))]$. Suppose that
$m\neq 0$, otherwise the statement is trivial.
Let ${\alpha}\in H^{1}(M)$ be an arbitrary element. Then
we compute that
$$\displaystyle\Big{<}F(\xi)\cup PDmA,PD{\alpha}\Big{>}=$$
$$\displaystyle\Big{<}F(\xi)\cup{\alpha},mA\Big{>}=$$
$$\displaystyle\Big{<}\partial_{\xi}[{\omega}]\cup{\alpha},[ev_{1}({\cal M}_{g,1%
}(A,J))]\Big{>}=$$
$$\displaystyle\Big{<}\partial_{\xi}([{\omega}]\cup{\alpha}),[ev_{1}({\cal M}_{g%
,1}(A,J))]\Big{>}=$$
$$\displaystyle\Big{<}[{\omega}]\cup{\alpha},\partial_{\xi}[ev_{1}({\cal M}_{g,1%
}(A,J))]\Big{>}=0.$$
The assumption on $[ev_{*}(\xi)]$ implies that the Wang
homomorphism is trivial on $H^{1}(M)$ which gives the
third equality.
$\Box$
Remark 5.3
The other assumptions which imply
the hypothesis of the above theorem are discussed in [K].
More informations about the
flux groups can be found in [LMP1].
5.2 A proof of Theorem 1.4
For convenience of the reader we recall the statement:
Theorem 1.4
Let $(M,\omega)$ be a compact symplectic manifold.
Then the
rank of its flux group ${\Gamma}_{{\omega}}$ satisfies the following
estimate:
${\hbox{\em rank }}G(M,\omega)\leq{\hbox{\em rank }}{\Gamma}_{{\omega}}\leq\dim
G%
_{Q}(M,\omega)+\dim\big{[}\ker(\cup[{\omega}]^{n-1})\cap\ker(\cup PDA)\big{]},$
for any flux free class $A\in H_{2}(M;\mathbb{Z})$
such that
$\dim{\cal M}_{g,1}(A,J)=2$.
Proof: We use the commutativity of the following diagram:
$$\xymatrix@1{\pi_{1}(Symp(M,\omega))\ar[r]^{-}{ev_{*}}\ar[dr]^{F}&\pi_{1}(M)\ar%
[r]&H_{1}(M;\mathbb{Z})\ar[r]^{-}{PD}&H^{2n-1}(M;\mathbb{Z})\ar[d]^{\cdot n}\\
&H^{1}(M;\mathbb{R})\ar[rr]^{\cup[{\omega}]^{n-1}}&&H^{2n-1}(M;\mathbb{R})}$$
which is easy to prove. Let us consider the first inequality.
Take elements
$\xi_{1},\dots,\xi_{k}$ ($k={\hbox{\em rank }}G(M,\omega)$) such that
$\sum a_{i}F(\xi_{i})=0$ for some nonzero $a_{i}\in\mathbb{Z}$.
This means that the element $\sum a_{i}\xi_{i}\in\pi_{1}(Symp(M,\omega))$
may be represented by a Hamiltonian loop. Hence their
evaluation $ev_{*}(\sum a_{i}\xi_{i})=\sum a_{i}ev_{*}(\xi_{i})=0$
due to results from Floer theory and we obtain the first inequality.
It may be confusing that we use addition in $\pi_{1}(M)$,
but it is in fact in the image of $\pi_{1}(Symp(M,\omega))$ which is abelian.
The proof of the second inequality requires more advanced
argument which relies on so called topological rigidity of
Hamiltonian loops [LMP2]:
If $\xi\in\pi_{1}(Diff(M))$ can be represented by loops
in $\pi_{1}(Symp(M,\omega))$ and $\pi_{1}(Symp(M,{\omega}^{\prime}))$,
then $\partial_{\xi}[{\omega}]=0$ iff $\partial_{\xi}[{\omega}^{\prime}]=0$
Let $K:=\ker ev_{*}:H_{1}(Symp(M,\omega);\mathbb{Q})\to H_{1}(M;\mathbb{Q})$. Then
we obtain the extension
$0\to F(K)\to{\Gamma}_{{\omega}}\to{\Gamma}_{{\omega}}/F(K)$,
where ${\Gamma}_{{\omega}}/F(K)$ has no torsion. Indeed,
if $F(\xi)\cdot F(K)$ were of finite order, say $k$ then it would mean
that $0=ev_{*}(k\xi)=kev_{*}(\xi)$ but $ev_{*}(\xi)\neq 0$ in
$H_{1}(M;\mathbb{Q})$ which is impossible. Hence we get that
$$\text{rank}{\Gamma}_{{\omega}}=\text{rank}F(K)+\text{rank}({\Gamma}_{{\omega}}%
/F(K)).$$
It follows from the above diagram that
$\text{rank}({\Gamma}_{{\omega}}/F(K))=\text{rank}G_{Q}(M,\omega).$ Thus we have only
to show that
$$\text{rank}F(K)\leq\dim\ker(\cup[{\omega}]^{n-1})\cap\ker(\cup PDA).$$
The argument is similar to that in [LMP2] (Theorem 2.D).
Let ${\omega}_{{\varepsilon}}$ be a rational symplectic form which is
a small perturbation of ${\omega}$. Then obviously
$\dim\ker\cup[{\omega}]^{n-1}\leq\dim\ker\cup[{\omega}_{{\varepsilon}}]^{n-1}.$
Moreover, the Gromov-Witten invariant $[ev_{1}({\cal M}_{g,1})]$ is
the same for both symplectic forms.
Suppose that there exist elements $\xi_{1},...,\xi_{l}\in\pi_{1}(Symp(M,\omega))$
such that $F(\xi_{1}),...,F(\xi_{l})\in F(K)$
are linearly independent over $\mathbb{Z}$ and
$l>$ dim $\ker\cup[{\omega}^{n-1}]\cap\ker\cup PDA$.
Let
$\xi_{1}^{\epsilon},...,\xi_{l}^{\epsilon}\in\pi_{1}(Symp(M,{\omega}_{\epsilon})$
are represented by perturbations of loops representing
$\xi_{1},...,\xi_{l}\in\pi_{1}(Symp(M,\omega))$.
Then the fluxes
$F_{\epsilon}(\xi_{j}^{\epsilon})$
are rational classes and
are contained in
$\ker\cup[{\omega}_{\epsilon}^{n-1}]\cap\ker\cup PDA$,
according to Proposition 5.2.
It follows that some of their
nontrivial combination over $\mathbb{Z}$ equals zero:
$\sum_{j}m_{j}F_{\epsilon}(\xi_{j}^{\epsilon})=0$
, $m_{j}\in\mathbb{Z}$, $j=1,...,k$.
Due to the topological rigidity of Hamiltonian loops, we get that
$\sum_{j}m_{j}F(\xi_{j})=0$
which gives the contradiction and completes the proof
$\Box$
5.3 Hamiltonian fibrations
The aim of this section is to prove Theorem 1.5.
The idea of the proof is to show that
an element ${\alpha}\in H^{*}(M)$ cannot be in the image
of the Wang homomorphism, provided that certain Gromov-Witten
invariant is nonzero. If any ${\alpha}\in H^{*}(M)$
has this property, then c-splitting holds for $(M,\omega)$.
Theorem 1.5
Let $(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ be a compact Hamiltonian fibration.
Suppose that for any
${\alpha}\in H^{i}(M)$, where $i\leq 2n-4$,
there exist ${\beta}_{0},{\beta}_{1},\dots,{\beta}_{k}\in\text{im \!}i^{*}$ such
that
$$\Phi_{A}({\alpha}\cup{\beta}_{0},{\beta}_{1},\dots,{\beta}_{k})\neq 0$$
for some Gromov-Witten invariant $\Phi_{A}$.
Then the spectral sequence associated to
$(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ collapses at $E_{2}$.
Proof: We use the induction on the dimension of the base.
Due to Lalonde and McDuff [LM] we know that
the spectral sequence collapses at $E_{2}$ for any
Hamiltonian fibration over the base of dimension
at most 3. This implies that $E_{2}=E_{3}=E_{4}$ in the
spectral sequence associated to an arbitrary fibration.
Suppose the fibration is does not c-split and $E_{m}$ is the
first nontrivial term in the spectral sequence.
Then there exists some $\eta\in H^{*}(M)$ such that
$\partial_{m}(1\otimes\eta)\neq 0$. We restrict
the fibration over an $m$-dimensional CW-complex
with one dimensional top cohomology such that
$\partial_{m}(1\otimes\eta)\neq 0$ in the induced
spectral sequence.
Let’s consider
the associated Wang homomorphism for which we have
that $\partial\eta\neq 0$.
First notice that
$\deg(\partial\eta)\leq 2n-4$.
Indeed, it
follows from the fact that
the Wang homomorphism coming from the first possible nontrivial
differential decreases the degree by 3 and
$\partial:H^{2n}(M)\to H^{2n-r+1}(M)$
is zero since the top cohomology is generated by $[{\omega}]^{n}$.
Since ${\beta}_{0},\dots,{\beta}_{k}\in\ker\partial$ then
Theorem 4.4 implies that
$$\Phi_{A}(\partial\eta\cup{\beta}_{0},{\beta}_{1},\dots,{\beta}_{k})=\Phi_{A}(%
\partial(\eta\cup{\beta}_{0}),{\beta}_{1},\dots,{\beta}_{k})=0,$$
which contradicts the assumption.
$\Box$
As a corollary we can recover the Blanchard theorem which
states that c-splitting holds for manifolds satisfying the
Hard Lefschetz condition.
Corollary 5.4 (Blanchard [Bl])
If $(M,\omega)$ satisfies the Hard Lefschetz condition (i.e.
$\cup[{\omega}]^{k}:H^{n-k}\to H^{n+k}$ is an isomorphism for
$k=1,...,n$, $\dim M=2n$), then any Hamiltonian fibration
$(M,\omega)\stackrel{{\scriptstyle i}}{{\longrightarrow}}P\stackrel{{%
\scriptstyle\pi}}{{\longrightarrow}}B$ over a compact manifold $B$ c-splits.
Proof: Consider the spherical ($g=0$) invariant
$\Phi_{0}:H^{*}(M)\times H^{*}(M)\times H^{*}(M)\to\mathbb{Q}$
for trivial homology class A=0 (see Section 3).
Clearly, it is
defined by the usual cup product, namely
$\Phi_{0}({\alpha},{\beta},{\gamma})=\left<{\alpha}\cup{\beta}\cup{\gamma},[M]\right>$
[MS2].
As usual we restrict ourselves to fibrations
with the only one possibly nontrivial term in the spectral
sequence.
Consider the Wang homomorphism $\partial$ in this case.
Suppose that $\partial\eta={\alpha}$, where ${\alpha}\in H^{k}(M)$ and $k<n$
is the least degree of nonzero element in the image of $\partial$.
The assumption $k\leq n$ is due to Remark 1.6 (2).
Let ${\beta}\in H^{2n-k}$ be such that
${\alpha}\cup{\beta}\neq 0$ in $H^{2n}(M)$.
Then
${\beta}={\gamma}\cup[{\omega}]^{n-k}$, due to Hard Lefschetz condition, and
$$\Phi_{0}({\alpha},[{\omega}]^{k},{\gamma})={\beta}\cup[{\omega}]^{k}\cup{%
\gamma}\neq 0.$$
Note that $\partial({\gamma})=0$ because $\deg(\partial({\gamma}))<k=deg({\alpha})$.
According to Theorem 1.5 and
Remark 1.6 (2),
we get the statement.
$\Box$
5.4 An explicit example
Let $(V,{\omega}_{V})$ be a compact 4-dimensional symplectic manifold.
According to results of Gromov [Gr] and
Tischler [Ti], there exists a symplectic
embedding $V\to{\mathbb{C}\mathbb{P}}^{5}$. Let’s consider a symplectic blow-up
$(M,\omega):=(\widetilde{{\mathbb{C}\mathbb{P}}^{5}}_{V},{\omega})$ of ${\mathbb{C}\mathbb{P}}^{5}$ along $V$.
The aim of this section is to prove the following
Proposition 5.5
Let $B$ be a compact manifold. Then, any Hamiltonian
fibration $(M,\omega)\to P\to B$ is c-split.
Remark 5.6
1.
The c-splitting conjecture is established for symplectic
manifolds satisfying the hard Lefschetz condition (e.g. Kähler
manifolds) [Bl]. Also, as we already have mentioned,
it is true for 4-dimensional and simply connected 6-dimensional
manifolds. The above proposition proves the c-splitting conjecture
for the family of symplectic manifolds which contains
the easiest examples which don’t satisfy the hard Lefschetz
condition [M1].
2.
Recall that the cohomology of the above blow-up has the following
form [M1, Gi]
$$H^{*}(M)=H^{*}({\mathbb{C}\mathbb{P}}^{5})\oplus H^{*}(V)[u]/u^{3}.$$
Proof of Proposition 5.5:
Throughout the proof we adopt the notation from the
appendix.
Consider the Leray-Serre spectral sequence associated
to the fibration $(M,\omega)\to P\to B$. Due to Lalonde and
McDuff [LM], we know that the differentials $\partial_{2}=\partial_{3}=0$.
Suppose that $\partial_{4}\neq 0$. We can restrict the fibration
over a compact 4-dimensional manifold, such that $\partial_{4}=\partial$ is
also nonzero.
Claim: $\partial:H^{5}(M)\to H^{2}(M)$ is zero.
Suppose that is there exists ${\alpha}\in H^{5}(M)$
such that $\partial({\alpha})=ax+bu$, where
$x=p^{*}{\omega}_{0}$, $u:=t^{*}(\tau)$ and $a,b\in\mathbb{R}$.
Since the symplectic structure om $M$ away from the
blow-up locus is the same as the standard structure
${\omega}_{0}$ on ${\mathbb{C}\mathbb{P}}^{5}$,
then
$$\Phi_{L}([{\omega}^{5}],[{\omega}^{5}])\neq 0,$$
where $L\in H_{2}(M)$ is the class of line.
For $a\neq 0$ we get that
$\Phi_{L}((ax+bu)\cup{1\over a}x^{4},[{\omega}^{5}])\neq 0$,
because $u\cup x^{4}=0$. It follows from Theorem 4.4
that for $a\neq 0$ $ax+bu\notin\text{im \!}\partial$.
Next we show that $u$ cannot be in the image of $\partial$.
To see this consider the following Gromov-Witten invariant
$$\Phi_{A}(u\cup t^{*}(\tau_{{\omega}V}),u\cup t^{*}(\tau_{{\omega}V}))\neq 0.$$
Since $\partial(t^{*}(\tau_{{\omega}V})=0$ because $M$ is simply connected,
then we obtain that $u\notin\text{im \!}\partial$. Thus we have proven
the claim.
The rest is easy now. By the Poincaré duality, it follows
from the claim that $\partial:H^{8}(M)\to H^{5}(M)$ is zero.
Notice that $H^{8}(M)$ is generated by $x^{4}$ and
$u\cup t^{*}(\tau_{{\omega}_{V}^{2}})$ hence we get that
$$\partial(t^{*}(\tau_{{\omega}_{V}^{2}}))=0,$$
which implies that $\partial H^{6}(M)\to H^{3}(M)$ is zero.
Again by the Poincaré duality we get that
$\partial H^{7}(M)\to H^{4}(M)$ is zero, which finishes
the proof.
The case of higher dimensional bases go through
in the same way.
$\Box$
Appendix A Appendix (by Jarosław Kȩdra and Kaoru Ono):
Simple examples of nontrivial Gromov-Witten invariants
A.1 The symplectic blow-up construction
Let $(M^{2n},{\omega}_{M})$ and $(V^{2m},\omega_{V})$ be compact
symplectic manifolds such that there exists
a symplectic embedding $(V,{\omega}_{V})\to(M,{\omega}_{M})$ of codimension
$2k+2$. We consider a symplectic blow-up of $(M,\omega)$
along $V$ denoted by $\widetilde{M}_{V}$.
It is obtained by cutting
out a small tubular neighborhood of $V$ in $M$ and
glue in a symplectic disc bundle over projectivized
normal bundle of $V$. More precisely, there exist a small
tubular neighborhood ${\cal N}_{V}$ of $V$ in $M$, which is symplectomorphic
to an ${\varepsilon}$-disc subbundle $\nu_{V}({\varepsilon})$
in the normal bundle $\nu_{V}$
of $V$. Since $\nu_{V}$ is a complex bundle then
then we have that
$\nu_{V}({\varepsilon})=P\times_{U(k+1)}D^{2k+2}_{{\varepsilon}}$,
where $P\to V$ is a $U(k+1)$-principal bundle.
We cut out the neighborhood ${\cal N}_{V}$ and glue back in
$P\times_{U(k+1)}\widetilde{D_{{\varepsilon}}}$, where
$\widetilde{D_{{\varepsilon}}}$ is the ${\varepsilon}^{\prime}$-blow-up
of the standard 2k+2-ball of radius ${\varepsilon}>{\varepsilon}^{\prime}$
(see [MS1] page 250 for further details).
It is easy to see that
$P\times_{U(k+1)}\widetilde{D_{{\varepsilon}}}$ is
symplectomorphic to the tautological disc bundle over
$\mathbb{P}_{V}$ the
projectivization of the normal bundle $\nu_{V}$.
Thus there is a natural embedding $\mathbb{P}_{V}\to\widetilde{M}_{V}$.
and we have the following commutative diagram in which
the horizontal arrows are symplectic embeddings.
$$\xymatrix{{\mathbb{C}\mathbb{P}}^{k}\ar[d]&\\
\mathbb{P}_{V}\ar[d]^{\pi}\ar[r]^{f}&\widetilde{M}_{V}\ar[d]^{p}\\
V\ar[r]^{i}&M}$$
The blow-up diagram
A.2 The choice of a homology class and an
almost complex structure
In order to define Gromov-Witten invariants we have to
chose a second
homology class and
an almost complex structure on $\widetilde{M}_{V}$. Moreover,
by the choice of this data we want to ensure that
some Gromov-Witten invariant is nontrivial. The idea is
to find a $J$-holomorphic stable map and to show that
this is the only element in its homology class intersecting
generic representatives of certain homology
classes.
The Choice of a homology class:
Let $u:{\mathbb{C}\mathbb{P}}^{1}\to\widetilde{M}_{V}$ be the composition of the
canonical embedding of the line into ${\mathbb{C}\mathbb{P}}^{k}$ with the
maps ${\mathbb{C}\mathbb{P}}^{k}\to\mathbb{P}_{V}\to\widetilde{M}_{V}$ of the above
diagram. Then we define $A:=u_{*}[{\mathbb{C}\mathbb{P}}^{1}]$.
The choice of an almost complex structure:
First we define an almost complex structure
$J(\mathbb{P}_{V})$ on $\mathbb{P}_{V}$ such that it is standard when
restricted to the fibers and the projection
$\pi:\mathbb{P}_{V}\to V$ is holomorphic with respect to
compatible almost complex structure on $V$.
We decompose the tangent bundle of $\mathbb{P}_{V}$ into the tangent bundle
along fibers $\hbox{\text{T}}_{vert}\mathbb{P}_{V}$ and the horizantal subbundle $Hor$
which is the symplectic complement of $\hbox{\text{T}}_{vert}\mathbb{P}_{V}$. Then we
define the
almost complex structure $J(\mathbb{P}_{V})$ as the direct sum of
standard complex structures on $\hbox{\text{T}}_{vert}\mathbb{P}_{V}$
and the pull back of a compatible almost
complex structure on $V$ to $Hor=\pi^{*}\hbox{\text{T}}V$.
With respect to this almost complex
structure, the projection $\pi$ becomes holomorphic.
Finally, the needed almost complex structure $J$
on $\widetilde{M}_{V}$ is an extension of $J(\mathbb{P}_{V})$ compatible
with the symplectic structure.
Lemma A.1
Let $v:{\mathbb{C}\mathbb{P}}^{1}\to\widetilde{M}_{V}$ be a $J$-holomorphic
curve representing the class $A=u_{*}[{\mathbb{C}\mathbb{P}}^{1}]$. Then the image
$v({\mathbb{C}\mathbb{P}}^{1})$ is contained in a fiber ${\mathbb{C}\mathbb{P}}^{k}$ of the
bundle $\pi:\mathbb{P}_{V}\to V$.
Proof: First we
show that the image of $v$ is contained in
$\mathbb{P}_{V}$. Start with the
observation that the intersecion pairing
$[\mathbb{P}_{V}]\circ A=-1$.
Indeed,
$$\displaystyle[\mathbb{P}_{V}]\circ A$$
$$\displaystyle=$$
$$\displaystyle\left<eu(\nu_{\mathbb{P}_{V}}),A\right>$$
$$\displaystyle=$$
$$\displaystyle\left<c_{1}(\nu_{\mathbb{P}_{V}}),u_{*}[{\mathbb{C}\mathbb{P}}^{1%
}]\right>$$
$$\displaystyle=$$
$$\displaystyle\left<c_{1}(u^{*}\nu_{\mathbb{P}_{V}}),[{\mathbb{C}\mathbb{P}}^{1%
}]\right>=-1$$
because the normal bundle of $\mathbb{P}_{V}$ restricted
to ${\mathbb{C}\mathbb{P}}^{k}\subset\mathbb{P}_{V}$ is the tautological
line bundle.
According to
the positivity of intersection of $J$-holomorphic
submanifolds, every such curve must be
cointained in $\mathbb{P}_{V}$.
The above agrument shows that $v:{\mathbb{C}\mathbb{P}}^{1}\to\mathbb{P}_{V}$
is a $J(\mathbb{P}_{V})$-holomorphic curve.
Hence, if $v:{\mathbb{C}\mathbb{P}}^{1}\to\mathbb{P}_{V}$ is a $J(\mathbb{P}_{V})$-holomorphic map, then
$\pi\circ v:{\mathbb{C}\mathbb{P}}^{1}\to V$ is also
holomorphic with respect to almost complex structure on $V$.
Since we know that a line contained in a fiber is
one of holomorphic curves in our homology class, $\pi\circ f$ is
null-homotopic,
especially, null-homologous. The only pseudo-holomorphic map in a symplectic
manifold, which is null-homologous, is constant.
Hence $\pi\circ v$ is a constant map, so $v$ is a map to one of fibers
of $\pi$.
$\Box$
A.3 Regularity and compactness
Let $u:{\mathbb{C}\mathbb{P}}^{1}\to(\widetilde{M}_{V},{\omega})$ be an inclusion of
the line as desribed above.
We prove the regularity of the almost complex structure
$J$ at $u$, which is stated in the following
Theorem A.2
The linearized operator
$$D_{u}=D\overline{\partial}_{J}(u):C^{\infty}(u^{*}\hbox{\text{T}}\widetilde{M}%
_{V})\to{\Omega}^{0,1}(u^{*}\hbox{\text{T}}\widetilde{M}_{V})$$
is surjective.
Proof: Let $u:{\mathbb{C}\mathbb{P}}^{1}\to(\widetilde{M}_{V},{\omega})$ be as above.
There is the following splitting of the pull back
of the tangent bundle. Recal that $\dim V=2m$
$$\displaystyle u^{*}(\hbox{\text{T}}\widetilde{M}_{V})$$
$$\displaystyle=$$
$$\displaystyle u^{*}(\hbox{\text{T}}\mathbb{P}_{V}\oplus\nu_{\mathbb{P}_{V}})$$
$$\displaystyle=$$
$$\displaystyle u^{*}({\cal E}^{m}\oplus\hbox{\text{T}}_{vert}\mathbb{P}_{V}%
\oplus\nu_{\mathbb{P}_{V}})$$
$$\displaystyle=$$
$$\displaystyle\underbrace{{\cal E}^{m}\oplus\hbox{\text{T}}{\mathbb{C}\mathbb{P%
}}^{k}|_{{\mathbb{C}\mathbb{P}}^{1}}}_{E}\oplus\underbrace{u^{*}(\nu_{\mathbb{%
P}_{V}})}_{N}$$
Here ${\cal E}^{n}$ denotes the trivial complex vector bundle of
rank $n$, $\hbox{\text{T}}_{vert}$ denotes the subbundle tangent
to the fibers of the fibration and $\nu_{\mathbb{P}_{V}}$
is the normal bundle to $\mathbb{P}_{V}$ in $\widetilde{M}_{V}$.
With respect to the above splitting, the operator
$D_{u}:C^{\infty}(E\oplus N)\to{\Omega}^{0,1}(E\oplus N)$
has the following matrix form
$$\left[\begin{array}[]{cc}D_{u}|_{E}&*\\
0&Du|_{N}\end{array}\right].$$
Now we have to show that appropriate restricted operators
are surjective.
To deal with the operator $D_{u}|_{N}$ we use the
result of Hofer, Lizan and Sikorav [HLS] (Theorem $1^{\prime}$ ).
It states that if $L\to{\Sigma}$ is a holomorphic line bundle
over a Riemannian surface of genus $g$, then a generalized
$\overline{\partial}$-operator is surjective, provided that
$c_{1}(L)\geq 2g-1$. Thus we have to check that
$c_{1}(\nu_{\mathbb{P}_{V}})\geq-1$. But this is clearly satisfied,
because the normal bundle restricted to the fiber of
$\mathbb{P}_{V}$ is just the tautological line bundle
over ${\mathbb{C}\mathbb{P}}^{k}$. Hence its first Chern class is -1.
To show surjectivity of $D_{u}|_{E}$ we again argue with
the direc decomposition of $E$ into ${\cal E}^{m}$ and
$\hbox{\text{T}}{\mathbb{C}\mathbb{P}}^{k}|_{{\mathbb{C}\mathbb{P}}^{1}}$.
Since $u:{\mathbb{C}\mathbb{P}}^{1}\to{\mathbb{C}\mathbb{P}}^{k}$ is the standard
holomorphic embedding and the structure $J$ restricted to
${\mathbb{C}\mathbb{P}}^{k}$ is the stanard integrable one,
then the factor $\hbox{\text{T}}{\mathbb{C}\mathbb{P}}^{k}|_{{\mathbb{C}\mathbb{P}}^{1}}$ is preserved.
Moreover, the restriction of
the operator $D_{u}$ is the usual Cauchy-Riemann
derivative in this case. According to Lemma 3.5.1 in [MS2] we obtain
that $D_{u}$ restricted to $\hbox{\text{T}}{\mathbb{C}\mathbb{P}}^{k}|_{{\mathbb{C}\mathbb{P}}^{1}}$ is surjective.
The surjectivity on the trivial factor ${\cal E}^{m}$ again follows
from [HLS].
$\Box$
A.4 The main result
In this section we prove the nontriviality of certain
Gromov-Witten invariant of the blow-up $\widetilde{M}_{V}$.
Theorem A.3
Let ${\alpha},{\beta}\in H^{*}(\widetilde{M}_{V};\mathbb{Z})$ be cohomology classes.
Suppose that
$$\pi_{*}(\hbox{\text{PD}}f^{*}({\alpha}))\circ\pi_{*}(\hbox{\text{PD}}f^{*}({%
\beta}))=1\in H_{0}(V).$$
Then the Gromov-Witten invariant
$\Phi_{A}({\alpha},{\beta})\neq 0$.
Proof: First note that
$$\hbox{\text{PD}}f^{*}({\alpha})=f_{!}(\hbox{\text{PD}}{\alpha})=[\mathbb{P}_{V%
}\cap C_{{\alpha}}],$$
where
$f_{!}:H_{*}(\widetilde{M}_{V})\to H_{*-2}(\mathbb{P}_{V})$ is the homology
transfer map and $C_{{\alpha}}$ denotes a cycle representing
the homology class Poincare dual to ${\alpha}$.
Thus the latter is the homology class in $\mathbb{P}_{V}$ obatined by
the intersection of the cycle $C_{{\alpha}}$ with
$\mathbb{P}_{V}$.
The condition
$\pi_{*}(\hbox{\text{PD}}f^{*}({\alpha}))\circ\pi_{*}(\hbox{\text{PD}}f^{*}({%
\beta}))=1$
implies that for generic cycles representing the
homology classes $\hbox{\text{PD}}f^{*}({\alpha})$ and $\hbox{\text{PD}}f^{*}({\beta})$
there exists an odd number of fibers, say
$\pi^{-1}(x_{1}),...,\pi^{-1}(x_{2l-1})$,
intersected by each of the cycles in exactly one
point.
Let $v:{\mathbb{C}\mathbb{P}}^{1}\to\widetilde{M}_{V}$ be a $J$-holomorphic
curve in the class $A$ that contributes to the Gromov-Witten
invariant $\Phi_{A}({\alpha},{\beta})$.
Since any $J$-holomorphic curve
in the class $A$ has to be contained in some fiber
of $\pi$ (Lemma A.1), then we have that
$v({\mathbb{C}\mathbb{P}}^{1})\subset\pi^{-1}(x_{i})={\mathbb{C}\mathbb{P}}^{k}$ for some
$i=1,2,...,2l-1$.
Thus it follows that $\Phi_{A}({\alpha},{\beta})\neq 0$, because
in each fiber $\pi^{-1}(x_{i})={\mathbb{C}\mathbb{P}}^{k}$ there is exactly
one line passing through two generic points and the
number of the fibers is odd.
$\Box$
Example A.4
Let ${\alpha}=a^{k}\in H^{2k}(\widetilde{M}_{V})$, where $a\in H^{2}(\widetilde{M}_{V})$ is the
class Poincaré dual to $[\mathbb{P}_{V}]$. Then
$f^{*}(a)=c_{1}(\nu_{\mathbb{P}_{V}})$ is the first Chern class of the normal
bundle of $\mathbb{P}_{V}$.
Recall that this normal bundle when restricted to the fibre
${\mathbb{C}\mathbb{P}}^{k}$ of $\mathbb{P}_{V}$ is the tautological line bundle
over projective space.
This implies that $f^{*}(a)^{k}=f^{*}({\alpha})$ restricts
to the positive generator in $H^{2k}({\mathbb{C}\mathbb{P}}^{k})$.
We claim that $\pi_{*}(\hbox{\text{PD}}f^{*}({\alpha})=[V]$ the fundamental
class of $V$. This follow from the following computation.
Let ${\cal V}$ denote the positive
generator of the top cohomology of $V$.
$$\displaystyle\left<\pi_{*}(\hbox{\text{PD}}f^{*}({\alpha}),{\cal V}\right>$$
$$\displaystyle=$$
$$\displaystyle\left<(\hbox{\text{PD}}f^{*}({\alpha}),\pi^{*}({\cal V})\right>$$
$$\displaystyle=$$
$$\displaystyle\left<f^{*}({\alpha})\cup\pi^{*}({\cal V}),[\mathbb{P}_{V}]\right>$$
$$\displaystyle=$$
$$\displaystyle 1.$$
Let ${\beta}\in H^{2n-2}(\widetilde{M}_{V})$ be the class Poincare
dual to $-A\in H^{2}(\widetilde{M}_{V})$. The computation in the proof
of Lemma A.1 shows that
$-A\cdot[\mathbb{P}_{V}]=1$. Note that
$\hbox{\text{PD}}f^{*}({\beta})=f_{!}(-A)=1\in H_{0}(\mathbb{P}_{V})$ hence
$\pi_{*}(\hbox{\text{PD}}f^{*}({\beta}))=1\in H_{0}(V)$.
Finally we obtain that
$$\pi_{*}(\hbox{\text{PD}}f^{*}({\alpha}))\circ\pi_{*}(\hbox{\text{PD}}f^{*}({%
\beta}))=1\in H_{0}(V)$$
and according to Theorem A.3 we get that
$\Phi_{A}({\alpha},{\beta})\neq 0$.
Now we want to give more general version of the
above example. In order to do this,
consider the following composition of maps
$$\begin{CD}\mathbb{P}_{V}@>{f}>{}>\widetilde{M}_{V}@>{t}>{}>Th(\nu_{\mathbb{P}_%
{V}}),\end{CD}$$
where $Th(\nu_{\mathbb{P}_{V}})$ is the Thom space.
Let $\tau_{a}\in H^{*}(Th(\nu_{\mathbb{P}_{V}}))$ be an element
corresponding to given
$a\in H^{*}(\mathbb{P}_{V})$ under the Thom isomorphism.
Corollary A.5
Let $a=\pi^{*}(v)\cup c_{1}(\nu_{\mathbb{P}_{V}})^{k-1}$ and
$b=\pi^{*}(w)\cup c_{1}(\nu_{\mathbb{P}_{V}})^{k-1}$, where
$v\cup w={\cal V}$ the positive generator of the top cohomology
of $V$. Then
$$\Phi_{A}(t^{*}(\tau_{a}),t^{*}(\tau_{b}))\neq 0.$$
Proof: First observe that
$$f^{*}(t^{*}(\tau_{a}))=a\cup c_{1}(\nu_{\mathbb{P}_{V}}),$$
because first Chern class equals the Euler class in this case.
This implies the first equality in the followin computation.
$$\displaystyle\pi_{*}(f^{*}(\tau_{a}))\circ\pi_{*}(f^{*}(\tau_{b}))=$$
$$\displaystyle\pi_{*}[\hbox{\text{PD}}(\pi^{*}(v)\cup c_{1}(\nu_{\mathbb{P}_{V}%
})^{k})]\circ\pi_{*}[\hbox{\text{PD}}(\pi^{*}(w)\cup c_{1}(\nu_{\mathbb{P}_{V}%
})^{k})]=$$
$$\displaystyle\left<\pi^{!}(\pi^{*}(v)\cup c_{1}(\nu_{\mathbb{P}_{V}})^{k})\cup%
\pi^{!}(\pi^{*}(w)\cup c_{1}(\nu_{\mathbb{P}_{V}})^{k}),[V]\right>=$$
$$\displaystyle\left<v\cup w,[V]\right>=1.$$
The second one is the definition of the transfer and
the third follows from the fact that the transfer is
a homomorphism of modules. The statement follows by
the direct application of the Theorem A.3.
$\Box$
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Authors addresses:
Jarosław Kȩdra
Institute of Mathematics
University of Szczecin
ul.Wielkopolska 15
70-451 Szczecin
Poland
e-mail: [email protected]
Kaoru Ono
Department of Mathematics
Hokkaido University
Sapporo 060-0810
Japan
email: [email protected] |
Effect of pore-size disorder on the electronic properties of semiconducting graphene nanomeshes
Sarah Gamal${}^{1}$
Mohamed M. Fadlallah${}^{2,3}$
[email protected]
Lobna M. Salah${}^{1}$
Ahmed A. Maarouf${}^{4}$
[email protected]
${}^{1}$Department of Physics, Faculty of Science, Cairo University, Giza 12613, Egypt
${}^{2}$Physics Department, Faculty of Science, Benha University, Benha 13518, Egypt
${}^{3}$Center for Computational Energy Research, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands
${}^{4}$Department of Physics, Institute for Research and Medical Consultations, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia
(November 18, 2020)
Abstract
Graphene nanomeshes (GNMs) are novel materials that recently raised a lot of interest. They are fabricated by forming a lattice of pores in graphene. Depending on the pore size and pore lattice constant, GNMs can be either semimetallic or semiconducting with a gap large enough ($\sim$ 0.5 eV) to be considered for transistor applications. The fabrication process is bound to produce some structural disorder due to variations in pore sizes. Recent electronic transport measurements in GNM devices (ACS Appl. Mater. Interfaces 10, 10362, 2018) show a degradation of their bandgap in devices having pore-size disorder. It is therefore important to understand the effect of such variability on the electronic properties of semiconducting GNMs. In this work we use the density functional-based tight binding formalism to calculate the electronic properties of GNM structures with different pore sizes, pore densities, and with hydrogen and oxygen pore edge passivations. We find that structural disorder reduces the electronic gap and the carrier group velocity, which may interpret recent transport measurements in GNM devices. Furthermore the trend of the bandgap with structural disorder is not significantly affected by the change in pore edge passivation. Our results show that even with structural disorder, GNMs are still attractive from a transistor device perspective.
ÙTight binding density functional theory, graphene, graphene nanomeshes, antidot graphene, disordered pores
I Introduction
Graphene is probably one of the most studied materials in the past decade Geim (2007); Novoselov et al. (2004), due to its extraordinary optical, mechanical, and electronic properties. Its high mobility (15000 cm${}^{2}$/Vs) Novoselov et al. (2004); Castro Neto et al. (2009) makes it a potential replacement for silicon in electronics Castro Neto et al. (2009), but a zero bandgap prevents its utilization as a field effect transistor (FET). Many approaches have been used to open a bandgap ($E_{g}$) in graphene based structures, such as placing a graphene bilayer in a vertical electric field Ohta et al. (2006), controlled interactions with a substrate Zhou et al. (2007), and quantum confinement in graphene nanoribbons Celis et al. (2016).
Another way for opening a bandgap in the graphene spectrum is through the creation of a lattice of pores, thus forming a graphene nanomesh (GNM) Lopata et al. (2010); Jippo and Ohfuchi (2013). GNMs have been attracting research interest due to their advantages over graphene, namely their electronic and chemical properties. GNMs with specific pore symmetries and size range possess a fractional eV electronic bandgap Maarouf et al. (2013), which is very attractive to the transistor industry. Chemical functionalization is possible through controlled pore edge passivation, which can make GNMs highly and selectively reactive Boukhvalov and Katsnelson (2008). Pore passivation by species different from carbon creates pore edge dipoles resulting from the mismatch in electronegativity between carbon and the passivating species. On the other hand, the gap opening in GNMs is not as simple, and certain symmetry considerations have to be satisfied for a gap to open. GNMs have been synthesized with pore sizes in the range of 5-20 nm Wallace (1947); Ding et al. (2014); Bai et al. (2010); Zhang et al. (2016).
Many studies have focused on the electronic Pedersen et al. (2008); Ouyang et al. (2011); Oswald and Wu (2012); Şahin and
Ciraci (2011), electronic transport Nguyen et al. (2012); Jippo et al. (2011a, b), and thermoelectric transport Eldeeb et al. (2018); Fadlallah et al. (2017); Nobakht et al. (2017); Feng and Ruan (2016) properties of perfect (disorder free) GNMs. The bandgap depends on the geometrical details of the pore lattice, including the pore shape, size, and pore lattice constant Pedersen et al. (2008); Liu et al. (2009a). Generally for hexagonal pores, the bandgap is found to be proportional to the ratio between the number of removed carbon atoms at the total number of carbon atoms in the unit cell Pedersen et al. (2008). For example, a hexagonal lattice of 0.8 nm pores with a pore-to-pore distance of about 2.3 nm has a 0.5 eV bandgap Eldeeb et al. (2018); Fadlallah et al. (2017). Highly stable p- and n-doping of GNM semiconducting structures have been predicted, where controlled pore edge passivation facilitates the trapping of electron accepting and donating species Maarouf et al. (2013); Eldeeb et al. (2018). This promoted GNMs to be considered in some applications, such as FETs Xia et al. (2010); Lin et al. (2009), molecular sensors Eldeeb et al. (2018); Cao et al. (2019); Afzali et al. (2014a), nanoparticles support templates Afzali et al. (2014b), hydrogen storage Fadlallah et al. (2019), spintronic devices Fadlallah et al. (2017), and supercapacitors Chi et al. (2020); Peng et al. (2015).
A few techniques have been used to fabricate GNMs, including nanosphere lithography Sinitskii and Tour (2010), nanoimprint lithography Liang et al. (2010), block copolymer lithography Bai et al. (2010); Kim et al. (2010), irradiation etching Fischbein and DrndiÄ (2008); Wang et al. (2020), atomic force microscopy etching Zhang et al. (2012), and oxygen reactive ion etching Liu et al. (2014); Tang et al. (2017). GNMs fabricated with these methods will most likely have structural defects, such as variations in the size and shape of the pores, as well as the pore lattice structure and symmetry. Structural defects can severely alter the electronic and transport properties of semiconductors. If GNMs are to be seriously considered as candidates for graphene-based transistors, it becomes crucial to understand the extent to which structural defects degrade their electronic properties, especially the bandgap size, and the carrier velocity.
There are numerous tight binding (TB) studies of graphene based structures. The popularity of the TB method resides on its successful reproduction of the low energy spectrum of graphene and carbon nanotubes. The electronic properties of GNMs have been thoroughly studied using TB Pedersen et al. (2008); Fürst et al. (2009); Ouyang et al. (2011). Disorder in the pore shape have even been considered, where it is found that for strong disorder, the bandgap varies inversely with the average pore lattice constant Hung Nguyen et al. (2013), a trend that is in qualitative agreement with some experimental results Liang et al. (2010). Despite these efforts, a DFT-based method is required to quantitatively investigate the disorder arising from various geometrical and chemical factors affecting the electronic and transport properties of GNMs. Such approach allows for a more realistic treatment of various defects arising from the fabrication process, such as local stresses and pore edge passivations, and hence provides a more accurate prediction of the bandgap and carrier velocity.
In this work, we study disorder in the pore size and pore lattice constant of hydrogen and oxygen passivated GNMs, as would be induced by the fabrication processes, using a density functional theory based TB (DFTB) approach. We study the electronic properties of GNMs with pores of multiple sizes: 0.36 nm, 0.83 nm, and 1.29 nm, and two pore lattice constants: 1.47 nm, and 2.21 nm. We calculate the density of states (DOS), band structure, and carrier group velocity. This allows us to investigate the dependence of the bandgap on various GNM parameters (pore size, lattice constant, edge passivation), as well as different structural permutations of the same pore configuration.
Section II describes the methodology which is used in this work. To understand the electronic properties of disordered pores in graphene, we start in III-A with the results of GNMs having one pore size, using 3 base GNM supercells: 12$\times$12 system with 4 pores, 18$\times$18 system with 4 pores, and 18$\times$18 system with 9 pores. The results include the DOS, band structures, the dependence of the bandgap on the pore radius and the pore lattice constant. In addition, we compare the results of DFTB to two sets of corresponding results obtained by DFT, so as to validate the use of the DFTB method with our GNM structures. GNMs, with pore size and pore lattice constant disorder, and with two passivations, are discussed in III-B. In certain cases, all pore permutations of the same disordered GNM are studied. In this work, we consider a total of 56 GNM systems.
II Methodology
To calculate the electronic properties of the considered large systems, we employ the self-consistent charge (SCC) density functional tight binding scheme (SCC-DFTB), as implemented in the DFTB+ package Aradi et al. (2007), where the total energy is a second-order expansion of the DFT energy in the charge density fluctuations Frauenheim et al. (2000); Koskinen and Mäkinen (2009). The SCC formalism describes the Coulombic interaction between atomic charges. It has been successfully used to calculate the electronic properties of bulk, surface, and nanowire systems Hellström et al. (2013); Guo et al. (2017), molecular materials Dolgonos et al. (2010), as well as some biological systems Elstner (2006). The success of DFTB to describe the electronic properties of many systems has been demonstrated through numerous comparisons with DFT, for example to calculate the structural and electronic properties of 2D phosphorous carbide polymorphs Heller et al. (2018), or to study the effect of structural changes on the electronic properties of GaN nanowires Ming et al. (2016). Generally, DFTB agrees well with DFT and can therefore be utilized to study large-scale systems that are impossible to study with DFT.
The mio Slater-Koster parameter library contains all possible atom-atom interactions in our GNM structures Elstner et al. (1998). The conjugate gradient algorithm is used to perform the structural relaxation of the studied GNM structures, with atomic forces less than $10^{-4}$ eV/Å. Convergence with respect to the k-points grid was achieved for various GNM systems. Table 1 shows the grid for each GNM system. A 12 Å vacuum distance is used to avoid image interaction in the $z$-direction.
III Electronic structure of disordered GNMs
There are two players when it comes to gap opening in graphene. First, there is the mixing of the distinct $K$ and $K^{\prime}$ points. One can think of the pore lattice as an external potential acting on the graphene electrons. For a bandgap to open up at the Dirac points, the external potential associated with the pore lattice has to mix states at the two distinct $K$ points. For a triangular lattice of pores with a lattice parameter which is a multiple of 3 of the graphene lattice parameter, there will be a gap that increases with the pore radius. One third of these GNMs will be semiconducting Tang et al. (2014). Second, there are the pore edge effects, which are akin to the bandgaps of graphene nanoribbons Son et al. (2006). Below, we first discuss the electronic properties of H-GNMs with pores of equal size. We then move to disordered GNMs having pores of different size.
III.1 GNMs with one pore size - disorder-free case
We begin with H-GNMs having hydrogen passivated pores. We consider 3 pore sizes, which we term ”small”, ”medium”, and ”large”. Figure 1a shows a 12$\times$12 unit cell with 4 small pores, while figure 1b shows an 18$\times$18 unit cell with 9 small pores. When all pores are equal, the two systems reduce to a simpler one with the 6$\times$6 unit cell shown in Fig. 1c. Figures 1d,e and f are the corresponding systems with the medium pore. The small pore H-GNM has 8.3% of its carbon atoms removed, while the medium pore H-GNM loses 33.3% of its carbon atoms. Structural relaxation of the 6$\times$6 unit cells gives pore diameters of 0.36 nm and 0.83 nm. The 12$\times$12 and 18$\times$18 configurations allow us to investigate the effect of the pore size and the pore density on the electronic properties of H-GNMs.
The DOS and the band structures of 6$\times$6 H-GNMs are shown in Fig. 2. Both H-GNMs are semiconducting with electronic bandgaps of 0.75 eV, and 1.55 eV, for small and medium pore H-GNMs, respectively. The electronic structures and bandgap values agree well with previous DFT calculations Maarouf et al. (2013). A comparison of DFTB and DFT results is discussed below. For the small pore H-GNM, the states in the vicinity of the bandgap are delocalized, and the DOS grows almost linearly away from the valence and conduction band edges. The medium pore H-GNM has two flat bands surrounding its bandgap, indicating the presence of states localized at pore edge.
We also consider cases with a different pore density (Fig. 3a, c, and e), where we have 18$\times$18 systems with 4 pores, and with 3 different pore sizes. When all 4 pores are equal, the unit cells reduce to the 9$\times$9 cells shown in Fig. 3b, d, and f. The percentages of the removed carbon atoms are 3.7%, 14.8%, and 33.3%, for the small, medium, and large pore H-GNM, respectively. Structural optimization of these H-GNMs gives pore diameters of 0.36 nm, 0.83 nm, and 1.21 nm. The corresponding electronic bandgaps are 0.35 eV, 0.65 eV, and 0.82 eV, respectively.
The DOS, the band structures (Fig. 4), and the bandgap values agree well with previously published DFT results (Wang et al., 2020). The small and medium pore H-GNMs have qualitatively similar DOS and band structures, characterized by a low energy linear DOS, with delocalized states. The bandgap of the large pore H-GNM is bounded by flat bands arising from states localized at the pore edge. It should be noted here that flat bands are observed in the 6$\times$6 medium pore H-GNM, and in the 9$\times$9 large pore H-GNM, where both H-GNMs have 33.3% of their carbon atoms removed. The presence of the localized states in these two cases can be explained by edge effects Tang et al. (2014).
In addition to the systems with 6$\times$6 and 9$\times$9 unit cells discussed so far, we have also studied 12$\times$12, 15$\times$15, and 18$\times$18 systems, with each of the 3 considered pores. This allows us to further confirm the suitability of the DFTB method for studying these H-GNM systems. The dependence of the electronic bandgap on the pore diameter in each of these systems is shown in Fig. 5a, where we get the expected trend for the bandgap Liu et al. (2009b); Tang et al. (2014). The same results can be analyzed with respect to the pore lattice constant (Fig. 5b), and the bandgap can be fitted to the form: $E_{g}=(8.27/M^{2})\exp(x)$ Tang et al. (2014), where $x=0.40R_{a}-0.03R_{a}^{2}$, $R_{a}=(2r/a_{0}-1)$, and $r$ is the pore radius, $a_{0}=2.64$ Å is the graphene lattice constant, and $M$ indicates the periodicities of the unit cell of the GNM. It should be noted here that the 15$\times$15 and 18$\times$18 systems are almost impossible to be studied with DFT calculations due to their huge size and number of electrons.
The pore radius and the pore lattice parameter define the neck width of the GNM, which is the smallest graphene thickness between two adjacent pore edges. For a given pore radius, GNMs with certain neck widths will have their bandgaps surrounded by flat bands resulting from states localized at the pore edge Tang et al. (2014). We see examples of this case in Figs. 2d and 4f.
Electronic properties of graphene based systems using the DFTB method have been shown to agree well with those obtained from DFT-based methods Zobelli et al. (2012). Nevertheless, we to further validate our approach, we still compare the electronic properties of GNM basic systems used in this work, using DFTB, and DFT with two types of bases. Figure 6 shows the DOS of three GNMs, 6$\times$6 small pore, 9$\times$9 medium pore, and 12$\times$12 large pore, calculated using DFTB+, SIESTA (DFT, localized orbitals basis) Kasry et al. (2019), and Quantum Espresso (DFT, plane wave basis) Maarouf et al. (2013); Sinitskii and Tour (2010); Liang et al. (2010). Table 2 shows the bandgap values obtained using these packages. As we see, the DFTB DOS results and the bandgap values agree well with those of DFT, which demonstrates the advantage of the DFTB method to study relatively larger systems that are formidable to DFT calculations. Although hybrid functionals usually provide more accurate bandgap estimates, the system sizes considered in our study are beyond the reach of those functionals. Nevertheless, the bandgap of the 6$\times$6 small pore system calculated using the HSE functional was found to be 0.88 eV, i.e. slightly larger than the GGA value of 0.75 eV (supplementary information of Ref. Maarouf et al. (2013)).
III.2 GNMs with different pore sizes - Structural disorder
Now that we have established a baseline that we can compare to, we move to systems with unit cells having more than one pore size. As explained before, these systems are built from the base 6$\times$6 and 9$\times$9 systems. We begin by considering H-GNMs with small and medium pores. We have two supercell cases, 12$\times$12 (Fig. 1a,d), and 18$\times$18 (Fig. 1b,e).
In the 12$\times$12 case, we will assume we have two small pores and two medium pores. There are 3 different configurations for such a H-GNM, arising from the different permutations of the pores in the supercell (insets, Fig. 7a). As we see, all three H-GNMs have virtually equal bandgaps ($E_{g}\sim$ 0.21 eV), and identical DOS, which suggests that the electronic properties of H-GNMs might not be highly dependent on the fine details of their structural disorder.
To further test this result, we study the case of 18$\times$18 H-GNM with 7 small and 2 medium pores. Initially one may think that there are many permutations, but the triangular symmetry of the H-GNM supercell renders many of them identical. Figure 7b shows the 6 different configurations of this H-GNM system and their DOS. The bandgap ranges between 0.12 eV and 0.18 eV for all those systems, and their electronic spectra are very similar. This further suggests that, as a first approximation, one may safely assume that the fine details of the structural disorder do not significantly affect the electronic properties of disordered H-GNMs. Therefore, we will consider only one configuration when varying the structural disorder of each H-GNM system
We first consider H-GNMs systems with 4 pores, and with two pore sizes. Pore size can be small, medium, or large. We begin by systems where all 4 pores have one size. Different disorder configurations are obtained by continuously replacing one of the 4 pores with a pore of a different size, until we reach the other limiting case of 4 pores of the second size, with a total of 5 configurations. Each disordered system is structurally optimized, and its band structure and DOS are calculated. The disorder may not only affect the size of the electronic bandgap, and consequently the ON/OFF ratio of an H-GNM-based field effect transistor (FET) device Bai et al. (2010); Berrada et al. (2013), but it can also lead to a degradation of the electronic mobility through a diminished carrier group velocity Maarouf et al. (2013). To investigate this effect, we calculated the group velocity of carriers in the linear region of the conduction band of various studied H-GNMs.
Figure 8 shows the variation of $E_{g}$ (left $y$-axis) and $v_{g}$ (right $y$-axis) of the 12$\times$12 and 18$\times$18 H-GNMs with different disorder configurations. We design the plots such that the two disorder free structures are the limiting cases on the far left and far right of the $x$-axis. Figure 8a shows $E_{g}$ and $v_{g}$ of the structurally disordered 12$\times$12 H-GNM system, with small and medium pores. The bandgap decreases between the two disorder-free values of 1.55 eV and 0.72 eV, with the smallest bandgap of 0.15 eV for the H-GNM system with 3 small pores and one medium pore (structure D). The group velocity follows a trend that is roughly proportional to the bandgap, and ranges between 0.2 and 0.8 of the velocity of pristine graphene, which is still attractive from a FET device perspective.
Figure 8b shows $E_{g}$ and $v_{g}$ of the disordered 18$\times$18 H-GNM system made from medium and large pores. Structural disorder causes the bandgap to decrease below the two disorder-free limiting cases (0.8 eV and 0.65 eV), with the smallest bandgap (0.07 eV) being for the H-GNM with one medium and three small pores (structure C). For the two other disordered systems (B and D), $E_{g}\sim 0.2$ eV and $v_{g}\sim 0.1$. Figure 8c shows that the bandgap of the 18$\times$18 H-GNM with medium and small pores follows a similar trend, where it decreases between the two limiting cases of 0.65 eV and 0.34 eV. The maximum decrease is for the H-GNM system having one medium and three small pores, with $E_{g}\sim 0.1$ eV, and a $v_{g}\sim 0.1$ (structure D). The last 18$\times$18 4 pore H-GNM system is that with large and small pores (Fig. 8d). $E_{g}$ maintains a monotonic decrease between the disorder-free values of 0.8 eV and 0.34 eV, and $v_{g}\geq 0.2$ for two of the three disorder cases.
We now move to the 18$\times$18 H-GNM systems with 9 pores, and two pore sizes, small and medium. We introduce disorder by following a procedure similar to that of the 4-pore H-GNM systems. The larger number of pores allows for a larger number of disordered H-GNM configurations. The disorder axis (Fig. 9, $x$-axis) has the H-GNM with 9 medium pore on the far left (point A), and the H-GNM with the 9 small pores on the far right (point J), with 8 disordered configurations in between. These configurations are obtained by continuously replacing one medium pore with one small pore. The $y$-axis is as before. $E_{g}$ exhibits a trend similar to that of the 12$\times$12 case, decreasing between the two disorder free values of 1.55 eV and 0.72 eV, and vanishing for the structure with 3 medium and 6 small pores (point G). For all other disordered cases, $E_{g}$ is between 0.2 and 0.8 eV, and $v_{g}$ is between 0.2 and 0.7, both are very lucrative from a device perspective.
Our calculations may help interpret the recent experimental transport measurements made on GNMs by Schmidt et al Schmidt et al. (2018). They used helium ion beam milling to pattern suspended graphene, creating pores with diameters of about 3.5 nm, and with a pitch of about 18 nm. They measured the I/V characteristics of suspended GNMs at room temperature, reporting a gap of about 0.45 eV for one device. Their results include another GNM device with the same pore parameters, but with some missing pores, was found to have no gap. This agrees well with our theoretical predictions, and provides experimental evidence that disorder in pore lattice may cause a reduction in the bandgap of a GNM.
All systems presented so far had their pore edges passivated with hydrogen. In general, and irrespective of how the GNM is fabricated, the pores will be passivated by some chemical species. The most common passivations considered are hydrogen, oxygen, and nitrogen Maarouf et al. (2013); Eldeeb et al. (2018). For this, we studied a set of oxygen-passivated GNMs. Figure 10 compares the dependence of the electronic bandgap on structural disorder for an oxygen-passivated and the hydrogen passivated 12$\times$12 GNM systems. As we see, the trend is not altered by the change in passivation, and the variations in the values of the bandgap can be attributed to the local deformations occurring near the passivation sites. The group velocity is generally between 0.2 and 0.9 of that in graphene, which is very pleasing from a device perspective. In certain cases though flat bands exist due to states localized at the pore edge, in which case the group velocity is very small.
Other forms of disorder that may be considered in future work include pores with unequal number of removed $A$ and $B$ sublattices carbon atoms. This may arise in many situations, such as with irregularly shaped pores Tang et al. (2014); Sun et al. (2019), triangular pores Yang et al. (2011), or with circular/hexagonal pores centered on carbon atoms. This results in the presence of midgap localized states, as well as magnetism with a magnetic moment that is proportional to the difference between the $A$ and $B$ atoms. This is not unique to graphene, but rather to bipartite lattices, as predicted by Lieb’s theorem Lieb (1989).
IV Conclusion
The electronic properties of structurally disordered graphene nanomeshes (GNMs) were studied using the density functional based tight binding method. We defined structurally disordered GNM as one whose unit cell has two different pore sizes. We calculated band structures and densities of states for 56 GNM systems, with different pore sizes, different permutations of the pores, and different passivations (hydrogen and oxygen). We found that the disorder in the pore size causes the energy gap to decrease, albeit to a value that is still technologically attractive. In addition, the carriers group velocity is typically between 0.2 and 0.9 of the velocity in pristine graphene, again suggesting the plausibility of a GNM-based field effect transistor. Furthermore, changing the location of pores in the unit cell seems to have little effect on the electronic properties of a GNM. These results apply for the two considered pore passivations. Our results indicate that the utilization of GNMs for nanoelectronic devices is possible, even with some pore size variability resulting during the fabrication process.
Acknowledgment
The authors would like to acknowledge the support of the
supercomputing facility at the Bibliotheca Alexandrina, Alexandria,
Egypt. We would also like to thank B. Aradi and Ben Hourahine for fruitful
discussions.
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Correlation Functions
of the integrable higher-spin
XXX and XXZ spin chains through the fusion method
Tetsuo Deguchi${}^{1}$111e-mail [email protected]
and Chihiro Matsui${}^{2,3}$
222e-mail [email protected]
( )
Abstract
For the integrable higher-spin XXX and XXZ spin chains
we present multiple-integral representations for
the correlation function of an arbitrary product of
Hermitian elementary matrices in the massless ground state.
We give a formula expressing it by a single term of multiple integrals.
In particular, we explicitly derive
the emptiness formation probability (EFP).
We assume $2s$-strings for the ground-state solution
of the Bethe ansatz equations for the spin-$s$ XXZ chain,
and solve the integral equations for the spin-$s$ Gaudin matrix.
In terms of the XXZ coupling $\Delta$ we define $\zeta$
by $\Delta=\cos\zeta$,
and put it in a region $0\leq\zeta<\pi/2s$
of the gapless regime: $-1<\Delta\leq 1$ ($0\leq\zeta<\pi$),
where $\Delta=1$ ($\zeta=0$) corresponds to the antiferromagnetic point.
We calculate the zero-temperature correlation functions
by the algebraic Bethe ansatz, introducing the Hermitian elementary
matrices in the massless regime, and
taking advantage of the fusion construction of
the $R$-matrix of the higher-spin representations of the
affine quantum group.
${}^{1}$
Department of Physics, Graduate School of Humanities and Sciences,
Ochanomizu University
2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan
${}^{2}$
Department of Physics, Graduate School of Science, the University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
${}^{3}$ CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan
1 Introduction
The correlation functions of the spin-1/2 XXZ spin chain have
been studied extensively through the algebraic Bethe-ansatz
during the last decade [1, 2, 3, 4, 5, 6].
The multiple-integral representations
of the correlation functions for the infinite lattice at zero temperature
first derived through the affine quantum-group symmetry
[7, 8] and
also by solving the $q$-KZ equations [9, 10]
have been rederived and then generalized into those
for the finite-size lattice under non-zero magnetic field.
They are also extended into those at finite temperatures [11].
Furthermore, the asymptotic expansion of a correlation function
has been systematically discussed [12].
Thus, the exact study of the correlation functions of the XXZ spin chain
should be not only very fruitful but also quite fundamental
in the mathematical physics of integrable models.
Recently, the correlation functions and form factors of the
integrable higher-spin XXX spin chains and
the form factors of the integrable higher-spin XXZ spin chains
have been derived by the algebraic Bethe-ansatz method
[13, 14, 15].
In the spin-1/2 XXZ chain the Hamiltonian
under the periodic boundary conditions is given by
$${\cal H}_{\rm XXZ}={\frac{1}{2}}\sum_{j=1}^{L}\left(\sigma_{j}^{X}\sigma_{j+1}%
^{X}+\sigma_{j}^{Y}\sigma_{j+1}^{Y}+\Delta\sigma_{j}^{Z}\sigma_{j+1}^{Z}\right%
)\,.$$
(1.1)
Here $\sigma_{j}^{a}$ ($a=X,Y,Z$) are the Pauli matrices defined
on the $j$th site and $\Delta$ denotes the XXZ coupling.
We define parameter $q$ by
$$\Delta=(q+q^{-1})/2\,.$$
(1.2)
We define $\eta$ and $\zeta$ by $q=\exp\eta$ and $\eta=i\zeta$,
respectively. We thus have $\Delta=\cos\zeta$.
In the massless regime: $-1<\Delta\leq 1$,
we have $0\leq\zeta<\pi$ for the spin-1/2 XXZ spin chain (1.1).
At $\Delta=1$ (i.e. $q=1$), the Hamiltonian (1.1)
corresponds to the antiferromagnetic Heisenberg (XXX) chain.
The solvable higher-spin generalizations of the XXX and XXZ spin chains
have been studied by the fusion method in several references
[16, 17, 18, 19, 20, 21, 22, 23]. The spin-$s$ XXZ Hamiltonian is derived from the spin-$s$ fusion
transfer matrix (see also section 2.6).
For instance, the Hamiltonian of the integrable
spin-$1$ XXX spin chain is given by
$${\cal H}^{(2)}_{\rm XXX}={\frac{1}{2}}\sum_{j=1}^{N_{s}}\left({\vec{S}}_{j}%
\cdot{\vec{S}}_{j+1}-({\vec{S}}_{j}\cdot{\vec{S}}_{j+1})^{2}\right)\,.$$
(1.3)
Here ${\vec{S}}_{j}$ denotes the spin-1 spin-angular momentum
operator acting on the $j$th site among the $N_{s}$ lattice
sites of the spin-$s$ chain.
For the general spin-$s$ case, the integrable
spin-$s$ XXX and XXZ Hamiltonians denoted
${\cal H}^{(2s)}_{\rm XXX}$ and ${\cal H}^{(2s)}_{\rm XXZ}$, respectively,
can also be derived systematically.
The correlation functions of integrable higher-spin XXX and XXZ spin chains
are associated with various topics of mathematical physics.
For the integrable spin-1 XXZ spin chain
correlation functions have been derived
by the method of $q$-vertex operators
through some novel results of
the representation theory of the quantum algebras
[24, 25, 26, 27, 28].
They should be closely related to the higher-spin solutions
of the quantum Knizhnik-Zamolodchikov equations [10].
For the fusion eight-vertex models,
correlation functions have been discussed
by an algebraic method [29].
Moreover, the partition function of the six-vertex model
under domain wall boundary conditions have been extended
into the higher-spin case [30].
In a massless region $0\leq\zeta<\pi/2s$,
the low-lying excitation spectrum at zero temperature
of the integrable spin-$s$ XXZ chain
should correspond to the level-$k$ $SU(2)$ WZWN model with $k=2s$.
By assuming the string hypothesis
it is conjectured that the ground state
of the integrable spin-$s$ XXX Hamiltonian
is given by $N_{s}/2$ sets of $2s$-strings [31].
It has also been extended into the XXZ case [32].
The ground-state solution of $2s$-strings
is derived for the spin-$s$ XXX chain
through the zero-temperature limit of
the thermal Bethe ansatz [18].
The low-lying excitation spectrum
is discussed in terms of spinons for the spin-$s$ XXX and XXZ
spin chains [31, 32].
Numerically It was shown that
the finite-size corrections to the ground-state energy
of the integrable spin-$s$ XXX chain are consistent
with the conformal field theory (CFT) with $c=3s/(s+1)$
[33, 34, 35, 36].
Here $c$ denotes the central charge of the CFT.
It is also the case with the integrable spin-$s$ XXZ chain
in the region $0\leq\zeta<\pi/2s$ [37, 38, 39].
The results are consistent with the conjecture
that the ground state of the integrable spin-$s$
XXZ chain with $0\leq\zeta<\pi/2s$
is given by $N_{s}/2$ sets of $2s$-strings
[22, 32, 37, 38, 39, 40, 41, 42].
Furthermore, it was shown analytically that the low-lying
excitation spectrum of the integrable spin-$s$ XXZ chain in the
region $0\leq\zeta<\pi/2s$
is consistent with the CFT of $c=3s/(s+1)$ [41, 42].
In fact, the low-lying excitation spectrum of spinons
for the spin-$s$ XXX chain
is described in terms of the level-$k$ $SU(2)$ WZWN model
with $k=2s$ [43].
In the paper we calculate zero-temperature correlation functions
for the integrable higher-spin XXZ spin chains
by the algebraic Bethe-ansatz method.
For a given product of elementary matrices
we present the multiple-integral representations of the
correlation function in the region $0\leq\zeta<\pi/2s$
of the massless regime near the antiferromagnetic point ($\zeta=0$).
For an illustration, we derive the multiple-integral representations
of the emptiness formation probability (EFP) of the
spin-$s$ XXZ spin chain, explicitly.
Here the spin $s$ is given by an arbitrary positive integer or half-integer.
Assuming the conjecture that
the ground-state solution of the Bethe ansatz equations
is given by $2s$-strings for the regime of $\zeta$,
we derive the spin-$s$ EFP for a finite chain
and then take the thermodynamic limit. We solve the integral
equations associated with the spin-$s$ Gaudin matrix
for $0\leq\zeta<\pi/2s$,
and express the diagonal elements
in terms of the density of strings.
Here we remark that the integral
equations associated with the spin-$s$ Gaudin matrix have not been
explicitly solved, yet, even for the case of the integrable
higher-spin XXX spin chains [13].
We also calculate the spin-$s$ EFP for the homogeneous chain where
all inhomogeneous parameters $\xi_{p}$ are given by zero.
Here we shall introduce inhomogeneous parameters $\xi_{p}$
for $p=1,2,\ldots,N_{s}$, in §2.4.
Furthermore, we take advantage of the fusion construction
of the spin-$s$ $R$-matrix in the algebraic Bethe-ansatz
derivation of the correlation functions [15].
Given the spin-$s$ XXZ spin chain on the $N_{s}$ lattice sites,
we define $L$ by $L=2sN_{s}$ and consider the spin-1/2 XXZ spin chain
on the $L$ sites with inhomogeneous parameters $w_{j}$
for $j=1,2,\ldots,L$. In the fusion method
we express any given spin-$s$ local operator
as a sum of products of operator-valued elements
of the spin-1/2 monodromy matrix in the limit of
sending inhomogeneous parameters $w_{j}$ to
sets of complete $2s$-strings as shown in Ref. [15].
Here, we apply the spin-1/2 formula
of the quantum inverse scattering problem [4],
which is valid at least for generic inhomogeneous parameters.
Therefore, sending inhomogeneous parameters $w_{j}$
into complete $2s$-strings,
we can evaluate the vacuum expectation values
or the form factors of spin-$s$ local operators which are expressed
in terms of the spin-1/2 monodromy matrix elements
with generic inhomogeneous parameters $w_{j}$.
Here, the rapidities of the ground state satisfy
the Bethe ansatz equations with inhomogeneous parameters $w_{j}$.
We assume in the paper that the Bethe roots are
continuous with respect to inhomogeneous parameters $w_{j}$,
in particular, in the limit of sending $w_{j}$ to complete $2s$-strings.
We can construct higher-spin transfer matrices
by the fusion method [22, 23].
Here we recall that the spin-1/2 XXZ Hamiltonian (1.1)
is derived from the logarithmic derivative
of the row-to-row transfer matrix of the six-vertex model.
We call it the spin-1/2 transfer matrix and denote it
by $t^{(1,1)}(\lambda)$.
Let us express by $V^{(\ell)}$ an $(\ell+1)$-dimensional vector space.
We denote by $T^{(\ell,\,2s)}(\lambda)$
the spin-$\ell/2$ monodromy matrix acting on
the tensor product of the auxiliary space
$V^{(\ell)}$ and the $N_{s}$th tensor product of the quantum spaces,
$(V^{(2s)})^{\otimes N_{s}}$. We call it of type
$(\ell,(2s)^{\otimes N_{s}})$, which we express ($\ell,2s$) in the
superscript. Taking the trace of
the spin-$\ell/2$ monodromy matrix $T^{(\ell,\,2s)}(\lambda)$
over the auxiliary space
$V^{(\ell)}$, we define the spin-$\ell/2$ transfer matrix,
$t^{(\ell,\,2s)}(\lambda)$.
For $\ell=2s$, we have the spin-$s$ transfer matrix
$t^{(2s,\,2s)}(\lambda)$,
and we derive the spin-$s$ Hamiltonian from its logarithmic derivative.
We construct the ground state $|\psi_{g}^{(2s)}\rangle$ of the spin-$s$
XXZ Hamiltonian by the $B$ operators of the $2$-by-$2$ monodromy matrix
$T^{(1,2s)}(\lambda)$.
As shown by Babujan, the spin-$s$ transfer matrix $t^{(2s,2s)}(\lambda)$
commutes with the spin-1/2 transfer matrix $t^{(1,2s)}(\lambda)$
due to the Yang-Baxter relations,
and hence they have eigenvectors in common [18].
The ground state $|\psi_{g}^{(2s)}\rangle$ of the spin-$s$ XXZ spin chain
is originally an eigenvector of the spin-$s$ transfer matrix
$t^{(2s,2s)}(\lambda)$, and consequently it is also
an eigenvector of the spin-$1/2$ transfer matrix $t^{(1,2s)}(\lambda)$.
Therefore, the ground state $|\psi^{(2s)}_{g}\rangle$
of the spin-$s$ XXZ spin chain can be constructed
by applying the $B$ operators of the $2$-by-$2$ monodromy matrix
$T^{(1,2s)}(\lambda)$ to the vacuum.
We can show that the fusion $R$-matrix corresponds
to the $R$-matrix of the affine quantum group $U_{q}(\widehat{sl_{2}})$.
We recall that by the fusion method, we can construct the $R$-matrix acting on
the tensor product $V^{(\ell)}\otimes V^{(2s)}$
[16, 17, 18, 19, 20, 21, 22, 23]. We denote it by $R^{(\ell,2s)}$.
In the affine quantum group, the $R$-matrix
is defined as the intertwiner of
the tensor product of two representations $V$ and $W$
[44, 45, 46].
Due to the conditions of the intertwiner
the $R$-matrix of the affine quantum group is determined uniquely
up to a scalar factor [47],
which we denote by $R_{V,W}$.
Therefore, showing that the fusion $R$-matrix satisfies
all the conditions of the intertwiner, we prove that the fusion $R$-matrix
coincides with the $R$-matrix of the quantum group, $R_{V,W}$.
Consequently, for $\ell=2s$ the fusion $R$-matrix, $R^{(2s,2s)}(\lambda)$,
becomes the permutation operator when spectral parameter $\lambda$
is given by zero. This property of the $R$-matrix
plays a central role in the derivation of the integrable
spin-$s$ Hamiltonian. It is also fundamental
in the inverse scattering problem in the spin-$s$ case [48].
There are several relevant and interesting studies
of the integrable spin-$s$ XXZ spin chains.
The expression of eigenvalues of the spin-$s$ XXX transfer matrix
$t^{(2s,2s)}(\lambda)$ was derived by Babujan
[17, 18, 21]
through the algebraic Bethe-ansatz method.
It was also derived by solving the series
of functional relations
among the spin-$s$ transfer matrices [22].
The functional relations are systematically generalized to
the $T$ systems [49].
Recently, the algebraic Bethe ansatz
for the spin-$s$ XXZ transfer matrix
has been thoroughly reviewed and reconstructed
from the viewpoint of the algebraic Bethe ansatz
of the $U(1)$-invariant integrable model
[50, 51].
Quite interestingly, it has also been applied to
construct the invariant subspaces associated with the
Ising-like spectra of the superintegrable chiral Potts
model [52].
The content of the paper consists of the following. In section 2,
we introduce the $R$-matrix for the spin-1/2 XXZ spin chain.
We then introduce conjugate basis vectors in order to formulate
Hermitian elementary matrices $\widetilde{E}^{m,\,n}$
in the massless regime where $|q|=1$.
We define the massless higher-spin monodromy matrices
$\widetilde{T}^{(\ell,\,2s)}(\lambda)$
in terms of the conjugate vectors,
after reviewing the fusion construction of the massive
higher-spin monodromy matrices
$T^{(\ell,\,2s)}(\lambda)$
and higher-spin XXZ transfer matrices,
$t^{(\ell,\,2s)}(\lambda)$, for $\ell=1,2,\ldots$, as follows.
We express the matrix elements of
$T^{(\ell,\,2s)}_{0,12\cdots N_{s}}(\lambda)$
in terms of those of the spin-1/2 monodromy matrix
$T^{(1,\,1)}_{0,12\cdots L}(\lambda)$.
Here $T^{(1,\,1)}_{0,12\cdots L}(\lambda)$ is defined on
the tensor product of the two-dimensional auxiliary space
$V_{0}^{(1)}$ and the $L$th tensor product
of the 2-dimensional quantum space, $(V^{(1)})^{\otimes L}$.
Here we recall $L=2s\times N_{s}$.
In the fusion construction [15],
monodromy matrix $T^{(1,\,2s)}_{0,12\cdots N_{s}}(\lambda)$
acting on the $N_{s}$ lattice sites is
derived from monodromy matrix
$T^{(1,\,1)}_{0,12\cdots L}(\lambda)$ acting
on the $2sN_{s}$ lattice sites by setting inhomogeneous parameters $w_{j}$
to $N_{s}$ sets of complete $2s$-strings and by multiplying it by
the $N_{s}$th tensor product of projection operators which project
$(V^{(1)})^{\otimes 2s}$ to $V^{(2s)}$.
In section 3, we explain the method for
calculating the expectation values of given
products of spin-$s$ local operators.
We express the local operators in terms of global operators with
inhomogeneous parameters $w_{j}^{(2s;\,\epsilon)}$, which are defined to be
close to complete $2s$-strings with small deviations of $O(\epsilon)$,
and evaluate the scalar products and the expectation values
for the Bethe state with the same
inhomogeneous parameters $w_{j}^{(2s;\,\epsilon)}$.
Then, we obtain the expectation values, sending $\epsilon$ to 0.
Here we note that the projection operators
introduced in the fusion construction
commute with the matrix elements of monodromy
matrix $T^{(1,\,1)}$ of inhomogeneous parameters
$w_{j}^{(2s;\,\epsilon)}$ with $O(\epsilon)$ corrections.
In section 4 we calculate
the emptiness formation probability (EFP) for the spin-$s$ XXZ spin chain
for a large but finite chain, and then evaluate the
matrix $S$ which is introduced for expressing
the EFP of an infinite spin-$s$ XXZ chain
in the massless regime with $\zeta<\pi/2s$.
Here we solve explicitly the integral equations for the spin-$s$
Gaudin matrix,
expressing the $2s$-strings of the ground-state solution
systematically in terms of the string centers.
In section 5, we present explicitly
the multiple-integral representation of the spin-$s$ EFP.
We also derive it for the inhomogeneous chain where
all the inhomogeneous parameters $\xi_{p}$ are given by 0.
For an illustration, we calculate the multiple-integral representation
of spin-1 EFP for $m=1$, $\langle\widetilde{E}^{2,\,2}\rangle$, explicitly.
In the XXX limit, the value of
$\langle\widetilde{E}^{2,\,2}\rangle$ approaches 1/3,
which is consistent
with the XXX result of Ref. [13].
In section 6, we present the multiple-integral representations of the
integrable spin-$s$ XXZ correlation functions.
We express the correlation function
of an arbitrary product of elementary matrices by
a single term of multiple integrals.
For instance, we calculate the multiple-integral representation
of the spin-1 ground-state expectation value,
$\langle\widetilde{E}^{1,\,1}\rangle$, explicitly,
and show that it is consistent with
the value of spin-1 EFP in section 5, i.e. we show
$\langle\widetilde{E}^{1,\,1}\rangle+2\langle\widetilde{E}^{2,\,2}\rangle=1$.
Finally in section 7, we give concluding remarks.
2 Fusion transfer matrices
2.1 $R$-matrix and the monodromy matrix of type $(1,1^{\otimes L})$
Let us introduce the $R$-matrix of the XXZ spin chain
[1, 3, 4, 5].
We denote by $e^{a,\,b}$ a unit matrix that has only one nonzero element
equal to 1 at entry $(a,b)$ where $a,b=0,1$.
Let $V_{1}$ and $V_{2}$ be two-dimensional vector spaces.
The $R$-matrix acting on $V_{1}\otimes V_{2}$ is given by
$${R}^{+}(\lambda_{1}-\lambda_{2})=\sum_{a,b,c,d=0,1}R^{+}(u)^{a\,b}_{c\,d}\,\,e%
^{a,\,c}\otimes e^{b,\,d}=\left(\begin{array}[]{cccc}1&0&0&0\\
0&b(u)&c^{-}(u)&0\\
0&c^{+}(u)&b(u)&0\\
0&0&0&1\\
\end{array}\right)\,,$$
(2.1)
where $u=\lambda_{1}-\lambda_{2}$,
$b(u)=\sinh u/\sinh(u+\eta)$ and
$c^{\pm}(u)=\exp(\pm u)\sinh\eta/\sinh(u+\eta)$.
In the massless regime, we set $\eta=i\zeta$ by a real number
$\zeta$,
and we have $\Delta=\cos\zeta$. In the paper we mainly consider
the region $0\leq\zeta<\pi/2s$.
In the massive regime, we assign $\eta$ a real nonzero number
and we have $\Delta=\cosh\eta>1$.
Here we remark that the $R^{+}(\lambda_{1}-\lambda_{2})$
is compatible with the homogeneous grading
of $U_{q}(\widehat{sl}_{2})$, which is explained in Appendix A
[15].
We denote by $R^{(p)}(u)$ or simply by $R(u)$ the symmetric $R$-matrix
where $c^{\pm}(u)$ of (2.1) are replaced by
$c(u)=\sinh\eta/\sinh(u+\eta)$ [15]. The symmetric $R$-matrix
is compatible with the affine quantum group $U_{q}(\widehat{sl}_{2})$
of the principal grading [15].
Let $s$ be an integer or a half-integer. We shall mainly
consider the tensor product
$V_{1}^{(2s)}\otimes\cdots\otimes V_{N_{s}}^{(2s)}$
of $(2s+1)$-dimensional vector spaces $V^{(2s)}_{j}$
with $L=2sN_{s}$.
In general, we consider the tensor product
$V_{0}^{(2s_{0})}\otimes V_{1}^{(2s_{1})}\otimes\cdots\otimes V_{r}^{(2s_{r})}$ with $2s_{1}+\cdots+2s_{r}=L$,
where $V_{j}^{(2s_{j})}$ have spectral parameters
$\lambda_{j}$ for $j=1,2,\ldots,r$.
For a given set of matrix elements
$A^{a,\,\alpha}_{b,\,\beta}$ for $a,b=0,1,\ldots,2s_{j}$ and
$\alpha,\beta=0,1,\ldots,2s_{k}$,
we define operator $A_{j,k}$ by
$$\displaystyle A_{j,k}$$
$$\displaystyle=$$
$$\displaystyle\sum_{a,b=1}^{\ell}\sum_{\alpha,\beta}A^{a,\,\alpha}_{b,\,\beta}I%
_{0}^{(2s_{0})}\otimes I_{1}^{(2s_{1})}\otimes\cdots\otimes I_{j-1}^{(2s_{j-1})}$$
(2.2)
$$\displaystyle\quad\otimes e^{a,b}_{j}\otimes I_{j+1}^{(2s_{j+1})}\otimes\cdots%
\otimes I_{k-1}^{(2s_{k-1})}\otimes e_{k}^{\alpha,\beta}\otimes I_{k+1}^{(2s_{%
k+1})}\otimes\cdots\otimes I_{r}^{(2s_{r})}\,.$$
We now consider the $(L+1)$th tensor product of spin-1/2 representations,
which consists of the tensor product of auxiliary space $V_{0}^{(1)}$ and
the $L$th tensor product of quantum spaces $V_{j}^{(1)}$
for $j=1,2,\ldots,L$, i.e.
$V_{0}^{(1)}\otimes\left(V_{1}^{(1)}\otimes\cdots\otimes V_{L}^{(1)}\right)$.
We call it the tensor product of type $(1,1^{\otimes L})$
and denote it by the following symbol:
$$(1,1^{\otimes L})=(1,\overbrace{1,1,\ldots,1}^{L})\,.$$
(2.3)
Applying definition (2.2)
for matrix elements $R(u)^{ab}_{cd}$ of a given $R$-matrix,
we define $R$-matrices
$R_{jk}(\lambda_{j},\lambda_{k})=R_{jk}(\lambda_{j}-\lambda_{k})$
for integers $j$ and $k$ with $0\leq j<k\leq L$.
For integers $j,k$ and $\ell$ with $0\leq j<k<\ell\leq L$,
the $R$-matrices satisfy the Yang-Baxter equations
$$R_{jk}(\lambda_{j}-\lambda_{k})R_{j\ell}(\lambda_{j}-\lambda_{\ell})R_{k\ell}(%
\lambda_{k}-\lambda_{\ell})=R_{k\ell}(\lambda_{k}-\lambda_{\ell})R_{j\ell}(%
\lambda_{j}-\lambda_{\ell})R_{jk}(\lambda_{j}-\lambda_{k})\,.$$
(2.4)
We define the monodromy matrix of type $(1,1^{\otimes L})$
associated with homogeneous grading by
$$T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda_{0};w_{1},w_{2},\ldots,w_{L})=R_{0L}^{+}%
(\lambda_{0}-w_{L})\cdots R_{02}^{+}(\lambda_{0}-w_{2})R_{01}^{+}(\lambda_{0}-%
w_{1})\,.$$
(2.5)
Here we have set $\lambda_{j}=w_{j}$ for $j=1,2,\ldots,L$,
where $w_{j}$ are arbitrary parameters.
We call them inhomogeneous parameters. We have expressed the symbol of
type $(1,1^{\otimes L})$ as $(1,\,1)$ in superscript.
The symbol $(1,\,1\,+)$ denotes that it is consistent with
homogeneous grading.
We express operator-valued matrix elements of
the monodromy matrix as follows.
$$T^{(1,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}\}_{L})=\left(\begin{array}[]{cc}A^%
{(1+)}_{12\cdots L}(\lambda;\{w_{j}\}_{L})&B^{(1+)}_{12\cdots L}(\lambda;\{w_{%
j}\}_{L})\\
C^{(1+)}_{12\cdots L}(\lambda;\{w_{j}\}_{L})&D^{(1+)}_{12\cdots L}(\lambda;\{w%
_{j}\}_{L})\end{array}\right)\,.$$
(2.6)
Here $\{w_{j}\}_{L}$ denotes the set of $L$ parameters,
$w_{1},w_{2},\ldots,w_{L}$.
We also denote the matrix elements of
the monodromy matrix by
$[T^{(1,1+)}_{0,12\cdots L}(\lambda;\{w_{j}\}_{L})]_{a,b}$
for $a,b=0,1$.
We derive
the monodromy matrix consistent with principal
grading, $T^{(1,1\,p)}_{0,12\cdots L}(\lambda;\{w_{j}\}_{L})$,
from that of homogeneous grading via similarity transformation
$\chi_{01\cdots L}$ as follows [15].
$$\displaystyle T^{(1,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}\}_{L})=\chi_{012%
\cdots L}T^{(1,1\,p)}_{0,12\cdots L}(\lambda;\{w_{j}\}_{L})\chi_{012\cdots L}^%
{-1}$$
$$\displaystyle=\left(\begin{array}[]{cc}\chi_{12\cdots L}A^{(1\,p)}_{12\cdots L%
}(\lambda;\{w_{j}\}_{L})\chi_{12\cdots L}^{-1}&e^{-\lambda_{0}}\chi_{12\cdots L%
}B^{(1\,p)}_{12\cdots L}(\lambda;\{w_{j}\}_{L})\chi_{12\cdots L}^{-1}\\
e^{\lambda_{0}}\chi_{12\cdots L}C^{(1\,p)}_{12\cdots L}(\lambda;\{w_{j}\}_{L})%
\chi_{12\cdots L}^{-1}&\chi_{12\cdots L}D^{(1\,p)}_{12\cdots L}(\lambda;\{w_{j%
}\}_{L})\chi_{12\cdots L}^{-1}\end{array}\right)\,.$$
(2.7)
Here $\chi_{01\cdots L}=\Phi_{0}\Phi_{1}\cdots\Phi_{L}$ and
$\Phi_{j}$ are given by diagonal two-by-two matrices
$\Phi_{j}={\rm diag}(1,\exp(w_{j}))$ acting on $V_{j}^{(1)}$
for $j=0,1,\ldots,L$, and we set $w_{0}=\lambda_{0}$.
In Ref. [15] operator
$A^{(1\,+)}(\lambda)$ has been written as $A^{+}(\lambda)$.
Hereafter we shall often abbreviate the symbols $p$
in superscripts which shows the principal grading, and denote
$(2s\,p)$ simply by $(2s)$.
Let us introduce useful notation for expressing
products of $R$-matrices as follows.
$$\displaystyle R_{1,23\cdots n}^{(w)}$$
$$\displaystyle=$$
$$\displaystyle R_{1n}^{(w)}\cdots R_{13}^{(w)}R_{12}^{(w)}\,,$$
$$\displaystyle R_{12\cdots n-1,n}^{(w)}$$
$$\displaystyle=$$
$$\displaystyle R_{1n}^{(w)}R_{2n}^{(w)}\cdots R_{n-1\,n}^{(w)}\,.$$
(2.8)
Here $R_{ab}^{(w)}$ denote the $R$-matrix
$R_{ab}^{(w)}=R_{ab}^{(w)}(\lambda_{a}-\lambda_{b})$
for $a,b=1,2,\ldots,n$, where $w=+$ and $w=p$ in superscripts
show the homogeneous and the principal grading, respectively.
Then, the monodromy matrix of type $(1,1^{\otimes L}\,w)$
is expressed as follows.
$$\displaystyle T_{0,\,12\cdots L}^{(1,\,1\,w)}(\lambda_{0};\{w_{j}\}_{L})$$
$$\displaystyle=$$
$$\displaystyle R_{0,\,12\cdots L}^{(\,w)}(\lambda_{0};\{w_{j}\}_{L})$$
(2.9)
$$\displaystyle=$$
$$\displaystyle R_{0L}^{(w)}R_{0L-1}^{(w)}\cdots R_{01}^{(w)}\,.$$
For instance we have
$B_{12\cdots L}^{(1\,w)}(\lambda_{0};\{w_{j}\}_{L})=[R_{0,12\cdots L}^{(1\,w)}(%
\lambda_{0};\{w_{j}\}_{L})]_{0,1}\,.$
2.2 Projection operators and the massive fusion $R$-matrices
Let $V_{1}$ and $V_{2}$ be $(2s+1)$-dimensional vector spaces.
We define permutation operator $\Pi_{1,\,2}$ by
$$\Pi_{1,\,2}\,v_{1}\otimes v_{2}=v_{2}\otimes v_{1}\,,\quad v_{1}\in V_{1}\,,\,%
v_{2}\in V_{2}\,.$$
(2.10)
In the case of spin-1/2 representations,
we define operator ${\check{R}}_{12}^{+}(\lambda_{1}-\lambda_{2})$ by
$${\check{R}}_{12}^{+}(\lambda_{1}-\lambda_{2})=\Pi_{1,\,2}\,R_{12}^{+}(\lambda_%
{1}-\lambda_{2})\,.$$
(2.11)
We now introduce projection operators $P_{12\cdots\ell}^{(\ell)}$ for
$\ell\geq 2$.
We define $P_{12}^{(2)}$ by $P_{12}^{(2)}={\check{R}}_{1,2}^{+}(\eta)$.
For $\ell>2$ we define projection operators
inductively with respect to $\ell$ as follows
[46, 23].
$$P_{12\cdots\ell}^{(\ell)}=P_{12\cdots\ell-1}^{(\ell-1)}{\check{R}}^{+}_{\ell-1%
,\,\ell}((\ell-1)\eta)P_{12\cdots\ell-1}^{(\ell-1)}\,.$$
(2.12)
The projection operator $P_{12\cdots\ell}^{(\ell)}$
gives a $q$-analogue of the full symmetrizer
of the Young operators for the Hecke algebra [46].
We shall show the idempotency:
$\left(P_{12\cdots\ell}^{(\ell)}\right)^{2}=P_{12\cdots\ell}^{(\ell)}$
in Appendix B.
Hereafter we denote $P_{12\cdots\ell}^{(\ell)}$ also by
$P_{1}^{(\ell)}$ for short.
Applying projection operator $P_{a_{1}a_{2}\cdots a_{\ell}}^{(\ell)}$
to vectors in the tensor product $V_{a_{1}}^{(1)}\otimes V_{a_{2}}^{(1)}\otimes\cdots\otimes V_{a_{\ell}}^{(1)}$,
we can construct
the $(\ell+1)$-dimensional vector space
$V_{a_{1}a_{2}\cdots a_{\ell}}^{(\ell)}$ associated with
the spin-$\ell/2$ representation of $U_{q}(sl_{2})$.
For instance, we have
$P_{a_{1}a_{2}}^{(2)}|+-\rangle_{a}=(q/[2]_{q})||2,1\rangle_{a}$,
where we have introduced
$|+-\rangle_{a}=|0\rangle_{a_{1}}\otimes|1\rangle_{a_{2}}$.
The symbols such as $q$-integers are defined in Appendix C.
Moreover, the basis vectors $||\ell,n\rangle$ ($n=0,1,\ldots,\ell$)
and their dual vectors $\langle\ell,n||$
are given for arbitrary nonzero integers $\ell$
in Appendix C.
We denote $V_{a_{1}a_{2}\cdots a_{\ell}}^{(\ell)}$
also by $V_{a}^{(\ell)}$ or $V_{0}^{(\ell)}$ for short.
Since
$P^{(\ell)}_{12\cdots\ell}$ is consistent
with the spin-$\ell/2$ representation of $U_{q}(sl(2))$
(see (C.6)),
we have
$$P^{(\ell)}_{12\cdots\ell}=\sum_{n=0}^{\ell}||\ell,n\rangle\,\langle\ell,n||\,.$$
(2.13)
Applying projection operator
$P_{2s(j-1)+1\cdots 2s(j-1)+2s}^{(2s)}$ to tensor product
$V_{2s(j-1)+1}^{(1)}\otimes\cdots\otimes V_{2s(j-1)+2s}^{(1)}$,
we construct the spin-$s$ representation
$V_{2s(j-1)+1\cdots 2s(j-1)+2s}^{(2s)}$. We denote it also by
$V_{j}^{(2s)}$, briefly.
In the tensor product of quantum spaces
$V^{(2s)}_{1}\otimes\cdots\otimes V_{N_{s}}^{(2s)}$,
we define $P_{12\cdots L}^{(2s)}$
by
$$P_{12\cdots L}^{(2s)}=\prod_{i=1}^{N_{s}}P^{(2s)}_{2s(i-1)+1}\,.$$
(2.14)
Here we recall $L=2sN_{s}$.
We have put $2s$ in place of $\ell$.
We now introduce the massive fusion $R$-matrix
$R^{(\ell,\,2s\,+)}_{0,\,j}$
on the tensor product
$V_{0}^{(\ell)}\otimes V_{j}^{(2s)}$ ($j=1,2,\ldots,N_{s}$).
It is valid in the massive regime with $\Delta>1$.
We first set rapidities $\lambda_{a_{j}}$
of auxiliary spaces $V_{a_{j}}^{(1)}$
by $\lambda_{a_{k}}=\lambda_{a_{1}}-(k-1)\eta$ for
$k=1,2,\ldots,\ell-1$, and then
rapidities $\lambda_{2s(j-1)+k}$
of quantum spaces $V_{2s(j-1)+k}^{(1)}$ by
$\lambda_{2s(j-1)+k}=\lambda_{2s(j-1)+1}-(k-1)\eta$
for $k=1,2,\ldots,2s$ and $j=1,2,\ldots,N_{s}$.
We define the massive fusion $R$-matrix
$R^{(\ell,\,2s\,+)}_{0,\,j}$
as follows.
$$\displaystyle R_{0\,j}^{(\ell,\,2s\,+)}(\lambda_{a_{1}}-\lambda_{2s(j-1)+1})=P%
_{a_{1}\cdots a_{\ell}}^{(\ell)}P_{2s(j-1)+1}^{(2s)}\,R^{+}_{a_{1}\cdots a_{%
\ell},\,2s(j-1)+1\cdots 2sj}\,P_{a_{1}\cdots a_{\ell}}^{(\ell)}P_{2s(j-1)+1}^{%
(2s)}$$
$$\displaystyle\qquad=P_{a_{1}\cdots a_{\ell}}^{(\ell)}P_{2s(j-1)+1}^{(2s)}\,R^{%
+}_{a_{1}\cdots a_{\ell},\,2sj}\cdots R^{+}_{a_{1}\cdots a_{\ell},\,2s(j-1)+2}%
R^{+}_{a_{1}\cdots a_{\ell},\,2s(j-1)+1}\,P_{a_{1}\cdots a_{\ell}}^{(\ell)}P_{%
2s(j-1)+1}^{(2s)}\,.$$
2.3 Conjugate vectors and the massless
fusion $R$-matrices
In order to construct Hermitian elementary matrices
in the massless regime where $|q|=1$,
we now introduce vectors $\widetilde{||\ell,n\rangle}$
which are Hermitian conjugate to $\langle\ell,n||$ when
$|q|=1$
for positive integers $\ell$ with $n=0,1,\ldots,\ell$.
Setting the norm of $\widetilde{||\ell,n\rangle}$
such that
$\langle\ell,n||\,\widetilde{||\ell,n\rangle}=1$,
we have
$$\widetilde{||\ell,n\rangle}=\sum_{1\leq i_{1}<\cdots<i_{n}\leq\ell}\sigma_{i_{%
1}}^{-}\cdots\sigma_{i_{n}}^{-}|0\rangle q^{-(i_{1}+\cdots+i_{n})+n\ell-n(n-1)%
/2}\left[\begin{array}[]{cc}\ell\\
n\end{array}\right]_{q}\,q^{-n(\ell-n)}\left(\begin{array}[]{cc}\ell\\
n\end{array}\right)^{-1}\,.$$
(2.16)
Here we have denoted the binomial coefficients
for integers $\ell$ and $n$ with $0\leq n\leq\ell$ as follows.
$$\left(\begin{array}[]{cc}\ell\\
n\end{array}\right)={\frac{\ell!}{(\ell-n)!n!}}\,.$$
(2.17)
The $q$-binomial coefficients are defined in Appendix C.
Dual vectors $\widetilde{\langle\ell,n||}$, which are
conjugate to $||\ell,n\rangle$, are defined in Appendix C,
and we have
$$\widetilde{\langle\ell,n||}\,\widetilde{||\ell,n\rangle}=\left[\begin{array}[]%
{cc}\ell\\
n\end{array}\right]_{q}^{2}\,\left(\begin{array}[]{cc}\ell\\
n\end{array}\right)^{-2}\,.$$
(2.18)
They are determined by
the action of $X^{\pm}$ with opposite coproduct:
$\Delta^{op}=\tau\circ\Delta$. For instance, we have
$\widetilde{||\ell,n\rangle}={\rm const.}\,\Delta^{op}(X^{-})^{n}||\ell,0\rangle$. Here $X^{\pm}$ and $\Delta^{op}$ are defined in Appendix A.
For an illustration,
in the spin-1 case, the basis vectors $||2,n\rangle$ ($n=0,1,2$)
are given by [15]
$$\displaystyle||2,0\rangle$$
$$\displaystyle=$$
$$\displaystyle|++\rangle\,,$$
$$\displaystyle||2,1\rangle$$
$$\displaystyle=$$
$$\displaystyle|+-\rangle+q^{-1}|-+\rangle\,,$$
$$\displaystyle||2,2\rangle$$
$$\displaystyle=$$
$$\displaystyle|--\rangle\,.$$
(2.19)
Here $|+-\rangle$ denotes $|0\rangle_{1}\otimes|1\rangle_{2}$,
briefly.
The conjugate vectors $\widetilde{||2,n\rangle}$ ($n=0,1,2$)
are given by
$$\displaystyle\widetilde{||2,0\rangle}$$
$$\displaystyle=$$
$$\displaystyle|++\rangle\,,$$
$$\displaystyle\widetilde{||2,1\rangle}$$
$$\displaystyle=$$
$$\displaystyle\left(|+-\rangle+q|-+\rangle\right){\frac{[2]_{q}}{2q}}\,,$$
$$\displaystyle\widetilde{||2,2\rangle}$$
$$\displaystyle=$$
$$\displaystyle|--\rangle\,.$$
(2.20)
In the massless regime,
operator $\widetilde{||2,1\rangle}\langle 2,1||$ is Hermitian
while $||2,1\rangle\langle 2,1||$ is not.
Let us now introduce another set of projection operators
$\widetilde{P}_{1\cdots\ell}^{(\ell)}$
as follows.
$$\widetilde{P}_{1\cdots\ell}^{(\ell)}=\sum_{n=0}^{\ell}\widetilde{||\ell,\,n%
\rangle}\langle\ell,\,n||\,.$$
(2.21)
Projector $\widetilde{P}_{1\cdots\ell}^{(\ell)}$ is idempotent:
$(\widetilde{P}_{1\cdots\ell}^{(\ell)})^{2}=\widetilde{P}_{1\cdots\ell}^{(\ell)}$.
In the massless regime where $|q|=1$,
it is Hermitian:
$\left(\widetilde{P}_{1\cdots\ell}^{(\ell)}\right)^{\dagger}=\widetilde{P}_{1%
\cdots\ell}^{(\ell)}$.
From (2.13) and (2.21),
we show the following properties:
$$\displaystyle P_{12\cdots\ell}^{(\ell)}\widetilde{P}_{1\cdots\ell}^{(\ell)}$$
$$\displaystyle=$$
$$\displaystyle P_{12\cdots\ell}^{(\ell)}\,,$$
(2.22)
$$\displaystyle\widetilde{P}_{1\cdots\ell}^{(\ell)}P_{12\cdots\ell}^{(\ell)}$$
$$\displaystyle=$$
$$\displaystyle\widetilde{P}_{1\cdots\ell}^{(\ell)}\,.$$
(2.23)
In the tensor product of quantum spaces,
$V^{(2s)}_{1}\otimes\cdots\otimes V_{N_{s}}^{(2s)}$,
we define $\widetilde{P}_{12\cdots L}^{(2s)}$ by
$$\widetilde{P}_{12\cdots L}^{(2s)}=\prod_{i=1}^{N_{s}}\widetilde{P}^{(2s)}_{2s(%
i-1)+1}\,.$$
(2.24)
Here we recall $L=2sN_{s}$ such as for (2.14).
We define the massless fusion $R$-matrix
$\widetilde{R}^{(\ell,\,2s\,+)}_{0,\,j}$,
applying projection operators $\widetilde{P}$
consisting of conjugate vectors to
the product of $R$-matrices, as follows.
$$\displaystyle\widetilde{R}_{0\,j}^{(\ell,\,2s\,+)}(\lambda_{a_{1}}-w_{2s(j-1)+%
1})=\widetilde{P}_{a_{1}\cdots a_{\ell}}^{(\ell)}\widetilde{P}_{2s(j-1)+1}^{(2%
s)}\,R^{+}_{a_{1}\cdots a_{\ell},\,2s(j-1)+1\cdots 2sj}\,\widetilde{P}_{a_{1}%
\cdots a_{\ell}}^{(\ell)}\widetilde{P}_{2s(j-1)+1}^{(2s)}$$
$$\displaystyle\qquad=\widetilde{P}_{a_{1}\cdots a_{\ell}}^{(\ell)}\widetilde{P}%
_{2s(j-1)+1}^{(2s)}\,R^{+}_{a_{1}\cdots a_{\ell},\,2sj}\cdots R^{+}_{a_{1}%
\cdots a_{\ell},\,2s(j-1)+2}R^{+}_{a_{1}\cdots a_{\ell},\,2s(j-1)+1}\,%
\widetilde{P}_{a_{1}\cdots a_{\ell}}^{(\ell)}\widetilde{P}_{2s(j-1)+1}^{(2s)}\,.$$
We should remark that the massless fusion $R$-matrix
$\widetilde{R}^{(\ell,\,2s)}$
and the massive fusion $R$-matrix ${R}^{(\ell,\,2s)}$
have the same matrix elements.
Some examples are shown in Appendix D.
2.4 Higher-spin monodromy matrix of type
$(\ell,\,(2s)^{\otimes N_{s}})$
We now set the inhomogeneous parameters $w_{j}$ for $j=1,2,\ldots,L$,
as $N_{s}$ sets of complete $2s$-strings [15].
We define $w_{(b-1)\ell+\beta}^{(2s)}$ for $\beta=1,\ldots,2s$,
as follows.
$$w_{2s(b-1)+\beta}^{(2s)}=\xi_{b}-(\beta-1)\eta\,,\quad\mbox{for}\quad b=1,2,%
\ldots,N_{s}.$$
(2.26)
We shall define the monodromy matrix of type $(1,(2s)^{\otimes N_{s}})$
associated with homogeneous grading.
We first define the massless monodromy matrix by
$$\displaystyle\widetilde{T}^{(1,\,2s\,+)}_{0,\,12\cdots N_{s}}(\lambda_{0};\{%
\xi_{b}\}_{N_{s}})$$
$$\displaystyle=$$
$$\displaystyle\widetilde{P}_{12\cdots L}^{(2s)}R_{0,\,1\ldots L}^{(1,\,1\,+)}(%
\lambda_{0};\{w_{j}^{(2s)}\}_{L})\widetilde{P}_{12\cdots L}^{(2s)}$$
(2.27)
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{cc}\widetilde{A}^{(2s+)}(\lambda;\{\xi_{b}%
\}_{N_{s}})&\widetilde{B}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s}})\\
\widetilde{C}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s}})&\widetilde{D}^{(2s+)}(%
\lambda;\{\xi_{b}\}_{N_{s}})\end{array}\right)\,.$$
Here, the (0,0) element is given by
$\widetilde{A}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s}})=\widetilde{P}_{12\cdots L}^%
{(2s)}A^{(1+)}(\lambda;\{w_{j}^{(2s)}\}_{L})\widetilde{P}_{12\cdots L}^{(2s)}$.
We then define the massive monodromy matrix by
$$\displaystyle T^{(1,\,2s\,+)}_{0,\,12\cdots N_{s}}(\lambda_{0};\{\xi_{b}\}_{N_%
{s}})$$
$$\displaystyle=$$
$$\displaystyle{P}_{12\cdots L}^{(2s)}R_{0,\,1\ldots L}^{(1,\,1\,+)}(\lambda_{0}%
;\{w_{j}^{(2s)}\}_{L}){P}_{12\cdots L}^{(2s)}$$
(2.28)
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{cc}{A}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s}})%
&{B}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s}})\\
{C}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s}})&{D}^{(2s+)}(\lambda;\{\xi_{b}\}_{N_{s%
}})\end{array}\right)\,.$$
Let us introduce a set of $2s$-strings with small deviations from
the set of complete $2s$-strings.
$$w_{2s(b-1)+\beta}^{(2s;\,\epsilon)}=\xi_{b}-(\beta-1)\eta+\epsilon r_{b}^{(%
\beta)}\,,\quad\mbox{for}\quad b=1,2,\cdots,N_{s},\,\mbox{and}\quad\beta=1,2,%
\ldots,2s.$$
(2.29)
Here $\epsilon$ is very small and
$r_{b}^{(\beta)}$ are generic parameters.
We express the elements of the monodromy matrix $T^{(1,1)}$ with
inhomogeneous parameters given by $w_{j}^{(2s;\,\epsilon)}$
for $j=1,2,\ldots,L$ as follows.
$$T^{(1,\,1\,+)}_{0,\,12\cdots L}(\lambda;\{w_{j}^{(2s;\epsilon)}\}_{L})=\left(%
\begin{array}[]{cc}A^{(2s+;\,\epsilon)}_{12\cdots L}(\lambda)&B^{(2s+;\,%
\epsilon)}_{12\cdots L}(\lambda)\\
C^{(2s+;\,\epsilon)}_{12\cdots L}(\lambda)&D^{(2s+;\,\epsilon)}_{12\cdots L}(%
\lambda)\end{array}\right)\,.$$
(2.30)
Here we recall that
$A^{(2s+;\,\epsilon)}_{12\cdots L}(\lambda)$ denotes
$A^{(1+)}_{12\cdots L}(\lambda;\{w_{j}^{(2s;\,\epsilon)}\}_{L})$.
We also remark the following.
$$\widetilde{A}^{(2s+)}_{12\cdots N_{s}}(\lambda;\{\xi_{p}\}_{N_{s}})=\lim_{%
\epsilon\rightarrow 0}\widetilde{P}_{12\cdots L}^{(2s)}A^{(2s+;\,\epsilon)}_{1%
2\cdots L}(\lambda;\{w_{j}^{(2s;\,\epsilon)}\}_{L})\widetilde{P}_{12\cdots L}^%
{(2s)}\,.$$
(2.31)
Let us express the tensor product
$V_{0}^{(\ell)}\otimes\left(V_{1}^{(2s)}\otimes\cdots\otimes V_{N_{s}}^{(2s)}\right)$, by the following symbol
$$(\ell,\,(2s)^{\otimes N_{s}})=(\ell,\,\overbrace{2s,2s,\ldots,2s}^{N_{s}})\,.$$
(2.32)
Here we recall that $V_{0}^{(\ell)}$ abbreviates
$V_{a_{1}a_{2}\ldots a_{\ell}}^{(\ell)}$.
In the case of auxiliary space $V_{0}^{(\ell)}$
we define the massless monodromy matrix of type
$(\ell,\,(2s)^{\otimes N_{s}})$ by
$$\widetilde{T}^{(\ell,\,2s\,+)}_{0,\,12\cdots N_{s}}=\widetilde{P}^{(\ell)}_{a_%
{1}a_{2}\cdots a_{\ell}}\,\widetilde{T}_{a_{1},\,12\cdots N_{s}}^{(1,\,2s\,+)}%
(\lambda_{a_{1}})\widetilde{T}_{a_{2},\,12\cdots N_{s}}^{(1,\,2s\,+)}(\lambda_%
{a_{1}}-\eta)\cdots\widetilde{T}_{a_{\ell},\,12\cdots N_{s}}^{(1,\,2s\,+)}(%
\lambda_{a_{1}}-(\ell-1)\eta)\,\widetilde{P^{(\ell)}}_{a_{1}a_{2}\cdots a_{%
\ell}}\,,$$
(2.33)
and the massive monodromy matrix of type
$(\ell,\,(2s)^{\otimes N_{s}})$ by
$$T^{(\ell,\,2s\,+)}_{0,\,12\cdots N_{s}}=P^{(\ell)}_{a_{1}a_{2}\cdots a_{\ell}}%
\,T_{a_{1},\,12\cdots N_{s}}^{(1,\,2s\,+)}(\lambda_{a_{1}})T_{a_{2},\,12\cdots
N%
_{s}}^{(1,\,2s\,+)}(\lambda_{a_{1}}-\eta)\cdots T_{a_{\ell},\,12\cdots N_{s}}^%
{(1,\,2s\,+)}(\lambda_{a_{1}}-(\ell-1)\eta)\,P^{(\ell)}_{a_{1}a_{2}\cdots a_{%
\ell}}\,.$$
(2.34)
For instance, the (0, 1) element of
the massive monodromy matrix $T^{(2,\,2s\,+)}(\lambda)$ is given by
$$\langle 2,0||T^{(2,\,2s\,+)}_{a_{1}a_{2},\,12\cdots N_{s}}(\lambda)||2,1%
\rangle=A_{a_{1}}^{(2s\,+)}(\lambda)B_{a_{2}}^{(2s\,+)}(\lambda-\eta)+q^{-1}B_%
{a_{1}}^{(2s\,+)}(\lambda)A_{a_{2}}^{(2s\,+)}(\lambda-\eta)\,.$$
(2.35)
2.5 Series of commuting higher-spin transfer matrices
Suppose that $|\ell,m\rangle$
for $m=0,1,\ldots,\ell$, are the orthonormal
basis vectors of $V^{(\ell)}$,
and their dual vectors are given by $\langle\ell,m|$
for $m=0,1,\ldots,\ell$.
We define the trace of operator $A$ over the space $V^{(\ell)}$ by
$${\rm tr}_{V^{(\ell)}}A=\sum_{m=0}^{\ell}\langle\ell,m|A|\ell,m\rangle\,.$$
(2.36)
The trace of $A$ over $V^{(\ell)}$ is equivalent to the trace
of $A$ over the $\ell$th tensor product of $V^{(1)}$,
$(V^{(1)})^{\otimes\ell}$, multiplied by a projector $P^{(\ell)}$
(or $\widetilde{P}^{(\ell)}$) as follows.
$$\displaystyle{\rm tr}_{V^{(\ell)}}A$$
$$\displaystyle=$$
$$\displaystyle{\rm tr}_{(V^{(1)})^{\otimes\ell}}\left(P^{(\ell)}A\right)$$
(2.37)
$$\displaystyle=$$
$$\displaystyle\sum_{a_{1},\ldots,a_{\ell}=0,1}\left(P^{(\ell)}A\right)^{a_{1}%
\cdots a_{\ell}}_{a_{1}\cdots a_{\ell}}\,.$$
It follows from (2.13) that the trace with respect to
$V^{(\ell)}$ is given by (2.36).
We define the massive transfer matrix of type
$(\ell,(2s)^{\otimes N_{s}})$ by
$$\displaystyle t^{(\ell,\,2s\,+)}_{12\cdots N_{s}}(\lambda)={\rm tr}_{V^{(\ell)%
}}\left(T^{(\ell,\,2s\,+)}_{0,\,12\cdots N_{s}}(\lambda)\right)$$
$$\displaystyle\,=\sum_{n=0}^{\ell}{}_{a}\langle\ell,n||T^{(1,\,2s\,+)}_{a_{1},%
\,12\cdots N_{s}}(\lambda)T^{(1,\,2s\,+)}_{a_{2},\,12\cdots N_{s}}(\lambda-%
\eta)\cdots T^{(1,\,2s\,+)}_{a_{\ell},\,12\cdots N_{s}}(\lambda-(\ell-1)\eta)%
\,||\ell,n\rangle_{a}\,,$$
(2.38)
and the massless transfer matrix of type
$(\ell,(2s)^{\otimes N_{s}})$ by
$$\displaystyle\widetilde{t}^{(\ell,\,2s\,+)}_{12\cdots N_{s}}(\lambda)={\rm tr}%
_{V^{(\ell)}}\left(\widetilde{T}^{(\ell,\,2s\,+)}_{0,\,12\cdots N_{s}}(\lambda%
)\right)$$
(2.39)
$$\displaystyle=$$
$$\displaystyle\sum_{n=0}^{\ell}{}_{a}\langle\ell,n||\,\widetilde{T}^{(1,\,2s\,+%
)}_{a_{1},\,12\cdots N_{s}}(\lambda)\widetilde{T}^{(1,\,2s\,+)}_{a_{2},\,12%
\cdots N_{s}}(\lambda-\eta)\cdots\widetilde{T}^{(1,\,2s\,+)}_{a_{\ell},\,12%
\cdots N_{s}}(\lambda-(\ell-1)\eta)\,\widetilde{||\ell,n\rangle}_{a}\,.$$
It follows from the Yang-Baxter equations that
the higher-spin transfer matrices commute in the tensor product space
$V_{1}^{(2s)}\otimes\cdots\otimes V_{N_{s}}^{(2s)}$,
which is derived by applying
projection operator $P^{(2s)}_{12\cdots L}$
to $V^{(1)}_{1}\otimes\cdots\otimes V_{L}^{(1)}$.
For instance, for the massless
transfer matrices, making use of (2.22) and (2.23)
we show
$$P^{(2s)}_{12\cdots L}{[}\widetilde{t}^{(\ell,\,2s\,+)}_{12\cdots N_{s}}(%
\lambda),\,\,\widetilde{t}^{(m,\,2s\,+)}_{12\cdots N_{s}}(\mu){]}=0\,,\quad%
\mbox{for}\,\,\ell,m\in{\bf Z}_{\geq 0}.$$
(2.40)
Therefore, for the massless transfer matrices,
the eigenvectors
of $\widetilde{t}^{(1,\,2s\,+)}_{12\cdots N_{s}}(\lambda)$
constructed by applying $\widetilde{B}^{(2s\,+)}(\lambda)$
to the vacuum $|0\rangle$
also diagonalize the higher-spin transfer matrices, in particular,
$\widetilde{t}^{(2s,\,2s\,+)}_{12\cdots N_{s}}(\lambda)$.
Thus, we construct the ground state
of the higher-spin Hamiltonian in terms of operators
$\widetilde{B}^{(2s\,+)}(\lambda)$,
which are the (0, 1) element of the monodromy matrix
$\widetilde{T}^{(1,\,2s\,+)}$.
2.6 The integrable higher-spin Hamiltonians
We now discuss the integrable massless spin-$s$ XXZ Hamiltonian.
For $(2s+1)$-dimensional vector spaces $V_{1}^{(2s)}$ and $V_{2}^{(2s)}$,
we can show that the massive spin-$s$ fusion $R$-matrix
$R^{(2s,\,2s\,+)}_{1\,2}(u)$ at $u=0$ becomes
the permutation operator $\Pi_{1,\,2}$
for $V_{1}^{(2s)}\otimes V_{2}^{(2s)}$. Furthermore,
operator ${\check{R}}^{(2s,\,2s\,+)}_{1,\,2}(u)=\Pi_{1,\,2}{R}_{1\,2}^{(2s,\,2s\,+)}(u)$
has the following spectral decomposition:
$${\check{R}}^{(2s,\,2s\,+)}_{1,\,2}(u)=\sum_{j=0}^{2s}\rho_{4s-2j}(u)\,\left({P%
}_{4s-2j}^{2s,\,2s}\right)_{1,\,2}\,,$$
(2.41)
where operator
$({P}_{4s-2j}^{2s,\,2s})_{1,\,2}$
projects $V_{1}^{(2s)}\otimes V_{2}^{(2s)}$
to spin-$(2s-j)$ representation for $j=0,1,\ldots,2s$.
Functions $\rho_{4s-2j}(u)$ are given by [45]
$$\rho_{4s-2j}(u)=\prod_{k=2s-j+1}^{2s}{\frac{\sinh(k\eta-u)}{\sinh(k\eta+u)}}\,.$$
(2.42)
The massless spin-$s$ $R$-matrix is thus given by
$$\widetilde{{\check{R}}}^{(2s,\,2s\,+)}_{i,\,i+1}(u)=\sum_{j=0}^{2s}\rho_{4s-2j%
}(u)\,\widetilde{P}^{(2s)}_{2s(i-1)+1}\widetilde{P}^{(2s)}_{2si+1}\,\cdot\,%
\left({P}_{4s-2j}^{2s,\,2s}\right)_{i,\,i+1}\,.$$
(2.43)
It is easy to show that the massless spin-$s$ $R$-matrix
$\widetilde{R}_{1\,2}^{(2s,\,2s\,+)}(u)$
becomes the permutation operator at $u=0$:
$\widetilde{R}_{1\,2}^{(2s,\,2s\,+)}(0)=\Pi_{1,\,2}$.
Therefore,
putting inhomogeneous parameters $\xi_{p}=0$ for $p=1,2,\ldots,N_{s}$,
we show that
that the transfer matrix
$\widetilde{t}^{(2s,\,2s\,+)}_{12\cdots N_{s}}(\lambda)$
becomes the shift operator at $\lambda=0$.
We thus derive the massless spin-$s$ XXZ Hamiltonian
by the logarithmic derivative of the massless spin-$s$ transfer matrix,
similarly as for the massive case.
$$\displaystyle{\cal H}^{(2s)}_{\rm XXZ}$$
$$\displaystyle=$$
$$\displaystyle\left.{\frac{d}{d\lambda}}\log\widetilde{t}^{(2s,\,2s\,+)}_{12%
\cdots N_{s}}(\lambda)\right|_{\lambda=0,\,\xi_{j}=0}=\sum_{i=1}^{N_{s}}\left.%
\frac{d}{du}\widetilde{\check{R}}_{i,i+1}^{(2s,2s)}(u)\right|_{u=0}$$
(2.44)
$$\displaystyle=$$
$$\displaystyle\sum_{i=1}^{N_{s}}\sum_{j=0}^{2s}{\frac{d\rho_{4s-2j}}{du}}(0)\,%
\widetilde{P}^{(2s)}_{2s(i-1)+1}\widetilde{P}^{(2s)}_{2si+1}\,\cdot\,\left({P}%
_{4s-2j}^{2s,\,2s}\right)_{i,\,i+1}\,.$$
3 Higher-spin expectation values
3.1 Algebraic Bethe ansatz
In terms of the vacuum vector $|0\rangle$
where all spins are up,
we define functions $a(\lambda)$ and $d(\lambda)$ by
$$\displaystyle A^{(1\,p)}(\lambda;\{w_{j}\}_{L})|0\rangle$$
$$\displaystyle=$$
$$\displaystyle a(\lambda;\{w_{j}\}_{L})|0\rangle\,,$$
$$\displaystyle D^{(1\,p)}(\lambda;\{w_{j}\}_{L})|0\rangle$$
$$\displaystyle=$$
$$\displaystyle d(\lambda;\{w_{j}\}_{L})|0\rangle\,.$$
(3.1)
We have $a(\lambda;\{w_{j}\}_{L})=1$ and
$$d(\lambda;\{w_{j}\}_{L})=\prod^{L}_{j=1}b(\lambda,w_{j})\,.$$
(3.2)
Here $b(\lambda,\mu)=b(\lambda-\mu)$.
For the homogeneous grading ($w=+$)
and the principal grading ($w=p$),
it is easy to show the following relations:
$$\displaystyle A^{(2s\,w)}(\lambda)|0\rangle$$
$$\displaystyle=$$
$$\displaystyle\widetilde{A}^{(2s\,w)}(\lambda)|0\rangle=a^{(2s)}(\lambda;\{\xi_%
{k}\})|0\rangle\,,$$
$$\displaystyle D^{(2s\,w)}(\lambda)|0\rangle$$
$$\displaystyle=$$
$$\displaystyle\widetilde{D}^{(2s\,w)}(\lambda)|0\rangle=d^{(2s)}(\lambda;\{\xi_%
{k}\})|0\rangle\,,$$
(3.3)
where $a^{(2s)}(\lambda;\{\xi_{k}\})$ and
$d^{(2s)}(\lambda;\{\xi_{k}\})$ are given by
$$\displaystyle a^{(2s)}(\lambda;\{\xi_{k}\})$$
$$\displaystyle=$$
$$\displaystyle a(\lambda;\{w_{j}^{(2s)}\})=1\,,$$
$$\displaystyle d^{(2s)}(\lambda;\{\xi_{k}\})$$
$$\displaystyle=$$
$$\displaystyle d(\lambda;\{w_{j}^{(2s)}\})=\prod^{N_{s}}_{p=1}b_{2s}(\lambda,%
\xi_{p})\,.$$
(3.4)
Here we have defined $b_{t}(\lambda,\mu)$ by
$b_{t}(\lambda,\mu)={\sinh(\lambda-\mu)}/{\sinh(\lambda-\mu+t\eta)}$ .
Here we recall $b(u)=b_{1}(u)=\sinh u/\sinh(u+\eta)$.
In the massless regime, we define the Bethe vectors
$|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,w)}\,\rangle$
for $w=+$ and $p$, and their dual vectors
$\langle\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,w)}|$
for $w=+$ and $p$, as follows.
$$\displaystyle|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,w)}\,\rangle$$
$$\displaystyle=$$
$$\displaystyle\prod_{\alpha=1}^{M}\widetilde{B}^{(2s\,w)}(\lambda_{\alpha})|0%
\rangle\,,$$
(3.5)
$$\displaystyle\langle\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,w)}\,|$$
$$\displaystyle=$$
$$\displaystyle\langle 0|\,\prod_{\alpha=1}^{M}\widetilde{C}^{(2s\,w)}(\lambda_{%
\alpha})\,.$$
(3.6)
Here we recall
$\widetilde{B}^{(2s\,+)}(\lambda_{\alpha})=\widetilde{P}^{(2s)}_{1\cdots L}B^{(%
1\,+)}(\lambda_{\alpha},\{w_{k}\}_{L})\widetilde{P}^{(2s)}_{1\cdots L}$.
The Bethe vector (3.5)
gives an eigenvector of the massless transfer matrix
$$\widetilde{t}^{(1,\,2s\,w)}(\mu;\{\xi_{p}\}_{N_{s}})=\widetilde{A}^{(2s\,w)}(%
\mu;\{\xi_{p}\}_{N_{s}})+\widetilde{D}^{(2s\,w)}(\mu;\{\xi_{p}\}_{N_{s}})$$
(3.7)
for $w=+$ and $w=p$ with the following eigenvalue:
$$\Lambda^{(1,{2s}\,w)}(\mu)=\prod_{j=1}^{M}{\frac{\sinh(\lambda_{j}-\mu+\eta)}{%
\sinh(\lambda_{j}-\mu)}}+\prod_{p=1}^{N_{s}}b_{2s}(\mu,\xi_{p})\,\cdot\,\prod_%
{j=1}^{M}{\frac{\sinh(\mu-\lambda_{j}+\eta)}{\sinh(\mu-\lambda_{j})}}\,,$$
(3.8)
if rapidities $\{\lambda_{j}\}_{M}$ satisfy the Bethe ansatz equations
$$\prod_{p=1}^{N_{s}}b_{2s}^{-1}(\lambda_{j},\xi_{p})=\prod_{k\neq j}{\frac{b(%
\lambda_{k},\lambda_{j})}{b(\lambda_{j},\lambda_{k})}}\quad(j=1,\ldots,M)\,.$$
(3.9)
Let us denote by
$|\{\lambda_{\alpha}(\epsilon)\}_{M}^{(2s\,w;\,\epsilon)}\rangle$
the Bethe vector of $M$ Bethe roots $\{\lambda_{j}(\epsilon)\}_{M}$
for $w=+,p$:
$$|\{\lambda_{\alpha}(\epsilon)\}_{M}^{(2s\,w;\,\epsilon)}\rangle=B^{(2s\,w;%
\epsilon)}(\lambda_{1}(\epsilon))\cdots B^{(2s\,w;\epsilon)}(\lambda_{M}(%
\epsilon))|0\rangle\,,$$
(3.10)
where rapidities $\{\lambda_{j}(\epsilon)\}_{M}$ satisfy the Bethe ansatz
equations with inhomogeneous parameters $w_{j}^{(2s;\epsilon)}$ as follows.
$$\frac{a(\lambda_{j}(\epsilon);\{w_{k}^{(2s;\,\epsilon)}\}_{L})}{d(\lambda_{j}(%
\epsilon);\{w_{k}^{(2s;\,\epsilon)}\}_{L})}=\prod_{k=1;k\neq j}^{M}{\frac{b(%
\lambda_{k}(\epsilon),\lambda_{j}(\epsilon))}{b(\lambda_{j}(\epsilon),\lambda_%
{k}(\epsilon))}}\,.$$
(3.11)
It gives an eigenvector of the transfer matrix
$$t^{(1,1\,w)}(\mu;\{w_{j}^{(2s;\,\epsilon)}\}_{L})=A^{({2s\,w;\,\epsilon})}(\mu%
;\{w_{j}^{(2s;\,\epsilon)}\}_{L})+D^{({2s\,w;\,\epsilon})}(\mu;\{w_{j}^{(2s;\,%
\epsilon)}\}_{L})$$
(3.12)
with the following eigenvalue:
$$\Lambda^{(1,1\,w)}(\mu;\{w_{j}^{(2s;\,\epsilon)}\}_{L})=\prod_{j=1}^{M}{\frac{%
\sinh(\lambda_{j}(\epsilon)-\mu+\eta)}{\sinh(\lambda_{j}(\epsilon)-\mu)}}+%
\prod_{j=1}^{L}b(\mu,w_{j}^{(2s;\,\epsilon)})\,\cdot\,\prod_{j=1}^{M}{\frac{%
\sinh(\mu-\lambda_{j}(\epsilon)+\eta)}{\sinh(\mu-\lambda_{j}(\epsilon))}}\,.$$
(3.13)
Let us assume that in the limit of $\epsilon$ going to 0,
the set of Bethe roots $\{\lambda_{j}(\epsilon)\}_{M}$ is given by
$\{\lambda_{j}\}_{M}$. Then, we have
$$P_{12\cdots L}^{(2s)}\,|\widetilde{\{\lambda_{j}\}}_{M}^{(2s\,+)}\rangle=\lim_%
{\epsilon\rightarrow 0}P_{12\cdots L}^{(2s)}\,|\{\lambda_{j}(\epsilon)\}_{M}^{%
(2s\,+;\epsilon)}\rangle\,.$$
(3.14)
3.2
Hermitian elementary matrices ${\widetilde{E}}_{i}^{m\,,\,n\,(2s\,+)}$
in the massless regime
We define massless elementary matrices
$\widetilde{E}^{m,\,n\,(2s+)}$ for $m,n=0,1,\ldots,2s$,
in the spin-$s$ representation of $U_{q}(sl_{2})$ as follows.
$$\widetilde{E}^{m,\,n\,(2s\,+)}=\widetilde{||\ell,m\rangle}\langle\ell,n||\,.$$
(3.15)
In the tensor product space,
$(V^{(2s)})^{\otimes N_{s}}$,
we define $\widetilde{E}^{m,\,n\,(2s\,+)}_{i}$
for $i=1,2,\ldots,N_{s}$ by
$$\widetilde{E}^{m,\,n\,(2s\,+)}_{i}=(I^{(2s)})^{\otimes(i-1)}\otimes\widetilde{%
E}^{m,\,n\,(2s\,+)}\otimes(I^{(2s)})^{\otimes(N_{s}-i)}\,.$$
(3.16)
Elementary matrices $\widetilde{E}^{n,\,n\,(2s\,+)}$
for $n=0,1,\ldots,2s$,
are Hermitian in the massless regime.
In fact, when $|q|=1$, for $m,n=0,1,\ldots,2s$, we have
$$\left(\widetilde{E}^{m,\,n\,(2s\,+)}\right)^{\dagger}=\left[\begin{array}[]{c}%
2s\\
m\end{array}\right]_{q}^{2}\,\left[\begin{array}[]{c}2s\\
n\end{array}\right]_{q}^{-2}\,\left(\begin{array}[]{c}2s\\
m\end{array}\right)^{-1}\,\,\left(\begin{array}[]{c}2s\\
n\end{array}\right)\,\widetilde{E}^{n,\,m\,(2s\,+)}\,.$$
(3.17)
We can express any given spin-$s$ local operator
of the massless case
in terms of the spin-1/2 global operators
by a method similar to the massive case [15].
For $m=n$, we have
$$\displaystyle\widetilde{E}_{i}^{n,\,n\,(2s+)}$$
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}2s\\
n\end{array}\right)\,\widetilde{P}^{(2s)}_{1\cdots L}\,\prod_{\alpha=1}^{(i-1)%
2s}(A^{(1+)}+D^{(1+)})(w_{\alpha})\prod_{k=1}^{n}D^{(1+)}(w_{(i-1)2s+k})$$
(3.18)
$$\displaystyle\quad\times\,\prod_{k=n+1}^{2s}A^{(1+)}(w_{(i-1)2s+k})\prod_{%
\alpha=i2s+1}^{2sN_{s}}(A^{(1+)}+D^{(1+)})(w_{\alpha})\,\,\widetilde{P}^{(2s)}%
_{1\cdots L}\,.$$
Formulas expressing $\widetilde{E}^{m,\,n\,(2s+)}$
for $m>n$ or $m<n$ are given in Appendix E.
When we evaluate expectation values,
we want to remove the projection operators
introduced in order to express the spin-$s$ local operator
in terms of spin-1/2 global operators such as in (3.18).
Then, we shall make use of the following lemma.
Lemma 3.1.
Projection operators $P^{(2s)}_{12\cdots L}$ and
$\widetilde{P}^{(2s)}_{12\cdots L}$
commute with the matrix elements of the monodromy matrix
$T^{(1,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}^{(2s;\epsilon)}\}_{L})$
such as $A^{(2s\,+;\epsilon)}(\lambda)$
in the limit of $\epsilon$ going to 0.
$$\displaystyle P_{12\cdots L}^{(2s)}T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda;\{w_{%
j}^{(2s;\,\epsilon)}\}_{L})\,P_{12\cdots L}^{(2s)}$$
$$\displaystyle=$$
$$\displaystyle P_{12\cdots L}^{(2s)}\,T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda;\{w%
_{j}^{(2s;\,\epsilon)}\}_{L})+O(\epsilon)\,,$$
(3.19)
$$\displaystyle P_{12\cdots L}^{(2s)}T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda;\{w_{%
j}^{(2s;\,\epsilon)}\}_{L})\,\widetilde{P}_{12\cdots L}^{(2s)}$$
$$\displaystyle=$$
$$\displaystyle P_{12\cdots L}^{(2s)}\,T^{(1,1)}_{0,12\cdots L}(\lambda;\{w_{j}^%
{(2s;\,\epsilon)}\}_{L})+O(\epsilon)\,.$$
(3.20)
For instance we have
$P_{12\cdots L}^{(2s)}B^{(2s\,+;\,\epsilon)}(\lambda)P_{12\cdots L}^{(2s)}=P_{1%
2\cdots L}^{(2s)}B^{(2s\,+;\,\epsilon)}(\lambda)+O(\epsilon).$
Proof.
Taking derivatives with respect to inhomogeneous parameters $w_{j}$,
we can show
$$T^{(1,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}^{(2s\,+;\,\epsilon)}\}_{L})=T^{(1,%
1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}^{(2s)}\}_{L})+O(\epsilon)\,,$$
(3.21)
where $T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}^{(2s)}\}_{L})$
commutes with the projection operator $P_{12\cdots L}^{(2s)}$
as follows [15].
$$P_{12\cdots L}^{(2s)}T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}^{(2s)}\}_{L%
})=P_{12\cdots L}^{(2s)}T^{(1,\,1\,+)}_{0,12\cdots L}(\lambda;\{w_{j}^{(2s)}\}%
_{L})P_{12\cdots L}^{(2s)}\,.$$
(3.22)
We show (3.20) making use of (2.22).
∎
3.3 Expectation value of a local operator
through the limit: $\epsilon\rightarrow 0$
In the massless regime,
we define the expectation value of product of operators
$\prod_{k=1}^{m}\widetilde{E}_{k}^{i_{k},\,j_{k}\,(2s\,+)}$
with respect to an eigenstate
$|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,+)}\rangle$ by
$$\langle\prod_{k=1}^{m}\widetilde{E}_{k}^{i_{k},\,j_{k}\,(2s\,+)}\rangle\left(%
\{\lambda_{\alpha}\}_{M}^{(2s\,+)}\right)={\frac{\langle\widetilde{\{\lambda_{%
\alpha}\}}_{M}^{(2s\,+)}|\,\prod_{k=1}^{m}\widetilde{E}_{k}^{i_{k},\,j_{k}\,(2%
s\,+)}|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,+)}\rangle}{\langle%
\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,+)}|\widetilde{\{\lambda_{\alpha}\}%
}_{M}^{(2s\,+)}\rangle}}\,.$$
(3.23)
We evaluate the expectation value of a given spin-$s$ local operator
for a Bethe-ansatz eigenstate $|\{\lambda_{\alpha}\}_{M}^{(2s)}\rangle$,
as follows.
We first assume that the Bethe roots
$\{\lambda_{\alpha}(\epsilon)\}_{M}$
are continuous with respect to small parameter $\epsilon$.
We express the spin-$s$ local operator
in terms of spin-1/2 global operators
such as formula (3.18) with generic
inhomogeneous parameters $w_{j}^{(2s;\epsilon)}$.
Applying (3.19) and (3.19)
we remove the projection operators
out of the product of global operators.
We next calculate the scalar product for the Bethe state
$|\{\lambda_{k}(\epsilon)\}_{M}^{(2s;\,\epsilon)}\rangle$
which has the same inhomogeneous parameters $w_{j}^{(2s;\epsilon)}$,
making use of the formulas of the spin-1/2 case.
Then we take the limit of sending $\epsilon$ to 0, and obtain
the expectation value of the spin-$s$ local operator.
For an illustration,
let us consider the expectation value of
$\widetilde{E}_{1}^{n,\,n\,(2s\,+)}$.
First, applying projection operator $P^{(2s)}_{12\cdots L}$ to
$|\widetilde{\{\lambda_{\alpha}\}}^{(2s\,+)}_{M}\rangle=\prod_{\alpha=1}^{M}%
\widetilde{B}^{(2s\,+)}(\lambda_{\alpha})|0\rangle$
we show
$$\displaystyle P^{(2s)}_{1\cdots L}|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,%
+)}\rangle$$
$$\displaystyle=$$
$$\displaystyle P^{(2s)}_{1\cdots L}\,\prod_{\alpha=1}^{M}B^{(2s\,+;\epsilon)}(%
\lambda_{\alpha}(\epsilon))\,|0\rangle+O(\epsilon)$$
(3.24)
$$\displaystyle=$$
$$\displaystyle e^{-\sum_{\alpha=1}^{M}\lambda_{\alpha}(\epsilon)}\,P^{(2s)}_{1%
\cdots L}\,\chi_{12\cdots L}\,\prod_{\alpha=1}^{M}B^{(2s;\,\epsilon)}(\lambda_%
{\alpha}(\epsilon))\,|0\rangle+O(\epsilon)\,.$$
Second, making use of the relation
$\langle 0|=\langle 0|P_{12\cdots L}^{(2s)}$,
we show
$$\displaystyle\langle\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s+)}|$$
$$\displaystyle=$$
$$\displaystyle\langle 0|\,\prod_{\alpha=1}^{M}C^{(2s\,+;\epsilon)}(\lambda_{%
\alpha}(\epsilon))\,P^{(2s)}_{1\cdots L}+O(\epsilon)$$
(3.25)
$$\displaystyle=$$
$$\displaystyle\langle 0|\,\prod_{\alpha=1}^{M}C^{(2s;\,\epsilon)}(\lambda_{%
\alpha}(\epsilon))\,\chi_{12\cdots L}^{-1}\,P^{(2s)}_{1\cdots L}\,e^{\sum_{%
\alpha=1}^{M}\lambda_{\alpha}(\epsilon)}+O(\epsilon)\,.$$
Making use of (3.18) we have
$$\displaystyle\langle\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,+)}|\,%
\widetilde{E}^{n\,n\,(2s\,+)}_{1}\,|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s%
\,+)}\rangle$$
(3.26)
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}2s\\
n\end{array}\right)\,\langle 0|\prod_{\alpha=1}^{M}C^{(2s+;\,\epsilon)}(%
\lambda_{\alpha}(\epsilon))P^{(2s)}_{1\cdots L}\,\underline{\widetilde{P}_{12%
\cdots L}^{(2s)}}\,\prod_{k=1}^{n}D^{(2s+;\,\epsilon)}(w_{k}^{(2s;\,\epsilon)}%
)\prod_{k=n+1}^{2s}{A}^{(2s+;\,\epsilon)}(w_{k}^{(2s;\,\epsilon)})$$
$$\displaystyle\quad\times\,\prod_{\alpha=2s+1}^{2sN_{s}}(A^{(2s\,+;\,\epsilon)}%
+D^{(2s\,+;\,\epsilon)})(w_{\alpha}^{(2s;\,\epsilon)})\,\,\underline{%
\widetilde{P}^{(2s)}_{1\cdots L}}\,\cdot\,\prod_{\alpha=1}^{M}\widetilde{B}^{(%
2s\,+)}(\lambda_{\alpha})|0\rangle\,+O(\epsilon)\,.$$
Here we have $\prod_{j=1}^{2sN_{s}}(A^{(2s+;\,\epsilon)}+D^{(2s+;\,\epsilon)})(w_{j}^{(2s;\,%
\epsilon)})=I^{\otimes L}$
for generic $\epsilon$.
We apply projection operators $P^{(2s)}$ to $\widetilde{P}^{(2s)}$
from the left,
which are underlined in (3.26),
and make use of (2.22).
We then move
the projection operators $P^{(2s)}$ in the leftward direction,
making use of (3.19).
Thus, the right-hand side of (3.26)
is now given by the following:
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}2s\\
n\end{array}\right)\,\langle 0|\prod_{\alpha=1}^{M}C^{(2s+;\,\epsilon)}(%
\lambda_{\alpha}(\epsilon))\,\,\prod_{k=1}^{n}D^{(2s+;\,\epsilon)}(w_{k}^{(2s;%
\,\epsilon)})\,\prod_{k=n+1}^{2s}A^{(2s+;\,\epsilon)}(w_{k}^{(2s;\,\epsilon)})$$
(3.27)
$$\displaystyle\times\prod_{j=2s+1}^{2sN_{s}}(A^{(2s+;\,\epsilon)}+D^{(2s+;\,%
\epsilon)})(w_{j}^{(2s;\,\epsilon)})\,\,\prod_{\beta=1}^{M}B^{(2s+;\,\epsilon)%
}(\lambda_{\beta}(\epsilon))|0\rangle+O(\epsilon).$$
After applying the gauge transformation
$\chi_{1\cdots L}^{-1}$ inverse to (2.7)
[15], we obtain
$$\displaystyle\langle\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,+)}|\widetilde{%
E}^{n\,n\,(2s\,+)}_{1}|\widetilde{\{\lambda_{\alpha}\}}_{M}^{(2s\,+)}\rangle$$
(3.28)
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}2s\\
n\end{array}\right)\,\lim_{\epsilon\rightarrow 0}\,\langle 0|\prod_{\alpha=1}^%
{M}C^{(2s;\,\epsilon)}(\lambda_{\alpha}(\epsilon))\,\,\prod_{k=1}^{n}D^{(2s;\,%
\epsilon)}(w_{k}^{(2s;\,\epsilon)})\,\prod_{k=n+1}^{2s}A^{(2s;\,\epsilon)}(w_{%
k}^{(2s;\,\epsilon)})$$
$$\displaystyle\quad\times\,\prod_{j=2s+1}^{2sN_{s}}(A^{(2s;\,\epsilon)}+D^{(2s;%
\,\epsilon)})(w_{j}^{(2s;\,\epsilon)})\,\,\prod_{\beta=1}^{M}B^{(2s;\,\epsilon%
)}(\lambda_{\beta}(\epsilon))|0\rangle\,.$$
Here $A^{(2s;\,\epsilon)}$ and $D^{(2s;\,\epsilon)}$ denote
matrix elements $A^{(2s\,p;\,\epsilon)}$
and $D^{(2s\,p;\,\epsilon)}$
of the monodromy matrix
with principal grading, respectively.
In the last line of (3.27),
we have evaluated the eigenvalue of transfer matrix
$A^{(2s;\,\epsilon)}(w_{j}^{(2s;\,\epsilon)})+D^{(2s;\,\epsilon)}(w_{j}^{(2s;\,%
\epsilon)})$
on the eigenstate
$|\{\lambda_{\beta}(\epsilon)\}_{M}^{(2s;\,\epsilon)}\rangle$
as follows.
$$\displaystyle\prod_{j=2s+1}^{2sN_{s}}\left(A^{(2s;\,\epsilon)}(w_{j}^{(2s;\,%
\epsilon)})+D^{(2s;\,\epsilon)}(w_{j}^{(2s;\,\epsilon)})\right)|\{\lambda_{%
\beta}(\epsilon)\}_{M}^{(2s;\,\epsilon)}\rangle$$
$$\displaystyle=\left(\prod_{j=2s+1}^{2sN_{s}}\prod_{\alpha=1}^{M}b^{-1}(\lambda%
_{\alpha}(\epsilon)-w_{j}^{(2s;\,\epsilon)})\right)\,|\{\lambda_{\beta}(%
\epsilon)\}_{M}^{(2s;\,\epsilon)}\rangle\,.$$
(3.29)
Before sending $\epsilon$ to 0,
we expand the products of $C$ operators multiplied by operators
$A$ and $D$ by the commutation relations between $C$ and $A$ as well as
$C$ and $D$, respectively. We then
evaluate the scalar product of $B$ and $C$ operators
with inhomogeneous parameters $w^{(2s;\epsilon)}_{j}$.
Finally, we derive the expectation value in
the limit of sending $\epsilon$ to 0.
Sending $\epsilon$ to 0, we calculate
the expectation value of $A^{(2s)}(\lambda)+D^{(2s)}(\lambda)$
at $\lambda=w_{2}^{(2s)}$.
For instance, we calculate
$A^{(2s;\,\epsilon)}(w_{2}^{(2s;\,\epsilon)})+D^{(2s;\,\epsilon)}(w_{2}^{(2s;\,%
\epsilon)})$
on the vacuum $|0\rangle$ as follows.
$$\displaystyle\lim_{\epsilon\rightarrow 0}\langle 0|\left(A^{(2s;\epsilon)}(w_{%
2}^{(2s;\epsilon)})+D^{(2s;\epsilon)}(w_{2}^{(2s;\epsilon)})\right)|0\rangle$$
(3.30)
$$\displaystyle=$$
$$\displaystyle\lim_{\epsilon\rightarrow 0}\langle 0|\left(A(w_{2}^{(2s;\epsilon%
)};\{w_{j}^{(2s;\epsilon)}\}_{L})+D^{(2s)}(w_{2}^{(2s;\epsilon)};\{w_{j}^{(2s;%
\epsilon)}\}_{L})\right)|0\rangle$$
$$\displaystyle=$$
$$\displaystyle\lim_{\epsilon\rightarrow 0}\,\left(1+\prod_{j=1}^{\ell N_{s}}b(w%
_{2}^{(2s;\epsilon)}-w_{j}^{(2s;\epsilon)})\right)\langle 0|0\rangle$$
$$\displaystyle=$$
$$\displaystyle(1+0)\,\langle 0|0\rangle\,.$$
If we put $\lambda=w_{2}^{(2s)}$ after sending $\epsilon$ to 0,
the result is different from (3.30) as follows.
$$\lim_{\lambda\rightarrow w^{(2s)}_{2}}\langle 0|\left(A^{(2s)}(\lambda;\{w_{j}%
^{(2s)}\}_{L})+D^{(2s)}(\lambda;\{w_{j}^{(2s)}\}_{L})\right))|0\rangle=\left(1%
+\prod_{p=1}^{N_{s}}b_{\ell}(w^{(2s)}_{2}-\xi_{p})\right)\langle 0|0\rangle\,.$$
(3.31)
4 Derivation of matrix $S$
4.1 The ground-state solution of $2s$-strings
We shall introduce $\ell$-strings for an integer $\ell$.
Let us shift rapidities $\lambda_{j}$ by $s\eta$ such as
$\tilde{\lambda}_{j}=\lambda_{j}+s\eta$. Then,
the Bethe ansatz equations (3.9) are given by
$$\prod_{p=1}^{N_{s}}{\frac{\sinh(\tilde{\lambda}_{j}-{\xi}_{p}+s\eta)}{\sinh(%
\tilde{\lambda}_{j}-{\xi}_{p}-s\eta)}}=\prod_{\beta=1;\beta\neq\alpha}^{n}{%
\frac{\sinh(\tilde{\lambda}_{j}-\tilde{\lambda}_{\beta}+\eta)}{\sinh(\tilde{%
\lambda}_{j}-\tilde{\lambda}_{\beta}-\eta)}}\,,\quad\mbox{for}\,\,j=1,2,\ldots%
,n\,.$$
(4.1)
We define an $\ell$-string by the following set of rapidities.
$${\tilde{\lambda}}_{a}^{(\alpha)}=\mu_{a}+({\ell}+1-2\alpha){\frac{\eta}{2}}+%
\epsilon_{a}^{(\alpha)}\quad\mbox{for}\quad\alpha=1,2,\ldots,\ell.$$
(4.2)
We call $\mu_{a}$ the center of the $\ell$-string
and $\epsilon_{a}^{(\alpha)}$ string deviations.
We assume that $\epsilon_{a}^{(\alpha)}$ are very small for large $N_{s}$:
$$\lim_{N_{s}\rightarrow\infty}\epsilon_{a}^{(\alpha)}=0.$$
(4.3)
If they are zero, then we call the set of rapidities of
(4.2) a complete $\ell$-string.
The string center $\mu_{a}$ corresponds
to the central position among the $\ell$ complex
numbers: ${\tilde{\lambda}}_{a}^{(1)},{\tilde{\lambda}}_{a}^{(2)},\ldots,{\tilde{\lambda%
}}_{a}^{(\ell)}$.
Furthermore we assume that $\mu_{a}$ are real.
If inhomogeneous parameters, $\xi_{p}$, are small enough,
then the Bethe ansatz equations should have
an $\ell$-strings as a solution.
In terms of rapidities ${\lambda}_{j}$ which are not shifted,
an $\ell$-string is expressed in the following form:
$$\lambda_{a}^{(\alpha)}=\mu_{a}-(\alpha-1/2)\eta+\epsilon_{a}^{(\alpha)}\quad%
\mbox{for}\quad\alpha=1,2,\ldots,\ell\,.$$
(4.4)
We denote ${\lambda}_{a}^{(\alpha)}$
also by ${\lambda}_{(a,\alpha)}$.
Let us now introduce the conjecture that the ground state
of the spin-$s$ case $|\psi_{g}^{(2s)}\rangle$ is
given by $N_{s}/2$ sets of $2s$-strings:
$$\lambda_{a}^{(\alpha)}=\mu_{a}-(\alpha-1/2)\eta+\epsilon_{a}^{(\alpha)}\,,%
\quad\mbox{for}\,\,a=1,2,\ldots,N_{s}/2\,\,\mbox{and}\,\,\alpha=1,2,\ldots,2s.$$
(4.5)
In terms of $\lambda_{a}^{(\alpha)}$s in the massless regime,
for $w=+$ and $p$, we have
$$|\psi_{g}^{(2s\,w)}\rangle=\prod_{a=1}^{N_{s}/2}\prod_{\alpha=1}^{2s}%
\widetilde{B}^{(2s\,w)}(\lambda_{a}^{(\alpha)};\{\xi_{p}\})|0\rangle.$$
(4.6)
Hereafter we set $M=2sN_{s}/2=sN_{s}$.
According to analytic and numerical studies
[40, 41, 42],
we may assume the following
properties of string deviations $\epsilon_{a}^{(\alpha)}$s.
When $N_{s}$ is very large, the deviations are given by
$$\epsilon_{a}^{(\alpha)}=i\,\delta_{a}^{(\alpha)}\,,$$
(4.7)
where $i$ denotes $\sqrt{-1}$, and
$\delta_{a}^{(\alpha)}$ are real. Moreover,
$\delta_{a}^{(\alpha)}-\delta_{a}^{(\alpha+1)}>0$
for $\alpha=1,2,\ldots,2s-1$, and
$|\delta_{a}^{(\alpha)}|>|\delta_{a}^{(\alpha+1)}|$
for $\alpha<s$,
while $|\delta_{a}^{(\alpha)}|<|\delta_{a}^{(\alpha+1)}|$
for $\alpha\geq s$.
In the thermodynamic limit: $N_{s}\rightarrow\infty$,
the Bethe ansatz equations for the ground state of the
higher-spin XXZ chain
become the integral equation for the string centers,
as shown in Appendix F [53].
The density of string centers, $\rho_{\rm tot}(\mu)$, is given by
$$\rho_{\rm tot}(\mu)={\frac{1}{N}_{s}}\sum_{p=1}^{N_{s}}{\frac{1}{2\zeta\cosh(%
\pi(\mu-\xi_{p})/\zeta)}}$$
(4.8)
Thus, the sum over all the Bethe roots of the ground state
is evaluated by integrals
in the thermodynamic limit, $N_{s}\rightarrow\infty$,
as follows.
$$\displaystyle\frac{1}{N_{s}}\sum_{A=1}^{M}f(\lambda_{A})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{N_{s}}\sum_{\alpha=1}^{2s}\sum_{a=1}^{N_{s}/2}f(\lambda_%
{(a,\alpha)})$$
(4.9)
$$\displaystyle=$$
$$\displaystyle\sum_{\alpha=1}^{2s}\int_{-\infty}^{\infty}f(\mu_{a}-(\alpha-1/2)%
\eta+\epsilon_{a}^{(\alpha)})\,\rho_{\rm tot}(\mu_{a})\,d\mu_{a}+O(1/N_{s})\,.$$
For the homogeneous chain where $\xi_{p}=0$ for $p=1,2,\ldots,N_{s}$,
we denote the density of string centers by $\rho(\lambda)$.
$$\rho(\lambda)={\frac{1}{2\zeta\cosh(\pi\lambda/\zeta)}}\,.$$
(4.10)
Let us introduce useful notation of the suffix of rapidities.
For rapidities $\lambda_{a}^{(\alpha)}=\lambda_{(a,\alpha)}$
we define integers $A$ by $A=2s(a-1)+\alpha$ for
$a=1,2,\ldots,N_{s}/2$ and for $\alpha=1,2,\ldots,2s$.
We thus denote $\lambda_{(a,\alpha)}$ also
by $\lambda_{A}$ for $A=1,2,\ldots,sN_{s}$, and
put $\lambda_{(a,\alpha)}$
in increasing order with respect to $A=2s(a-1)+\alpha$
such as $\lambda_{(1,1)}=\lambda_{1},\lambda_{(1,2)}=\lambda_{2},\ldots,\lambda_{(N_{s}%
/2,2s)}=\lambda_{sN_{s}}$.
In the ground state
rapidities $\lambda_{A}$ for
$A=1,2,\ldots,M$, are now expressed by
$$\lambda_{2s(a-1)+\alpha}=\mu_{a}-(\alpha-1/2)\eta+\epsilon_{a}^{(\alpha)}\quad%
\mbox{for}\,\,a=1,2,\ldots,N_{s}/2\,\,\mbox{and}\,\,\alpha=1,2,\ldots,2s.$$
(4.11)
For a given real number $x$,
let us denote by $[x]$ the greatest integer less than or equal to $x$.
When $A=2s(a-1)+\alpha$ with $1\leq\alpha\leq 2s$,
integer $a$ is given by $a=[(A-1)/2s]+1$, and integer $\alpha$
is given by $\alpha=A-2s[(A-1)/2s]$.
4.2 Derivation of the spin-$s$ EFP for a finite chain
We define the emptiness formation probability (EFP)
for the spin-$s$ case by
$$\tau^{(2s\,+)}(m)={\frac{\langle\psi_{g}^{(2s\,+)}|\widetilde{E}_{1}^{2s,2s\,(%
2s\,+)}\cdots\widetilde{E}_{m}^{2s,2s\,(2s\,+)}|\psi_{g}^{(2s\,+)}\rangle}{%
\langle\psi_{g}^{(2s\,+)}|\psi_{g}^{(2s\,+)}\rangle}}\,.$$
(4.12)
We shall denote $\tau^{(2s\,+)}(m)$ by $\tau^{(2s)}(m)$.
Let us assume that
Bethe roots $\{\lambda_{\alpha}(\epsilon)\}_{M}$
with inhomogeneous parameters $w_{j}^{(2s;\,\epsilon)}$
($j=1,2,\ldots,L$; $L=2sN_{s}$)
become the ground-state solution
of the spin-$s$ XXZ spin chain, $\{\lambda_{\alpha}\}_{M}$,
in the limit of sending $\epsilon$ to 0.
We denote the Bethe vector with Bethe roots
$\{\lambda_{\alpha}(\epsilon)\}_{M}$ by
$$\displaystyle|\psi_{g}^{(2s\,+;\,\epsilon)}\rangle$$
$$\displaystyle=$$
$$\displaystyle\prod_{\alpha=1}^{M}B^{(2s;\,\epsilon)}(\lambda_{\alpha}(\epsilon%
))|0\rangle=e^{-\sum_{\alpha=1}^{M}\lambda_{\alpha}(\epsilon)}\,\chi_{12\cdots
L%
}\,\cdot\,\prod_{\alpha=1}^{M}B^{(2s\,p;\,\epsilon)}(\lambda_{\alpha}(\epsilon%
))|0\rangle$$
(4.13)
$$\displaystyle=$$
$$\displaystyle e^{-\sum_{\alpha=1}^{M}\lambda_{\alpha}(\epsilon)}\,\chi_{12%
\cdots L}\,|\psi_{g}^{(2s;\,\epsilon)}\rangle\,.$$
Here we recall the transformation inverse to (2.7).
We now calculate the norm of the spin-$s$ ground state
from that of the spin-1/2 case through
the limit of sending $\epsilon$ to 0 as follows.
$$\displaystyle{\langle\psi_{g}^{(2s\,+)}|\psi_{g}^{(2s\,+)}\rangle}=\lim_{%
\epsilon\rightarrow 0}\,\langle\psi_{g}^{(2s;\,\epsilon)}|\psi_{g}^{(2s;\,%
\epsilon)}\rangle$$
(4.14)
$$\displaystyle=$$
$$\displaystyle\lim_{\epsilon\rightarrow 0}\,\langle 0|\prod_{k=1}^{M}C^{(2s;\,%
\epsilon)}(\lambda_{k})\prod_{j=1}^{M}B^{(2s;\,\epsilon)}(\lambda_{j})|0\rangle$$
$$\displaystyle=$$
$$\displaystyle\lim_{\epsilon\rightarrow 0}\,\sinh^{M}\eta\prod_{j,k=1;j\neq k}^%
{M}b^{-1}(\lambda_{j}(\epsilon),\lambda_{k}(\epsilon))\,\cdot\,{\rm det}\Phi^{%
(1)^{\prime}}\left(\{\lambda_{k}(\epsilon)\}_{M};\{w_{j}^{(2s;\,\epsilon)}\}_{%
L}\right)$$
$$\displaystyle=$$
$$\displaystyle\sinh^{M}\eta\prod_{j,k=1;j\neq k}^{M}b^{-1}(\lambda_{j},\lambda_%
{k})\,\cdot\,{\rm det}\Phi^{(2s)^{\prime}}\left(\{\lambda_{k}\}_{M};\{\xi_{p}%
\}_{N_{s}}\right)$$
where matrix elements of the spin-$s$ Gaudin matrix
for $j,k=1,2,\ldots,M$,
are given by
$$\displaystyle\Phi^{(2s)\,^{\prime}}_{j,k}(\{\lambda_{l}\}_{M};\{\xi_{p}\})$$
(4.15)
$$\displaystyle=$$
$$\displaystyle-{\frac{\partial}{\partial\lambda_{k}}}\log\left({\frac{a^{(2s)}(%
\lambda_{j})}{d^{(2s)}(\lambda_{j})}}\,\prod_{t\neq j}{\frac{\sinh(\lambda_{t}%
-\lambda_{j}+\eta)}{\sinh(\lambda_{t}-\lambda_{j}-\eta)}}\right)$$
$$\displaystyle=$$
$$\displaystyle\delta_{j,k}\left(\sum_{p=1}^{N_{s}}\frac{\sinh(2s\eta)}{\sinh(%
\lambda_{j}-\xi_{p})\sinh(\lambda_{j}-\xi_{p}+2s\eta)}-\sum_{C=1}^{M}{\frac{%
\sinh 2\eta}{\sinh(\lambda_{j}-\lambda_{C}+\eta)\sinh(\lambda_{j}-\lambda_{C}-%
\eta)}}\right)$$
$$\displaystyle+{\frac{\sinh 2\eta}{\sinh(\lambda_{j}-\lambda_{k}+\eta)\sinh(%
\lambda_{j}-\lambda_{k}-\eta)}}\,.$$
By applying formula (3.18) with $n=2s$,
the numerator of (4.12) is given by
$$\displaystyle\langle\psi_{g}^{(2s\,+;\,\epsilon)}|\widetilde{E}_{1}^{2s,2s\,(2%
s\,+)}\cdots\widetilde{E}_{m}^{2s,2s\,(2s\,+)}|\psi_{g}^{(2s\,+;\,\epsilon)}%
\rangle=\lim_{\epsilon\rightarrow 0}\langle\psi_{g}^{(2s;\,\epsilon)}|\prod_{k%
=1}^{m}E_{k}^{2s,2s\,(2s)}\,|\psi_{g}^{(2s;\,\epsilon)}\rangle$$
$$\displaystyle\quad=\lim_{\epsilon\rightarrow 0}\langle\psi_{g}^{(2s;\,\epsilon%
)}|P^{(2s)}_{12\cdots L}\prod_{i=1}^{m}\Bigg{(}\prod_{\alpha=1}^{2s(i-1)}\left%
(A^{(2s;\,\epsilon)}+D^{(2s;\,\epsilon)}\right)(w_{\alpha}^{(2s;\,\epsilon)})%
\cdot\prod_{k=1}^{2s}D^{(2s;\,\epsilon)}(w_{2s(i-1)+k}^{(2s;\,\epsilon)})$$
$$\displaystyle\qquad\qquad\qquad\cdot\prod_{\alpha=1}^{2sN_{s}}\left(A^{(2s;\,%
\epsilon)}+D^{(2s;\,\epsilon)}\right)(w_{\alpha}^{(2s;\,\epsilon)})\Bigg{)}P^{%
(2s)}_{12\cdots L}|\psi_{g}^{(2s)}\rangle$$
$$\displaystyle\quad=\prod_{j=1}^{m}\prod_{\alpha=1}^{M}b_{2s}(\lambda_{\alpha},%
\xi_{j})\,\lim_{\epsilon\rightarrow 0}\langle\psi_{g}^{(2s;\,\epsilon)}|\,D^{(%
2s;\,\epsilon)}(w_{1}^{(2s;\,\epsilon)})\cdots D^{(2s;\,\epsilon)}(w_{2sm}^{(2%
s;\,\epsilon)})\,|\psi_{g}^{(2s;\,\epsilon)}\rangle\,.$$
(4.16)
Let us set $\lambda_{M+j}(\epsilon)=w_{j}^{(2s;\,\epsilon)}$
for $j=1,2,\ldots,2sm$.
Applying formula (G.1) to (4.16)
we have
$$\displaystyle\langle 0|\prod_{\alpha=1}^{M}C^{(2s;\,\epsilon)}(\lambda_{\alpha%
}(\epsilon))\prod_{j=1}^{2sm}D^{(2s;\,\epsilon)}(\lambda_{M+j}(\epsilon))\,%
\prod_{\beta=1}^{M}B^{(2s;\,\epsilon)}(\lambda_{\beta}(\epsilon))\,|0\rangle$$
(4.17)
$$\displaystyle=$$
$$\displaystyle\sum_{c_{1}=1}^{M}\sum_{c_{2}=1;c_{2}\neq c_{1}}^{M}\cdots\sum_{c%
_{2sm}=1;c_{2sm}\neq c_{1},\ldots,c_{2sm-1}}^{M}G_{c_{1}\cdots c_{2sm}}(%
\lambda_{1}(\epsilon),\cdots,\lambda_{M+2sm}(\epsilon);\{w_{j}^{(2s;\,\epsilon%
)}\}_{L})$$
$$\displaystyle\times\langle 0|\prod_{k=1;k\neq c_{1},\ldots,c_{2sm}}^{M+2sm}C^{%
(2s;\,\epsilon)}(\lambda_{k}(\epsilon))\,\prod_{\alpha=1}^{M}B^{(2s;\,\epsilon%
)}(w_{j}^{(2s;\,\epsilon)})|0\rangle\,,$$
where
$$G_{c_{1}\cdots c_{2sm}}(\lambda_{1},\cdots,\lambda_{M+2sm};\{w_{j}\}_{L})=%
\prod_{j=1}^{2sm}\left(d(\lambda_{c_{j}};\{w_{j}\}_{L}){\frac{\prod_{t=1;t\neq
c%
_{1},\ldots,c_{j-1}}^{M+j-1}\sinh(\lambda_{c_{j}}-\lambda_{t}+\eta)}{\prod_{t=%
1;t\neq c_{1},\ldots,c_{j}}^{M+j}\sinh(\lambda_{c_{j}}-\lambda_{t})}}\right)\,.$$
(4.18)
We remark that from (4.18)
the set of integers $c_{1},\ldots,c_{2sm}$
of the most dominant terms in (4.17) are given by
$m$ sets of $2s$-strings.
If they are not, the numerator of (4.18)
and hence the right-hand-side of (4.17)
becomes smaller at least by the order of $1/N_{s}$ in the large $N_{s}$ limit.
However, each of the most dominant terms diverges with respect to $N_{s}$
in the large-$N_{s}$ limit,
and they should cancel each other so that the final result becomes finite.
We therefore calculate all possible contributions
with respect to the set of integers, $c_{1},c_{2},\ldots,c_{2sm}$.
Let us take a sequence of distinct integers $c_{j}$
satisfying $1\leq c_{j}\leq M$ for $j=1,2,\ldots,2sm$.
We denote it by $(c_{j})_{2sm}$, i.e.
$(c_{j})_{2sm}=(c_{1},c_{2},\ldots,c_{2sm})$.
Let us denote by $\Sigma_{M}$ the set of integers,
$1,2,\ldots,M$: $\Sigma_{M}=\{1,2,\ldots,M\}$.
We then consider the complementary set of integers
$\Sigma_{M}\setminus\{c_{1},\ldots,c_{2sm}\}$, and
put the elements in increasing order such as
$z_{1}<z_{2}<\cdots<z_{M-2sm}$.
We then extend the sequence $z_{n}$ of $M-2sm$ integers into
that of $M$ integers by setting $z_{j+M-2sm}=c_{j}$
for $j=1,2,\ldots,2sm$.
We shall denote $z_{n}$
also by $z(n)$ for $n=1,2,\ldots,M$.
In terms of sequence $(z_{n})_{M}$
we express the scalar product in the last line of
(4.17) as follows.
$$\displaystyle\langle 0|\prod_{k=1;k\neq c_{1},\ldots,c_{2sm}}^{M+2sm}C^{(2s;\,%
\epsilon)}(\lambda_{k}(\epsilon))\,\prod_{\alpha=1}^{M}B^{(2s;\,\epsilon)}(%
\lambda_{\alpha}(\epsilon))|0\rangle$$
$$\displaystyle\quad=\langle 0|\prod_{k=1}^{M-2sm}C^{(2s;\,\epsilon)}(\lambda_{z%
(k)}(\epsilon))\prod_{j=1}^{2sm}C^{(2s;\,\epsilon)}(w_{j}^{(2s;\,\epsilon)})\,%
\prod_{i=1}^{M-2sm}B^{(2s;\,\epsilon)}(\lambda_{z(i)}(\epsilon))\prod_{j=1}^{2%
sm}B^{(2s;\,\epsilon)}(\lambda_{c_{j}}(\epsilon))|0\rangle.$$
We evaluate scalar product (LABEL:eq:scalar),
sending $\nu_{j}$ to $\lambda_{z(j)}(\epsilon)$
for $j\leq M-2sm$ and to $w_{j-M+2sm}^{(2s;\,\epsilon)}$ for $j>M-2sm$
in the following matrix:
$$H^{(1)}((\lambda_{z(k)}(\epsilon))_{M},(\nu_{z(1)},\ldots,\nu_{z(M-2sm)},\nu_{%
M-2sm+1},\ldots,\nu_{M});(w_{j}^{(2s;\,\epsilon)})_{L})\,.$$
(4.20)
Here we define the matrix elements
$H_{ab}^{(2s)}(\{\lambda_{\alpha}\}_{n},\,\{\mu_{j}\}_{n};\,\{\xi_{k}\}_{N_{s}})$
for $a,b=1,2,\ldots,n$, by
$$\displaystyle H_{ab}^{(2s)}(\{\lambda_{\alpha}\}_{n},\,\{\mu_{j}\}_{n};\,\{\xi%
_{k}\}_{N_{s}})$$
$$\displaystyle={\frac{\sinh\eta}{\sinh(\lambda_{a}-\mu_{b})}}\left({\frac{a(\mu%
_{b})}{d^{(2s)}(\mu_{b};\{\xi_{k}\})}}\prod_{k=1;k\neq a}^{n}\sinh(\lambda_{k}%
-\mu_{b}+\eta)-\prod_{K=1;k\neq a}^{n}\sinh(\lambda_{k}-\mu_{b}-\eta)\right)\,.$$
Let us denote $M-2sm$ by $M^{{}^{\prime}}$.
We write the composite of two sequences
$(a(i))_{M}$ and $(b(j))_{N}$ as $(a(i))_{M}\#(b(j))_{N}$.
Explicitly we have
$$(a(i))_{M}\#(b(j))_{N}=(a(1),\ldots,a(M),b(1),\ldots,b(N))\,.$$
(4.22)
For $j>M^{{}^{\prime}}=M-2sm$, we have
$$\displaystyle\lim_{\nu_{j}\rightarrow w_{j-M^{{}^{\prime}}}^{(2s;\,\epsilon)}}%
d(\nu_{j};\{w_{j}^{(2s;\,\epsilon)}\}_{L})\,H^{(1)}_{i,\,j}((\lambda_{z(k)}(%
\epsilon))_{M},(\nu_{k})_{M^{{}^{\prime}}}\#(\nu_{k+M^{{}^{\prime}}})_{2sm};(w%
_{j}^{(2s;\,\epsilon)})_{L})$$
(4.23)
$$\displaystyle=$$
$$\displaystyle\prod_{\alpha=1}^{M}\sinh(\lambda_{\alpha}(\epsilon)-w_{j-M^{{}^{%
\prime}}}^{(2s;\,\epsilon)}+\eta)\Bigg{(}\displaystyle{\frac{\sinh\eta}{\sinh(%
\lambda_{z(i)}(\epsilon)-w_{j-M^{{}^{\prime}}}^{(2s;\,\epsilon)})\sinh(\lambda%
_{z(i)}(\epsilon)-w_{j-M^{{}^{\prime}}}^{(2s;\,\epsilon)}+\eta)}}$$
$$\displaystyle\quad-d(w_{j-M^{{}^{\prime}}}^{(2s;\,\epsilon)};\{w_{j}^{(2s;\,%
\epsilon)}\}_{L})\prod_{t=1}^{M}{\frac{\sinh(\lambda_{t}(\epsilon)-w_{j-M^{{}^%
{\prime}}}^{(2s;\,\epsilon)}-\eta)}{\sinh(\lambda_{t}(\epsilon)-w_{j-M^{{}^{%
\prime}}}^{(2s;\,\epsilon)}+\eta)}}$$
$$\displaystyle\qquad\times\,\displaystyle{\frac{\sinh\eta}{\sinh(\lambda_{z(i)}%
(\epsilon)-w_{j-M^{{}^{\prime}}-1}^{(2s;\,\epsilon)})\sinh(\lambda_{z(i)}(%
\epsilon)-w_{j-M^{{}^{\prime}}-1}^{(2s;\,\epsilon)}+\eta)}}\Bigg{)}\,.$$
The second term of (4.23) for matrix element $(i,j)$
vanishes since we have
$d(w_{j-M^{{}^{\prime}}}^{(2s;\,\epsilon)};\{w_{k}^{(2s;\,\epsilon)}\}_{L})=0$.
Here we remark that if we directly evaluate matrix $H^{(2s)}$
at $\epsilon=0$,
the second term of (4.23) for matrix element $(i,j)$
for $j\neq 2s(n-1)+1+M^{{}^{\prime}}$ with $n=1,2,\ldots,m$,
does not vanish,
although it is deleted by subtracting column $j$ by column $j-1$,
as discussed for the XXX case in Ref. [13].
We thus have
$$\displaystyle\lim_{\epsilon\rightarrow 0}{\rm det}H^{(1)}((\lambda_{z(k)}(%
\epsilon))_{M},(\lambda_{z(1)}(\epsilon),\ldots,\lambda_{z(M-2sm)}(\epsilon),w%
_{1}^{(2s;\,\epsilon)},\ldots,w_{2sm}^{(2s;\,\epsilon)});(w_{j}^{(2s;\,%
\epsilon)})_{2sN_{s}})$$
(4.24)
$$\displaystyle=$$
$$\displaystyle(-1)^{M-2sm}\,\prod_{b=1}^{M-2sm}\prod_{k=1}^{M}\sinh(\lambda_{k}%
-\lambda_{z_{b}}-\eta)\prod_{j=1}^{2sm}\prod_{k=1}^{M}\sinh(\lambda_{k}-w_{j}^%
{(2s)}+\eta)$$
$$\displaystyle\times$$
$$\displaystyle{\rm det}\Psi^{(2s)^{\prime}}((\lambda_{z(i}))_{M},(\lambda_{z(i}%
))_{M^{{}^{\prime}}}\#(w_{j}^{(2s)})_{2sm};\{\xi_{p}\}_{N_{s}})$$
where $(i,j)$ element of
$\Psi^{(2s)^{\prime}}((\lambda_{z(k)})_{M},(\lambda_{z(k)})_{M^{{}^{\prime}}}\#%
(w_{k}^{(2s)})_{2sm};\{\xi_{p}\}_{N_{s}})$
for $i=1,2,\ldots,M$,
are given by
$$\displaystyle\Psi^{(2s)^{\prime}}_{i,\,j}((\lambda_{z(1)},\ldots,\lambda_{z({M%
-2sm})},\lambda_{c_{1}},\ldots,\lambda_{c_{2sm}}),(\lambda_{z(1)},\ldots,%
\lambda_{z({M-2sm})},w_{1}^{(2s)},\ldots,w_{2sm}^{(2s)});(\xi_{p})_{N_{s}})$$
$$\displaystyle\quad=\left\{\begin{array}[]{cc}\Phi^{(2s)^{\prime}}_{z(i),\,z(j)%
}(\left(\lambda_{k}\right)_{M};\{\xi_{p}\})&\mbox{for}\quad j\leq M-2sm,\\
&\\
\displaystyle{\frac{\sinh\eta}{\sinh(\lambda_{z(i)}-w_{j-M^{{}^{\prime}}}^{(2s%
)})\sinh(\lambda_{z(i)}-w_{j-M^{{}^{\prime}}}^{(2s)}+\eta)}}&\mbox{for}\quad j%
>M-2sm\,.\\
\end{array}\right.$$
(4.25)
Therefore, for $i,j=1,2,\ldots,M-2sm$, we have
$$\left(\left(\Phi^{(2s)^{\prime}}((\lambda_{z(k)})_{M};\{\xi_{p}\})\right)^{-1}%
\,\Psi^{(2s)^{\prime}}((\lambda_{z(k)})_{M},(\lambda_{z(k)})_{M^{{}^{\prime}}}%
\#(w_{j}^{(2s)})_{2sm};\{\xi_{p}\})\right)_{i,\,j}=\delta_{i,\,j}\,.$$
(4.26)
In terms of sequence $(c_{j})_{2sm}$,
we express the dependence of matrix
$(\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}$
on the sequence of Bethe roots $(\lambda_{z(i)})_{M}$ etc.,
briefly, as follows.
$$(\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}((c_{j})_{2sm},\{\xi_{p}\})=(%
\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}\left((\lambda_{z(j)})_{M},(%
\lambda_{z(j)})_{M^{{}^{\prime}}}\#(w_{j}^{(2s)})_{2sm};\{\xi_{p}\}\right)\,.$$
(4.27)
Recall $M^{{}^{\prime}}=M-2sm$.
Similarly, we define ${\Phi}^{(2s)\,^{\prime}}((c_{j})_{2sm},\{\xi_{p}\})$
and $\Psi^{(2s)^{\prime}}((c_{j})_{2sm},\{\xi_{p}\})$ by
$$\displaystyle{\Phi}^{(2s)\,^{\prime}}_{i,\,j}((c_{l})_{2sm},\{\xi_{p}\})$$
$$\displaystyle=$$
$$\displaystyle\Phi^{(2s)\,^{\prime}}_{i,\,j}(\{\lambda_{z(k)}\}_{M};(\xi_{p})_{%
N_{s}})=\Phi^{(2s)\,^{\prime}}_{z(i),\,z(j)}(\{\lambda_{k}\}_{M};(\xi_{p})_{N_%
{s}})\,,$$
$$\displaystyle\Psi^{(2s)^{\prime}}_{i,\,j}((c_{k})_{2sm},\{\xi_{p}\})$$
$$\displaystyle=$$
$$\displaystyle\Psi^{(2s)^{\prime}}_{i,\,j}((\lambda_{z(k}))_{M},(\lambda_{z(k})%
)_{M^{{}^{\prime}}}\#(w_{k}^{(2s)})_{2sm};(\xi_{p})_{N_{s}})\,,$$
(4.28)
for $i,j=1,2,\ldots,M$.
Here we remark again that sequence
$(z(i))_{M}$ is determined by sequence
$(c_{1},\ldots,c_{2sm})$ by the definition that
$\{z(1),z(2),\ldots,z(M^{{}^{\prime}})\}=\{1,2,\ldots,M\}\setminus\{c_{1},%
\ldots,\,c_{2sm}\}$,
and $z(1)<\cdots<z(M^{{}^{\prime}})$ while $z(j+M^{{}^{\prime}})=c_{j}$ for
$j=1,2,\ldots,2sm$.
From property (4.26)
we define a $2sm$-by-$2sm$ matrix
$\phi^{(2s;\,m)}((c_{j})_{2sm};\{\xi_{p}\})$ by
$$\phi^{(2s;\,m)}((c_{j})_{2sm};\{\xi_{p}\})_{j,\,k}=\left(\left(\Phi^{(2s)^{%
\prime}}\right)^{-1}\Psi^{(2s)^{\prime}}((c_{j})_{2sm};\{\xi_{p}\})\right)_{j+%
M^{{}^{\prime}},\,k+M^{{}^{\prime}}}\quad\mbox{for}\,\,j,k=1,2,\ldots,2sm.$$
(4.29)
Making use of (G.2),
we obtain the spin-$s$ EFP for the finite-size chain as follows.
$$\displaystyle\tau^{(2s)}_{N_{s}}(m)=\frac{1}{\prod_{1\leq j<r\leq 2s}\sinh^{m}%
(r-j)\eta}\times\frac{1}{\prod_{1\leq k<l\leq m}\prod^{2s}_{j=1}\prod^{2s}_{r=%
1}\sinh(\xi_{k}-\xi_{l}+(r-j)\eta)}$$
$$\displaystyle\times\sum_{c_{1}=1}^{M}\sum_{c_{2}=1;c_{2}\neq c_{1}}^{M}\cdots%
\sum_{c_{2sm}=1;c_{2sm}\neq c_{1},\ldots,c_{2sm-1}}^{M}H^{(2s)}(\lambda_{c_{1}%
},\cdots,\lambda_{c_{2sm}};\{\xi_{p}\})\,{\rm det}\left(\phi^{(2s;\,m)}((c_{j}%
)_{2sm};\{\xi_{p}\})\right)$$
where $H^{(2s)}(\lambda_{c_{1}},\ldots,\lambda_{c_{2sm}};\{\xi_{p}\})$
is given by
$$\displaystyle H^{(2s)}(\lambda_{c_{1}},\ldots,\lambda_{c_{2sm}};\{\xi_{p}\})$$
$$\displaystyle\quad=\frac{1}{\prod_{1\leq l<k\leq 2sm}\sinh(\lambda_{c_{k}}-%
\lambda_{c_{l}}+\eta)}\times\prod^{2sm}_{j=1}\prod^{m}_{b=1}\prod^{2s-1}_{%
\beta=1}\sinh(\lambda_{c_{j}}-\xi_{b}+\beta\eta)$$
$$\displaystyle\quad\times\prod^{m}_{l=1}\prod^{2s}_{r_{l}=1}\left(\prod^{l-1}_{%
b=1}\sinh(\lambda_{c_{2s(l-1)+r_{l}}}-\xi_{b}+2s\eta)\prod^{m}_{k=l+1}\sinh(%
\lambda_{c_{2s(l-1)+r_{l}}}-\xi_{b})\right)\,.$$
(4.31)
4.3 Diagonal elements of the spin-$s$ Gaudin matrix
Let us define $K_{n}(\lambda)$ for $\eta=i\zeta$
with $0<\zeta<\pi$ by
$$K_{n}(\lambda)={\frac{1}{2\pi i}}\,{\frac{\sinh(n\eta)}{\sinh(\lambda-n\eta/2)%
\sinh(\lambda+n\eta/2)}}\,.$$
(4.32)
Lemma 4.1.
For $0<\zeta<\pi/2s$, we have
$$K_{1}(\lambda+n\eta)=\int_{-\infty}^{\infty}K_{2}(\lambda-\mu+n\eta+i\epsilon)%
\rho(\mu)d\mu\quad(n=1,2,\ldots,2s-1)\,,$$
(4.33)
and
$$K_{1}(\lambda-n\eta)=\int_{-\infty}^{\infty}K_{2}(\lambda-\mu-n\eta-i\epsilon)%
\rho(\mu)d\mu\quad(n=1,2,\ldots,2s-1).$$
(4.34)
Here we recall $\eta=i\zeta$.
Proof.
We first consider the case of positive $n$.
Let us recall the Lieb equation
$$\rho(\lambda)=K_{1}(\lambda)-\int_{-\infty}^{\infty}K_{2}(\lambda-\mu)\rho(\mu%
)d\mu\,.$$
(4.35)
Shifting variable $\lambda$ analytically to
$\lambda+i\zeta-i\epsilon$ in (4.35)
we have
$$\rho(\lambda+i\zeta-i\epsilon)=K_{1}(\lambda+i\zeta-i\epsilon)-\int_{-\infty}^%
{\infty}K_{2}(\lambda-\mu+i\zeta-i\epsilon)\rho(\mu)d\mu$$
(4.36)
Using
$${\frac{1}{\sinh(\lambda-\mu-i\epsilon)}}={\frac{1}{\sinh(\lambda-\mu+i\epsilon%
)}}+2\pi i\delta(\mu-\lambda)$$
(4.37)
we have
$$\int_{-\infty}^{\infty}K_{2}(\lambda-\mu+i\zeta-i\epsilon)\rho(\mu)d\mu=\int_{%
-\infty}^{\infty}K_{2}(\lambda-\mu+i\zeta+i\epsilon)\rho(\mu)d\mu+\rho(\lambda%
)\,.$$
(4.38)
Combining $\rho(\lambda+i\zeta)=-\rho(\lambda)$ we obtain eq.
(4.33) for $n=1$. Making analytic continuation
with respect to $\lambda$ we derive eq. (4.33)
for $n=2,3,\ldots,2s-1$. Similarly, we can show
(4.34).
∎
Proposition 4.2.
When $0<\zeta<\pi/2s$,
matrix elements $(A,A)$ of the spin-$s$ Gaudin matrix
with $A=2s(a-1)+\alpha$ are evaluated by
$${\frac{1}{2\pi i\,N_{s}}}\,\Phi^{(2s)\,^{\prime}}_{A,\,A}(\{\lambda_{j}\}_{M})%
=\rho_{\rm tot}(\mu_{a})+O(1/N_{s})\,.$$
(4.39)
Relations (4.39) are expressed
in terms of integrals as follows.
$$\rho_{\rm tot}(\mu_{a})={\frac{1}{N_{s}}}\sum_{p=1}^{N_{s}}K_{2s}(\mu_{a}-(%
\alpha-1/2)\eta-\xi_{p}+s\eta)-\sum_{\gamma=1}^{2s}\int^{\infty}_{-\infty}K_{2%
}(\mu_{a}-\mu_{c}-(\alpha-\gamma)\eta+\epsilon^{(\alpha,\gamma)})\rho_{\rm tot%
}(\mu_{c})d\mu_{c}\,,$$
(4.40)
where
$\epsilon^{(\alpha,\gamma)}=\epsilon^{(\alpha)}_{a}-\epsilon^{(\gamma)}_{c}$.
We recall that $C$ corresponds to $(c,\gamma)$ with $C=2s(c-1)+\gamma.$
Proof.
Let us first show
$$K_{2s}(\mu_{a}-(\alpha-1/2)\eta+s\eta)-\sum_{\gamma=1}^{2s}\int^{\infty}_{-%
\infty}K_{2}(\mu_{a}-\mu_{c}+(\gamma-\alpha)\eta+\epsilon^{(\alpha,\gamma)})%
\rho(\mu_{c})d\mu_{c}=\rho(\mu_{a})$$
(4.41)
for $\alpha=1,2\ldots,2s$.
Making use of the following relations
$$K_{n}(\lambda)={\frac{1}{2\pi i}}\left({\frac{\cosh(\lambda-n\eta/2)}{\sinh(%
\lambda-n\eta/2)}}-{\frac{\cosh(\lambda+n\eta/2)}{\sinh(\lambda+n\eta/2)}}\right)$$
(4.42)
we have
$$K_{2s}(\lambda+(2s-1)\eta/2)=\sum_{n=0}^{2s-1}K_{1}(\lambda+n\eta)\,.$$
(4.43)
We thus obtain (4.41) as follows.
$$\displaystyle K_{2s}(\mu_{a}-(\alpha-1/2)\eta+s\eta)-\sum_{\gamma=1}^{2s}\int^%
{\infty}_{-\infty}K_{2}(\mu_{a}-\mu_{c}+(\gamma-\alpha)\eta+\epsilon^{(\alpha,%
\gamma)})\rho(\mu_{c})d\mu_{c}$$
(4.44)
$$\displaystyle=$$
$$\displaystyle\sum_{\gamma=1}^{2s}\left(K_{1}(\mu_{a}+(\gamma-\alpha)\eta)-\int%
_{-\infty}^{\infty}K_{2}(\mu_{a}-\mu_{c}+(\gamma-\alpha)\eta+\epsilon^{(\alpha%
,\gamma)})\rho(\mu_{c})d\mu_{c}\right)$$
$$\displaystyle=$$
$$\displaystyle K_{1}(\mu_{a})-\int_{-\infty}^{\infty}K_{2}(\mu_{a}-\mu_{c})\rho%
(\mu_{c})d\mu_{c}$$
$$\displaystyle=$$
$$\displaystyle\rho(\mu_{a})\,.$$
Here, in the second line of (4.44),
the summands for $\gamma\neq\alpha$ vanish due to lemma 4.1.
We then apply the Lieb equation (4.35) to show
the last line of (4.44).
We obtain (4.40) from (4.41).
∎
Corollary 4.3.
Let us take a sequence of integers,
$c_{1},c_{2},\ldots,c_{2sm}$,
which satisfy $1\leq c_{j}\leq M$ for $j=1,2,\ldots,2sm$,
and determine a sequence $(z(n))_{M}$ by the conditions
that $\{z(1),z(2),\ldots,z(M^{{}^{\prime}})\}=\{1,2,\ldots,M\}\setminus\{{c_{1}},%
\ldots,{c_{2sm}}\}$,
with $z(1)<\cdots<z(M^{{}^{\prime}})$ and $z(j+M^{{}^{\prime}})=c_{j}$ for
$j=1,2,\ldots,2sm$. Here we recall $M^{{}^{\prime}}=M-2sm$.
In the region: $0<\zeta<\pi/2s$,
diagonal elements $(j,j)$ of the spin-$s$ Gaudin matrix
$\Phi^{(2s)^{\prime}}((c_{k})_{2sm})$ are evaluated as
$${\frac{1}{2\pi i\,N_{s}}}\,\Phi^{(2s)\,^{\prime}}_{j,\,j}((c_{k})_{2sm})=\rho_%
{\rm tot}(\mu_{a})+O(1/N_{s})\,,\quad\mbox{for}\,\,j=1,2,\ldots,M.$$
(4.45)
Here integer $a$ satisfies $z(j)=2s(a-1)+\alpha$ for an integer $\alpha$
with $1\leq\alpha\leq 2s$.
4.4 Integral equations
We calculate matrix elements of
$\left((\Phi^{(2s)^{\prime}})^{-1}\,\Psi^{(2s)^{\prime}}\right)((c_{j})_{2sm})$
through the spin-$s$ Gaudin matrix.
For $j,k=1,2,\ldots,M$,
we have
$$\displaystyle\left(\Psi^{(2s)^{\prime}}((c_{j})_{2sm})\right)_{j,\,k}$$
$$\displaystyle=$$
$$\displaystyle\left(\Phi^{(2s)^{\prime}}(\Phi^{2s^{\prime}})^{-1}\Psi^{(2s)^{%
\prime}}\right)_{j,\,k}$$
(4.46)
$$\displaystyle=$$
$$\displaystyle\sum_{t=1}^{M}\left(\Phi^{(2s)^{\prime}}((c_{j})_{2sm})\right)_{j%
,\,t}\left((\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}((c_{j})_{2sm})%
\right)_{t,\,k}\,.$$
We remark that matrix elements $(A,B)$ of $\Psi^{(2s)^{\prime}}((c_{l})_{2sm})$
with $A=j+M^{{}^{\prime}}$ and $B=k+M^{{}^{\prime}}$
are expressed in terms of $K_{1}(\lambda)$ as
$$\Psi^{(2s)^{\prime}}_{j+M^{{}^{\prime}},\,k+M^{{}^{\prime}}}((c_{j})_{2sm})/2%
\pi i=K_{1}(\lambda_{c_{j}}-w_{k}^{(2s)}+\eta/2)\quad\mbox{for}\quad j,k=1,2,%
\ldots,2sm.$$
(4.47)
Suppose that we have a sequence $(z(n))_{M}$ for
a given sequence $(c_{i})_{2sm}$
satisfying $1\leq c_{i}\leq M$ for $i=1,2,\ldots,2sm$.
Let us take a pair of integers $j,k$ with $1\leq j,k\leq M$.
We denote $z(j)$ by $A$, and we introduce $a$ and $\alpha$ by
$A=2s(a-1)+\alpha$ with $1\leq a\leq N_{s}/2$ and $1\leq\alpha\leq 2s$.
Applying proposition 4.2
and corollary 4.3 to (4.46)
we have
$$\displaystyle\hskip 28.452756pt\sum_{t=1}^{M}\Phi^{(2s)^{\prime}}_{j,\,t}((c_{%
l})_{2sm})/2\pi i\,\left((\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}((c_{l%
})_{2sm})\right)_{t,\,k}\\
\displaystyle=\sum_{t=1}^{M}\Biggl{\{}\left(\sum^{N_{s}}_{p=1}{\frac{1}{2\pi i%
}}\,\frac{\sinh(2s\eta)}{\sinh(\lambda_{A}-\xi_{p})\sinh(\lambda_{A}-\xi_{p}+2%
s\eta)}-\sum^{M}_{D=1}K_{2}(\lambda_{A}-\lambda_{D})\right)\delta_{A,\,z(t)}\\
\displaystyle\hskip 170.716535pt+K_{2}(\lambda_{A}-\lambda_{z(t)})\Biggr{\}}%
\left((\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}((c_{l})_{2sm})\right)_{t%
,\,k}\\
\displaystyle=N_{s}\rho_{\rm tot}(\mu_{a})\left((\Phi^{(2s)^{\prime}})^{-1}%
\Psi^{(2s)^{\prime}}((c_{j})_{2sm})\right)_{j,\,k}+\sum_{t=1}^{M}K_{2}(\lambda%
_{A}-\lambda_{z(t)})\,\left((\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}((c%
_{l})_{2sm})\right)_{t,\,k}+O(1/N_{s})\,.$$
(4.48)
Let us discuss the order of magnitude of
the correction term in (4.48).
It follows from (4.45) that
if the density of string centers $\rho(\mu_{a})$ is $O(1)$
in the large $N_{s}$ limit,
then the diagonal element $(A,A)$ of $\Phi^{(2s)^{\prime}}$ is
$O(N_{s})$.
We thus suggest that the matrix elements
of $\phi^{(2s;\,m)}((c_{l})_{2sm})$ should be at most
of the order of $1/N_{s}$,
and hence the correction term in (4.48)
should be at most $O(1/N_{s})$.
Let us now define $\varphi_{A,\,B}(\{\xi_{p}\})$
by the following relations for $j,k=1,2,\ldots,M$:
$$\varphi_{z(j),\,k}=N_{s}\rho_{\rm tot}(\mu_{a})\,\left((\Phi^{(2s)^{\prime}})^%
{-1}\Psi^{(2s)^{\prime}}((c_{l})_{2sm})\right)_{j,\,k}\,,$$
(4.49)
where integer $a$ is given by $a=[(z(j)-1)/2s]+1$.
In terms of function $a(z)=[(z-1)/2s]+1$ we have
$$\displaystyle\sum_{t=1}^{M}K_{2}(\lambda_{A}-\lambda_{z(t)})\,\left((\Phi^{(2s%
)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}((c_{l})_{2sm})\right)_{t,\,k}$$
(4.50)
$$\displaystyle=$$
$$\displaystyle\sum_{t=1}^{M}K_{2}(\lambda_{A}-\lambda_{z(t)}){\frac{\varphi_{z(%
t),\,k}}{N_{s}\,\rho_{\rm tot}(\mu_{a(z(t))})}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\gamma=1}^{2s}{\frac{1}{N_{s}}}\sum_{c=1}^{N_{s}/2}(\rho_{%
\rm tot}(\mu_{c}))^{-1}\,K_{2}(\lambda_{A}-\lambda_{(c,\gamma)})\,\varphi_{2s(%
c-1)+\gamma,\,k}\,.$$
Here we have replaced the sum over $t$ by the sum over
$c$ and $\gamma$ where $z(t)=C=2s(c-1)+\gamma$ with $1\leq\gamma\leq 2s$.
Expressing $z(j)$ and $k$ by $A$ and $B$, respectively,
we have the following equations:
$$\varphi_{A,\,B}(\{\xi_{p}\})+\sum_{\gamma=1}^{2s}{\frac{1}{N_{s}}}\sum_{c=1}^{%
N_{s}/2}(\rho_{\rm tot}(\mu_{c}))^{-1}\,K_{2}(\lambda_{A}-\lambda_{(c,\gamma)}%
)\varphi_{C,\,B}(\{\xi_{p}\})\,={\frac{1}{2\pi i}}\,\Psi^{(2s)^{\prime}}_{A,\,%
B}((c_{l})_{2sm})+O(1/N_{s})\,.$$
(4.51)
Let us introduce $b$ and $\beta$ by $k=2s(b-1)+\beta+M^{{}^{\prime}}$ with
$1\leq\beta\leq 2s$ and $1\leq b\leq N_{s}/2$.
In terms of string center $\mu_{a}$ we express (or approximate)
$\varphi_{z(j),\,k}(\{\xi_{p}\})$ by a continuous function
of $\mu_{a}$ and $\xi_{b}$, as follows.
$$\varphi_{z(j),\,k}(\{\xi_{p}\})=\varphi_{\alpha}^{(\beta)}(\mu_{a},\xi_{b})+O(%
1/N_{s})\,.$$
(4.52)
By taking the large-$N_{s}$ limit,
the discrete equations (4.51) are now expressed as follows.
$$\varphi_{\alpha}^{(\beta)}(\mu_{a},\xi_{b})+\sum_{\gamma=1}^{2s}\int_{-\infty}%
^{\infty}K_{2}(\mu_{a}-\mu_{c}+(\gamma-\alpha)\eta+\epsilon_{AC})\varphi_{%
\gamma}^{(\beta)}(\mu_{c},\xi_{b})d\mu_{c}=K_{1}(\lambda_{c_{j-M^{{}^{\prime}}%
}}-w_{k}^{(2s)}+\eta/2)\,.$$
(4.53)
Here $\epsilon_{AC}=\epsilon_{a}^{(\alpha)}-\epsilon_{c}^{(\gamma)}$.
We recall that for $j>M^{{}^{\prime}}$ we have set $z(j)=c_{j-M^{{}^{\prime}}}$.
Lemma 4.4.
In the region $0<\zeta<\pi/2s$,
a solution to the integral equations (4.53)
for integers $A=2s(a-1)+\alpha$ (i.e. $(a,\alpha)$)
and $B=2s(b-1)+\beta+M^{{}^{\prime}}$ with
$1\leq\alpha,\beta\leq 2s$ and $1\leq b\leq m$ is given by
$$\varphi_{A,\,B}=\varphi_{\alpha}^{(\beta)}(\mu_{a},\xi_{b})=\,\,\rho(\mu_{a}-%
\xi_{b})\,\delta_{\alpha,\,\beta}.$$
(4.54)
Proof.
(i) In $(\alpha,\alpha)$ case, i.e. when integers
$A=2s(a-1)+\alpha$ and $B=2s(b-1)+\alpha$
correspond to indices $(a,\alpha)$ and $(b,\alpha)$,
respectively, assuming that
$\varphi_{\gamma}^{(\alpha)}(\mu_{c},\xi_{b})=0$ for $\gamma\neq\alpha$,
we reduce integral equations (4.53)
to the Lieb equation (4.35).
Therefore, we have
$\varphi_{\alpha}^{(\alpha)}(\mu_{a},\xi_{b})=\rho(\mu_{a}-\xi_{b})$.
(ii) In $(\alpha,\beta)$ case, i.e. when $A=(a,\alpha)$
and $B=(b,\beta)$ with $\beta\neq\alpha$,
assuming $\varphi_{\gamma}^{(\beta)}(\mu_{c},\xi_{b})=0$
for $\gamma\neq\beta$, we have from (4.53)
$$\int_{-\infty}^{\infty}K_{2}(\mu_{a}-\mu_{c}+(\beta-\alpha)\eta+\epsilon_{AB})%
\varphi_{\beta}^{(\beta)}(\mu_{c},\xi_{b})d\mu_{c}={\frac{1}{2\pi i}}\,\Psi^{(%
2s)}_{A,\,B}\,.$$
(4.55)
For $\alpha<\beta$, we have $\epsilon_{AB}=i\epsilon$.
Shifting $\mu_{a}$ analytically such as
$\mu_{a}\rightarrow\mu_{a}-(\beta-\alpha-1)\eta$, we have
$$\int_{-\infty}^{\infty}K_{2}(\mu_{a}-\mu_{c}+\eta+i\epsilon)\varphi_{\beta}^{(%
\beta)}(\mu_{c},\xi_{b})d\mu_{c}={\frac{1}{2\pi i}}\,{\frac{\sinh\eta}{\sinh(%
\mu_{a}+\eta-\xi_{b}-\eta/2)\sinh(\mu_{a}+\eta-\xi_{b}+\eta/2)}}\,.$$
(4.56)
Making use of (4.37) we reduce it essentially
to the Lieb equation. We thus obtain
$\varphi_{\beta}^{(\beta)}(\mu_{a},\xi_{b})=\rho(\mu_{a}-\xi_{b})$.
For $\alpha>\beta$, we have $\epsilon_{AB}=-i\epsilon$, and
show it similarly, shifting $\mu_{a}$ analytically as
$\mu_{a}\rightarrow\mu_{a}-(\alpha-\beta+1)\eta$.
∎
Proposition 4.5.
Let us take a set of integers, $c_{1},\ldots,c_{2sm}$,
satisfying $0<c_{j}\leq M$ for $j=1,2,\ldots,2sm$.
Suppose that the number of $c_{j}$ which satisfy
$c_{j}-2s[(c_{j}-1)/2s]=\alpha$
is given by $m$ for each integer $\alpha$ satisfying
$1\leq\alpha\leq 2s$.
Then, when $0<\zeta<\pi/2s$,
the solution to integral equations (4.53)
for $A=c_{j}$ with $j=1,2,\ldots,2sm$ and
for $B=B^{{}^{\prime}}+M^{{}^{\prime}}$ where $B^{{}^{\prime}}=1,2,\ldots,2sm$,
is given by
$$\varphi_{\alpha}^{(\beta)}(\mu_{a_{j}},\xi_{b})=\,\,\rho(\mu_{a_{j}}-\xi_{b})%
\,\delta_{\alpha,\,\beta}\,,$$
(4.57)
where $a_{j}=[(c_{j}-1)/2s]+1$, $\alpha=c_{j}-2s[(c_{j}-1)/2s]$
and $B^{{}^{\prime}}=2s(b-1)+\beta$ with
$1\leq\beta\leq 2s$.
Proof.
It follows from lemma 4.54 that
$\varphi_{c_{j},\,B}$ of (4.57) gives a solution to the
integral equations. Taking the Fourier transform of (4.53),
we show in §5.5 that the set of integral equations
(4.53) for $A=c_{j}$ for $j=1,2,\ldots,2sm$ and
$B^{{}^{\prime}}=1,2,\ldots,2sm$ has a unique solution.
Thus we obtain the unique solution (4.57).
∎
Let us recall the assumption
that function $\varphi_{\alpha}^{(\beta)}(\mu_{a},\xi_{b})$
is continuous with respect to $\mu_{a}$ and $\xi_{b}$.
Then, for any given set of integers, $c_{1},\ldots,c_{2sm}$
satisfying $0<c_{j}\leq M$ for $j=1,2,\ldots,2sm$, we may
approximate the matrix elements of
$(\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}$ as follows.
For integers $j$ and $k$ with $1\leq j,k\leq 2sm$,
we define $a_{j}$, $\alpha_{j}$, $b_{k}$ and $\beta_{k}$ as follows.
$$\displaystyle a_{j}$$
$$\displaystyle=$$
$$\displaystyle[(c_{j}-1)/2s]+1\,,\quad\alpha_{j}=c_{j}-2s[(c_{j}-1)/2s]\,,$$
$$\displaystyle b_{k}$$
$$\displaystyle=$$
$$\displaystyle[(k-1)/2s]+1\,,\quad\beta_{k}=k-2s[(k-1)/2s]\,.$$
(4.58)
Then, we have
$$\left((\Phi^{(2s)^{\prime}})^{-1}\Psi^{(2s)^{\prime}}\left(({c_{j}})_{2sm}%
\right)\right)_{j+M^{{}^{\prime}},\,k+M^{{}^{\prime}}}={\frac{1}{N_{s}}}\,{%
\frac{\rho(\mu_{a_{j}}-\xi_{b_{k}})}{\rho_{\rm tot}(\mu_{a_{j}})}}\delta_{%
\alpha_{j},\,\beta_{k}}+O({1/{N_{s}^{2}}})\,.$$
(4.59)
Here we recall $M^{{}^{\prime}}=M-2sm$.
For a given $2s$-string, $\lambda_{(a,\alpha)}$,
with $\alpha=1,2,\ldots,2s$, we define
$\lambda_{(a,\alpha)}^{{}^{\prime}}$ by the ‘regular part’
of $\lambda_{(a,\alpha)}$:
$$\lambda_{(a,\alpha)}^{{}^{\prime}}=\mu_{a}-(\alpha-1/2)\eta.$$
(4.60)
Let us introduce a $2sm$-by-$2sm$ matrix $S$ by
$$S_{j,\,k}(c_{1},\ldots,c_{2sm};(\xi_{p})_{N_{s}})=\rho(\lambda^{{}^{\prime}}_{%
c_{j}}-w_{k}^{(2s)}+\eta/2)\,\delta_{\alpha_{j},\,\beta_{k}}\quad\mbox{for}\,%
\,\,j,k=1,2,\ldots,2sm.$$
(4.61)
Here $a_{j}$, $\alpha_{j}$, $b_{k}$ and $\beta_{k}$ are given by (4.58).
Then, we obtain
$$\phi^{(2s;\,m)}_{j,\,k}((c_{k})_{2sm};\{\xi_{p}\})={\frac{1}{N}_{s}}{\frac{1}{%
\rho_{\rm tot}(\mu_{a})}}\,S_{j,\,k}((c_{k})_{2sm};(\xi_{p})_{N_{s}})+O({1/{N_%
{s}^{2}}})\,.$$
(4.62)
and we have
$${\rm det}\left(\phi^{(2s;\,m)}((c_{k})_{2sm};\{\xi_{p}\})\right)=\prod_{j=1}^{%
2sm}\left({\frac{1}{N_{s}}}\,{\frac{1}{\rho_{\rm tot}(\mu_{a_{j}})}}\right)\,%
\cdot\,\Big{(}{\rm det}S((c_{j})_{2sm};(\xi_{p})_{N_{s}})\,+O(1/N_{s})\Big{)}.$$
(4.63)
4.5 Fourier transform in the cases of spin-$1$ and general spin-$s$
The integral equations (4.53)
for the spin-1 case are given by
$$\left\{\begin{array}[]{l}\displaystyle\varphi^{(1)}_{1}(\mu,\xi_{p})+\int^{%
\infty}_{-\infty}K_{2}(\mu-\lambda)\varphi^{(1)}_{1}(\lambda,\xi_{p})\,d%
\lambda+\int^{\infty}_{-\infty}K_{2}(\mu-\lambda+\eta+\epsilon^{(1,2)})\varphi%
^{(1)}_{2}(\lambda,\xi_{p})d\lambda\\
\hskip 227.622047pt\displaystyle={\frac{1}{2\pi i}}\,\frac{\sinh(\eta)}{\sinh(%
\mu-\xi_{p}+{\frac{\eta}{2}})\sinh(\mu-\xi_{p}-{\frac{\eta}{2}})}\\
\displaystyle\varphi^{(2)}_{1}(\mu,\xi_{p})+\int^{\infty}_{-\infty}K_{2}(\mu-%
\lambda)\varphi^{(2)}_{1}(\lambda,\xi_{p})d\lambda+\int^{\infty}_{-\infty}K_{2%
}(\mu-\lambda+\eta+\epsilon^{(1,2)})\varphi^{(2)}_{2}(\lambda,\xi_{p})d\lambda%
\\
\hskip 227.622047pt\displaystyle={\frac{1}{2\pi i}}\,\frac{\sinh(\eta)}{\sinh(%
\mu-\xi_{p}+\frac{\eta}{2})\sinh(\mu-\xi_{p}+{\frac{3\eta}{2}})}\\
\displaystyle\varphi^{(1)}_{2}(\mu,\xi_{p})+\int^{\infty}_{-\infty}K_{2}(\mu-%
\lambda-\eta+\epsilon^{(2,1)})\varphi^{(1)}_{1}(\lambda,\xi_{p})d\lambda+\int^%
{\infty}_{-\infty}K_{2}(\mu-\lambda)\varphi^{(1)}_{2}(\lambda,\xi_{p})d\lambda%
\\
\hskip 227.622047pt\displaystyle={\frac{1}{2\pi i}}\,\frac{\sinh(\eta)}{\sinh(%
\mu-\xi_{p}-\frac{3\eta}{2})\sinh(\mu-\xi_{p}-\frac{\eta}{2})}\\
\displaystyle\varphi^{(2)}_{2}(\mu,\xi_{p})+\int^{\infty}_{-\infty}K_{2}(\mu-%
\lambda-\eta+\epsilon^{(2,1)})\varphi^{(2)}_{1}(\lambda,\xi_{p})d\lambda+\int^%
{\infty}_{-\infty}K_{2}(\mu-\lambda)\varphi^{(2)}_{2}(\lambda,\xi_{p})d\lambda%
\\
\hskip 227.622047pt\displaystyle={\frac{1}{2\pi i}}\,\frac{\sinh(\eta)}{\sinh(%
\mu-\xi_{p}-\frac{\eta}{2})\sinh(\mu-\xi_{p}+\frac{\eta}{2})}\end{array}\right..$$
(4.64)
We solve integral equations (4.53)
via the Fourier transform.
Let us express the Fourier transform of function
$\varphi^{(\beta)}_{\alpha}(\mu,\xi)$ by
$$\widehat{\varphi}^{(\beta)}_{\alpha}(\omega,\xi)=\int^{\infty}_{-\infty}e^{i%
\mu\omega}\varphi_{\alpha}^{(\beta)}(\mu,\xi)d\mu,\quad\mbox{for}\,\,\alpha,%
\beta=1,2,\ldots,2s\,.$$
(4.65)
We denote by $\widehat{K}_{n}(\omega)$ the Fourier transform of
kernel $K_{n}(\lambda)$.
We define matrix ${\cal M}^{(2s)}_{\hat{\varphi}}$ by
$$\left({\cal M}^{(2s)}_{\hat{\varphi}}\right)_{\alpha\beta}=\widehat{\varphi}^{%
(\beta)}_{\alpha}(\omega,\xi)\quad\mbox{for}\quad\alpha,\beta=1,2,\ldots,2s\,.$$
(4.66)
We introduce a $2s$-by-$2s$ matrix ${\cal M}^{(2s)}_{K_{2}}$. We define
matrix element $(j,k)$ for $j,k=1,2,\ldots,2sm$, by
$$\left({\cal M}^{(2s)}_{K_{2}}\right)_{j,k}=\left\{\begin{array}[]{cc}%
\displaystyle{1+\int^{\infty}_{-\infty}e^{i\mu\omega}K_{2}(\mu)\,d\mu}&\mbox{%
for}\,\,j=k\,,\\
&\\
\displaystyle{\int^{\infty}_{-\infty}e^{i\mu\omega}K_{2}(\mu+(k-j)\eta+i0)\,d%
\mu}&\mbox{for}\,\,j<k\,,\\
&\\
\displaystyle{\int^{\infty}_{-\infty}e^{i\mu\omega}K_{2}(\mu-(j-k)\eta-i0)\,d%
\mu}&\mbox{for}\,\,j>k\,.\\
\end{array}\right.$$
(4.67)
When $0<\zeta<\pi/2s$,
we calculate the matrix elements of ${\cal M}^{(2s)}_{K_{2}}$ as follows.
$$\left({\cal M}^{(2s)}_{K_{2}}\right)_{j,k}=\delta_{j,k}(1+{\widehat{K}}_{2}(%
\omega))+(1-\delta_{j,k})e^{(k-j)\zeta\omega}\left({\widehat{K}}_{2}(\omega)-e%
^{{\rm sgn}(j-k)\,\zeta\omega}\right)\quad\mbox{for}\,\,j,k=1,2,\ldots,2s.$$
(4.68)
Here we define ${\rm sgn}(j-k)$ by the following:
${\rm sgn}(j-k)=-1$ for $j-k<0$, and ${\rm sgn}(j-k)=+1$
for $j-k>0$. Here, ${\widehat{K}}_{2}(\omega)$, is given by
$${\widehat{K}}_{2}(\omega)=\int_{-\infty}^{\infty}e^{i\omega\mu}K_{2}(\mu)d\mu=%
{\frac{\sinh(\displaystyle{\frac{\pi}{2}}-\zeta)\omega}{\sinh(\displaystyle{%
\frac{\pi\omega}{2}})}}\,.$$
(4.69)
Similarly, we define a $2s$-by-$2s$ matrix ${\cal M}^{(2s)}_{K_{1}}$ by
$$\left({\cal M}^{(2s)}_{K_{1}}\right)_{j,k}=\int_{-\infty}^{\infty}e^{i\omega%
\mu}K_{1}(\mu-\xi_{b}+(k-j)\eta)d\mu\quad\mbox{for}\,\,j,k=1,2,\ldots,2s.$$
(4.70)
When $0<\zeta<\pi/2s$, we can show
$$\left({\cal M}^{(2s)}_{K_{1}}\right)_{j,k}=e^{i\xi_{b}\omega}\left\{\delta_{j,%
k}{\widehat{K}}_{1}(\omega)+(1-\delta_{j,k})\,e^{(k-j)\zeta\omega}\left({%
\widehat{K}}_{1}(\omega)-e^{{\rm sgn}(j-k)\zeta\omega/2}\right)\right\}$$
(4.71)
for $j,k=1,2,\ldots,2s$. Here
${\widehat{K}}_{1}(\omega)$ is given by
$${\widehat{K}}_{1}(\omega)=\int_{-\infty}^{\infty}e^{i\omega\mu}K_{1}(\mu)d\mu=%
{\frac{\sinh\left(\displaystyle{{\frac{\pi}{2}}-{\frac{\zeta}{2}}}\right)%
\omega}{\sinh(\displaystyle{\frac{\pi\omega}{2}})}}\,.$$
(4.72)
Taking the Fourier transform of integral equations (4.53)
we have the following matrix equation.
$${\cal M}^{(2s)}_{K_{2}}\,{\cal M}^{(2s)}_{\hat{\varphi}}={\cal M}^{(2s)}_{K_{1%
}}\,.$$
(4.73)
For the spin-1 case, from (4.64) we have
$$\displaystyle\left(\begin{array}[]{cc}1+\widehat{K}_{2}(\omega)&e^{\zeta\omega%
}\widehat{K}_{2}(\omega)-1\\
e^{-\zeta\omega}\widehat{K}_{2}(\omega)-1&1+\widehat{K}_{2}(\omega)\\
\end{array}\right)\left(\begin{array}[]{cc}\widehat{\varphi}^{(1)}_{1}(\omega)%
&\widehat{\varphi}^{(2)}_{1}(\omega)\\
\widehat{\varphi}^{(1)}_{2}(\omega)&\widehat{\varphi}^{(2)}_{2}(\omega)\\
\end{array}\right)$$
$$\displaystyle\quad=e^{i\xi_{b}\omega}\,\left(\begin{array}[]{cc}\widehat{K}_{1%
}(\omega)&e^{\zeta\omega}\widehat{K}_{1}(\omega)-e^{\zeta\omega/2}\\
e^{-\zeta\omega}\widehat{K}_{1}(\omega)-e^{-\zeta\omega/2}&\widehat{K}_{1}(%
\omega)\\
\end{array}\right)\,.$$
(4.74)
It is easy to show that matrix $M^{(2)}({\widehat{\varphi}})$
is given by the following:
$$\left(\begin{array}[]{cc}\widehat{\varphi}^{(1)}_{1}(\omega)&\widehat{\varphi}%
^{(2)}_{1}(\omega)\\
\widehat{\varphi}^{(1)}_{2}(\omega)&\widehat{\varphi}^{(2)}_{2}(\omega)\\
\end{array}\right)=\left(\begin{array}[]{cc}\displaystyle{\frac{e^{i\xi_{b}%
\omega}}{2\cosh(\zeta\omega/2)}}&0\\
0&\displaystyle{\frac{e^{i\xi_{b}\omega}}{2\cosh(\zeta\omega/2)}}\\
\end{array}\right)\,.$$
(4.75)
We calculate the determinant of ${\cal M}^{(2)}_{K_{2}}$ (the spin-1 case)
as follows.
$$\displaystyle{\rm det}\left(\begin{array}[]{cc}1+\widehat{K}_{2}(\omega)&e^{%
\zeta\omega}\widehat{K}_{2}(\omega)-1\\
e^{-\zeta\omega}\widehat{K}_{2}(\omega)-1&1+\widehat{K}_{2}(\omega)\end{array}\right)$$
$$\displaystyle=$$
$$\displaystyle{\rm det}\left(\begin{array}[]{cc}1+\widehat{K}_{2}(\omega)&e^{%
\zeta\omega}\widehat{K}_{2}(\omega)-1\\
-(1+e^{-\zeta\omega})&1+e^{-\zeta\omega}\end{array}\right)$$
(4.76)
$$\displaystyle=$$
$$\displaystyle{\rm det}\left(\begin{array}[]{cc}\widehat{K}_{2}(\omega)(1+e^{%
\zeta\omega})&e^{\zeta\omega}\widehat{K}_{2}(\omega)-1\\
0&1+e^{-\zeta\omega}\end{array}\right)$$
$$\displaystyle=$$
$$\displaystyle\widehat{K}_{2}(\omega)(1+e^{-\zeta\omega})(1+e^{\zeta\omega})\,.$$
Here, we first subtract the 2nd row by the 1st row multiplied by
$e^{-\zeta\omega}$. We next add the 2nd column to the 1st column.
Finally, the determinant is factorized and we have the result.
By the same method we can calculate the determinant of
${\cal M}^{(2s)}_{K_{2}}$ for
the spin-$s$ case. We have
$${\rm det}{\cal M}^{(2s)}_{K_{2}}=\left(2\cosh(\zeta\omega/2)\right)^{2s}\,%
\widehat{K}_{2s}(\omega)\,.$$
(4.77)
The determinant is nonzero generically,
and hence the solution to matrix equation (4.73) is unique.
Therefore, we obtain
the solution of integral equation (4.53).
5 The EFP of the spin-$s$ XXZ spin chain near AF point
5.1 Multiple-integral representations of the spin-$s$ EFP
Let us derive multiple-integral representations
for the emptiness formation probability of the
spin-$s$ XXZ spin chain.
We shall take the large $N_{s}$ limit of
the EFP (LABEL:eq:EFP-finite) for a finite-size system,
and we replace rapidity $\lambda_{c_{j}}$ with
complex variable $\lambda_{j}$ for $j=1,2,\ldots,2sm$, as follows.
For a given rapidity of $2s$-string,
$\lambda_{A}=\mu_{a}-(\alpha-1/2)\eta+\epsilon_{A}$,
we define its regular part $\lambda_{A}^{{}^{\prime}}$ by
$\lambda_{A}^{{}^{\prime}}=\mu_{a}-(\alpha-1/2)\eta$.
In the large-$N_{s}$ limit, we first replace
$\lambda_{c_{k}}=\lambda_{c_{k}}^{{}^{\prime}}+\epsilon_{c_{k}}$
by $\lambda_{k}^{{}^{\prime}}+\epsilon_{c_{k}}$
where $\lambda_{k}^{{}^{\prime}}$ are complex integral variables
corresponding to complete strings such as $\lambda_{k}^{{}^{\prime}}=\mu_{k}-(\beta-1/2)\eta$ for some integer $\beta$
with $1\leq\beta\leq 2s$ where $\eta=i\zeta$ with $0<\zeta<\pi$
and $\mu_{k}$ is real.
We express $\lambda_{k}^{{}^{\prime}}$ and $\epsilon_{c_{k}}$
simply by $\lambda_{k}$ and $\epsilon_{k}$,
respectively, and then we obtain multiple-integral representations.
Applying (4.63)
we derive the emptiness formation probability for arbitrary spin-$s$
in the thermodynamic limit $N_{s}\rightarrow\infty$, as follows.
$$\begin{split}\displaystyle\tau^{(2s)}_{\infty}(m;\{\xi_{p}\})=&\displaystyle%
\frac{1}{\prod_{1\leq j<r\leq 2s}(\sinh(r-j)\eta)^{m}}\times\frac{1}{\prod_{1%
\leq k<l\leq m}\prod^{2s}_{j=1}\prod^{2s}_{r=1}\sinh(\xi_{k}-\xi_{l}+(r-j)\eta%
)}\\
&\displaystyle\times\prod^{2sm}_{l=1}\left(\sum^{2s}_{k=1}\int^{\infty+(-k+%
\frac{1}{2})\eta}_{-\infty+(-k+\frac{1}{2})\eta}d\lambda_{l}\right)H^{(2s)}(%
\lambda_{1},\cdots,\lambda_{2sm})\,\textstyle{\det}S(\lambda_{1},\ldots,%
\lambda_{2sm})\end{split}$$
(5.1)
where $H^{(2s)}((\lambda_{l})_{2sm})$ is given by
$$\begin{split}\displaystyle H^{(2s)}((\lambda_{l})_{2sm})=&\displaystyle\frac{1%
}{\prod_{1\leq l<k\leq 2sm}\sinh(\lambda_{k}-\lambda_{l}+\eta+\epsilon_{k,l})}%
\\
&\displaystyle\times\prod^{2sm}_{l=1}\prod^{m}_{k=1}\prod^{2s-1}_{p=1}\sinh(%
\lambda_{l}-\xi_{k}+(2s-p)\eta)\\
&\displaystyle\times\prod^{m}_{l=1}\prod^{2s}_{r_{l}=1}\left(\prod^{l-1}_{k=1}%
\sinh(\lambda_{2s(l-1)+r_{l}}-\xi_{k}+2s\eta)\prod^{m}_{k=l+1}\sinh(\lambda_{2%
s(l-1)+r_{l}}-\xi_{k})\right)\end{split}$$
(5.2)
and matrix elements of $S(\lambda_{1},\ldots,\lambda_{2sm})$ are given by
$$S_{j,\,2s(l-1)+k}=\left\{\begin{array}[]{ll}\rho(\lambda_{j}-\xi_{l}+(k-{\frac%
{1}{2}})\eta)&\mbox{if}\quad\lambda_{j}-\mu_{j}=(\frac{1}{2}-k)\eta\\
0&otherwise.\end{array}\right.$$
(5.3)
Here $\mu_{j}$ denotes
the center of the $2s$-string in which
$\lambda_{j}$ is the $k$th rapidity.
Explicitly, we have $\lambda_{j}=\mu_{j}-(k-1/2)\eta$.
In the denominator, we have set $\epsilon_{k,l}$ associated
with $\lambda_{k}$ and $\lambda_{l}$ as follows.
$$\epsilon_{k,\,l}=\left\{\begin{array}[]{cc}i\epsilon&\mbox{for}\quad{\cal I}m(%
\lambda_{k}-\lambda_{l})>0\,,\\
-i\epsilon&\mbox{for}\quad{\cal I}m(\lambda_{k}-\lambda_{l})<0.\end{array}\right.$$
(5.4)
In the homogeneous case we have $\epsilon_{p}=0$ for $p=1,2,\ldots,N_{s}$.
We have thus defined inhomogeneous parameters $\xi_{p}$.
We recall that in the homogeneous case, the spin-$s$ Hamiltonian is derived
from the logarithmic derivative of the spin-$s$ transfer matrix.
Here we should remark that an expression of the matrix elements
of $S$ similar to (5.3)
has been given in eq. (6.14) of Ref. [13]
for the correlation functions of the integrable spin-$s$ XXX spin chain.
5.2 Symmetric expression of the spin-$s$ EFP
We shall express the spin-$s$ EFP (5.1)
in a simpler way, making use of permutations
of $2sm$ integers, $1,2,\cdots,2sm$,
and the formula of the Cauchy determinant.
Let us take a set of integers $a_{(j,k)}=a_{2s(j-1)+k}$
satisfying $1\leq a_{(j,k)}\leq N_{s}/2$
for $j=1,2,\ldots,m$ and $k=1,2,\ldots,2s$.
Here we remark that indices $a_{(j,k)}$ correspond
to the string centers $\mu_{(j,k)}=\mu_{2s(j-1)+k}$.
In order to reformulate the sum
over integers, $c_{1},\ldots,c_{2sm}$, in eq. (LABEL:eq:EFP-finite)
in terms of indices $a_{(j,k)}$,
let us introduce ${\hat{c}}_{1},\ldots,{\hat{c}}_{2sm}$ by
$${\hat{c}}_{2s(j-1)+k}=2s(a_{(j,k)}-1)+k\,,\quad\mbox{for}\quad j=1,2,\ldots,m%
\quad\mbox{and}\,k=1,2,\ldots,2s\,.$$
(5.5)
We also define $\beta(z)$ by $\beta(z)=z-2s[(z-1)/2s]$.
Then, ${\hat{c}}_{j}$ are expressed as follows.
$${\hat{c}}_{j}=2s(a_{j}-1)+\beta(j)\,,\quad\mbox{for}\,\,j=1,2,\ldots,2sm.$$
(5.6)
We decompose the sum over $c_{j}$ into $2s$ sums over $a_{j}$ as follows.
$$\sum_{c_{j}=1}^{M}g(c_{j})=\sum_{k=1}^{2s}\sum_{a_{j}=1}^{N_{s}/2}g(2s(a_{j}-1%
)+k)\,.$$
(5.7)
Let us consider such a function $f(c_{1},c_{2},\ldots,c_{2sm})$
of sequence of integers $(c_{j})_{2sm}$
that vanishes unless $c_{j}$’s are distinct.
We also assume that $f(c_{1},c_{2},\ldots,c_{2sm})$ vanishes
unless the number of $c_{j}$’s satisfying
$\beta(c_{j})=\alpha$ is given by $m$ for each integer $\alpha$
satisfying $1\leq\alpha\leq 2s$. Here we recall that the two properties
are in common with the summand of (LABEL:eq:EFP-finite), in particular,
with ${\rm det}(\phi^{(2s;\,m)}((c_{j})_{2sm};\{\xi_{p}\})$.
Then, we have
$$\displaystyle\sum_{c_{1}=1}^{M}\sum_{c_{2}=1}^{M}\cdots\sum_{c_{2sm}=1}^{M}f(c%
_{1},\cdots,c_{2sm})={\frac{1}{(m!)^{2s}}}\sum_{a_{1}=1}^{{N_{s}}/2}\sum_{a_{2%
}=1}^{{N_{s}}/2}\cdots\sum_{a_{2sm}=1}^{{N_{s}}/2}\sum_{P\in{\cal S}_{2sm}}f({%
\hat{c}}_{P1},\cdots,{\hat{c}}_{P(2sm)})$$
$$\displaystyle\qquad=\sum_{a_{1}=1}^{{N_{s}}/2}\sum_{a_{2}=1}^{{N_{s}}/2}\cdots%
\sum_{a_{2sm}=1}^{{N_{s}}/2}\sum_{\pi\in{\cal S}_{2sm}/(S_{m})^{2s}}f({\hat{c}%
}_{\pi 1},\cdots,{\hat{c}}_{\pi(2sm)})\,.$$
(5.8)
Here an element $\pi$ of ${\cal S}_{2sm}/({\cal S}_{m})^{2s}$
gives a permutation of integers $1,2,\ldots,2sm$,
where $\pi j$’s such that $\pi j\equiv k$ (mod $2s$)
are put in increasing order in the sequence
($\pi 1,\pi 2,\ldots,\pi(2sm))$ for $k=0,1,\ldots,2s-1$.
Reformulating the sum over $c_{j}$ s in (LABEL:eq:EFP-finite)
in terms of $a_{j}$’s, in the large $N_{s}$ limit we have
$$\displaystyle\tau^{(2s)}_{\infty}(m)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\prod_{1\leq\alpha<\beta\leq 2s}(\sinh(\beta-\alpha)\eta%
)^{m}}\times\frac{1}{\prod_{1\leq k<l\leq m}\prod^{2s}_{\alpha=1}\prod^{2s}_{%
\beta=1}\sinh(\xi_{k}-\xi_{l}+(\alpha-\beta)\eta)}$$
(5.9)
$$\displaystyle\times\prod_{k=1}^{2s}\prod_{j=1}^{m}\int_{-\infty}^{\infty}d\mu_%
{(j,k)}\,\sum_{P\in{\cal S}_{2sm}}{\frac{1}{(m!)^{2s}}}{\rm det}S(\lambda_{P1}%
,\ldots,\lambda_{P(2sm)})\,H^{(2s)}(\lambda_{P1},\ldots,\lambda_{P(2sm)})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{\prod_{1\leq\alpha<\beta\leq 2s}(\sinh(\beta-\alpha)\eta%
)^{m}}\times\frac{1}{\prod_{1\leq k<l\leq m}\prod^{2s}_{\alpha=1}\prod^{2s}_{%
\beta=1}\sinh(\xi_{k}-\xi_{l}+(\alpha-\beta)\eta)}$$
$$\displaystyle\times$$
$$\displaystyle\prod_{j=1}^{2sm}\int_{-\infty}^{\infty}d\mu_{j}\,\sum_{\pi\in{%
\cal S}_{2sm}/({\cal S}_{m})^{2s}}{\rm det}S(\lambda_{\pi 1},\ldots,\lambda_{%
\pi(2sm)})\,H^{(2s)}(\lambda_{\pi 1},\ldots,\lambda_{\pi(2sm)})\,,$$
where symbols $\lambda_{j}$ denote the following
$$\lambda_{j}=\mu_{j}-(\beta(j)-{\frac{1}{2}})\eta\,\,\quad\mbox{for}\,\,j=1,2,%
\ldots,2sm.$$
(5.10)
We calculate ${\rm det}S$
applying the Cauchy determinant formula
$$\det\left(\frac{1}{\sinh(\lambda_{a}-\xi_{k})}\right)=\frac{\prod^{m}_{k<l}%
\sinh(\xi_{k}-\xi_{l})\prod^{m}_{a>b}\sinh(\lambda_{a}-\lambda_{b})}{\prod^{m}%
_{a=1}\prod^{m}_{k=1}\sinh(\lambda_{a}-\xi_{k})}\,,$$
(5.11)
and we obtain the symmetric expression of the spin-$s$ EFP as follows.
$$\begin{split}\displaystyle\tau^{(2s)}_{\infty}(m;\{\xi_{p}\})&\displaystyle=%
\frac{1}{\prod_{1\leq\alpha<\beta\leq 2s}\sinh^{m}(\beta-\alpha)\eta}\,\prod_{%
1\leq k<l\leq m}\frac{\sinh^{2s}(\pi(\xi_{k}-\xi_{l})/\zeta)}{\prod^{2s}_{j=1}%
\prod^{2s}_{r=1}\sinh(\xi_{k}-\xi_{l}+(r-j)\eta)}\\
&\displaystyle\times\,{\frac{i^{2sm^{2}}}{(2i\zeta)^{2sm}}}\,\left(\prod^{2sm}%
_{j=1}\int^{\infty}_{-\infty}d\mu_{j}\right)\,\prod^{2s}_{\gamma=1}\prod_{1%
\leq b<a\leq m}\sinh(\pi(\mu_{2s(a-1)+\gamma}-\mu_{2s(b-1)+\gamma})/\zeta)\\
&\displaystyle\times\left(\prod^{2sm}_{j=1}{\frac{\prod^{m}_{b=1}\prod^{2s-1}_%
{\beta=1}\sinh(\lambda_{j}-\xi_{b}+\beta\eta)}{\prod_{b=1}^{m}\cosh(\pi(\mu_{j%
}-\xi_{b})/\zeta)}}\right)\,\sum_{\sigma\in{\cal S}_{2sm}/({\cal S}_{m})^{2s}}%
({\rm sgn}\,\sigma)\\
&\displaystyle\times\frac{\prod^{m}_{l=1}\prod^{2s}_{r_{l}=1}\left(\prod^{l-1}%
_{k=1}\sinh(\lambda_{\sigma(2s(l-1)+r_{l})}-\xi_{k}+2s\eta)\prod^{m}_{k=l+1}%
\sinh(\lambda_{\sigma(2s(l-1)+r_{l})}-\xi_{k})\right)}{\prod_{1\leq l<k\leq 2%
sm}\sinh(\lambda_{\sigma(k)}-\lambda_{\sigma(l)}+\eta+\epsilon_{\sigma(k),%
\sigma(l)})}\,.\end{split}$$
(5.12)
Here (${\rm sgn}\,\sigma$)
denotes the sign of permutation
$\sigma\in{\cal S}_{2sm}/({\cal S}_{m})^{2s}$.
5.3 The spin-$s$ EFP for the homogeneous chain
Sending $\xi_{b}$ to zero for $b=1,2,\ldots,m$, we derive
the spin-$s$ EFP in the homogeneous limit.
Here we remark that the spin-$s$ Hamiltonian is derived from the logarithmic
derivative of the spin-$s$ transfer matrix
$t^{(2s,\,2s)}(\lambda;\{\xi_{b}\}_{N_{s}})$
in the homogeneous case where $\xi_{b}=0$ for $b=1,2,\ldots,N_{s}$.
Sending $\xi_{b}$ to zero for $b=1,2,\ldots,m$,
we have the following:
$$\begin{split}&\displaystyle\lim_{\xi_{1}\rightarrow 0}\lim_{\xi_{2}\rightarrow
0%
}\cdots\lim_{\xi_{m}\rightarrow 0}\,\left(\tau^{(2s)}_{\infty}(m;\{\xi_{p}\}_{%
m})\right)\\
&\displaystyle={\frac{\left(\pi/\zeta\right)^{sm(m-1)}}{\prod_{1\leq\alpha<%
\beta\leq 2s}\sinh^{m^{2}}((\beta-\alpha)\eta)}}\\
&\displaystyle\times\,{\frac{i^{2sm^{2}}}{(2i\zeta)^{2sm}}}\,\,\prod^{2sm}_{j=%
1}\int^{\infty}_{-\infty}d\mu_{j}\,\prod^{2s}_{\beta=1}\prod_{1\leq b<a\leq m}%
\sinh(\pi(\mu_{2s(a-1)+\beta}-\mu_{2s(b-1)+\beta})/\zeta)\\
&\displaystyle\times\left(\prod^{2sm}_{j=1}{\frac{\prod^{2s-1}_{\beta=1}\sinh^%
{m}(\lambda_{j}+\beta\eta)}{\cosh^{m}(\pi\mu_{j}/\zeta)}}\right)\times\sum_{%
\sigma\in{\cal S}_{2sm}/({\cal S}_{m})^{2s}}({\rm sgn}\,\sigma)\\
&\displaystyle\times{\frac{\prod^{m}_{l=1}\prod^{2s}_{r_{l}=1}\left(\sinh^{l-1%
}(\lambda_{\sigma(2s(l-1)+r_{l})}+2s\eta)\sinh^{m-l}(\lambda_{\sigma(2s(l-1)+r%
_{l})})\right)}{\prod_{1\leq l<k\leq 2sm}\sinh(\lambda_{\sigma(k)}-\lambda_{%
\sigma(l)}+\eta+\epsilon_{\sigma(k),\sigma(l)})}}\,.\end{split}$$
(5.13)
Here we recall definition (5.10) of $\lambda_{j}$.
Let us discuss that expression (5.13) gives
the spin-$s$ EFP for the homogeneous chain.
First, we remark that
$\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})$ does not depend on
$\xi_{p}$ with $p>m$. Hence we may consider that
inhomogeneous parameters $\xi_{p}$ with $p>m$ are all set to be zero,
after computing the EFP: $\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})$.
We now show that the order of the homogeneous limit: $\xi_{p}\rightarrow 0$
and the thermodynamic limit $N_{s}\rightarrow\infty$ can be reversed.
We can show the following relation:
$$\prod_{p=1}^{m}\lim_{\xi_{p}\rightarrow 0}\,\left(\lim_{N_{s}\rightarrow\infty%
}\,\left(\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})\right)\right)=\lim_{N_{s}%
\rightarrow\infty}\left(\prod_{p=1}^{m}\lim_{\xi_{p}\rightarrow 0}\,\left(\tau%
^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})\right)\right)\,.$$
(5.14)
In fact, when $N_{s}$ is very large, it follows from
(4.63) that we have
$$\tau^{(2s)}_{\infty}(m;\{\xi_{p}\}_{m})=\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})%
+O(1/N_{s}).$$
(5.15)
Furthermore, we can explicitly show that
$\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})$ is continuous
with respect to $\xi_{p}$ at $\xi_{p}=0$ for $p=1,2,\ldots,m$.
We first reformulate the sum over $c_{j}$ in (LABEL:eq:EFP-finite)
into the sum over $a_{(j,k)}$ by relation (5.8).
$$\displaystyle\sum_{c_{1}=1}^{M}\cdots\sum_{c_{2sm}=1}^{M}{\rm det}S(({c}_{j})_%
{2sm})H^{(2s)}((\lambda_{c_{j}})_{2sm})$$
(5.16)
$$\displaystyle=$$
$$\displaystyle\prod_{j=1}^{m}\prod_{k=1}^{2s}\left({\frac{1}{m!}}\sum_{a_{(j,k)%
}=1}^{N_{s}/2}\right)\sum_{P\in{\cal S}_{2sm}}{\rm det}S(({\hat{c}}_{Pj})_{2sm%
})H^{(2s)}((\lambda_{{\hat{c}}_{j}})_{2sm})$$
$$\displaystyle=$$
$$\displaystyle\prod_{j=1}^{m}\prod_{k=1}^{2s}\left({\frac{1}{m!}}\sum_{a_{(j,k)%
}=1}^{N_{s}/2}\right){\rm det}S(({\hat{c}}_{j})_{2sm})\sum_{P\in{\cal S}_{2sm}%
}({\rm sgn}P)\,H^{(2s)}((\lambda_{{\hat{c}}_{j}})_{2sm})\,.$$
We then apply the Cauchy determinant formula to
evaluate ${\rm det}S({\hat{c}}_{j})$ as follows.
$$\displaystyle{\rm det}S(({\hat{c}}_{j})_{2sm})={\rm det}S((\lambda_{{\widehat{%
c}}_{j}})_{2sm})=\prod_{\alpha=1}^{2s}{\rm det}S^{(\alpha)}((\lambda_{2s(a_{(j%
,\alpha)}-1)+\alpha})_{m})$$
$$\displaystyle=$$
$$\displaystyle{\frac{\left(\prod_{j<k}\sinh\pi(\xi_{j}-\xi_{k})/\zeta\right)^{2%
s}\cdot\prod_{\alpha=1}^{2s}\prod_{j<k}\sinh\left\{\pi(\mu_{2s(a_{(j,\alpha)}-%
1)+\alpha}-\mu_{2s(a_{(k,\alpha)}-1)+\alpha})/\zeta\right\}}{(2i\zeta)^{2sm}%
\prod_{j=1}^{2sm}\prod_{b=1}^{m}\sinh\pi(\mu_{j}-\xi_{b}-\eta/2)/\zeta}}\,.$$
Making use of (LABEL:eq:detS) we show that such factors in
the denominator of (LABEL:eq:EFP-finite)
that vanish in the limit of sending $\xi_{p}$
to zero are canceled by the factors in the numerator of
(LABEL:eq:detS).
We thus have shown that the EFP for the finite system,
$\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}\}_{m})$,
is continuous with respect to $\xi_{p}$ at $\xi_{p}=0$.
Therefore, expression (5.13) gives
the spin-$s$ EFP for the homogeneous chain.
That is, we have the following equality:
$$\lim_{\xi_{1}\rightarrow 0}\lim_{\xi_{2}\rightarrow 0}\cdots\lim_{\xi_{m}%
\rightarrow 0}\,\left(\tau^{(2s)}_{\infty}(m;\{\xi_{p}\}_{m})\right)=\lim_{N_{%
s}\rightarrow\infty}\tau^{(2s)}_{N_{s}}(m;\{\xi_{p}=0\}_{N_{s}})\,.$$
(5.18)
5.4 The spin-1 EFP with $m=1$
Let us calculate $\tau^{(2s)}(m)$ for $s=1$ and $m=1$.
From formula (5.1) we have
$$\displaystyle\tau^{(2)}(1)={\frac{1}{\sinh\eta}}\left(\int^{\infty-\eta/2}_{-%
\infty-\eta/2}d\lambda_{1}+\int^{\infty-3\eta/2}_{-\infty-3\eta/2}d\lambda_{1}%
\right)\left(\int^{\infty-\eta/2}_{-\infty-\eta/2}d\lambda_{2}+\int^{\infty-3%
\eta/2}_{-\infty-3\eta/2}d\lambda_{2}\right)$$
$$\displaystyle\qquad\times H^{(2)}(\lambda_{1},\lambda_{2})\,{\rm det}S(\lambda%
_{1},\lambda_{2})$$
$$\displaystyle=$$
$$\displaystyle-{\frac{1}{i\sin\zeta}}\,{\frac{1}{4\zeta^{2}}}\int^{\infty}_{-%
\infty}d\mu_{1}\int^{\infty}_{-\infty}d\mu_{2}\frac{\sinh(\mu_{1}-\xi_{1}+\eta%
/2)\sinh(\mu_{2}-\xi_{1}-\eta/2)}{\sinh(\mu_{1}-\mu_{2}+i\epsilon)\,\cosh\left%
(\pi(\mu_{1}-\xi_{1})/\zeta\right)\cosh\left(\pi(\mu_{2}-\xi_{1})/\zeta\right)}$$
$$\displaystyle-$$
$$\displaystyle{\frac{1}{i\sin\zeta}}\,\frac{(-1)}{4\zeta^{2}}\int^{\infty}_{-%
\infty}d\mu_{1}\int^{\infty}_{-\infty}d\mu_{2}\frac{\sinh(\mu_{1}-\xi_{1}-\eta%
/2)\sinh(\mu_{2}-\xi_{1}+\eta/2)}{\sinh(\mu_{1}-\mu_{2}-2\eta)\,\cosh\left(\pi%
(\mu_{1}-\xi_{1})/\zeta\right)\cosh\left(\pi(\mu_{2}-\xi_{1})/\zeta\right)}\,.$$
Here we note that ${\rm det}S(\lambda_{1},\lambda_{2})=0$ for
$\lambda_{1}=\mu_{1}-\eta/2$ and $\lambda_{2}=\mu_{2}-3\eta/2$,
or for $\lambda_{1}=\mu_{1}-3\eta/2$ and $\lambda_{2}=\mu_{2}-\eta/2$.
Showing the following relations of integrals
$$\displaystyle\int_{-\infty}^{\infty}d\mu{\frac{1}{\cosh\left(\pi\mu/\zeta%
\right)}}{\frac{\sinh(\mu-\eta/2)}{\sinh(\lambda-\mu+i\epsilon)}}$$
$$\displaystyle=$$
$$\displaystyle-\int_{-\infty}^{\infty}d\mu{\frac{1}{\cosh\left(\pi\mu/\zeta%
\right)}}{\frac{\sinh(\mu+\eta/2)}{\sinh(\lambda-\mu-\eta)}}-2\pi i{\frac{%
\sinh(\lambda-\eta/2)}{\cosh(\pi\lambda/\zeta)}}\,,$$
$$\displaystyle\int_{-\infty}^{\infty}d\lambda{\frac{1}{\cosh\left(\pi\lambda/%
\zeta\right)}}{\frac{\sinh(\lambda+\eta/2)}{\sinh(\lambda-\mu-\eta)}}$$
$$\displaystyle=$$
$$\displaystyle-\int_{-\infty}^{\infty}d\lambda{\frac{1}{\cosh\left(\pi\lambda/%
\zeta\right)}}{\frac{\sinh(\lambda-\eta/2)}{\sinh(\lambda-\mu-2\eta)}}\,,$$
(5.20)
we thus have
$$\tau^{(2)}(1)={\frac{\pi}{4}}{\frac{1}{\zeta\sin\zeta}}\left(\int_{-\infty}^{%
\infty}dx\,{\frac{\cosh 2\zeta x-\cosh\eta}{\cosh^{2}\pi x}}\right)\,.$$
(5.21)
Evaluating the integral we obtain the spin-1 EFP with $m=1$ as follows.
$$\tau^{(2)}(1)=\frac{\zeta-\sin\zeta\cos\zeta}{2\zeta\sin^{2}\zeta}\,.$$
(5.22)
Let us denote by $\langle E_{1}^{a,\,b}\rangle$ the ground-state
expectation value of operator $E_{1}^{a,\,b}$. For the spin-1 case,
we have $\widetilde{E}_{1}^{0,\,0\,(2\,+)}+\widetilde{E}_{1}^{1,\,1\,(2\,+)}+\widetilde%
{E}_{1}^{2,\,2\,(2\,+)}=\widetilde{P}_{1}^{(2)}$,
and hence we have
$$\langle\widetilde{E}_{1}^{0,\,0\,(2\,+)}\rangle+\langle\widetilde{E}_{1}^{1,\,%
1\,(2\,+)}\rangle+\langle\widetilde{E}_{1}^{2,\,2\,(2\,+)}\rangle=1\,.$$
(5.23)
Due to the uniaxial symmetry we have
$\langle\widetilde{E}_{1}^{0,\,0\,(2\,+)}\rangle=\langle\widetilde{E}_{1}^{2,\,%
2\,(2\,+)}\rangle$.
Thus, we obtain
$$\langle\widetilde{E}_{1}^{1,\,1\,(2\,+)}\rangle={\frac{\cos\zeta\,(\sin\zeta-%
\zeta\cos\zeta)}{\zeta\sin^{2}\zeta}}\,.$$
(5.24)
In the XXX limit, we have
$$\lim_{\zeta\rightarrow 0}\frac{\zeta-\sin\zeta\cos\zeta}{2\zeta\sin^{2}\zeta}=%
{\frac{1}{3}}\,.$$
(5.25)
The limiting value 1/3 coincides with the spin-1 XXX result
obtained by Kitanine [13].
As pointed out in Ref. [13], $\langle E_{1}^{22}\rangle=\langle E_{1}^{11}\rangle=\langle E_{1}^{00}\rangle=%
1/3$ for the XXX case since it has
the rotational symmetry.
In the symmetric expression of the spin-$1$ EFP with $m=1$,
putting $\lambda_{1}=\mu_{1}-\eta/2$ and $\lambda_{2}=\mu_{2}-3\eta/2$
in (5.13),
we directly obtain the following:
$$\displaystyle\tau^{(2s)}(1)$$
$$\displaystyle=$$
$$\displaystyle{\frac{1}{i\sin\zeta}}\,{\frac{1}{4\zeta^{2}}}\int^{\infty}_{-%
\infty}d\mu_{1}\int^{\infty}_{-\infty}d\mu_{2}\frac{\sinh(\mu_{1}+\eta/2)\sinh%
(\mu_{2}-\eta/2)}{\cosh\left(\pi\mu_{1}/\zeta\right)\cosh\left(\pi\mu_{2}/%
\zeta\right)}$$
(5.26)
$$\displaystyle\quad\times\left({\frac{1}{\sinh(\mu_{2}-\mu_{1}-i\epsilon)}}-{%
\frac{1}{\sinh(\mu_{1}-\mu_{2}+2\eta)}}\right)\,.$$
In the second line of (5.26) the first term
corresponds to the first term of (LABEL:eq:EFPs=1),
while the second term to
the second term of (LABEL:eq:EFPs=1) with
$\mu_{1}$ and $\mu_{2}$ being exchanged.
6 Spin-$s$ XXZ correlation functions near AF point
6.1 Finite-size correlation functions of
the integrable spin-$s$ XXZ spin chain
We now calculate correlation functions other than EFP
for the spin-$s$ XXZ spin chain by the method of §3.3,
making use of the formulas of Hermitian
elementary matrices such as (3.18).
We define the correlation function of the spin-$2s$ XXZ spin chain
for a given product of $(2s+1)\times(2s+1)$ elementary matrices
such as $\widetilde{E}_{1}^{i_{1},\,j_{1}\,(2s+)}\cdots\widetilde{E}_{m}^{i_{m},\,j_{m}%
\,(2s+)}$ as follows.
$$F^{(2s\,+)}(\{i_{k},j_{k}\})=\langle\psi_{g}^{(2s\,+)}|\prod_{k=1}^{m}%
\widetilde{E}_{k}^{i_{k},\,j_{k}\,(2s\,+)}|\psi_{g}^{(2s\,+)}\rangle/\langle%
\psi_{g}^{(2s\,+)}|\psi_{g}^{(2s\,+)}\rangle\,.$$
(6.1)
By the method of expressing
spin-$s$ local operators in terms of spin-1/2 global operators
[15],
we express the $m$th product of $(2s+1)\times(2s+1)$ elementary matrices
in terms of a $2sm$th product of $2\times 2$ elementary matrices
with entries $\{\epsilon_{j},\epsilon_{j}^{{}^{\prime}}\}$ as follows.
$$\prod_{k=1}^{m}\widetilde{E}_{k}^{i_{k},\,j_{k}\,(2s\,+)}=C(\{i_{k},j_{k}\};\{%
\epsilon_{j}^{{}^{\prime}},\,\epsilon_{j}\})\,\widetilde{P}_{12\ldots L}^{(2s)%
}\,\cdot\,\prod_{k=1}^{2sm}e_{k}^{\epsilon_{k}^{{}^{\prime}},\,\epsilon_{k}}\,%
\cdot\,\widetilde{P}_{12\ldots L}^{(2s)}.$$
(6.2)
We evaluate the spin-$2s$ XXZ correlation function
$F^{(2s\,+)}(\{i_{k},j_{k}\})$ by
$$F^{(2s\,+)}(\{i_{k},j_{k}\})=C(\{i_{k},j_{k}\};\{\epsilon_{j}^{{}^{\prime}},\,%
\epsilon_{j}\})\,\langle\psi_{g}^{(2s+)}|\,\widetilde{P}_{12\ldots L}^{(2s)}\,%
\cdot\,\prod_{j=1}^{2sm}e_{j}^{\epsilon_{j}^{{}^{\prime}},\,\epsilon_{j}}\,%
\cdot\,\widetilde{P}_{12\ldots L}^{(2s)}|\psi_{g}^{(2s+)}\rangle/\langle\psi_{%
g}^{(2s+)}|\psi_{g}^{(2s+)}\rangle\,.$$
(6.3)
We denote the right-hand side of (6.3) by
$F^{(2s\,+)}(\{\epsilon_{j},\,\epsilon_{j}^{{}^{\prime}}\})$.
Let us introduce some symbols.
We denote by ${\mbox{\boldmath$\alpha$}}^{+}$
the set of suffices $j$ such that $\epsilon_{j}=0$,
and by ${\mbox{\boldmath$\alpha$}}^{-}$
the set of suffices $j$ such that $\epsilon_{j}^{{}^{\prime}}=1$:
$${\mbox{\boldmath$\alpha$}}^{+}=\{j;\,\epsilon_{j}=0\}\,,\quad{\mbox{\boldmath$%
\alpha$}}^{-}=\{j;\,\epsilon_{j}^{{}^{\prime}}=1\}\,.$$
(6.4)
We denote by $\alpha_{+}$ and $\alpha_{-}$ the number
of elements of the set ${\mbox{\boldmath$\alpha$}}^{+}$
and ${\mbox{\boldmath$\alpha$}}^{-}$, respectively. Due to charge conservation,
we have
$$\alpha_{+}+\alpha_{-}=2sm.$$
(6.5)
We denote by $j_{\rm min}$ and $j_{\rm max}$ the smallest element and
the largest element of ${\mbox{\boldmath$\alpha$}}^{-}$, respectively.
We also denote by $j^{{}^{\prime}}_{\rm min}$ and $j^{{}^{\prime}}_{\rm max}$
the smallest element and the largest element
of ${\mbox{\boldmath$\alpha$}}^{+}$, respectively.
Let $c_{j}$ ($j\in{\mbox{\boldmath$\alpha$}}^{-}$) and
$c_{j}^{{}^{\prime}}$ ($j\in{\mbox{\boldmath$\alpha$}}^{+}$)
be integers such that
$1\leq c_{j}\leq M$ for $j\in{\mbox{\boldmath$\alpha$}}^{-}$
and $1\leq c_{j}^{{}^{\prime}}\leq M+j$ for $j\in{\mbox{\boldmath$\alpha$}}^{+}$.
We define sequence $(b_{\ell})_{2sm}$ by
$$(b_{1},b_{2},\ldots,b_{2sm})=(c_{j_{\rm max}^{{}^{\prime}}}^{{}^{\prime}},%
\ldots,c^{{}^{\prime}}_{j_{\rm min}^{{}^{\prime}}},c_{j_{\rm min}},\ldots,c_{j%
_{\rm max}})\,.$$
(6.6)
Here sequence
$(c_{j_{\rm max}^{{}^{\prime}}}^{{}^{\prime}},\ldots,c^{{}^{\prime}}_{j_{\rm min%
}^{{}^{\prime}}},c_{j_{\rm min}},\ldots,c_{j_{\rm max}})$ is given by the
composite of
sequence of $c^{{}^{\prime}}_{j}$’s in decreasing order with respect to suffix $j$,
and sequence of $c_{j}$’s in increasing order with respect to suffix $j$.
Let us introduce the following symbols:
$$\prod_{j\in{\mbox{\boldmath$\alpha$}}^{-}}\left(\sum_{c_{j}=1}^{M}\right)\prod%
_{j\in{\mbox{\boldmath$\alpha$}}^{+}}\left(\sum_{c_{j}^{{}^{\prime}}=1}^{M+j}%
\right)=\sum_{c_{j_{\rm min}}=1}^{M}\cdots\sum_{c_{j_{\rm max}}=1}^{M}\sum_{c^%
{{}^{\prime}}_{j^{{}^{\prime}}_{\rm min}}=1}^{M+j^{{}^{\prime}}_{\rm min}}%
\cdots\sum_{c^{{}^{\prime}}_{j^{{}^{\prime}}_{\rm max}}=1}^{M+j^{{}^{\prime}}_%
{\rm max}}\,.$$
(6.7)
Extending the derivation of the spin-$s$ EFP
we can rigorously derive the following expression of the spin-$s$ XXZ
correlation functions in the massless regime with $0\leq\zeta<\pi/2s$.
$$\displaystyle F^{(2s\,+)}(\{\epsilon_{j},\epsilon_{j}^{{}^{\prime}}\})=\prod_{%
j\in{\mbox{\boldmath$\alpha$}}^{-}}\left(\sum_{c_{j}=1}^{M}\right)\prod_{j\in{%
\mbox{\boldmath$\alpha$}}^{+}}\left(\sum_{c_{j}^{{}^{\prime}}=1}^{M+j}\right)%
\quad{\rm det}M^{(2sm)}((b_{\ell})_{2sm})$$
$$\displaystyle\,\,\times(-1)^{\alpha_{+}}{\frac{\prod_{j\in{\mbox{\boldmath$%
\alpha$}}^{-}}\left(\prod_{k=1}^{j-1}\varphi(\lambda_{c_{j}}-w_{k}^{(2s)}+\eta%
)\prod_{k=j+1}^{2sm}\varphi(\lambda_{c_{j}}-w_{k}^{(2s)})\right)}{\prod_{1\leq
k%
<\ell\leq 2sm}\varphi(\lambda_{b_{\ell}}-\lambda_{b_{k}}+\eta)}}$$
$$\displaystyle\,\,\times{\frac{\prod_{j\in{\mbox{\boldmath$\alpha$}}^{+}}\left(%
\prod_{k=1}^{j-1}\varphi(\lambda_{c_{j}^{{}^{\prime}}}-w_{k}^{(2s)}-\eta)\prod%
_{k=j+1}^{2sm}\varphi(\lambda_{c_{j}^{{}^{\prime}}}-w_{k}^{(2s)})\right)}{%
\prod_{1\leq k<\ell\leq 2sm}\varphi(w_{k}^{(2s)}-w_{\ell}^{(2s)})}}+O(1/N_{s})\,.$$
(6.8)
Here $\varphi(\lambda)=\sinh\lambda$.
We have defined
the $2sm\times 2sm$ matrix $M^{(2sm)}((b_{j})_{2sm})$ as follows.
For $\ell,k=1,2,\ldots,2sm$, the matrix element of $(\ell,k)$
is given by
$$\left(M^{(2sm)}((b_{j})_{2sm})\right)_{\ell,\,k}=\left\{\begin{array}[]{cc}-%
\delta_{b_{\ell}-M,\,k}&{\rm for}\,\,b_{\ell}>M\\
\delta_{\beta(b_{\ell}),\,\beta(k)}\,\cdot\,\rho(\lambda_{b_{\ell}}-w_{k}^{(2s%
)}+\eta/2)/{N_{s}\rho_{\rm tot}(\mu_{a(b_{\ell})})}&{\rm for}\,\,b_{\ell}\leq M%
\end{array}\right.$$
(6.9)
where $\mu_{j}$ denote the centers of $\lambda_{j}$ as follows.
$$\lambda_{j}=\mu_{j}-(\beta(j)-\frac{1}{2})\eta\qquad j=1,2,\ldots,2sm.$$
(6.10)
We recall that $a(j)$ and $\beta(j)$
have been defined in terms of the Gauss’ symbol $[\cdot]$ by
$a(j)=[(j-1)/2s]+1$ and
$\beta(j)=j-2s[(j-1)/2s]$, respectively.
We remark that under the limit of sending $\epsilon$ to zero,
the sum over variable $c_{j}$ is restricted up to $M$.
6.2 Multiple-integral representations of the spin-$s$ XXZ correlation
function for an arbitrary product of elementary matrices
Let us formulate matrix $S$ for the correlation function
of an arbitrary product of elementary matrices.
We define the $(j,k)$ element of matrix
$S=S\left((\lambda_{j})_{2sm};(w_{j}^{(2s)})_{2sm}\right)$ by
$$S_{j,k}=\rho(\lambda_{j}-w_{k}^{(2s)}+\eta/2)\,\delta(\alpha(\lambda_{j}),%
\beta(k))\,,\quad{\rm for}\quad j,k=1,2,\ldots,2sm\,.$$
(6.11)
Here $\delta(\alpha,\beta)$ denotes the Kronecker delta. We define
$\alpha(\lambda_{j})$ by $\alpha(\lambda_{j})=\gamma$
if $\lambda_{j}=\mu_{j}-(\gamma-1/2)\eta$ or $\lambda_{j}=w_{k}^{(2s)}$
where $\beta(k)=\gamma$ ($1\leq\gamma\leq 2s$).
We remark that $\mu_{j}$ correspond to the centers
of complete $2s$-strings $\lambda_{j}$.
We also remark that
the above definition of matrix $S$ generalizes that of
(5.3) since
$\alpha(\lambda_{j})$ is now also defined
also for $\lambda_{j}=w_{k}^{(2s)}$.
Let $\Gamma_{j}$ be a small contour rotating counterclockwise
around $\lambda=w_{j}^{(2s)}$.
Since the ${\rm det}S$ has simple poles at $\lambda=w_{j}^{(2s)}$
with residue $1/2\pi i$,
we therefore have
$$\int_{-\infty+i\epsilon}^{\infty+i\epsilon}{\rm det}S((\lambda_{k})_{2sm})\,d%
\lambda_{1}=\int_{-\infty-i\epsilon}^{\infty-i\epsilon}{\rm det}S((\lambda_{k}%
)_{2sm})\,d\lambda_{1}-\oint_{\Gamma_{1}}{\rm det}S((\lambda_{k})_{2sm})\,d%
\lambda_{1}\,.$$
(6.12)
For sets ${\mbox{\boldmath$\alpha$}}^{-}$ and ${\mbox{\boldmath$\alpha$}}^{+}$
we define ${\tilde{\lambda}}_{j}$ for $j\in{\mbox{\boldmath$\alpha$}}^{-}$
and ${\tilde{\lambda}}^{{}^{\prime}}_{j}$ for $j\in{\mbox{\boldmath$\alpha$}}^{+}$,
respectively,
by the following sequence:
$$({\tilde{\lambda}}^{{}^{\prime}}_{j^{{}^{\prime}}_{max}},\ldots,{\tilde{%
\lambda}}^{{}^{\prime}}_{j^{{}^{\prime}}_{min}},{\tilde{\lambda}}_{j_{min}},{%
\tilde{\lambda}}_{j_{max}})=(\lambda_{1},\ldots,\lambda_{2sm})\,.$$
(6.13)
Thus, from the expression of
the correlation function in terms of a finite sum (6.8)
we obtain the multiple-integral representation as follows.
$$\displaystyle F^{(2s+)}(\{\epsilon_{j},\epsilon_{j}^{{}^{\prime}}\})$$
(6.14)
$$\displaystyle=$$
$$\displaystyle\left(\int_{-\infty+i\epsilon}^{\infty+i\epsilon}+\cdots+\int_{-%
\infty-i(2s-1)\zeta+i\epsilon}^{\infty-i(2s-1)\zeta+i\epsilon}\right)d\lambda_%
{1}\cdots\left(\int_{-\infty+i\epsilon}^{\infty+i\epsilon}+\cdots+\int_{-%
\infty-i(2s-1)\zeta+i\epsilon}^{\infty-i(2s-1)\zeta+i\epsilon}\right)d\lambda_%
{\alpha_{+}}$$
$$\displaystyle\left(\int_{-\infty-i\epsilon}^{\infty-i\epsilon}+\cdots+\int_{-%
\infty-i(2s-1)\zeta-i\epsilon}^{\infty-i(2s-1)\zeta-i\epsilon}\right)d\lambda_%
{\alpha_{+}+1}\cdots\left(\int_{-\infty-i\epsilon}^{\infty-i\epsilon}+\cdots+%
\int_{-\infty-i(2s-1)\zeta-i\epsilon}^{\infty-i(2s-1)\zeta-i\epsilon}\right)d%
\lambda_{m}$$
$$\displaystyle\quad\times Q(\{\epsilon_{j},\epsilon_{j}^{{}^{\prime}}\};\lambda%
_{1},\ldots,\lambda_{2sm})\,{\rm det}S(\lambda_{1},\ldots,\lambda_{2sm})\,.$$
Here we have defined
$Q(\{\epsilon_{j},\epsilon_{j}^{{}^{\prime}}\};\lambda_{1},\ldots,\lambda_{2sm})$
in terms of small numbers
$\epsilon_{\ell,k}$ of (5.4) by
$$\displaystyle Q(\{\epsilon_{j},\epsilon_{j}^{{}^{\prime}}\};\lambda_{1},\ldots%
,\lambda_{2sm}))$$
$$\displaystyle=$$
$$\displaystyle(-1)^{\alpha_{+}}{\frac{\prod_{j\in{\mbox{\boldmath$\alpha$}}^{-}%
}\left(\prod_{k=1}^{j-1}\varphi({\tilde{\lambda}}_{j}-w_{k}^{(2s)}+\eta)\prod_%
{k=j+1}^{2sm}\varphi({\tilde{\lambda}}_{j}-w_{k}^{(2s)})\right)}{\prod_{1\leq k%
<\ell\leq 2sm}\varphi(\lambda_{\ell}-\lambda_{k}+\eta+\epsilon_{\ell,k})}}$$
(6.15)
$$\displaystyle\times$$
$$\displaystyle{\frac{\prod_{j\in{\mbox{\boldmath$\alpha$}}^{+}}\left(\prod_{k=1%
}^{j-1}\varphi({\tilde{\lambda}}^{{}^{\prime}}_{j}-w_{k}^{(2s)}-\eta)\prod_{k=%
j+1}^{2sm}\varphi({\tilde{\lambda}}^{{}^{\prime}}_{j}-w_{k}^{(2s)})\right)}{%
\prod_{1\leq k<\ell\leq 2sm}\varphi(w_{k}^{(2s)}-w_{\ell}^{(2s)})}}\,.$$
Thus, correlation functions (6.1) are expressed
in the form of a single term of multiple integrals.
Similarly as the symmetric spin-$s$ EFP,
we can show the symmetric expression of the multiple-integral
representations of the spin-$s$ correlation function
as follows.
$$\begin{split}&\displaystyle F^{(2s\,+)}(\{\epsilon_{j},\,\epsilon_{j}^{{}^{%
\prime}}\})=\frac{1}{\prod_{1\leq\alpha<\beta\leq 2s}\sinh^{m}(\beta-\alpha)%
\eta}\,\prod_{1\leq k<l\leq m}\frac{\sinh^{2s}(\pi(\xi_{k}-\xi_{l})/\zeta)}{%
\prod^{2s}_{j=1}\prod^{2s}_{r=1}\sinh(\xi_{k}-\xi_{l}+(r-j)\eta)}\\
&\displaystyle\sum_{\sigma\in{\cal S}_{2sm}/({\cal S}_{m})^{2s}}\prod^{\alpha_%
{+}}_{j=1}\left(\int^{\infty+i\epsilon}_{-\infty+i\epsilon}+\cdots+\int^{%
\infty-i(2s-1)\zeta+i\epsilon}_{-\infty-i(2s-1)\zeta+i\epsilon}\right)d\mu_{%
\sigma j}\,\prod^{2sm}_{j=\alpha_{+}+1}\left(\int^{\infty-i\epsilon}_{-\infty-%
i\epsilon}+\cdots+\int^{\infty-i(2s-1)\zeta-i\epsilon}_{-\infty-i(2s-1)\zeta-i%
\epsilon}\right)d\mu_{\sigma j}\\
&\displaystyle\times({\rm sgn}\,\sigma)\,Q(\{\epsilon_{j},\epsilon_{j}^{{}^{%
\prime}}\};\lambda_{\sigma 1},\ldots,\lambda_{\sigma(2sm)}))\,\left(\prod^{2sm%
}_{j=1}{\frac{\prod^{m}_{b=1}\prod^{2s-1}_{\beta=1}\sinh(\lambda_{j}-\xi_{b}+%
\beta\eta)}{\prod_{b=1}^{m}\cosh(\pi(\mu_{j}-\xi_{b})/\zeta)}}\right)\\
&\displaystyle\times\,{\frac{i^{2sm^{2}}}{(2i\zeta)^{2sm}}}\,\prod^{2s}_{%
\gamma=1}\prod_{1\leq b<a\leq m}\sinh(\pi(\mu_{2s(a-1)+\gamma}-\mu_{2s(b-1)+%
\gamma})/\zeta)\,.\end{split}$$
It is straightforward to take the homogeneous limit:
$\xi_{k}\rightarrow 0$.
Here we recall that (${\rm sgn}\,\sigma$)
denotes the sign of permutation
$\sigma\in{\cal S}_{2sm}/({\cal S}_{m})^{2s}$.
6.3 An example of the spin-1 correlation function
Applying formula (3.18) to the spin-1 case with $m=n=1$,
we have
$$\displaystyle\widetilde{E}_{1}^{1,\,1\,(2+)}$$
$$\displaystyle=$$
$$\displaystyle 2\,\widetilde{P}^{(2s)}_{1\cdots L}\,D^{(1+)}(w_{1})A^{(1+)}(w_{%
2})\prod_{\alpha=3}^{2N_{s}}(A^{(1+)}+D^{(1+)})(w_{\alpha})\,\,\widetilde{P}^{%
(2s)}_{1\cdots L}\,.$$
(6.16)
Therefore, we evaluate it sending
$\epsilon$ to zero, as follows.
$$\displaystyle\langle{\psi}_{g}^{(2\,+)}|\widetilde{E}_{1}^{1,\,1\,(2+)}|{\psi}%
_{g}^{(2\,+)}\rangle/\langle{\psi}_{g}^{(2\,+)}|{\psi}_{g}^{(2\,+)}\rangle$$
(6.17)
$$\displaystyle=$$
$$\displaystyle 2\,\lim_{\epsilon\rightarrow 0}\langle\psi_{g}^{(2\,+;\,\epsilon%
)}|D^{(2+;\epsilon)}(w_{1}^{(2;\epsilon)})A^{(2+;\epsilon)}(w_{2}^{(2;\epsilon%
)})\prod_{\alpha=3}^{2N_{s}}(A^{(2+;\epsilon)}+D^{(2+;\epsilon)})(w_{\alpha}^{%
(2;\epsilon)})|\psi_{g}^{(2+;\epsilon)}\rangle/\langle\psi_{g}^{(2+;\epsilon)}%
|\psi_{g}^{(2+;\epsilon)}\rangle$$
$$\displaystyle=$$
$$\displaystyle 2\,\left(\int_{-\infty+i\epsilon}^{\infty+i\epsilon}+\int_{-%
\infty-i\zeta+i\epsilon}^{\infty-i\zeta+i\epsilon}\right)d\lambda_{1}\,\left(%
\int_{-\infty-i\epsilon}^{\infty-i\epsilon}+\int_{-\infty-i\zeta-i\epsilon}^{%
\infty-i\zeta-i\epsilon}\right)d\lambda_{2}\,Q(\lambda_{1},\lambda_{2})\,{\rm
det%
}S(\lambda_{1},\lambda_{2})$$
where $Q(\lambda_{1},\lambda_{2})$ is given by
$$Q(\lambda_{1},\lambda_{2})=(-1){\frac{\varphi(\lambda_{2}-\xi_{1}+\eta)\varphi%
(\lambda_{1}-\xi_{1}-\eta)}{\varphi(\lambda_{2}-\lambda_{1}+\eta+\epsilon_{2,1%
})\varphi(\eta)}}$$
(6.18)
and matrix $S(\lambda_{1},\lambda_{2})$ is given by
$$S(\lambda_{1},\lambda_{2})=\left(\begin{array}[]{cc}\rho(\lambda_{1}-w_{1}^{(2%
)}+\eta/2)\delta(\lambda_{1},1)&\rho(\lambda_{1}-w_{2}^{(2)}+\eta/2)\delta(%
\lambda_{1},2)\\
\rho(\lambda_{2}-w_{1}^{(2)}+\eta/2)\delta(\lambda_{2},1)&\rho(\lambda_{2}-w_{%
2}^{(2)}+\eta/2)\delta(\lambda_{2},2)\end{array}\right)\,.$$
(6.19)
We thus note note that
the correlation function is now expressed
in terms of a single product of the multiple-integral
representation.
Let us now evaluate the double integral (6.17), explicitly.
The integral over $\lambda_{1}$ is decomposed into the following:
$$\displaystyle\left(\int_{-\infty+i\epsilon}^{\infty+i\epsilon}+\int_{-\infty-i%
\zeta+i\epsilon}^{\infty-i\zeta+i\epsilon}\right)d\lambda_{1}\,Q(\lambda_{1},%
\lambda_{2})\,{\rm det}S(\lambda_{1},\lambda_{2})$$
(6.20)
$$\displaystyle=$$
$$\displaystyle\left(\int_{-\infty-i\zeta/2}^{\infty-i\zeta/2}+\int_{-\infty-i3%
\zeta/2}^{\infty-i3\zeta/2}\right)d\lambda_{1}\,Q(\lambda_{1},\lambda_{2})\,{%
\rm det}S(\lambda_{1},\lambda_{2})$$
$$\displaystyle-\oint_{\Gamma_{1}}d\lambda_{1}\,Q(\lambda_{1},\lambda_{2})\,{\rm
det%
}S(\lambda_{1},\lambda_{2})-\oint_{\Gamma_{2}}d\lambda_{1}\,Q(\lambda_{1},%
\lambda_{2})\,{\rm det}S(\lambda_{1},\lambda_{2})\,.$$
Thus, the integral (6.17) is calculated as
$$\displaystyle\langle\widetilde{\psi}_{g}^{(2\,+)}|\widetilde{E}_{1}^{11\,(2\,+%
)}|\widetilde{\psi}_{g}^{(2\,+)}\rangle/\left(2\langle\widetilde{\psi}_{g}^{(2%
\,+)}|\widetilde{\psi}_{g}^{(2\,+)}\rangle\right)$$
(6.21)
$$\displaystyle=$$
$$\displaystyle-2\pi i\int_{-\infty}^{\infty}{\frac{\sinh(x-\eta/2)\sinh(x-3\eta%
/2)}{\sinh\eta}}\rho^{2}(x)\,dx+2\cosh\eta\,\int_{-\infty}^{\infty}{\frac{%
\sinh(x-\eta/2)}{\sinh(x+\eta/2)}}\rho(x)\,dx$$
$$\displaystyle-\int_{-\infty}^{\infty}\rho(x)dx+(-1)2\cosh\eta\,\int_{-\infty}^%
{\infty}{\frac{\sinh(x-\eta/2)}{\sinh(x+\eta/2)}}\rho(x)\,dx$$
$$\displaystyle=$$
$$\displaystyle{\frac{\cos\zeta(\sin\zeta-\zeta\cos\zeta)}{2\zeta\sin^{2}\zeta}}\,.$$
We have thus confirmed (5.24) directly evaluating
the integrals.
7 Concluding remarks
In the paper we have explicitly shown
the multiple-integral representation of the emptiness formation
probability for the integrable spin-$s$ XXZ spin chain
in a region of the massless regime
of $\eta=i\zeta$ with $0\leq\zeta<\pi/2s$.
We have also calculated the emptiness formation
probability for the homogeneous case
of the integrable spin-$s$ XXZ spin chain.
In the XXX limit where we send $\zeta$ to zero,
the expression of EFP for the spin-$s$ XXZ case reduces to
that of the spin-$s$ XXX case.
Moreover, we have presented a formula
for the multiple-integral representation
of the spin-$s$ XXZ correlation function
of an arbitrary product of elementary
matrices in the massless regime
where $\eta=i\zeta$ with $0\leq\zeta<\pi/2s$.
We have also presented the symmetric expression
of the multiple-integral representations of the
spin-$s$ XXZ correlation functions.
Finally, we have introduced conjugate vectors
$\widetilde{||2s,n\rangle}$ in order to formulate
Hermitian elementary matrices $\widetilde{E}^{m,\,n\,(2s\,+)}$
and Hermitian projection operators $\widetilde{P}^{(\ell)}$
in the massless regime.
We have also defined the massless fusion $R$-matrices
$\widetilde{R}^{(\ell,\,2s\,w)}$ for $w=+$ and $p$.
Acknowledgment
One of the authors (T.D.) would like to thank B.M. McCoy
and other participants of the workshop
on the integrable chiral Potts model, organized by M.T. Batchelor,
Kioloa, NSW, Australia, Dec. 7-11, 2008, for many useful comments.
He is grateful to K. Motegi, J. Sato, and Y. Takeyama
for bringing him useful references.
Furthermore, the authors would like to thank
S. Miyashita for encouragement and keen interest in this work.
This work is partially supported by
Grant-in-Aid for Scientific Research (C) No. 20540365.
Appendix A Affine quantum group
with homogeneous grading
The affine quantum algebra $U_{q}(\widehat{sl_{2}})$
is an associative algebra over ${\bf C}$ generated by
$X_{i}^{\pm},K_{i}^{\pm}$ for $i=0,1$ with the following relations:
$$\displaystyle K_{i}K_{i}^{-1}$$
$$\displaystyle=$$
$$\displaystyle K^{-1}_{i}K_{i}=1\,,\quad K_{i}X_{i}^{\pm}K_{i}^{-1}=q^{\pm 2}X_%
{i}^{\pm}\,,\quad K_{i}X_{j}^{\pm}K_{i}^{-1}=q^{\mp 2}X_{j}^{\pm}\quad(i\neq j%
)\,,$$
$$\displaystyle{[}X_{i}^{+},X_{j}^{-}{]}$$
$$\displaystyle=$$
$$\displaystyle\delta_{i,j}\,{\frac{K_{i}-K_{i}^{-1}}{q-q^{-1}}}\,,$$
$$\displaystyle(X_{i}^{\pm})^{3}X_{j}^{\pm}$$
$$\displaystyle-$$
$$\displaystyle[3]_{q}\,(X_{i}^{\pm})^{2}X_{j}^{\pm}X_{i}^{\pm}+[3]_{q}\,X_{i}^{%
\pm}X_{j}^{\pm}(X_{i}^{\pm})^{2}-X_{j}^{\pm}(X_{i}^{\pm})^{3}=0\quad(i\neq j)\,.$$
(A.1)
Here the symbol $[n]_{q}$ denotes the $q$-integer of an integer $n$:
$$[n]_{q}={\frac{q^{n}-q^{-n}}{q-q^{-1}}}\,.$$
(A.2)
The algebra $U_{q}(\widehat{sl_{2}})$ is also a Hopf algebra over ${\bf C}$
with comultiplication
$$\displaystyle\Delta(X_{i}^{+})$$
$$\displaystyle=$$
$$\displaystyle X_{i}^{+}\otimes 1+K_{i}\otimes X_{i}^{+}\,,\quad\Delta(X_{i}^{-%
})=X_{i}^{-}\otimes K_{i}^{-1}+1\otimes X_{i}^{-}\,,$$
$$\displaystyle\Delta(K_{i})$$
$$\displaystyle=$$
$$\displaystyle K_{i}\otimes K_{i}\,,$$
(A.3)
and antipode:
$S(K_{i})=K_{i}^{-1}\,,S(X_{i})=-K_{i}^{-1}X_{i}^{+}\,,S(X_{i}^{-})=-X_{i}^{-}K%
_{i}$, and counit:
$\varepsilon(X_{i}^{\pm})=0$ and $\varepsilon(K_{i})=1$ for $i=0,1$ .
The algebra $U_{q}(sl_{2})$ is given by the Hopf subalgebra of
$U_{q}(\widehat{sl_{2}})$ generated by
$X_{i}^{\pm}$, $K_{i}$ with either $i=0$ or $i=1$. Hereafter
we denote by $X^{\pm}$ and $K$ the generators of $U_{q}(sl_{2})$.
For a given complex number $\lambda$ we define
a homomorphism of algebras $\varphi_{\lambda}$:
$U_{q}(\widehat{sl_{2}})\rightarrow U_{q}({sl_{2}})$.
$$\varphi_{\lambda}(X_{0}^{\pm})=e^{\pm 2\lambda}\,X^{\mp}\,,\quad\varphi_{%
\lambda}(X_{1}^{\pm})=X^{\pm}\,,\quad\varphi_{\lambda}(K_{0})=K^{-1}\,,\quad%
\varphi_{\lambda}(K_{1})=K\,\,.$$
(A.4)
Map (A.4) is associated
with homogeneous grading [8].
For a representation $(\pi,V^{(\ell)})$ of $U_{q}(sl_{2})$
we have a representation of $U_{q}(\widehat{sl_{2}})$ by
$\pi(\varphi_{\lambda}(a))$ for $a\in U_{q}(\widehat{sl_{2}})$.
We call it the spin-$\ell/2$ evaluation representation
with evaluation parameter $\lambda$,
and denote it
by $(\pi_{\lambda},V^{(\ell)}(\lambda))$ or $V^{(\ell)}(\lambda)$.
We define opposite coproduct
$\Delta^{op}$ by
$$\Delta^{op}(a)=\tau\circ\Delta(a)\quad\mbox{\rm for}\,\,a\in U_{q}(sl_{2})\,,$$
(A.5)
where $\tau$ denotes the permutation
operator: $\tau(a\otimes b)=b\otimes a$ for $a,b\in U_{q}(sl_{2})$.
Appendix B Fusion projection operators being idempotent
We give the derivation [23] of
$(P_{12\cdots\ell}^{(\ell)})^{2}=P_{12\cdots\ell}^{(\ell)}$
making use of the Yang-Baxter equations.
Lemma B.1.
Operators $P_{12\cdots\ell}^{(\ell)}$
defined by (2.12) have
the following two expressions:
$$P_{1\,2\cdots\ell-1}^{(\ell-1)}{\check{R}}^{+}_{\ell-1,\,\ell}((\ell-1)\eta)P_%
{1\,2\cdots\ell-1}^{(\ell-1)}=P_{2\,3\cdots\ell}^{(\ell-1)}{\check{R}}_{1,\,2}%
^{+}((\ell-1)\eta)P_{2\,3\cdots\ell}^{(\ell-1)}\,.$$
(B.1)
Proof.
Applying notation (2.2) to permutation operator $\Pi_{1,2}$
we define permutation operators $\Pi_{j,\,k}$ for integers $j$ and $k$
satisfying $0\leq j<k\leq L$.
The form of the left-hand side of (B.1)
is expressed in terms of $R$-matrices as follows
(see also eq. (3.7) of [15]).
$$P_{1\cdots\ell}^{(\ell)}=\prod_{j=1}^{[\ell/2]}\Pi_{j,\,\ell-j+1}\,\cdot\,R^{+%
}_{\ell-1\,\ell}\cdots R^{+}_{2,\,3\cdots\ell}R^{+}_{1,\,2\cdots\ell}\,.$$
(B.2)
Making use of the Yang-Baxter equations (2.4)
we reformulate (B.2) as follows.
$$P_{1\cdots\ell}^{(\ell)}=\prod_{j=1}^{[\ell/2]}\Pi_{j,\,\ell-j+1}\,\cdot\,R^{+%
}_{1\,2}R^{+}_{12,\,3}\cdots R^{+}_{1,\,2\cdots\ell}\,$$
(B.3)
which gives the expression of the right-hand side of (B.1).
∎
From (B.1) we show that
$P_{j+1\,j+2\cdots\,j+\ell}^{(\ell)}$ is expressed as follows.
$$P_{j+1\,j+2\cdots j+\ell-1}^{(\ell-1)}{\check{R}}^{+}_{j+\ell-1,\,j+\ell}((%
\ell-1)\eta)P_{j+1\,j+2\cdots j+\ell-1}^{(\ell-1)}=P_{j+2\,j+3\cdots j+\ell}^{%
(\ell-1)}{\check{R}}^{+}_{j+1,\,j+2}((\ell-1)\eta)P_{j+2\,j+3\cdots j+\ell}^{(%
\ell-1)}\,.$$
(B.4)
Lemma B.2.
Operator $P_{12\cdots\ell}^{(\ell)}$ projects
operator ${\check{R}}_{\ell-1,\,\ell}(u)$ to 1 as follows.
$$P_{12\cdots\ell}^{(\ell)}{\check{R}}^{+}_{\ell-1,\,\ell}(u)=P_{12\cdots\ell}^{%
(\ell)}\,.$$
(B.5)
Proof.
Due to the spectral decomposition of the $R$-matrix, we have
$P^{(2)}_{\ell-1,\ell}{\check{R}}^{+}_{\ell-1,\,\ell}(u)=P^{(2)}_{\ell-1,\ell}$.
Applying (B.1) and (B.4),
we thus obtain (B.5).
∎
Proposition B.3.
Operators $P_{12\cdots\ell}^{(\ell)}$
defined by (2.12)
are idempotent:
$\left(P_{12\cdots\ell}^{(\ell)}\right)^{2}=P_{12\cdots\ell}^{(\ell)}$ .
Proof.
We show it from (B.5) by induction on
$\ell$. Suppose that it is idempotent
for $\ell$. We have
$$\displaystyle\left(P_{12\cdots\ell+1}^{(\ell+1)}\right)^{2}$$
$$\displaystyle=$$
$$\displaystyle P_{12\cdots\ell}^{(\ell)}\,R^{+}_{\ell\,\ell+1}(\ell\eta)\,\cdot%
(P_{12\cdots\ell}^{(\ell)})^{2}\,\cdot R^{+}_{\ell\,\ell+1}(\ell\eta)\,P_{12%
\cdots\ell}^{(\ell)}$$
(B.6)
$$\displaystyle=$$
$$\displaystyle P_{12\cdots\ell}^{(\ell)}\,R^{+}_{\ell\,\ell+1}(\ell\eta)\,\cdot%
\,P_{12\cdots\ell}^{(\ell)}\,R^{+}_{\ell\,\ell+1}(\ell\eta)\,\cdot\,P_{12%
\cdots\ell}^{(\ell)}$$
$$\displaystyle=$$
$$\displaystyle P_{12\cdots\ell}^{(\ell)}\,R^{+}_{\ell\,\ell+1}(\ell\eta)\,P_{12%
\cdots\ell}^{(\ell)}\cdot\,P_{12\cdots\ell}^{(\ell)}=P_{12\cdots\ell}^{(\ell)}%
\,R^{+}_{\ell\,\ell+1}(\ell\eta)\,P_{12\cdots\ell}^{(\ell)}\,.$$
∎
Appendix C Basis vectors of spin-$\ell/2$ representation of $U_{q}(sl_{2})$
In terms of
the $q$-integer $[n]_{q}$ defined in (A.2),
we define the $q$-factorial $[n]_{q}!$ for integers $n$ by
$$[n]_{q}!=[n]_{q}\,[n-1]_{q}\,\cdots\,[1]_{q}\,.$$
(C.1)
For integers $m$ and $n$ satisfying $m\geq n\geq 0$
we define the $q$-binomial coefficients as follows
$$\left[\begin{array}[]{c}m\\
n\end{array}\right]_{q}={\frac{[m]_{q}!}{[m-n]_{q}!\,[n]_{q}!}}\,.$$
(C.2)
We now define the basis vectors of the $(\ell+1)$-dimensional
irreducible representation of $U_{q}(sl_{2})$,
$||\ell,n\rangle$ for $n=0,1,\ldots,\ell$ as follows.
We define $||\ell,0\rangle$ by
$$||\ell,0\rangle=|0\rangle_{1}\otimes|0\rangle_{2}\otimes\cdots\otimes|0\rangle%
_{\ell}\,.$$
(C.3)
Here $|\alpha\rangle_{j}$ for $\alpha=0,1$
denote the basis vectors of the spin-1/2 representation defined
on the $j$th position in the tensor product. We define
$||\ell,n\rangle$ for $n\geq 1$ and evaluate them as follows [15] .
$$\displaystyle||\ell,n\rangle$$
$$\displaystyle=$$
$$\displaystyle\left(\Delta^{(\ell-1)}(X^{-})\right)^{n}||\ell,0\rangle\,{\frac{%
1}{[n]_{q}!}}$$
(C.4)
$$\displaystyle=$$
$$\displaystyle\sum_{1\leq i_{1}<\cdots<i_{n}\leq\ell}\sigma_{i_{1}}^{-}\cdots%
\sigma_{i_{n}}^{-}|0\rangle\,q^{i_{1}+i_{2}+\cdots+i_{n}-n\ell+n(n-1)/2}\,.$$
We define the conjugate vectors explicitly by the following:
$$\langle\ell,n||=\left[\begin{array}[]{c}\ell\\
n\end{array}\right]_{q}^{-1}\,q^{n(\ell-n)}\,\sum_{1\leq i_{1}<\cdots<i_{n}%
\leq\ell}\langle 0|\sigma_{i_{1}}^{+}\cdots\sigma_{i_{n}}^{+}\,q^{i_{1}+\cdots%
+i_{n}-n\ell+n(n-1)/2}\,.$$
(C.5)
It is easy to show the normalization conditions [15]:
$\langle\ell,n||\,||\ell,n\rangle=1$.
In the massive regime where $q=\exp\eta$ with real $\eta$,
conjugate vectors $\langle\ell,n||$ are Hermitian conjugate to vectors
$||\ell,n\rangle$.
Through the recursive construction (2.12)
of $P^{(\ell)}$s,
it is easy to show the following [15]:
$$\displaystyle P^{(\ell)}_{12\cdots\ell}||\ell,n\rangle$$
$$\displaystyle=$$
$$\displaystyle||\ell,n\rangle\,,$$
$$\displaystyle\langle\ell,n||P^{(\ell)}_{12\cdots\ell}$$
$$\displaystyle=$$
$$\displaystyle\langle\ell,n||\,.$$
(C.6)
Thus, the fusion projector $P^{(\ell)}$
is consistent with the spin-$\ell/2$ representation
of $U_{q}(sl_{2})$.
In order to define Hermitian elementary matrices,
we now introduce another set of dual basis vectors.
For a given nonzero integer $\ell$ we define
$\widetilde{\langle\ell,n||}$ for $n=0,1,\ldots,n$, by
$$\widetilde{\langle\ell,n||}=\left(\begin{array}[]{c}\ell\\
n\end{array}\right)^{-1}\,\sum_{1\leq i_{1}<\cdots<i_{n}\leq\ell}\langle 0|%
\sigma_{i_{1}}^{+}\cdots\sigma_{i_{n}}^{+}\,q^{-(i_{1}+\cdots+i_{n})+n\ell-n(n%
-1)/2}\,.$$
(C.7)
They are conjugate to $||\ell,n\rangle$:
$\widetilde{\langle\ell,m||}\,||\ell,n\rangle=\delta_{m,n}$ .
In the massless regime where $|q|=1$,
matrix $||\ell,n\rangle\widetilde{\langle\ell,n||}$ is
Hermitian: $(||\ell,n\rangle\widetilde{\langle\ell,n||})^{\dagger}=||\ell,n\rangle%
\widetilde{\langle\ell,n||}$.
However, in order to define projection operators $\tilde{P}$ such that
$P\tilde{P}=P$, we define another set of vectors
$\widetilde{||\ell,n\rangle}$ in section 2.4. They are conjugate to
the dual vectors $\langle\ell,n||$.
Appendix D The massless fusion $R$-matrices of the spin-1 case
Let us evaluate the matrix elements of the massless monodromy matrix
$\widetilde{T}^{(1,\,2\,+)}_{0,\,1}(\lambda_{0};\xi_{1})$,
i.e. the massless $L$ operator of the spin-1 representation.
$$\widetilde{T}^{(1,\,2\,+)}_{0,\,1}(\lambda_{0};\xi_{1})=\widetilde{P}^{(2)}_{1%
2}R^{(1,\,1\,+)}_{0,\,1\,2}(\lambda_{0};\{w_{j}^{(2)}\}_{2})\widetilde{P}^{(2)%
}_{12}\,.$$
(D.1)
Here we have set inhomogeneous parameters $w_{1}^{(2)}=\xi_{1}$ and
$w_{2}^{(2)}=\xi_{1}-\eta$.
Let us recall $R_{0,12}^{+}=R_{0,2}^{+}R_{0,1}^{+}$.
For instance, we have $A_{12}^{+}=A_{2}^{+}A_{1}^{+}+B_{2}^{+}C_{1}^{+}$.
In terms of $b_{0j}=b(\lambda_{0}-w_{j}^{(2)})$ and
$c_{0j}=c(\lambda_{0}-w_{j}^{(2)})$ for $j=1,2$,
the (1,1) element of $\widetilde{A}_{1}^{(2+)}$ is given by
$$\langle 2,1||A^{(2+)}(\lambda_{0})\widetilde{||2,1\rangle}=(b_{01}+b_{02}+q^{-%
2}c_{01}c_{02})/2\,.$$
(D.2)
Thus, setting $u=\lambda_{0}-\xi_{1}$,
all the non-zero matrix elements of
$\widetilde{T}^{(1,\,2\,+)}(\lambda_{0})$
are given by
$$\displaystyle\langle 2,0||A^{(2+)}(\lambda_{0})\widetilde{||2,0\rangle}$$
$$\displaystyle=$$
$$\displaystyle\langle 2,2||D^{(2+)}(\lambda_{0})\widetilde{||2,2\rangle}=1\,,$$
$$\displaystyle\langle 2,1||A^{(2+)}(\lambda_{0})\widetilde{||2,1\rangle}$$
$$\displaystyle=$$
$$\displaystyle\langle 2,1||D^{(2+)}(\lambda_{0})\widetilde{||2,1\rangle}=\sinh(%
u+\eta)/\sinh(u+2\eta)\,,$$
$$\displaystyle\langle 2,2||A^{(2+)}(\lambda_{0})\widetilde{||2,2\rangle}$$
$$\displaystyle=$$
$$\displaystyle\langle 2,0||D^{(2+)}(\lambda_{0})\widetilde{||2,0\rangle}=\sinh u%
/\sinh(u+2\eta)\,,$$
$$\displaystyle\langle 2,1||B^{(2+)}(\lambda_{0})\widetilde{||2,0\rangle}$$
$$\displaystyle=$$
$$\displaystyle e^{-u}\sinh\eta/\sinh(u+2\eta)\,,$$
$$\displaystyle\langle 2,2||B^{(2+)}(\lambda_{0})\widetilde{||2,1\rangle}$$
$$\displaystyle=$$
$$\displaystyle[2]_{q}\,q^{-1}e^{-u}\sinh\eta/\sinh(u+2\eta)\,,$$
$$\displaystyle\langle 2,0||C^{(2+)}(\lambda_{0})\widetilde{||2,1\rangle}$$
$$\displaystyle=$$
$$\displaystyle[2]_{q}\,e^{u}\sinh\eta/\sinh(u+2\eta)\,,$$
$$\displaystyle\langle 2,1||C^{(2+)}(\lambda_{0})\widetilde{||2,2\rangle}$$
$$\displaystyle=$$
$$\displaystyle qe^{u}\sinh\eta/\sinh(u+2\eta)\,.$$
(D.3)
We should remark that the massive monodromy matrix
${T}^{(1,\,2\,+)}_{0,\,1}(\lambda_{0};\xi_{1})$ has
the same matrix elements as the massless monodromy matrix
$\widetilde{T}^{(1,\,2\,+)}_{0,\,1}(\lambda_{0};\xi_{1})$.
For instance, we calculate the (1,1) element of operator
$A^{(2+)}_{1}$ as follows.
$$\langle 2,1||A^{(2+)}(\lambda_{0})||2,1\rangle=b_{01}q^{-2}+b_{02}+q^{-2}c_{01%
}c_{02}=\sinh(u+\eta)/\sinh(u+2\eta)\,.$$
(D.4)
Let us define the matrix elements of the massless
fusion $R$ matrix of type (2, 2)
as follows.
$$\widetilde{R}^{(2,\,2\,+)}_{0,\,1}(\lambda_{0}-\xi_{1})^{b_{0}\,b_{1}}_{c_{0}%
\,c_{1}}=\,_{1}\langle 2,b_{1}||_{a}\langle 2,b_{0}||R^{(2,\,2\,+)}_{0,\,1}(%
\lambda_{0}-\xi_{1})\widetilde{||2,c_{0}\rangle_{a}}\widetilde{||2,c_{1}%
\rangle_{1}}\,.$$
(D.5)
Here $||2,c_{0}\rangle_{a}$ and $||2,c_{1}\rangle_{1}$
denote vectors in the auxiliary space $V_{0}^{(2)}$ and
the quantum space $V_{1}^{(2)}$, respectively.
Making use of matrix elements of
the monodromy matrix of type (1, 2)
we derive the fusion $R$ matrix of type (2,2).
For an illustration, let us calculate
$\widetilde{R}^{(2,\,2\,+)}_{0,1}(u)^{1\,0}_{0\,1}$.
$${}_{a}\langle 2,1||R^{(2\,2\,+)}_{0,\,1}(\lambda_{0}-\xi_{1})\widetilde{||2,0%
\rangle_{a}}=\left(A_{1}^{(2+)}(\lambda)C^{(2+)}_{1}(\lambda-\eta)+q^{-1}C_{1}%
^{(2+)}(\lambda)A^{(2+)}_{1}(\lambda-\eta)\right)q/[2]_{q}\,.$$
(D.6)
Evaluating operators $A^{(2+)}_{1}$ and $C^{(2+)}_{1}$
in the quantum space $V_{1}^{(2)}$, we have
$$R^{(2\,2\,+)}_{0,1}(u)^{1\,0}_{0\,1}={\frac{[2]_{q}e^{u}\sinh\eta}{\sinh(u+2%
\eta)}}\,.$$
(D.7)
The fusion $R$-matrix becomes permutation $\Pi_{0,\,1}$ at $u=0$.
In fact, we have
$R^{(2,\,2\,+)}_{0,1}(0)^{1\,0}_{0\,1}=1$.
Appendix E Spin-$s$ elementary matrices
in global operators
For integers $i_{k}$ and $j_{k}$ with
$1\leq i_{1}<\cdots<i_{m}\leq\ell$ and
$1\leq j_{1}<\cdots<j_{n}\leq\ell$,
we have
$$\displaystyle\widetilde{||\ell,m\rangle}\langle\ell,n||$$
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{c}\ell\\
n\end{array}\right)\,\left[\begin{array}[]{c}\ell\\
m\end{array}\right]_{q}\,\left[\begin{array}[]{c}\ell\\
n\end{array}\right]_{q}^{-1}\,q^{-(i_{1}+\cdots+i_{m})+(j_{1}+\cdots+j_{n})+m(%
m+1)/2-n(n+1)/2}$$
(E.1)
$$\displaystyle\times\,\widetilde{P}_{1\cdots\ell}^{(\ell)}\left(\prod_{k=1}^{m}%
e_{i_{k}}^{1,\,0}\cdot\prod_{p=1;p\neq i_{1},\ldots,i_{m},j_{1},\ldots,j_{n}}^%
{\ell}e_{p}^{0,\,0}\cdot\prod_{q=1}^{n}e_{j_{q}}^{0,\,1}\right)\widetilde{P}_{%
1\cdots\ell}^{(\ell)}\,.$$
Applying the spin-1/2 formulas of
QISP [4] to (E.1), we can express
any given spin-$s$ local operator
in terms of the spin-1/2 global operators.
It is parallel to the massive case [15].
Let us set $i_{1}=1,i_{2}=2$, …, $i_{m}=m$ and
$j_{1}=1,j_{2}=2$, …, $j_{n}=n$ in (E.1). For $m>n$
we have
$$\displaystyle\widetilde{E}_{i}^{m,\,n\,(\ell\,+)}=\left(\begin{array}[]{c}\ell%
\\
n\end{array}\right)\,\left[\begin{array}[]{c}\ell\\
m\end{array}\right]_{q}\,\left[\begin{array}[]{c}\ell\\
n\end{array}\right]_{q}^{-1}\,\widetilde{P}^{(\ell)}_{1\cdots L}\,\prod_{%
\alpha=1}^{(i-1)\ell}(A^{(1+)}+D^{(1+)})(w_{\alpha})\prod_{k=1}^{n}D^{(1+)}(w_%
{(i-1)\ell+k})$$
$$\displaystyle\times\prod_{k=n+1}^{m}B^{(1+)}(w_{(i-1)2s+k})\,\prod_{k=m+1}^{%
\ell}A^{(1+)}(w_{(i-1)\ell+k})\prod_{\alpha=i\ell+1}^{\ell N_{s}}(A^{(1+)}+D^{%
(1+)})(w_{\alpha})\,\,P^{(\ell)}_{1\cdots L}\,.$$
(E.2)
For $m<n$ we have
$$\displaystyle\widetilde{E}_{i}^{m,\,n\,(\ell\,+)}=\left(\begin{array}[]{c}\ell%
\\
n\end{array}\right)\,\left[\begin{array}[]{c}\ell\\
m\end{array}\right]_{q}\,\left[\begin{array}[]{c}\ell\\
n\end{array}\right]_{q}^{-1}\,\widetilde{P}^{(\ell)}_{1\cdots L}\,\prod_{%
\alpha=1}^{(i-1)\ell}(A^{(1+)}+D^{(1+)}(w_{\alpha})\prod_{k=1}^{m}D^{(1+)}(w_{%
(i-1)\ell+k})$$
$$\displaystyle\times\prod_{k=m+1}^{n}C^{(1+)}(w_{(i-1)2s+k})\,\prod_{k=m+1}^{%
\ell}A^{(1+)}(w_{(i-1)\ell+k})\prod_{\alpha=i\ell+1}^{\ell N_{s}}(A^{(1+)}+D^{%
(1+)})(w_{\alpha})\,\,P^{(\ell)}_{1\cdots L}\,.$$
(E.3)
Appendix F Derivation of the density of string centers
In terms of shifted rapidities ${\tilde{\lambda}}_{A}$ with
$A=2s(a-1)+\alpha$ for $a=1,2,\ldots,N_{s}/2$
and $\alpha=1,2,\ldots,2s$,
the Bethe ansatz equations for the homogeneous chain
are given by
$$\left({\frac{\sinh(\tilde{\lambda}_{A}+s\eta)}{\sinh(\tilde{\lambda}_{A}-s\eta%
)}}\right)^{N_{s}}=\prod_{B=1;B\neq A}^{M}{\frac{\sinh(\tilde{\lambda}_{A}-%
\tilde{\lambda}_{B}+\eta)}{\sinh(\tilde{\lambda}_{A}-\tilde{\lambda}_{B}-\eta)%
}}\,,\quad\mbox{for}\,\,A=1,2,\ldots,M\,.$$
(F.1)
Putting $\lambda_{A}=\mu_{a}-(2s+1-2\alpha)$
and taking the product over $\alpha$ for $\alpha=1,2,\ldots,2s$,
for the left-hand side of (F.1)
and for the right-hand side of (F.1), we have
$$\displaystyle(-1)^{2s}\left\{\prod_{k=1}^{2s}\left({\frac{\sinh((k-1/2)\eta-%
\mu_{a})}{\sinh((k-1/2)\eta+\mu_{a})}}\right)^{N_{s}}\right\}^{-1}$$
(F.2)
$$\displaystyle=$$
$$\displaystyle(-1)^{2s+N_{s}/2}\prod_{b=1}^{{N_{s}}/2}\left\{{\frac{\sinh(2s%
\eta-(\mu_{a}-\mu_{b}))}{\sinh(2s\eta+(\mu_{a}-\mu_{b}))}}\prod_{k=1}^{2s-1}%
\left({\frac{\sinh(k\eta-(\mu_{a}-\mu_{b}))}{\sinh(k\eta+(\mu_{a}-\mu_{b}))}}%
\right)^{2}\right\}^{-1}\,.$$
Taking the logarithm of (F.2) and
making use of the following relation
$$K_{2k}(\lambda)={\frac{d}{d\lambda}}\frac{1}{2\pi i}\log\left({\frac{\sinh(k%
\eta-\lambda)}{\sinh(k\eta+\lambda)}}\right)$$
(F.3)
we have the integral equation
for the density of string centers, $\rho(\lambda)$, as follows.
$$\rho(\lambda)=\sum_{j=1}^{\ell}K_{2j-1}(\lambda)-\int_{-\infty}^{\infty}\left(%
K_{4s}(\lambda-\mu_{b})+\sum_{k=1}^{2s-1}2K_{2k}(\lambda-\mu_{b})\right)\rho(%
\lambda)d\lambda\,.$$
(F.4)
For $0<\zeta\leq\pi/m$ we have the following Fourier transform:
$$\int_{-\infty}^{\infty}e^{i\mu\omega}K_{m}(\mu)d\mu={\frac{\sinh((\pi-m\zeta)%
\omega/2)}{\sinh(\pi\omega/2)}}\,.$$
(F.5)
Taking the Fourier transform of (F.4) we have
the Fourier transform $\widehat{\rho}(\omega)$ of $\rho(\lambda)$ as
follows.
$$\widehat{\rho}(\omega)=\left(\sum_{k=1}^{2s}\widehat{K}_{2k-1}(\omega)\right)/%
\left(1+\widehat{K}_{4s}(\omega)+2\sum_{k=1}^{2s-1}\widehat{K}_{2k}(\omega)%
\right)\\
=\frac{1}{2\cosh(\zeta\omega/2)}\,.$$
Taking the inverse Fourier transform
we obtain $\rho(\lambda)=1/2\zeta\cosh(\pi\lambda/\zeta)$.
Appendix G Some formulas of the algebraic Bethe ansatz
Applying the commutation relations between $C$ and $D$ operators
we have
$$\langle 0|\prod_{\alpha=1}^{M}C(\lambda_{\alpha})\prod_{j=1}^{m}D(\lambda_{M+j%
})=\sum_{a_{1}=1}^{M+1}\sum_{a_{2}=1;a_{1}\neq a_{1}}^{M+2}\cdots\sum_{a_{m}=1%
;a_{1}\neq a_{1},\ldots,a_{m}}^{M+m}G_{a_{1}\cdots a_{m}}(\lambda_{1},\cdots,%
\lambda_{M+m})$$
where
$$G_{a_{1}\cdots a_{m}}(\lambda_{1},\cdots,\lambda_{M+m})=\prod_{j=1}^{m}\left(d%
(\lambda_{a_{j}};\{w_{j}\}_{L}){\frac{\prod_{b=1;b\neq a_{1},\ldots,a_{j-1}}^{%
M+j-1}\sinh(\lambda_{a_{j}}-\lambda_{b}+\eta)}{\prod_{b=1;b\neq a_{1},\ldots,a%
_{j}}^{M+j}\sinh(\lambda_{a_{j}}-\lambda_{b})}}\right)\,.$$
(G.1)
Let $\{\lambda_{k}\}_{M}$ be a set of Bethe roots.
We have [4, 5]
$$\displaystyle\langle 0|\prod_{k=1;k\neq a_{1},\ldots,a_{m}}^{M}C(\lambda_{k})%
\prod_{j=1}^{m}C(w_{j})\prod_{\gamma=1}^{M}B(\lambda_{\gamma})\,|0\rangle/%
\langle 0|\,\prod_{k=1}^{n}C(\lambda_{k})\prod_{\gamma=1}^{M}B(\lambda_{\gamma%
})\,|0\rangle$$
(G.2)
$$\displaystyle=$$
$$\displaystyle{\rm det}\left(\left(\Phi^{{}^{\prime}}(\{\lambda_{\alpha}\})%
\right)^{-1}\,\Psi^{{}^{\prime}}(\{\lambda_{\alpha}\}\setminus\{\lambda_{a_{1}%
},\ldots,\lambda_{a_{m}}\}\cup\{w_{1},\ldots,w_{m}\})\right)\times\prod_{j=1}^%
{m}$$
$$\displaystyle\times$$
$$\displaystyle\prod_{\alpha=1;\alpha\neq a_{1},\ldots,a_{m}}^{M}{\frac{\sinh(%
\lambda_{\alpha}-w_{j}+\eta)}{\sinh(\lambda_{\alpha}-\lambda_{a_{j}}+\eta)}}%
\prod_{j=1}^{m}\prod_{\alpha=1}^{M}{\frac{\sinh(\lambda_{\alpha}-\lambda_{a_{j%
}})}{\sinh(\lambda_{\alpha}-w_{j})}}\prod_{1\leq j<k\leq m}{\frac{\sinh(%
\lambda_{a_{j}}-\lambda_{a_{k}})}{\sinh(w_{j}-w_{k})}}\,.$$
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